Thermo-Electrically Pumped
Semiconductor Light Emitting Diodes
by
Parthiban Santhanam
A0
B.S., University of California at Berkeley (2006)
S.M., Massachusetts Institute of Technology (2009)
Submitted to the Department of
Electrical Engineering and Computer Science
in partial fulfillment of the requirements for the degree of
MASSCHU$ETfS
APR 10 201
LIBRARIES
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2014
© Massachusetts Institute of Technology 2014. All rights reserved.
.......................
...........
Department of
Electrical Engineering and Computer Science
January)4, 2014
C ertified by ....................................
.........
Rajeev J. R-am
Professor of Electrical Engineering
Thesis Supervisor
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Accepted by ..................................
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Doctor of Philosophy in Electrical Engineering
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Le
A. Kolodziej ski
Chair, Department Committee on Graduate Theses
Thermo-Electrically Pumped
Semiconductor Light Emitting Diodes
by
Parthiban Santhanam
Submitted to the Department of
Electrical Engineering and Computer Science
on January 14, 2014, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Electrical Engineering
Abstract
Thermo-electric heat exchange in semiconductor light emitting diodes (LEDs) allows
these devices to emit optical power in excess of the electrical power used to drive
them, with the remaining power drawn from ambient heat. In the language of semiclassical electron transport, the electrons and holes within the device absorb lattice
phonons as they diffuse from their respective contacts into the LED's active region.
There they undergo bimolecular radiative recombination and release energy in the
form of photons. In essence the LED is acting as a thermodynamic heat pump operating between the cold reservoir of the lattice and the hot reservoir of the outgoing
photon field.
In this thesis we report the first known experimental evidence of an LED behaving as a
heat pump. Heat pumping behavior is observed in mid-infrared LEDs at sub-thermal
forward bias voltages, where electrical-to-optical power conversion at arbitrarily high
efficiency is possible in the limit of low optical output power. In this regime, the
basic thermal physics of an LED differs from that seen at conventional higher voltage
operating points. We construct a theoretical model for entropy transport in an LED
heat pump and examine its consequences both theoretically and experimentally. We
use these results to propose a new design for an LED capable of very high efficiency
power conversion at power densities closer to the limit imposed by the Second Law
of Thermodynamics. We then explore the potential application of these thermophotonic heat pumps as extremely efficient sources for low-power communication and
high-temperature absorption spectroscopy.
Thesis Supervisor: Rajeev J. Ram
Title: Professor of Electrical Engineering
3
4
Acknowledgments
The work described in this thesis represents the collective efforts of a number of
people. I'd like to take a minute to recognize a few of them.
I feel I should begin where everything I've done has, with my family. Over the
course of my formal education, I have slowly come to realize the incredible impact
that the attitudes of my parents toward knowledge and learning have had on me.
As long as I can remember, they have woven the process of learning with the other
joys of life, and have thereby contributed to the quality of my life immeasurably. I
remember vividly the emphasis my father placed on the fundamentals as he taught
me math on weekends. I believe there is a direct connection from those experiences to
my approach to research and for that he deserves my thanks. In more recent times, I
have looked to them for help and guidance more often than I could have anticipated.
In response they have been more understanding than I thought possible and were
always generous with their unwavering love and support. My sister and her family
have been the closest family members within driving distance for several years now.
They have served as a constant reminder that the often myopic mindset of graduate
school is not all that life has to offer. In a very concrete sense, I could not have
reached the point I'm at without them. I only hope I can return the favor someday.
In good faith I cannot omit the countless friends, roommates, classmates, and
nontrivial combinations thereof who have supported my growth through conversation, cohabitation, cooperation, commemoration, and occasionally commiseration.
My former roommates Shawn Henderson and Matt McFall, both of whom I have
been lucky enough to call friends for more than half my life, have been two of my
closest companions and I hope they will continue to be in the coming phases of life.
My friend Rachel VanCott has been a constant presence in a time of fluctuation; Mike
Rosenberg has shared many of the interests I have carried since childhood and helped
in the dissipation of my cravings to watch and play sports. Laura Dargus has always
had an open seat, a free minute, and plenty of empathy, and I won't soon forget the
chats we've had in her office. David Hucul and Nabil Iqbal have been remarkable
5
catalysts for getting out and doing fun stuff. Donny Winston's zest for life has left
me with some unbelievable stories and a friend whom I can always count on. The
Cookie Monday regulars, my Intramural sports teammates, my fellow Wichita transplants, the WAKA Kickballers, the many easygoing RLE admins, the VP crew, and
my Ashdown/Sid-Pac friends have all given me countless happy memories and played
a real role in making my twenties what they have been.
Several professional relationships deserve mention here. First and foremost, my
work would not have been possible without the generous funding I have received from
the EECS Department, the Office of Naval Research, the NDSEG Fellowship Program,
and Weatherford Int'l. Of the many MIT faculty members whose classes I hope never
to forget, I was fortunate to have on my thesis committee four of the professors I've
most admired. Professor Mehran Kardar and Professor Lizhong Zheng, from whom I
took Statistical Mechanics and Information Theory respectively, rank highly on that
list. I was delighted to have them on my thesis committee, through which I was able
to get feedback from points of view outside the semiconductor device community. I
was also lucky to have Professor Vladimir Bulovic, whose enthusiasm for academic
research has luminesced brightly as a research advisor and as the Director of MTL, on
my committee; his interest in applying our thinking to organic LEDs was instrumental
in clarifying the assumptions underlying our theoretical framework. In a similar vein,
my discussions with collaborators including Prof. Ali Shakouri, Dr. Je-Hyeong Bahk,
Dr. Mona Zebarjadi, Prof. Boris Matveev, Dr. Jess Ford, Dr. Ligong Wang were
necessary parts of the work described in this thesis.
Many of my fellow students have also contributed significantly. From Prof. Qing
Hu's group, David, Ivan, Qi, Wilt, and Sushil were always ready to discuss new ideas,
lend equipment and teaching time, and generally foster an enjoyable and productive
atmosphere for research. Prof. Ben Williams, Dr. Alan Lee, and Dr. Tom Liptay
were senior figures when I first came to MIT, and I probably took away more advice
from each of them than they know. I owe a special thanks to Prof. Dave Weld for
the time he took from his postdoc and first year as junior faculty at UCSB to provide
feedback and walk me through my first article submission to Physical Review Letters.
6
It was an important point in graduate school for me, and someday I hope to emulate
his genuine and patient encouragement.
As part of Rajeev Ram's Physical Optics and Electronics Group, several of my
labmates have been so many things to me- role models, coffee buddies, friends, sources
of advice, and sanity checks. I have shared so much of my experience in the last five
years with Dodd Gray- both professionally and personally. He was the yin to my
yang during our early work with low-biased LEDs and was an absolute rock of moral
support in the years before our work was published. Duanni Huang's persistence in
building the communication experiment was admirable and working with him provided me with important lessons in mentorship. More recently, Bill Herrington and
Priyanka Chatterjee have brought the lab to life with their fresh perspectives and
I look forward to working with them going forward. When Karan Mehta came to
our group, we immediately bonded over our interest in physics and the conversations
we shared during walks and over coffee have shaped many of the physical pictures I
rely on daily. Jason Orcutt was the consummate professional in lab, or at least as
much as a graduate student can be without losing their street cred. Over the years
I have often asked myself "What Would Jason Do?" and I continue to emulate him
in many ways. I will remember Reja Amatya for her seemingly effortless work ethic
and her choice to pursue the kind of research project that makes the world a better
place. Kevin Lee's humor and high spirits brightened the atmosphere in the group,
and his amazing nose bubble video will live on in the lab's lore. Tauhid Zaman was
a one-man minority in his appreciation of the ten-page handouts on Second Quantization that I may never live down, and from what I remember, he was never bashful
about anything really. Johanna Chong raised my opinion of the MIT undergraduate
experience and has always been a good friend.
Shireen Goh's organizational skills
remain a model for me, and I wish her the best in her new life in Singapore. Evelyn
Kapusta was a hoot. I only hope that I can retain my "cloud person" status forever.
I'd also like to thank William Loh for his technical perspective and willingness to sit
down and explain things with patience.
During my grad school years my research advisor Prof. Rajeev J. Ram had an
7
enormous influence on me. As a teacher, mentor, role model, and finally a colleague,
I have been the beneficiary of his attitudes toward many things in research and
in life. During the first week of graduate school, I attended a welcome lecture by
some senior academic official at which the ideal of an advisor's role was likened to
"academic fatherhood." Aside from the unnecessarily gendered word choice, I felt that
description fit my goal as well. I had been told by many of my fellow grad students
that such a relationship was overly idealistic and these days impossible. Perhaps it
is because I was fortunate enough to work with Rajeev, but in retrospect this view
strikes me as cynical, and I consider myself lucky to have avoided it.
I still remember many of the conversations I've had with Rajeev.
He shared
his views on the importance of role models, how to find the right research project,
and why so many people struggle with their twenties these days. One of the more
memorable methods he employed was to tell a Buddhist parable. Here I'd like to
approximate returning the favor.
There once was an American living in Japan, who while hiking in a forest came
across an old man outside his secluded home. As he was keen to practice his Japanese,
he began a conversation. The old man said he was a martial arts instructor and offered
to teach the American a lesson in karate. The American accepted the offer and worked
hard to be a good student. At the end of the lesson, the old man offered to teach
him again the next day, and the American accepted the gracious offer. That night,
the American went back to the city and told some of his American friends about his
new sensei and one of them asked to tag along. The next day two Americans came to
the old man, and he taught them both. Again at the end of the lesson he offered to
teach them again the next day. For weeks this pattern continued, with the American
students increasing in number until the sensei had a full class. One day at the end
of class, the students got together and decided they should offer to pay the old man
for teaching them. They approached him with their offer, but the old man declined.
When the students insisted that his teaching was so good that they felt like they
should be paying for it, the old man replied: "if I decided to charge you, you couldn't
afford me."
8
In the same way, the lessons Rajeev has taught me are valuable, but since he has
so much to give the world, so is his time. From my perspective, the dedication he
shows toward his graduate students seems beyond compensation. He must do it for
better reasons. My plan is to pay it forward. Thanks again, Rajeev, for your time
and energy.
9
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10
Contents
1
Background
1.1
LED Efficiency and Heat ......
1.2
The LED as a Thermodynamic Heat Engine . . . . . . . . . . . . . .
21
1.3
Previous Work Toward Unity Efficiency . . . . . . . . . . . . . . . . .
27
1.4
Efficient Communication with a Photonic Heat Pump . . . . . . . . .
30
1.5
Potential Practical Applications . . . . . . . . . . . . . . . . . . . . .
35
1.5.1
Infrared Spectroscopy . . . . . . . . . . . . . . . . . . . . . . .
35
1.5.2
Lighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
1.5.3
Other Applications . . . . . . . . . . . . . . . . . . . . . . . .
40
Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
1.6
2
15
................
.......
16
LEDs as Heat Pumps
43
2.1
Electron Transport and Entropy Flow in LEDs . . . . . . . . . . . . .
43
2.1.1
Current Continuity . . . . . . . . . . . . . . . . . . . . . . . .
44
2.1.2
Quasi-Equilibrium
. . . . . . . . . . . . . . . . . . . . . . . .
46
2.1.3
Thermally-Assisted Injection . . . . . . . . . . . . . . . . . . .
47
2.1.4
Recombination
50
2.1.5
Continuity of Entropy Flux
. . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
52
2.2
The Heat Pump Picture
. . . . . . . . . . . . . . . . . . . . . . . . .
54
2.3
LEDs in the Low-Bias Regime . . . . . . . . . . . . . . . . . . . . . .
58
2.4
Carnot-Efficient LEDs and Real LEDs
. . . . . . . . . . . . . . . . .
64
2.4.1
Carnot Efficiency . . . . . . . . . . . . . . . . . . . . . . . . .
64
2.4.2
Non-Ideality of Existing LEDs . . . . . . . . . . . . . . . . . .
66
11
The Power-Efficiency Trade-Off . . . . . . . . . . . . . . . . .
67
2.5
Design of LEDs for Heat Pumping . . . . . . . . . . . . . . . . . . . .
69
2.6
Circuits are Cycles
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
2.7
Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . .
81
2.4.3
3
Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . .
85
3.1.1
Current-Biased Lock-In Technique . . . . . . . . . . . . . . . .
85
3.1.2
Temperature Control . . . . . . . . . . . . . . . . . . . . . . .
88
3.1.3
Thermal Shock of LED Packaging . . . . . . . . . . . . . . . .
90
3.1.4
Optical Design
. . . . . . . . . . . . . . . . . . . . . . . . . .
93
3.2
Demonstration of r/ > 1: A = 2.5pm . . . . . . . . . . . . . . . . . . .
94
3.3
High Power Attempt: A
. . . . . . . . . . . . . . . . . . . .
99
3.4
Lower Emitter Temperatures: A = 3.4pm . . . . . . . . . . . . . . . .
101
3.4.1
Exclusion of Emissivity Modulation . . . . . . . . . . . . . . .
101
3.4.2
Unity Efficiency at Room Temperature . . . . . . . . . . . . .
108
3.4.3
Does Voltage Determine Brightness?
. . . . . . . . . . . . . .
114
Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . .
116
3.1
3.5
4
83
Experiments on Existing Emitters
=
4.7pm
119
Communication with a Thermo-Photonic Heat Pump
. . . . . . . . . . . . .
119
4.1.1
Sample Calculation . . . . . . . . . . . . . . . . . . . . . . . .
120
4.1.2
Extrapolation to Low Power . . . . . . . . . . . . . . . . . . .
125
4.1.3
Extrapolation to Carnot-efficient LEDs . . . . . . . . . . . . .
129
. . . .
131
4.2.1
The Entropy Trade-Off . . . . . . . . . . . . . . . . . . . . . .
131
4.2.2
Calculation of the kBT ln(2) Limit . . . . . . . . . . . . . . . .
133
4.3
A Thermo-Photonic Link . . . . . . . . . . . . . . . . . . . . . . . . .
142
4.4
Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . .
152
4.1
4.2
Power Measurements as Slow Communication
Limits of Energy-Efficient Communication with a Heat Pump
12
5
6
High-Temperature mid-IR Absorption Spectroscopy
155
. . . .
156
. . . .
157
High-Temperature Sources for Spectroscopy.....
. . . .
160
5.4
High-Temperature Infrared Photo-Detection
. . . .
164
5.5
High-Temperature Emitter-Detector Compensation
.
. . . .
171
5.6
Summary and Conclusions . . . . . . . . . . . . . . .
. . . .
176
5.1
Motivation . . . . . . . ..
. . . . . . . . . . . . . . .
5.2
Mapping Spectroscopy onto Communication
5.3
. . . . .
. . . . .
Conclusions and Future Work
179
6.1
Thesis Summary and Conclusions . . . . . . . . . . .
180
6.2
Further Scientific Questions
184
. . . . . . . . . . . . . .
6.2.1
Entropy and Information in Photons
. . . . .
184
6.2.2
Entropy and Information in Electrons . . . . .
189
6.3
Further Applied Directions . . . . . . . . . . . . . . .
192
6.4
Engineering Toward Second Law Bounds . . . . . . .
195
A Entropy and Temperature of Light
201
B Maximum Efficiency at 1 Sun
213
References
227
13
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14
Chapter 1
Background
In the last two decades opto-electronic devices such as diode lasers, photo-voltaics,
and light-emitting diodes (LEDs) have been developed with improved capabilities at
drastically reduced costs. As a result, widespread use of these devices is no longer exclusive to the traditional applications that have historically driven their development
[1]. Beyond their historical use as indicator lights, LEDs have been widely adopted
for displays [2], sources in spectroscopic applications [3, 4, 5], automotive applications [6], outdoor lighting [7], and increasingly the markets for indoor commercial
and residential lighting [8].
In this thesis we consider LEDs as thermodynamic heat pumps. In
lish the basic thermal physics of traditional LED operation. In
§ 1.1 we estab-
§ 1.2 we demonstrate
that this behavior stands in contradiction to what should be expected from a heat
pump. In
§ 1.3 we review the literature on LED heat-pumping in anticipation of
presenting its experimental observation in Chapter 3. Some theoretical and practical
consequences of the heat-pumping regime are motivated in
§ 1.4 and § 1.5 respec-
tively, before they are analyzed more fully in later chapters. In
short outline of the thesis as a whole.
15
§ 1.6 we provide a
1.1
LED Efficiency and Heat
As the roles of light-emitting diodes expand, the variety of operating conditions they
are subjected to is broadening and the demands on their performance are rising.
Their performance in high-temperature environments remains a ubiquitous challenge,
as suggested by Figure 1-1. The efficiency of LEDs depends strongly on the thermal
environment in which they operate. To explain the physical origin of this dependence
in both the traditional and heat-pumping regimes, we begin by briefly reviewing the
physics of a conventional double-hetero-junction LED. A simplified band diagram of
such a device has been adapted from [9] and appears in Figure 1-2.
The wall-plug efficiency q of an LED is defined as the ratio of emitted optical
power L to the supplied electrical power IV. Since each electron that passes through
the device has some probability of emitting a photon of energy hw, the efficiency 77
may be decomposed in terms of this probability (here denoted
iEQE):
(hwy
qV
n7EQE
()
qV
(Rradiative) active
(RSRH)active + (Rradiative)active
±
(RAuger)active
Here (lw) is the average energy of the emitted photons, q is the magnitude of the
electron's charge, and V is the applied voltage; the external quantum efficiency
EQE
is the ratio of the rate at which photons exit the device to the rate at which electrons
pass through it as current. As shown,
EQE
may be further decomposed into the
efficiency with which generated photons are extracted from the device
(qextract),
the
efficiency with which injected electrons from the cathode and injected holes from
the anode fall into the narrow-gap active region and remain confined there until
recombination (7inject), and the fraction of that active-region recombination which is
radiative.
Here we have also assumed that all recombination events outside the active region
do not contribute to useful light (these events are accounted for by
Tinject <
1) and
that all recombination events inside the active region are of one of three types [1, 14]:
16
0
Q
A=1.9m
1
Xz4.7im
0-
X=2.1pm
X=3.4pm
50
100
150
200
250
300
350
Temperature (K)
400
450
500
Figure 1-1: Electrical-to-optical power conversion efficiency at typical operating currents versus temperature for several modern LEDs emitting at various wavelengths.
The green dashed line shows the performance of an InGaN LED (h
~ 2eV) from
2011 [7]; the blue squares and the black dashed line are from two near-infrared In-
GaAsSb LEDs (hw
600meV) from 2006 [10] and 2009 [11] respectively; the black
circles are from a mid-infrared InAs LED (hw
350meV) from 2002 [12]; the solid red
line is from a long-wavelength InAsSb LED (hw
250meV) from 2009 [13]. In spite
of the range of wavelengths and the variety of material systems in which they were
fabricated, for each of these devices, the efficiency clearly decreases with increasing
temperature.
17
Ep
tico
0M
a)
.
-
-
F
EFp
n
E
Position
Figure 1-2: Simplified band diagram of a conventional double-hetero-junction LED.
The solid lines indicate the edges of the conduction and valence bands (labeled Ec
and Ev respectively). The dashed lines indicate the Fermi level in the metal contacts
and the electron and hole quasi-Fermi levels in the semiconductor regions (labeled
EFn and EFp respectively). The wavy line denotes an exiting photon. The solid
double-line between the diamonds represents the imaginary boundary which is crossed
exactly once for each quantum of charge that flows as current. Charge may cross the
double-line by either thermionic emission of minority carriers (i.e. carrier leakage) or
active region recombination events. The double-hetero-junction structure is generally
designed to confine carriers to minimize leakage and thereby increase overall efficiency.
Figure adapted from [9].
18
trap-assisted Shockley-Read-Hall (SRH) [15, 16], radiative, or Auger [17]. Their rates
(typically expressed in units of cm- 3 s- 1 ), are denoted above as RSRH,
RAuger
respectively; the symbol
(-
)atjve
Rradiative,
and
denotes an average over the active region
volume.
Both the injection efficiency
RAuger
Tinject
and the non-radiative Auger recombination rate
are strongly dependent on temperature. Together they are responsible for the
decreased efficiency of LEDs with operating temperature [17].
In Figure 1-2, the solid double-line between the diamonds represents the imaginary
boundary which is crossed exactly once for each quantum of charge that flows as
current. Charge may cross the double-line in one of 3 ways: (1) by the net thermionic
emission of an electron over the p-side conduction band hetero-barrier at left, (2) by
a recombination event in the active region in the middle, or (3) by the net thermionic
emission of a hole over the n-side valence band hetero-barrier at right. The doublehetero-junction structure is generally designed to confine carriers so that current of
type (2) dominates over (1) and (3), leading to high
minject.
Since the parasitic leakage
processes (1) and (3) are thermionic emission processes over finite barriers, their rates
in typical operating regimes are exponentially dependent on temperature.
The rate of non-radiative Auger recombination is also exponentially dependent
on temperature in a similar way. The Auger process may be visualized as the timereversed version of impact ionization. In impact ionization, a high-energy electron
collides with an electron in the valence band to promote it to the conduction band.
The final states of both electrons must also have the same total momentum as the
initial electron states. In Auger recombination, an electron recombines with a hole of
different momentum (i.e. a non-vertical inter-band transition), and gives this energy
to another free electron which subsequently relaxes non-radiatively. Because of the
momentum-difference required of the original electron-hole pair, states very near the
band-edge of direct-gap semiconductors are not sufficient. Instead, the Auger process
requires the carriers undergoing recombination to inhabit excited initial states. This
requirement causes the temperature-dependence of
RAuger,
as it is dependent on the
presence of carriers with kinetic energy above some threshold energy which depends
19
on the bandstructure.
In summary, elevated temperatures traditionally cause LEDs to be less efficient, as
excited carriers undergo more non-radiative Auger recombination and the quality of
carrier confinement is reduced. Degraded device performance may be seen empirically
in Figure 1-1 to hold across virtually the entire range of commercially-available LED
emission wavelengths.
Moreover, even at room temperature the inefficiency of the LED itself leads to heat
generation which may further degrade performance.
State-of-the-art visible LEDs
fabricated from InGaN achieve high internal quantum efficiency at low power density,
but at higher current density the portion of input power which is not emitted as
light results in substantial self-heating.
This heating contributes to the so-called
"efficiency droop" [18, 191, thereby reducing the potential for energy savings from
solid-state lighting [7]. It also decreases bulb lifetime, thereby increasing amortized
capital and installation costs of LED lighting solutions [7].
LEDs designed to emit photons in the spectroscopically-valuable mid- and farinfrared wavelength ranges also face major thermal challenges. In the mid-infrared
(A=2-8pm), state-of-the-art LEDs are at most 1-3% efficient [12, 11, 13, 20]. The remaining 97-99% of the electrical drive power results in self-heating; the consequences
for efficiency and lifetime frequently motivate these devices to be driven by pulsed currents. In the far-infrared, LEDs are again highly inefficient and sufficiently sensitive
to junction temperature to require external thermo-electric cooling [21].
In short, regardless of emission wavelength, the basic thermal physics of an LED
is the same:
" Imperfect wall-plug efficiency leads to self-heating that increases the device's
operating temperature.
" Elevated temperatures lead to decreased efficiency, regardless of whether selfheating or ambient conditions are responsible for them.
On the other hand, if the light-emitting diode is examined as a thermodynamic
device, the exact opposite would be expected. Since the LED is driven by entropy20
free electrical power and results in the emission of entropy-carrying incoherent light,
it is possible for the device to absorb entropy from the ambient (i.e. self-cooling).
Moreover, since for a given spectral intensity of incoherent light output the outgoing
photon modes are occupied at some finite temperature, increased junction temperatures should reduce the thermal gradient against which an LED must pump heat and
thereby permit higher efficiency.
The observed behavior of modern LEDs differs from these thermodynamic behaviors because even state-of-the-art emitters are far from their ideal limits. In this
work, we offer a theoretical framework to explain this discrepancy, present experimental and numerical results to support it, and explore practical changes to device
designs to make LEDs more ideal.
1.2
The LED as a Thermodynamic Heat Engine
In Statistical Mechanics, the word "heat" is used to refer to any form of energy
which possesses entropy [22]. This usage applies equally to forms of energy referred
to colloquially as "heat," such as the kinetic energy in the relative motion of the
molecules in a gas or the constant vibrations of atoms in a crystal lattice, as well as
those for which the entropy is frequently less relevant, such as the kinetic energy in the
relative motion of electrons and holes in a semiconductor or the thermal vibrations
of the electromagnetic field in free space. Critically, the Laws of Thermodynamics
which govern the flow of heat are formulated independently of the Laws which govern
the deterministic trajectories of mechanical systems, be they classical or quantum.
As a result concepts such as the Carnot limits for the efficiency of various energy
conversion processes apply equally well to the gases and solid cylinder walls of an
internal combustion engine as to the electrons, holes, and photons in a modern LED.
An LED is an electronic device which takes entropy-free electrical work as input
and emits incoherent light which carries entropy. Instead of irreversibly generating
the entropy that it ejects into the photon reservoir, an LED may absorb it from
another reservoir at finite temperature, such as the phonon bath. As the diagram in
21
Figure 1-3 suggests, the device may absorb heat from the phonon bath and deposit
it into the photon field in much the same way as a Thermo-Electric Cooler (TEC)
absorbs heat from the cold side of the module and deposits it on the hot side [23, 24].
In the reversible limit the flows of energy and entropy are highly analogous for an LED
and a TEC. Moreover, in both the LED and TEC, the Peltier effect is responsible
for the absorption of heat from the reservoir being cooled into the electronic system
[25, 26, 27, 28]. Electrical work is being used to pump entropy from one reservoir to
another instead of simply creating it though irreversible processes. The LED, like the
TEC, is a thermodynamic machine.
For each bit of entropy JS absorbed on net from the phonon reservoir at finite
temperature, an amount of heat TatticeJS comes with it. Since input and output power
must balance in steady-state, the rate at which this heat and the input electrical work
enter the system (both measured in Watts) must exactly equal the rate at which heat
is ejected into the photon reservoir (also measured in Watts). That is to say, when
lattice heat is being absorbed an LED's wall-plug efficiency r7 (or equivalently, its
heating coefficient of performance), defined as the ratio of output optical power to
input electrical power, must exceed unity.
Additionally, in this picture the lattice remains slightly cooled compared to its
surroundings, so that heat is continuously conducted into the device from the environment in steady-state. Rather than self-heating, the LED is experiencing self-cooling.
The Second Law of Thermodynamics (i.e. non-deletion of entropy) places a clear
limit on the maximum efficiency of an LED in this framework. To understand this
limit, we must first understand the thermodynamics of photon gases at finite temperature.
Incoherent electromagnetic radiation which originates in an LED is equally capable
of carrying entropy with it as electromagnetic radiation from a hot blackbody. All
incoherent light is therefore, in the statistical-mechanical sense described above, a
type of heat. The ratio of the rate at which radiation carries away energy to the rate
22
4
Irreversible
Entropy
Generation
Joule Heating
&Thermal
Conduction
TEC cold side
Photon Field
Phntnn FiPId
Irreversible
Entropy
Generation
I
native
Recobntion
Phonon Field
Pnonon iela
Figure 1-3: Diagrams depicting energy and entropy flows in two types of thermodynamic heat pumps: TECs (top row) and LEDs (bottom row). The left column shows
the theoretical energy and entropy flows in Carnot-efficient devices. The right column
shows the same in devices with common sources of irreversibility.
23
at which it carries away entropy gives its flux temperature [29]:
TF
-
dUldt
U(1-3)
S
dS/dt
Although this notion of temperature may be used to calculate the thermodynamic
limits of power-conversion efficiency, the rate of entropy flux in light is difficult to
measure directly. Fortunately a more intuitive definition of the temperature of light
is presented in Figure 1-4.
Consider two bodies that are each perfectly thermally isolated from their environments (i.e. by adiabatic walls) and similarly isolated from each other. Suppose
body 1 has energy U1 and entropy S, and likewise the second body has energy U2
and entropy S2. If the insulating boundary between bodies 1 and 2 is replaced with
one which permits the flow of energy, the total energy U + U2 will flow to rearrange
itself in the way which maximizes the total entropy. The flow will stop only when
the addition of a differential amount of energy 6U to either body results in the same
fractional increase in the number of available micro-states for that body (i.e. the
same increase in its entropy). Equivalently, we may say that the flow of energy stops
when the bodies have equal temperature [22]:
aS1
1
1
OU1
T1
T2
_S
2
OU 2
(14
Now consider a similar scenario in which body 1 is an LED and body 2 is a perfect
blackbody radiator. To begin, both bodies are adiabatically isolated from their environments and each other. In the case of the LED, this means that the walls must
be perfect mirrors, such that each photon emitted eventually returns to generate a
quantum of reverse-current.
Assume no non-radiative recombination occurs. The
LED is "on," but is in steady-state and consumes no power. Assume that the bodies
have no means of exchanging energy other than through photons and that to begin
the boundary between them is also a perfectly-reflective mirror.
If the mirror is modified to permit transmission over a narrow range of wavelengths
24
Body
1: LED
Body 2: Hot blackbody
Body 2
Body I
Perfect Mirr
jAdiabatic Boundary
(no heat exchange)
Adiabatic Boundar
(no heat exchnge)]
Body 2
Body 1
Body 1: LED
Bodies re-arrange energy U1+U, to maximize total entropy Si+S,Equilibrium reached when:
TI IS dU
Body
A-Selective Mirror
1i1
100%
-
-
1
2
Transmission %
Reflection %
2
9U2
1
Body 2: Hot Blackbody
Equilibrium: Zero Net Photon Flux
Body 2
%1t
0
Mrahiw
n
Bodies re-arrange energy U1+U2 to maximize total enrropy !S+S2.
Equilibrium reached when:
-1
dS
j
SaU
du,
=
adl
a2
-1
2
Figure 1-4: The brightness temperature of an incoherent source (here, an LED) may
be defined as follows. At each optical frequency, consider the temperature at which
a perfect blackbody would emit with the same spectral intensity (i.e. power per unit
area per unit frequency). This temperature indicates the ratio of the rate of energy
flux to the rate of entropy flux carried by the radiation in that band. The weighted
average of these temperatures over the intensity spectrum of the emitter gives the
brightness-temperature of the source.
25
around A0 , energy will flow on net from the body with higher spectral power density
normal to the boundary (i.e. I(A) in W m-
2
nm- 1 ) to the body with lower I(A)
at A0 . If we assume the LED is perfectly incoherent, the flow of photons in either
direction is equally capable of carrying entropy, and therefore equally justified in
being termed 'heat.'
Since heat may only flow from high temperature to low, the
equilibrium condition for the two bodies may only be satisfied when 11 (Ao)
=
12 (Ao).
Since the relationship between intensity and temperature for a perfect blackbody is
given by the Planck radiation law, we may define the brightness temperature TB of
any completely incoherent source as the temperature of blackbody whose spectral
intensity equals that of the emitter in the wavelength range of interest [29, 30]:
4h7r2 C2
Iemitter(AO) = Iblackbody (AO
;TB)
0 exp
I
h(27rc/Ao)
(h(kBT/A)
kBTB/
-
1
(1.5)
Note that unlike the color temperature of radiation commonly used in the lighting and display industries, a longer-wavelength emitter is not necessarily cooler than
a short-wavelength emitter. Both the linewidth and intensity of the source matter and
may result in thermodynamically-cold emission from a blue LED or thermodynamicallyhot emission from a red one. The flux temperature TF and brightness temperature
TB of a source may be cool, even when the radiation is blue.
A note to the reader: a more detailed discussion of the distinction between the
flux (TF) and brightness (TB) photon temperatures can be found in Appendix A.
Since the temperature of an incoherent photon flux is essentially a measure of
its spectral intensity I(A), the Second Law places a different efficiency constraint
on emitters of different spectral intensity. As a function of lattice temperature and
emitter intensity, the Carnot limit may be expressed compactly as follows:
77
For bright sources (I(A)
77Carnot =
Tphoton (I)
Tphoton (I) - Tattice
1.6
> Iblackbody(A; Tattice)), the LED must pump heat against
the large temperature difference between the lattice and the outgoing photon field.
26
This results in a maximum efficiency, even for a perfect Carnot-efficient LED, which
exceeds unity but only slightly.
For dim sources (I(A)
-
Iblackbody(A;
Tiattice)
<
Iblackbody (A; Tattice)), the LED must only pump heat against a small temperature
difference. As a result, efficiencies far in excess of unity are possible.
Examination of Equation 1.6 at fixed spectral intensity I reveals another counterintuitive aspect of the heat-pump regime.
As Tattice is increased, the temperature
difference against which the LED must pump becomes smaller, and the maximum
allowable efficiency increases.
Thus, the basic thermal physics of an LED in the heat pump regime is the reverse
of the conventional thermal physics:
" Above-unity efficiency results in self-cooling that decreases the device's operating temperature.
" For a desired spectral intensity, a higher lattice temperature means that the
device can be more efficient.
These differences may result in practical consequences for both the device-level
design of LED active regions (explored in
§ 2.5) and the thermal design of their
packaging (which we discuss briefly in Chapter 6).
1.3
Previous Work Toward Unity Efficiency
For several decades it has been theoretically understood that the presence of entropy
in incoherent electromagnetic radiation theoretically permits semiconductor lightemitting diodes (LEDs) to emit more optical power than they consume in electrical
power [31, 29, 32, 33]. Moreover, starting very early on the phenomenon has drawn
the attention of the applied research community. In 1959 a US Patent was granted
for a refrigeration device based on the principle [34]. In the last decade, the applied
literature on the subject has expanded to include more realistic modeling and more
recent advances in device fabrication technologies [14, 35, 36, 37, 38, 39] and at least
one attempt to demonstrate practical cooling is currently underway [40]. Nevertheless,
27
prior to this work, the basic phenomenon of electrically-driven light emission above
unity efficiency had never been experimentally verified.
The experimental literature on electro-luminescent cooling stretches back more
than five decades, beginning before even the early work of Tauc [31] in 1957 and
Weinstein [291 in 1960. A summary of this work appears in Table 1.1 alongside data
for experiments described in this thesis.
Year
Author(s)
Vmin
qVmin/kBT
e~z/kBT
Max Reported q
1953
Lehovec, et al. [41]
1.8 V
70
< 2.5 x 10-34
Not Published
1964
Dousmanis, et al. [42]
1.25 V
186
2.8 x 10-90
16% [43]
1966
Nathan, et al. [44]
1.1 V
6380
10-36o
6 %
2005
Wang, et al. [4.5]
0.36 V
14.2
3.8 x 10-
2011
Oksanen, et al. [40, 46]
0.5 V*
19.3*
4 x 10-13
2011
THIS WORK (§ 3.2, [47])
70 uV
0.002
8.4 x 10-
Not Published
Not Published
7
231 ± 37 %
Table 1.1: Summary of previous experiments towards electro-luminescent cooling
(i.e. electro-luminescence with q >1). The asterisk (*) indicates that these figures
were taken from simulation data. The quantity qVmin/kBT highlights the primary
difference between the approach taken in this work and previous experiments. The
quantity e-h/kBT provides a scale for the optical power available in the low-bias
regime.
As early as 1953, Lehovec et al. speculated on the role of thermo-electric heat
exchange in SiC LEDs [41].
The authors were motivated by their observation of
light emission with photon energy hw on the order of the electrical input energy per
electron, given by the product of the electron's charge q and the bias voltage V.
In 1964, Dousmanis et al. demonstrated that a GaAs diode could produce electroluminescence with an average photon energy 3% greater than qV [42].
Still, net
cooling was not achieved due to non-radiative recombination [43] and the authors
concluded that the fraction of current resulting in escaping photons, typically called
the external quantum efficiency
1
7EQE,
must be large to observe net cooling. They
wrote:
28
"Diodes with high quantum yield are required for direct experimental
observation of the cooling effect."
DOUSMANIS,
ET. AL.
PHYSICAL REVIEW, 1964
A similar observation was made two years later in a cryogenic GaAs LED (hw =1.44eV)
by Nathan, et al [44]. Then after several decades of minimal experimental activity,
recent modeling and design efforts have indicated that
EQE could be raised toward
unity by maximizing the fraction of recombination that is radiative [14, 35, 38] and
employing photon recycling to improve photon extraction [14, 35, 37]. As a result,
at least one experiment was performed by Wang, et al. in 2005 [45], but no optical power or wall-plug efficiency data was published. At least one effort to observe
electro-luminescent cooling with
JEQE
above 50% continues to be active [40], although
early results suggest problems with shunts in the emitting diode [46].
All of these experiments followed the logic of the quote above from Dousmanis,
et al. by attempting to raise %EQEtoward unity. In contrast, q > 1 was observed in
this work with nEQE ~ 3 x 10- 4 . Since the wall-plug efficiency q of a diode may be
expressed as follows:
,(1.7)
S=--EQE
qV
in order to achieve above-unity q with small
7
7EQE
requires V <
hw/q. Multiple
authors have dismissed such operating regimes in the past because of the low output
power available in this regime, but the present work has found it's consideration
worthwhile for 3 main reasons:
" Regardless of the power requirements for a practical cooling system, lower power
may be sufficient for specific applications and/or experimental confirmation.
" The greatest deviations from conventional 7 < 1 operation (i.e. highest coefficients of performance) always occur at low power. This is a general property of
endo-reversible heat pumps.
" The decrease in power from lowering V can be compensated by increasing the
29
ratio kBT hw.
The third observation above was made in 1985, when Paul Berdahl presented an
analysis of semiconductor diodes as radiant heat engines [43]. In that work, he showed
that the available cooling power decreased exponentially with the ratio of the diode
materials bandgap Egap to the thermal energy kBT, in accordance with the blackbody
emission power integrated over the absorptive/emissive band.
1.4
Efficient Communication with a Photonic Heat
Pump
The experimental result presented in Chapter 3 not only realizes photon generation
with wall-plug efficiency in excess of unity (i.e. net cooling), but further demonstrates
that arbitrarily high wall-plug efficiency is available at infinitesimal power. Data for
the generation of 2.47pm photons (w
~ 500 meV) in a 423 K environment (kBT
36 meV) appears in Figure 1-5.
At the low-power, high-efficiency endpoint of this data set, the LED consumes
just 8.8 meV of work per photon to create an optical signal which may be directly
electrically modulated. In principle, such a device could be used as the source in a
simple on-off-keying (OOK) communication link. If the emission of one such photon
were used to indicate a '1' and the lack thereof to indicate a '0', on average just 4.4
meV of work would be required per bit transmitted. This figure is well below the
accepted limit [48, 49] for efficient electromagnetic communication of kBT ln(2) per
bit (about 25 meV/bit at 423K).
This simple communication architecture ignores the substantial increase in biterror-rate (BER) that such a scheme would suffer due to thermal noise (i.e. blackbody
radiation), even with perfect collimation and a perfect receiver node. Unsurprisingly,
the kBT ln(2) limit for all electromagnetic systems is fundamentally connected to this
thermal noise; the limit and the power density of this noise source both vanish as T -+
0. In Chapter 4, we explore theoretically the limits of energy-efficient communication
30
with a Carnot-efficient heat pump in the presence of noise due to blackbody radiation.
Before that, however, it is constructive to review a few basic results from the extensive
literature on this topic.
Photoneryk...
C
0
0
0.101
~~0
kBT-In(2)
CL
I
0.01
CL
0.001
10
10
10
Photon Emission Rate (ifs)
Figure 1-5: At low power, a conventional LED may generate a photon with an arbitrarily small amount of work. As with any endo-reversible heat pump, the efficiency
scales inversely with the output power resulting in the trade-off between photon emission rate and per-photon work consumption. For low photon emission rates, the
per-photon work has been experimentally observed to fall below kBT - log(2), raising
interesting questions about the limits of efficient communication.
In 1948, Claude Shannon published a paper in the Bell System Technical Journal
entitled A Mathematical Theory of Communication [50]. The manuscript is often said
to have laid the conceptual groundwork for the digital revolution by proving that all
forms of digital and analog information could essentially be measured in the same
units- typically bits. In this same paper, Shannon proved that for a known physical
channel with known noise properties, one could calculate a maximum capacity for the
transmission of information per unit bandwidth.
In his paper, Shannon considered the problem of communication in the presence
of Additive White Gaussian Noise (AWGN). Interestingly, this noise distribution
corresponds to the thermal noise distribution for field variables (i.e. voltage V or
electric field E) in the quantum degenerate limit hw <
31
kBT where most electronic
circuits and radio-frequency links operate [51].
For this type of noise source, the
following formula for channel capacity may be proven [50]:
C = Af log 2 ( +
(1.8)
where C is the channel capacity in bits per second, Af is the bandwidth of the
channel, P is the average power of the signal, and N is the average noise power per
unit frequency within the channel's bandwidth. For a given noise power density (per
unit frequency), the formula indicates how much power must be present in the signal
field to communicate at a given rate C.
This result is typically associated with discussions of the fundamental energy requirements for any physical process of communication.
To see why, consider the
question of linear electro-magnetic communication using a single channel (i.e. a single transverse mode with a single polarization state) in the presence of blackbody
radiation.
Assume the noise power N comes from thermal fluctuations of the electromagnetic
field and the frequencies of interest are assumed to be in the quantum degenerate
limit. Since the quanta become irrelevant in this limit, the field may be described
by classical statistical fluctuations so that for each mode, the average energy of the
Then if we consider a channel of length
fluctuating field is kBT by equipartition.
L
> c/f, the density of forward-traveling modes is simply 1/(2ir/L) in k-space or
L/c in
f.
Combining this information, we arrive at the thermal energy density per
unit frequency in the channel of length L:
L
U
-- = - kBT
Af
C
Since this channel empties its thermal energy at the receiver end in time At
(1.9)
=
L/c,
the noise power is simply:
U
N =A-=
Af
L
-kBT
c
32
c
- = kBT.
L
1.10
Substituting this expression into the channel capacity formula above allows us to
relate the rate of information flow C to the rate of energy flow in our signal P:
C = Af log 2 1 + k
).
kBTA f
(1.11)
The maximum ratio of C to P appears at low power, where the logarithm can be
expanded to give the minimum energy per bit transmitted under these assumptions:
min
(-
=
kBT ln(2).
(1.12)
It has been pointed out by numerous authors [49, 52, 53] that this formula does
not imply that there is a minimum energy cost to communication.
The canonical
example is mailing a hard drive. Considering this example recasts communication as
a choice of reference frame rather than a physical process. In contrast, many authors
have come to the conclusion that the operation of erasure does appear to carry with
it an unavoidable energy cost [49].
Over the years, several authors have used specific examples to point out the relationship between the kBT hn(2) result and the assumptions that went into its derivation.
One commonly pointed out assumption is that of the field's linearity with
respect to the addition of noise to the signal [49, 52].
In this work, we point out another assumption which we believe may not have
been previously raised. We point out that there is a distinction between the rate of
flow of entropy-free work into a source and the outflow of electromagnetic energy.
One immediate question of interest presents itself: is there a limit to the ratio of an
emitter's work expenditure rate to the information flow rate it may encode:
min
C
-
=
mm
m
bit
- ?
(1.13)
For the emitter involved in the electro-luminescent cooling experiment, operation
below the cooling power peak allows the ratio of power to work consumption rate P/W
to be arbitrarily large. As a result, it presents a surprisingly accessible platform for
33
Classic AWGN Symbol Space
-- 1 Bit-1B
0.1
-0
Heat Pump Symbol Space
0.1
BIt
0co
CD
0
0
0
-0.1
-0.1
0
0.2
0.4
0.6
Time (s)
0.8
1
0
0.2
0.4
0.6
0.8
Time (s)
I B#
1
-1 Bit
04
04
-82 -0.1
0
Voltage (V)
0.1
..8.2
0.2
-0.
0
Voitage (V)
.
0.2
Figure 1-6: Typical members of the symbol spacedormee
nunication in the presence
of thermal electromagnetic noise. The left column shows two representations of a pair
of symbols for communication with a conventional signal. The right column shows
two representations of a pair of symbols for communication with a heat pump.
experimentally exploring the limits of energy-efficient electromagnetic communication in the non-degenerate noise hw > kBT limit. Naive interpretation of this fact
combined with Equation 1.8 suggests that arbitrarily efficient communication should
be possible using a heat pump.
Upon closer examination, however, the signal generated by a heat pump may
be arbitrarily efficient in the power-conversion sense (i.e.
many symbols per unit
energy), but the '1' symbol produced in the efficient regime is less distinguishable
from the '0' symbol and therefore leads to less information flow (i.e.
fewer bits
communicated per symbol transmitted). This is because the '1' symbols it transmits
are composed of a different distortion of the photon field from thermal equilibrium,
as shown in Figure 1-6.
Interestingly, this result suggests a fundamental trade-off
between the disorder required for efficient heat-pumping and the distinguishability of
the symbols in the codebook, leading to a direct connection between the informationtheoretic entropy of a source and the physical entropy exiting the apparatus used
to communicate it.
A thorough information-theoretic analysis of this trade-off is
34
presented in Chapter 4.
1.5
Potential Practical Applications
The basic result of an LED operating as a heat pump also holds consequences for
several potential practical applications.
1.5.1
Infrared Spectroscopy
As seen in Figure 1-7, several common molecules have distinct absorption features in
the mid-infrared wavelength range. For this reason, substantial attention has been
given to developing sources for absorption spectroscopy here [54, 12, 5, 35, 10, 56].
wavenumber (cmn )
2223
NO
Methan
CO 2
I.00
NN2
wavelength (pfr)
Figure 1-7: Numerous abundant molecules have absorption features in the midinfrared wavelength range, making it a valuable band for spectroscopy. This figure is
taken directly from Figure la of Reference [54].
A review of the available emitters appears in Table 1.2. Two main types of emitters are available: thermal emitters and light-emitting diodes. Thermal emitters are
efficient, but emit over a wide wavelength range and carry long thermal time constants which limit their direct switching speeds. Mid-IR LEDs may be switched at
35
Type
Producer
Model No.
Wall-Plug
Efficiency
Emitted
Power
Modulation
Frequency
Thermal
HelioWorks
EP3872
0.15 %
3.5 mW
2 Hz
Thermal
HawkEye Tech-
IR-55 R
0.29 %
2.7 mW
10 Hz
NL8LNC
0.25 %
5.6 mW
5 Hz
Tun IR
0.20 %
0.27 mW
1 Hz
nology
Thermal
IonOptics (ICX
Photonics)
Thermal
IonOptics (ICX
Photonics)
Thermal
Heimann Sensor
HSL EMIR2000R
0.31 %
1.4 mW
10 Hz
Thermal
Intex
MTRL-17900R
0.39 %
3.8 mW
15 Hz
LED
ICO
(RMT)
LED-42
0.01 %
0.01 mW
100 kHz
Ltd
www.optico.ru
LED
IoffeLED
LED42Sc
0.15 %
0.03 mW
10 MHz
LED
Roithner
LED-43
0.013 %
0.01 mW
10 MHz
Table 1.2: Comparison of existing sources for spectroscopy around A = 4.25pm. Note
that wall-plug efficiency and emitted power consider only the power within a spectral
band of AA = 0.45pm and an angular cone of 300. Table adapted from [4].
36
hundreds of MHz, allowing the use of lock-in techniques to improve the Limit-ofDetection (LOD) [57, 55], but are relatively inefficient in terms of power conversion.
However, since LED emission is concentrated at photon energies just above the material bandgap, the so-called "spectral wall-plug efficiency" (i.e. wall-plug efficiency
considering only emitted photons in a narrow band of spectroscopic interest) of an
LED can be competitive with that of a thermal source. Analysis of the characteristics
required for spectroscopy suggests that a reasonable figure-of-merit for an opto-pair
(emitter-detector pair) system is the so-called "normalized LOD," measured in parts
per million per mW of source drive power per
Is of lock-in time constant [4], and
suggests that LED-based spectroscopy systems are substantially superior to those
that use thermal emitters.
High-Temperature Environments
Although infrared LEDs can be designed to emit at a variety of wavelengths of spectroscopic interest [54, 21] and may be directly modulated at the high frequencies
employed by lock-in techniques [4], conventional devices suffer from carrier leakage
and Auger recombination which limit their utility at high temperatures. Unfortunately many of the largest applications for such spectroscopy tools are tied to such
harsh environments.
For example, radiation near A =3.3ptm is strongly absorbed by methane, so LEDs
at this wavelength could be used for downhole oilfield spectroscopy. As the pace of
discovery of new oilfields has diminished, oil companies have been forced to focus on
upgrading existing ones to meet rising global demand. To accomplish this efficiently,
they must avoid costly errors in the design of surface extraction facilities and capital
misallocation caused by inaccurate reserve estimates. As a result, renewed focus has
fallen on developing platforms for in-situ determination of the gas-to-oil ratio (GOR)
in the hydrocarbon-rich fluids present downhole in an oilfield [58]. However, spectral
data in the mid-infrared has been unavailable downhole due to a lack of sources and
detectors for this purpose [59].
An efficient, fast-switching source at 3.3pm could
benefit such a spectroscopy system if it were capable of operating at temperatures of
37
175-200'C and pressures >100 MPa.
Radiation near A =4.2[pm and A =4.7pim are strongly absorbed by carbon dioxide
and carbon monoxide respectively. Spectroscopic analysis in these bands could be
used to determine the composition of combustion products.
The extreme temper-
atures found in vehicle exhaust and industrial flue gases (as well as the machinery
around them) may require high-temperature performance for sources intended to perform these operations in situ [601.
Ultra-Low-Power Systems
Recent advances in the efficiency of micro-electronic circuits have enabled a new
generation of ultra-low-power sensor and display systems based on LEDs.
Here,
the LEDs are frequently the primary load, and therefore constrain the mobility and
lifetime of the overall systems.
For example, the power budget of an ultra-low-power pulse oximetry system developed in 2010 appears in Figure 1-8 [3]. Here, the differential absorption of two LEDs
(one at 660nm and another at 940) is used to detect the oxygen concentration of the
blood in a patient's finger. Over 90% of the total power in this system is devoted
to the LEDs and their associated switching control circuits. The authors consider
this to be practically valuable because it permits a single set of 4 AAA batteries
to operate the sensor for up to 60 days. While this is more than 10x longer than
other implementations, if the 660nm and 940nm photons could be generated twice as
efficiently, the operating lifetime between charges could be increased to 120 days.
The requirements for the brightness and wavelength of the source in this pulseoximetry system are significantly less demanding than in general-purpose indoor illumination. It therefore seems likely that any improvements to state-of-the-art LED
efficiency resulting from design for low-voltage operation will benefit ultra-low-power
systems before they are relevant for solid-state lighting.
38
Power Consumption per Block
Value
Oscillator/LED & Switching Control
4.4mW
Two Transimpedance Amplifiers
80pW
Two Low-pass Fitters
300AW
Less than 4001W of
Ratio Computation
2.2pW
processing power
Rferene Otnerator/Bias Circuitry
1 1.5pW
Total
4.8mW
Figure 1-8: Recent advances in efficient amplification circuitry leave state-of-the-art
ultra-low-power pulse oximetry systems with power budgets dominated by inefficient
red and infrared LEDs. Figure taken from Table II of Reference [3].
1.5.2
Lighting
Recent advances in solid-state lighting have made available LEDs capable of converting electrical power into white visible optical power above 50% wall-plug efficiency
[61, 62] with further improvement anticipated in the coming years [63]. These results,
however, are typically achieved with pulsed operation, where the emitting diode does
not heat up. So-called "hot" steady-state testing leads to substantially diminished
efficiency [7]. As discussed in
§ 1.1, the ubiquitous loss mechanisms of non-radiative
Auger recombination and carrier leakage are largely responsible [18, 19].
Experimental confirmation of electrical-to-optical power-conversion efficiency in
excess of unity raises the possibility of building electrical light bulbs with no net
waste heat generation [37]. Not only would such bulbs be highly efficient, but they
could result in large cost savings from the removal of heat sinks that dominate material
costs and improvements in bulb lifetime due to the abatement of thermally-accelerated
failures of driver components such as electrolytic capacitors
[73.
Although the work in this thesis focuses on devices which emit outside the visible
39
spectrum, we discuss the future of solid-state lighting technology in light of our results
in § 6.4.
1.5.3
Other Applications
The net absorption of heat from the emitter's lattice combined with the ease of
achieving long ballistic path lengths for infrared photons in semiconductors makes
electro-luminescence interesting as a solid-state cooling technology [36, 37, 38, 39, 40,
34].
Less widely-discussed but conceptually related is a generalization of ThermoPhoto-Voltaic (TPV) electrical power generation known as Thermophotonics (TPX)
[64, 65, 66]. Here, the passive narrow-band-emitting surface of the TPV is replaced
with an active device, an LED. When V = 0, the passive and active emitters have
identical performance, but as a forward bias is applied, the emitted power rises more
rapidly than the input power. In fact, since this ratio diverges as V -+ 0 and the
efficiency of extracting work from those photons is nonzero (Temitter
> Tabsorber),
the
maximum net output power (i.e. electrical power from the photo-voltaic minus LED
drive power) is guaranteed to take place at V > 0. Nevertheless, emitter surface materials are chosen based on other criteria, for example their high-temperature stability
and ease of patterning into photonic crystals, and new constraints would be placed
on them by the need to make the emitting surface a direct-gap semiconductor inside
a diode structure. In light of these constraints, it is likely that high %QE emitters
would be required to improve TPV performance.
Finally, LEDs with extremely high wall-plug efficiency may be useful for free-space
communication by satellites. When these power-constrained satellites send signals to
the ground, the power consumed in reconstructing the signal is much cheaper than
the power consumed in transmitting it. As a result, the constraints on these systems
closely approximate the problem of encoding information into the outgoing electromagnetic field with a minimum of electrical power consumption.
In fact, schemes
such as Pulse-Position Modulation (PPM), which allows multiple bits of information to be communicated per photon [67], find application here. Moreover, the long
40
wavelengths of certain "atmospheric windows" [56] for which efficient lasers are not
available correspond to wavelengths at which heat pumping LEDs theoretically emit
more power at high efficiency. As a result, the LED technologies developed in this
work may prove useful for such niche communication systems.
1.6
Thesis Outline
In Chapter 2 we use various simplified device models to explain LED operation above
unity efficiency and explore device design concepts intended to push LED performance toward the limits imposed by The Second Law.
In Chapter 3 we validate
aspects of this framework through a series of experiments on existing devices.
In
Chapter 4, we explore the ultimate consequences of these design improvements for
photonic communication by exploring the physical limits of energy-efficient communication with a heat pump.
In Chapter 5, we consider the practical applications
of these thermo-electrically pumped LEDs to power-constrained infrared absorption
spectroscopy systems operating in high-temperature environments.
41
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42
Chapter 2
LEDs as Heat Pumps
In this chapter we explore the thermodynamic behavior of LEDs and construct a
framework for their analysis as heat pumps. In
§ 2.1, we review electron transport
in an LED with emphasis on the flow of entropy. Then in
§ 2.2, we organize these
flows within a basic model of an LED as a thermodynamic heat pump. In
§ 2.3,
we explain why all LEDs should in theory act as heat pumps at low forward bias
voltage. In
§ 2.4 we analyze an idealized reversible LED and discuss its relationship
to existing devices. We find that an ideal LED achieves the Carnot efficiency and that
both ideal and non-ideal LEDs face the same trade-off between power and efficiency
that all thermodynamic machines operating at nonzero power experience.
Then in
§ 2.5 we present initial work on the design of LEDs for efficient operation in the heat
pumping regime. Finally, in
§ 2.6,
we generalize this framework to describe the flow
of electrons around a closed circuit as the flow of a working fluid through a closed
thermodynamic cycle.
2.1
Electron 'ransport and Entropy Flow in LEDs
Although the claim of steady-state electrical-to-optical energy conversion at aboveunity efficiency may appear to violate the Laws of Thermodynamics, it is not only
consistent with them, it's presence at low power is a fundamental property of any
LED. The issue of energy conservation in q > 1 operation (i.e. the First Law issue)
43
is resolved by the inclusion of lattice heat absorption within the diode.
This explanation immediately raises a question about consistency with the Second
Law of Thermodynamics. Because the vibrational energy of the lattice is heat, the
net absorption of energy from the lattice must be associated with a net absorption
of entropy as well. The issue of entropy non-deletion (i.e. the Second Law issue) is
resolved by the entropy associated with the emitted incoherent photons. That is to
say, for a bounding surface drawn around an LED operating at r7 > 1, the net inflow
of entropy due to lattice heat absorption is offset by an outflow of at least as much
entropy through the photons. Furthermore we may calculate the inflow and outflow
of entropy to the electron-hole subsystem due to thermally-assisted carrier injection
and radiative recombination, and thereby provide a more mechanistic explanation of
how the device transports the absorbed entropy from the lattice to the photon field.
By calculating these entropy flows we arrive at a more complete model of device
operation which complies with a continuity equation for entropy flux. That is to say,
we may show that our model of LED operation is not only globally consistent with
The Second Law, it is locally consistent as well.
2.1.1
Current Continuity
As depicted in Figure 2-1, a conventional double hetero-junction light-emitting diode
consists of a layer of narrow-bandgap intrinsic semiconductor sandwiched between a
pair of wider-gap layers. The wider-gap layers are doped p and n-type and have metal
contacts attached to form the positive and negative electrical terminals of the device
respectively. When a forward voltage is applied, electrons from the n side and holes
from the p side are injected into the active region. There they undergo recombination
through various mechanisms, connecting the electron-type current from the n side
with hole-type current on the p side to satisfy current continuity. Although some
leakage does occur (i.e. minority carriers diffusing across the double-line boundary
in Figure 2-1), the hetero-structures are designed to minimize this. Since the basic
transport processes can still be understood while neglecting leakage, we will do so
here in § 2.1. In our simplified picture then, for each quantum of charge that flows
44
E*
LUE)
Position x
Figure 2-1: Simple band diagram for a double hetero-junction LED at low forward
bias. The basic transport processes described in § 2.1 are overlaid for the reader's
convenience. The double-line with diamonds is a fictitious boundary that we assume
is crossed only by recombining carriers in this simplified analysis.
45
between the terminals of a diode at sub-bandgap forward bias voltage, the following
three processes must take place:
" One electron must escape the cathode, traverse the n-doped quasi-neutral region, enter the intrinsic active region, and climb a potential energy barrier to
the recombination site.
" One hole must escape the anode, traverse the p-doped quasi-neutral region,
enter the intrinsic active region, and climb a potential energy barrier to the
recombination site.
" The electron and the hole must recombine by some process which conserves
energy and momentum.
The first two processes are referred to as the thermally-assisted injection of electrons and holes respectively. The last is recombination. After a short introduction to
the concept of quasi-equilibrium, we will proceed to analyze these two processes to
complete our picture of electron transport in this simplified model.
2.1.2
Quasi-Equilibrium
Electronic transport in these devices is typically described in the framework of quasiequilibrium. In quasi-equilibrium, the single-particle states in a given band and region
of the device are taken to be in sufficiently close contact to be occupied according to
some Fermi-Dirac distribution, with some Fermi level EF and some temperature T.
Typically this assumption is justified by the fast phonon scattering present in common
semiconductors at room temperature. Typical timescales for carrier momentum and
energy relaxation are on the order of picoseconds and nanoseconds respectively, while
the timescales for carrier diffusion processes connecting different regions of devices
with micron-scale features are much slower [68]. Thus, under the assumption of quasiequilibrium, specification of EF,e(X), Te(x), EF,h(X), Th(x) at each point constitutes
a complete description of transport within a device.
46
Moreover, fast phonon scattering typically limits differences between the carrier
temperatures and the temperature of lattice. As a result, it is common to see band
diagrams which depict only EF,e(v) and EF,h(x) across the device, and assume
Te (X) = Th (x) = Tlattice
(2.1)
-
The concept of quasi-equilibrium is useful in great part because (in isothermal
systems), carriers only flow in response to differences in EF.
When two adjacent
points in space have different electrical potential, an electric field is present a drift
flux of carriers occurs in response to it. Likewise, when two adjacent points in space
have a different number density of carriers, diffusion leads to a flux from high density
to low. The quasi-Fermi level combines these two processes in such a way that a
flat
EF,e(X)
indicates that the electron drift and diffusion fluxes are balanced and
offsetting. That is to say, the conduction states at the points in space where
EFe(X)
is flat may be considered to be in equilibrium. And of course the same is true for
holes when EF,h(x) is flat.
On the other hand, when EF,e(x) and EF,h(x) are not flat, adjacent positions are
not in equilibrium. Gradients of quasi-Fermi levels within the conduction band drive
electron flows, and likewise for the valence band and hole flows. A difference between
the Fermi levels for the two bands at the same point in space drives generation or
recombination.
VEF,e(x)
2.1.3
# 0
->
Electron Flux
(2.2)
VEF,h(x) 4 0
-
Hole Flux
(2.3)
EF,e - EF,h > 0
--
Recombination
(2.4)
Thermally-Assisted Injection
With no applied voltage, all Fermi levels at all positions remain equal. There is no
net injection to the active region. Still, the gas of electrons on the n side (and holes
on the p side) is perpetually emitting and absorbing phonons to exchange energy and
47
entropy with the lattice vibrational modes. These processes are in equilibrium when
the amount of entropy added by a small addition of energy to each system is equal
(i.e. they are at the same temperature).
When a small forward bias voltage is applied, the potential energy of electrons at
the n-contact becomes higher than those at the p contact. As a result, VEF,e(x) and
VEF,h(x) become nonzero and net flows of these carriers occur as shown in Figure 2-1.
In order for an electron to flow from the states relevant for conduction at the
n-contact (x = L) to those relevant at a recombination site (x =
XR),
it must climb
a potential barrier. The same is true for a hole from the p-contact (x = 0). As is
readily seen from the figure, the combined heights of these two barriers,
AVeiectrons
and AVholes, are simply related to the natural energy scales of the problem.
AVeectrons
+ AVholes = Egap,active - qV + O(kBT)
,
where
(2.5)
and
(2.6)
XR
AVholes = Hmetal,p
AVeectrons =
j
+
]R -VEv(x)dx
+ O(kBT)
VEc(x)dx + Rn,metal + O(kBT)
(2.7)
Here Ec (x) and Ev (x) denote the conduction and valence band energies respectively,
and
r1 a,b
denotes the Peltier coefficient at the metal-semiconductor interface with
material a at left and b at right. The terms of order kBT are present because electron transport occurs within the conduction band rather than at the band-edge, and
likewise for holes. Because kBT <
Egap,active,
they will not figure prominently in our
analysis here.
So where does an electron get the energy to climb this barrier? The answer is
that lattice heat is absorbed all along the x = 0 to x = xR path by means of the
Peltier effect. Typically the Peltier effect is described as thermo-electric effect at an
interface: when an electric current I crosses from some solid material a to another
material b, heat is removed from the lattice vibrations in the vicinity of the interface
at a rate
Q
proportional to the current:
Q
=
A,BI
48
.
(2.8)
Re-thermalization
4
G
e flux
G
D
u.-
-E-
F~e
LU
-2
0 0.5
1
Figure 2-2: The Peltier effect at an interface. Hot carriers are thermionically emitted
over a hetero-junction barrier. A re-thermalization process ensures that the electrons
in material a remain in a thermal distribution. This process requires the absorption
of approximately AEc of lattice thermal energy per electron, resulting in so-called
"Peltier cooling."
A more physical picture of the Peltier effect is found in Figure 2-2, and may be
readily generalized to conduction away from interfaces. In quasi-equilibrium, conduction between two points in a given band of a given solid can be ascribed to a difference
in EF between those points. Consider Figure 2-3a. If we discretize space and consider
adjacent points, we see that in a region with an electric field (i.e. VEc
# 0), there is
both a finite difference in the Fermi energy as well as a finite difference in the energy
of the conduction states available for transport. Using this procedure, we can see
that the transport in Figure 2-3b will also lead to lattice heat absorption.
Generally speaking, whenever the direction of carrier flow f/q opposes the electric
force qE, the Peltier effect causes the carrier population to absorb heat from the
lattice. This is exactly the situation in Figure 2-1.
The amount of heat absorbed
across the device during the thermally-assisted injection of each electron-hole pair is
49
Re-thermalization
Re-thermalization
4f
(
eflul
Ece
----------
EC
Ee
~
(a) Discrete model.
-_--....
(b) Continuous model.
Figure 2-3: Models for electron transport in a region where the Fermi level gradient
drives a net carrier flow against the direction of electric field drift. The continuousspace model at right is a generalized version of the Peltier effect shown in Figure 2-2.
The generalization is intuitive when the quasi-equilibrium concept is applied, as is
done in the model at left.
equal to the height of the potential barrier they must climb. In fact, a Peltier term
I corresponding to the thermal energy absorbed per pair may be substituted into
Equation 2.5 to give:
Egap,active -
2.1.4
qV
AVeectrons + AVhoes
(2.9)
Recombination
The final transport process required to maintain current continuity is the recombination of injected electrons and holes. Although some leakage happens in any real
device, for simplicity we consider only recombination sites in the active region.
As with the majority carriers in the doped regions, even when the device is off the
electrons and holes in the active region are perpetually experiencing generation and
recombination as the result of their interaction with other reservoirs. These processes
can be thought of in terms of the following chemical reaction equation:
e~ + h+ (
50
) Ubandgap
(2.10)
where e- is an electron, h+ is a hole, and Ubandgap denotes some excitation with
energy (and other conserved quantities) equal to that of the electron-hole pair. As
with any chemical reaction, the reactants and products are in equilibrium at some
concentrations.
values (i.e.
When the concentration of electrons n and holes p exceeds these
when np exceeds the squared intrinsic carrier concentration n?), net
recombination occurs and the reaction in Equation 2.10 is driven from left to right.
Likewise, when n and p are below their equilibrium values, net generation occurs and
the reaction is driven backwards.
Each time that an electron-hole pair is annihilated, both energy and entropy are
removed from the electron and hole gases. That is to say, the number of microscopic
configurations in which the conduction and valence bands can be occupied is reduced.
This entropy, however, cannot disappear entirely. Doing so would violate The Second
Law.
Instead, the entropy which is removed from the electronic sub-system (i.e.
the degrees of freedom from excitations of the conduction and valence band states)
is transported to another sub-system at the same location in the device. Which
sub-system that is depends on where the electron-hole pair's energy went. For nonradiative recombination, the destination is the lattice. For radiative recombination,
the destination is the photon field.
When a non-radiative recombination event occurs, Ubandgap is deposited into the
lattice excitation spectrum. These new excitations allow the lattice to inhabit a larger
space of microscopic configurations and thereby increase the entropy of the phonon
field. The amount of entropy (AS)iattice may be calculated simply by making use of
the lattice temperature Tattice:
(AS)Iattice = Ubandgap
Tiattice
(.1
The same is true for the photon field. The number of possible microscopic configurations of the photon field also increases. In fact, the definition of brightness temperature given in Equation 1.5 was designed specifically to quantify the additional
51
entropy (AS)phtOnl that appears in the photon field with Ubandgap:
(AS)photon
2.1.5
Ubangap
(2.12)
Tphoton
Continuity of Entropy Flux
When an LED is put into a forward bias condition such as shown in Figure 2-1, it is
taken out of equilibrium. After a short time, its sub-systems approach a condition of
quasi-equilibrium and the LED operates in steady-state. In this steady state, current
flows in the direction of bias, electrical power is drawn from the power supply, some
light is emitted, the device heats up and loses heat to the environment, and the
net power entering the device through electrons, phonons, and photons reaches zero.
Although such low voltages are not commonly utilized, these operating points are
easily measured on existing devices, as seen in Figure 2-4.
Temp. Model Exper.
25*C
o
-e
84*C
--
3. ---
10-6
0.8
A
-
C
.. ...
~10
---
0.6
0.4
10
0.2a)
0
102
0 >
1-s
106
-
1e
A
-2
0
Current (A)
Figure 2-4: I-V and L-I curves for an existing infrared LED emitting at A = 2 .15ptm.
Current flows and easily detectable levels of light are emitted even when the applied
voltage qV is far below the bandgap energy.
The steady-state condition is characterized by steady flows that obey continuity
equations. Since charge is conserved, a complete description of steady-state operation
52
obeys the following continuity equation at each point in space:
(-q)V.Je
where Je and
Jh
(2.13)
+ qV.h = G-R
are the electron and hole fluxes, G is the local rate of electron-hole
pair generation, and R is the local rate of electron-hole recombination. A solution
may be visualized as in Figure 2-5.
S Recombinationi
Mins Genraion
Charge Flow
Figure 2-5: Charge flow in our simplified model obeys current continuity.
After considering the thermodynamics of thermally-assisted injection and recombination, entropy flow may also be considered. The Second Law permits the generation
of entropy, so the analogous continuity equation is:
V - is,e + V - is,h + V
where
JS,e,
JS,h,
J,lattice, and
JS,photon
-
S,lattice
+ V - JS,photon =
S
(2.14)
are the entropy flux carried by the electrons,
holes, phonons, and photons respectively, and S is the irreversible entropy generation
rate. A solution may be visualized as in Figure 2-6. It is worth noting that such a
picture may be drawn for any electronic device, and that all inefficiencies in their operation can be accounted for by some $, including those with significant consequence
for the engineering systems they compose.
53
Photon
Field
-----
Electronic
- -
Block Arrows Denote
Entropy Flow
Phonon
Field
r
Figure 2-6: Cartoon depicting entropy flux in a simple double hetero-junction LED
structure.
2.2
The Heat Pump Picture
Let us now abstract away the internal dynamics of the electronic system and consider
just the flows of entropy and energy between the three sub-systems in Figure 2-6. For
each quantum of charge that flows through the device, one net recombination event
occurs. We would like to know how much entropy enters and leaves each system.
Knowledge of the energy flows between the sub-systems combined with Equation 2.11
and Equation 2.12 determines the entropy flows in and out of the lattice and photon
fields respectively. However, because the electronic sub-system is not in equilibrium
at any fixed temperature, we must examine it more closely.
We begin with a simple model for the electronic degrees of freedom at a single point
in space. Consider the statistical two-level system shown in Figure 2-7. Define
be the probability of occupancy for the higher energy state,
fv
fc
to
to be the occupancy
of the lower state, and take the states to be separated by energy AE. In terms of
these quantities then, we may write expressions for the total energy and entropy of
54
AE
Figure 2-7: A statistical two-level system.
the system:
U=fc-AE + (fc+fv).Eo
and
(2.15)
S = -kB [(f, In fc +1(I - fc) ln(1 - fc)) + (fv - fc)]
.(2.16)
If we define a degree of freedom corresponding to excitation from the lower state to
the upper state, we may find the amount of entropy change in the system per unit
energy change for changes of this type. This ratio can be expressed conveniently as
the inverse temperature T 1 of the electronic system:
T-
OS
OU
dS
dfc
dS
dS
f
U
dfc
dfv
dU
(2.17)
df,
= -kB [ln fc + 1 - ln(1 -kB
n
-kB
S
_
T - = -- =
OU
n
(
fc)
-1]
(2.18)
c
) +
AE
(2.19)
kB
(2
In (
f
If we constrain the probability for occupancy of either state fc
.-
(2.20)
+ fv to be 1 so that
the Fermi level EF falls halfway between the states in energy, the equation above can
be rearranged to recover the expression for Fermi-Dirac occupancy in equilibrium at
55
temperature T:
exp
exp
(
EB)
(kBT
(
=
/2
1f
1
I-fv
--
fc= (exp
fc
=
(
fc)
2
(2.21)
fc
1
(2.22)
Eupperstate - EF
(2.23)
-1
The preceding result is unsurprising, but clarifies an important point. The inverse
temperature of a Fermionic system, meaning the amount of entropy that is added to
it when a unit of energy is added, can be calculated purely from the occupation of the
states. That is to say, two situations which are described differently must still have
the same temperature if their occupancies are the same. To see how this applies to
the thermodynamics of electrons and holes in the active region of an LED, consider
the following slightly more concrete example.
---------
Ege,EFh
-
~
--- - AEFY=
Ea--p
- - Ege
---
EF,e, EF~h
- - - - - - - - - EF~h
Tlatte = 300K
(a) Two-level
equilibrium.
r
= 300K
system in
Tttie = 300K
T* = 600K
(b) Two-level system excited electrically.
Tttice = 600K
T* = 600K
(c) Two-level system excited thermally.
Figure 2-8: Two-level systems that exhibit different types of excitations which lead to
the same occupation of states have the same effective temperature T*. In Figure 2-8a,
the electronic system is in equilibrium with a 300K lattice. In Figure 2-8b, the
electronic system is not in equilibrium with the lattice. A Fermi-level separation has
increased the occupancy of the higher-energy state and decreased the occupancy of the
lower-energy state. Although the lattice temperature in Figure 2-8b is still 300K, the
effective temperature T* that indicates the ratio of entropy to energy in the electronic
system is 600K. In Figure 2-8c, the electronic system is again in equilibrium with the
lattice, but the lattice is now at 600K. The occupancies fc and fv in Figure 2-8b and
Figure 2-8c are identical, so their values of T* are the same.
56
Consider an ensemble of homogeneous quantum dots, each with a single relevant
low-energy electron state and a single relevant high-energy state. Again let the total
charge between the states be
fc +
fv = 1 to ensure charge neutrality. If the lattice
of these dots is kept at 300K and no electrical excitation is applied, the statistical
two-level system will have a Fermi level at exactly halfway between the two states
and the occupancies
fc and fv can be determined by the Fermi-Dirac distribution.
This situation is described by the diagram in Figure 2-8a.
Now let us excite this system. Since a recombination event removes an electron
from a higher energy state and places it in a lower energy state (and vice versa for
a generation event), let us again focus on the degree of freedom corresponding to
fc
-+
fc
+
6f and fv -+ fv - 6f. Note that this is the same degree of freedom that
we used in Equation 2.17 and corresponds to excitations that conserve total charge.
Figure 2-8b and Figure 2-8c show two physically different types of excitations that
result in the same values of
f, and fv.
In Figure 2-8b, the electrical system has
been taken out of equilibrium with the lattice by an applied voltage qV = AE/2.
In Figure 2-8c, the absolute temperature of the lattice has been doubled. In both
situations, the number of kBT'S of between each state and its quasi-Fermi level has
been halved. As a result, the Fermi-Dirac occupation of the states in both situations is
equivalent (i.e.
fc and fv are the same in both). Since the total entropy S and energy
U of the system is determined solely by
fc and fv, the procedure from Equation 2.17
1
yields the same temperature T = (S/U)
for either situation. From now on we
will refer to this temperature as the effective temperature T* seen by the inter-band
processes like radiative recombination.
From these examples, we may follow [43] to a general expression for T* in a
semiconductor whose quasi-Fermi levels are separated by an energy qV in a region
with bandgap energy
Egap:
T* = Tiattice
(V_-_
1 _E
.
(2.24)
egap
From here, we may significantly simplify the internal dynamics of the electronic
57
system from a picture like Figure 2-6. For inter-band processes in which the electronic
system loses energy to another reservoir (e.g. recombination), the corresponding loss
of entropy is determined by T* from Equation 2.24.
By contrast, for intra-band processes like thermally-assisted injection, the twolevel system model is not necessary. For the 3D semiconductors in the simple LED
model we will use going forward, the distribution of carriers at a given position and
within a given band is approximately thermal.
At a position in the device where
a forward bias causes the carriers in a specific band to flow "uphill" toward higher
electrostatic potential energy, the injection process involves an inflow of carriers at
low energy and an outflow of carriers at high energy. As described in § 2.1.3, in
steady-state the energy absorbed via Peltier heat exchange with the lattice supplies
the energy for the re-thermalization of these carriers by moving carriers from more
occupied low-energy states into less-occupied higher-energy states. As they do so,
the carriers move from portions of phase space which are more densely populated
to portions that are more sparsely populated. This flow of carriers thus increases
the number of microscopic configurations of the electronic states at this position;
the carriers thus absorb entropy along with energy from the lattice. The amount of
entropy absorbed is determined by the same temperature that determines the spread
of carriers in phase space in that location within that single band.
As a result,
for intra-band processes in which the electronic system gains energy from another
reservoir (e.g.
thermally-assisted injection), the corresponding amount of entropy
added to the electronic system is given by the local lattice temperature TiatticeIf we modify Figure 2-6 by consolidating all flows of entropy together, and we also
include the corresponding flows of energy from the various sources, the picture becomes the canonical diagram for a thermodynamic heat pump as shown in Figure 2-9.
2.3
LEDs in the Low-Bias Regime
As described in Chapter 1, it has long been known that at low output power an LED
may in principle operate with wall-plug efficiency q; far in excess of unity [29, 31].
58
Photon Field
Irreversible
Entropy
Generation
Non-radlative
Recombination
Phonon Field
Phonon Field
Figure 2-9: The flows of entropy and energy between various sub-systems in an LED
can be organized in the canonical picture of a thermodynamic heat pump. At left is
an idealized picture. The irreversible contributions shown in the picture at right can
be quantified for any real LED using the arguments from § 2.2.
That is, its optical output power (L, measured in Watts) may be a large multiple of
its input electrical power (IV, also measured in Watts) in steady-state. In fact, the
Second Law of Thermodynamics permits an arbitrarily large value of 7.
This is the
situation in the low-bias regime we will discuss here.
We pause briefly to address a question of terminology. Typically the ratio of the
rate at which heat (in this case, photons) is emitted by a heat pump to the rate
at which it consumes work is called the pump's heating coefficient of performance
COPH, but in this work we refer to this quantity as the wall-plug efficiency q. We
note that in other electrically-driven sources of incoherent light for which q < 1, the
output energy also has entropy associated with it, so that L/(IV) would be most
appropriately termed COPH in this case as well. Nevertheless, convention dictates
that L/(IV) is referred to as the wall-plug efficiency q. For this reason, we follow
several previous authors [31, 29, 14, 351 in referring to this quantity as the wall-plug
efficiency (or simply efficiency) q, which we allow to exceed unity.
Recall the expression for the wall-plug efficiency of an LED from
(OW)
q=
qV
59
7
7EQE
-
§ 1.1:
(2.25)
Although different device structures and material systems lead to various types
of recombination whose rates (both relative and absolute) can vary widely, here we
will consider three processes: trap-assisted Shockley-Reed-Hall recombination, bimolecular radiative recombination, and Auger recombination.
The rates of SRH,
bimolecular, and Auger recombination are typically expressed in terms of the electron and hole concentrations, n and p respectively, while all other dependences are
captured by some phenomenological rate constant (here A, B, and C). It is worth
noting that these constants are intended to be independent of the magnitude of the
local electrical excitation; the n and p dependences capture that physics. The most
common form of these expressions appears below:
(n + n
(np - ni)
r)T + (p + pi)(r.
A(n - no) or A(p - po)
Rrad = B
(2.27)
(2.28)
(np - n2)
RAuger = C (n (np - ni2) + (np - n )p)
(2.29)
Instead of the carrier concentrations n and p, these rates can be rewritten in terms of
the Fermi level separation, taken to be equal to the applied voltage qV. In the dilute
Boltzmann limit, the product np rises exponentially as with qV so that
(2.30)
np = ni (eevkBT)
Where doping is used to create a large majority carrier population at equilibrium,
the increase in the product np in response to a small forward bias is due to the
increased minority carrier density. That is to say, the quasi-Fermi level of the majority
species is relatively fixed while the quasi-Fermi level of the minority species is moved
closer to the minority band edge, increasing that carrier density. Thus, to a good
approximation
P = Po
and
n = no (eqV/kBT
60
where p > n and
(2.31)
n = no
p = po (eqv/kBT)
and
where n
Substituting these expressions into Equation 2.26, where p
n2 (eqv/kBT
RSRil
(Po + p1)rn
Cpo n2
RAuger =
2-33)
2TnoL Tn
-
Rrad = B ni (eqV/kBT
-
2
-
= Ano (eqv/kBT
(2.32)
> n we have
n (eqv/kBT
_
> p.
1)
(2.34)
i)
(2.35)
-
(2.36)
(eqv/kBT
where we have assumed the states contributing to SRH recombination are near the
zero-bias equilibrium Fermi level and the trap lifetimes
order.
A similar expression can be derived for n
mr
and Tp were on the same
> p. At some point along the
junction, n is on the order of p. Here we can write simple expressions for n and p
which are valid when qV/kBT < 1 in terms of the carrier asymmetry x = no/(no+po).
P = Po (exqV/kBT)
and
n = no (e(l-x)qv/kBT)
where p
-
(2.37)
n.
Note that for larger voltages, the effects of carrier asymmetries will wash out in the
same way as doped regions experience at much higher bias. Beyond this point, if
both species remain in the Boltzmann limit, both Fermi levels move toward their
respective band edges symmetrically (i.e. n and p grow with qV like when x = 1/2
in Equation 2.37). Substituting as before, for regions with p ~ n, we have:
RSRH
(no
Rrad = B n
2i (eqvlkBT
(v/B
1
(e(1-x)qV/kBT) + po (exqV/kBT)
(eqv/kBT
-
(2.38)
+ 2ni) - T
1)
(2.39)
RAuger = C - [no (e(1x)qv/kBT) ±po (exqv/kB)
(V/kBT
-
1)
.
(2.40)
Now let us imagine a device in which the active region extends from x = 0 to
x
=
L and consider three separate regions: p
and n
> n over (0,Xp), p ~ n over (Xp,Xn),
> p over (XnL). From the expression for EQE from Equation 1.1, if we
61
assume the extraction and injection efficiencies to be independent of applied bias, we
can capture the voltage dependence of the quantum efficiency:
77EQE C)C
(RSRH)active
±
(Rradiative)active
(Rradiative)active ± (RAuger)active
(2.41)
From the above equations for RSRH, Rrad, and RAuger, it is clear that all three recombination processes have nonzero contributions at linear order in qV/kBT.
This may at first seem counter-intuitive, because we typically think of defect-based
SRH recombination as a one-particle process, radiative bimolecular recombination as
a two-particle process, and non-radiative Auger recombination as a three-particle
process. While this is true, not all of the particles in these processes need to be excess
particles. Some can be thermally-generated equilibrium carriers that exist when the
device is off but at finite temperature. In fact, if we were to ignore the thermallygenerated equilibrium carriers, we should not expect q > 1 operation to be possible,
since the low-temperature reservoir would be at T = OK and have no entropy.
The fact that radiative bimolecular recombination has a finite contribution at
linear order in the dimensionless electrical excitation qV/kBT implies that the external
quantum efficiency of a very general class of LEDs remains a nonzero constant as
V -+ 0:
lim 7EQE # 0
V-+O
(2.42)
Experimental evidence of this behavior is presented primarily in Chapter 3, but
the basic fact is readily apparent in Figure 2-10. From this it follows that arbitrarily
high wall plug efficiency is achievable at low voltage:
lim 'q =
V -+0
lim O
v--+o qV
7EQE = 00
(2.43)
This type of behavior, where unbounded coefficient of performance for heat pumping is
available at arbitrarily low power is a general feature of thermodynamic heat engines.
We discuss this trade-off futher in
§ 2.4.3.
62
10
kIT/q @ 135"C
0
0
G)
10
1350 C
IkT/q @ 84"C
P
-
Lii
0
uJ
15
H
Ed
H
H
84 0 C
.
.
/
k T/q @ 25-C
---±
---j
10-
*0
-
. a-m- z W-O
10 -
-
2
10
. A
10 -
10-2
104
Voltage (V)
10~
100
Figure 2-10: The quantum efficiency of a conventional LED approaches a constant as
the applied voltage falls below kBT/q (~ 25 meV at room temperature). The discrete
markers represent experimental data while the lines represent simulation results based
on the equations presented in this chapter.
63
Carnot-Efficient LEDs and Real LEDs
2.4
2.4.1
Carnot Efficiency
I
Operating
Point
2 CO
by
Figure 2-11: I-V curve for an ideal LED. Input electrical power is represented
represented
is
power
output
the red box between the origin an the point (V,I) while
to
by the larger box between the origin and (hw/q,I). As the operating point moves
lower voltage, the ratio of these areas (i.e. the wall-plug efficiency) diverges.
Consider the I-V curve of an LED with unity quantum efficiency as shown in
Figure 2-11. As usual, the electrical input power into the diode is given by IV. Here,
this quantity is represented by a box between the origin and the operating point
(V,I). Now, since this LED has unity quantum efficiency, the rate at which photons
exit the device is equal to the rate at which charge flows through it, and each photon
carries away hw worth of energy, the output power is represented by a box between
the origin and the point (hw/q,I). From this picture, several simple features can be
seen.
64
First, for any forward bias voltage V < hw, output power exceeds input power.
This means the device is cooling. Subtracting the box corresponding to input power
from the box corresponding to output power gives the cooling power. Also, since all
these boxes are the same height, the ratio of output to input power q can be easily
visualized as hw/qV. Finally, we can see that this ratio diverges as V becomes small.
As we will discuss shortly in § 2.4.3, it is also apparent that as this happens, the
amount of current flowing is also reduced and the power flowing through the system
becomes small as well.
Recall now that in
§ 1.1 and § 1.2, we examined two simple but very different
expressions for the efficiency of an LED. The first expressed that each electron that
flowed through the device could result in the emission of a photon of energy hw with
probability
EQE.
Since qV of electrical energy is required to drive this current, we
wrote:
qV
Then in
w=
- EQE
-
(2.44)
§ 1.2, the maximum efficiency permitted by The Second Law was expressed in
terms of the lattice temperature of the device
Tattice
and the temperature of outgoing
photon field Tphtn..:
7
1Carnot
Thoton (.)
(2.45)
Tphoton (I) - Tattice
As we will see shortly, these two expressions lead to a singular concept of an ideal,
Carnot-efficient LED.
Returning to the ideal LED whose I-V curve is shown in Figure 2-11, let us
determine the output power at an operating voltage V. When a voltage V is applied,
the quasi-Fermi levels of the active region separate by an energy AEF = qV. The
conduction band states are then occupied with more electrons than at equilibrium;
the valence band states also contain more holes. Recalling the logic from Figure 2-8
and Equation 2.24, the occupation of these states is roughly equal to the occupation
at equilibrium at the elevated temperature T* = Tattice(1 - (qV/hw))-.
Thus we
might expect the output power to match the spectral intensity of a blackbody at T*.
65
If the active region of the device is many absorption-lengths thick at the emission
wavelength, it should radiate with unity emissivity.
If the intensity I from an ideal device is just the blackbody intensity at temperature T* over the relevant spectrum, then
= T* in the expression for the
Tphoton(I)
Carnot efficiency. Substituting the expression for T* into Equation 2.45 gives:
Tphoton (Iideal)
U'Carnot
Warnot =
Tphoton (lideal)
-(.6
T*
T*
-
(2.47)
Tattice
T attice (
Tattice (1 qV
2.4.2
(2.46)
Tattice
-
ideal
-
r '
)
-
(2.48)
Tattice
77
(2.49)
Non-Ideality of Existing LEDs
For real LEDs, of course, the effects of non-radiative recombination are substantial
and
EQE
< 1. Although the descriptions of these devices can be quite complex, the
relationship between them and a Carnot-efficient device is captured entirely by
IJEQE
when the voltage is well below V = Egap/q and the active region carriers are in the
Boltzmann limit.
At higher voltages corresponding to conventional operating points, diodes can
reach transparency and inversion, so that both our approximation of an opticallythick active region and the Boltzmann approximation become invalid. If the electronic
system is inverted, for example, then even recombination events that result in a final
photon state with no entropy (i.e. lasing) can be thermodynamically preferred. If we
naively applied the equations above to such a situation, we would predict an infinite
photon flux corresponding to an infinite brightness temperature as V -+ Egap/q. Since
this is neither physical nor in agreement with observations, we should expect that at
some point the radiative transition becomes saturated and the intensity falls below
that of the effective temperature T* from Equation 2.24.
66
2.4.3
The Power-Efficiency Trade-Off
Even a Carnot-efficient LED faces a fundamental constraint on its spectral intensity
due to the finite phase-space density of photon modes and the speed of light. As described in @ 1.2, a given spectral intensity of a light source, 1(A), requires a particular
minimum temperature
Tph o t o n
of the outgoing photon field.
2whc2 A
Itotal (A) =
exp
'background
(A) =
AkB
2hc
exp
L
'Itotal -
'background
Oc
5
(2.50)
photon
2
A
5
(2.51)
ABT..bient
exp
exp
h
exp
onlp
1
-
[
e
AkBTambientI
(2.52)
1
)
Meanwhile, the outgoing photon field temperature Tphoton limits the efficiency by the
Second Law:
7
Carnot =
Tphoton
-
Tambient
(2.53)
For a given wavelength and lattice temperature, a fundamental connection can
therefore be made between power and efficiency.
In this way, the Carnot efficient
LED is analogous to other endo-reversible heat engines, but with its finite thermal
conductance from the electron-hole system of the active region to the photon field set
by the Planck radiation law. This trade-off between power and efficiency is depicted
for various wavelengths of interest in Figure 2-12.
A regime of special interest may be found when AT
67
= Tphoton -Tambient
< Tambient-
10
10
0
0
103
102
C)
>-10
LU 0
10
-15
10
-10
10
-5
10
0
10
Spectral Intensity (W/m 2/nm)
Figure 2-12: Efficiency versus spectral intensity of the electrically-driven optical power
for Carnot-efficient LEDs emitting at various wavelengths of interest. From left to
right they are 555 nm (peak response of the human eye), 1104 nm (Silicon absorption
edge at 300 K), 1550 nm (SiO 2 fiber loss minimum), and 2600 nm (emission wavelength from experiment in § 3.2). At all wavelengths, there is a low-power regime
in which the outgoing optical field is barely brighter than the blackbody background
and efficiency scales inversely with power. For longer wavelengths, this corresponds
to a higher intensity. Note: calculations assume ambient temperature of 273K.
68
Defining
Tphoton/Tambient
L oc
=1 + x and expanding for small x, we see:
I(2.54)
exp
kBTambien
AkBTarnbint(1+X
L oc
exp
B
B ambientl
amintBabin
(2.55)
[x
L oc
hc
arnbient
exp
hex
he
Ak h
(2.56)
oc
B arnbient
-
Thus, since
WCarnot =
Tphoton
=
-
Tambient
small x corresponds to high efficiency q
(1 + X)
-
+ = ±- 1
X
I
(2.57)
> 1, where Warnot oc 1/L. This behavior
can be seen readily in Figure 2-12. At each wavelength, below some power level the
slope of the Carnot bound becomes 1/L. Because infrared wavelengths carry more
blackbody radiation at typical ambient temperatures, this transition occurs at higher
power for these wavelengths than visible wavelengths. As we will see in Chapter 3,
this will lead us to focus experimental efforts on infrared emitters.
2.5
Design of LEDs for Heat Pumping
As we saw in
§ 2.4.2, although existing LEDs share certain qualitative features with
ideal LEDs, non-radiative recombination, leakage, and imperfect photon extraction
act as significant sources of irreversibility and cause real LEDs to operate far from the
Carnot efficiency bound. This is particularly true for infrared LEDs, for which lower
material quality in the active region leads to shorter trap-assisted non-radiative lifetimes and smaller bandgap energies lead to increased Auger recombination and carrier
leakage. In order to redesign devices with improved optical power and conversion efficiency at low voltage, Dodd Joseph Gray, Jr. and I have created and experimentally
validated a numerical model of charge and heat transport using the commercial soft-
69
ware package Sentaurus Device distributed by Synopsys. The material parameters
in this simulation (and the simulations used to fit experimental data in
§ 3.2) were
modeled with the equations in Table 2.1 which reference constants in Table 2.2 and
Table 2.3.
Starting with the structure of an existing Gao.s 5 Ino.15Aso.13 Sbo. 8 7/GaSb
2.15pm LED designed for high-bias room temperature operation, we alter the active
layer thickness, active layer doping, operating temperature, and active material SRH
lifetime to improve on this design at low-bias. The results reported here mirror those
reported in Ref. [69].
Model
Formula
Bandgap
Egap = Egap,o - #(T - 300) - AjNj 3
Density
of States
Ni
Mobility
SRH
Recombination
=NO
T
Bimolecular
BzN
- C-N /
3
/2
300
i= Pi 0
300 )
RSRHnp
- ni
TSRH,h,O
Surface SRH
Recombination
-
(n+ni)+- TSRH,e,O
3/2
(300
(p ni)
np - n2
RSRH,surf = Vsurf(n +
Rr
=
p + 2ni)
B (T)-3/2
gap
(np - n?) ,where
Recombination
B = BO f (a) = BO x 0.15
Auger
Recombination
Ru,=C(
RAuger =
C(n
(See pp. 67-74 of Ref. [9])
)(p-n2
ip r )
-
Table 2.1: Equations describing phenomenology of material parameters. The constants which were used with these equations are found in Table 2.2 and Table 2.3. In
these tables, T refers to the absolute temperature in Kelvin and when i appears as a
subscript of a capital letter it stands in for the carrier species or dopant type.
70
Parameter Name
Intrinsic bandgap
Symbol
Egapo
For GaInAsSb
(85% Ga, LM)
0.583 eV [70]
For GaSb
0.726 eV [71]
Thermal bandgap
3.78 x10- 4 eV K-' [71]
narrowing
3.78 x10-
4
eV K-
1
[71]
parameter
Jain-Roulston n-type
bandgap narrowing
An
parameters
Cfor
Jain-Roulston p-type
bandgap narrowing
parameters
ASame
Bn
Same as
GaSb
as
Bp
CP
1.36x10- 8 eV-cm [72]
1.66x10- 7 eV.cm 3/ 4
1.19x10- 1 0 eV.cm 3/ 2
for GaSb
8.07x10-9 eV-cm [72]
2.80 x 10-7 eV-cm 3/ 4
4.12x10- 1 2 eV-cm 3 /2
for
as
Electron SRH lifetime
TSRIH,e,O
variable
iOns [73]
Hole SRH lifetime
TSPH,h,O
variable
600ns [73]
Vsr
1900 cm/s [74]
1900 cm/s [74]
Radiative constant
BO
3x10" 1 cm 3 /s [75]
8.5x10- 1 1 cm 3 /s [73]
Auger constant
C
Absorption
a
Surface SRH
recomb. velocity
2.3x10-
28
cm 6 /s [74]
4000 cm-
1
[74]
5x10-
30
cm 6 /s [73]
Not Used
Electron mobility
pe,O
5000 cm 2 /Vs [76]
3150 cm 2 /Vs [73]
Hole mobility
ph,o
850 cm 2 /Vs [77]
640 cm 2 /Vs [73]
Electron mobility
temp. exponent
'e
1.9 [78
0.9 [79
2.3 [78]
1.5 [73]
Hole mobility
temp. exponent
Conduction band
density of states
Nco
1.9x10 17 cm- 3 [80, 81]
2.1x10 1 7 cm
3
[81]
Valence band
density of states
N,,
1.5x10 19 cm
1.8x10 1 9 cm
3
[81]
3
[80, 81]
Table 2.2: Material parameters associated with the electrons and holes. Values are for
material at 300K unless otherwise specified. Note that the figure given for absorption
ce refers to the value approximately kBT above the absorption edge.
71
1
Experiments (undoped)
108
NND2 X 101
7
m
~10
_j
1-
- 1
3
- -4 010
'P000ND
280
300
3
320
X 1014
Cm-3
N1=6 X 10cm1
340 360 380 400
Temperature (K)
420
-
440
460
106
TSRI
95ns
=1S
10 8
--
p-type
E.
E
10-10
..
95ns
'I'S
*SRM
n-type
_j
1012
10-1
280
300
320
340
360
380
Temperature (K)
400
420
440
460
Figure 2-13: Output optical power density at unity wall-plug efficiency L,= 1 versus
operating temperature. At top, L7= 1 is plotted for three n-type dopant densities
of ND = 3 x 10
17
1
cm- 3 (blue dashed line), ND = 2 x 10
3
16
cm- 3 (red solid line) and
ND = 6 x 10 cm- (black dot-dashed line), demonstrating that an optimal dopant
density exits for low bias operation at 300K. Hollow squares denote experimental data
from Chapter 3. At bottom, we plot L.= 1 as a function of temperature for n- (red)
and p-type (blue) doping. Solid lines denote calculations with the GaInAsSb SRH
lifetime T = 95ns and dashed lines denote T = lys.
72
For GaInAsSb
Fo% Ga, LM)
(85% Ga, LM)
For GaSb
Lattice Thermal
Codtivit
Thermalattice
Conductivity
14 W/mK [82]
3 W/mK [82]
Static Dielectric
15.64 [80,81]
15.7 [81]
Parameter Name
Symbol
Constant
Series
Rseries
0.779 Q
7collection
24.5%
Resistance
Light
Collection
(fit in Ref. [47])
(fit in Ref. [47])
Efficiency
Table 2.3: Remaining material parameters not included in Table 2.2.
In Figure 2-13 we compare the power density at unity wall-plug efficiency versus
temperature across various designs of a 2.15pm LED. At top the plot compares devices with differing levels of n-type doping in the active region. Near 300K, designs
with ND -2 x 10
16
cm-
result in more than a 10x improvement over both nominally-
undoped and heavily-doped designs. Results across the range of dopant densities and
temperatures (not plotted here) point to the existence of an optimal doping concentration at this temperature. At higher temperatures, the intrinsic carrier concentration
ni is higher so that the importance of doping is diminished. Above about 400K, the
optimal dopant concentration is small so the original structure is nearly optimal. At
these temperatures, experimental data from Chapter 3 matches our numerical results
closely.
Intuitively, doping improves device operation at low bias by increasing the internal
quantum efficiency [14, 47]. In the low bias regime at low temperature, excess minority
carriers experience non-radiative trap-based SRH recombination as well as radiative
bimolecular recombination. Since the rate of bimolecular recombination is linear in
the concentrations of both electrons and holes, for a given minority carrier density, a
doped structure with greater majority carrier density will experience more bimolecular
recombination.
Meanwhile, the SRH recombination rate is linear in the minority
73
carrier density and the trap density, but is nearly unchanged with majority carrier
concentration (i.e.
doping).
Increasing the bimolecular recombination rate which
determines optical output power relative to the typically dominant SRH process leads
to an increase in quantum efficiency. This explains the initial increase in power at
unity wall-plug efficiency with dopant concentration.
At very high doping, Auger recombination becomes relevant even for voltages below the thermal voltage, and the linear increase in bimolecular recombination rate is
outweighed by the quadratic increase in Auger recombination with dopant concentration. For example, when CCCH-type Auger is the dominant Auger process, the
Auger recombination rate may be expressed as
RAuger
Crn2 p. In an n-type material,
this leads to the aforementioned quadratic dependence on doping. If the material is
highly p-doped, other processes like CHHS or CHHL replace the CCCH process in
the logic above, but the quadratic dependence remains. Combining this general result
with the previous result relating bimolecular and SRH recombination rates, we find
there exists an intermediate dopant concentration at which the quantum efficiency at
low voltage is optimized. In keeping with the result given by Heikkila, et. al. in Ref.
[14], the low-bias quantum efficiency is optimized when the dopant concentration is
V7C, where
T
is the SRH lifetime and C is the Auger coefficient.
The preceding analysis was done assuming an excitation characterized by a constant excess minority carrier population. However, as doping is changed, the forward
bias voltage corresponding to this density changes.
If we translate the logic above
into terms at constant voltage, we find that doping the active region suppresses SRH
recombination by reducing the equilibrium minority carrier population while the bimolecular recombination rate, which is proportional to n?, is unchanged due to the
law of mass action (i.e. nopo
=
n2). The Auger recombination rate is increased with
doping because the quadratic dependence on doping described above is not completely offset by the decreased equilibrium minority carrier population. Thus we find
that the same conclusions hold. The low-bias quantum efficiency and power density
at unity efficiency are maximized at a finite optimal dopant concentration which reflects a balance between the parasitic SRH and Auger non-radiative recombination
74
pathways.
The lower plot of Figure 2-13 shows the power density at unity wall-plug efficiency
versus temperature for four different devices.
The two solid curves correspond to
devices with active region SRH lifetime 7 = 95ns while the dotted curves correspond
to
T =
1[ts. The devices with longer active region SRH lifetime have higher quantum
efficiency and thus higher L, 1 . The blue curves indicate the results of simulating
structures with p-type optimally doped active regions while the red curves correspond
to n-type doping. The p-doped devices have almost an order of magnitude higher L.
1
at a given temperature as their n-type counterparts. This asymmetry can be explained
by the differences in SRH recombination rates near the hetero-junction between the
n-GaSb region and the intrinsic InGaAsSb active region. At this interface, a narrow
region exists in which the electron density is very high, due to the difference in
electron affinity between GaSb (4.06eV) and the quaternary alloy (4.18eV) in our
model. Excess electrons in this region can experience SRH recombination or undergo
spatially indirect transitions to valence states at adjacent locations where the hole
density is also high. A more thorough analysis including experimental data for band
alignments would likely enhance the predictive power of this model, but as with many
facets of simulations in the InGaAsSb material system, conclusive experimental data
remains scarce. As a result, predictions about the relative values of quantum efficiency
for n-doped and p-doped active region designs are less firm than other predictions like
the existence and magnitude of an optimal doping level.
Figure 2-14 presents a breakdown of the various recombination processes contributing to conduction through an optimally p-doped diode as a function of temperature. At all temperatures, SRH recombination in the active region is the dominant
pathway. At higher temperature, the relative strengths of the active region processes
increase for the multi-particle Auger process while the relative strength of the oneparticle SRH process decreases.
This is in keeping with the explanation provided
above. Furthermore, higher temperature leads to an increase in leakage current. This
is to be expected, since as temperature increases, there is an exponential increase in
the fraction of carriers able to thermionically emit over the hetero-barriers and escape
75
1014 I
,,.--eaka
Leakage,
SRH in Active
-
SO"
1012
-
Auger inActive
-0000
E
0
.2
0
0
a)
Radiative in Active
1010
300
400
350
Temperature (K)
I
450
Figure 2-14: Radiative (red solid line), SRH (black dashed line), and Auger (black
dot-dashed line) recombination rates per unit area in an optimally p-doped structure
plotted as a function of device lattice temperature. The leakage curve (blue dashed
line) combines all recombination processes outside the active region and may be seen
as a parasitic component of the current density flowing in response to an applied
voltage. The data shown here represents the results of a diode with active region
SRH lifetime r = yIps at the unity efficiency operating point.
76
the active region to undergo recombination in the quasi-neutral regions or near the
contacts.
x 10-1
NA 2 X 1ol cm- 3
E
NA= 3 X 10 cm-3
1.5
61017
E0
0.5
.=
0.5
NA=
6 X 1017 CM-3
II
I
11n
0
1
Thickness (pm)
2
3
Thickness (gm)
4
Figure 2-15: Power at unity wall-plug efficiency versus active region thickness for
structures with three different levels of p-type doping. The inset is a plot of the
extraction efficiency versus thickness, which is a decaying exponential due to reabsorption of photons generated within the active region. At all three doping levels, we
find an optimal thickness which reflects the trade-off between reabsorption and the
need for substantial active region volume to outweigh the effects of leakage.
The results of varying the active region thickness of these structures is shown in
Figure 2-15. For each of the three p-type doping levels examined, as well as others
not shown here, there exists an optimal thickness which maximizes the power at
unity efficiency. At very small thicknesses, the total fraction of current which passes
through a recombination pathway in the active region is proportional to thickness.
In essence, a thicker active region diminishes the importance of leakage current by
increasing the total current, and thereby increases the quantum efficiency and thus
Lni.
At large thicknesses, the majority of photons generated through radiative
77
recombination undergo reabsorption before they can escape the active region. The
probability of escape, or equivalently the extraction efficiency
7
7ext, was calculated
from a 3-dimensional model assuming a uniform distribution of photon generation
in position and angle and a distribution in energy proportional to the density of
electron-hole pairs connected by vertical transitions at low bias at 300K [9].
The
results of this calculation differ only by a constant factor from what one would expect
from a one-dimensional model, namely an exponential decay of
xext with thickness.
Given this extraction efficiency, since the fraction of active region recombination that
is radiative (i.e. the ratio of recombination rates expressed in Equation 1.1) is small,
the external quantum efficiency also drops exponentially with thickness. Combining
these two mechanisms results in curves of the shape seen in Figure 2-15, with an
optimum active region thickness around 1.5pm.
Figure 2-16 shows the results of the final redesign. This design includes optimal
p-type doping and optimal active region thickness, both chosen to maximize L.=
at
298K. The Carnot limit shown here is calculated by combining the experimentallymeasured spectrum at 300K with a given power density to find the brightness temperature for the average photon Tphoton, which is subsequently used in the well-known
expression for the maximum efficiency of a heat pump from Equation 2.45.
The
redesigned structure's overall behavior of achievable efficiency versus power density
resembles the Carnot efficiency more closely than the original design. Since the remaining difference is directly connected to the external quantum efficiency, which is
limited primarily by non-radiative SRH recombination in the active region, the design is likely within a few percent of optimality given the assumed SRH lifetime and
bimolecular recombination coefficient.
2.6
Circuits are Cycles
When a battery is connected to a diode, current flows in a loop. Electrons flow from
the negative terminal of the battery through a wire to the device's cathode, across
the device from cathode to anode, back to the battery's positive terminal, and finally
78
100
ai)
1
0)
Redesign
.011
E
D
Existing Device
-(
ee
f~~~4I
10
11
-11
010
--10
10
-9
10
-8
10
-7
10
-6
-5
10~ 10
-4
10
-3
Power Density (W/mm2)
Figure 2-16: Results of the redesign of the 2 .15pm LED for low bias. The plot shows
wall-plug efficiency as a function of power density for the existing device characterized
in § 3.2 (dotted red line represents simulations while the discrete markers represent
experimental data), as well as the simulation results from the redesigned device (solid
blue line). Also included is an estimate of the Carnot limit (black dashed line) for
wall-plug efficiency as a function of power density. The redesigned device has orders of
magnitude better performance at brightnesses on the nanowatt per square mm level.
The overall behavior resembles the Carnot efficiency more closely than the original
design, but is still limited by non-radiative SRH recombination.
79
back through the battery to its negative terminal. At each point along this path, a
given electron experiences a range of environments. Although these environments are
typically described using various transport frameworks based primarily on statistical
mechanics, they may also be described by thermodynamic state functions.
When
described in this way, the simple battery-diode circuit is analogous to an internal
combustion engine as described in Figure 2-17.
Internal Combustion Engine
S3
@@@(D@@
Figure 2-17: The path of electrons through a circuit is a closed loop and may be
described by a thermodynamic cycle. Along this path electrons may exchange energy
and entropy with other reservoirs such phonons or photons just as the working fluid in
a more conventional thermodynamic machine may exchange energy and entropy with
a condenser plate, heat sink, or mechanical subsystem with few degrees of freedom.
The electrons in different parts of the circuit at left are at different stages in the same
cycle. The circuit is analogous to the internal combustion engine at right in that
different portions of the working fluid are at different phases of the same cycle.
The most common descriptions of circuits are based on statistical mechanics.
Although a macroscopic circuit involves many microscopic degrees of freedom, we
typically care only about the aggregated similarities among the collection of degrees
of freedom. For example, when current flows through a wire, we care primarily about
the resistance of the wire. That resistance is a measure of the average momentum
of electrons down a small electro-chemical potential (i.e. Fermi level) gradient. The
current does not care about what distribution of electron momenta give rise to that
average.
The descriptions we have offered in this chapter are different. By considering the
flow of entropy within an electronic device, we are asking not about the similarities
80
between the dynamics of the microscopic degrees of freedom, but about their differences. In this chapter we have focused on the flows of entropy between different
sub-systems through interactions (e.g. entropy flow from the lattice to the electrons
in thermally-assisted injection). We found this most useful because the purpose of an
LED is to transport energy from one domain to another. For other types of devices,
mapping the flow of entropy within a given sub-system (e.g. within the lattice, or
within the band-edge states of relevance to charge transport) may prove useful. Moreover, since any closed circuit is also a closed cycle for the electrons that flow through
it, in addition to single devices like transistors, even complex integrated circuits may
be amenable to thermodynamic analysis of this type.
By modeling entropy flow directly, we can directly identify the origin of any irreversible entropy generation which must underly any differences between real electronic
machines and their idealized conceptions. In much the same way as the thermodynamic analysis of combustion engines allowed the development of new cycles and new
engines based on them, it seems plausible that a systematic study of the entropy
flow in electronic devices could yield practical design improvements. We discuss this
subject again in Chapter 6.
2.7
Summary and Conclusions
In this chapter we have assembled a theoretical framework for the thermodynamic
analysis of transport in a semiconductor light-emitting diode.
We began with a
detailed description of the entropy flows involved in the basic electron transport processes in an LED, with emphasis on the case when the applied forward bias voltage
is less than the bandgap energy and both the electron and hole populations in the
active region are in the dilute Boltzmann limit. In this case, we found that the electrons absorb entropy from the lattice during injection and release entropy with the
outgoing photons.
We recognized that this behavior is in close analogy with con-
ventional mechanical heat pumps, with the electrons acting as a working fluid to
transport entropy and energy from one reservoir (the lattice phonon bath) to another
81
(the outgoing photon field).
Given these tools, we were able to generate a heat pump diagram from the conventional description of electron transport in a semiconductor device which identifies
carriers as flowing across an band diagram while experiencing generation and recombination at various points. Next we considered what an ideal Carnot-efficient LED
would look like, and found that such a device corresponds to the case of perfect external quantum efficiency. We then saw that even such ideal devices would face a
fundamental trade-off between efficiency and power density, but that the constraint
was less strict for LEDs emitting at longer wavelengths. The latter observation will
serve as the motivating factor behind the experimental design in the next chapter.
We ended our theoretical discussion by applying the framework we built to redesign an existing 2.15pm LED for more efficient electrical-to-optical power conversion at low forward bias voltages. Our simulation results, which were based on an
experimentally validated model, indicate that existing growth capabilities are sufficient to realize unity wall-plug efficiency at room temperature in an LED at this
wavelength. We closed our theoretical discussion by arguing briefly that the space of
problems amenable to this type of thermodynamic analysis is quite wide and in principle includes any electronic device or combination of devices which forms a closed
circuit and operates in steady-state.
82
Chapter 3
Experiments on Existing Emitters
Although the prospect of very high efficiency (i.e. q
pressed in
>> 1) electro-luminescence ex-
§ 2.4.3 was theoretically predicted by Jan Tauc as early as 1957 [31], the
phenomenon of q > 1 photon generation had remained experimentally unconfirmed
until the present work.
In this chapter, we present a series of experimental mea-
surements of electrically-driven light emission from devices which were designed and
fabricated outside the scope of this work. These devices, mostly infrared LEDs, were
designed for conventional high-current (A/cm2 -scale and above) operation at room
temperature.
By investigating their performance at high temperatures and small
currents, new physics was observed.
In addition to the first experimental evidence of above-unity electrical-to-optical
power conversion, this series of experiments provides empirical evidence to corroborate
the theory from Chapter 2. To confirm that
1
7EQE
becomes voltage-independent in this
regime and that q therefore scales inversely with power in agreement with
§ 2.3, we
perform optical power measurements in the low-bias regime. We further expand this
measurement to include very high efficiency points
> 1 to show that a single photon
with energy hw > kBT can be generated for less than kBT in electrical input work
as suggested by
§ 2.4.3. To provide further evidence that the observed optical signal
comes from heat pumping as described in
§ 2.2 and not emissivity modulation, we
also conduct experiments in which the LED temperature is held above and below the
detector and ambient laboratory temperatures. We then expand these measurements
83
to include the first experimental evidence of 7 > 1 operation at room temperature
and combine these results with measurements on other LEDs to support the effective
temperature concept.
The chapter is organized as follows. We begin in
§ 3.1 with a review of the basic
experimental techniques that will be required. These include lock-in measurements
of detector photo-current and emitter voltage, temperature control and minimizing
thermal shock, and the use of passive optical elements to improve photon collection.
Next, in
§ 3.2 we describe experiments that establish the basic physics of the low-bias
regime. Our goal is to show that the external quantum efficiency becomes a constant
for qV <
kBT, that efficiency therefore scales inversely with optical output power,
and that this behavior continues beyond the conventional limit of unity wall-plug
efficiency. To this end, light-current-voltage (L-I-V) measurements are made on a
heated LED emitting at
2 5
. pm. Since the optical power available from the LED at
unity efficiency is much less than the blackbody background, the lock-in technique
is necessary; because the optical power also increases rapidly with emitter temperature, thermal control is required to keep the device at an elevated temperature. In
§ 3.3, we attempt to reach higher values of Lusity by increasing the emission wavelength. Despite the wavelength-scaling of the Carnot limit we derived in
§ 2.4.3, we
find that increased non-radiative Auger recombination in the measured 4.7ptm LED
restricts Lunity. For this experiment, a photo-detector sensitive to longer wavelengths
is required. To maintain a low noise floor while decreasing the detector bandgap, a
smaller-area detector is used; it also incorporates a hyper-hemispherical optical immersion lens that limits photo-detection to a small acceptance angle. As a result,
additional passive optics were required to achieve reasonable collection efficiency. Finally, in
§ 3.4, similar measurements are reported on an LED emitting at 3.4pum using
the same detector. Measurements on this intermediate-wavelength LED do show increased power at unity efficiency, and permit the observation of q > 1 operation at
room temperature. We end our discussion by using data at all three wavelengths to
examine the direct connection between bias voltage V and optical power L suggested
by the effective temperature model from the previous chapter.
84
3.1
Experimental Techniques
Current-Biased Lock-In Technique
3.1.1
In order to measure the low optical power levels emitted by the LEDs in these experiments, a lock-in technique was used. Such a procedure is necessary because at
sufficiently low voltages, electrically-driven light emission is smaller than the background blackbody radiation incident on the detector. Since the arrival rate of blackbody photons and the corresponding current generated in the detector are fluctuating
quantities, optical signals resulting from low forward bias voltages must be somehow
distinguished from blackbody radiation to be measured with useful accuracy. Modulating the LED allows us to separate the photo-current it produces from most of
the noise in the detector circuit. By looking specifically for a photo-current signal
with the same frequency and phase as the excitation over a long integration time,
arbitrarily small optical power signals can be measured.
In particular, in order to measure optical power from an LED in the low-bias
regime (V <
kBT/q) with a signal-to-noise ratio above 1, a lock-in technique is
necessary. This is because the spectral intensity emitted from an LED is equal to
double its equilibrium blackbody intensity when a forward bias of qV
=
kBT - ln(2)
is applied. This may be shown in several ways, but the simplest is to calculate the
effective temperature at which the active region must glow to double the blackbody
intensity, then solve for the corresponding applied voltage:
11
Cw
exp
(
1
-=
)
J(f; T*) =21I(f;T) =2 Cw
1
exp
exp (~h*)
kBT-
2
=Tln
k
T* =T (I - k
85
exp
+
kBT)
h
kBT
(
(3.1)
-1
(3.2)
(3.3)
(3.4)
Comparing this expression with Equation 2.24, we see that the blackbody spectral
intensity at w is doubled when the forward voltage is qV
=
kBT ln(2). Recall now that
even with perfect optics, the best one can do is to exclude all intervals of phase space
which do not contain the signal and include only and all of the desired volume which
does contain the signal. As a result, if we take the mean of the blackbody-induced
photo-current to be noise, the preceding logic indicates that without modulating the
excitation, the best signal-to-noise ratio one can measure at qV = kT
ln(2) is 1:1.
Since we want to measure optical power from LEDs at qV < kBT, we will use lock-in.
In these experiments, the LED was placed electrically in series with an unheated
resistor (5MQ, 500kQ, 50kQ, or 5kQ depending on the magnitude of current required)
and the combined load was biased with a 1013 Hz on-off voltage square wave. For the
measurements on the heated 2.1pm LED, the inverse slope of the diode's I-V curve
around the origin (i.e. it's zero-bias resistance) varied from 6kQ at low temperature
to 168Q at high temperature. Thus for the low-bias measurements at high temperature, the series resistor could be chosen to dominate the load across the function
generator so that the LED was approximately current biased. The optical power was
detected by various free-space infrared photo-detectors whose photo-current signal
was amplified and measured by a trans-impedance gain stage connected to a digital
lock-in amplifier.
The gain stage was composed of a commercial trans-impedance
amplifier (SRS model SR570 Low-Noise Current Preamplifier) operating in low-noise
mode with gain 2pA/V, followed by a second voltage-to-voltage amplification stage
with gain between 2 and 20. The analog filters built into these two amplifiers were
configured to form a bandpass filter with one-pole or two-pole roll-offs around 100 Hz
and 10 kHz, so that power outside this band did not cause output or input overloads
at any stage. Within the digital lock-in amplifier (Perkin Elmer model 7280 Wide
Bandwidth DSP Lock-In Amplifier), only the notch filters at 60 Hz and its harmonics
were used, and the analog gain stage before the ADC was not used. The values for
the optical power L reported here are related to the raw voltage Vr
86
read out from
the digital lock-in amplifier by the following equation:
L=
1
1
Rphoto-diode
GTIA
7r
V
Vraw
-
(3.5)
,
where Rphoto-diode is the detector responsivity in A/W, GTIA is the total trans-impedance
gain of the amplifier(s) in V/A, and the dimensionless factor of 7r/V1 is necessary
because Vraw indicates the root mean square of the first harmonic (1003 Hz) contributing to the square wave whereas L refers to the height of the square wave whose
low-end value is zero.
X
2.
W
Low-Power Light Measurements
10-
-
-E
a2.5
s
12--
-2.
I
votaerobg
E
>-
I
-.
.
-5
SmSample
*.
-
Raw X (In-Phase
Component)
.15.
....
x 13-
Figure 3-1: Raw X and Y quadrature components of the observed photo-current generated in the detector when a small forward bias is applied to the LED at 135C. The
data is not scaled to account for the detector's responsivity or intermediate amplification; instead the data here is the raw output from the digital lock-in amplifier. Note
that data recorded when the LED was not driven indicate that the background noise
had no preferential phase relationship to the excitation signal. For all measurements
in this particular figure, a time constant of T=10s was used; the raw measurements
under a given excitation condition were recorded at intervals of At=l0s. The data
set labeled 'EMI Test' refers to measurements taken when the current source was
disconnected from the LED; its proximity to the origin confirms that the recorded
current was not the result of electromagnetic interference between the source loop
and the detector loop.
As shown in Figure 3-1, the noise was observed to be zero-mean, with no preferential phase relationship to the excitation signal. The figure includes several raw
0
data points from the lowest-voltage lock-in measurements of optical power at 135 C.
87
Although the blackbody background in DC measurements was substantial, the spectral power of the noise around the lock-in frequency was observed to be independent
of the source temperature. Instead, the temperature of the detector element correlated with the noise power, suggesting that the dominant noise source could be the
current-noise of thermal generation processes in the photo-diode. Another explanation could be the dependence of the detector's shunt resistance on its temperature.
Closer consideration suggests that these two explanations in fact reflect the same
physics, as the former refers to the microscopic processes that give rise to the latter
macroscopic phenomenon. This subject is explored in greater detail in § 5.4, where a
more complete analysis of the detector noise is presented.
For each experiment, overlapping power measurements were made at higher optical power using a simplified version of the setup. The AC current source (which
includes the series resistor) was replaced by a DC voltage source and the lock-in amplifier was replaced with a digital multimeter. In general, the lock-in optical power
measurements were in agreement with the DC to within the experimental uncertainty.
Some variation between data sets was expected due to imperfect feedback control of
the LED temperature combined with the extreme sensitivity of various measurements
to this parameter. Data acquired by both methods in the overlapping power range
appears in the figures in later sections.
3.1.2
Temperature Control
In order to measure the efficiency of photon generation by these devices as a function of elevated lattice temperature, a temperature control circuit was constructed.
The commercial LED21Sr, LED34Sr, and LED47Sr devices were manufactured in
threaded M5xO.5 metal cans (roughly a 5mm long cylinder, 5mm in diameter). For
each measurement, the can was placed inside of a copper cylinder with a recess in one
end for the LED and a recess in the other for a cartridge heater capable of heating at
around 1kW. The LED recess was not close-fit and tapped due to thermal expansion
issues, so thermal paste was necessary to ensure reasonably high thermal conductance
between this copper housing and the outer can.
88
Figure 3-2: Experimental setup for thermal feedback control of 2.1pm LED during
efficiency measurements.
89
Power to the cartridge heater was drawn from standard 120V 60Hz wall-power,
with a duty cycle set by a high-power FET whose gate voltage was controlled by a
digital P-I-D temperature controller. Input to the P-I-D controller originally came
from a single thermistor placed near the LED end of the copper housing. However,
this led to long thermal time constants and ringing of device temperature on long
timescales. To address this issue, the single thermistor was replaced by two thermistors as shown in Figure 3-2. The first thermistor was placed near the heater for tight
feedback control; the second was placed near the LED for more accurate temperature
measurement.
The measured device temperature was highly uncertain. In spite of the use of thermal paste, variations due to pressure applied to holding the thermistor near against
the copper surface led to differences in temperature measurement of up to 10'C at
the 125 to 135'C range. During the acquisition of the data presented in § 3.2, the
thermal impedance between the copper cylinder and the thermistor was relatively
high and the measured temperature was 125'C. Subsequent experimentation using
a metal clamp to hold the thermistor in place suggested that the actual temperature of the copper housing during this experiment was 135'C, so the experimental
temperature was reported as such. Regardless, the finite thermal resistance between
the copper housing and the lattice temperature of the semiconductor p-n junction
remains a substantial systematic uncertainty of order 10 0 C. This was deemed acceptable because the central quantities in the high-temperature experiments, namely the
input and output power of the LED, were measured independently of this parameter.
3.1.3
Thermal Shock of LED Packaging
A major initial obstacle to repeatability of these experiments was thermal shock to
the LED packaging.
Several early attempts to replicate the phenomenon resulted
in the discrete, irreversible changes to the electrical response indicative of new shunt
resistances, with the device eventually becoming an electrical short-circuit at elevated
temperatures.
Three working hypotheses were formulated, one of which was ruled out. The first
90
430 qm
7
Fig. F. S-egric :
or stheincreier
f
ai
''mictn
ate
r
'
(b) Packaged LED (back).
es s
tp ot gfpr-chap diodt f
k.ed
i t Ahen-maSb
adstvatey(2)isahe
re
t-buGed.sSb aiet. (3) ti jn p-hiCeSb wyi.(4 is
u-aSb aye (rrsion
sis e Sdid carrier a(6)s the
ontact
ir
UteAppcdt
cabeed
ande
(d nd (9)F
a3-3t.
coclactpads wis the deposited Sio +Pb coating.
()Dagof
(a) Packaged LED (front).
1
LEout
(c) Diagram of LED mount.
(Taken from [10])
Figure 3-3: Images depicting possible locations of device failures due to thermal shock.
Figure 3-3a and Figure 3-3b show the outer LED packaging. From the backside of
the device, an epoxy filler is seen surrounding the leads for strain relief. A thermallyinduced strain field near the junction between the wire leads and the silicon carrier
wafer may be responsible for the observed device failures. Alternatively, thermal
expansion differences between the die and carrier wafer may have resulted in contact
failures at the points labeled '6 or '9' in the diagram in Figure 3-3c.
91
hypothesis was that the contact metal at high temperatures was diffusing into the
semiconductor material. This seemed feasible because the device failures were taking
place above the maximum permitted operating and storage temperatures presented on
the datasheet and the softer materials used to make long-wavelength opto-electronics
often have lower threshold temperatures for the diffusion of metals. However, because
device failures happened at various temperatures, this seemed unlikely. After raising
the matter with the growers of the device, a research group led by Prof. Matveev
at the loffe Physico-Technical Institute in St. Petersburg, we were informed that the
typical temperature for metal diffusion in the quaternary found in the cap layers of the
LED21Sr was 180 to 185'C. This was commensurate with the 43% reduction in output
power seen when the copper housing temperature was raised from 190'C to 195'C
during a measurement with a Fourier Transform Infra-Red (FTIR) spectrometer.
Because most early device failures happened at temperatures far below this, typically
between 80 and 135'C, alternative hypotheses were developed.
A second working hypothesis was the failure of solder junctions between the deposited contact layers and the Silicon carrier wafer shown in Figure 3-3. Attempts
were made to examine these bonds directly by machining open the M5xO.5 can of a
failed device, but the tools used were not sufficiently precise and the carrier wafer
with device was lost as scrap.
The third hypothesis remains the most likely.
The epoxy designed for strain
relief of the lead wires entering the backside of the device (see Figure 3-3) could
experience an internal strain field due to the elevated temperatures and cause shunt
paths between bare wire leads buried within it to become significant.
Since very slow increases of temperature still resulted in stable light emission
at temperatures much higher than typical failure temperatures, the observed failure
mode was more characterized by thermal shock than harsh steady-state thermal conditions. This is commensurate with the third hypothesis as polymer materials can
experience internal strain fields as they are heated, but slowly relax on the timescale
of minutes or hours. Temperature slew rates of less than 5K per hour were sufficient to avoid this effect altogether. By incorporating these limits on slew rates into
92
the experimental protocol, the result was eventually highly reproducible, with the
remaining variations small enough to be explained by differences due to variations in
growth and fabrication processes which were present before any thermal cycling.
3.1.4
Optical Design
We have also used passive optical elements to increase the photon collection efficiency
of optical power measurements made with detectors of various sizes. For measurements of optical power at A < 2.6pm, an InGaAs photo-diode from Hamamatsu
with a relatively large 3mm-diameter active area was used. At longer wavelengths,
immersion-lens photo-diodes manufactured by Vigo System were used. The effective
area of the Vigo detectors were at most 1mm 2 . Because of the die size, working
distance, and divergent emission cone of the LEDs, only a fraction of the emitted
photons could be collected even by placing these Vigo detectors directly up against
the packaged LEDs.
To improve collection efficiency, a pair of lenses was used. The first, a 2"-diameter
Germanium lens with
detector.
f =50mm,
was placed 2f from the source and 2f from the
Because the acceptance cone of the photo-detectors were > 300, further
reduction of the spot size could be achieved using a second lens. A smaller 12mmdiameter CaF 2 lens was placed near the detector-side beam waist for this purpose.
For experiments using the Vigo PVI-3TE-6 detector with 1mm 2 active area, the lenses
were observed to improve collection efficiency by roughly a factor of 4.
Once this optical engineering had been done, it was expected that switching detectors to the Vigo PVI-3TE-4 with (0.25mm) 2 would result in 10x lower noiseequivalent power because of the shorter cutoff wavelength (4[pm instead of 6pm).
However, the large reduction in the signal magnitude due to the reduced detector
area dominated (i.e. the signal reduction was more than a factor of 10), even with
the use of passive optics, suggesting that the final spot size achieved is far from ideal
and further optical engineering should improve results.
One more set of experiments was designed to quantify the collection efficiency of
the detectors. A lensless planar 2mm x 2mm photo-conductor was used to map the in-
93
tensity field. The goal was to acquire data on the intensity as a function of transverse
position, which could then be de-convolved by the 2 x 2mm area aperture function
to reconstruct the beam profile. Initial results showed that the photo-detector was
collecting roughly 1/8 of the light from the 3.4pm LED and 1/4 of the light from
the 4.7pm LED. These measurements, however, relied on accurate knowledge of the
photo-conductor's response spectrum, which was highly uncertain near 4.7pm due
to its proximity to the detector's red cutoff. An attempt was made to use a pinhole to compare photo-conductor measurements against photo-diode measurements
for an intensity profile which both should collect nearly ideally.
This experiment
was unsuccessful because the total optical power through the pinhole was very small
and revealed a significant noise source due to electromagnetic interference and/or a
ground loop connecting the function generator on the source side with the lock-in
amplifier on the detector side. These observations of phase-locked noise casted doubt
on all measurements made with the photo-conductor, but since no such observations
occurred with the photo-diode, this set of experiments was temporarily abandoned.
3.2
Demonstration of r7 > 1: A = 2.5pm
The first experiment we report here was the first demonstration of an LED operating above unity efficiency. As mentioned in
§ 1.3, the operating regime in which this
phenomenon was observed differed substantially from that of previous work. This difference is captured concisely by three characteristic energies: the electrical energy qV,
the thermal energy kBT, and the bandgap energy of the semiconductor from which
the photons are emitted
Egap.
The phenomenon was observed by applying a very
small forward bias of 70pV, so that qV was several hundred times smaller than kBT.
In the low-bias regime, V < kBT/q, the experimentally-measured external quantum
efficiency
nEQE
oc L/I was observed to become voltage-independent and further re-
ductions in voltage increased the wall-plug efficiency q = L/(IV).
Previously, the
low-bias regime had been dismissed [31, 14] as producing impractically little power.
However, by moving to narrow bandgap materials and raising the ambient temper-
94
ature as Berdahl originally suggested[43] in 1985, the power available in this regime
was increased by several orders of magnitude.
This series of experiments was performed on an existing commercial device, the
LED21Sr made by loffeLED, Ltd. As described in § 3.1.1, a current-biased lock-in
technique was employed to source a small current square wave into the LED. An
uncooled Hamamatsu G5853-23 long-wavelength (A < 2.6pm) InGaAs p-i-n photodiode (Rpeak=1.3 A/W) with a circular active area 3mm in diameter was placed a few
millimeters away from the LED's emitting surface so that most emitted photons were
captured by the detector. Lock-in measurements of the photo-current signal and the
voltage across the LED were performed. As shown in Figure 3-4, the LED voltage
measurements were found to be in agreement with DC measurements of the zero-bias
resistance. As shown in Figure 3-1, optical power measurements at low power were
zero to within uncertainty when the source current was off, and increased linearly
with source current in the low-bias regime as expected.
Measurements of voltage and optical power were performed at various temperatures using the thermal control scheme described in
§ 3.1.2. Power measurements
at higher current were in agreement with DC measurements to fair accuracy; both
AC and DC power measurements appear side-by-side in Figure 3-5. The uncertainty
in the optical power measurements at DC was quantified by measuring the fluctuating photo-current with the LED off but the source stage at temperature. However,
because the optical power at fixed voltage is in theory very sensitive to emitter temperature, and imperfect thermal control resulted in ringing of as much as 5YC during
the measurements, the data should be expected to contain significant fluctuations of
the optical power not captured by zero-signal measurements in this case. This source
of error does not reduce the accuracy of the results, however because variations among
distinct measurements at different excitation levels would cause this uncertainty to
be reflected in the position of the data points in the final measurements as shown.
Furthermore, the uncertainty in power due to this effect would be much less important when measurements are taken at fixed current, as in the AC measurements at
low power. We note also that because our lock-in setup was not able to source more
95
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CI
25'
L..
0
1350 C
kT/q @ 135"C
106
-
k,T/q @ 84-C
kBT/q @ 25C
10~4
10-3
10-2
101
Voltage (V)
Figure 3-4: The lock-in voltage measurements in the low-bias regime were in agreement with the values for zero-bias resistance extracted through DC measurements.
The discrete markers indicate pairs of voltage and current that were measured by
lock-in. The lines indicate the zero-bias I-V curve given by DC measurements. The
lines are dashed above 10mV, where significant deviations from linearity are expected.
96
than 2mA and we could include only DC optical measurements above 100nW and
still have meaningful data (i.e. signal-to-noise ratio > 10), measurements using the
AC and DC techniques existed over a finite range of input and output power levels.
a
b
10-2
10
3
/ A
Temp. Model Exper.
o
25*C ---84*C -------A
135*C
4) 10-
o
104)135'C
Temp. Model Exper.
250C
84*C ..----A
--
o
,
-2
us10
010-
10
10
100% Wall-Plu Efficiency
10
104
-
10
10-2
10
10~5
l-
Light Power (W)
Current (A)
Figure 3-5: Efficiency measurements of the LED21Sr infrared LED at various temperatures. Sub-figure (a) shows the external quantum efficiency as a function of
electrical current. Sub-figure (b) shows the wall-plug efficiency as a function of detected optical power. In each case, the lines denote the results of a numerical model
and the discrete markers denote experimental data. Where error bars are not visible,
the measured uncertainties are too small to show.
When a 2.1V square wave was sourced across the LED at 135'C, a 0.41PA current
square wave was driven through the LED. During the on-phase of the square wave,
the forward bias voltage across the LED was just 72.5±4pV. During this phase, the
emission of 69±11pW of optical power was detected by the aforementioned lock-in
photo-detection technique with time constant r=10s. As seen in the top left of subfigure (b) in Figure 3-5, since just 29.9±0.lpW of was used to drive the LED source,
the wall-plug efficiency of the device at this operating point was q =2.31±0.37, and
constituted experimental confirmation of / > 1. This single measurement represents
the high-temperature low-power endpoint of a larger data set characterizing the supplied current and voltage along with the resulting optical output power as the LED's
97
temperature was varied between 25'C and 135'C.
Since only the detected photons were considered as output power, corrections due
to imperfect collection could only further raise this efficiency. Furthermore, as seen
in Figure 3-6, the LED's emission spectrum gradually red-shifts out of the responsive
band of the photo-diode at high temperatures. The optical power measurements were
calculated using the detector's peak responsivity of
Rphoto-diode =
1.3 A/W so that
reported optical power figures again serve as a lower bound.
1
0j
/
0- 0.8
/
CO
a)
Detector
Responsivity
CE 0.6
0.4
C
cc
25 0C
0.2
190*C
0
1500
2500
2000
Wavelength (nm)
3000
Figure 3-6: Relative intensity spectra of the LED21Sr device at various temperatures. Also shown is a piecewise-linear approximation to the relative responsivity
spectrum as presented in the photo-diode's data-sheet. The peak responsivity of the
Hamamatsu G5853-23 long-wavelength InGaAs p-i-n photo-diode, which exists from
approximately 1900 to 2400nm, is 1.3 A/W. This value was used to compute optical
power from raw lock-in measurements at all temperatures because the spectra could
not be easily acquired simultaneously.
By examining the dimensionless quantum efficiency as a function of voltage across
these measurements as presented in Figure 3-7, we can also confirm our prediction
98
that
7
becomes voltage-independent below kBT/q.
7EQE
a
104
kT/q @ 135'C
C4
b
10~3 - 1350C
k,T/q @ q84"OC
C
wE
/0
104
84 0 C
./0
kT/q @ 25-C
25'C
00
~EE.
105
10~4
10-
-.
100-2
Voltage (V)
-
-U
-
10~1
100
Figure 3-7: External quantum efficiency versus LED voltage for the LED21Sr at
various temperatures. The quantum efficiency of the LED in this experiment was
observed to be voltage-independent for voltages less than V = kBT/q as expected
from § 2.3.
3.3
High Power Attempt: A = 4.7pm
While the preceding electrical and optical power measurements on the emission of
2.5pm photons demonstrated that high efficiency was possible in spite of significant
irreversibility (i.e. low
'qEQE),
the application space for the phenomenon is strongly
limited by the lack of power available in this regime. As we showed in
§ 2.4.3, this is to
some extent a fundamental trade-off: for a given wavelength and quantum efficiency,
lower intensity light requires less voltage and allows a larger fraction of the electronpumping energy to come from the surrounding lattice vibrations and thereby permits
99
higher efficiency.
This observation suggests that concerted efforts to improve the
quantum efficiency and moving to longer wavelengths could lead to higher power.
Because the former requires substantially more effort, the latter was the explored
first.
As seen in Figure 2-12, the Carnot limit for longer wavelength LEDs permits
higher power densities in the low-bias regime. However, as with the 2.5pm emitter,
substantial deviations from Carnot-efficient operation were observed in initial tests.
The results of low-bias efficiency measurements at various temperatures appear in
Figure 3-8.
0
010
-4
_
__
_
_
_
_
_
_
_
_
_
0
1_0
U
-5
10
C 10
0
E
Curn)()
w0pt-4ialPwe
W
-6
10 -310
10908
-210-
Current (A)
Output Optical Power (W)
Figure 3-8: Initial efficiency measurements on 4.7pm LED. At left is a plot of quantum
efficiency (?EQE) versus current (I)and at right is a plot of efficiency (,q) versus output
power (L) as a function of temperature for a 4.7pm LED. The dashed lines are best
fit curves that follow the expected q oc 1/L power law.
SWdeo Veaim ofAMLAB
The lack of monotonic temperature-dependence in these observations suggests
that irreversible changes may have taken place during the measurements. Sentaurusbased transport simulations have suggested that temperature-dependences may in
fact change sign, but those results did not qualitatively fit the observed data either.
In those simulations, SRH recombination was the primary non-radiative pathway at
low temperature, and at high temperature Auger became the primary non-radiative
pathway. As a result, a maximum of quantum efficiency was seen with respect to
temperature rather than a minimum, suggesting an alternative explanation is required
to explain the observations here.
After these measurements were taken, tests on other long-wavelength devices re100
vealed that high currents may result in irreversible damage and the measurement
protocol was changed to avoid this. At the time of the collection of this data, however, this failure mode was unknown. If such an irreversible change was responsible
for the reduction in efficiency between the 30'C and 60'C measurements, the reproducible aspect of the temperature-dependence of device performance would be
indicated by the differences between the 60'C and 100'C data.
In this case, the
observations suggest that temperature is indeed still increasing both
EQE
and
77.
Although our investigations into these devices remains incomplete, the low observed quantum efficiency suggests that shorter-wavelength sources may in fact generate higher intensity light at unity efficiency. Observations to date suggest this is
due to improved material quality and the improved low-bias quantum efficiency that
results from it.
3.4
Lower Emitter Temperatures: A = 3.4ptm
Various experiments were performed on another commercial device (LED34Sr), this
time at lower emitter temperatures.
Some experiments were designed to refute al-
ternative explanations for the observations at 2.1pim, while others were intended to
confirm that the phenomenon was observable with the emitter at room temperature.
The work described here formed the basis for a journal article [83] published in 2013,
and the structure of our discussion mirrors that of the article.
3.4.1
Exclusion of Emissivity Modulation
One of the chief criticisms of the experimental technique at 2.5pm was the possibility
of the detected signal originating in a modulation of emissivity for blackbody radiation
rather than thermo-electric pumping of the device active region.
By attempting
to observe the same phenomenon in a configuration in which the emitter was not
much hotter than the detector or the other surfaces surrounding the experiment,
we attempted to exclude this alternative explanation. Since lower temperatures are
required to thermally generate carriers in a smaller bandgap material, but much
101
smaller bandgap materials appeared to have too high of defect densities to permit
sufficient
?7EQE,
an LED emitting at an intermediate wavelength of 3.4pm was chosen
for this task.
In Figure 3-5, the q oc 1/L scaling is observed over 3 orders of magnitude in
output power (6 orders in input power) at 135'C, extending all the way down to
the Noise-Equivalent Power (NEP) limit of the photo-detector circuit. While this
does combine with the modeling results presented alongside the data and the theory
offered in Chapter 2 to present strong evidence for the scaling law to continue to
arbitrarily low power, any physical effect that might result in a phase-locked photocurrent whose magnitude is linear in the excitation current could in principle create
this effect. In particular, the possibility has been raised that the signal may result
from a small modulation of the emitter's emissivity with current.
previously in
As mentioned
§ 3.1.1, the observed signal at low bias is necessarily much smaller
than the blackbody radiation power flowing out from the LED surface and onto
the detection surface. When V = 70p-V, this ratio is just a few parts in a thousand,
meaning that even a (spectrally flat) 1% modulation of surface emissivity with current
would be more than sufficient to explain the observation. Nevertheless, modeling and
theory suggest this is not the explanation, as do the following results.
Precisely what is meant by "emissivity" in this context is not entirely clear because
the term is a macroscopic property while our description to this point has been
primarily microscopic. For the purposes of this discussion, we regard a microscopic
model as one which refers to discrete particles and which describes a physical state in
reference to the quantum state of the complete many-body system, even if that state
is not a pure state. When discussing a macroscopic quantity like emissivity alongside
such microscopic models, we must be careful to define the terminology explicitly.
We use the term "emissivity" to refer to the degree of energetic coupling between
a body at thermal equilibrium and outgoing radiation modes. A body's emissivity,
when combined with a body's temperature, determines the power density of thermal
radiation it emits, and unless otherwise stated the presumption is that the spectral
intensity of the emitted radiation is proportional to a blackbody radiator of the same
102
temperature (i.e. the emissivity is a constant with respect to wavelength).
The emissivity c is defined with respect to radiating body which is at thermal
equilibrium at some temperature T, and so we must specify how we will generalize
these concepts to a non-equilibrium body such as an LED under nonzero applied
voltage. The emissivity is conventionally a constant independent of the excitation
of the system while the temperature contains all the information about its level of
thermal excitation. For this reason, we choose to generalize the concepts by using
the emissivity to refer to any change which is not an excitation of the body itself.
For example, applying a voltage which brings the electron-hole subsystem out of
equilibrium with the lattice would not refer to an emissivity change, but a change in
the surface reflectivity would. Using this definition, we now examine the compatibility
of the alternative interpretation of emissivity modulation with experimental results.
We organize the changes of the LED state corresponding to an emissivity change
into two categories. Neither a perfectly transparent body nor a body with a perfectly
reflective surface permit energy to flow out of the internal degrees of freedom of a
finite-temperature body into radiation modes in free space. Thus either a modulation
of the LED's transparency or its surface reflectivity at the wavelengths at which the
photo-detector is sensitive would constitute a modulation of emissivity.
First we consider a transmission modulation. Since the LED in the high-temperature
experiment was housed in opaque packaging, which was in turn held in a recess within
a heated copper rod, all of the bodies behind the device whose emission would be seen
in place of the LED's when the LED's transmission was increased were at the same
temperature as the device.
The experimental procedure could not ensure that the
temperature was exactly the same, but if the temperatures were equal, there would be
no signal at the photo-detector. If the housing seen through the device were taken to
be slightly higher than the LED, such an effect could produce a photo-current of the
sign that was seen. However, the temperature of the setup was always ringing around
the set-point temperature under the control of the Wavelength Electronics LFI-3751
P-I-D feedback controller, so the magnitude of this signal (increasing with the temperature difference between LED and its housing) would vary in sync with this ringing.
103
If the ringing brought the housing to a temperature below that of the active region,
that would even cause a sign change. Since no behavior of this type was observed
in the high-temperature experiment, the transmission modulation interpretation is
inconsistent with experiments.
The reflectivity modulation possibility must be considered in each of two subcases: those in which the device's surface reflectivity is primarily specular and those
in which it is primarily diffusive. These two possibilities are addressed by the following
experiments.
Of these we first consider specular reflection.
In this case the temperature of
the absorptive detector surface would affect the photon flux returning to it from the
emitter and thereby affect the measured photo-current. In this case, the rays which
would depart the surface of the device on a trajectory which eventually lands on the
absorptive photo-responsive surface of the detector could originate in modes of identical transverse position at the emitter surface, identical transverse momentum, and
longitudinal momentum of the opposite sign. Since the emitter and detector structures, as well as the intermediate optics, possess symmetry under inversion in the
transverse dimensions (i.e. they are all circles or squares), under perfect alignment
these light rays would all originate at the detector surface itself. Thus the temperature of the detector would be relevant to the signal seen if this reflectivity were being
modulated by our current source. If the detector temperature were colder than the
emitter temperature, then replacing photon flux from the LED with reflected photon flux from the detector would lead to a decrease in measured photon flux with
increasing reflectivity. Since the lock-in measurement of the photo-current indicated
an increase in photon flux with voltage, the sign of the reflectivity's dependence on
applied voltage would need to be negative (i.e.
MRsurf/V < 0).
Likewise, if the
detector temperature exceeded the LED temperature, the phase of the photo-current
signal should flip sign.
This situation was tested with the 3. 4 pm LED at room temperature and the TEcontrolled photo-diode held above and below room temperature. In Figure 3-9, raw
data from two nearly-identical experiments are shown side-by-side. The displacement
104
-10
<
4
X 10
470C PD
OFF ON
01.......
-0
4
.
0O
a
00
-c0
.L
-L *CP
(/3
-4
2
0
2
4
6
8
In-Phase Photo-Current (A) x 1o o
Figure 3-9: In-phase and out-of-phase components of the photo-current signal for two
similar measurements, one with the detector hotter than the emitter and one with it
colder. As the detector temperature was varied from above the emitter temperature
to below, the phase of the optical signal did not undergo a 1800 shift, indicating that
the observed signal is not due to a modulation of a specular surface reflectivity. The
measurement was taken with an excitation voltage of 4.4mV and a current of 2pA,
placing it clearly in the low-bias regime. The solid markers denote measurements
of the amplified photo-current signal with the LED off; the open markers denote
measurements of the same signal with the LED on. The the phase of the optical power
signal is near zero for both detector temperatures. The magnitude difference results
from the temperature dependence of the detector's responsivity, which we explore in
greater detail in § 5.4. As explained in the text, this is compatible with an electroluminescent cooling signal but not with a specular surface-reflectivity modulation
signal under good optical alignment.
105
of the solid blue squares from the open blue squares near the origin indicates the
presence of a phase-locked photo-current signal when the detector temperature was
well below ambient (-50 C). The displacement of the solid red circles from the open
red circles near the origin indicates the detection of a phase-locked photo-current
0
signal when the detector temperature was well above ambient (+47 C). Clearly the
sign of these two signals is the same. Since the phase-locked photo-current signal did
not in fact flip sign, we conclude that the preceding explanation is incompatible with
the observations in Figure 3-9.
Lens
Copper
Thermocouple /Housing
Uncooled (297K)
Photo-detector
LED
TEC-..............
-.......................
®r
*
U.
LLZ
294KIED
..
.......
......
E 0 297K LED
0* 3WOKLED
-16
16
8
0
-8
In-Phase Photo-Current (pA)
24
Figure 3-10: At top: a diagram depicting the experimental setup for the experiment
in which the temperature of the LED was heated above and cooled below ambient.
At bottom: the in-phase and out-of-phase components of the resulting photo-current
signal. The meaning of the different markers is stated explicitly in the legend to the
left of the plot and follows the same conventions as Figure 3-9. Results indicate that
the sign of the photo-current did not change with the sign of the LED-to-ambient
temperature difference, refuting the interpretation of the electro-luminescent cooling
measurements presented throughout this chapter as instead originating in a voltagecontrolled modulation of the surface reflectivity of the LED.
106
Next we consider the case of diffusive reflection, which also covers the case of specular reflection under poor alignment. In the case of diffusive reflection, the rays which
would land on the absorptive detector surface could originate from ray incident on the
surface. Since the environment is roughly in equilibrium at 297K (and the blackbody
flux averaged over the detector's photo-responsive band was not significantly higher
than this in other observations), in analogy with the previous test, we performed similar measurements in which the sign of the temperature-difference of relevance was
changed. In this case, because a changing of the diffusive reflection coefficient would
affect the fraction of the emerging photon flux which originated in the device or the
surrounding environment, we chose to raise and lower the temperature of the LED.
As shown in the plot at the bottom of Figure 3-10, we see that once again the sign
of the photo-current signal did not flip with the sign of
(TLED
-
Tambient).
From this
observation, we infer that the measurements reported throughout this chapter are
not compatible with originating in the modulation of the LEDs' diffusive surface reflectivity or specular surface reflectivity under conditions of poor alignment. In fact,
since the detector in these measurements was not cooled, the data in Figure 3-10
alone stands in contradiction to the effect of any type of surface reflectivity regardless
of alignment.
We now pause to clarify the role of the preceding arguments concerning emissivity modulation within the broader experimental effort described in this chapter. We
regard the preceding arguments as reasonably strong evidence against interpreting
our measurements as emissivity modulation and supporting evidence for the observation of electro-luminescent cooling. It is not easy to entirely exclude any family of
interpretations with a few data sets like the ones found in Refs. [47] and [83], let alone
prove a single interpretation beyond doubt. The consistency of the data with predictions from theoretical models, combined with the reproducibility of the measurement
using LEDs in various material systems, with different detectors, and at different
temperatures is also a significant contribution to our confidence in the interpretation. More specifically, the theoretical models quite clearly predict that the quantum
efficiency of an LED should become independent of voltage for qV < kBT.
107
The
value for
qEQE
indicated by the data was approximately 3.30x 10-4 and 3.35x 104
for the 2.15pm LED at 135'C and the 3.4ptm LED at room temperature respectively
(statistical uncertainty for both
?7EQE
values was between 1 and 2 x 10-). In both of
these experiments, literature data suggests that SRH recombination is the dominant
recombination process by orders of magnitude at low-bias. As a result the external
quantum efficiency depends primarily on the SRH lifetime r, the bimolecular recombination coefficient B, and the efficiency of photon collection. Since literature values
of the first two quantities and reported values for the third are commensurate with the
observed
7EQE,
we see it as unlikely that not only would this calculation be inaccurate,
but that some unspecified physical mechanism would lead to measurements with not
only the correct 7 oc
scaling over 3+ orders of magnitude, but with the a constant
factor very nearly equal to what one would predict for the electro-luminescent cooling
effect value using figures from literature. Further independent measurements of the
effect, such as spectral shift due to band gap narrowing or lattice cooling of surrounding matter, would provide further evidence. We take up this and related topics in
Chapter 6.
3.4.2
Unity Efficiency at Room Temperature
This section presents the results of room temperature measurements of mid-infrared
LEDs at low bias voltages. We find that the results mirror those of
@3.2
and further
support the theory presented in Chapter 2.
We begin by describing the LEDs used in the experiment. Two devices were tested,
4
one emitting with a center wavelength near 3. pm and the other near 4.7pm. The
devices were grown and fabricated by the research group of Professor Boris A. Matveev
at the Ioffe Physico-Technical Institute in St. Petersburg, Russia. The devices were
originally acquired through a North American distributor named Boston Electronics,
but the labeling of individual devices allowed the Ioffe group to provide details of
the device fabrication beyond those made available to commercial customers.
For
completeness, we provide the original text provided by the loffe group, with minor
changes to the language to enhance readability:
108
A = 3.4um
X = 4.7pm
more
InAsSb
N
more 1nP
t
n-type InAs (111):
,n=2e16cm
p-InAs,, Sb(,()P,,:
P=2e17 to 5e]-7 cm
Zn-doped
5 pm
Zn-dopod p-InAsSb: p=5e17
50-60 pm
Nmminally
35OiimbP
.:el
ndoped
nAs (100):
n=3el8 to 6e18 cm
m
7 m
=I~
Sb-doped W
350
3.5 pm
200
m
Figure 3-11: Layer stacks for the 4.7pm LED (left) and the 3.4pm LED (right). Data
in these figures is derived primarily from communications with Prof. B. A. Matveev
found in the main text.
"The 4.7pm light source (grown on wafer #236) was made from an
80 pm thick narrow gap InAsSbP/InAs hetero-structure. Thin (350pm)
4
3
n-InAs wafers (n = 2x1016 cm , initial dislocation density Nd 10
cm-2) with (111)-oriented surfaces were used as substrates. Due to the
high Phosphorus segregation coefficient, the InP concentration diminished
within a 50-60pm-thick 'undoped' n-InAsSbP layer providing the energy
gap decrease along the growth direction with energy gap gradient VEgap
of about 1-2 meV/pm. Zn was used as a p-dopant for p-n junction formation at the final stages of the growth with the resulting distance from the
p-InAsSb surface of about 5[tm. At the hetero-junction, the layer lattice
constant a was nearly the same as for the InAs substrate (lattice mismatch Aa/a < 0.05%) while the narrow band part of the structure (i.e.
the p-InAsSb region) was lattice mismatched with respect to the InAs
substrate. Due to high InAs plasticity at the growth temperature (6503
720 C) the p-InAsSb(Zn)/n-InAsSb-InAsSbP (p ~5 x1017 cm~ , n ~
'7
cm-3) graded structure formation was accompanied by stress relaxation
via substrate bending providing an 'inverse' dislocation distribution across
the hetero-structure. That is, when the plastically deformed/bent InAs
substrate was finally incorporated with the bent graded layer of high crys2
talline quality, the epi-layer dislocation density Nd didn't exceed 105 cm~
2
while Nd in the substrate was as high as >107 cm- [84]."
PROF. BORIS
A. MATVEEV
PRIVATE COMMUNICATION ON AUGUST 21, 2012
109
At left in Figure 3-11 is a visual representation of the device's LPE-grown (Liquid
Phase Epitaxy-grown) layer stack. After growth, the diode was packaged with an
immersion lens in the same way as the 2.15jpm LED in § 3.1.3. Wet photo-lithography
was used to etch a square 150pmx150pm mesa structure (roughly 25-30Qm deep).
A Cr-Au(Zn)-Ni-Au reflective (R=0.6) anode contact was used to reflect light back
through the substrate. To improve photon extraction and create a narrower beam
profile, a nearly hyper-hemispherical Silicon lens was attached using a chalcogenide
glue with an index m2.4.
Regarding the 3.4pm source, the following description was provided:
"The layer stack of the 3.4pm light source (#6341) was a single heterojunction structure consisting of a 200pm-thick heavily doped n+-InAs,
(100)-oriented transparent substrate doped with Sn to n ~ (3 - 6) x 1018
cm- 3 , followed by two epitaxial layers. These two layers included a 7 Jtmthick n-InAs active region and a 3.5pim-thick wide-gap p-type Zn-doped
(p = (2 - 5) x 1017 cm- 3 ) InAsSbP cap layer.
The alloy composition
of the cap layer was approximately 73% As, 9% Sb, and 18% P. Further
information on this growth is described in Ref. [85]."
PROF. BORIS A. MATVEEV
PRIVATE COMMUNICATION ON AUGUST 21, 2012
At right in Figure 3-11 is a visual representation of this layer stack, designed to
become an LED emitting at 3.4[tm. This growth was subsequently processed in much
the same way as the previous 4.7pm growth, except that the dimensions of the square
mesa were 230x230pm.
Both the 3.4 and 4.7pm LEDs were studied across a wide range of operating
points at room temperature.
The forward bias voltage, current, and light output
were measured for each device across five orders of magnitude in current, extending
from conventional operating points where the applied bias voltage qV is on the order
of the bandgap energy Egap down to the low-bias regime. The results are shown in
Figure 3-12.
110
10--6
10
oc
0L
-10
cc
2
**
/A
-8
3.4pm
440A4g
4.
0
o
A
k?0.05
F
A22.15p±m
0
0
-12
.4.4
8--
10-0
10
10
10~-
A
10~-
Current(pA)
0.1
0.15
-
10~4
10-2
Input Electrical Power (W)
Figure 3-12: Output optical power versus input electrical power for three room tem2
perature mid-infrared LEDs. For the device emitting at 3.4pm (area 5.29x 104 cm ,
2
4
wafer #6341) and 4.7pm (area 2.25x10- cm , wafer #236), the power at unity efficiency was high enough to be directly observed in our lock-in measurements. For the
device emitting at 2.15pm, it was not. Note: Data for the 2.15pim LED is from § 3.2.
Insets: (top left) Relative intensity spectra for the three devices at room temperature;
(bottom right) cooling power versus current for the 3.4pim LED at room temperature.
111
The experiments on these devices were quite similar to those previously performed
on the heated 2.15[tm LED. For current levels up to 2 mA, a 1013 Hz square wave voltage source was used in combination with a series resistor of magnitude greater than
either device's zero-bias resistance (sometimes called the 'shunt resistance' though we
avoid that term here because this conduction is necessarily not due to shunts alone).
Optical power was measured via lock-in zero-bias photo-detection with time constants
ranging from 500 milliseconds to 500 seconds. For higher current measurements, a
DC source-meter was used along with zero-bias DC photo-detection. DC measurements of current, voltage, and optical output power were in fair agreement with AC
measurements; both types of measurements appear together in Figure 3-12.
For comparison, Figure 3-12 also includes a theoretical curve representing the
Carnot limit for an emitter with the same wavelength, active area, and temperature
(298K) as the 3.4pm LED data presented.
We take the idealized emitter to be
optically thick at the emission wavelength, so that the theory in
§ 2.4.1 may be
used to relate optical power density to Carnot efficiency.
From
§ 2.4, we know that for mid-infrared LEDs with a given
7
7EQE,
Lunity should
increase as the photon energy decreases or the lattice temperature increases. Lockin power measurements on the 3.4pm LED showed that Lusity increased with Tiattice
from 300 K up to around 420 K. Above 420 K Lunity decreased with Tiattice, suggesting
that the increases in power at fixed voltage were likely outweighed by decreases in
quantum efficiency from non-radiative recombination and leakage, and the increased
importance of parasitic effects from contact resistance.
As shown in Figure 3-13,
this temperature dependence indicates that for the 3. 4 pm LED, Lusity is maximized
when hw/kBT is around 10. Although this does not seem to be fundamental, various
authors have argued for much smaller[43] and much larger[42, 14] values of hw/kBT
without experimental realization, so this phenomenological observation may serve as
a guide for further experiments.
112
A=2.5pm
10-9
A=3.4pm
w 10
-10
00
c10~
4---.A=4.7pm
0
0.
6
15
12
9
hw / kBT (dimensionless)
18
Figure 3-13: Optical power density at unity wall-plug efficiency versus the dimensionless ratio Egap/kBT. The triangles labeled 2.5ptm correspond to the high-temperature
results from § 3.2; the circles labeled 3.4pm correspond to results from § 3.4 and
similar experiments at elevated temperatures; the square labeled 4 .7 pm corresponds
to experiments on an LED of the same model as the one characterized in § 3.3.
113
Does Voltage Determine Brightness?
3.4.3
0
0
10
01
0
=4.7pm
00
0.
b 10
10-121
a t
A~M.4m
A=2.15pm.
-3
10~4
10~5
10~-
10-3
10-2
1-
100
Voltage (V)
Figure 3-14: Optical power density versus applied forward bias voltage for three midinfrared LEDs. The discrete markers denote experimental data for voltages up to half
the bandgap energy per electronic charge q. The solid lines correspond to numerical
calculations based on Equation 2.45 and Equation 2.24.
In Chapter 2, we presented a transport model for thermo-electrically pumped
LEDs at voltages well below the bandgap. In this model, the Fermi level separation
in the active region leads to an excess population of electrons and holes, which can
also be described by a temperature at each above-gap transition energy. This temperature, T*, is the thermodynamic temperature seen by fields which interact through
inter-band transitions, so that the spectral power density of photon emission may be
directly related to this value. In Equation 2.24, we made the simplifying assumption
that AEF ~ qV (true when the junction resistance dominates over parasitic series
resistances, as in most LEDs with reasonably large bandgaps and low voltages). In
114
this way, for an emitter with a known bandgap energy and lattice temperature, the
power density above the blackbody background should be fully determined by
T*,
and therefore V. Here we seek to test this hypothesis experimentally.
We also note that in contrast with the optical power density, the current density is
not entirely determined by the bandgap energy and voltage. The presence of material
defects leads to a device-specific quantity of current flowing through trap-assisted nonradiative recombination pathways in parallel with the known quantity of net radiative
recombination we have just described.
In Figure 3-14 we compare the results of these room temperature power measurements with calculations based on Equation 2.45 and Equation 2.24.
For each
LED, experimental data is shown for voltages from zero up to half the bandgap energy per electronic charge. Across this range, the data is in qualitative agreement
with numerical calculations. We note that the active area of the photo-diode used
at 3.4 and 4.7[pm was significantly smaller (1 x 1mm) than that used at
2
.15pm (3
mm outside diameter). Thus the longer-wavelength measurements may include the
effect of imperfect collection efficiency by the detector; we have not corrected for this
possibility in any of the data in Figure 3-14, but initial measurements with a lensless
photo-conductor indicated this effect may have reduced the signal by a factor of 6.7.
At higher voltages, series resistances and other rate-limiting transport processes
cause L to fall short of the calculations based on V rather than AEF, including those
based on Equation 2.24. We note that our simple model must break at some voltage,
since as qV approaches Egap and the band-edge states approach inversion, T* and L
diverge in a non-physical way. We take up this discussion briefly in Chapter 6.
115
3.5
Summary and Conclusions
We began this chapter by detailing various experimental techniques required to investigate the wall-plug efficiency for photon generation by mid-infrared LEDs at low
intensity. We then reported a number of experimental results related the general phenomenon of thermo-electric pumping in these devices. These results included the first
known experimental confirmation of electrically-driven light emission from a diode in
excess of the electrical power used to drive it.
We began in
§ 3.1 by presenting a series of hurdles encountered during these
experiments and the solutions developed to address them. Some of these techniques
were fundamentally needed to execute the desired experiments, such as the lock-in
photo-detection technique explained in
of the emitter diode developed in
§ 3.1.1 and high-temperature feedback control
§ 3.1.2. However other techniques were developed
in response to unexpected hurdles, including limiting the temperature slew rate to
avoid irreversible damage to the emitting diodes from thermal shock and introducing
extra free-space optical elements to enhance the photon collection efficiency of our
smaller detectors, developed in
§ 3.1.3 and § 3.1.4 respectively.
We then used these these techniques in
§ 3.2 through § 3.4 to examine the effi-
ciency of three mid-infrared LEDs across a range of temperatures from 300 to 400
K. Although data across several orders of magnitude in current density was acquired,
the focus of our experiments was on the behavior of these devices in the low-bias
regime, where qV < kBT. The results of these experiments confirmed the primary
hypotheses from Chapter 2, that an LED behaving as a thermodynamic heat pump
would have wall-plug efficiency that increases with temperature and would emit more
optical power than the electrical power used to drive it. We saw that for the longestwavelength emitter, the indium arsenide antimonide 4.7ptm LED from
§ 3.3, the
benefits of increasing exp(-Egap/kBT) appeared to be outweighed by increases in
Auger recombination and leakage as well as the increased relevance of parasitic series
116
resistances in diodes with large saturation currents.
A cursory meta-analysis sug-
gested that the maximum power at unity efficiency was found in devices whose ratio
of bandgap energy to thermal energy was approximately 10.
Despite the low power density at which the thermodynamic behavior was observed
in these light-emitting diodes, these experiments served to establish a new direction
in research on the phenomenon of electro-luminescent cooling. The space of operating points explored here, those with bias voltage much less than the thermal energy
(i.e. the low-bias regime qV
< kBT), not only provides a platform for experimental
demonstration, but is where the greatest deviations from conventional q < 1 behavior
are found. We found that not only are very high efficiencies (q
> 1) only possible
at low voltages, but that in the low voltage limit the efficiency is required to diverge.
While the fundamental trade-off between power and efficiency demands that very
high efficiencies be associated with correspondingly low power densities, the experiments reported in this chapter suggest that thermo-electrically pumped LEDs may
be more naturally suited to applications in which efficiency is more important than
power density. In Chapter 4 we will explore one such application: using LEDs operating at very high efficiency to explore the limits of energy-efficient classical photonic
communication.
117
THIS PAGE INTENTIONALLY LEFT BLANK
118
Chapter 4
Communication with a
Thermo-Photonic Heat Pump
In this chapter we explore the implications of heat-pumping behavior in LEDs for the
energy-efficiency limits of classical photonic communication. In § 4.1 we revisit data
from high-efficiency measurements from our setup in Chapter 3 to motivate the topic.
In
§ 4.2 we calculate the minimum amount of work required for a Carnot-efficient heat
pump to encode a bit into the electromagnetic field at finite temperature. In
§ 4.3,
we present an experimental demonstration of a low-biased LED communication link
in which the source consumes just a few tens of femtojoules per bit.
4.1
Power Measurements as Slow Communication
In order for the photo-detector in the power measurements in Chapter 3 to detect an
optical signal with nonzero signal-to-noise ratio (SNR), the information about whether
or not the LED under test is on must be shared across the optical path. In essence,
we may relabel the LED whose power is being measured as the transmitter and the
photo-diode with amplification and analog-to-digital conversion as the receiver, and
call the entire setup a communication link.
119
At low power, where above-unity efficiency is seen, long time constants are required
to achieve an SNR above 1:1. As a result, we may expect that the bitrate for any
communication across this link would be very low. Nevertheless, the rate of electrical
power consumption is also quite low, suggesting that the amount of electrical work
required per bit could be small enough to motivate certain practical applications.
In
§ 4.1.1 we begin by performing a sample calculation on an actual above-unity
efficiency power measurement.
In
§ 4.1.2 we extrapolate these results to the low
power limit to find the minimum work required per bit of information detected by the
receiver. Finally, since the devices are still far from Carnot-efficient (i.e. r/EQE < 1),
in
§ 4.1.3 we modify these calculations to extrapolate results for a theoretical device
with perfect quantum efficiency r7EQE
4.1.1
=
1-
Sample Calculation
For this sample calculation, we will use the second-lowest-power data point from the
150'C measurements of the LED emitting around A
=
2.5[tm. Note that this is also
the data set behind Figure 1-5. The raw data for that point and the two around it
appears in Table 4.1.
Optical Power L (pW)
Std Dev L, (pW)
SNR
Electrical Power IV (pW)
11.367
3.1838
3.5703
0.19875
36.336
3.0214
12.026
1.7028
123.38
2.5508
48.369
18.940
Table 4.1: Selected power measurement data from 150'C LED emitting around 2.5pm.
These are the three lowest-power measurements from this data set. The time constant
for all three lock-in measurements was Is.
From the data in the second row, we see that an optical signal with SNR~12 can
be generated using just 1.7028 pW. We may estimate the electrical work W required
120
to send this signal by the product of the input power and the time constant; this
yields W = 1.7 pJ.
To calculate the number of bits of information transmitted, we must know the
probability of an error.
That is, the probability that a sent '1' will turn into a
detected '0' or vice versa. Consider the histogram of the 36 pW data point found in
Figure 4-1. The 'off' measurements are clearly separate from the 'on' measurements,
so the number of data points available is insufficient to find an error. Nevertheless,
by fitting these histograms to Gaussians and defining a decision boundary that makes
error rates symmetric, we may estimate the probability of such errors. In this case,
the two Gaussians are separated by roughly 12 standard deviations. For a simple
on-off keying (OOK) scheme in which the '0' and '1' symbols are equally likely to
be sent, the optimal decision boundary falls halfway between the means of the two
Gaussians, or just under 6 standard deviations away from each. Thus we can calculate
the probability of seeing a '1' when a '0' is sent (or vice versa) as the integral of the
normal distribution's tail starting from 6 standard deviations out. In Table 4.2 and
the adjoined caption, we express the resulting joint probability mass function (PMF)
describing this scenario. The joint PMF P is a function of the transmitted symbol x
and the received symbol y, and contains the probabilities of all possible outcomes of
a single symbol transmission event.
From the joint PMF in Table 4.2, the amount of information shared across this
channel may be calculated as the mutual information I. In standard information
theoretic notation, I is defined as the Kullback-Leibler (K-L) divergence
DKL
between
Px,y(x, y) and it's product-of-marginals Qx,y(x, y) = Px(x) - Py(y). Intuitively, the
K-L divergence is like a measure of distance between two probability distributions
(although it lacks basic properties like symmetry). The K-L divergence between two
probability mass functions fA(a) and fB(b) (defined over the same set of events {i})
121
15
105
00
7
IV=1.7pW, L=36pW
off
oLn
0
--------------
1
2
Raw Lock-In Signal
3
x 10
Figure 4-1: Histogram of the raw 'R'-values from the 36 pW output power data
point. The blue histogram blocks at left are from measurements with the LED off.
The blocks at right are from measurements with the LED on. The two red curves
represent Gaussian fits to this data which we use to calculate the information content
of the signal. Twice as many off measurements were made as on measurements.
However, the standard deviations of the best-fit Gaussians were similar, suggesting
that sufficient data was available for a fit.
122
X=1
x=0
(Send '0', LED off)
(Send '1', LED on)
y=O
Pxy (0, 0) =
Pxy (1, 0) =
(Receive '0', LED looks off)
0.5 - -I(-5.895)
i4(-5.895)
Px'Y (I, I) =
(0, 1) =
y1Px'y
(Receive '1', LED looks on)
!D(-5.895)
0.5 - -J'(-5.895)
Table 4.2: Joint probability mass function (PMF) for communication at 36 pW. Here
1(x) denotes the cumulative distribution function of a Gaussian distribution with
mean 0 and standard deviation 1. The probability mass in one tail of a Guassian
distribution starting 5.895 standard deviations from the mean is D(-5.895) ~ 1.9 x
10-9.
is defined as:
DKL(fAI
fA(ai) log (AfB(bi)
IB)
(4.1)
Vi
where a2 and bi are the values of A and B for each event i.
The joint PMF P expresses the probability of every event in the sample space,
where events are defined by what is sent and received; the product-of-marginals
Q
expresses the combination of two distributions formed from events defined in terms
of what is sent or received, but not both. That is to say, P describes our channel,
while
Q
describes a channel which looks the same from either side (transmitter or
receiver), but through which random noise prohibits any information from being
communicated. The divergence (i.e. the difference) between these two situations is
defined as the mutual information I for the channel:
I
=
DKL(PQ)
E
PX,Y (X, y) - log
PXY
(4.2)
Qx'Y (X, y)
Xy=O,1}
where the logarithms in the above expression are base 2 and I is in units of bits. The
result of applying Equation 4.2 to Table 4.2 is very nearly 1, indicating that almost
one full bit of information is conveyed with each symbol transmitted. A more exact
123
calculation follows.
Defining 6 = (D(-5.895)/2, we find that the diagonal terms contribute to I as
follows:
Px y(0, 0) - log
Px,y(0, 0)
Qx'y (0, 0)
=
(0.5
-
0.5
6) - log
0.25
=(0.55-6)
= (0.5 -6)
-log(2)
L
= (0.5 - 6)
6
-
+ log0.-6)
(0.5
]
[log(2) + log (1 - 26)).
Using the common expansion of base-2 logarithms log(1 + x)
=
1
X
+
(x 2 ),
= 0.5 - 6 + 0.5 log(1 - 26) - 6 log(1 - 26)
=
1
0.5 - 6 + 0.5 -n(2)
= 0.5 -
I+
n 2)
In(2)/
.
(-26) +...
6+ O(62).
The contributions of the off-diagonal terms may similarly be ordered in powers of 6:
Px y(0, 1) - log
'
Qx'y (0, 1)
=6
- log
(.2)
6 log(6) + 26.
Thus,
I = 1 - 26log(1/6)
1
-2 n (2)
S1) 6 + 0(62).
(4.3)
Using this expansion for the 36 pW measurement, the information contained in a
power measurement is roughly 1 - 5 x 10- 9 bits. Direct numerical evaluation yields a
124
similar result. Thus the amount of work required to communicate a bit, W/I ~ 1.7
pJ or about 3 x 108 times the thermal energy kBT.
4.1.2
Extrapolation to Low Power
As we saw in the derivation of the Landauer limit (kBTln(2) per bit) in § 1.4, the
most energy-efficient communication happens at low power.
This fact can be intuitively expected from considering communication with an
irreversible (i.e. not heat-pumping) transmitter over a time slice sufficient to send
just one symbol. As the signal power P becomes much larger than the noise power N,
the amount of energy consumed in this time slice scales linearly with P. Meanwhile,
the number of distinguishable quantization levels at a given bit error rate scales
linearly with P, so the number of bits of information scales as log(P).
expect communication protocols with P
Thus we
>> N (i.e. high SNR) will be further from
the minimum energy consumption per bit of mutual information shared across the
channel under consideration.
With that in mind, let us re-examine the result from the previous section. Consider
an experiment performed in the same configuration as the 36 pW data point discussed
above, but with much lower drive current through the LED. Since the output power
and photo-current are linearly related to the drive current, they would also be much
lower. However, the noise-equivalent power of the receiver is set by thermal processes
within the photo-diode, so for a fixed amplifier configuration and lock-in time constant
the uncertainty in power measurement should be fixed. As a result, the SNR should
decrease along with the input power.
To find the minimum electrical energy per bit for this link then, we must perform
the following calculations:
* For an arbitrary SNR r, find the amount of electrical work W consumed over
the time constant for the theoretical power measurement At.
125
"
For an arbitrary SNR r, find the mutual information I shared across the channel.
" Find limr o
. (Note: This amounts to a communication protocol in which
the signal-to-noise ratio is much less than 1, so much less than one bit is communicated with each symbol. Since such a protocol would need to be repeated
to communicate any practical amount of information, it may have too low of
a bitrate for practical systems. Ve consider it here primarily for its scientific
value.)
Recalling that in the low-bias regime n
-
1/L, we may write a simple expression
for W in terms of the output power at unity efficiency Lunity:
L
W = -
L2
. At =
77
. At
.(4.4)
Lunity
In terms of the signal-to-noise ratio r = L/L, (where L, is the standard deviation in
the light power measurements, as it was in Table 4.1), then we have:
W =
r
2
L2
" - At
.
(4.5)
Lunity
For the mutual information calculation, consider an analogous version of Figure 4-1
for arbitrary r. If we consider the 'on' and 'off' states to see the same uncertainty
L, as before (reasonable because the noise physically originates from an additive process), we may again place the symmetric decoding boundary halfway between the two
means. As a result, the off-diagonal elements of the corresponding joint probability
distribution will be given by the probability mass in the tails a Gaussian, this time
starting from r/2 standard deviations from mean. Thus, we arrive at Table 4.3.
Since we will be evaluating (D(x) near x = 0, we should first Taylor expand this
126
Send '0' (LED off)
}A(-r/2)
Receive '1' (LED looks on)
}I(-r/2)
}b(-r/2)
0.5 -
Receive '0' (LED looks off)
Send 'I' (LED on)
0.5 -
}((-r/2)
Table 4.3: Joint probability mass function (PMF) for communication with arbitrary
signal power L = rL,.
function around that point:
@(x) = @(x) L
= 0.5 +
-0.5+
+
d@(x)
dx
d [
dx- - -_
0
1
exp
-0.5+
+0(x
x + O(x 2 )
xo
i1
e
(
2
(2
e xp
~2}
2
X=O
dy]
x=O
x + O(x 2 )
X + 0(X2
)
Substituting this expression for D gives us Table 4.4, which is valid only in the
low-power limit r < 1.
Send '0' (LED off)
Receive '0' (LED looks off)
0.25 +
Receive '1' (LED looks on)
0.25 -
Send '1' (LED on)
1 r
0.25 -
1r
r
0.25+
Lr
1
Table 4.4: Joint probability mass function (PMF) for communication with low signalto-noise ratio r = L/L, < 1.
As with the 36 pW data point before, we can compute the contribution to the
mutual information I from the diagonal terms and off-diagonal terms separately, then
sum them. For convenience, let us define a new small parameter a =
127
1r.
The diagonal contribution is as follows:
Pxy(0, 0) - log
(Px'Y(0,
0)
QxY ( 0, 0)]
0.25
a
= (0.25 + a)
log
= (0.25 + a)
log (1 + 4a)
= (0.25 + a) - (4a - (4a)2
2
-
(a - 22 +4a2 + O(a3 ))
In
(2))
a+
±
(3))
ln(2)
n(2)
In
(2)
And the off-diagonal contribution is:
Pxy(0, 1) - log
(Px'y
(0, 1)\
Qx'Y (0, 1)]
= (0.25 - a)
log
= (0.25 - a)
log (1 - 4a)
= (0.25 - a) -(4a
0.25-
-4
= (-a-2a2+ 4a22+
h (2)
)e+
( In2)(2)J
(
a2 + O(a3))
O(a3))
ln(2)
n (2)
az2 + 0(ae3 )
Combining these terms, we get:
82
Sa2 + O(a3 )
in (2)a
(4.6)
Thus we arrive at the general expression for maximum energy efficiency of com-
128
munication given a power measurement of the type from Chapter 3:
min
-
lim
T
["
2
r-+O
L2
-
Lo-
At - 47rln(2)
(4.7)
(4.8)
Lunity
In § 4.1.1 we saw that the 36 pW data point could be interpreted as communicating
about one bit per 1.7 pJ. If instead we had operated the LED at much lower power,
more efficient communication should have been possible.
Since from Table 4.1 we see that 36.336 pW of optical power could be generated
for just 1.7028 pW of electrical input power, we may use the q ~ 1/L scaling law
to infer that the power available at unity efficiency was Laity = 775.4 pW. (Note: a
best fit from the all of the above-unity efficiency data points [86] gives Luity ~ 764
pW, in good agreement with this figure.) Thus, using just the data from the second
row of Table 4.1, we find that the minimum work required per bit for this link is:
min
W _(3.214pW) 2
- s - 47ln(2) = 103 fJ/bit
1
775.37 pW
(4.9)
For context, please note that the effective data rate for the channel at this operating
point is less than one bit per second.
In § 4.3, we will address this issue using
orthogonal frequency-division multiplexing (OFDM).
4.1.3
Extrapolation to Carnot-efficient LEDs
As we discussed in § 2.4, in theory if all non-radiative recombination is eliminated
from an LED, but the device's active region remains optically thick, then the device
acts very nearly as a Carnot-efficient heat pump. Here we use the term "Carnotefficient" to mean thermodynamically reversible, without any entropy generation but
129
possibly with entropy transport, and as efficient as possible given the Second Law of
Thermodynamics. Recall now the results from
§ 2.3 and § 2.4. Since in such a device
the net amount of radiative recombination (i.e.
recombination minus generation)
should still be given by Rrajiative = B(np - n?) = Bn?(eqV/kBT
-
1), at low bias this
still contains a term linear in V. Thus we expect the idealized LED to have a finite
zero-bias resistance. Again, therefore we have:
L
I
IV -Oc 12
1 x -'R q
R ZB
1
L
(4.10)
However, if we write 77 = Lusity/L, we must be careful to note that L..ity is defined
by the low-bias behavior; the actual unity efficiency point will be where V
> kBT/q
and so will be much larger than Lusity.
With this in mind, we have performed a numerical calculation using Equation 2.24
for a diode with area A =0.0616mm 2 and red cutoff wavelength A = 2.6pm at temperature T
=
423K. We find that Lunity = 20.6 W/m
2
x A = 1.27pW, or about 1600x
larger than our measured result. Since the minimum work per bit scales inversely
with Lusity, our calculation suggests that for an LED with unity quantum efficiency,
roughly 63 aJ/bit should be required. Note that this is about 4 orders of magnitude
away from kBT ln(2) ~ 4 x 10-2 1 .
We know that in principle reductions in L, can also be made by decreasing the area
and temperature of the photo-diode.
Consideration of such idealizations, however,
leads to an interesting question: What happens if we reduce the area and temperature
of the diode until L, falls by more than a factor of 125? Doing so would decrease the
work per bit to below kBT ln(2). At first this might seem possible because the the
input-referred current noise that leads to L, is inversely proportional to the photodiode's resistance, and the resistance across a p-i-n junction can be made extremely
large at cryogenic temperatures.
An important hole in this analysis is that as the
130
thermal noise in the photo-diode is reduced, at some point other sources of noise in the
power measurement may dominate. In the following section, we consider a different
noise source: shot noise in the arrival of blackbody photons.
we described in
In the experiments
§ 3.2, thermal noise due to lattice vibrations in the photo-diode
dominated over this contribution.
As we will see in
§ 4.2, the presence of this unavoidable noise source imposes a
lower bound on the work per bit required by a thermo-photonic heat pump. Although
we explore this problem theoretically, in principle it may also be measured experimentally. We will return to this proposition as part of Future Work in Chapter 6.
Limits of Energy-Efficient Communication with
4.2
a Heat Pump
The Entropy Trade-Off
4.2.1
Consider a single photonic mode occupied thermally at a temperature To. Take the
expected number of photons in the mode to be E[N] = No. Now imagine that two
devices are capable of increasing the number of photons E[N] in the mode in two
different ways.
" Device A increases E[N] to No + 1 by deterministically adding a single photon
to the mode, making it a non-thermal distribution.
" Device B increases E[N] to No + 1 by increasing the temperature of the mode,
so that it remains a thermal distribution with some temperature T > To.
The photon number distributions for these scenarios are depicted in Figure 4-2. The
state of the photon field that results from the distortion by Device A is described
by a photon number probability distribution PN(n) which is the same as the original
131
distribution, except shifted to the right by 1. Since the entropy of a distribution (or
the von Neumann entropy of the corresponding density matrix) does not depend on
the labels of the outcomes, this state has exactly the same entropy as the original
distribution at To. Meanwhile the state of the photon field produced by Device B has
an increased expected number of photons by further spreading out the distribution.
This state has more entropy than the original distribution.
0.4
0.4
0.4
>0.3
.go3E[N]=
S0.2
E[3
3
(Device A)
0.3 1
[
E[N]= 2
.Z,.
100.2
I
>0.3j
.go3E[N]
0 0.2
[J3
=3
(Device B)
02
0
0.1
0.1
0.1
0
1 2 3 4 5 6
Number of Photons N
0123456
Number of Photons N
1 2 3 4 5 6
Number of Photons N
0
Figure 4-2: Photon distributions for the initial thermal state (middle), as well as the
final non-thermal state produced by Device A and the final thermal state produced
by Device B. Both Device A and Device B increase the number of photons in the
mode, but only Device B increases the mode's physical entropy.
The extra entropy which Device B adds to the output state changes the lower
bound on how much work must be consumed imposed by the Second Law.
The
Second Law does not permit the destruction of entropy, but since the photon field's
final configuration has more entropy, some of this entropy can in principle be drawn
from another thermal reservoir. If we presume the existence of another reservoir at
temperature To (in our case, the phonon bath), each bit of entropy AS which is drawn
from this reservoir brings with it energy TOAS. In essence, both devices are distorting
the initial state of the photon field, but the Second Law permits Device B to create
this distortion more efficiently. That is to say, in order to increase E[N] by 1, Device
A must consume hw of work, while Device B may consume less. Moreover, for very
small distortions of the original thermal state, Device B is effectively pumping heat
from some reservoir at To to a mode occupied at T = To
efficiency diverges: rCarnot
T/6T -+ oo.
132
+ 6T. Here the Carnot
Now consider the possibility of using Devices A and B as transmitters, along with
a detector that counts the quanta of the photon mode in question, to form a simple
communication link. The fact that Device B can more efficiently increase the number
of quanta in the mode raises the interesting question of whether kBT ln(2) per bit
limit we derived in § 1.4 may be overcome by heat pumping.
In this section we will see that closer examination of such a link reveals a fundamental aspect of communication that we have thus far neglected. Device A and
Device B can both change the original occupation of the photon modes by adding the
same amount of energy, but the resulting final states are still different. In particular,
the final state that Device B produces is fundamentally less distinguishablefrom the
original state than the final state Device A creates. In essence, Device B makes efficient use of the mode's capacity to store entropy with its efficiency improvements on
the transmitter side while Device A makes efficient use of that capacity to make the
final state more distinguishable on the detector side.
Thus the calculation of the efficiency limit for communication across a link made
with Device B is interesting not only as a generalization of a basic problem in communication and information theory, but suggests a fundamental connection between the
notions of entropy in thermodynamics and entropy in digital systems, a subject which
may become increasingly relevant as more efficient digital systems are developed.
4.2.2
Calculation of the kBTln(2) Limit
To calculate the theoretical minimum work-per-bit required to communicate with a
thermodynamic heat pump, we rely on two different bounds in combination:
1. The Carnot bound imposed by the Second Law of Thermodynamics. We use
the Second Law to place a lower bound on the amount of work required to pump
heat into the electromagnetic field at finite temperature.
133
2. The information-theoretic Shannon Limit derived from the Channel Coding
Theorem. We use the Theorem to place an upper bound on the amount of
information (measured in bits) which can be reliably sent across a noisy channel.
We begin by calculating the arrival rate of blackbody photons at a detector with
perfect quantum efficiency above it's band gap energy Egap and zero quantum efficiency below it. For an incoming electromagnetic field occupied at finite temperature
T, we can calculate the number of above-gap photons per unit volume from the
density of modes in reciprocal k-space, the dispersion relation hw = hck giving the
photon energy in each mode with wave-vector k, and the Bose-Einstein distribution
giving the expected number of photons in each mode. Integrating over wave-vectors
corresponding to above-gap photon energies, we have:
N
d3
2
V
(27)3 Egap/(hc)
(kBTh) 3
3
72C
eXp
(4.11)
1
(
J
Egap/(kBT)
k
2dx
ex
-
which leads to a particle flux of:
3
(kBTh)
2 2
47 c
JN
f0
fEgap/(kBT)
2dx
ex
-
1
From here we can simplify our calculation by assuming the above-gap photons are in
the dilute Boltzmann limit, so that:
3
JN JN=(kBT/h)
4 2 2 (-I -e-x
472
C2
JN
3
(kBT/h)
4J c2
(X 2 + 2x + 2)) o
or
Egap/(kBT)
+ 2x ±
where
g2)
134
= E2
Egap
kBT
(4.14)
(4.15)
The number of above-gap photons incident on an illuminated detector of area A in
time At is therefore:
A = AAt(kBT/h) 3
(X +
2
(4.16)
xg + 2) - ex9
This dimensionless number A can also be thought of as the number of photons occupying a finite volume of phase space. We note that these modes are in thermal
equilibrium by construction.
From this result, we can construct a cost function which represents the amount of
work required to pump heat from a reservoir at ambient temperature TA to a finitesize system (i.e. the phase space volume flowing through the detector surface in time
At). As the energy and entropy from the reservoir are pushed into the system, its
temperature will rise. Since we are interested in the low-power limit, we consider the
case in which the temperature of the system begins at TA and ends at TB > TA.
The quantity of work required to pump each unit of heat into the system depends
on the temperature T of the system. For a Carnot-efficient heat pump, we may define
a function of temperature i(T) = dU/dW = T/(T - TA), and use it to express the
total work required to raise the system temperature from TA to TB:
fU(TB)
W=]f dW =
dU
I
fTdU
=
U(TA)
135
T
T(T)
JTAdT
T--T
T
dT
(4.17)
The expression in Equation 4.16 can be differentiated to find the heat capacity dU/dT
of the small system. Keeping terms to leading order in
dU
d
-
xg
> 1:
(AEgap)
=Egap
(4.18)
AAt (kBT/h)
3
9
4r2c2
-
1[(X2 +
Txg
AAt (kBT/h)
SEgap
47r2c2
3
[xg (2xge-xg
2xg + 2)
-
e~
j
]
+ -
AEgap
T
X~e-xg) + 3 X2 eX9]
~ AEgap = kB A X2
(4.19)
(4.20)
(4.21)
Thus since the heat capacity is finite, if we switch variables to T' = T - TA and only
keep terms to leading order in T', we may integrate to find:
W = JdW
(L
T,
[.
-
-
dU 1
[kB [xg(TA+
TB -TA
-
T'
TA±TdT'
JTB-T
dT
k*[g(TkB
T')] 2 A(TA ±T). TA T'] dT'
T' d T
T B
0(T
g A)
2'TA+
-i
SdTB
Ig
A
dU"~
2
TA
(4.23)
(4.24)
T
TBA
=kB
(4.22)
-A)2
2
(4.25)
{(TB - TA
-T=TA-
(4.26)
TA.
In terms of the change in energy of the small system f dU = AU = AAEgap, we have:
W=AU
TB -TA
TA
1
2
(4.27)
or one half the typical Carnot expression for pumping heat between two reservoirs
at TA and TB. Intuitively this is because some portion of the heat that is pumped
into the small system is pumped across a small temperature difference compared with
TB - TA and some portion of the heat is pumped across almost the entire difference
136
TB - TA.
Since we are considering small distortions to the equilibrium field (i.e.
we have linearized the energy in the small system as a function of temperature) the
average portion of heat sees half the temperature difference, or
(TB
-
TA).
Using this result, we now look to construct an expression for W written solely
in terms of A. Since our measurements of the arrival of blackbody photons are over
time intervals ft much longer than h/6Ephton, where 6Ephotn is the range of photon
energies measured by the detector, the volume of phase space which we are probing
includes many longitudinal modes. The arrival statistics of the photons are therefore
given by the sum of many independent random variables representing the number of
photons measured in each mode. As a result, the arrival process of blackbody photons
counted by the detector is to good approximation a Poisson process; A is the expected
number of arrivals over time At. Because the probability mass function (PMF) of the
random variable N, which expresses the number of arrivals during a time interval At,
is more easily parameterized in terms of A than the photon temperature, we return
to Equation 4.16 to construct an expression for the work W in terms of A.
Differentiating A(T) and again keeping terms to leading order in xg:
dA
1 dU
dT
EgapdT
_
AEgap
kBT
(4.28)
2
or equivalently
dA
A
dT Egap
T kBT
(4.29)
From this we can express the work W(A, AA) required to pump a volume of phase
space from a state with an expected number of photons A to a state with an expected
number of photons A + AA.
AA~-
dA
-AT
dT T=T,
AE
kBTA
137
TB
-TA
(A4ga.3
(4-0)
Thus we have
AA
kBTA
W(A, AA) = (AXEgap)
AgaP
1
1
-
-
2
TA
2
AA 2
kBTA -
(4.31)
A
Now that we have a cost function, we must calculate the minimum cost W per bit.
This calculation was first performed numerically by the author (P Santhanam) and
analytically by Dr. Ligong Wang. Here we present both results.
Count m Photons
(1 - a) e-
a e(A+AA) (A+AA)rm
Count 2 Photons
(1 - a) e-A
a e-(A+AA)
Count 1 Photon
(1 - a) e-A A
2
a e-(\+A) (A + AA)
a e-(A+AA)
(1 - a) e-
Count 0 Photons
(A+AA)
Send Y=O (LED off)
Send Y=1 (LED on)
Table 4.5: Joint probability mass function (PMF) for communication with Poisson
symbols.
Consider the following joint PMF fym(y, m), constructed using the probability
distributions in Figure 4-3 as conditionals, which represents communication over a
noisy channel consisting of a source which modulates the Poisson arrival rate of photons at a photon-counting receiver.
fYM(y,
m) =
oy, (1 - a)
[+
IM!)
eA] +
-y,.a [ ((A +AA)
M!
e(A+AA)
(4.32)
Here Y indicates whether a symbol with nonzero cost (i.e. the LED 'on' state) or zero
cost ('off' state) is sent by the source in a given time slice; M is the number of photons
counted by the receiver as a result. From the PMF in Table 4.5, it is clear that for a
small but finite AA, the bit of data encoded in Y cannot be reliably transmitted in
just one time slice. We may convey this fact by quantifying the maximum amount
138
W
0. I
E[N] = X = 10
4.
0.11-
0
0
5
20
15
10
Number of Photons N
25
30
0.21
E[N] =X+AX= 12
CO
0
CL
0.11-
0
5
20
15
10
Number of Photons N
25
30
Figure 4-3: Poisson distributions with mean A (top) and A + AA (bottom). These
distributions form two conditionals, fm(mjY = 0) and fM(mIY = 1) respectively, of
the two-dimensional probability mass function fyi(y, m) representing the channel.
139
of information (represented as a real, non-integer number of bits) which may be
communicated reliably per time slice over repeated experiments [50].
This fractional number of bits, the mutual information I shared across the channel,
may be calculated from the joint PMF in Equation 4.32. Using Equation 4.2, we can
write an expression for I in this problem:
I[fyM] = DKL(fNlfy - fM)
(4.33)
where fy and fM are the Y- and M-marginal PMFs of fyu respectively. Again we
will take the logarithm in Equation 4.1 to be in base 2, so that I has units of bits.
From here we could in principle compute the marginal distributions from Equation 4.32 and find the maximum value of the ratio of I[fym] to the expected cost in
work &W(A, AA). This is the procedure followed in the numerical calculation. However, to expedite the analytical solution, we employ the general result from Verddi
[87] which states that when your codebook contains one zero-cost symbol (Y = 0)
and one nonzero-cost symbol (Y = 1), the channel capacity per unit cost (i.e. the
inverse of the minimum cost per bit) is:
D(fm(mJY = 1)HfM(mY = 0))
c(Y = 1)
(4.34)
where c(Y = y) is the cost of the nonzero-cost symbol. In our case, the denominator
is our expression for the work W consumed during a time-slice in which the LED is
on'.
Denominator = c(Y = 1) =
140
1
AX 2
kBTA 2
A
-
(4.35)
3
2.5
C
u.2
0
o1
3:O.5
10
10-2
100
101
AX (number of photons)
101
Figure 4-4: Results of numerical calculation of work per bit required for a Carnotefficient heat pump. The solid red curve represents the work per bit as a function
of the average number of excess photons AA. (Note that AA is proportional to the
power of the optical signal.) The horizontal dashed line represents a consumption of
kBT In 2 of work, while the vertical dashed line represents a signal power equal to the
noise power (i.e. the signal-to-noise ratio is 1:1). Flash signaling (i.e. infrequent use
of symbols with non-zero cost) was modeled using a = 10-4. Ambient temperature
was taken to be 300K, the detector bandgap energy was taken to match Silicon at 1.12
eV, and the detector area was taken as 1mm 2 . The time-slice duration of each symbol
At was taken to be 2.16 s so that the expected number of photons counted during an
off-state time-slice was very nearly 10, and the signal-to-noise ratio is ~ AA/10.
141
The numerator is:
Numerator
ff0-
((
Y
±))e(
)lg[(A+AA)M)
(A+ AA)M) e-(A+AA)
M!
.log
(m
M!
M=0
2~ ((A+ AA)M
e-(A+AA)
M=O
\0(
+
Am!
I~)
M
log
e-(A+AA) - M
log
(4.36)
-
+ A
AA
I + AA)
AnA2
1
(4.37)
(4.38)
.M=O
(A+AA)log1+
A
AA
=(A+A
1AA2
2-
2 A
A
1
A
2
2
(4.39)
In2
)
A
1 -AA
In2 1n2
3
-O(AA
+
In 2
(4.40)
(4.41)
Combining these results, we arrive at an analytical solution to the question posed in
Equation 1.13. In the low-power limit where AA -+ 0, we find
min
(W'
bi
=
kBT In 2
(4.42)
This result matches the numerical results plotted in Figure 4-4.
4.3
A Thermo-Photonic Link
The calculations we presented in
§ 4.1 suggested that in the very low power limit,
in which the signal to noise ratio of the power measurements was less than 1, the
measurement represented the conveyance of some nonzero information (i.e. less than
1 bit) about whether or not the source was turned on. We then noted that if we
calculate the energy consumed to drive the source LED during the measurement, we
could find that the energy efficiency with which this "communication" was taking
place reached an asymptote at 103 femtojoules per bit. That is, in the same regime
142
where we found the Landauer limit in
§ 1.4, the minimum energy per bit found in that
calculation could also be found at low signal-to-noise ratio but had a higher value of
around 100 fJ/bit.
While 100 fJ/bit may be a reasonable energy efficiency for a low-power channel, the
corresponding effective data rate (i.e. the maximum rate at which it is theoretically
possible to use coding to communicate error-free [50]) of less than one bit per second
was highly impractical.
However, because the aforementioned measurements only
used a very narrow band of frequencies of this linear channel, multiplexing could in
principle compensate for this. In fact, because the width of the frequency band which
the lock-in measurement uses is Af ~ 1/r, where r is again the integration time, a
densely frequency-division multiplexed channel can achieve the same symbol rate as
an OOK channel. Put simply, the rate at which O's and l's can be transmitted is
only constrained by the physical bandwidth which the hardware can achieve.
Here we describe a communication channel [88] constructed from an LED-photodiode
configuration closely resembling the setup from the power measurements. The hardware control and data acquisition elements of the experimental channel were built
primarily by Duanni Huang, while the experimental design and execution were done
in collaboration with the author. The theoretical calculations used to extrapolate
from the final results were developed primarily by the author.
The first step to constructing a working channel was to replace the function generator current source and the lock-in amplifier with digital-to-analog (DAC) and
analog-to-digital (ADC) converters respectively. The ADC was chosen to have a high
sample-rate and bit depth to ensure the remaining hardware was functioning as anticipated. A series of low-biased LED power measurements using a single modulation
frequency were performed with both the old and new hardware configurations; as seen
in Figure 4-5, measurements were in close agreement, indicating that the new hardware had not introduced any large systematic errors and that the dominant source of
143
Lens
Copper
Housing
LED (167 0 C)
Photodioc e (-20*C)
(a)
[
(k,T 1n2) Joules per photon
S
----
0
--
!F 10
DAC-board
0 Lock-in amplifier
----
-- - - - - - -
Ih.Unity wall-plug efficiency
0
-J
z 2.47sm
T = 1670C
%%q
X
(b)
-12
107
,0
1
Output Light Power (W)
Figure 4-5: (top) A depiction of the hardware setup for the experimental link which
includes the components relevant for the transmission of the signal in the optical
domain. (bottom) A plot of LED wall-plug efficiency versus optical power. Measurements taken with the lock-in amplifier and the analog-to-digital converter both match
the theory from Chapter 2 and each other quite well.
144
noise across the channel remained outside of the DAC and ADC elements.
During the operation of the communication channel, 2.5pim photons were emitted
by the LED at 167'C and detected by a photo-diode at -20'C with red cutoff wavelength near
2 .6p-m.
The observed quantum efficiency was 2xIO-I and the power at
unity wall-plug efficiency was 533 pW.
(a)
'Pre-processing
I
-
Encoder
_
H
8-PSK
_D/A
Phase
Tracking
* Bit Decoder
Optical
Channel
16-bit
OFDM
Gain
16-bit
ADStages
FFT
Post-processing
L -I---------------
I-------------------------
10
10
(b
)
.
* ce**
n(c)0i&_
2
syymbolsj
2.5,
001
2:U
044.110
15
000
E1
10*
1
0.5
-2
-2
-1
0
Real
1
2
3
995
x 10"
1000
1005
Frequency (Hz)
1010
1015
Figure 4-6: Subfigure a (top) is a block diagram of the experimental channel. The
diagram represents the flow of information from the input bit stream to the output bit
stream. Subfigure b (bottom left) is a plot of complex amplitudes Bi which emerged
from the Fast Fourier Transform (FFT) block on the detector side of the channel. The
observed points are clumped into 8 regions which correlate very strongly with the 3bit sequence used to encode the relevant Fourier component. Subfigure c (bottom
right) shows the magnitude of the FFT of a different sample signal for which only 4
of the 22 frequencies were intentionally excited. Because the time block over which
the signal was sent was 1 second, the frequency spacing in this plot is 1 Hz.
To create a low-power communication channel using the LED-photodiode pair at
low power, as seen in the block diagram at the top of Figure 4-6, we used a technique
145
known as Orthogonal Frequency Division Multiplexing or OFDM. In an OFDM channel, all of the information in a fixed-size block of data is sent simultaneously over a
fixed length of time T. During that time, one bit (or more generally a packet of bits)
is encoded into the amplitude of sine wave at with frequency f
=
1/T which persists
over that block of time. The second bit is encoded into a sine wave with frequency
2/T and added to the contribution from the first sine wave. Since these two sine waves
are orthogonal over any interval of duration T in the sense that the normal notion of
the inner product of two functions is zero (i.e. when (f (t), g(t))
=
J[ f(t')g(t')dt' =
0
we say f(t) and g(t) are orthogonal over [0, T]). In this way, the remaining bits in
the block of data are encoded into the higher harmonics, chosen to be orthogonal to
all the other sine waves in this interval. The resulting waveform, which contains the
information in the block of data, is used by to drive a current ILED(t) through the
source:
M
ILED (t)
B3l - Iocos (27fit + arg(Bi))
=
,
(4.43)
i=
where Bi is a complex number which encodes one or more bits into the ith harmonic
frequency,
fi
= i/T, where T is the analog block length, I0 is a coefficient with dimen-
sions of current which is varied in these experiments, M is the number of frequencies
multiplexed together, and we have used cos(.) instead of sin(-) here without loss of
generality owing to the variable phase of Bi. A plot of one such signal's Fourier transform appears in the bottom right of Figure 4-6. In the example waveform, the time
block T is 1 second, so that the orthogonal waves are spaced 1 Hz apart. The Bi for
most of the frequencies in the plot is zero while the magnitude of the components at
4 of the frequencies take on the same nonzero value. Note that because the y-axis
is power, the plot does not show the phase information which is obtained in the decoding process. This phase degree of freedom is necessary for the codebook used in a
number of the experiments, which we explain next.
In order to further decrease the per-bit energy consumption of the link, in several
146
experiments we employed a form of phase-shift keying to encode three bits into each
of the {Bi} above. To do so, we designed our codebook to have 8 distinct symbols:
seven symbols were at the same amplitude but had equally spaced phases on the
interval [0, 2-r) while the one remaining symbol was taken as zero. The plot at the
bottom left of Figure 4-6 shows a series of measurements resulting from the use of all
8 symbols in this codebook with equal frequency- a choice we make for the practical
reason of wanting a simple mapping from bit sequence to symbol sequence.
It is worth noting that this codebook is counter-intuitive in light of the result
from [87], which suggests that the optimal codebook in the presence of a zero-cost
symbol involves only the zero-cost symbol and a single nonzero-cost symbol. A better
understanding of which assumption is invalid in our particular case remains a subject
of interest going forward. A working hypothesis is that both codebooks of the type
described here and those with a single nonzero-cost symbol converge to the same
minimum value for energy per bit in the low power limit (i.e. the solution is not
unique), but the optimal solution at finite bit rate in our channel reflects the specific
structure of our symbol space and cost function. Essentially, while any solution of
our form can be improved upon in the sense of having a lower cost, the practical need
for bit rate makes our solution preferable because it gives a constant factor increase
in rate with a cost increase that appears only at quadratic order in time-averaged
signal power.
The final technique we wish to discuss was developed to compensate for a frequencydependent phase lag which, because they only appear at higher frequencies, we suspect
originated from the detector-side amplifiers which were operating at high gain. To correct for this issue, we simply sent a pre-specified trial signal with nonzero-amplitude
symbols at every frequency (i.e. no frequencies were encoding all zeros), then calibrated out the effect by inverting this phase lag with a software phase advance just
after computing the FFT.
147
Once a bit sequence has been sent through the link, we characterize the fidelity of
the channel using the fraction of those bits which were incorrectly decoded- the Bit
Error Rate (BER). Once the signal is digitized on the decoder side, the signal is passed
through an FFT which generates a complex amplitude for each frequency representing
the real amplitude and phase of the relevant component of the signal's waveform. This
complex amplitude is then processed through a Maximum Likelihood (ML) decoder,
whose goal is to map that value to the most likely candidate symbol for its value
at the source. This task is greatly simplified because our protocol involves sending
each of the code-words with equal frequency. It is further simplified because the noise
source has a probability density which is a monotonically decreasing function of the
distance from the complex amplitude at the source (the noise distribution is Gaussian
empirically, which is to be expected because the noise is thermal and determined by
the temperature of the photo-diode lattice). Our ML decoder then has the following
simple geometric interpretation: we simply find the symbol whose complex value is
closest to the measured value in the complex plane.
Practical time constraints limited the number of bits we could test with any given
protocol, so in order to extrapolate to low BER values, we found it useful to also
model the errors. Furthermore this served as a model with which to compare experiments. In our model, we assume the conditional probability distributions given a
symbol with complex amplitude Bk to be a two-dimensional Gaussian in the complex
plane, centered around Bk with some standard deviation ao. Since the noise is taken
to be additive, the conditional distributions for each symbol were taken as Gaussians with the same standard deviation uo but centered on the point in the complex
plane corresponding to the source-side amplitude for that symbol. Combining these
conditional distributions with the uniform distribution of symbols emerging from the
source, we arrived at the joint PDF for our channel. We then used our simple geometric interpretation of the decoding process to segment the space of measured complex
148
amplitudes into nearest-neighbor regions surrounding each symbol and identifying the
probability for error given a particular source-side amplitude of Bk as the integral over
the regions which were outside the decoding region centered on the point Bk. The
overall expected error rate was simply the inner product of the vector of conditional
probabilities of error with the uniform symbol frequency distribution. Since this operation only involved a single two-dimensional integral for each of the 8 symbols, it
was not computationally infeasible to model low values of BER; if we had elected to
randomly generate noisy measurements via a Monte Carlo method and decode them,
this would not be the case.
The final calculation relevant to the final results was the amount of energy consumed by the channel. We operated the detector in photo-voltaic mode so that no
power was consumed in reverse-biasing the device.
The power at the source was
calculated from a simple expression derived from Equation 4.43 as follows:
E
=
1 ET2- \ILED (t)
2R
1I2R
dt'
2
-T.Z
M
1
Bi| 2
,
444)
where R is the zero-bias resistance of the source LED. Note that the cross-terms of
the integral in Equation 4.44 disappear only when both of the following are true:
" the LED operates in the low-bias regime so that the current and voltage are
linearly scaled versions of the same waveform, and
" the waveforms at each frequency have no DC component, meaning that our
source is operating half in forward bias and half in reverse bias; the signal
remains visible by the linearity of the response in the low bias regime qV < kBT.
The results of our experiments appear in Figure 4-7. Our measurements indicate
that the channel was able to communicate using just 40 fJ/bit with a bit error rate of
3 x 10-3. As expected, decreasing the amplitude of the source waveform decreases the
energy consumption of the channel per bit transmitted Ebit, but also simultaneously
149
100
-.
o 2.9kbps 8-PSK
o
10-1
--- Theory 2.9kbps
-Theory 29kbps
04
CU
29kbps 8-PSK
88kbps 4-PSK
Theory 88kbps
0
I-
10-2
+0
iB
0 .,
10-3
(a)
10-4,
1 0-16
10-15
10-12
10 13
10-14
10-11
Energy per bit (J)
CL)
0
101
-
-
- -
100
-
Current Device Exp.
--Current Device Th.
1 and
= 1EQE
matched detector area
a
102
L_
C
LU
4-J
LU
I103
C:
0
0
4-J
(b)
(b)
1041
10-22
10-20
10-18
10-16
10-14
10-12
10-10
Energy per bit (J)
Figure 4-7: Subfigure (a) [top] plots the experimental results as paired values of Bit
Error Rate (BER) and energy per bit alongside theory curves using the signal-tonoise ratio as a fitting parameter. The experiment labeled '4PSK' utilized a very
similar protocol to the '8PSK' experiments described in detail in the main text.
Subfigure (b) [bottom] shows the extrapolations of our model calculations based on
idealizations of the LED and photo-diode. The portion of the curves to the left of
the line denoting of kBT in 2 per bit do not necessarily represent values beyond the
Landauer Limit because at such high BER the amount of mutual information carried
across the channel by each '0' or '1' is significantly than 1 bit. The proximity of these
model curves to the kBT In 2 line, however, does suggest that further work on channels
using improved LED-photodiode pairs could serve as a platform for investigating the
thermodynamic limits of classical photonic communication.
150
increases the probability that each bit will be decoded erroneously. Furthermore, the
relationship between Ebit and the BER is in good agreement with the model for the
most part. At high bit rates, where higher frequencies must be used, the presence
of greater noise at these high frequencies leads to increases in BER not captured by
the model here. We note that these deviations from the model are compatible with
originating in the limitations of the amplifiers used in our experiment rather than
inherent limitations of the LED-photodiode segment of the link.
The lower plot from Figure 4-7 shows extrapolations from these results under
idealizations similar to those we considered in
§ 4.1.3. The rightmost curve is the
result of the model calculation using signal-to-noise ratio as a fitting parameter. The
next curve to the left shows the results of a nearly identical channel, but using an
LED with 100% quantum efficiency. The leftmost curve represents the modeled results
given both 100% quantum efficiency from the source LED and a decreased noise level
from using a smaller-area photo-detector (matched to the emitter area) with the same
detectivity.
In spite of the very low quantum efficiency (2 x 10-)
of the LED used as a source,
the channel described here saw performance around two orders of magnitude away
from that of state-of-the-art low-power laser communication channels using nanophotonic techniques to minimize power consumption [89, 90].
While most of these
channels can transmit information with a higher maximum bit rate, our implementation is different because it does not require any fixed power consumption like a laser
which must reach threshold. As a result, we can transmit at low bit rates with low
energy consumption while other systems can only achieve low energy per bit when
both power and bit rate are far above the values reported here.
Although we have constructed a channel capable of genuinely transmitting data
with very little power at the source, some caveats apply. First, the source and detector
of the channel were not at the same temperature. However, the results of Chapter 2
151
and
§ 3.4 strongly suggest that this condition is not fundamental and that similar
results could be achieved using an isothermal configuration. Second, several aspects
of the channel's encoding and decoding were treated as exogenous for the energy
analysis; only the electrical power used to drive the source LED (recall that the
photo-diode did not consume power) was considered. In particular, the amount of
power required for trans-impedance amplification may be greater for a technique in
which the photo-current signal is very small. Nevertheless, for systems which need to
transmit data at kilobits per second with minimal power consumption on the source
side, this type of channel may be of practical interest.
Furthermore, because this
channel takes a different approach to efficient communication which relies on efficient
photon generation, further experimentation may reveal new insights into the ultimate
limits of energy-efficient optical communication.
4.4
Summary and Conclusions
In this chapter we studied, both experimentally and theoretically, the minimum energy requirements for a communication channel whose source is a thermo-photonic
heat pump.
We began in
§ 4.1.1 by analyzing experimental results from Chapter 3 to determine
how much information about the state of the LED under test was being captured by
the detector circuit. We found that the amount of work consumed by the LED per
bit of information captured by the detector reached its maximum value in the limit of
low power. In this limit, we found that our existing setup could transmit information
for approximately 100 fJ per bit, and that such experiments on more ideal LEDs
could reduce this figure to less than 100 aJ per bit. In the case of an idealized lowtemperature detector, we found that the presence of thermal blackbody radiation
constitutes a source of noise with which any information-carrying optical signal must
152
compete. A theoretical analysis of this situation was presented in
§ 4.2 and showed
that the presence of thermal blackbody radiation emerging from the source (which
is required for LED heat-pumping) imposes a lower bound on the work per bit of
kBT ln(2).
We ended the chapter with
§ 4.3, in which we presented an experimental thermo-
photonic link capable of communicating at kilobit data rates while consuming just
40 fJ per bit in the source and detector diodes together. Extrapolations of this result
based on the physics in Chapter 2 and Chapter 3 suggest that it may be feasible to
develop a channel in which the dominant noise is from blackbody radiation. As LEDs
with high quantum efficiency at low-bias are developed, their use in such a channel
should enable communication with power consumption approaching the lower bound
imposed by the Second Law of Thermodynamics.
153
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154
Chapter 5
High-Temperature mid-IR
Absorption Spectroscopy
In this chapter we explore the potential for using LEDs in the low-bias regime for hightemperature infrared absorption spectroscopy. In
§ 5.1 we outline the motivation for
this work, focusing on the particular problem of developing a platform for the analysis
of the complex fluids found downhole in oil wells. In
§ 5.2 we connect the problem of
extracting spectroscopic information from a sample under analysis to the problem of
communication and make use of relevant results from Chapter 4. In
§ 5.3 we present
experimental data on high-temperature LEDs that constitutes a proof-of-principle for
using them for absorption spectroscopy, and subsequently evaluate the suitability of
these sources for a downhole spectroscopy system with specific targets. In
§ 5.4 we
explain the limitations of infrared photo-diodes caused by decreasing shunt resistance
with temperature. Finally, in
§ 5.5 we present results from an experiment in which
both the source and detector operate at high temperature and demonstrate that
thermo-electric pumping can be used to compensate for increased detector noise and
maintain signal-to-noise ratio at elevated temperatures.
155
5.1
Motivation
Thus far our study of thermo-electrically pumped LEDs has been motivated primarily
by scientific questions regarding the nature of the phenomenon and the constraints it
places on quantities of practical interest, such as power and efficiency. In Chapter 2
we saw that more power is available at a given efficiency when the ratio hw/kBT
is small. Furthermore, in Chapter 3 we found that hw/kBT ~ 10 characterized the
most experimentally-accessible regime for observing high efficiency LED operation. In
Chapter 4 we quantified the limits on information transmission that appear at the low
power levels where high efficiency LED operation is observed. Taken together, these
basic observations about thermo-electrically pumped LEDs point to applications of
LEDs at infrared wavelengths and high temperatures in which relatively low power
is required. One such application is downhole infrared absorption spectroscopy. Here
we pursue it primarily as an engineering problem.
Although the evaluation and extraction of crude oil from underground formations
benefits heavily from the in-situ analysis of the extracted material, the environment
downhole in an oil well is harsh and presents many simultaneous challenges
[581.
Downhole analysis systems must operate at temperatures ranging from colder surface
temperatures around 00 C up to 200'C and pressures up to 20,000 psi [91]. Moreover,
physical the size of the borehole limits not only the size of the platform, but also
the electrical power and communication bandwidth available to these systems. Nevertheless, downhole fluid analysis systems have been recently developed by multiple
oilfield services companies and can provide valuable information from oil fields at
various stages of the extraction life cycle [58, 92, 93, 941.
Current downhole platforms like the one currently employed by Weatherford International, Ltd. perform spectroscopy at visible and near-IR wavelengths, but do
not gather information from mid-IR wavelengths beyond about 2 Pm [59]. However
several valuable target analytes can be detected by their mid-infrared absorption,
156
including H2 S, C0
2
, and hydrocarbons of various lengths. Much of the reason for
the lack of mid-IR capabilities owes to the poor performance of both the sources (see
Figure 1-1) and detectors at these wavelengths and temperatures [59].
Our goal in this chapter is to assess the feasibility of using low-bias LEDs to add
infrared spectroscopy capabilities to an existing visible/near-IR downhole platform.
To do so we will first investigate the high-temperature operation of sources and detectors, then later analyze them together to evaluate the feasibility of meeting the
constraints on temperature and power consumption faced by the existing downhole
spectroscopy platform.
5.2
Mapping Spectroscopy onto Communication
The purpose of a spectroscopy system is to extract information about certain properties of a sample or analyte. In the case of mid-infrared absorption spectroscopy,
the property of relevance is the absorption length of the sample under analysis for
photons in a certain band of mid-infrared wavelengths. Often the goal is to extract
this information as quickly and energy-efficiently as possible, much as the goal of a
communication system is for the receiver to extract the information contained in a
digital bit-stream from the transmitter.
Consider the logic of Figure 5-1.
The top diagram is a simplified depiction of
an absorption spectroscopy system. We are free to think of the combination of the
photon source and sample as together as encoding the information of interest (i.e.
the sample's transmission coefficient) into the portion of the photon field carrying
light rightward out of the sample. Since the transmission coefficient is a real number
between 0 and 1, we are free to represent it in binary with finite precision.
For
example, three-bit sequence '011' could refer to the interval of possible transmission
coefficients from 0.375 to 0.500, while the three-bit sequence of '100' could refer to
157
Photon
Source
T
Transmitter
Photon
Detector
Sample
=
37.5%
Digital
Receiver
Digital Source which
Encodes {0,1) * Power
Receiver
r-
P=
0 11
xP---
-a
I7xj
P
P401
P = 0.375xP.
Figure 5-1: Depiction of the mapping between absorption spectroscopy and digital
communication. All three diagrams are meant to depict the same physical situation,
but described with different language. The top diagram is labeled as an absorption spectroscopy system. The diagram at the bottom left is labeled as an analog
communication channel. The diagram at the bottom right is labeled as a digital communication channel. The information extracted by the user at the detector/receiver
side (i.e. the digital bitstream or transmission coefficient) is the same in each case as
well.
158
the adjacent interval of 0.500 to 0.625.
We note that in practical systems which
employ digital lock-in amplifiers to measure transmission of a sample, the bitstream
described here is closely analogous to the actual bitstream which results from the
discrete Fourier transform of the digitized photo-current signal.
From this type of mapping, we can make a few general observations. First, the
precision of our representation is determined by the length of the finite bit-stream.
If the transmission coefficient is known more precisely, more bits will be required to
represent it with that precision.
Secondly, while in an information-theoretic sense,
each bit should carry the same amount of information, the leftmost bits are the most
significant bits and the significance of bits decreases from left to right. Thus if the
sequence of bits we use is longer than can be accurately decoded, then the rightmost
bits will not contain any real information about our quantity of interest.
The information-theoretic model of absorption spectroscopy outlined here may
help us debug real systems with multiple potential sources of noise. For example, if
an ensemble of measurements taken under identical conditions produces different bit
sequences for which the most significant bits are equally likely to contain errors as the
least significant bits, this may indicate the presence of electromagnetic interference
introducing noise in the electrical signal after digitization. If the ensemble produces
bit sequences which correspond to transmission coefficients that drift periodically in
time with the frequency of another environmental variable (e.g. the mechanical pump
frequency, the 60Hz wall power frequency), that may indicate an addressable design
flaw. If however the system is operating properly, we should see that the measured
bit sequences correspond to a transmission coefficient between far from 0 or 1, with
a standard deviation greater than the quantization limit.
Furthermore, this viewpoint combines with the intuition from Chapter 4 to offer
direction on future system designs. First of all, the model shows that the technique
employed in
§ 4.3 does not easily map to a technique for efficient low-power absorption
159
spectroscopy. If we use orthogonal frequency-division multiplexing to probe the sample over multiple lock-in channels simultaneously, we find that each frequency channel
detects a few bits of information, but that the channels contain duplicate information. As a result, many such channels cannot be used to reconstruct the information
gathered by a single frequency channel with their combined power. If however, we
developed a technique by which multiple channels could acquire information about
bits with different significance, then their information could be used in this way. For
example, a setup using a bank of interferometers with different arm lengths could
be designed to probe changes in the real index of refraction of a sample at different
scales. An interferometer with a short arm would be sensitive to large changes in
index, while an interferometer with a long arm would be sensitive to small changes of
index but only able to measure the index modulo the amount required to change the
output intensity by one complete fringe. In this way, the information from different
levels of precision can be acquired simultaneously with parallel channels. Information
about the real index could then be used to find the imaginary index of refraction (i.e.
the absorption) via the Kramers-Kronig relations.
5.3
High-Temperature Sources for Spectroscopy
The same model of 2.1pm LED from § 3.2 was heated to 150'C, where above-unity
efficiency operation was easily detectable.
Here, partially-absorbing Parafilm wax
paper samples were stacked between the emitter and detector, creating an optical
path with variable transmission. The results [86] are shown in Figure 5-2.
The measured transmission coefficients were in agreement with the Beer-Lambert
Law, indicated by the dashed fit line. These measurements offer confirmation that
sources operating in the / > 1 regime can produce sufficient signals to perform absorption spectroscopy at high temperature.
160
"*100
cn65g
CO) 0
~30
-
0
300
200
100
Sample Thickness (jm)
400
0
Figure 5-2: Results of an absorption spectroscopy measurement performed on 150 C
LED operating above unity efficiency. Note that transmission scales exponentially
with the thickness of the partially-absorbing material placed in the optical path as
expected. This represents confirmation that useful sample information may be acquired in the above-unity efficiency emitter regime.
161
As discussed in
§ 5.1, three targets of interest for oil analysis in a high-temperature
downhole environment are the determination of absolute concentrations of CO 2 and
H2 S, and the relative concentrations of hydrocarbons at various lengths. All three of
these targets are potentially accessible via mid-infrared spectroscopy, if good enough
sources and detectors can be developed. CO 2 has a strong absorption feature near
A = 4.2pm, while H2 S has a strong absorption feature near A = 3.7pm. LED sources
at both of these wavelengths have been developed in the InAsSbP:InAs material
system. As seen in Figure 5-3, sources at 3.7pum have been tested at high temperature
and exhibit the same temperature-dependence as other thermo-electrically pumped
LEDs: their low-power wall-plug efficiency increases with temperature. The efficiency
at fixed output power is more than 10x greater at 100'C than at 25'C. As evident
from Figure 5-3, this increase is more than sufficient to compensate for the spectral
redshift up to this temperature and beyond.
Due to the common presence of Carbon-Carbon single and double bonds and
Carbon-Hydrogen bonds, many hydrocarbons absorb infrared light around A
=
3.4pm.
However, the shape of the absorption line in this range differs between chains of different lengths because of the different distributions of these types of bonds [95]. As
a result, sufficiently sensitive spectroscopic analysis of this absorption line can be
used to "fingerprint" a mixture of hydrocarbons to find their relative concentrations.
The capability to perform such analysis in the harsh downhole environment could
prove industrially applicable by improving estimates of oil-to-gas ratio for valuation
of existing wells and feedback into geophysical models used to plan extraction operations. The LEDs examined in
§ 3.4 emit at the relevant wavelengths and could be
suitable for a high-temperature downhole system since they exhibit this same type of
improvement with temperature as well.
The results of initial experiments on existing crude oil samples are shown in Figure 5-4. At left, Fourier-Transform Infra-Red (FTIR) spectroscopy reveals a clear,
162
-6
10-
100 0c 0
100
-~
-9
10
25 0 C
0
10
10~11
"
0
0-2
100-6 10-5
10 10-4
10 100-3 10
Voltage (V)
3.6811m
3.41pm
0-1
10
4.2
.91Pm
E
..
~.....
~~
4
Ceted
3.8
0.6
C3.6
0.
Bnm/K
0.4
3.
~3nm/K
C
3
3.2
3.4
.
3.8
Wavelength (pm)
4
1
4.2
40
60
80
Temperature (C)
100
Figure 5-3: Data for an InAsSbP:InAs light-emitting diode. Top: Output Power versus Voltage at two temperatures. Bottom left: Emission spectrum at 25*C. Bottom
right: Estimate of spectrum's redshift with temperature. The LED delivers significant optical power around the 3.7 pm H 2S line across the range of temperatures
investigated.
163
spectrally-isolated absorption line from a 1Optm-long optical path through a sample
of crude oil. This short attenuation length renders conventional optical geometries
impractical due to the presence of particulates in crude oil. Instead, an Attenuated
Total-internal-Reflection (ATR) geometry is being considered, and a prototype instrument is currently in development. At right, significant differences between two
samples from different oil wells can be resolved.
The clarity of these differences
suggests that fingerprinting with sufficient accuracy to resolve typical hydrocarbon
length distributions in crude oil should be possible with just a handful of filtered
optical channels. We note that these measurements were made with the oil samples
at standard temperature and pressure, but blurring of these lines at high temperature and pressure [95] could present a significant challenge for any type of downhole
hydrocarbon fingerprinting.
006
WWavelength
(am)
Figure 5-4: Left: Fourier-Transform Infra-Red (FTIR) spectroscopy reveals a clear,
spectrally-isolated absorption line from a 10 pim-long optical path through a sample
of crude oil. Right: FTIR spectroscopy of two crude samples from two different wells
reveals significantly distinguishable line shapes in this range.
5.4
High-Temperature Infrared Photo-Detection
As noted by Fujisawa, et.
al. in Ref.
[59], poor room-temperature performance
of mid-infrared photo-detectors compared to near-infrared and visible, and further
164
degradations in signal-to-noise ratio of these systems with increasing temperature
often prohibit downhole analysis systems from using mid-infrared spectroscopy. This
decreased performance is captured by the decrease in shunt resistance (i.e. zero-bias
resistance of the photo-detector) with temperature, as seen in the I-V characteristics
of the photo-diode shown at left in Figure 5-5.
This decrease is fundamentally due to the increase in concentration of thermallyexcited carriers in the diode. This increase in carrier population leads to increased
recombination at a given voltage, and ultimately more current flow. Note that this
physical explanation is in close analogy with the decrease in zero-bias resistance of
an LED; this forms the basis of the emitter-detector compensation concept described
in
5 5.5.
120
100
1.5
--
1
LED (Emitter)
..
.
0.5
250K
-0.5
-1
-1.5
-0.1
.O
- .
-.
..
~40-
350K
.(Detector)
-0.05Voltage (V) 0
20
0.05
Photo-Diode
a
310
500
320
330
340
Diode Temperature (K)
350
Figure 5-5: Left: Current-voltage characteristics for a HgCdTe p-i-n photo-diode
at various temperatures. The detector's cutoff wavelength ~ 6pm. Note that the
slope of the I-V curve around the origin (i.e. R) increases with temperature.
Right: Shunt resistance of a photo-diode and zero-bias resistance of an LED with
temperature. Both decrease exponentially with temperature. The LED's emission
wavelength is ~ 3.7pum.
We have constructed a model for the noise in our photo-detection circuit. It is
depicted as a circuit diagram in Figure 5-6. In this model, we include two sources of
current noise, one from thermal vibrations in the diode and one from shot noise, in
parallel with the resistance of the diode at the input to the trans-impedance amplifi-
165
/
Photo-diode
I
I
I
I
I
shot
0
:R B
I
I
I
I
+
I
I
Trans-Impedance
Amplifier (TIA)
Figure 5-6: Circuit diagram representing the basic noise model developed for our
photo-detector circuit. Here RZB stands for the zero-bias resistance of the photodiode (or equivalently the shunt resistance), Vth stands for the zero-mean voltage
noise generated by thermal motion of electrons within the photo-diode and the metal
for
contacts and wires which connect the diode to other circuit elements, I, stands
RGain
and
flux,
photon
incident
the
in
the zero-mean current noise due to shot noise
in
stands for the trans-impedance gain of the combined amplification stages measured
Ohms (Q). Note that we have omitted the analog-to-digital conversion in our model
because we assume any noise introduced at this stage is negligible compared to the
other noise sources included in the model.
166
cation (TIA) stage. Since we are interested in the outcome of a lock-in measurement,
we would like to express the root mean square -)rms of the Johnson-like thermal voltage noise [96] (Note: we use the term "Johnson-like" because Johnson noise typically
refers to noise from a resistor rather than a diode) in terms of the bandwidth Af as
follows [97].
KVthrms = 4 kBT Af
Rshunt
(5.1)
(Vth)rms =
4 kBT Af
shunt
For lock-in measurements with a pass-band width of 1 Hz (the time constant
T =
1 second) and responsivity RPD = 2.96 A/W, we then expect the standard
deviation of our lock-in measurements at low detector temperatures to correspond to
a noise equivalent power (NEP) for this noise source of:
NEPth =
V4kBT AfRshunt
x
PD)
--
(5.2)
59.49
x (2.96)-
= 5.64 pW
Likewise if we perform the same calculation using the absolute temperature, zerobias resistance, and responsivity of the photo-diode across the range of temperatures
tested, we can find the thermal contribution to NEP. In Figure 5-7, the results of
these calculations are presented beside experimental data.
Also included in Figure 5-7 is an estimate of the noise caused by thermal photons
from the 300K ambient environment being absorbed by the photo-diode. When the
temperature of the photo-diode is above 300K, the interactions between the incident
thermal photon field and the photo-diode's active region doesn't serve to create a
significant non-equilibrium population of free carriers, so we should not expect this
167
102
Standard Deviation
in Power Measurements
L
0 10
0.
Shot Noise from
Johnson-like
Ambient Photons
JoTh nsr lie
Thermal Noise
10
210
240
270
300
330
360
Temperature (K)
Figure 5-7: Noise-equivalent power (in pico-watts) of the lock-in measurements of
optical power emitted from a mid-infrared LED and detected by a photo-diode versus the temperature of the diode. Note that the thermal and electrical conditions
of the source were constant across these measurements. The hollow square markers represent experimental measurements; the hollow circles represent a model of
the noise-equivalent power from Johnson-like thermal noise which uses experimental
measurements of the photo-diode's shunt resistance; the dark dotted line represents
a model of shot noise in the incident photons in the limit of low detector temperature. The shot noise calculation assumes the photo-current from incident photons is
dominated by those produced by the 300 K ambient in the laboratory. Qualitative
agreement is reasonable, but observed levels exceed the modeled values by about a
factor of 3. Likely sources of model errors include constant factors from the digital
signal processing of the lock-in amplifier, uncertainty in the responsivity of the photodiode near its cutoff wavelength of 6 pm, and the omission of other possible noise
sources caused by the trans-impedance amplifier.
168
noise source to be relevant for detector temperatures above ambient temperature.
Below ambient temperature, however, incident thermal photons within the responsive
band of the detector can generate electron-hole pairs in excess of the equilibrium
population that are swept out and appear as photo-current.
Because the radiation in the responsive band is strongly multi-mode on the timescales
of our measurements, we begin our analysis with the assumption that the arrival of
thermal photons behaves as a memoryless Poisson arrival process. Based on the detector's data sheet [98], the detector used here had an effective area of 1mm2 and
was responsive out to 6pm. A quick calculation of the equilibrium blackbody flux
through this area of photons with wavelength 6pm and shorter yields about 16pW. If
we assume this flux is dominated by photons near the 6pm edge, and we use the detector's responsivity to these photons is perfect (RPD = 4.84A/W), this corresponds
to an arrival rate of about A =4.9x10" detected photons per second.
We now use the basic result that the time-derivative S of the random variable N
representing the accumulated arrivals from a Poisson process has an autocorrelation
whose Fourier transform is flat with a value of A as a function of angular frequency
w away from base-band.
We find that the "power spectral density" of the noise
integrated over a band Af is 2 x 2wRA x Af, where the extra factor of 2 accounts for
only consider positive frequencies
f.
Because A has no units, we should be careful
with the units in this expression. Note that the "power spectral density" in this case
refers to the absolute square of the Fourier transform S(f) of the time-dependent
random variable S(t), which itself has units of photons per second. To find the units
of S(f), we find that the units of the "signal energy" f IS(t')J2 dt' are photons 2 per
second. Parseval's relation then indicates that the "signal energy" calculated in the
Fourier domain,
Af
IS(f') 2df'
=
47rA - Af, has units of photons 2 per second as well.
As a result S(f) has units of photons. Including just the Fourier components within
the 1 Hz band of our lock-in photo-current measurement, we should therefore expect
169
the "signal energy" of a time interval of 1 second to be 47A or 6.2 x 1015. We therefore
conclude that the effect of ambient photons on the cooled detector should be to add
current noise Ishot with zero mean and
(fshot)rms
= 12.6 pA.
Since this root mean square current fluctuation is interpreted as noise in the arrival
of 3.7pm photons, to convert this figure back to noise equivalent power to compare
with experimental measurements, we must now use the same responsivity we assumed
when converting our photo-current measurements back to power originally (RPD
=
2.96 A/W). We therefore arrive at an estimated noise equivalent power for the shot
noise due to ambient photons of:
NEPshot = 4.26 pW
.
(5.3)
As seen in Figure 5-7, this figure is roughly 3 times smaller than the average
uncertainty in our low-temperature power measurements at -13'C and -53 0 C of about
12 pW. The preceding calculation was essentially an estimate, and because of its
sensitivity to certain parameters like the cut-off wavelength of the detector and the
ambient temperature as seen through the acceptance cone of the detector's immersion
optics, we should not expect very high accuracy. Based on the detector's data sheet
[98] we estimate the width of the Urbach tail to be around 30 meV; an uncertainty
in the cutoff wavelength of this magnitude would change our estimated NEP by
approximately a factor of 1.5. Although the ambient environment in the laboratory
was roughly 300K, it is also plausible that thermal radiation emitted by the heat
sink on the heat rejection side of the photo-diode's thermo-electric cooler could be
dominant; an increase in the temperature of incident radiation by 5 K would increase
our estimated NEP by 15%.
Furthermore, a more careful calculation may consider the following effects: constant factors in the noise bandwidth for a given time constant from the lock-in ampli-
170
fier's digital signal processing (we assumed Af = 1/T where
T
is the time constant),
the effects of wavelength-dependent responsivity with special attention to the bandedge of the detector, and the finite acceptance angle of the immersion lens abutting
the responsive area of the photo-diode. Considering the uncertainties in the input
quantities, the author believes the incidence of thermal photons within the responsive
band of the detector cannot be excluded as the dominant noise source in the measurements with detector temperature < 300K. Further experimentation and modeling
could yield a more complete analysis of the noise in our power measurements, and
may in fact be a necessary step for building a spectroscopy system at this wavelength
which is designed to operate in a high temperature environment. We will return to
this topic, as well as the potential relevance of ambient thermal photons in the context
of communication with a heat pump, in our discussion of future work in Chapter 6.
5.5
High-Temperature Emitter-Detector Compensation
Although the increase in noise associated with decreased photo-diode shunt resistance
is a robust consequence of operating in an elevated temperature environment, the
logic of Chapter 2 offers an equally robust mechanism to compensate for this. As
the resistance of an LED around the origin
the emitter's quantum efficiency
RZB,LED
decreases with temperature, if
remains fixed a given voltage results in more
fEQE
proportionally more light emission in the low-bias regime.
Since the signal at the
photo-detector is proportional to the light output from the LED, we see that the
signal Vsignal emerging after trans-impedance amplification also scales inversely with
RZB,LED:
Vsignal = RGain
'
RPD
'
7EQE -RB,LED
171
.
VLED
-
(5.4)
Although the photo-diode's responsivity
7EQE
RPD
and the LED's quantum efficiency
both change significantly with temperature, we will primarily focus on the
temperature-dependence of the RLED term here. In our present experiment as
the detector temperature is raised from 300 K to 350 K, the responsivity falls by a
factor of 3 and the quantum efficiency falls by just over 10%. Meanwhile the LED's
zero-bias resistance decreases by a factor of 9.2.
By comparison, if the noise in our measurements above ambient temperature is
dominated by Johnson-like thermal voltage noise, then the standard deviation in our
lock-in measurements of the voltage signal after trans-impedance amplification VNoise
can be expressed as follows:
VNoise
- RGain
'
V4 kBT Af
(5.5)
s/shunt
As the temperature is raised from 300 K to 350 K, the explicit temperature dependence in this expression increases by 17% while the square root of the inverse shunt
resistance increases by 42%.
Combining the expressions for Vsigna, and VNoise from Equation 5.4 and Equation 5.5 respectively, the signal-to-noise ratio (defined here as the ratio of the stan-
dard deviation to the mean of a lock-in optical power measurement with 1 Hz of noise
bandwidth) can be expressed in a way that reveals its temperature dependence.
SNR
__ Vsignai
VNoise
_
RPD 7 EQE VLED
/4
kBT Af
/Rshunt
(5.6)
RZB,LED
We have arranged the above equation so that the dominant temperature-dependent
terms (i.e. the resistances of the source and detector diodes near zero voltage) appear
at the end. Recall now that the resistance of a diode decreases exponentially with the
ratio of the thermal energy kBT to the bandgap energy Egap (i.e. oc e-Egp/kBT). If we
assume the bandgap of both the source and detector diodes are similar in magnitude,
172
we find that under conditions of fixed source side voltage amplitude the SNR of
the combined system actually increases with temperature. Note also that at shorter
wavelengths, the diode resistances become exponentially more sensitive to changes
in temperature, while the competing effects of reduced responsivity and quantum
efficiency may not. Thus our model suggests that this low-bias spectroscopy system's
signal-to-noise ratio should increase with temperature because of the combined effect
of both diode resistances, and that this should remain true for similar near-infrared
systems as well.
As the resistance becomes small enough, the voltage noise at the input of the transimpedance amplifier could replace the thermal Johnson-like noise. Let us now consider
the temperature-dependence of the SNR when this noise source is dominant.
The
relevant circuit diagram for this noise model is identical to the one in Figure 5-6 except
with the thermal voltage source replaced by one with a magnitude (VTIA)rrns which
is determined by the first stage of the trans-impedance amplifier. The corresponding
expression for the noise in the final measurement is:
VNoise
(5.7)
VTIA)rms
RGain
Rshunt
and the signal-to-noise ratio is:
SNR
-
Vsignai = RPD
-
EQE * VLED
X
shunt(5.8)
RZB,LED
VNoise
In this case, the scaling of SNR with the diode resistances depends equally on both
diodes. However, since Egap for the emitter must be at least as large as Egap for the
detector in order for the emitted photons from the LED to fall within the responsive
band of the photo-diode, the oc
e-Eap/kBT
scaling of the resistance is more sensitive
for the emitter. Thus even when the TIA's voltage noise is dominant, the SNR of the
combined system should increase with temperature.
173
Finally, we consider the case where current noise is dominant. Whether the dominant current noise is from the shot noise in the incident thermal photons or results
from the first stage of the TIA, the shunt resistance in our model does not affect the
noise in the final measurement:
VNoise = RGain - (In)rms
(5.9)
,
and the increased signal strength again dominates:
SNR
=
Vignal
VNoise
-
RPD 77EQE VLED
(In)rms
1(510)
RZB,LED
To summarize, we have presented a model for the signal-to-noise ratio of a midinfrared absorption spectroscopy system which uses a small AC voltage to drive an
LED and performs a lock-in measurement on the amplified photo-current signal from
an unbiased photo-diode.
By assuming that the temperature dependences of the
source LED's quantum efficiency 7EQE and the detector's responsivity
RPD
are neg-
ligible compared to the temperature dependences of the zero-bias resistance in either
diode, we found that the combined system's signal-to-noise ratio should actually improve with temperature. This result holds whether the dominant noise source in the
system is the Johnson-like thermal noise in the photo-diode, the shot noise of incident
thermal photons, or either current- or voltage-noise from the input stage of the TIA
which provides trans-impedance gain to the photo-current signal.
We have also performed an experiment to test this hypothesis. A basic absorption
spectroscopy system was built using an LED emitting at 3.7pm (see Figure 5-3 and
related discussion) and a photo-diode responsive to light at wavelengths from 2 to
6
pm (see
§ 5.4) and tests were performed at various temperatures for both devices.
To isolate our measurements from effects related to the red-shift of the source and the
wavelength-dependent transmission of a sample, no sample was introduced into the
174
TLED
0
-
10
2
Rshunt'
Tphoto-diode
PD'
& ZB,LED
C/)
Z
0
TLED
fixed at 300K
L10
1
10,
Co
Rshu
and RPD
10
220
240
260
280
300
320
Photo-Diode Temperature (K)
340
Figure 5-8: Signal-to-noise ratio of a basic low-bias lock-in mid-infrared spectroscopy
system versus photo-diode temperature. The hollow red squares indicate measurements in which the LED source was held at 300 K; the hollow blue circles represent
measurements in which the LED source was matched to the photo-diode temperature; the black dotted line represents a model in which the source is fixed but the
shunt resistance and responsivity of the detector take experimental values; the green
long-dashed line represents a similar model in which the zero-bias resistance of the
source LED also takes on values from experiments with the source LED at elevated
temperature. Note that no sample was placed between the source and detector for
these measurements.
175
optical path for these tests. The results appear in Figure 5-8 and confirm empirically
that the SNR of the combined system does in fact increase as the temperature of both
the source and detector diodes are simultaneously increased from 300 K to 350 K. In
a practical high-temperature spectroscopy system, increases in signal-to-noise ratio of
this type may allow compensation for other issues, such as red-shifting of the source
away from the target wavelength, which are outlined in
§ 5.3. We briefly discuss the
practical potential of this type of spectroscopy system in Chapter 6.
5.6
Summary and Conclusions
In this chapter we have conducted experiments on a basic absorption spectroscopy
system implemented using a 3.7pm LED driven by a small AC voltage, a photo-diode
sensitive from 2 to 6pm, a trans-impedance amplifier, and a digital lock-in amplifier. In keeping with the observations of Chapter 3, the decreased performance of the
source-side LED at conventional operating points (where qV is on the order of Egap)
is reversed in the low-bias regime. In the low-bias regime, the increased output power
of the LED at constant input voltage is shown to be sufficient to compensate for the
decreased performance of the detector photo-diode at elevated temperatures. Models
are developed for the temperature-dependence of the noise in the detector circuit and
reasonable agreement with experiments is observed. The noise models on the detector side are then combined with the LED models from Chapter 2 to create a larger
model for the spectroscopy system's overall signal-to-noise ratio. These models suggest that for a variety of potential noise sources, the improvements on the source-side
should outweigh the decreased performance on the detector-side, leading to a signalto-noise ratio which increases with temperature. From this work we conclude that
the exponentially decreasing performance with temperature of mid-infrared LEDs
and photo-diodes at conventional operating points may not, as previous authors have
176
suggested [59], prohibit the development of mid-infrared absorption spectroscopy systems capable of operating in high-temperature environments provided they employ a
zero-bias lock-in photo-detection technique along the lines described in
177
§ 3.1.1.
THIS PAGE INTENTIONALLY LEFT BLANK
178
Chapter 6
Conclusions and Future Work
In this chapter we present a high-level summary of the work detailed in this thesis,
then using these ideas we describe several potential research directions going forward.
In
§ 6.1 we combine the results of Chapters 2 and 3 to construct a physical picture
of optoelectronic device operation, and apply that thinking to understand the results
from Chapters 4 and 5 regarding communication and spectroscopy respectively. Our
subsequent discussion of related research directions, some of which the author expects
to actually pursue in the near term, will be organized as follows. In
§ 6.2 we will
outline some questions of scientific interest which have been raised in the course of
this work.
In doing this, we will first focus on problems associated with physical
entropy and information in photonic systems followed by problems related to the
physical entropy and information of electrons in semiconductors. In
§ 6.3 we describe
a number of future directions for applied work, some of which we have alluded to
in Chapters 2 through 5.
Finally, in
§ 6.4 we discuss the long-term prospects for
energy-efficient solid-state lighting and photonic communication by considering the
constraints imposed by the Second Law of Thermodynamics in light of this work.
179
6.1
Thesis Summary and Conclusions
In this thesis we have presented theoretical and experimental results in support of a
thermodynamic interpretation of incoherent light generation by light-emitting diodes.
From the thermodynamic analysis of charge and entropy transport in a forwardbiased diode from
§ 2.1, we found that the carrier injection process necessarily requires
the absorption of lattice heat through the Peltier effect whenever the bias voltage qV
is less than the bandgap energy Egap. In
§
2.2, we directly computed the entropy
removed from the electron-hole system by a single radiative recombination event. By
dividing the energy of the resulting photon by this quantity, we arrived at a simple
expression for the effective temperature T* seen by inter-band processes in terms of
the voltage V, the photon energy hw, and the lattice temperature
T*Tlattice
I -
Tiattice:
.
(6.1)
Using an analogy common in statistical mechanics, we saw that T* serves as a sort
of "exchange rate" between entropy and energy for distortions of the electron-hole
system which conserve charge. From the Second Law constraint disallowing the deletion of entropy, we saw that T* serves as an upper bound for the temperature of the
outgoing optical field, and thus an upper bound on optical spectral power density. In
practical terms, Equation 6.1 connects voltage directly to brightness.
In
§ 2.3 we considered non-ideal LEDs, including those with low external quan-
tum efficiency. We found that although these devices could not achieve net electroluminescent cooling at high bias, at very low voltages they could. In fact, the linearity
of the diode's response to application of a small bias voltage showed that in theory
every light-emitting diode should experience cooling at sufficiently low voltage. However, since this low voltage constrains the outgoing optical field to a temperature
barely above ambient, observation of this phenomenon requires measurement of very
180
small amounts of light.
Next, in
§ 2.4 we saw that since a device with no non-radiative recombination
has no sources of irreversible entropy generation, as an LED's quantum efficiency approaches unity it behaves increasingly like a Carnot-efficient heat pump. Nevertheless,
since there is a fundamental limit on spectral power density imposed by the Second
Law limit on the maximum temperature of outgoing photons, there is a Carnot bound
on efficiency at fixed power density for an LED with a known low-voltage emission
spectrum.
The preceding results substantiate our interpretation of an LED as a thermodynamic heat pump. They moreover justify two basic counter-intuitive aspects of the
thermal physics of highly efficient LEDs at voltages V < Egap/q:
" Instead of discharging waste heat into the device's lattice, an efficient LED cools
its lattice by pumping heat into outgoing photon modes which carry the energy
(and entropy) away from the device.
" Since heat can be pumped with a higher coefficient of performance against a
smaller temperature difference, an efficient LED source with a given absolute
spectral intensity (which determines the outgoing photon temperature) becomes
more efficient in a higher-temperature environment.
As a first step in developing devices closer to the Carnot limit, in
§ 2.5 we used
an experimentally validated computational model of a 2.15 ym InGaAsSb emitter
to design a new layer stack for operation at sub-bandgap voltages. We found that
significant improvements should be attainable using existing technology and that the
optimized device's behavior should exhibit a monotonic power-efficiency trade off
qualitatively resembling that of a Carnot-efficient device.
In
§ 2.6, we closed our
theoretical discussion of thermodynamic device behavior by noting that analysis of
this nature is quite general and that in fact the flow of electrical current through
181
any semiconductor device (or combination thereof forming a closed circuit) can be
analyzed as a closed thermodynamic cycle.
In Chapter 3, we presented experimental evidence to test these theoretical predictions. We began by providing a short description of some major techniques required
for the efficiency measurements that followed. These included the use of an AC LED
drive current with phase-locked photo-detection, feedback thermal control with limited temperature slew rates to avoid thermal shock, and some basic optical design
for efficient collection of photons from an imperfectly collimated LED source.
In
§ 3.2 we presented the first demonstration of electroluminescent cooling by observing
electrically-driven optical power in excess of the electrical power required to drive a
2.5 pm LED at 135'C. Next, in
§ 3.3 we documented an unsuccessful attempt to make
a similar observation from a 4.7 Mm LED. The experiment was expected to achieve
unity efficiency at higher power density due to the longer emission wavelength, but
increased non-radiative recombination, leakage, and contact resistance offset the anticipated increases. In
§ 3.4.1 we used LEDs emitting at 3.4 Pm to obtain further
evidence that the optical power measurements found throughout this chapter were not
the result of linear emissivity modulation. Finally in
§ 3.4.2 we present observations
of LEDs at 3.4 and 4.7 pm operating above unity wall-plug efficiency.
In Chapter 4 and Chapter 5 we explored the application of efficient LEDs at
low forward bias to low-power digital communication and high-temperature infrared
absorption spectroscopy respectively.
In
§ 4.1 we motivated our consideration of these LEDs as a source for optical
communication by reinterpreting the power measurements from Chapter 3 as communication over a very slow channel.
For existing measurements of a 150'C LED
emitting 2.5 Mm photons onto a 3mm-diameter photo-diode at 25'C, the corresponding channel required just 1.7 pJ per bit. Our subsequent discussion of extrapolating
these measurements to the low power and high quantum efficiency limits indicated
182
that future experiments could approach the well-known Landauer limit [48, 49, 52]
of kBT In 2 per bit. In
§ 4.2 we analyzed an idealized system with no sources of irre-
versibility and no noise other than the presence of equilibrium blackbody radiation
at the optical frequencies which contain the signal.
Using a codebook which was
optimized for energy efficiency, we solved for both the mutual information shared
across the channel and the work required to generate the signal in the low power
limit. Using both analytical and numerical methods, we found that in this limit, the
work required by this idealized channel is exactly kBT In 2 per bit. Finally in
§ 4.3 we
investigated orthogonal frequency-division multiplexing as a means of increasing the
data rate of such a channel without sacrificing its per-bit energy efficiency. We described our experimental realization of a multiplexed 3 kbps low-biased LED channel
which consumed just 40 femtojoules of electrical energy per bit with a bit error rate
of 3x10
3
.
Our discussion of the potential for thermo-electrically pumped LED sources in
spectroscopy began in
§ 5.1 with a brief discussion of the technological need they
could fill. We saw that while applications such as downhole fluid analysis in oil wells
and combustion exhaust gas analysis possess valuable spectroscopic information at
mid-infrared wavelengths, the inefficiency of existing sources and detectors at room
temperature and above are often prohibitive. However, since the low-biased LEDs
discussed in this work are highly efficient and can become more efficient at higher temperature, they could be used for absorption spectroscopy in these high-temperature
applications. After a brief aside in
§ 5.2 to establish a framework to analyze a spec-
troscopy system using the same information-theoretic tools as in the previous chapter,
we looked more closely at the high-temperature behavior of LEDs and photo-diodes
in
§ 5.3 and § 5.4 respectively. We found that while the performance of the photo-
diode-based detector circuit decays exponentially with temperature, the performance
of the LED sources simultaneously improves exponentially with temperature. Finally
183
in
§ 5.5, we concluded our discussion with a promising observation that the primary
reason for both the improvement of the source diode and the degradation of the detector diode is their decreased resistance to current flow at low voltages. Furthermore
since the detector diode in such a pair must have a bandgap energy equal to or below
that of the source to absorb its emitted photons, the improvements on the source
side can effectively compensate for the detector's decreased shunt resistance to create
a spectroscopy system whose signal-to-noise ratio actually improves with increasing
temperature.
6.2
Further Scientific Questions
The work described in this thesis spans a range from the basic to the applied. Most
of the work described in Chapter 2 and Chapter 3 was aimed at characterizing and
understanding the thermodynamics of light-emitting diodes under sub-bandgap bias
conditions. The latter chapters focused primarily on the application of such devices in
systems with exogenous goals like transmitting information with high energy efficiency
or extracting spectral information from a fluid sample at high temperature.
Although our scientific work on thermo-electric pumping in LEDs enabled certain classes of applications, several interesting scientific questions remain. Broadly
speaking we categorize them into those related to the role of entropy and information in photons and those related to their role in electronic degrees of freedom in
semiconductor devices.
6.2.1
Entropy and Information in Photons
In Chapter 2 we chose to represent the excitation of the outgoing photon field as
a thermal state with an effective temperature T*. With the LED at forward bias,
T* would be greater than the ambient temperature, leading to greater occupation of
184
these outgoing modes compared to background thermal radiation, and thus an out
flux of optical power. This is not the only valid way to describe the state of these
modes as they carry completely incoherent radiation away from the device. In fact, an
alternate description in which the temperature is fixed and the field is taken to have
nonzero chemical potential p is more common. The same occupation, and therefore
spectral intensity can be described either way:
f =
1
or
e kBT* -1
1
f=
e kBT
.
(6.2)
-1
In fact, since the density matrix of a thermal field is geometric, it is determined
entirely by the ratio of probabilities of the mode containing n and n + 1 photons.
Since either the inclusion of At > 0 and T* > T serve to parameterize this same ratio,
these two descriptions correspond to the same density matrix, and therefore represent
physically identical states:
p =
In) (nI (1 - ),"
(6.3)
n
where
hw
r = ekBTr
______
or
r
e kBT
.
(6.4)
In Chapter 2, we elected to use the T* description because this quantity could be
used in the expressions for entropy and energy so as to result in familiar and intuitive expressions for the Carnot limit of a thermo-photonic heat pump. Both of these
descriptions, however, fail in the degenerate limit. As p -+ hw or as T* -+ 00, the
ratio of probabilities approaches one, and the expected occupation sees a non-physical
divergence. This situation corresponds to the electron-hole system approaching transparency, where our many of our assumptions break down including the assumption
that our sample is "optically thick."
185
An interesting direction for further work may be to characterize LEDs at voltages
just below the band-gap to see when this description breaks down.
At inversion,
the thermodynamic theory has little to say- the electrons and holes have a negative
temperature and so eliminating a pair generates entropy regardless of the final state
of the photonic system. However, when real diode lasers reach threshold, the current
noise in the electronic system can pass through to the photon field. Since this mechanism for the introduction of disorder into the photon field is not accounted for in the
present theory, it is unclear to the author whether or not such a mechanism would
begin to dominate even below inversion. In essence this amounts to characterizing
the domain of validity of the theoretical picture presented in Chapter 2.
Measurements of the average intensity and the intensity autocorrelation could
reveal a better understanding of the breakdown of this theory, and ultimately the
practical limits of using conventional semiconductor diodes for photonic heat pump-
ing.
First, the average spectral intensity emerging from an LED close to transparency
could be compared against the relationship between V and L from the low-bias limit.
From Chapter 2, our prediction would be for a spectral intensity given by the Planck
formula but suppressed by a factor of the absorption through the active region at
this bias condition.
Since this quantity approaches zero as the device approaches
transparency, our prediction can be expected to diverge from reality.
Second, experiments similar to the one carried out by Hanbury Brown and Twiss
in 1956 [99] can be used to characterize the degree of coherence of a photon source. If
the light emerging from an LED at low bias were subjected to such an experiment, the
intensity autocorrelation could be directly measured and compared against existing
models of current noise in diodes to determine if and when a noise source other than
the fundamental noise from recombination of electrons and holes carrying entropy
becomes dominant. Further theoretical study of the problem would be required to
186
properly connect these experimental results to measurements of photon entropy given
that the signal contains both equilibrium radiation and radiation driven by externally
supplied electrical work.
In addition to the use of quantum optical techniques to characterize the domain of
validity for the theoretical predictions from Chapter 2, the work in Chapter 4 raises
interesting questions about the limits of efficient photonic communication.
The experimental link described in
§ 4.3 immediately raises the question of efficient
communication when the source and detector are not at the same temperature. If a
high-temperature emitter is connected to a low-temperature detector by an optical
path that interacts with matter on both sides, energy will flow from hot to cold in
the form of a net flux of thermal radiation even when the channel is not in use.
Since it is possible to extract work from such a temperature difference, that work
could in principle be used to encode information on the field leaving the emitter. At
first glance, it seems this strategy could be used to consume less than kBT ln 2 per
transmitted bit, or even to net generate power during communication. However, it
is not immediately obvious how one would extract the exergy from the net photon
flux while also allowing the signal to be recovered at the detector. Furthermore, since
the energy efficiency limit established theoretically in
§ 4.2 presumes the existence
of a perfect detector (i.e. all uncertainty at the detector was due to the entropy in
the incoming photon field), that result may be more logical to interpret as a limit on
the energy required to transduce known information from the electrical input signal
onto the optical output of the finite-temperature source, regardless of the state of the
detector.
Such an interpretation of communication evokes images of a familiar model from
statistical mechanics known as Maxwell's Demon. The Maxwell Demon tries to open
and shut a boundary between two halves of a container of gas at equilibrium.
His
goal is to generate a temperature difference by preferentially allowing the fast-moving
187
particles transmit from left to right while letting the slow-moving particles transmit
from right to left.
If such a Demon could perform this operation with negligible
power consumption, the final state of the gas could be used to drive a heat engine
and extract work in violation of the Second Law of Thermodynamics.
In this situation, the Demon is encoding known information about the state of
the individual particles composing the gas into the degrees of freedom describing
the particles' motion. From these degrees of freedom, a heat engine is then able to
extract work. In order to comply with the Second Law, the amount of work required
for the encoding process must be greater than or equal to the amount which could
be extracted from the final state. In the same sense, the LED in the link from
§ 4.3
is attempting to encode known information into the outgoing photon field.
Since
the final state of that outgoing photon field has a higher temperature T* > Tattice,
the requirement of consuming kBT In 2 per bit encoded could be seen as the Maxwell
Demon's analog for electrical-to-optical conversion.
This interpretation in turn raises a possibility of eventual practical importance.
The exergy in the photons which comprise the signal in our LED link could be used to
drive a photo-current at the detector side. If the detector were operating as a perfect
photo-voltaic, the power recovered could be used to drive the source or to physically
represent the electrical signal that emerges at the receiver. Such a link doesn't consume any power in the traditional sense; rather it allows a physical signal to flow
from the electrical domain at the source into the optical domain for transmission,
then back into the electrical domain at the receiver.
We note that this conceptual configuration shares characteristics with more abstract models for zero-net-power communication emphasized by Landauer [49] in response to misunderstandings he ascribed to overgeneralization of his original analysis
leading to the kBT ln 2 limit.
188
6.2.2
Entropy and Information in Electrons
The results described in this thesis also raise interesting questions related to the flow
of entropy and information through electronic degrees of freedom (i.e.
motion of
electrons and holes). In the basic description from § 2.1 of electron transport through
a double hetero-junction LED with bias voltage V < Egap/q we saw that the electrons
and holes absorbed entropy from the lattice during injection and released entropy into
another reservoir during recombination. As we saw in § 2.2, the electrons are in effect
a working fluid for the heat pump. That is, just as in the case of a macroscopic,
mechanical refrigerating heat pump, a closed thermodynamic sub-system internal to
the pump (typically a two-phase refrigerant fluid such as R-134A [100]) is used to
absorb entropy from the reservoir being cooled and eject entropy into the reservoir
being heated.
Furthermore, as we briefly discussed in § 2.6, if we step into the "frame" of an
electron as it passes through the device, the local environment follows some path in
the space of the relevant thermodynamic state variables (i.e. temperature, specific
entropy, electro-chemical potential, and number density). When the device is combined with a source of work to form a closed circuit, the corresponding path returns
to its original position and forms a closed cycle.
For macroscopic, mechanical heat engines and heat pumps, the development of
new cycles like the Sterling and Brayton cycles led to significant practical improvements.
Today a turbine engine using a Brayton cycle can be 60-65% efficient, as
much as double that of an engine running a simpler diesel cycle [100]. Similarly, new
device-level designs of LEDs designed to operate as thermo-photonic heat pumps may
be able to mold the flow of electrons into new, improved thermodynamic cycles. In
this way, the basic building block of semiconductor physics, the p-n diode, can serve
a similar role for semiconductor engines as the single-piston reciprocating engine did
for mechanical engines almost two centuries ago.
189
EE
EE
V
TS
X
S
T,,t,,x(I-qV/E g)1
Titfc
T*
Figure 6-1: Two representations of electron transport in a forward-biased lightemitting diode, vertically aligned to emphasize the connections between the models.
At top is a familiar band diagram, which is essentially a statistical model because it
attempts to describe the state of the system in terms of micro-states. For example,
the probability for occupancy of a particular conduction band state can be calculated
from the electron quasi-Fermi level and temperature at that point. At bottom is
a thermodynamic model for the same physical device. Here the variable T* refers
to the temperature seen by inter-band processes and Seiecton and Shole refer to the
per-particle entropy of the electrons and holes respectively. These quantities can be
used to calculate the per-particle heat and entropy fluxes in the electronic degrees of
freedom. Since the bottom figure does not indicate any specific values for the presence of accessible states or their average occupation (i.e. there are no bands or Fermi
levels drawn), we say the description is thermodynamic rather than statistical.
190
We note that developments along this line are already underway for related technologies. For thermo-electric heat pumps and heat engines, which are typically composed primarily of semiconducting materials, new cycles leading to significant systemlevel improvements have already been developed [101, 102]. Researchers working to
improve the efficiency of solar photo-voltaics have begun to measure their progress
in terms of the suppression of a series of free energy losses [103], and thermo-photovoltaics have been reported which utilize thermo-electric heat exchange at heterojunctions to enhance their open-circuit voltage [104].
Finally we end our description of the outlook for thermodynamic considerations
in electronics by noting that even circuits with multiple devices may be subject to
similar thermodynamic analyses.
For example, a basic CMOS inverter (composed
of a single NMOS and a single PMOS transistor) within a network of logic gates
can be analyzed in this way. As electrons flow from ground up to
Vdd,
they traverse
four metal-semiconductor junctions and four semiconductor p-n junctions in series. In
doing so, they irreversibly generate on average ASinverter = ( CVa)/T of entropy each
time they transport the information about the charge state of the gate electrodes from
their input to their output. Here C is the input capacitance of the subsequent stage
and T is the ambient temperature surrounding the larger logic network. In essence,
the gate is generating
ASinverter
to perform a reversible transformation on a single bit
of information and move a copy of the result to its output. By applying a similar
procedure to other logic gates and ever larger networks of gates, we can ultimately
build accurate thermodynamic models of entire computing machines. These models
may provide insights into energy-efficient computation along the lines of adiabatic
computation [48, 105] and computational sprinting [106], or lead to new capabilities
related to thermal physics like physical random number generation [107].
191
6.3
Further Applied Directions
The results described in this thesis also point to several areas of applied work. The
first, most obvious direction is to use the design developed in
§ 2.5 to grow, fabricate,
and test an infrared LED.
Despite their wider bandgap, similar projects in the indium phosphide and gallium
arsenide material systems could also hold promise if properly designed for deep-subbandgap operating voltages (i.e. Egap - qV
> kBT). In particular, the strategy of
doping the active region could lead to LEDs with very high quantum efficiencies in this
regime due to the low defect densities achievable today. Although the power density
in this regime will suffer from the wider bandgap, detectors at these wavelengths
should also have lower noise equivalent power for the same reason. For applications
like communication and spectroscopy, where signal-to-noise ratio can be more relevant
than total power, this strategy would let us experimentally test certain fundamental
limits which we have thus far relied on idealized extrapolations to explore.
Furthermore the general study of LEDs made from wider bandgap semiconductors
such as InGaN could help us understand the physics of these devices at sub-bandgap
voltages. In theory many of the phenomena described in Chapter 2 rely primarily
on both carrier species being in the Boltzmann regime.
Thus another interesting
direction would be to characterize visible devices at voltages which are on the order of
the bandgap energy but whose carriers remain in the Boltzmann regime. In particular,
the extent to which the connection between voltage and light intensity from
§ 3.4.3
remains valid for these devices could lead to new understanding of the temperaturedependence of efficiency for wide bandgap emitters.
Another interesting direction is to use the thermal physics expressed in
§ 6.1 above
to re-engineer an LEDs thermal design. Since the efficiency of an LED at qV < Egap
can be an increasing function of temperature, thermal packaging which maximizes
operating temperature could improve efficiency. As we alluded to in the publication
192
reporting the first demonstration of above-unity efficiency electrical-to-optical power
conversion:
"[For LEDs at sub-bandgap operating voltages,] self-heating may offer a convenient solution for sources with subunity [wall-plug efficiency].
Here, purposeful concentration of internally generated heat, such as in an
incandescent filament, should allow phonons to be recycled to thermally
pump the emitter."
SANTHANAM,
GRAY, AND RAM
PHYSICAL REVIEw LETTERS, 2012
For reasons related to a patent application which awaits examination (Provisional US
Patent Application No. 61/684315 filed August 17, 2012; full US Patent Application
filed August 16, 2013), we limit our discussion here.
Another very interesting direction to consider is the use of modern nano-photonic
and plasmonic techniques to further enhance the quantum efficiency of LEDs.
the transport models from
In
§ 2.1, the coefficient B parameterizes the rate of radia-
tive recombination (cm- 3 s- 1 ) for a given concentration of electrons and holes. The
microscopic physics contained in this figure in most cases can be obtained rather
straightforwardly using Fermi's Golden Rule. In such a calculation, the transition
rate is proportional to the absolute square of the matrix element connecting the
initial and final states of the combined electron-photon system and is linearly proportional to the density of final photon states available for emission. Using nano-scale
structures composed of dielectrics and metals, the local mode structure of the photon
field can be distorted to allow a particular region of space and optical frequency to
have a density of states much higher than in vacuum.
Via the Purcell effect, the
rate of radiative recombination can therefore be enhanced, and with it the device's
quantum efficiency.
While other opto-electronic devices like visible LEDs ultimately seek to interact
193
with photon modes in vacuum, solid-state refrigerators using thermo-photonic effects
to pump heat from the lattice of one diode to that of another diode do not. As a
result, metamaterials with very high densities of states can be used to increase the
radiative thermal conductance, which determines the heat pump's power density. Recent experiments have confirmed that the evanescent tails of photon modes can be
used to enhance radiative heat transfer while minimizing conductive and convective
heat transfer across nano-scale gaps [108].
By exploiting this principle to control
heat transfer across a narrow gap separating two diodes which use the Purcell effect
as described above, thermo-photonic heat pumping with high power density may be
achievable [109]. We note also that similar strategies for thermo-photo-voltaics have
recently come under consideration [110, 111] including those using metamaterials capable of drastically increasing radiative heat transfer in the near-field [112, 113]. Early
work suggests enhancements of two to three orders of magnitude may be realizable
[114, 112, 115].
In parallel with the development of improved sources, their employment in hightemperature mid-infrared spectroscopy systems represents a near-term target to which
heat pumping LEDs could lead to real system-level benefits. For example, the use of
LEDs and photo-detectors in the InAsSb ternary alloy system can be used to perform
absorption spectroscopy around 3.4 pm. From the spectra in Figure 5-4, we saw that
if the optical path between source and detector includes an ATR crystal exposed to
a crude oil sample, spectroscopy at this wavelength could be used to fingerprint the
hydrocarbon chains present in it. Extrapolating from the exponential dependence of
LED efficiency and photo-diode shunt resistance with temperature would suggest such
a system would suffer from an extremely low signal-to-noise ratio in the photo-current
signal. However, making use of the source-detector compensation technique from
§ 5.5
should allow spectroscopic data around 3.4 ym to be acquired with reasonable lock-in
integration times.
194
Applied work on communication channels with very high source-side energy efficiency could also be pursued. The magnitude of the dominant noise in the channel
presented in
§ 4.3 is likely quite close to the magnitude of the noise expected from the
random arrival of thermal blackbody photons. Thus relatively simple modifications
to the detector, such as further reducing its temperature or using a similar detector with a smaller absorptive area, could allow us to develop a channel whose noise
is dominated by fundamental sources connected to the temperature of the source
LED. In fact, the data presented in
§ 5.5 suggest this may be accomplished simply
by using the source-detector diode pair from the spectroscopy experiments around
3.4 pm. Furthermore, if devices possessing high quantum efficiency in the low-bias
regime V < kBT/q can be developed in the InP or GaAs material systems, near-IR
experiments using low-noise cryogenic photo-detectors or visible experiments using
photon-counting avalanche photo-diodes could be used to form interesting channels.
According to the theory in
§ 4.2, a channel which is whose signal is encoded by a
low-bias LED that is nearly free of irreversibility and whose noise is dominated by
thermal blackbody photons should come close enough to the kBT ln2 limit to help
address the outstanding question of whether classical communication at optical frequencies faces a limit stricter than kBT ln(2) because discrete photons carry energy
hw > kBT [116].
6.4
Engineering Toward Second Law Bounds
In this thesis we have presented the first experimental verification that an LED can
emit optical power in excess of the electrical power used to drive it, a concept which
was first introduced theoretically in 1957 [31]. As with any energy-conversion technology, the ultimate limits for the efficient generation of white light are set by the
Second Law of Thermodynamics. Using the theoretical framework we developed in
195
Chapter 2 and supported empirically in Chapter 3, for any incoherent light field a
sufficiently ideal electrically-driven source can generate it with a wall-plug efficiency
above 100%. That is, the device will harvest ambient heat to provide a portion of the
power which drives the source in steady-state rather than consuming more electrical
power than it emits in optical power and releasing the remainder as waste heat.
Indoor Light
AM1.5
High-Power LED die
TCarnot
0
%125
W
75
min il for
50
10-6
10-3
1
103
I=1
106
White Light Intensity (W/m 2 )
Figure 6-2: The theoretical thermodynamic efficiency limit for generating white light
from a thermo-electrically pumped LED as a function of light intensity. The solid
curve marked with circles denotes the maximum efficiency permitted by the Second
Law in a 300K ambient. The dotted curve marked with squares represents the minimum quantum efficiency required by an LED (at the relevant sub-bandgap operating
voltage) to achieve white light generation at 100% wall-plug efficiency. Here "white"
is taken to have the relative spectral intensity of a 5800K blackbody between 380nm
and 780nm and no radiation outside that band; the absolute spectral intensity then
scales with the intensity on the plot's x-axis.
Although all incoherent light fields carry with them some entropy, they do not all
carry the same amount. Relatively dim fields, for instance, can carry much more en-
196
tropy than bright fields with the same spectrum. Likewise, broadband fields can carry
more entropy than narrow-band fields with the same optical power per unit area. As
seen in Figure 6-2, the light emerging from a modern high-power LED chip
(~ 106
W/m 2 ) faces more strict fundamental thermodynamic limits than the ambient light
present in an OSHA-compliant workplace (~
1 W/m 2 ) [1171. We may also interpret
this result in terms more relevant for the thermal engineering of waste heat management: in order to rid ourselves of the waste heat problem by achieving a wall-plug
efficiency of 100%, the bright LED die will require at minimum a quantum efficiency
of 90% in comparison to 75% for a source matched to the intensity appropriate for
human consumption.
By generating light for human use at a brightness higher than it will ultimately
be consumed at, we are paying for more coherence than we can make use of. In
an indoor lighting context, light from the bright high-power LED chip ultimately
undergoes entirely avoidable irreversible entropy generation as it scatters off diffusing
surfaces or being partly absorbed by dark surfaces, or else is finally consumed by a
human retina which is insensitive to its remaining coherence. By contrast, if the same
total optical power is delivered to the room but is generated over a large area, the
emitted photons are already maximally disordered. Moving to a wider area emitter
at constant power effectively removes the irreversible entropy generation step and
replaces it with entropy transport from the ambient environment. The decision to
continue engineering improved small, high-power LEDs instead of more efficient by
larger area panels is then essentially an economic one.
An economic analysis of the ongoing efforts to replace old, highly inefficient light
sources with efficient LED lighting falls outside the scope of this work. Nevertheless,
we choose to briefly present an argument for the long-term relevance of the preceding
conclusion. There exists an empirical law called Haitz's Law, modeled after Moore's
law for transistor density in integrated circuits, for the development of cheaper and
197
brighter LEDs. Haitz's Law states that the price per lumen of LEDs is cut in half every
28 months, and has held true for the last 40 years [118]. In recent years, the trend has
actually accelerated (in part due to market interventions by governments). Between
2007 and 2012, the price fell by nearly a factor of 10 to less than half of a cent per
lumen [119]. Meanwhile, the real (i.e. inflation-adjusted) cost of electricity required
to power these LEDs has remained virtually flat. As a result, the lifetime ownership
cost of is increasingly dependent on the operating expenses (power consumption) and
less dependent on initial capital costs. Thus, in the long run, a technology which uses
an expensive wide-area emitter but consumes less power could prove more economical
than a small bulb that produces requires less of the presently-expensive semiconductor
area per lumen of lighting capacity.
Since the earliest surviving formulations of the Second Law of Thermodynamics,
those of Lazare and Sadi Carnot in 1824, it has been used to calculate the fundamental
limits of machines whose primary purpose was energy conversion. The generality of
the constraints it applies to physical systems has allowed it to remain as relevant to
modern machines like LEDs as it was to steam-driven turbines in the 19th Century.
Following the early work of Maxwell and Boltzmann on the kinetic theory of gases,
a few decades later the notion of entropy began a parallel development in which
the Second Law could be formulated as a statement about information. In modern
statistical mechanics, instead of interpreting entropy as a substance attached to some
forms of energy which cannot be destroyed, we can interpret entropy as a measure of
unknown information which cannot be deleted. In the same way that thermodynamic
machines seek to mold the flow of energy for practical purposes but remain constrained
by the inability to destroy entropy, today's information processing machines seek to
mold the flow of information but remain constrained by the Second Law.
In Chapter 4 we discussed one of the simplest information processing machines: a
communication link. The goal of this machine is simply to transport information from
198
one physical system to another. Because information is physical, we found we were
unable to encode the information we wanted to transport without imposing some order
on some physical subsystem, in this case an interval of photonic phase space, that then
travels from transmitter to receiver. This order, or equivalently known information
about the physical state of this subsystem, required a certain nonzero amount of
energy to be added to the subsystem because imposing this order without moving
any energy between subsystems would constitute a violation of the Second Law in
that subsystem. In this sense, the kBT In 2 per bit limit for our channel's efficiency
is a consequence of the Second Law. The Landauer limit is not a statement about
communication, but a statement about the representation of known information in
physical systems. It applies equally to any physically realizable information processing
machine.
In short, all descriptions of physical state contain information and all real information must be represented physically. In the long run, we will inevitably invent
new technologies, but the constraint of the Second Law on all information processing
machines will remain. As a result we should expect our information technologies,
such as those which perform digital communication, to follow the same trajectory as
energy conversion machines in their inevitable march toward the bounds set by the
Second Law.
199
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200
Appendix A
Entropy and Temperature of Light
The basic result about classical LED communication which this project seeks to
express relies on a thermodynamic analysis of the low-biased LED. Without a proper
understanding of entropy and the effective temperature of light, we cannot consider
the electronic subsystem of an LED to be the working fluid of a heat engine operating
between one thermodynamic reservoir of phonons and another reservoir of photons,
and so cannot derive a Carnot bound for the efficient generation of thermal photons.
The papers summarized in this document, therefore, constitute an important part of
the literature supporting the communication result.
The process of defining a temperature for use in a thermodynamic analysis of an
incoherent light-emitting device has three basic steps.
Step One: Dividing Up Phase Space
The first step in analyzing the thermodynamic properties of light is to find chunks
of phase space which can be treated as individual quasi-equilibrium systems. To do
so, we look for intervals in 6-dimensional phase space in which the average photon
occupancy is roughly constant. The 6 dimensional interval is the intersection of an
interval in three spatial dimensions and one in three reciprocal-space dimensions;
201
the latter could be equivalently described by a frequency interval and an interval of
solid-angle for the propagation direction. Intervals within which the light intensity
is directly proportional to the photon density-of-states may be said to have constant
occupancy.
Within such an interval, thermodynamic state-variables for the photonic system
may be calculated from a single mode with the values for extensive state-variables
scaled with the number of modes.
In particular, for incoherent thermal light the
average occupancy is sufficient information to know the entire state of a mode for
reasons outlined in Step Two. Making things even more convenient for thermal light,
the average occupancy may be simply calculated in any situation as the average
energy per h 3 /2 of phase space, divided by hw.
In some publications which seek to calculate the effective temperature of the light
emitted in some specific situation, this breaking of phase space is the first step.
In 1959, while exploring the thermodynamic limits of efficiency for lamps, Weinstein [29] published basic calculations assigning an effective temperature to the light
from a green ZnS phosphor. In this calculation, he approximated the emitted light
with a gaussian emission spectrum of width Av around some center-frequency vo.
By doing so, Weinstein implicitly assumes that the only portion of phase-space of
relevance to the calculation is that around the primary fluorescence frequency. In
1980, Landsberg and Tonge [33] chose to treat analytically the case of the gaussian
spectrum modulated by a power-law and calculated the effective temperature in this
general case. It is assumed that the utility of this result lies in assuming that the photon density-of-states may be easily approximated by a power law within the relevant
frequency band, even in the case of Purcell or photonic crystal effects.
To simplify things further, some authors in fact choose to examine only one finite
interval of phase space within which they proceed with a statistical analysis, and
outside of which the photons are negligible.
202
For example, when Mungan [30] applies this basic framework to the first experimental observation of net-cooling in a solid (Epstein et al [120]) for pedagogical
purposes, he makes similar assumptions about both the incoming pump laser light
and the fluorescence emitted by the Yb3 +:ZBLAN(P?) glass being cooled. In particular, he describes the laser as having constant intensity within the entire phase-space
interval of relevance. Although laser light need not be thermal, Mungan's assumption that the entropy may be calculated in this way is justified by the conclusion he
reaches. Although Mungan does not explicitly address the issue of non-thermal photon populations, by showing that were the laser light thermal (a quantum-statistical
state with maximal entropy per unit energy) the entropy it would carry would still be
negligible and
Taser -+
oc could be assumed in the subsequent thermodynamic anal-
ysis. In effect, the thermal-light assumption lower-bounds the effective temperature
of the laser light. For the case of the emitted fluorescence, Mungan performs two
separate calculations: first he assumes that the light is of constant intensity within
a bandwidth given by the full-width at half-maximum of the measured spectrum,
then he goes ahead with the full average-temperature calculation for the real measured spectrum. In the first (flat power-spectral density assumption) case, he finds
the outgoing radiation to have TF = 1760K; in the second case, he finds the outgoing
radiation to have TF
=
1530K. This calculation suggests that for spectra typical of flu-
orescent ytterbium, the flat-power assumption results in effective temperatures with
a 10-20% error. The flat-band calculation over-estimates temperature because the
tails of the distribution are not included; if the effect of non-flatness within the band
were of dominant consequence, the effective temperature would be under-estimated.
This final observation relies on the concavity of entropy, to be discussed in Step 3.
On the other hand, Weinstein later showed [121] that the quantum-mechanical
inverse relationship between spontaneous-emission linewidth and emitting carrier lifetime is necessary to resolve the apparent breakage of the 2nd Law when a hot system
203
of oscillators relaxes by undergoing radiative transitions. The lesson appears to be
that while the effective temperature of light is typically not sensitive to the exact
shape of the spectral density, the characteristic linewidth for a given integrated intensity is critical to satisfying the Laws of Thermodynamics.
Step Two: Drawing the Entropy Function
Electromagnetic modes should be thermally occupied if their occupancy results from
interacting with some composite of microscopic electronic subsystems with nonzero
matrix elements for photon emission/absorption (i.e. oscillators) whose entropy and
energy are related by a single temperature. In the case of an LED, for example, our
subsystems are pairs of vertically-aligned conduction and valence band states whose
upper-radiative (UR) state occupancies are given by the Fermi level and temperature of the electrons and whose lower-radiative (LR) state occupancies are likewise
given by the hole quantities. If the two temperatures and Fermi levels are the same,
then it is clear that whether we slice the E-k diagram horizontally (define bands as
subsystems) or vertically (define oscillators as subsystems), every quantum state is
occupied in equilibrium with every other state, so any photon modes that have come
to equilibrium with this system should also be thermally occupied at the same temperature. An analysis of small deviations of the AEF- and AT-types should reveal
that as such light-emitters are infinitesimally turned on, the photon fields with which
they interact should be continuously deformed and the assumption of thermal light
should remain valid.
If an electromagnetic mode is said to be occupied with thermal light, then its
density matrix is diagonalized in the number basis and the ratio of probabilities
for occupancy by n + 1 photons to the probability for n photons is a fixed number
independent of n. We may think of this ratio, related to the temperature of the mode,
204
as fixed by the basic construction of the canonical ensemble from statistical mechanics.
In the canonical ensemble, our single photonic mode interacts with a reservoir whose
entropy (log of number of configurations) increases by the same amount with the
addition of each unit of energy, regardless of how much energy has been pulled from
or put into that reservoir by our mode. Because the ratio of probabilities is fixed,
the occupancy probability distribution (diagonal elements of the density matrix) is
geometric and so self-similar. The self-similar nature of this probability mass function
(PMF) leads to a simple recursion-relation to calculate its entropy:
H(P) = Hb(r)
+ rH(P)
r
where
= (n±1)
P(ri)
r log(r) + (1 - r) log(1 - r)
1r
H(P) = -1ro()+
and since
H(P) = -1
Nlog
I
=
-+
S(N)
N
= (f)p
+ log(
N + I
r
- r
=
N + 1
then
)
(A.1)
(N + 1) log(N + 1) - N log N
= kB
[(N + 1) log(N + 1) - N log N]
The entropy function is plotted as a function of occupancy N in A-1 and as a
function of the probability-ratio r in A-2.
While the basic mathematical structure of the preceding derivation relies on Bose
statistics, the actual formula appears to have been derived in several different physical
models since Planck [122, 123, 124, 125, 126]. In some cases, the authors have chosen
to treat the photons as a closed statistical system of bosons [123, 124, 125], as we
have in the preceding discussion. In other cases, the authors have chosen to define the
entropy of the photon field by the temperature of a coupled reservoir of electromagnetic oscillators which has exchanged energy and entropy to reach a detailed-balance
equilibrium state with the photons [122, 125], as in the canonical ensemble. Since
205
4
C1
.
CL
0
2
4
6
Occupancy
8
10
Figure A-1: Entropy of a thermally-occupied photon mode (blue line) as a function
of expected occupancy. Since the expectation value for the energy is just Nhw, which
is just a rescaling of the horizontal axis, the qualitative behavior of this function
S(N) is the same as S(U). The thick red line indicates an approximation to this
formula which comes from considering only the binary variable indicating whether or
not the first photon is present. Note that for average occupancies < 1, this Fermionlike entropy function approximates the full entropy. Successively thinner red lines
indicate inclusion of the second and third photons' presence or absence.
206
3
b
4
c\I
C
4
4
4
3
4
o
CL
0
0.2
0.4
1 - Probability(GroundState)
=
0.6
0.8
1
Probability(AnyExcitations)
Figure A-2: Entropy of a thermally-occupied photon mode (blue line) as a function
of the characteristic ratio r = P(n + 1)/P(n). The red lines approximate the full
Bosonic solution with simple binary random variables including the first, second, and
third photons. Note that the first photon's entropy is just the entropy of a binary-r
random variable, as we'd expect for a Fermionic mode occupied with probability r.
the statistical results for these oscillators are then derived microscopically by treating them as bosons, the result is identical. Finally, the most accessible derivation
of the entropy for a photon field comes from simply inverting the blackbody energydensity and using the 3rd Law of Thermodynamics to recover the entropy expression
[125, 126]. Although this formulation hides all of the quantum mechanics behind the
Planck blackbody formula, it is reproduced here because of its simplicity.
Starting with the expression for the energy density U of a blackbody within a
frequency band Aw, we have:
207
U(T)
=
Aw x (density of modes at w) x (# photons per mode at w) x
h
(e
7r20
=-
kB
T- 1 (U)
=
no
2
B3
( photon)
)
exp hw/kBT - I)
as
In hW3A+1
aU
I T2c3U
(A.2)
Now, since by the 3rd Law of Thermodynamics, S -+ 0 as U, T
grate dS
=
-+
0, we can inte-
dU/T to find the entropy-density S at finite temperature (and therefore
energy-density). Defining the dimensionless energy-density as
72C3
=U
FA
(A.3)
we have
I
dS =S=
U
UdU'
hw
kB
hw
kB
hW3 Aw
72C3
hw 3AW
2 3
ir c U'
hw 3AW
hw
('
v=1+ii
(vIn
hW3AW
{(ft+
3
hw 72
kB
1'
dii' ln(1 + V) -
hw
kB
+
vvI
=1
(fi
t
,0)
1)ln(ii + 1) - (ii+1) - IlnI+1-dnii+f±+0 -0}
hW3,Ao
7203
{(i + 1) ln(t +1)
- iln f}
(A.4)
With the entropy formula in hand, we examine the relevant concepts of temperature for light.
208
Step Three: Defining Temperatures
Using the entropy function from the previous section, two different notions of temperature are commonly defined when examining the thermodynamic limits on particular
photonic devices [29, 33, 30].
First is the brightness temperature,
TB
=(s-1S
-o
aU
kB
I
(1+k)
l
10g (1 +
)O
(A.5)
which intuitively generalizes the notion of temperature from the micro-canonical
ensemble of closed photon gas-systems.
Some authors, including Landau in 1946,
appear to have chosen to work only in terms of this temperature [123, 125, 126]
presumably because of its simple relationship to the Planck formula for blackbody
light intensity.
Second, there is the flux temperature
TF
U TN
~
hwN
(N + 1) log(N + 1) - N log N
k
(A.6)
which is useful for direct use in thermodynamic formulations of light-emitters whose
outgoing photon fields are far from equilibrium with their incident fields. Since the
chunks of phase-space that we broke our problem into in Step One are continuously
streaming at the speed of light in configuration-space while our physical apparatus
remains stationary, we are often faced with the question of how much total energy
and entropy has left or entered our device with a given pulse. In this case, we care
about the entropy contributed by each and every photon, not just the last photon
that was added to the pulse. For this reason, the quantity with units of temperature
(energy/entropy) which determines the entropy flux associated with a unbalanced,
unidirectional photon energy flux is defined as
209
TF.
Now let's examine the relationship between
TB
and TF. Since knowing
TB
deter-
mines the thermal occupancy N for a mode with given w, and N can in turn define S
and thereby TF, an explicit analytical relationship between
TF
directly by substitution.
By defining
ykBTB
k'
to be dimensionless
kB TF
XB
B
and XF
and
TB
can be found
inverse temperatures, we have
XF =
(
N
1) log(N + 1) - log(N)
exp XB
expXB-l
expxB -1
1
expB
-log
= exp XB log(exp XB) + (1 - exp
-log
expXB - 1expXBXB)
log (exp XB
-
1
g
/
(A.7)
1)
= exp XBXB + (1 - exp XB) log (exp XB - 1)
In the low-intensity (i.e.
low-TFB, high-w, high-xFB) regime, the log can be
approximated by its Taylor expansion, resulting in the simplified relationship
XF
eXp
XBXB
= XB -
+ (1
exp XB)
-
(XB -
exp
-XB)
(A.8)
exp -XB +1
~XB +l
so that in this limit, the notions of temperature converge (TF
TF=TB
1+ kB<
-4
TB) as
(A.9)
but that TF always remains below TB. This final fact remains true even outside of the
low-occupancy limit and is a consequence of the concavity of entropy to be discussed
shortly. All of these results regarding the relationship between TF and TB appear
where they are useful throughout the literature [29, 33, 30].
One last point to focus on is the importance of the concavity of the entropy
function.
Although it may be proven rigorously that the entropy function of any
210
probability distribution is concave (i.e. the entropy of any mixture of variables is
necessarily more than the sum of the entropies of the variables alone; the choice of
which variable to use itself contains entropy) here we only note that the statement is
true for the family of thermal photon-occupancy distributions. The concavity of our
distribution can be seen visually in A-1: the line connecting any two points on the
curve lies entirely below the function S(N).
As we mentioned before, concavity shows us that the brightness temperature TB
(inverse slope of tangent-line to S(N)) is always greater than the flux temperature
TF (inverse slope of line from the origin thru S(N)).
That is, each additional photon
that stacks up in a given volume of phase-space contributes less to the entropy than
the one before it. Additional power always brings additional entropy flux, but also
always suffers from a law of diminishing returns.
Concavity also helps us recover our intuition about linewidth and entropy. To see
this, consider two similar physical situations. In the first, M-many photons uniformly
occupy all of the states within a frequency range from wo to wo
modes outside this frequency interval are empty.
+ Aw; all photon
In the second, let the M-many
photons instead uniformly occupy all of the states within a frequency range from WO
to wo + 2Aw. To find the relative amounts of entropy contributed by the photons in
each situation, we can simply notice that each mode that matters in the second case
carries exactly half as many photons as each in the first case. Since the entropy of
a half-as-occupied mode is necessarily more than half that of a fully-occupied mode,
the entropy in the second situation is necessarily greater. Running the argument in
reverse, as as we decrease the linewidth (or any dimension of the phase-space for that
matter, including directionality or volume) but maintain the same number of photons,
the entropy-per-energy tends to zero. Consequently, the effective temperatures TF and
TB both diverge and the entropy of such light has no consequence thermodynamically-
the light energy might as well be work.
211
Photonic irreversibility, a topic explored by Planck nearly a century ago [127],
can likewise be seen from this perspective. The motion of electromagnetic fields in
any real situation are also constantly seeking the local maximization of entropy. For
thermal light, this is equivalent to attempting to spread out in phase-space, with
photons preferring less-occupied modes over highly-occupied ones, where they can
contribute a greater amount to the total entropy.
Interestingly, this behavior is the exact opposite of what photons experience in a
laser, where stimulated emission bunches photons into well-occupied modes. To lase,
however, the local photon field must interact with a population of inverted systems,
for which the release of energy is accompanied by an increased entropy. Since we
did not include this non-photonic entropy in our earlier analysis, the situation with
the laser does not break the 2nd Law nor our intuition for irreversible processes in
purely-photonic systems in which each mode is populated by thermal light.
212
Appendix B
Maximum Efficiency at 1 Sun
For the purposes of numerous applications involving biological organisms, the intensity of solar radiation at the Earth's surface is of particular interest. Although the
sun's radiation is well-approximated as coming from a 5700K blackbody, its intensity
diminishes as it spreads out from the surface of the sun (Area=47rR') to a spherical
shell with radius given by the Earth's orbit
Rorbit
(Area=47Rorbit).
The radiation
simultaneously becomes more collimated, thereby preserving the phase-space density
of photons and avoiding the associated entropy increase.
On the other hand, light-induced biological processes are frequently described in
terms that refer only to the longitudinal momentum distribution (i.e. power per unit
area per unit frequency) and ignore the angular distribution of the incoming light.
For these processes, the incident-angle-averaged power spectral density Io(f; T) of
the incoming solar radiation is apparently the relevant quantity. If the quantity of
interest is in fact Io(f; T), then permitting the angular distribution of a given flux
(such as that from the sun) to be more spread out (such as in an LED emitting at
1 sun in a given band) is equivalent to allowing the photons to explore more phase
space and carry more entropy. Thus the Carnot limit for the efficient generation of
such a flux should be looser than simply q 5 Qcarnot
213
=
Tsun/(Tsun - Tambient). Instead,
the limit for the angularly spread-out light should be characterized by the brightness
temperature of that radiation TB(f).
Our goal here is to find the Carnot limit for the efficient generation of "1 sun" of
incoherent radiation, as a function of frequency.
We begin by using the Planck blackbody formula for the frequency-dependence of
the light intensity at the surface of the sun:
Io
Isolar-surface(f) =
1
-_-
.
_
expkBTsun
(B.1)
-1
At the surface of the earth, where the term "1 sun" is defined, we have:
1
= Isolar-surface (f)
1-sun
I
-
(B.2)
,
where G is a geometrical factor that describes the degree of collimation of light from
the sun:
G =
Ror)it
2
(Rsun
(B.3)
~46000.
At each frequency w, we may define the brightness temperature TB(w) of the
incoming radiation as the temperature of blackbody whose angle-averaged power
spectral density matches I1isun(w).
IB (W)
1
=-
1/G
-
e
exp keB
eXp
-1
expkBi =
kBTB
(B.4)
sns
1
(B.5)
kBTsun-
( exp c$run -)
+ 1
(B.6)
(B.7)
=
In [G (expkarun -)
+
(7
From here, we may employ the well-known formula for the Carnot efficiency for
pumping heat from a Tambient ambient up to the brightness temperature TB at each
214
frequency:
77(f) <
'7Camnot(f)
TB(f)
=T
(B.8)
TB(f) - Tambient
25 0
UI
5000
20
0
7U
C
CL
4000
15
a)
0
3000
10 0
F-
-
fli
20001
0
U)
0
10001
300K AmbientJ
500
1000
1500
2000
Wavelength (nm)
2500
0
C
500
1000
1500
2000
Wavelength (nm)
2500
Figure B-1: Plots demonstrating the thermodyn~Tam~%"Ie vance of the distinction
between collimated solar light (red) and angularly-diffuse light of the same power
spectral density (green) at "1 sun" intensity. Left: Brightness temperature as a
function of photon wavelength (plot corresponds to analytical result in (B.4)). Right:
Corresponding Carnot efficiency (computed using (B.8)).
These analytical results are explicitly plotted as a function of photon wavelength
in Figure B-1, where we have assumed Tambient = 300K for the efficiency calculations.
Averaged over the 5700K blackbody spectrum, the maximum efficiency for generation
of angularly-diffuse "1 sun" light is ~ 129%.
215
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216
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