PHONON-ASSISTED TUNNELING IN
SILICON/SILICON-GERMANIUM RESONANT
INTERBAND TUNNEL DIODES
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the
Graduate School of The Ohio State University
By
Ronghua Yu, B.S., M.S.
*****
The Ohio State University
2007
Dissertation Committee:
Approved by
Paul R. Berger, Adviser
Thomas Gramila
John W. Wilkins
Linn Van Woerkom
Adviser
Graduate Program in
Physics
ACKNOWLEDGMENTS
First and foremost, I thank my advisor, Prof. Paul R. Berger. No words can
fully express my sincere thanks to Dr. Berger for giving me the opportunity to learn
semiconductor devices under his guidance. I am grateful for his guidance and encouragement, especially when I met problems in my research or even made mistakes in my experiments, and I will always be. I must also thank our collaborator,
Dr. Phillip Thompson at the Naval Research Laboratory in Washington, D.C. who
does the expert MBE growth which forms the basis of this Si/SiGe RITD research.
I gratefully thank Prof. Thomas Gramila for his help on performing part of the
low-temperature measurements. It was a pleasure working in Prof. Gramila’s lab
measuring some of the diodes at low temperature. Not only did I complete some of
my measurements, but I also learned a great deal about the experimental methods
and tools in Prof. Gramila’s low temperature lab.
I thank my thesis committee members, Prof. John Wilkins, Prof. Linn Van Woerkom, in addition to Prof. Paul Berger and Prof. Thomas Gramila. My thanks also
go to Prof. Paul Evans, the Graduate School Representative, for my oral exam.
I wish to thank past and present group members: Prof. Sean Rommel, Dr. Niu Jin,
Dr. Yifan Xu, Dr. Sung-Yong Chung, Aimee Bross, Woo-Jun Yoon, Si-Young Park,
and Anisha Ramesh for all the help I received in my research. I would like to especially
ii
thank Aimee for her help on the electron beam lithorgraphy patterning of my samples
and Anisha for her help on peak fitting.
My thanks also go to Prof. Gramila’s group members: Sanghun An, Yuko Shiroyanagi, Gokul Gopalakrishnan, and Dongkyun Ko for their help during my lowtemperature measurements in Prof. Gramila’s lab.
I thank James Jones for his help during my use of the ECE cleanroom. I thank
Rita Rokhlin for her help on AFM, James Burns for his help on wire bonding,
Robert Merritt for his help on liquid helium orders, and the Physics Computing
Facility staff, especially J.D. Wear and Tim Randles, for the support I received.
I thank my mother and father for giving birth to me, raising me, and giving me
the chance of education and support throughout all these years. I thank Yeye for all
that she did to help me grow up since birth, and for her wisdom and encouragement
along the way. I thank my brother and his wife for all the help I received, and for
all the happy time spent with me. I am especially indebted to my brother who took
the role of helping my parents, which should have been shared between him and me,
while I was studying away from home.
I sincerely thank my mother-in-law and father-in-law. The support I received from
them was immeasurable. I would not be where I am now without their, and their
only daughter – my wife’s, help.
Last but not the least, I sincerely thank my wife. I could not have been what I am
now without her support in these past few years. In addition to the immense support
that I have received, I am fortunate to have the opportunity to learn a lot from an
intelligent lady who understands life, and more, a lot better than me. Finally, I thank
my newborn son for bringing so much joy to my life.
iii
ABSTRACT
Electron tunneling spectroscopy was applied toward Si/SiGe resonant interband
tunnel diodes (RITDs) to ascertain the phonon spectra of the composite tunneling
barrier which consists of Si1−x Gex and Si layering. Two issues were identified: the
error in the second-order derivative which would reduce the measurement sensitivity, and the contact resistance which would affect phonon energy measurement. A
harmonic detection system was used to measure the first-order derivative with an accuracy of 10−4 , followed by first-order numerical differentiation to obtain high quality
second-order derivative. Contacts to silicon using nickel silicide (NiSi) and delta
doping technique were studied, and preliminary results indicate that the contact resistance was very low. With typical Si/SiGe RITD configurations, the phonon spectra
were found to be dominated by the phonon peaks of bulk Si. Of the Si/SiGe RITD
structures that were measured, only those with a relatively thick Si1−x Gex layer in the
composite tunneling barrier and a high Ge content x = 0.6 produced phonon spectra
in which the transverse acoustic (TA) phonon of the Si0.4 Ge0.6 were observable, but
the longitudinal acoustic (LA), longitudinal optical (LO) and transverse optical (TO)
phonons of Si1−x Gex were not observed. The energy of the TA phonon of Si0.4 Ge0.6 in
Si/SiGe RITDs grown on Si substrates was determined to be about 4±1 meV higher
than that measured using Esaki tunnel diodes. This increase is attributed to the
compressive strain in the Si0.4 Ge0.6 layer in the Si/SiGe RITDs.
iv
VITA
March 23, 1973 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Born - Shuangliu, Sichuan, P.R. China
1994 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.S. Physics, University of Electronic
Science and Technology of China,
Chengdu, Sichuan, P.R. China
1997 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.S. Physics, University of Science and
Technology of China, Hefei, Anhui,
P.R. China
1999 - present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graduate Fellowship,
Graduate Research Associate,
Graduate Teaching Associate,
The Ohio State University, Columbus,
OH, U.S.A.
PUBLICATIONS
Research Publications
Sung-Yong Chung, Niu Jin, Anthony T. Rice, Paul R. Berger, Ronghua Yu, Z.-Q.
Fang, and Phillip E. Thompson, Growth temperature and dopant species effects on
deep levels in Si grown by low temperature molecular beam epitaxy, J. Appl. Phys.
93, 9104 (2003).
Niu Jin, Sung-Yong Chung, Anthony T. Rice, Paul R. Berger, Ronghua Yu, Phillip E.
Thompson, and Roger Lake, 151 kA/cm2 peak current densities in Si/SiGe resonant
interband tunneling diodes for high-power mixed-signal applications, Appl. Phys.
Lett. 83, 3308 (2003).
Sung-Yong Chung, Niu Jin, Paul R. Berger, Ronghua Yu, Phillip E. Thompson, Roger
Lake, Sean L. Rommel, and Santosh K. Kurinec, Three-terminal Si-based negative
v
differential resistance circuit element with adjustable peak-to-valley current ratios
using a monolithic vertical integration, Appl. Phys. Lett. 84, 2688 (2004).
Sung-Yong Chung, Niu Jin, Ryan E. Pavlovicz, Paul R. Berger, Ronghua Yu, Zhaoqiang Fang, and Phillip E. Thompson, Annealing of defect density and excess currents
in Si-based tunnel diodes grown by low-temperature molecular-beam epitaxy, J. Appl.
Phys. 96, 747 (2004).
Niu Jin, Sung-Yong Chung, Roux M. Heyns, Paul R. Berger, Ronghua Yu, Phillip E.
Thompson, and Sean L. Rommel, Tri-state logic using vertically integrated Si-SiGe
resonant interband tunneling diodes with double NDR, IEEE Electron Device Lett.
25, 646 (2004).
N. Jin, S.Y. Chung, R. Yu, P.R. Berger, and P.E. Thompson, Improved vertically
stacked Si/SiGe resonant interband tunnel diode pair with small peak voltage shift
and unequal peak currents, Electronics Lett. 40, 1548 (2004).
Niu Jin, Sung-Yong Chung, Roux M. Heyns, Paul R. Berger, Ronghua Yu, Phillip E.
Thompson, and Sean L. Rommel, Phosphorus diffusion in Si-based resonant interband
tunneling diodes and tri-state logic using vertically stacked diodes, Materials Science
in Semiconductor Processing 8, 411 (2005).
N. Jin, S.-Y. Chung, R. Yu, P.R. Berger, and P.E. Thompson, Temperature dependent
DC/RF performance of Si/SiGe resonant interband tunnelling diodes, Electronics
Lett. 41, 559 (2005).
Niu Jin, Ronghua Yu, Sung-Yong Chung, Paul R. Berger, Phillip E. Thompson, and
Patrick Fay, High sensitivity Si-based backward diodes for zero-biased square-law
detection and the effect of post-growth annealing on performance, IEEE Electron
Device Lett. 26, 575 (2005).
Niu Jin, Sung-Yong Chung, Ronghua Yu, Sandro J. Di Giacomo, Paul R. Berger, and
Phillip E. Thompson, RF performance and modeling of Si/SiGe resonant interband
tunneling diodes, IEEE Trans. Electron Devices 52, 2129 (2005).
Sung-Yong Chung, Ronghua Yu, Niu Jin, Si-Young Park, Paul R. Berger, and Phillip
E. Thompson, Si/SiGe resonant interband tunnel diode with fr0 20.2 GHz and peak
current density 218 kA/cm2 for K-band mixed-signal applications, IEEE Electron
Device Lett. 27, 364 (2006).
vi
S.-Y. Park, S.-Y. Chung, P.R. Berger, R. Yu, and P.E. Thompson, Low sidewall damage plasma etching using ICP-RIE with HBr chemistry of Si/SiGe resonant interband
tunnel diodes, Electronics Lett. 42, 719 (2006).
Sung-Yong Chung, Si-Young Park, Jeffrey W. Daulton, Ronghua Yu, Paul R. Berger,
and Phillip E. Thompson, Integration of Si/SiGe HBT and Si-based RITD demonstrating controllable negative differential resistance for wireless applications, Solid
State Electronics 50, 973 (2006).
Niu Jin, Sung-Yong Chung, Ronghua Yu, Roux M. Heyns, Paul R. Berger, and Phillip
E. Thompson, The effect of spacer thicknesses on Si-based resonant interband tunneling diode performance and their application to low-power tunneling diode SRAM
circuits, IEEE Trans. Electron Devices 53, 2243 (2006).
Sung-Yong Chung, Niu Jin, Ryan E. Pavlovicz, Ronghua Yu, Paul R. Berger, and
Phillip E. Thompson, Analysis of the Voltage Swing for Logic and Memory Applications in Si/SiGe Resonant Interband Tunnel Diodes Grown by Molecular Beam
Epitaxy, IEEE Trans. on Nanotechnology 6, 158 (2007).
S.-Y. Park, R. Yu, S.-Y. Chung, P.R. Berger, P.E. Thompson, and P. Fay, Sensitivity
of Si-Based Zero-Bias Backward Diodes for Microwave Detection, Electronics Lett.
43, 53 (2007).
FIELDS OF STUDY
Major Field: Physics
vii
TABLE OF CONTENTS
Page
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
Chapters:
1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2.
Materials Properties and Physical Processes . . . . . . . . . . . . . . . .
3
2.1
2.2
2.3
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3
8
13
14
15
16
18
Si/SiGe Resonant Interband Tunnel Diodes . . . . . . . . . . . . . . . .
23
3.1
3.2
23
30
33
35
2.4
2.5
3.
Silicon and Germanium . . . . . . . . . . . . .
Si1−x Gex Alloys and Si/SiGe Heterojunctions .
Phonons . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Phonons in Si and Ge . . . . . . . . . .
2.3.2 Phonons in Si1−x Gex . . . . . . . . . . .
Electron Tunneling: Direct and Indirect . . . .
Phonon-Assisted Tunneling in Si and Si1−x Gex
Esaki Tunnel Diode . . . . . . . . . . .
Resonant Tunnel Diodes . . . . . . . . .
3.2.1 III-V Resonant Tunnel Diodes . .
3.2.2 Si/SiGe Resonant Tunnel Diodes
viii
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3.3
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Electron Tunneling Spectroscopy . . . . . . . . . . . . . . . . . . . . . .
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4.1
4.2
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72
3.4
4.
4.3
5.
Electron Tunneling Spectroscopy . . . . . .
Derivatives by Numerical Differentiation . .
4.2.1 First-Order Derivative . . . . . . . .
4.2.2 Second-Order Derivative . . . . . . .
Derivatives by Harmonics Detection Method
4.3.1 Resolution . . . . . . . . . . . . . .
4.3.2 Low Temperature . . . . . . . . . . .
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Second-order Derivative from Numerical Differentiation with Optional Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.1
5.2
5.3
5.4
5.5
5.6
6.
Resonant Interband Tunnel Diodes . . . . . . . . . . . . .
3.3.1 III-V Resonant Interband Tunnel Diodes . . . . . .
3.3.2 Si/SiGe Resonant Interband Tunnel Diodes . . . .
Fabrication of Si/SiGe Resonant Interband Tunnel Diodes
3.4.1 Molecular Beam Epitaxy . . . . . . . . . . . . . . .
3.4.2 Rapid Thermal Anneal . . . . . . . . . . . . . . . .
3.4.3 Device Fabrication . . . . . . . . . . . . . . . . . .
3.4.4 Packaging and Wire Bonding . . . . . . . . . . . .
Obtaining Numerical Derivatives . . . . . . . . . . . . . . . . . . . 73
5.1.1 Numerical Derivatives from Interpolating Polynomials . . . 73
5.1.2 Numerical Derivatives from Fitting Polynomials . . . . . . . 77
Errors in Numerical Derivatives . . . . . . . . . . . . . . . . . . . . 79
Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3.1 Least-Squares Fit with A Polynomial . . . . . . . . . . . . . 80
5.3.2 Smoothing by Local Least-Squares Fit and Savitzky-Golay
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Obtaining Measurement with the Least Error . . . . . . . . . . . . 88
Numerical Derivatives of Experimental Data: Example . . . . . . . 90
5.5.1 Step Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.5.2 Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.5.3 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . . . 105
Second-order Derivative from Harmonics Detection System . . . . . . . . 107
6.1
6.2
Harmonics Detection System . . . . . . . . . . . . . . . . . . . . . 107
Second-Order Derivative from Harmonics Detection System . . . . 109
ix
6.2.1
6.2.2
6.2.3
6.2.4
6.2.5
7.
7.3
7.4
113
116
117
118
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Metal-Semiconductor Contacts . . . . . . . . . . . . . . . . . . . . 120
7.2.1 Schottky Contacts and Ohmic Contacts . . . . . . . . . . . 120
7.2.2 Characterization of Contact Resistance: TLM and Cross Kelvin
Test Structures . . . . . . . . . . . . . . . . . . . . . . . . . 123
Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Preliminary Results Using Large Contacts . . . . . . . . . . . . . . 131
7.4.1 Attempt to Anneal Composite Metal Stacks for Silicidation 131
7.4.2 TLM Results with Ni as Inductively Coupled Plasma Reactive Ion Etch Mask . . . . . . . . . . . . . . . . . . . . . . . 133
Phonons in Si/SiGe Resonant Interband Tunnel Diodes . . . . . . . . . . 136
8.1
8.2
8.3
8.4
8.5
9.
111
Ohmic Contacts to Silicon using Delta Doping and Nickel Silicide . . . . 120
7.1
7.2
8.
d2 I/dV 2 : Converted from Measured d2 V /dI 2 vs Numerical
Derivative of Measured dV /dI . . . . . . . . . . . . . . . .
d2 I/dV 2 : First-Order Numerical Derivative of Measured FirstOrder Derivative vs Second-Order Numerical Derivative of
Measured I-V (after Uniformization in V ) . . . . . . . . . .
d2 I/dV 2 : Numerical Differentiation With Respect To V vs
Numerical Differentiation With Respect To I . . . . . . . .
dI/dV : Measured vs Calculated . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . .
Experimental Design . . . . . . . . . . . . . . . . . . . . .
Sample Fabrication . . . . . . . . . . . . . . . . . . . . . .
Measurement . . . . . . . . . . . . . . . . . . . . . . . . .
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5.1 Commercial Si Esaki Diodes: 1N2927 and 1N2930A
8.5.2 0/3/2 Structures with Zero to 44% Ge Content . .
8.5.3 -1/4/n Structures with 60% Ge Content . . . . . .
8.5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . .
8.5.5 Summary . . . . . . . . . . . . . . . . . . . . . . .
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136
137
139
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150
161
162
Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . 165
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
x
LIST OF TABLES
Table
2.1
Page
Summary of phonon energies in Si, Ge, and SiGe alloys as a function
of Si fraction measured in electron tunneling experiments by Logan et
al. [28]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
7.1
Results of inductively coupled plasma reactive ion etch (ICP-RIE) of
Si with Ni as etch mask. . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.1
Fit results of the major peak around 20 mV. a: +6.2 mV shift; b:
voltage shift unknown. . . . . . . . . . . . . . . . . . . . . . . . . . . 155
xi
LIST OF FIGURES
Figure
Page
2.1
Diamond structure. From [1]. . . . . . . . . . . . . . . . . . . . . . .
3
2.2
(a) The constant-energy surfaces near the conduction band minima in
the first Brillouin zone of Si. (c) Calculated band structure of Si. (b)
The constant-energy surfaces near the conduction band minima in Ge.
(d) Calculated band structure of Ge. Γ = (0,0,0); X (∆) is along the
<100> directions; L (Λ) is along the <111> directions; K (Σ) is along
the <110> directions. From [1, 3]. . . . . . . . . . . . . . . . . . . .
6
Electronic transition between the conduction band minimum and the
valence band maximum in: (a) a direct semiconductor; (b) an indirect
semiconductor in which the change in crystal momentum is provided
by a phonon. From [1]. . . . . . . . . . . . . . . . . . . . . . . . . .
8
Band gap of straind Si1−x Gex on Si(100) substrate and of unstrained
Si1−x Gex . From [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
Schematic energy band lineups of: (a) compressively strained Si1−x Gex
on relaxed Si; (b) tensially strained Si on relaxed Si1−x Gex . . . . . .
11
Calculated fundamental gaps in eV of (100) strained Si1−x Gex alloys
(active material) grown pseudomorphically on unstrained Si1−y Gey alloys (substrate material). From [9]. . . . . . . . . . . . . . . . . . . .
11
Calculated (a) conduction band offset EC (x) − EC (y) and (b) valence
band offset EV (x) − EV (y) in eV of (100) strained pseudomorphically
grown Si1−x Gex on unstrained Si1−y Gey alloys. From [9]. . . . . . . .
12
Measured and calculated phonon dispersion curves of Si. 1 THz =
1012 Hz = 4.14 meV. From [2]. . . . . . . . . . . . . . . . . . . . . .
14
2.3
2.4
2.5
2.6
2.7
2.8
xii
2.9
Measured phonon dispersion relations in Ge at 80 K. From [17]. . . .
15
2.10 A square potential barrier with a particle of energy E lower than the
barrier height, incident from the left. On the right side of the barrier,
there is non-zero transmitted wave according to quantum mechanics.
16
2.11 Curves of d2 I/dV 2 against bias for junctions of (a) Ge, (b) Ge-Si alloy
containing 46% Si and (c) Si at 0.8 K. From [28]. . . . . . . . . . . .
21
2.12 The phonon energies plotted against the atom percentage of Si in the
Ge-Si alloy. The points are experimentally measured values, and the
lines are joins of the points. From [28]. . . . . . . . . . . . . . . . . .
22
3.1
3.2
3.3
3.4
3.5
3.6
3.7
Band diagram of an Esaki tunnel diode, i.e. a degenerately doped pn
junction in which the Fermi level enters the valence band on the p side
and the conduction band on the n side, at zero bias. From [30]. . . .
24
The simplified energy band diagram of an Esaki diode at various important biases along the I-V characteristics. From [30]. . . . . . . . .
25
I-V characteristics of an typical Esaki tunnel diode, which include a
negative differential resistance (NDR) region. From [30]. . . . . . . .
27
The three components of the current through an Esaki tunnel diode:
the (band to band) tunneling current, the (thermal) diffusion current,
and the excess current. From [30]. . . . . . . . . . . . . . . . . . . .
28
Double square barrier potential profile with an incident particle from
the left with energy E lower than the barrier height. In the right-most
region, there is (only) transmitted wave traveling to the right. . . . .
30
Band diagrams resonant interband tunnel diodes and one example realization: (a) Type 3 heterostructure with spacer layers; (b) Type 3
heterostructure without spacer layers; (c) δp-i-δn structure; (d) δn-iδp-i-δn structure; (e) double quantum-well heterostructure. From [31].
39
(a) Basic Si RITD structure; (b) Si/SiGe RITD structure with two thin
Si1−x Gex layers surrounding the B delta-doping plane. After [57, 65].
42
xiii
3.8
The band diagram of a Si RITD using the basic structure with a
4 nm intrinsic Si tunneling barrier, calculated by solving the coupled effective-mass Schrödinger equation and the Poisson equation.
From [57]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
The I-V characteristics of a 75 µm diameter Si/SiGe RITD with a 1/4/4 structure, showing the highest PVCR of 4.02 of samples grown
by the MBE system at Naval Research Laboratory, obtained by inductively coupled reactive ion etching with HBr. From [74]. . . . . . . .
47
3.10 I-V characteristics of the Si/SiRITDs with the 1/4/n structure with a
Ge content of 40%, where n = 4, 6, 8, 10, 12, corresponding to a total
tunneling barrier thicknesses of 8, 10, 12, 14, and 16 nm. From [78].
47
3.11 Schematic illustration of the essential parts of a MBE growth system. The depicted region is located in an ultrahigh vacuum chamber.
Three zones where the basic processes of MBE take place are indicated.
From [97, 92]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
3.12 Schematic illustration of a semiconductor substrate and an epitaxial
film containing a delta-doping layer. A schematic lattice is shown
on the left with impurity atoms confined to a single atomic plane.
From [98]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
3.13 The cross section of a minimum Si/SiGe RITD fabricated by selfaligned process. The mesa top metal contact and the substrate back
side metal contact are indicated by the black areas. . . . . . . . . . .
55
3.14 The cross section of a Si/SiGe RITD with two bonding pads. The gray
area is SiO2 ; the black area is contact metal. . . . . . . . . . . . . .
56
3.9
4.1
Left: A hypothetical piece-wise linear (continuous but not smooth) I-V
curves (grey color) and a smoothed version in which the transition is
smooth (solid color) (the two curves are very close and are not easily
distinguishable from each other). Middle: The first order derivative
dI/dV , from which the change in the slope in the I-V curve is much
more obvious. Right: The second order derivative d2 I/dV 2 , in which
the changes in slope of the I-V appears at peaks and therefore can
be studies quantitatively in terms of peak location, width and height.
After [113]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiv
62
4.2
4.3
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
Block diagram of a first and second harmonics measurement system
using the source-voltage-measure-current (SVMC) method. . . . . .
68
First and second harmonics detection circuit using the source-currentmeasure-voltage (SCMV) method. . . . . . . . . . . . . . . . . . . .
69
I-V characteristics of a representative Si/SiGe RITD at 4.2 K at small
positive biases. Two inflections are visible. . . . . . . . . . . . . . .
91
The first order numerical derivative dI/dV using 3-point numerical
derivative formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
First-order numerical derivatives dI/dB from different numerical derivative formulas: 2-point forward, vs 3-point, vs 5-point. . . . . . . . .
92
Second-order numerical derivative by first-order numerical differentiation of first-order numerical derivative, both using the 3-point numerical derivative formula. It is observed that the error or noise gets
bigger/worse in the second-order numerical derivative. . . . . . . . .
93
The second-order numerical derivatives from different methods: 3point second-order numerical differentiation vs twice 3-point first-order
numerical differentiation. . . . . . . . . . . . . . . . . . . . . . . . .
94
The second-order numerical derivatives using different number of points:
5-point vs 3-point. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
The dependence of the error in the (second-order) derivative on the
step size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
For the data considered, averaging does not appreciably reduce the
error or noise in the second-order numerical derivative using 3-point
second-order numerical derivative formula. . . . . . . . . . . . . . . .
97
For the data considered, averaging does not appreciably reduce the
error or noise in the second-order numerical derivative obtained by
twice 3-point first-order numerical differentiation. . . . . . . . . . . .
97
5.10 Effect of 3-point moving averaging smoothing on the second-order
derivative obtained by twice 3-point first-order numerical differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
xv
5.11 Moving averaging smoothing: 3-point, vs 7-point, vs 11-point. . . . . 101
5.12 Smoothing by local polynomial fit (Savitzky-Golay method) using 11
points using polynomials of: first-degree, second-degree, and fourthdegree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.13 Savitzky-Golay smoothing with a second-order degree polynomial using: 11 points, 15 points, and 19 points. . . . . . . . . . . . . . . . . 103
5.14 Second-order numerical differentiation by the second-order derivative
of the fitted local second-degree polynomial (differentiation by SavitzkyGolay method) using: 7 points, 11 points, and 15 points. . . . . . . . 104
6.1
Comparison between the second harmonic signal from the harmonics
detection system used and the numerical derivative of the first harmonic signal using 3-point numerical derivative formula. Some difference in shape is observed. It is believed that the second-order derivative
from the (first-order) numerical derivative of the first harmonic signal
is more reliable and trustworthy. . . . . . . . . . . . . . . . . . . . . 111
6.2
Numerical experiment of converting V-I data to I-V data for numerical
differentiation with respect to voltage. . . . . . . . . . . . . . . . . . 115
6.3
Second-order derivative with respect to V from conversion of numerical differentiation with respect to current (of V-I data) compared to
numerical differentiation with respect to voltage after uniformization
in V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.4
First-order derivative dI/dV : measured (first harmonic) vs calculation
(numerical differentiation). . . . . . . . . . . . . . . . . . . . . . . . 118
7.1
The transfer length method (TLM) structure, and the plot of the resistance between two neighboring contacts as a function of the spacing
between the contacts. W − Z = δ should be as small as possible.
In this plot, the electrically long contact (L LT ) approximation is
used to obtain the x intercept. The general expression for the x intercept is −2LT coth(L/LT ). For electrically long contacts, L LT ,
coth(L/LT ) ≈ 1, −2LT coth(L/LT ) ≈ −2LT . From [123]. . . . . . . 124
xvi
7.2
Cross Kelvin contact resistance test structure. (a) Cross section through
section A-A; (b) top view. From [123]. . . . . . . . . . . . . . . . . . 126
7.3
Structures for ohmic contact experiments: (a) p+ -p-n structure; (b)
n+ -n-p structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.4
An optical microscope image of 150Å Ni/800Å Mo/1000Å Au dots
after rapid thermal anneal at 500◦ C for 30 seconds in forming gas
(5% H2 /95% N2 ). The top Au layer was melted during the anneal,
as determined by the surface profile obtained with a Dektak surface
profiler. The diameter of the dots was 100 µm. . . . . . . . . . . . . 132
7.5
Representative resistance vs separation data from the TLM structures
of the n+ -n-p sample, and the least-squares fit with a straight line.
The equations of the fitted straight lines are also displayed. About
half of the fitted straight lines have a negative y intercept. While a
negative y intercept means invalid TLM data, it also indicates low
contact resistance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.1
Si/SiGe RITDs with a 0/3/2 structure with varying Ge content: x = 0,
10%, 20%, 30%, 40%, and 50%. . . . . . . . . . . . . . . . . . . . . . 138
8.2
Si/SiGe RITDs with the -1/4/n structure with 60% Ge, where n =
4,6,8,10,11,12 in nm for the the thickness of SiGe layer. . . . . . . . 139
8.3
SEM images of a 10 µm diameter Si/SiGe RITD, top view: (a) overview;
(b) close-up of the circular mesa with the top contact and the bottom
contact semi-enclosing the bottom of the mesa. . . . . . . . . . . . . 140
8.4
(a) Packaged diodes; (b) Package mounted on the dipper stick.
8.5
The I-V characteristics at 300 K, at 4.2 K, and the d2 I/dV 2 from
first-order numerical differentiation of the measured conductance (or
resistance)by first harmonic detection for two commercial Si Esaki tunnel diodes: 1N2927 (left column) and 1N2930A (right column). . . . 144
8.6
The shape of the phonon spectrum of 1N2927 is compare to that of
1N2930A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.7
Phonon spectra of 5-nm spacer Si RITDs of various sizes.
xvii
. . . 141
. . . . . . 145
8.8
0/3/2 structure with 10% (top row), 20% (middle row), and 40% (bottom row) Ge content: I-V characteristics at 300 K and 4.2 K, and
second-order numerical derivative d2 I/dV 2 obtained from measured
4.2 K I-V data using savgol(15,15,2,2). . . . . . . . . . . . . . . . 147
8.9
I-V characteristics of the 18 µm RITDs of the -1/4/n 60% Ge structures
at 300 K: (a) -1/4/4; (b) -1/4/6; (c) -1/4/8; (d) -1/4/10; (e) -1/4/11;
(f) -1/4/12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.10 d2 I/dV 2 of the -1/4/n structures with 60% Ge content obtained by
numerical differentiation of the resistance (data) measured by detection
of the first harmonic. (a) -1/4/4; (b) -1/4/6; (c) -1/4/8; (d) -1/4/10.
152
8.11 Fit of the observed ∼20 mV peak with two peaks of Gaussian shape,
-1/4/4 structure with 60% Ge: (a) 10 µm; (b) 18 µm; (c) 50 µm; (d)
75 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
8.12 Fit of the observed ∼20 mV peak with two peaks of Gaussian shape,
-1/4/6 structure with 60% Ge: (a) 10 µm; (b) 18 µm; (c) 50 µm; (d)
75 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.13 Energy of the Si0.4 Ge0.6 TA phonon after estimated contact resistance
correction vs. current. In the case of the -1/4/4 structure, the dependence of the estimated Si0.4 Ge0.6 TA phonon on current indicates that
the contact resistance correction is incomplete, especially at higher
currents. This was most likely due to non-linear I-V characteristics of
the contacts, manifested especially at higher currents. In both -1/4/4
and -1/4/6 structures, the extrapolation of the trends to zero current
yields the same improved estimate of the Si0.4 Ge0.6 TA phonon energy
of about 15.8 meV, with an estimated total measurement error of about
±1 meV. Note that data from the -1/4/4 10 µm RITD is not included
in this plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
xviii
CHAPTER 1
INTRODUCTION
The objective of this thesis is to study the phonon-assisted tunneling processes in
Si/SiGe-based resonant interband tunnel diodes (RITDs). This thesis is organized as
follows.
Chapter 2 briefly reviews the properties of the materials and the fundamental
physical process involved in the Si/SiGe RITDs. This knowledge forms the basis for
the understanding the devices and their operations.
Chapter 3 is a review of various types of tunnel diodes that have been developed
prior to the development of the Si/SiGe RITDs, This provides the background for
the development of Si/SiGe RITDs, and explains the significance of the new type of
tunnel diodes.
Chapter 4 introduces the experimental technique, i.e. electron tunneling spectroscopy, that is used to obtained the phonon information.
For the purpose of
measuring the phonon energies, the second-order derivative of the current-voltage
characteristic is needed. Two methods of obtaining the needed derivatives, namely
numerical differentiation and direct measurement by harmonic detection, are briefly
reviewed, and their advantages and disadvantages are compared.
1
In the next two chapters, Chapters 5 and 6, the two methods are studied in detail,
using experimental data. Great effort has been made to obtain the derivatives with
the least possible error or noise. In Chapter 5, the numerical differentiation method,
and smoothing techniques which naturally arise in the application of numerical differentiation, are examined extensively. It is confirmed that numerical differentiation
in general results in a high level of error. Even though the error in the first-order numerical derivative, when properly processed, can be low and usable, the much higher
level of error in the second-order numerical derivative severely limits the usefulness
of the second-order numerical derivative in practice.
In Chapter 6, the first- and second-order derivative data measured by a harmonic
detection system are carefully examined. For this specific harmonic detection system used, it was concluded that a hybrid approach, namely, second-order derivative
by first-order numerical differentiation of the directly measured first-order derivative produced a higher quality second-order derivative data. Indeed, by measuring a
number of commercial tunnel diodes and comparing to the published data, it was confirmed that this hybrid approach produces a very high quality second-order derivative.
Therefore, this approach was adopted in this thesis.
The issue of contact resistance was identified. Chapter 7 presents some partial
results of my effort in an attempt to attack this problem. Although the results are
still preliminary, they are promising.
The experimentally collected data from the Si/SiGe RITDs are presented and
analyzed in Chapter 8. Findings and issues are discussed.
Chapter 9 summarizes the work of this thesis, and discusses possible further development of the topics studied in this thesis.
2
CHAPTER 2
MATERIALS PROPERTIES AND PHYSICAL
PROCESSES
2.1
Silicon and Germanium
Silicon (Si) crystallizes in the diamond structure, which consists of two interpenetrating face-centered cubic (fcc) lattices with one fcc lattice displaced along the body
diagonal of the conventional cubic cell by one quarter length of the body diagonal [1].
Fig. 2.1 shows the diamond lattice in the conventional cubic cell. The lattice constant
a of the diamond structure is the length of the side of the conventional cubic cell.
For Si, the lattice constant is 5.431 Å at 300 K [2]. The first Brillouin zone of a fcc
Figure 2.1: Diamond structure. From [1].
3
lattice is a truncated octahedron, as shown in Fig. 2.2(a).
Si is a semiconductor, having a finite energy band gap separating a valence band
which, at 0 K, is completely filled and a conduction band which is completely empty.
At 300 K, the band gap of Si is 1.12 eV. The electronic property of a semiconductor
typically involves only the energy states near the conduction band minima and those
near the valence band maxima. This is because, upon excitation, the electrons with
the highest energies (in the valence band) will be excited first, while the states with the
lowest energies (in the conduction band) will be occupied first. Near the conduction
band minima and the valence band maxima, the energy E vs wave vector k (or
momentum p = ~k) is in a quadratic form, similar to the kinetic energy of a moving
particle with a certain mass, i.e.
E=
~2 k 2
,
2m∗
(2.1)
where m∗ is the effective mass of the electrons in the conduction band or the holes in
the valence band. That is to say, the curvature of the E-k relationship near the band
extrema is related to the effective mass of the charge carriers in the corresponding
band.
The electronic energy band structure of Si was calculated by Chelikowsky et al.
and is shown in Fig. 2.2(c) [3]. In the conduction band, the conduction band minima
are six-fold degenerate, and are located about 80% toward the zone boundary along
the <100> directions. The constant-energy surfaces near the conduction band minima
are shown in Fig. 2.2(a) [1]. They are ellipsoids with axes of revolution along the
<100> directions, and are elongated along the <100> directions, corresponding to a
heavier longitudinal (electron) effective mass of mL = 0.98m and a lighter transverse
effective mass of mT = 0.19m, where m is the free electron mass.
4
In the valence band, the two-fold degenerate maxima occur at k = 0, i.e. center of
the first Brillouin zone (the Γ point). The two subbands, degenerate at the valence
band maxima at k = 0, have, near the valence band maxima, constant-energy surfaces that are spherically symmetrical but with different curvatures, corresponding to
effective masses of about 0.49m and 0.16m, for the heavy holes and light holes respectively. A third band, not shown in the figure, is slightly (about 0.044 eV) below the
valance band maxima at k = 0 due to spin-orbit coupling, and is called the split-off
band.
Germanium (Ge) crystal also has the diamond structure, as Si. Ge has a slightly
larger lattice constant of 5.658 Å, and correspondingly a smaller band gap of 0.664 V
at 300 K. The first Brillouin zone of Ge is the same as that of Si. However, Ge
has four-fold degenerate conduction band minima located at the zone boundaries in
the <111> directions, see Fig. 2.2(b) and Fig. (d). There are eight half-ellipsoids
located at the hexagonal faces of the first Brillouin zone. Because of the periodic
nature of the Brillouin zones, the opposite faces on a Brillouin zone represent the
same electronic states, and therefore there are four full ellipsoids in one primitive
cell which is equivalent to the first Brillouin zone but in a different range of crystal
momentum k. The constant-energy surfaces are ellipsoids with axes of revolution
along and elongated along the <111> directions, corresponding to effective masses
of mL = 1.64m and mT = 0.082m. The valence band maxima occur at k = 0 and
are two-fold degenerate, with spherical constant energy surfaces near the maxima,
corresponding to a heavy hole effective mass of 0.28m and a light hole effective mass
of 0.04m.
5
(a)
(b)
(c)
(d)
Figure 2.2: (a) The constant-energy surfaces near the conduction band minima in the
first Brillouin zone of Si. (c) Calculated band structure of Si. (b) The constant-energy
surfaces near the conduction band minima in Ge. (d) Calculated band structure of
Ge. Γ = (0,0,0); X (∆) is along the <100> directions; L (Λ) is along the <111>
directions; K (Σ) is along the <110> directions. From [1, 3].
6
In both Si and Ge, the conduction band minima and the valence band maxima
do not occur at the same k. Because of this, the transition of an electron between
the bottom of the conduction band and the top of the valence band involves a change
in crystal momentum, due to the crystal momentum conservation requirement. In
this case, an electronic transition involving only a photon cannot take place, because
energy conservation and momentum conservation cannot be satisfied simultaneously,
due to the small momentum carried by photons of energies in the range of the band
gap. In a perfect crystal in which there are no crystal defects or impurities, phonons
or lattice vibrations are the only means to provide the momentum required by the
momentum change. Because of the requirement of the the participation of phonons
in the photon absorption or emission process in this type of semiconductors, this
type of semiconductors are called indirect semiconductors, see Fig. 2.3. Both Si and
Ge are indirect semiconductors. Note, electronic transitions across the band gaps
are impossible with the participation of only phonons either, because phonons with
the momentum required by momentum conservation have much lower energies than
needed by energy conservation requirement. In direct semiconductors, such as gallium
arsenide (GaAs), the conduction band minimum and the valence band maximum
occur at the same k, usually at k = 0. Therefore, electronic transitions across the
band gap with photon absorption or emission does not require the involvement of
phonons.
7
Figure 2.3: Electronic transition between the conduction band minimum and the
valence band maximum in: (a) a direct semiconductor; (b) an indirect semiconductor
in which the change in crystal momentum is provided by a phonon. From [1].
2.2
Si1−x Gex Alloys and Si/SiGe Heterojunctions
Si and Ge have the same crystal structure of diamond structure. The lattice
constant of Ge is only about 4.2% larger with respect to Si at 300 K. Si and Ge are
completely miscible, i.e. homogenous Si1−x Gex alloys can be obtained for the full range
of x from 0 to 1. Si1−x Gex alloys are solid substitutional crystals with the Diamond
structure, in which the lattice sites are randomly occupied by Si or Ge atoms. The
equilibrium lattice constant is roughly a linear interpolation between that of Si and
of Ge. The addition of Ge to Si to form SiGe alloys results in the reduction of the
(indirect) band gap as compared to that of the Si. The band gap of Si1−x Gex over
the full range of x has been measured by Braunstein et al. by optical absorption [4].
The data is included in Fig. 2.4.
8
@ -0.075 [
0
1
1
20
1
1
,
1
60
40
o/o
,
80
1
1
~
100
Ge
Fig. 9. Hydrostaticcontributiontotheindirectbandgap
of coherently
strained bulk Ge,Si, --x alloys on Si(OO1) substrates.
UNSTRAINED
Fig. 1l), and iii) bandgapof coher
alloys for growth on (O01)-Gex,2S
offset curve for Fig. 11). These t
us with the necessary permutation
ing discussions of bandgaps and b
ious modulation doping experimen
I v . BAND ALIGNMENTS
FOR CO
(Ge,Si)/SiHETEROSTRUCTU
SUBSTRA
In order to determine the band a
(Ge, Si)/.Si heterointerface we req
strained layer constituents and ii) a
either A E, or A E,, including bot
and coherency strain. The strained b
timated as outlined in Section 11
AE, for a pseudomorphic Ge/Si h
0 60
0
20
40
60
80
100
cently obtained by Van de Walle
MOLE % Ge
(001) oriented interfaces A E, for
Fig. 10. Fundamental(lowestenergy)indirectbandgap
of coherently
for
threecases,correspondingto:i)gr
strained Ge,Sil --r alloys on Si(OO1) substrates.
strates, with in-plane lattice const
Si and strained Ge; ii) grow
Figure 2.4: Band gap of straind Si1−x Gex on Si(100) substrate andcubic
of unstrained
considerations we have estimated the fundamental (indi- = 5.66 , thus cubic Ge and str
Si1−x Gex . From [5].
rect) bandgap of (Ge, Si) alloys forthree cases which are onGe0,38Si0.62substrates, all =
of interest: i) bandgap of coherently strained Ge,Sil --x and Ge strained. Their results indi
alloys on Si(OO1) substrates (lower curve in Fig. 11; T = lence band edge lies above theSi
90 K, dataarefrom
[l l ] ) , ii)bandgap of coherently the above cases. Values of A E,
strained
Si
on
(OO1)-Ge,~Sil
(center curve Si
of 1−xFig.
uppercan
curve (solid circle
Because of the same crystal structure, a substrates
thin pseudomorphic
Gex12alloy
A
,
,
~
be epitaxially grown on Si (and Ge also), and vice versa, subject to critical thickness limitations. When a pseudomorphic Si1−x Gex layer is grown on relaxed Si, the
Si1−x Gex layer is compressively strained in both directions in the plane of growth, i.e.
biaxial compression. When a pseudomorphic Si layer is grown on a relaxed Si1−x Gex
(virtual substrate), the Si layer is under biaxial tensile strain. Knowledge of the band
gap of the strained Si1−x Gex layer on relaxed Si and that of the strained Si on relaxed Si1−x Gex and the corresponding band alignments of Si/SiGe heterojunctions are
essential to the understanding of the applications for Si/SiGe heterostructures [5, 6].
The change to the band structure of SiGe alloys due to a biaxial strain can be,
in one approximation, regarded as the sum of that due to a hydrostatic strain (equal
dilation or compression in all three directions) and that due to a uniaxial strain [7, 8].
For example, biaxial tensile stress corresponds to a hydrostatic tension with a uniaxial
9
compression in the direction perpendicular to the biaxial tension. The hydrostatic
stress causes a uniform shift of the conduction band and the valence band, and hence
the band gap, while the uniaxial stress breaks degeneracy that might be present in
the conduction band edge and valence band edge. Fig. 2.4 shows the calculated lowest
(fundamental) band gap of a coherently strained Si1−x Gex alloy on Si(100) at 300 K,
together with that of unstrained Si1−x Gex . The hashed region is due to uncertainties
in the parameters used in the calculation. Similarly, the fundamental band gaps of
strained Si on relaxed Si1−x Gex was calculated and plotted in [5].
Fig. 2.5 shows the schematic band lineups of two typical Si/SiGe heterojunctions:
(a) compressively strained Si1−x Gex on relaxed Si, and (b) tensially strained Si on
relaxed Si1−x Gex [5, 9, 10, 11]. It is important to note that in the (compressively)
strained Si1−x Gex on relaxed Si structure, the conduction band offset is very small,
and the majority of the band offset is located in the valence band [11]. Therefore,
this structure is not suitable for electron-type devices which require band offsets (for
quantum wells and/or quantum barriers) in the conduction band, but is suitable for
hole-type devices which utilizes the band offsets in the valence band. On the other
hand, if an electron-type device is desired, a structure such as (tensially) strained Si
on relaxed Si1−x Gex virtual substrate should be used.
A more general case of a strained Si1−x Gex layer grown pseudomorphically on a
relaxed Si1−y Gey substrate was considered, and the band gap, conduction band offset
and valence band offset were calculated by Rieger et al. [9, 10]. The calculated band
gap, conduction band offset and valence band offset are shown in Fig. 2.6 and Fig. 2.7,
respectively.
10
EC
Relaxed Si
EV
Compressively
Strained Si1−x Gex Relaxed Si1−x Gex
(a)
(b)
Tensially
Strained Si
EC
EV
Figure 2.5: Schematic energy band lineups of: (a) compressively strained Si1−x Gex
on relaxed Si; (b) tensially strained Si on relaxed Si1−x Gex .
Figure 2.6: Calculated fundamental gaps in eV of (100) strained Si1−x Gex alloys
(active material) grown pseudomorphically on unstrained Si1−y Gey alloys (substrate
material). From [9].
11
Figure 2.7: Calculated (a) conduction band offset EC (x)−EC (y) and (b) valence band
offset EV (x) − EV (y) in eV of (100) strained pseudomorphically grown Si1−x Gex on
unstrained Si1−y Gey alloys. From [9].
12
2.3
Phonons
Phonons are normal mode vibrations of a crystal. A small-amplitude vibration of
the atoms on the lattice sites of a crystal can, as a first approximation, be treated as
a linear superposition of normal mode vibrations [1].
In addition to the actual motion of the atoms in space, phonons are also characterized by their energy E vs wave vector k relationships. The E vs k relationships are
also called the dispersion curves of the phonons. The dispersion curves are divided
into two branches, called the acoustic branch and the optical branch. In the acoustic
branches, E approaches zero as k goes to zero with a certain slope, in the form of
E = ck, characteristic of sound waves with sound speed c, for small ks. The optical
branches are above the acoustic branches, and the energy E does not go to zero as k
goes to zero. At any value of crystal momentum k, the motion of the atoms at the
lattice sites can be along or perpendicular to the direction of k, giving rise to one
longitudinal mode and two transverse modes in a three-dimensional crystal.
Different crystals have different normal mode vibrations, and therefore different
phonons and their dispersion relations. The phonon modes and the dispersion curves
are characteristic of a material. The phonon dispersion curves are usually measured
by neutron scattering, but electromagnetic (X-ray and visible light) scattering is also
used [1]. The participation of phonons in certain processes may also be measured by
other methods, but these methods typically only yield limited information, because,
for example, only a limited range or certain phonon modes can be excited. They may
not be used to measure the full phonon dispersion curve over the entire first Brillouin zone. The measurement of phonon energies by electron tunneling spectroscopy
belongs to this case.
13
2.3.1
Phonons in Si and Ge
The phonon dispersion curves of Si and Ge were measured using inelastic neutron
scattering by Brockhouse et al. [12, 13, 14], Dolling [15], and Nilsson et al. [16, 17].
A compilation of the experimental data and theoretical calculations of the Si phonon
dispersion curves, published in 1982, is shown in Fig. 2.8. Fig. 2.9 shows the phonon
dispersion curves of Ge measured at 80 K. More recent theoretical calculations of
the phonon dispersion curves in Si and Ge include those by Giannozzi et al. [18] and
Aouissi et al. [19].
Figure 2.8: Measured and calculated phonon dispersion curves of Si. 1 THz = 1012 Hz
= 4.14 meV. From [2].
14
Figure 2.9: Measured phonon dispersion relations in Ge at 80 K. From [17].
2.3.2
Phonons in Si1−x Gex
No complete phonon dispersion data for Si1−x Gex , from either theoretical calculation or inelastic neutron scattering measurement, is available yet. Only certain modes
of the phonons at certain specific wave vectors k, and therefore energies E, are studied
experimentally using various other methods. So far, the lattice vibrations in Si1−x Gex
alloys are mainly studied by Raman scattering, in which mainly the Si-Si, Si-Ge, and
Ge-Ge optical modes are measured [20]. Lattice vibrations in Si1−x Gex alloys are also
studied by infrared absorption [21, 22]. Since Si1−x Gex alloys are indirect semiconductors, the phonons participating in the phonon-assisted tunneling processes can be
measured in electron tunneling experiments. Published data of phonons in Si1−x Gex
15
(and Si and Ge) measured in electron tunneling experiments are discussed in a later
section.
2.4
Electron Tunneling: Direct and Indirect
Tunneling is a quantum mechanical phenomenon, which has no correspondence
in classical physics. In quantum mechanics, a particle can tunnel through a potential
barrier of height higher than the energy of the incident particle. Consider the square
potential barrier shown in Fig. 2.10. When a particle of energy E is incident from the
E
c
eik1 x
Bek2 x
-
Deik1 x
-
Ae−ik1 x
Ce−k2 x
Figure 2.10: A square potential barrier with a particle of energy E lower than the
barrier height, incident from the left. On the right side of the barrier, there is non-zero
transmitted wave according to quantum mechanics.
.
left on a barrier of height V higher than E, a non-zero transmitted wave will appear
to the right of the barrier traveling to the right, according to quantum mechanics [23].
This is not allowed in classical physics, because of the existence of a forbidden region
in which E < V . When the barrier is not too high (compared to the energy of the
incident particle), and the thickness of the potential barrier is thin enough, the transmission coefficient, which describes the probability that the particle is transmitted,
16
or tunnels from one side of the barrier to the other side, is appreciable for the tunneling phenomenon to be observed experimentally. For a barrier of a general profile,
when the barrier is high enough and/or thick enough, the Wentzel-Kramers-Brillouin
(WKB) approximation can be used to obtain the transmission coefficient T ,
Z b p2m(V − E) dx ,
T ≈ exp −2
~
a
(2.2)
where a and b the “turning points” that define the classically forbidden region (E = V
when x = a or b, and E < V for a < x < b). The condition for the WKB approxiR b √2m(V −E)
mation to apply is a
dx 1, which means high (in energy) and/or thick
~
(in thickness) barrier. Therefore, when WKB approximation applies, the transmission coefficient is very small, much smaller than 1. For a constant potential barrier,
the transmission probability decreases exponentially with, apart from a factor of 2,
increasing barrier thickness.
In a semiconductor, an electron can tunnel through the band gap from the valence band to an empty state in the conduction band, or from the conduction band
to an empty state in the valence band, provided an empty state exists. Tunneling
in a crystalline semiconductor requires the conservation of both energy and crystal
momentum. In direct semiconductors, the bottom of the conduction band occurs at
the same momentum as the top of the valence band in the first Brillouin zone. Therefore, an electron tunneling between the bottom of the conduction band and the top
of the valence band does not need a change in momentum, and a tunneling process is
possible without requiring additional mechanisms. However, in indirect semiconductors, such as Si or Ge, the bottom of the conduction band and the top of the valence
band do not occur at the same k in the first Brillouin zone. Consequently, energy
and momentum conservation requires the involvement of an additional mechanism in
17
the electron tunneling process. This additional mechanism provides the needed momentum change while maintaining energy conservation in the process. Phonons are
the only means of providing such a mechanism in a perfect crystal. In an imperfect
crystal, scattering from lattice defects, lattice imperfections and impurity atoms can
provide additional mechanisms for momentum conservation.
2.5
Phonon-Assisted Tunneling in Si and Si1−x Gex
Tunneling of electrons in semiconductor structures was discovered by Esaki using
a degenerately doped pn junction [24]. Due to the degenerate doping, the depletion
layer is extremely thin, which makes the probability of tunneling of electrons between
the conduction band on the n side and the valence band on the p side through
the barrier, i.e. the depletion region, large enough for experimental observation. At
small biases, a current can flow through the junction because of electron tunneling.
A degenerately doped pn junction is called a tunnel diode, or Esaki tunnel diode.
A unique feature of the tunnel diode is that the tunneling current decreases and
eventually disappears as one continues to increase the forward bias, resulting in a
decrease in current with increasing voltage, i.e. a negative differential resistance region
exists. Conventional diffusion current will also start to flow as the forward bias is
further increased, generally beyond the tunneling current. Therefore, an N shaped
current-voltage (I-V) characteristic is obtained, having three voltages satisfied at
one current value. This special property of a tunnel diode has many novel circuit
applications.
18
The tunneling phenomenon can also be used to study the phonons in indirect
semiconductors through the phonon-assisted tunneling process. Because of the involvement of phonons in the electron tunneling process in indirect semiconductors,
tunneling experiments are one means to study the phonons in the material in which
the phonon-assisted tunneling process occurs. This can be studied by measuring the
I-V characteristic of a degenerately doped pn junction at cryogenic temperatures.
At low temperatures, the smearing of the phonon spectra by thermal broadening is
reduces, therefore the phonon signals can be more easily identified.
This was first discovered by Holonyak et al. who observed inflections in the IV characteristics of degenerately doped Si and Ge pn junction diodes measured at
4.2 K [25]. The inflections correspond to the onset of phonon participation in the
electron tunneling process resulting in a sudden increase in tunneling probability and
therefore tunneling current. Specifically, the inflections correspond to the excitation
of phonons by the applied bias. In the absence of voltage drop across the series
resistance, the product of the applied biasing voltage V and the electron charge e
is equal to the energy E of the phonons excited. Therefore, the biases at which the
inflections occur give a direct measurement of the energies of the phonons involved.
Even though the inflections in the low-temperature I-V curves are observable, they
are relatively weak. Later, it was realized that by taking the first derivative of the I-V
(by harmonic detection), i.e. dI/dV , and plotting it as a function of V , the ability to
detect weak inflections in the I-V was improved. The differentiation technique was
taken one step further, by Chynoweth et al., to obtain the second order derivative as
a function of V , resulting in a further increase in resolving power and sensitivity for
19
a more accurate measurement of the phonon energies [26]. The phonon spectrum of
Ge was also measured by Payne et al. [27].
The method was applied to tunnel diodes made of bulk SiGe alloys by Logan et
al. to obtain the phonon energies as a function of composition [28]. The secondorder derivative d2 I/dV 2 as a function of applied bias V is plotted in Fig. 2.11 for
Ge, Si, and SiGe alloy with 46% Si. Peaks due to four phonon modes are readily
observable. They are, from lower energy to higher energy, the transverse acoustic
(TA), longitudinal acoustic (LA), longitudinal optical (LO), and transverse optical
(TO) phonons. It can be observed form the plots that the high sensitivity of the
differentiation technique makes the detection of the weak LA phonon in Si possible.
The phonon energies as a function of Si concentration as measured in [28] are
listed in Table 2.1 and plotted in Fig. 2.12.
Atomic
Fraction of
Silicon
0
0 [12]
0.07
0.22
0.46
0.70
0.79
1.00
1.00 [14]
Lattice
Constant
(Å)
5.640
5.600
5.544
5.491
5.472
TA
7.6 ± 0.2
8.0±0.2
9.0
10.8
12.3
14.5
16.0
18.7±0.2
17.9±0.6
Phonon Energy (meV)
LA
LO
27.5 ± 0.2
31.1 ± 0.2
26.6±1.2
30.5±1.2
30.7
34.3
34.3
not observable
36.3
43.0
36.8
44.0
37.5
45.0
47±2
56.2±1.0
43.7±2.1
53.2±2.1
TO
36.3 ± 0.2
34.6±1.2
39.3
48.5
49.6
50.3
50.6
59.1±0.2
58.5±1/2
Table 2.1: Summary of phonon energies in Si, Ge, and SiGe alloys as a function of Si
fraction measured in electron tunneling experiments by Logan et al. [28].
20
Figure 2.11: Curves of d2 I/dV 2 against bias for junctions of (a) Ge, (b) Ge-Si alloy
containing 46% Si and (c) Si at 0.8 K. From [28].
21
Figure 2.12: The phonon energies plotted against the atom percentage of Si in the
Ge-Si alloy. The points are experimentally measured values, and the lines are joins
of the points. From [28].
A new type of tunneling diodes, the Si/SiGe-based resonant interband tunnel
diodes (RITDs), which utilizes the resonant tunneling of electrons between the quantum well in the conduction band and that in the valence band, was developed by
Rommel et al. [29]. Its performance and large scale integrate circuit production capability are superior to the Esaki diodes. The subject of this thesis is to study the
phonon-assisted tunneling process in Si/SiGe RITDs.
22
CHAPTER 3
SI/SIGE RESONANT INTERBAND TUNNEL DIODES
3.1
Esaki Tunnel Diode
An Esaki tunnel diode is simply a degenerately doped pn junction diode [24, 30,
31]. Because of the degenerate doping on the both the p side and the n side, the
Fermi level enters the valence band of the semiconductor on the p side, and the
conduction band of the semiconductor on the n side, see Fig. 3.1. As a result, there
are unoccupied electronic states in the valence band on the p side, and free electrons
in the conduction band on the n side. Because of the degenerate doping, the depletion
width of the junction is extremely narrow, making the transmission coefficient of the
depletion region barrier large enough for electron tunneling effect to be observable.
At absolute zero temperature and zero bias, no tunneling of electrons from either
side to the other side occurs because of the energy conservation requirement (in addition to momentum conservation; the relatively small energy of the phonons involved
in phonon-assisted band to band electron tunneling in indirect semiconductors is ignored for the current discussion) and the fact that there are no electrons above the
Fermi level and no empty states below the Fermi level in thermal equilibrium. At
finite temperature, there will be a certain number of electrons above the Fermi level,
and correspondingly the same number of unoccupied states below the Fermi level,
23
Figure 3.1: Band diagram of an Esaki tunnel diode, i.e. a degenerately doped pn
junction in which the Fermi level enters the valence band on the p side and the
conduction band on the n side, at zero bias. From [30].
according to Fermi-Dirac statistics. The electrons in the conduction band on the n
side can tunnel through the depletion region barrier to the valence band on the p
side, and vice versa, while maintaining conservation of energy: (1) for an electron on
one side, there may exist an unoccupied state of the same energy on the other side
into which the electron can tunnel; (2) an electron on one side can tunnel through
the barrier into the other side, first occupy an originally unoccupied real state or a
virtual state of the same energy, but eventually occupy an originally unoccupied state
of a (slightly) different energy, in the “finite temperature thermal equilibrium” sense.
At zero bias, the electrons in the conduction band on the n side tunnel through the
barrier to the valence band on the p side at the same rate as they tunnel from the
p side to the n side, resulting in zero total tunneling current through the junction.
When a small forward or reverse bias is applied across the junction, the Fermi level on
one side moves up or down with respect to that on the other side, making tunneling
24
of electrons from one side to the other side dominate over that in the other direction,
resulting in nonzero total tunneling current.
The basic operation of an Esaki diode can be understood in terms of the band
diagram at 0 K, see Fig. 3.2. When a reverse bias is applied, the Fermi level on the n
Figure 3.2: The simplified energy band diagram of an Esaki diode at various important
biases along the I-V characteristics. From [30].
side moves down with respect to the Fermi level on the p side, so that the electrons
just below the Fermi level on the p side have corresponding unoccupied states (above
the Fermi level on the n side) of the same energy available for tunneling. Appreciable
tunneling of such electrons in the valence band on the p side to the conduction
band on the n side does occur, because of the extremely thin barrier, i.e. depletion
region, in the Esaki diode. Once an electron tunnels into the n side and fills an
unoccupied state above the Fermi level, it is subsequently removed from the n side
by the external bias, making the state unoccupied again and ready to accept the
25
next electron tunneled from the p side. This process continues, and a DC current is
maintained. The magnitude of the DC tunneling current depends on the availability of
electrons for tunneling and the availability of the unoccupied states to accept tunneled
electrons, subject to the energy conservation requirement. In the band diagram, this
is indicated by the overlapping of the occupied electron states below the Fermi level,
i.e. availability of electrons, on one side and the unoccupied states above the Fermi
level on the other side. As the reverse bias increases, the overlapping increases, and
so does the (reverse) current.
When a small forward bias is applied, the (forward) current is determined by
the overlapping of the occupied states below the Fermi level on the n side and the
unoccupied states above the Fermi level on the p side. The overlapping, and hence
the current, increases with increasing bias until the Fermi level on the n side crosses
the top of the valence band on the p side, or the bottom of the conduction band
on the n side crosses the Fermi level on the p side, after which the overlapping and
hence the current decrease with increasing bias. When the bias is increased to the
point so that the bottom of the conduction band on the n side crosses the top of the
valence band on the p side, overlapping becomes zero, and consequently no electron
tunneling can occur, and the tunneling current drops to zero. For even higher biases,
the tunneling current stays at zero.
The tunneling current as described above is not the only component of the total
current flowing through an Esaki diode. The thermal diffusion current in an ordinary
moderately doped pn junction diode is also present in an Esaki diode. In typical Esaki
diodes, the diffusion current becomes significant after the tunneling current drops to
a very small value or even zero, as the forward bias increases from zero. Even though
26
the tunneling current, as described above, and the diffusion current account for the
overall shape of the observed Esaki diode I-V characteristics, there is a non-ignorable
difference. This difference is called the excess current. A complete understanding of
the excess current is still lacking, but it has been proposed that the majority of the
excess current is due to electron tunneling through the defect-related energy states
within the band gap [32].
Fig. 3.3 shows a sketch of the I-V characteristics of a typical Esaki diode. The
three components of the total current through an Esaki diode are indicated in Fig. 3.4.
A negative differential resistance (NDR) region is present in the forward bias region
Figure 3.3: I-V characteristics of an typical Esaki tunnel diode, which include a
negative differential resistance (NDR) region. From [30].
of the I-V characteristics. The peak and valley in the I-V curve are indicated, together
with the corresponding voltages (VP and VV ) and currents (IP and IV ). The peakto-valley current ratio (PVCR) is the basic figure-of-merit characterizing the DC
27
Figure 3.4: The three components of the current through an Esaki tunnel diode: the
(band to band) tunneling current, the (thermal) diffusion current, and the excess
current. From [30].
performance of a tunnel diode,
PVCR =
IP
.
IV
(3.1)
When surface leakage current is ignored, and good uniformity of the junction is assumed, the current through the junction is proportional to the cross sectional area of
the junction. Within this assumption, another (size-independent) figure-of-merit is
JP , the peak current density (PCD),
JP =
IP
,
A
(3.2)
where A is the cross sectional area of the junction. In addition to the NDR region,
the Esaki diode’s I-V characteristics are linear around zero bias with a finite conductance. This is another important difference between an Esaki diode and an ordinary
moderately doped pn junction diode.
28
Theoretical calculation of the tunneling current starts from the calculation of the
current due to tunneling of electrons from the conduction band on the n side to the
valence band on the p side IC→V and that from the valence band on the p side to the
conduction band on the n side IV →C . IC→V and IV →C are given by
Z
IC→V = A
IV →C = A
EV
EC
Z EV
EC
fC (E)ρC (E) TC→V 1 − fV (E) ρV (E) dE,
(3.3)
fV (E)ρV (E) TV →C 1 − fC (E) ρC (E) dE,
(3.4)
where A is a constant, TC→V and TV →C are the transmission coefficients for electron
tunneling in the two directions, fC (E) and fV (E) are Fermi-Dirac distribution functions, ρC (E) and ρV (E) are the densities of states, in the conduction band and the
valence band, respectively. The total tunneling current through the junction is given
by
It = IC→V − IV →C = A
Z
EV
EC
fC (E) − fV (E) T ρC (E)ρV (E) dE,
(3.5)
where the transmission coefficients in both directions are assumed to be equal, i.e.
TC→V = TV →C = T . At zero bias, the currents due to tunneling of electrons in the two
opposite directions cancel each other, and the total tunneling current is zero. Under
a non-zero bias, the currents due to electron tunneling in the two directions will not
cancel each other, resulting in a nonzero net current. In a first-order approximation,
Eq. (3.5) can be integrated to obtain
V
V
exp 1 −
,
It = IP
VP
VP
(3.6)
where IP and VP are the peak current and peak voltage, respectively [33]. IP and VP
can be calculated in terms of the doping levels and material properties [30].
29
Esaki tunnel diodes have been realized in Si, Ge, Si1−x Gex alloys, and III-V semiconductors including GaAs, InSb, InAs and GaSb [30, 31, 34].
The unique negative differential resistance of an Esaki tunnel diode enables novel
circuit designs which include circuits of reduced device count or increased functionality
per device, as compared to circuits consisting of only conventional devices. Quantum
mechanical tunneling is in general a fast process [35], making tunnel diodes high
speed devices suitable for many high speed applications. However, despite the many
advantages of Esaki diodes, they only find limited use today in niche applications.
One of the reasons is that most Esaki tunnel diodes are discrete devices made with the
ball alloy method or pulse bond method, and therefore incompatible with mainstream
planar integrated circuit technology.
3.2
Resonant Tunnel Diodes
Consider the double square barrier potential profile in Fig. 3.5. The Schrödinger
E
c
eik1 x
Bek2 x
Deik1 x
-
-
Ae−ik1 x
F ek2 x
Heik1 x
-
Ce−k2 x
Ee−ik1 x
Ge−k2 x
Figure 3.5: Double square barrier potential profile with an incident particle from the
left with energy E lower than the barrier height. In the right-most region, there is
(only) transmitted wave traveling to the right.
equation for this potential can be solved, by matching the boundary conditions at
the boundaries between neighboring regions. The transmission coefficient can then be
30
calculated, as a function of the energy of the incident particle. It turns out that, for
certain energies of the incident particle, the transmission coefficient is equal to one,
i.e. the double potential barrier is totally transparent for particle transmission. This
phenomenon is called resonant tunneling. It is interesting that while the transmission
coefficient of a potential barrier is always lower than one (and decreases with increasing barrier height and width), two barriers in a row can be completely transparent
for certain energies of the incident particle.
For a double barrier potential of a general shape, the WKB approximation may
be used to obtain an approximate solution to the Schrödinger equation. Peaks in the
transmission coefficient are also obtained [23].
It turns out that the condition for the peaks in the transmission coefficient is the
same as that for the bound state energies in the potential well [23].
The states
corresponding to these energies are called quasi -bound states (of the potential well)
of the double barrier potential profile. The wave functions have very large amplitudes
within the well region between the two potential barriers. They are called quasi-bound
states, because they are not true bound states, in that the energy of the particle is
higher than the potential energy at infinity, i.e. the particle is, in principle, allowed
to move away from the potential well to infinity. It turns out that a quasi-bound
state does have a finite lifetime, and the particle will eventually “leak out” from the
potential well. In the wave packet picture, when the wave packet is incident on a
double-barrier potential profile, the transmitted wave packet will experience a time
delay as it passes the double barriers [23]. This time delay ∆t is the lifetime of the
quasi-bound state, which is finite. A finite lifetime means that the particle or wave
packet will eventually “leak out” from the potential well. Consequently, the energy of
31
the quasi-bound state is not well defined, with an uncertainty given by Heisenberg’s
Uncertainty Principle,
∆E ≥
~
.
∆t
(3.7)
Resonant tunneling occurs in potential profiles with more than two barriers (or
more than one well) also [36].
The multiple-barrier potential profile needed for resonant tunneling may be realized using semiconductors by variation of doping in the same semiconductor material,
but the most successful approach is to use heterojunctions consisting of different types
of semiconductor materials. In the case of semiconductor heterojunctions, the conduction band offsets or valence band offsets, when appropriate, create the potential (barrier and well) profile in the conduction band or valence band, for resonant tunneling
of electrons or holes, respectively. The band offsets in a heterostructure also depend
on the strain in the semiconductors, if the semiconductors are not lattice-matched
(but still pseudomorphic to be high quality crystals for operation). This adds another parameter that can be used in the designing of heterostructure semiconductor
resonant tunneling structures. In semiconductor resonant tunneling heterostructures,
the semiconductors are typically undoped, i.e. intrinsic.
Negative resistance can result from the resonant tunneling effect in semiconductor
structures. This is due to the local current maxima in the I-V characteristics when the
voltage applied across the device is such that the Fermi level at the emitter electrode
is at an energy for a resonant tunneling peak in the transmission coefficient, i.e.
coinciding with a quasi -bound state in the potential well. Because there is more than
one peak in the transmission coefficient as a function of energy, resonant tunneling
occurs at multiple energy values, corresponding to multiple quasi-bound states in the
32
potential well(s) in the multiple-barrier potential profile. As a result, multiple peaks,
valleys, and NDRs are present in the I-V characteristics.
The simplest semiconductor device utilizing the resonant tunneling effect is a
two-terminal resonant tunnel diode (RTD) [31]. Noticing that resonant tunneling in
semiconductor structures occurs either in the conduction band (resonant tunneling
of electrons) or in the valence band (resonant tunneling of holes), but not between
the conduction band and the valence band, an RTD is an intraband resonant tunnel
diode, and is a unipolar device. RTD structures are often symmetric, resulting in
I-V characteristics that are symmetric with respect to zero bias. The multiple NDR
regions and symmetric I-V characteristics of RTDs are different from those of Esaki
tunnel diodes.
Resonant tunneling in semiconductor structures with multiple potential barriers
was first proposed by Tsu and Esaki in 1973 [36]. Fabrication of such structures was
made possible initially by the then-new molecular beam epitaxy (MBE) technique,
and now with ultra-high vacuum chemical vapor deposition (UHVCVD) also. RTDs
were first proposed and realized in the III-V compound semiconductor materials system.
3.2.1
III-V Resonant Tunnel Diodes
The almost lattice-matched AlGaAs/GaAs materials system was used to create
the first semiconductor resonant tunneling structures by Chang et al. in 1974 [37].
Two Al0.7 Ga0.3 As/GaAs/Al0.7 Ga0.3 As double barrier structures of different configurations were grown on n-type GaAs substrates, 80 Å/50 Å/80 Å and 40 Å/40 Å/40 Å,
and were capped with a thick n-type GaAs layer as the top electrode. This structure
33
resulted in two Al0.7 Ga0.3 As barriers surrounding a central GaAs quantum well in the
conduction band, with a barrier height of about 0.4 eV. The I-V characteristics were
almost symmetric about the zero bias, reflecting the symmetric structure of the device.
The observed asymmetry should be due to the asymmetry of the actual structure.
Current peaks and negative differential resistance were observed at 77 K. Voltages corresponding to local maxima in current were identified, which were distributed almost
symmetrically about the zero bias point. These voltages reasonably agree with an
estimate of the quasi-bound states using a simplified square-well model. The higher
voltages associated with the narrower well structure (40 Å/40 Å/40 Å) as compared
to the corresponding voltages from the wider well structure (80 Å/50 Å/80 Å) are
consistent with the dependence of the energy (separation) of the (quasi-)bound state
on the width of the potential well. Negative differential resistance was not observed
at room temperature, which should be due to thermal smearing effect, the relatively
low barrier height, and limited crystal purity/quality from the early days of epitaxial growth. The observed strong dependence of the current on barrier thickness and
weak dependence on temperature are characteristic of quantum mechanical tunneling
effect. The dependence of the observed resonant energy on the width of the potential
well is indictive of the resonant tunneling effect. In the AlGaAs/GaAs system, a
room temperature PVCR of 7.2 with a PCD of 104 A/cm2 was realized later with a
InGaAs quantum well for carrier injection by Brugger et al. in 1991 [38].
Due to the large selection of III-V heterojunctions, most of the RTD research
to date has been based on III-V materials. RTDs with the highest PVCR and
highest PCD to date were realized in III-V materials system. Room temperature
34
PVCR of up to 50 has been realized by a 9 monolayer (ML) of bottom AlAs barrier and 11 ML of top AlAs barrier surrounding a well which consisted of 3 ML of
lattice matched In0.53 Ga0.47 As, 6 ML of strained InAs, and 3 ML of lattice matched
In0.53 Ga0.47 As, with In0.53 Ga0.47 As buffer and spacer layers, grown a n+ InP(100)
substrate using MBE [39]. Very high PCDs were also obtained with the same InPbased In0.53 Ga0.47 As/AlAs double-barrier resonant tunneling structures. A PCD of
460 kA/cm2 , concurrently with PVCR over 4, was obtained with two 4-ML AlAs
barriers surrounding a 16-ML In0.53 Ga0.47 As well [40].
Even though RTDs based on III-V compound semiconductors have achieved high
PVCRs and PCDs, their application is limited by the fact that the III-V materials
are incompatible with the current mainstream Si complementary metal oxide semiconductor (CMOS) technology. Also, the high speed performance of high mobility
heterostructure transistors using III-Vmitigate the need to fully explore the circuit
level benefits gained from the addition of NDR devices in Si-based circuits.
3.2.2
Si/SiGe Resonant Tunnel Diodes
Resonant tunneling of electrons in conduction band and holes in valence band can
also be realized in SiGe materials system using (strained) Si/SiGe heterostructures.
However, the performance of Si/SiGe RTDs so far are still limited.
In the SiGe material system, resonant tunneling of holes was observed first, primarily because the valence band discontinuity dominates the overall band offset in
the Si/SiGe heterojunctions fabricated on Si substrates.
Resonant tunneling of holes in the SiGe materials system was first observed by Liu
et al. [41]. Double-barrier structures with strained Si1−x Gex quantum wells formed
35
between unstrained Si barriers (in the conduction band) grown by MBE. Negative
differential resistance has been obtained with a PVCR of 1.8 at 77 K and 2.2 at 4.2 K
from a sample with a 3.3 nm Si0.79 Ge0.21 well between 6.0 nm Si barriers.
Resonant tunneling of holes in Si/SiGe heterostructures was also reported by
Gennser et al. in [42]. The double barrier heterostructure consisted of an unstrained
Si0.5 Ge0.5 layer between two tensially strained Si layers, resulting in a SiGe well surrounded by two Si barriers in the conduction band. The unstrained Si0.5 Ge0.5 layers
were achieved by the use of a relaxed buffer layer consisting of a thick 7000 Å p+
Si0.4 Ge0.6 layer on a Si(100) substrate. Note that a higher concentration of Ge was
used in the buffer layer to compensate for the fact that SiGe layers of height Ge
content do not completely relax even at thicknesses many times beyond the critical thickness [42]. PVCRs of 2 at 4 K and 1.5 at 77 K were obtained with a 50 Å
unstrained Si0.5 Ge0.5 layer between two 35 Å tensially strained Si layers.
Even though resonant tunneling of holes was observed first, no room temperature
NDR from resonant tunneling of holes in Si/SiGe heterostructures has ever been
observed.
Resonant tunneling of electrons in SiGe materials system was first reported by
Ismail et al. [43] using the Si/SiGe/Si/SiGe/Si double barrier structure grown by
ultrahigh vacuum chemical vapor deposition (UHVCVD). The structure was 150 Å
Si / 75 Å Si0.7 Ge0.3 / 50 Å Si / 75 Å Si0.7 Ge0.3 / 150 Å Si, in which the Si0.7 Ge0.3
barrier layers were unstrained while the Si layers were under tensile strain.
In this
structure, the conduction band of SiGe lies above that of Si, resulting in a Si well
surrounded by two SiGe barrier in the conduction band. The unstrained Si0.7 Ge0.3
36
layers were realized by a ramped superlattice buffer layer on a Si substrate. PVCR
of 1.2 was obtained at 300 K, and 1.5 at 77 K.
By using compressively strained Si1−x Gex layer on a relaxed Si1−y Gey virtual
substrate with x > y, the conduction band discontinuity or the barrier height in
the SiGe/Si/SiGe double barrier heterostructure is expect to increase, as compared
to the unstrained SiGe barrier layers with x = y. The increased barrier height is
expected to enhance the resonant tunneling effect and hence the dc performance. A
room temperature PVCR of 2.9 with a PCD of 4.3 kA/cm2 was obtained with a
2 nm strained i-Si0.4 Ge0.6 / 3 nm strained i-Si / 2 nm strained i-Si0.4 Ge0.6 structure
grown on a Si0.8 Ge0.2 virtual substrate by UHVCVD by See et al. [44]. The Si0.8 Ge0.2
virtual substrate consisted of a 3 µm n-Si1−y Gey strain relaxed buffer graded from
y = 0 to 0.2 grown on a n-type Si(100) substrate using UHVCVD. It was observed
that the current in the RTDs made from this structure appeared to be independent
of lateral junction area, which was attributed to the thermal effect due to internal
heating resulting from current flow [45]. As a result, the PCD scales with the inverse
of the lateral junction area. A high PCD of 282 kA/cm2 was obtained with a PCVD
of 2.43 from a 5 × 5µm2 device. The PVCR of 2.9 and PCD of 282 kA/cm2 (not
simultaneously) represent the highest values obtained in SiGe RTDs to date.
Resonant tunneling structures with multiple (> 2) barriers have also been realized
in both III-V and SiGe materials systems [46, 47, 48].
37
3.3
3.3.1
Resonant Interband Tunnel Diodes
III-V Resonant Interband Tunnel Diodes
Resonant interband tunnel diodes (RITDs) combine the features of both intraband
resonant tunnel diodes (RTDs) and interband Esaki tunnel diodes [31]. Resonant
interband tunnel diodes were first proposed by Sweeny and Xu [49]. Three band
structures for RITDs were proposed by Sweeny and Xu in [49], corresponding to the
band diagrams shown in Fig. 3.6 (e), (essentially) (a), and (c), respectively.
The band diagram shown in Fig. 3.6 (a) utilizes a type-III band alignment heterostructure, in which the bottom of the conduction band of one semiconductor lies
below the top of the valence band of the other semiconductor. This type of heterostructure can be realized in the antimonide-based materials system. Specifically,
the bottom of the conduction band of InAs is below the top of the valance band of
GaSb. As a result, interband tunneling of electrons from the conduction band of InAs
to the valence band of GaSb can occur. In the n-InAs / AlSb (barrier) / GaSb (well)
/ AlSb (barrier) / n-InAs structure, resonant tunneling of electrons from the n+ -InAs
layer on one side to the n+ -InAs layer on the other side can take place, corresponding
to the quasi-bound states or quantized energy levels in the p-type potential well in
the valence band of the p-GaSb layer. In this structure, an i-AlSb layer is inserted between the n+ -InAs layer and the p-GaSb layer on both side of the p-GaSb well layer.
These two i-AlSb layers provide additional potential barriers for electron tunneling,
and likely decreases the tunneling probability when resonant tunneling condition is
not satisfied. The effect is to reduce the valley current, and hence increase the PVCR.
Using this structure, PVCRs as high as 20 and 88 were obtained at room temperature
and 77 K respectively by Soderstrom et al. [50]. Using the same n-InAs / i-AlSb /
38
Figure 3.6: Band diagrams resonant interband tunnel diodes and one example realization: (a) Type 3 heterostructure with spacer layers; (b) Type 3 heterostructure
without spacer layers; (c) δp-i-δn structure; (d) δn-i-δp-i-δn structure; (e) double
quantum-well heterostructure. From [31].
39
p-GaSb heterostructure, an RITD structure of a type opposite to that in Fig. 3.6 (a)
can be constructed. This is the p-GaSb / AlSb (barrier) / InAs (well) / AlSb (barrier)
/ p-GaSb structure, in which the quasi-bound states in the n-type potential well in
the InAs layer are involved in the resonant tunneling of holes from the p-GaSb region
on one side to the p-GaSb region on the other side. With this structure, room temperature PVCRs of 13 and 8.3 were obtained by Luo et al. [51] and Chow et al. [52],
respectively. With InAs as the potential well, it is possible to use very wide wells in
RITD structures. For instance, an InAs well as wide as 110 nm showed a PVCR as
high as 44 at 77 K, as reported by Beresford et al. [53]. Noticing these structures are
symmetric, the I-V characteristics of these structures are also symmetric.
The band diagram in Fig. 3.6 (b) is a variation of (a), in which the thickness of the
intrinsic AlSb barrier layers is reduced to zero. Using a n-InAs/(nominally undoped)
GaSb/n-InAs structure grown on InAs or GaAs substrate, room temperature PVCRs
up to 2 have been obtained by Luo et al. and Yu et al. [54, 55]. The type-reversed
structure, the p-GaSb/InAs/p-GaSb structure, was also studied on a GaSb substrate,
with a PCVR of 1.7 at 77 K and only a weak feature at room temperature, according
to Luo et al. [54].
In Fig. 3.6 (c), the delta doping technique is used to create a δp-i-δn structure.
The delta doping creates potential wells on both side of the intrinsic barrier layer,
i.e. in the conduction band on the n side and in the valence band on the p side.
Resonant interband tunneling of electrons is associated with the quasi-bound states
in the potential well in the conduction band on the n side and the potential well in
the valence band on the p side. Using two thin intrinsic InGaAs layers surrounding
each of the two delta-doping planes in the InGaAs/GaAs δp-i-δn structure, PCVR of
40
5 has been obtained by Su et al. [56]. The δp-i-δn structure is also realized in Si/SiGe
materials system by Rommel et al. [29, 57]. These Si/SiGe RITDs are described in
more detail in a separate section below, and the phonon-assisted tunneling processes in
these tunneling structures are studied in this thesis. It is worth noting that in deltadoped resonant interband tunneling structures, heterojunctions are not necessary
although they can be used, as will be discussed later.
The δn-i-δp-i-δn structure in Fig. 3.6 (d) can be considered as two δp-i-δn structures stacked back to back, sharing a common δn region. Using a GaAs δn-i-δp-i-δn
structure, a room temperature PVCR of 1.7 was obtained by Wang et al. [58], which
was then improved to 3 [59]. InGaAs/GaAs δn-i-δp-i-δn structures in which the delta
doping planes were surrounded by InGaAs layers have produced PVCRs up to 3.2, as
reported by Su et al. [56]. As a type-reversed structure, an InSb δp-i-n+ -i-δp structure
was fabricated, which produced a PVCR of 1.5 at 77 K, by Brunner et al. [60].
The double-well RITD structure was proposed to achieve both the high speed
operation of RTDs and high PVCR of Esaki diodes in one single device [49]. The
proposed band diagram, redrawn in Fig. 3.6 (e), incorporates quantum wells (in both
the conduction band and the valence band) and a pn junction diode structure. By
using interband tunneling in a pn junction, the valley current is significantly reduced,
resulting in higher PVCR as compared to RTDs. Free electrons and holes are induced
in the quantum wells without requiring degenerate doping in the tunneling region,
thus reducing the junction capacitance which degrades the high frequency operation.
The Type I heterostructure as indicated in the band diagram in Fig. 3.6 (e) can be
realized using the InAlAs/InGaAs materials system. Using a n-InAlAs / i-InGaAs /
41
i-InAlAs / i-InGaAs / p-InAlAs double-well structure, PVCRs of greater than 70
have been obtained by Day et al. [61].
3.3.2
Si/SiGe Resonant Interband Tunnel Diodes
Si/SiGe resonant interband tunnel diodes using delta-doping technique were developed by Rommel et al. [29, 57, 62, 63, 64, 65, 66]. The basic structure consists
of a p-type delta-doping plane and a n-type delta-doping plane surrounding a thin
intrinsic tunneling barrier layer, as schematically drawn in Fig. 3.7(a). In both struc-
n+ Si
intrinsic Si
p+ Si
n+ Si
intrinsic Si
intrinsic Si1−x Gex
p+ Si1−x Gex
P δ-doping
B δ-doping
P δ-doping
B δ-doping
p+ Si
p+ Si(100) substrate
p+ Si(100) substrate
(a)
(b)
Figure 3.7: (a) Basic Si RITD structure; (b) Si/SiGe RITD structure with two thin
Si1−x Gex layers surrounding the B delta-doping plane. After [57, 65].
tures shown in Fig. 3.7, the thickness of the intrinsic Si layer in tunneling barrier is
typically a few nm, but can exceed 10 nm for lower currents. The thickness of the
p+ Si1−x Gex layer below the B delta-doping layer in Fig. 3.7(b) is typically 1 nm,
while the intrinsic Si1−x Gex layer in the barrier region can be up to 4 nm, limited
by critical thickness of Si1−x Gex growth on Si substrates. The p+ Si buffer/electrode
layer under the B delta doping layer is typically 80 nm thick or more, while the n+ Si
42
electrode/cap layer above the P delta doping plane is typically 100 nm. The structures shown here are n on p structures grown on p+ type Si(100) substrates, while the
polarity reversed p on n structures grown on n-type Si substrates are also possible and
have been realized. For doping, Boron (B) was used as the p-type dopant, and Antimony (Sb) was initially used as the n-type dopant, but was switched to phosphorus
(P) in subsequent development for higher PVCRs and PCDs, due to less segregation
of P.
The n-type (Sb or P) delta-doping plane creates a quantum well in the conduction
band on the n side, and the p-type (B) delta-doping plane, along with the Si/SiGe
valence band discontinuity in the case of Si/SiGe RITD structure, creates a quantum
well in the valence band on the p side. Fig. 3.8 shows the calculated band diagram of
a Si RITD using the basic structure with a 4 nm intrinsic Si as the tunneling barrier.
The band diagram shows the potential well in the conduction band on the n side
and that in the valence band on the p side created by the delta-doping planes, and
the quantized energy levels in the potential wells. Resonant interband tunneling of
electrons between the conduction band on the n side and the valence band on the p
side is associated with the quantized energy levels in the two quantum wells. It is
worth noting that the delta-doped RITD structure does not require heterojunctions,
and hence the SiGe materials, as the potential wells are primarily created by deltadoping. A variation of the all-Si RITD was realized in [63], and detailed theoretical
modeling was presented in [67, 68].
Central to the realization of these Si/SiGe RITDs are the growth of the deltadoping planes by low-temperature molecular beam epitaxy (LT-MBE) and the postgrowth rapid thermal anneal (RTA) for activation of dopant atoms and reduction of
43
330
IEEE ELECTRON DEVICE LETTER
TABLE I
SUMMARY OF THE ROOM TEMPERATURE I V
RITD’s FOR SiTD1 AND SiTD2. RESULTS
DIODES FROM SUBSTRATES ANNEALED AT 1
ENTRY OF 1.00 INDICATES THAT THE DEVI
IN THE FORWARD DIRECTION RATHER T
0
oxide. The Al dots served as a mask
etched in a CF /O plasma for mesa
backside contact was thermally evap
device fabrication.
Fig. 2. Calculated energy band diagram and resonant states of SiTD1. The
Sb and B activation of the -doped regions is assumed to be 50%. Xz denotes
the conduction band minimums along the kz axis of the Brillouin zone, and
Figure 3.8: The Xband
diagram
ofband
a minimums
Si RITD
basic
the conduction
along using
the kx andthe
ky axes
where structure with a
xy denotes
is the growth direction.
denotes the split-off
band-edge.
4 nm intrinsic Si ztunneling
barrier,SOcalculated
by valence
solving
the coupled effective-mass
III. RESULTS AND
DIS
Schrödinger equation and the Poisson equation. From [57].
Room temperature current–voltag
therefore, is also treated as the tunnel barrier. As Si has an were measured both with an HP
indirect bandgap, the peak current in this structure is attributed Parameter Analyzer and a Textronix
to phonon assisted tunneling between the
electron and to the previous study, NDR behavi
light hole states [9]. The NDR in this structure results from control samples which were not heat
crystal defects created
during
LT-MBE
Due to
dopant
diffusion
a decrease
in the
tunneling[29].
probability
with
appliedsegregation
bias as wasand
only evident in the 5- and 10- m
these bands uncross.
control samples, and the observed PV
during MBE growth, a sharp delta-doping profile cannot be realized was
withbarely
regular
Si than one. Postgreater
II. EXPERIMENTAL SETUP
substantially improved the device per
MBE which uses a relatively
high was
substrate
Epitaxial growth
achievedtemperature.
with a speciallyHowever,
designed a (temperatureTable I summarizes the peak curr
MBE growth system [10] using elemental Si and Ge in e-beam
current density
and PVCR of
dependent) limited-thickness
crystalline
Si layer
can be
grown
by MBEresulting
using substrate
from each anneal temperatu
sources, elemental
Sb in a standard
Knudsen
cell
and elemental
B in a high-temperature Knudsen cell. The structures were and SiTD2. The largest combination o
–
1-min anneal for bo
after a MBE
600 C,(LT
grown
75-mm
B-doped ( Therefore,
temperatures as low
as on
room
temperature.
thecm)
lowSi(100)
tempearure
of 1.36 kA
wafers. Prior to growth, the substrates were prepared using PVCR of 1.38 with a
of
9.4
kA/cm
were
1.45
with
a
a
cleaning
technique
previously
described
[11].
The
growths
MBE) technique can be used to obtain the delta-doping profiles needed in Si/SiGe
were initiated with a 2-nm undoped Si buffer layer grown at SiTD2, respectively. Anneal tempera
The substrate temperature was then lowered to 540 C value led to lower values of
It sho
RITDs, which do700
notC.require
growth of thick layers. In practice, a substrate tem/cm layer. degradation of Si RITD’s occurred a
for the growth of a 70-nm B-doped p -Si
The substrate
temperature was further reduced to 370 C for 100 C below that of the Si/Si G
perature in the 320
370◦ C ofrange
is used
to grow
with typical
the to
remainder
the sample
growth.
This Si/SiGe
included a RITDs
B - possibly
because B
the diffusion of B i
/cm an undoped14tunneling
barrier
(see
Ge
alloys [12].
doped layer
than
in
Si
and P delta-doping
both
cm−2and
. Even
though Spacer
a (thin)
layer had a significant
layerat 1 × 10 /cm
a 100-nm
Fig.concentrations
1), an Sb -doped
thickness
/cm contact layer. All regions of Fig. 3 shows an overlay of the
Sb-doped n -Si
of crystalline Si can
be obtained
by low
temperature
the defect
leveldiodes
in the
the sample
were grown
at a rate
of 0.1 nm/s.MBE,
Furthermore,
diameter
from SiTD1 and SiT
with the exception of the lack of Ge in SiTD1 and SiTD2, temperature of 600 C for 1 min. The
kA/cm ) is almost an order of m
their growth
and doping
was essentially
that of (10.8
grown layer is higher
than that
in layers
grown atidentical
highertosubstrate
temperatures.
of SiTD1 (1.42 kA/cm Since the de
the previously reported Si/Si Ge /Si RITD’s.
85%making
N
A series create
of samples
were annealed
under
15% Hgap,
identical
conditions
The defects in the crystal
energy
states in
thea band
electron
tun- with the exception
ambient in an AG Associates Heatpulse 610 Rapid Thermal elevated current density of SiTD2 m
Annealing Furnace. The anneal time of all samples was fixed spacer. The fact that all entries from
neling through these
states possible and therefore increasing the excess current. The
and
values than corr
at one minute. Anneal temperatures of 500, 550, 600, 650, and smaller
700 C were employed in this study. Control samples without SiTD2 reinforces this conclusion. H
heat treatment were also fabricated for SiTD1 and SiTD2.
spacer thickness did not yield as sharp
Al dots with diameters of 44
5, 10, 18, 50, and 75 m were density as the authors originally susp
patterned on the surface of the wafer via a standard contact that the depletion region for these st
lithography/liftoff process. The samples were dipped in a the -doping planes. With proper ad
buffered oxide etch prior to metallization to remove the native levels and to growth parameters, this p
increase in defect in general, results in a relatively more significant increase in the
valley current than in the current at other biases, therefore reducing the PVCR of
a tunnel diode or even prohibiting obtaining NDR [69]. The elevated defect density,
mainly point defect density, resulting from LT-MBE may be reduced by a post-growth
anneal process, which can also activate the dopants [70]. However, a byproduct of
anneal is the lowering and broadening of delta-doping profile due to dopant diffusion
at elevated temperatures. The amount of dopant diffusion also depends on the time
duration of the anneal. Therefore, an anneal process which is limited both in time and
in temperature is needed. In practice, a post-growth 60-second rapid thermal anneal
with an optimum temperature in the range of 600 to 900◦ C obtained experimentally
(in terms of PVCR and/or PCD) is used.
The incorporation of Ge in the intrinsic barrier layer in the form of Si1−x Gex alloys
was found by Rommel et al. to raise both the PCD and PVCR as compare to all-Si
structures, which was explained by enhanced tunneling probability resulting from a
lower tunneling barrier due to the Ge content [71]. The same conclusion was also
obtained by another research group, as reported by Duschl et al. [72]. Placing the
intrinsic SiGe layer in the tunneling barrier region right next to the P delta-doping
plane was found to reduce both the PCD and PVCR, as compared to placing the SiGe
layer in the middle of two (thin) intrinsic Si layers or right next to the B delta-doping
plane [73]. An analysis of the effect of incorporating Ge and the dependence on the
location of the SiGe layer in the tunneling barrier was presented in [73]. It was later
found by Jin et al. that, by also adding another SiGe layer on the injector side of the
B delta-doping plane, i.e. two SiGe layers effectively clad the B delta-doping plane
as shown in Fig. 3.7(b), resulting in higher optimum RTA temperatures up to 825◦ C
45
with higher PVCRs as compared to structures without the two SiGe cladding layers
(grown in the same MBE chamber) [65]. This was attributed to the suppression of
B diffusion during the post-growth anneal by the two SiGe layers surrounding the
B delta doping plane [65]. The suppression of B diffusion resulted in reduction of
the lowering and widening of the B delta doping profile due to B diffusion during
anneal, thus allowing a higher optimum anneal temperature which is more effective
in reducing the defect density for higher PVCR [70].
With chemical solution-based Si etching processes (wet etching), PVCRs over 3
are routinely obtained. The highest PVCR from samples grown at the same MBE
system is 4.02 from a -1/4/4 structure by inductively coupled reactive ion etching
(ICP-RIE) with HBr, which was attributed to the low side wall damage in the ICPRIE process used [74]. The I-V characteristics of this Si/SiGe RITD is displayed
in Fig. 3.9. The Si/SiGe RITD with a B delta doping / 3 nm i-Si0.54 Ge0.46 / 1 nm
i-Si / P delta doping structure grown by Duschl et al. in a different MBE system has
reached a high PVCR of 6.0 [75].
The PCD of the Si/SiGe RITDs depends on the thickness and composition (i.e.
Ge content and location) of the intrinsic tunneling barrier. PCDs from as high as
218 kA/cm2 to as low as 20 mA/cm2 , spanning seven orders of magnitude, have been
realized [76, 77, 78]. A semilog plot of the I-V characteristics of the thicker-spacer
lower-current Si/SiGe RITDs is shown in Fig. 3.10. The wide tunability of the PCD
of Si/SiGe RITDs makes them suitable for both high speed applications which require
high driving currents and memory applications which require low operating currents.
46
evices is
control.
ly scaled,
lification
d surface
mage. In
Si=SiGe
d test bed
in many
xed-signal
ow power
istor-only
] coupled
d heteroexcellent
limitation
miconducd peak-tont density
he PVCR
t consists
d leakage
is due to
ch can be
sidewall
etching.
sidewall
ing (ICP-
8
100 nm n+ Si
Si/SiGe RITD (-1/4/4)
anneal temp: 800°C
P d-layer (100)
4 nm undoped Si0.6Ge0.4
B d-layer (100)
1 nm p+ Si0.6Ge0.4
240 nm
p+
4
2
Si
1 nm Si buffer
Si substrate
a
wet etched
PVCR = 2.81
VCD = 40 A/cm2
6
4 nm undoped Si
current, mA
of Si=SiGe
has been
haracterise
nd tunnel
compared
ent density
processed
t chemical
n of HBr
nductively
the better
using HBr
cm2 while
40 A=cm2.
play a role
ared to wet
current component in terms of valley current density (VCD) and
PVCR. A thick spacer layer reduces the desired tunnelling current
component while raising the excess current contribution to the VCD
owing to increased tunnelling barrier thickness [6].
dry etched
PVCR = 4.02, VCD = 32 A/cm2
0
0
0.2
0.4
0.6
voltage, V
0.8
b
Fig. 1Figure
(a) Schematic
of 4 nm Si=4
RITD (Si/SiGe
1=4=4)RITD with a -1/4/4
3.9: Thediagram
I-V characteristics
of nm
a 75Si=SiGe
µm diameter
test structure
1 nm the
SiGe
cladding
below
active
region; grown
(b) I-Vby the MBE system
structure,with
showing
highest
PVCR
of 4.02
of samples
characteristics
of
representative
Si=SiGe
RITD
etched
using
HBrreactive ion etching
at Naval Research Laboratory, obtained by inductively coupled
2246
IEEE TRANSACTIONS ON ELECTRON DEVICES, V
chemistry
ICP-RIE
compared with control RITD using wet
with in
HBr.
From process
[74].
etching, both with nominally 75 mm mesa diameters
The Si=SiGe ( 1=4=4) RITD structure was grown by lowtemperature molecular beam epitaxial (LT-MBE) growth, shown in
Fig. 1a following a general schematic described in ref. [9].
Prior to device fabrication, small portions from the whole wafer
grown by LT-MBE were annealed separately in a rapid thermal annealing (RTA) furnace at 800 C for 60 s under a nitrogen gas ambient (N2)
to reduce the point defect density induced by the LT-MBE process. Selfaligned devices were fabricated by photolithographic and electron beam
evaporator of metal contacts.
The ICP-RIE etching used an Oxford System100 with ICP180 tool
and wet etching was via a standard HF:H2O:HNO3 (1:100:100)
solution. The desired mesa structures were created on the wafer using
the Ti=Au=Cr ohmic contact as a self-aligned mask. The optimal
parameters used in this study included an HBr flow rate of 40 sccm
with 250 W of ICP power (to sustain the plasma), 10 W of substrate bias
power (to couple the plasma to the stage) at a 5 mtorr chamber
pressure
◦ C with the spacer thickness varied from 8 to 16 nm on a se
Fig.
5. (a)3.10:
I–V characteristics
of Si-basedofRITDs
annealed
at 825 with
Figure
I-V
the
Si/SiRITDs
the 1/4/n
characteristics
to a structure with a
and of
with
a nominal
20 C stage temperature. Samples were attached
PVCR
for these
Ge
content
of RITDs.
40%, where n = 4, 6, 8, 10, 12, corresponding to a total tunneling barrier
four-inch Si carrier wafer with high temperature grease facilitating a
thicknesses of 8, 10, 12, 14, and 16 nm. From [78].
better heatsink during the plasma process to maintain the substrate
as a function
spacer thickness
is superimposed
comnominally
at room of
temperature.
This resulted
in an etchingforrate
of about
parison.
The
solid
triangles
()
indicate
the
maximum
PVCR
47 was used to reduce
80 nm=min. No post-plasma wet chemical treatment
obtained
by
varying
the
annealing
temperature
for each RITD
intrinsic plasma-induced damage.
spacer thickness. The solid squares () show the corresponding Jp at that optimized PVCR. The open squares ()
Results:
The minimum
power temperatures
to strike a plasma
during
various
as the
illustrate
the spreadsubstrate
in Jp atbias
the plasma
process
was
investigated.
As
the
substrate
bias
power
annealing temperature is varied. For clarity, the PVCRs at was
In addition to the n-on-p Si/SiGe RITD structures grown on p-type Si substrates
as shown in Fig. 3.7, p-on-n structures grown on n-type Si substrates were also demonstrated [79]. A symmetric pnp Si RITD was also realized with symmetric NDR region
in both bias directions [80].
The radio frequency (RF) characterization and small signal modeling of Si/SiGe
RITDs have also been carried out, and a resistive cutoff frequency of 20.2 GHz has
been obtained [81, 82, 76]. The radiation tolerance of the Si/SiGe RITDs was also
characterized, and Si/SiGe RITDs were determined to be more immune to radiation
damage than RTD counterparts [83].
On the materials side, the effects of the MBE growth temperature and the postgrowth anneal have been studied using deep level transient spectroscopy (DLTS) [69,
70, 84].
In terms of circuit applications, tri-state logic has been demonstrated with vertically stacked Si RITDs which showed double NDR regions [85, 86, 87]. Low voltage
monostable-bistable transition logic element (MOBILE) operation with monolithically integrated Si/SiGe RITD and CMOS was more recently achieved [88]. The
Si/SiGe RITDs have also been successfully integrated with SiGe heterojunction bipolar transistors (HBTs) to realize adjustable PVCRs and tunable NDR for voltage
controlled oscillators [89, 90].
3.4
3.4.1
Fabrication of Si/SiGe Resonant Interband Tunnel Diodes
Molecular Beam Epitaxy
Molecular beam epitaxy (MBE) is a technique of using unidirectional flow of
molecules or atoms to grow a crystalline film on a crystalline substrate in the layer
48
by layer manner with the thickness control at the atomic level [91, 92, 93, 94]. The
control of impurities require an ultrahigh vacuum environment. A schematic of the
essential parts of a MBE system is shown in Fig. 3.11. Three zones in which the basic
processes takes place in MBE growth are indicated. The molecular beams are generated in the first zone from the effusion cells. The second zone is the mixing zone, where
the molecular beams may interact with each other. The third zone is the crystallization zone which is the substrate surface where crystalline growth occurs. A series of
processes take place on the substrate surface. The substrate is typically heated to a
high temperature. First, the impinging atoms or molecules arrive at the surface are
absorbed by the substrate surface. Second, surface migration and/or dissociation of
the adsorbed molecules occurs, due to the elevated substrate temperature. Third, the
constituent atoms arrive at certain lattice sites and are incorporated into the crystal
lattice of the substrate or the layer already grown. Finally, the species not incorporated into the crystal lattice are desorbed and leave the substrate surface [92]. Under
favorable conditions, a single atomic layer by layer crystalline growth is realized. Si
and SiGe alloys are among the many materials that can be grown by MBE [95, 96].
The ability of growing high quality crystalline films with a high degree of the control
of dimension and doping level of MBE is needed by the Si/SiGe RITD structures.
In Si MBE, due to the high melting point of Si, typically an electron-beam source is
used instead of a resistively heated effusion cell.
The term delta doping in epitaxially grown semiconductors comes from the Dirac
delta function δ(x), and refers to the zero-thickness idealization of an actual doping
profile of a thin but finite thickness [98]. In practice, the best delta-doping profile
is one in which all of the dopant atoms are confined to a single atomic plane in the
49
Figure 3.11: Schematic illustration of the essential parts of a MBE growth system.
The depicted region is located in an ultrahigh vacuum chamber. Three zones where
the basic processes of MBE take place are indicated. From [97, 92].
crystal, see Fig. 3.12. Assume the growth direction is z, and the delta-doping plane
is located at zd , the three-dimentional (3D) doing profile is given by
N 3D (x, y, z) = N 2D δ(z − zd ),
(3.8)
where N 2D is the two-dimensional (2D) doping density, i.e. the number of dopant
atoms per unit area. Integration of Eq. 3.8 over an region containing the deltadoping plane give the 2D doping density. Sometimes, the concept of “equivalent
3D doping density” of a delta-doping density is useful. Assuming a homogenously
doped semiconductor with the same mean distance between the dopant atoms, the
equivalent 3D doping density is related to the 2D doping density by [98]
N 3D = (N 2D )3/2 .
50
(3.9)
Figure 3.12: Schematic illustration of a semiconductor substrate and an epitaxial
film containing a delta-doping layer. A schematic lattice is shown on the left with
impurity atoms confined to a single atomic plane. From [98].
For example, a delta-doping concentration of N 2D = 1014 cm−2 is equivalent to a
3D concentration of of N 3D = 1021 cm−3 . Therefore, a very high effective 3D doing
concentration can be achieved using a high concentration delta-doping plane.
In principle, delta doping in MBE growth can be realized by a “stop growth”,
i.e. by closing the shutter of cell containing the semiconductor material and leaving
the dopant cell only on for a period of time (to grow a delta-doping layer of a certain doping density) followed by closing the dopant cell and resuming semiconductor
growth. However, delta doping in Si faces two problems: dopant diffusion and segregation. Dopant atoms diffuse due to the doping concentration gradient. At the
typical growth-temperature of ≥400◦ C, MBE growth of doped Si suffers from low
dopant incorporation probabilities and segregation of dopant atoms to the surface,
known as the “Si-doping problem” [99]. All common dopants used in Si MBE (Sb, Ga,
In, B) suffer from segregation under certain conditions [100]. Dopant diffusion and
segregation in Si pose series problems in realizing delta doping, which, by definition,
51
corresponds to very high volume dopant concentration and very large concentration
gradient. Both diffusion and segregation are thermally activated. Therefore, they can
be reduced by using a lower substrate temperature.
Eaglesham et al. found that, contrary to the traditional notion of a minimum
substrate temperature for crystalline growth, crystalline growth can take place at any
temperature, including room temperature, with the limitation that only a limitedthickness crystalline layer can be grown [101]. The problem of delta doping in Si was
solved by Gossmann et al. using low-temperature MBE [102, 99, 103]. A substrate
temperature of 320◦ C was used for the growth the the delta-doping layers of the
Si/SiGe RITD.
In the typical growth of a n-on-p Si/SiGe RITD structure, the growth starts with a
substrate temperature of 650◦ C for the growth of a 1 nm buffer layer, following loading
the p+ -Si(100) substrate into the MBE chamber after a surface cleaning process.
Then, the substrate temperature is gradually lowered to 500◦ C during which a 10 nm
B-doped p+ -Si is grown, with the B effusion cell at around 1800◦ C. The rest of the
p+ -Si layer, typically ranging from 100 to 200 nm thick, is grown with a substrate
temperature of 500◦ C. A 1 nm Si0.4 Ge0.6 layer is then grown by opening the Ge shutter
and reducing the Si growth rate (by reducing the power delivered to the electron beam
source). Next, the B delta-doping plane is grown by a stop growth, i.e. by closing
the Si and Ge shutters and allowing only the B dopant effusion cell shutter open.
The growth of a B delta-doping layer of a density of about 1 × 1014 cm−2 at a B
effusion cell temperature of about 1800◦ C takes about 200 seconds, during which
time the substrate temperature is lowered from 500◦ C to 320◦ C. The temperature of
the substrate is maintained at 320◦ C for the growth of the remaining layers. Following
52
the growth of the B delta-doping layer, a 4 nm intrinsic Si0.4 Ge0.6 layer is grown by
closing the B shutter and re-opening of the Si and Ge shutters. The a layer of intrinsic
Si of a few nanometers, depending on the design, is grown by closing the Ge shutter.
Following the growth of intrinsic layers, a P delta-doping layer is grown by closing
the Si shutter and opening the P shutter, allowing only P atoms to arrive at the
substrate. The P cell temperature is about 750◦ C, and the growth of a 1 × 1014 cm−2
P delta-doping plane takes about 200 seconds. After the growth of the P delta-doping
layer, a typically 100 nm thick P-doped n+ -Si layer is grown by opening the Si shutter
again. This completes the MBE growth of the n-on-p Si/SiGe RITD structure.
3.4.2
Rapid Thermal Anneal
The use of low-temperature MBE results in increased defect density in the grown
Si layers as compared to that from MBE at typical substrate temperatures. The
defect density can be reduced by a rapid thermal anneal (RTA), which is effective
in reducing the defect density and activating the dopant atoms without introducing
significant dopant diffusion [104, 105]. In the fabrication of Si/SiGe RITDs, a 60second rapid thermal anneal at 600 to 850◦ C, depending on the structure, is usually
carried out. A ramp-up time of 10 to 15 seconds from room temperature is typically
used.
3.4.3
Device Fabrication
Although other photoresists such as Shipley 1811 (from Rohm and Haas Electronic
Materials), with a LOR lift-off resist (from MicroChem Corp.) if lift-off is needed,
could be used to fabricate the devices, AZ 5214E (from AZ Electronic Materials) was
53
used in this thesis. Specifically, the AZ 5214E image reversal process was used to
accommodate the polarity of the available masks.
AZ 5214E Image Reversal Process AZ 5214E (from AZ Electronic Materials)
is a positive photoresist, when a normal exposure and development process is used.
With this process, the sidewall obtained has a positive profile. A positive sidewall
profile is unsuitable for metal lift-off process, which requires an undercut profile. With
AZ 5214E, an undercut profile can be achieved by the so-called image reversal process
in which an additional bake is performed after the first exposure with a mask with
patterns, followed by a second exposure without mask (flood exposure). These two
additional steps prior to development reverse the the polarity of the pattern, resulting
an undercut profile in the reversed pattern. With the image reversal process, the AZ
5214E behaves as if a negative photoresist is used in a normal exposure-and-develop
process.
Fabrication of Minimal RITDs with a One-Mask Level Self-Aligned Process
Typically, after a new MBE structure was grown, simple RITDs containing only
a mesa with a top contact and a wafer backside contact are fabricated for quick
assessment of the basic characteristics of the sample (in terms of, e.g. PCD and
PVCR). The structure of such a minimal Si/SiGe RITD is shown in Fig. 3.13.
Si/SiGe RITDs with this minimal structure are fabricated using a simple onemask level self-aligned process. First, 300 Å Ti / 1000 Å Au are deposited on the
backside of the wafer containing the MBE-grown layers, which serves as the backside
contact.
Then a metal lift-off process is used to define circular metal contacts of
54
n+ Si
p+ Si
p+ Si Substrate
Figure 3.13: The cross section of a minimum Si/SiGe RITD fabricated by self-aligned
process. The mesa top metal contact and the substrate back side metal contact are
indicated by the black areas.
various sizes using the AZ 5214E image reversal process. This contact is the top
contact for the individual RITDs. The metals deposited are 300 Å Ti / 1000 Å Au.
Next, the wafer is wet etched past the active region layers in an HF/HNO3 solution
with the patterned circular Ti/Au contacts as etch mask, which completes the sample
fabrication process. The RITDs are then tested on a probe station, in which a top
contact and a backside contact were used for each of the RITDs.
Fabrication of RITDs with Bonding Pads with a Simplified Three-Mask
Level Process
Si/SiGe RITDs for low-temperature measurement need to be packaged. Therefore
bonding pads are needed for wire bonding. The structure of Si/SiGe RITDs with two
bonding pads is shown in Fig. 3.14. A simplified three-mask level process can be used
to fabricate Si/SiGe RITDs with this structure.
In Level 1, circular mesas are defined using a self-aligned process with ohmic
metal as the etch mask. This is done by opening circular holes in the spin-coated
photoresist layer by optical lithography, achieving an undercut profile, followed by
55
n+ Si
p+ Si
p+ Si Substrate
Figure 3.14: The cross section of a Si/SiGe RITD with two bonding pads. The gray
area is SiO2 ; the black area is contact metal.
metal deposition (300Å Ti / 1000Å Au) and liftoff. The undercut profile needed for
metal lift-off process is achieved by the AZ 5214E image reversal process.
The MBE wafer piece is cleaned with acetone and isopropanol, followed by blow
dry with nitrogen prior to photoresist application. The AZ 5214E photoresist is then
spin coated on the wafer piece (3000 RPM, 40 second, ∼1.6 µm), and baked on a
hot plate (90◦ C 60 second). Next, the coated sample is exposed with a clear field
optical mask with Level 1 pattern which defines the mesa, and the alignment marks
for future alignment with other layers (∼22 mW/cm2 , 1.1 second, 405 nm on KarlSuss MJB-3 aligner). The exposed sample is then subject to a post-exposure bake on
a hot plate (120◦ C, 60 seconds). The sample is then subject to a second exposure, a
flood exposure without a mask (∼22 mV/cm2 , 6 seconds), which completes the image
reversal process. The exposed sample is developed in 1:1 AZ Developer diluted with
deionized water (DI H2 O) for about 40 seconds, or MF 319 without dilution for about
40 seconds. and checked under an optical microscope.
Then, 300Å Ti and 1000Å Au are deposited on the sample using an electron-beam
evaporator. The sample is then put in acetone for metal lift-off. The patterned Ti/Au
56
is then used as an etch mask in a wet etch process to define the mesa.
The etchant
is 100 ml DI H2 O : 100 ml HNO3 : 2 ml HF, with an etch rate of about 1000Å/min
at room temperature (calibration of etch rate is needed). The etched depth, or mesa
height, should be about 1500Å excluding the metal, as discussed before.
In Level 2, SiO2 is deposited using plasma-enhanced chemical vapor deposition
(PECVD) and via holes are opened in the SiO2 layer. About 3000Å SiO2 was deposited on the sample at a relatively low substrate temperature of about 250◦ C. The
contact holes are obtained by etching SiO2 using patterned photoresist as the etch
mask. The sample is first coated with hexamethyldisilizane (HMDS) which promotes
the adhesion between the SiO2 and the photoresist (2000 RPM, 30 seconds; 110◦ C,
2 minutes). Without HMDS, AZ 5214E does not stick to SiO2 well enough to obtain good patterning. Then the sample is coated with AZ 5214E (3000 RPM, 40
seconds), baked (110◦ C, 50 seconds), and exposed with mask Level 2 (∼22 mV/cm2 ,
11 seconds, 405 nm), and developed (MF 319, ∼ 40 seconds). At this point, holes
in the photoresist layer corresponding to the desired contact holes in the SiO2 layer
are opened. The sample is then put in buffered HF (BHF), with etches away SiO2
to form contact holes in SiO2 layer, using the photoresist as the etch mask. Contact
holes to both the mesa top contact and mesa bottom contact are opened in this step.
In Level 3, bonding pads for both the mesa top and mesa bottom contacts are
formed, using a lift-off process. AZ 5214E image reversal process is used to define
the bonding pads. After development, 300Å Ti and 6000Å Al are deposited on the
sample, followed by lift-off in acetone. The contact to the mesa top and mesa bottom
are made through the contact holes. This completes the device fabrication.
57
3.4.4
Packaging and Wire Bonding
After testing and cleaving, a small piece of the sample containing the device to
be measured was packaged in a ceramic 16-pin dual-in-line (DIP) package header.
Packaging consists of gluing the wafer piece to the header and wire bonding. The
piece was first glued into the cavity of the package header using epoxy.
Then wire
bonding was carried out on a wire bonder using 1 mil Al wires. The thin Al wires
connect the bonding pads of the devices on the wafer piece to the leads on the package
header. This finishes the sample packaging for, e.g., low-temperature measurement.
58
CHAPTER 4
ELECTRON TUNNELING SPECTROSCOPY
4.1
Electron Tunneling Spectroscopy
As discussed before, phonons are needed in the electron tunneling processes in indirect semiconductors such as Si and Ge. In thermal equilibrium, there is a distribution
of phonons at any finite nonzero temperature, some of which make phonon-assisted
tunneling possible, resulting in nonzero tunneling current at a nonzero bias, provided
other conditions for tunneling such as the availability of empty states are satisfied.
This is one source of phonons that makes phonon-assisted tunneling in indirect semiconductors possible, i.e. phonons due to finite temperature, or fundamentally, created
by the interactions between the crystal lattice and the heat reservoir. When a biasing
voltage is applied across a junction, the interaction between the electrons and the
crystal lattice may also create phonons. This is another possible source of phonons.
Assuming there is no series resistance and therefore no voltage drop due to series
resistance, the energy E of the phonons created in the crystal in the junction region
is eV , where e is the electron charge and V is the applied biasing voltage. If the
phonons excited by the applied bias are such that their momentum satisfies the momentum conservation requirement in phonon-assisted tunneling process, tunneling of
electrons involving these phonons occurs, in addition to that assisted by thermally
59
created phonons. The participation of additional phonons, excited by applied bias,
in the phonon-assisted tunneling process results in an increase of the rate of increase
of the current with respect to the increasing biasing voltage, i.e. the slope of the I-V
curve. This occurs at several biases, corresponding to (the energies of) the various
phonons of the same momentum, due to the existence of phonons of various modes
or branches in a crystal in three dimensions. The increases in slope as the bias passes
certain values from below manifest themselves as inflections in the I-V curve at the
the corresponding voltages. Therefore, the electron tunneling process can be used to
study the phonons in indirect semiconductors. This is an example of a more general
experimental technique known as electron tunneling spectroscopy, which has been used
in a wide range of applications including the study of the energy gaps and the phonon
spectra in superconductors, phonons in normal metals, molecular vibrations in organic
molecules, in addition to phonons in semiconductors [106, 107, 108, 109, 110]. In the
case of molecular vibrations, this method is commonly known as inelastic electron
tunneling spectroscopy (IETS) [111, 112, 109, 110].
While elastic electron tunneling spectroscopy mainly depends on the availability of
empty states for electrons to tunnel into (in addition to other requirements such as energy and momentum conservation), inelastic electron tunneling spectroscopy makes
use of electron-phonon interaction, electron-molecular vibration interaction via the
electrical dipole of molecules, and the interaction between electrons and impurity or
trap states in a tunneling barrier. In the electron tunneling spectroscopy of a general tunnel junction consisting of two electrodes separated by a tunneling barrier, the
measured spectrum depends on the (e.g. phonon) properties of the two electrodes (superconductors, normal metals, and doped semiconductors) and the properties of the
60
tunneling barrier (e.g. phonons in intrinsic semiconductors and insulators, molecular
vibrations of embedded organic molecules, and other impurity or trap states in the
tunneling barrier).
The broadening of the Fermi-Diract distribution function at the electrodes at
finite temperatures, together with the availability of phonons at finite temperatures,
makes the onsets of the participation of phonons in the phonon-assisted tunneling
processes, and therefore, the inflections in the I-V characteristic, more gradual. If
the temperature at which the I-V is measured is high, the inflections in the I-V can
be relatively insignificant and may not be experimentally detectable. The inflections
due to the phonons created by the applied bias, which participate in the phononassisted tunneling process in indirect semiconductors, are more pronounced at low
temperatures. Therefore, electron tunneling experiments for the study of phonons
are typically performed at liquid helium temperature (4.2 K) or even lower.
As having been mentioned before, the phonon signals and thus the inflections in
the I-V are weak. This makes identification and measurement of the phonon signals
from the I-V curve difficult. This difficulty is overcome largely by making use of the
first derivative of the I-V curve. As a simple illustration, consider a hypothetical
I-V curve consisting of two straight line segments with slightly different slope. This
slight change in slope is not easily observable in the curve itself. However, if one
takes the first derivative dI/dV of the curve, the change in slope in the I-V curve
becomes a step in the first derivative curve, which, though still small, is much more
obvious. If one continues to take the second derivative, the step function in the first
derivative becomes a delta function in the second derivative. This is illustrated as
the gray curves in Fig. 4.1. In reality, other effects, mainly thermal broadening, make
61
I
2
dI
dV2
dI
dV
V
V
V
Figure 4.1: Left: A hypothetical piece-wise linear (continuous but not smooth) I-V
curves (grey color) and a smoothed version in which the transition is smooth (solid
color) (the two curves are very close and are not easily distinguishable from each
other). Middle: The first order derivative dI/dV , from which the change in the slope
in the I-V curve is much more obvious. Right: The second order derivative d2 I/dV 2 ,
in which the changes in slope of the I-V appears at peaks and therefore can be studies
quantitatively in terms of peak location, width and height. After [113].
the transitions more gradual, so the I-V curve is smooth, the sharp step functions in
the first derivative become rounded, and the delta functions in the second derivative
become peaks with finite heights and widths, as illustrated as the solid curves in
Fig. 4.1. In the study of phonons using electron tunneling spectroscopy, usually the
second-order derivative d2 I/dV 2 is used. Typically, only the locations of the peaks in
the second-order derivative are used for the determination of phonon energies, while
the the peak height is not used.
Two methods are often used to obtain derivatives in an experiment: numerical
differentiation of the measured I-V data, and direct measurement of the derivatives
by the harmonics detection method. These two methods are described below, and
their advantages and disadvantages are discussed.
62
4.2
Derivatives by Numerical Differentiation
The first and second derivatives can be obtained by numerical differentiation of
the I-V data that are directly measured and digitized.
4.2.1
First-Order Derivative
For the first order derivative, the simplest method is to use the following 2-point
finite difference formula
f 0 (x1 ) =
f (x2 ) − f (x1 )
+ O(x2 − x1 )
x2 − x1
(4.1)
to approximate the derivative at x = x1 or x = x2 (forward and backward, respectively) with error to the first order in (x2 − x1 ).
Apart from the error from the O(x2 − x1 ) term which is inherent in this approximation formula, errors that may be present in the raw data (x1 , x2 , f (x1 ), f (x2 )) get
propagated to the final calculated derivative, and new errors may be introduced by
the numerical calculation process. For experimental data, the measured (raw) data
inevitably contain a measurement error. This is the type of error that is present
in x1 , x2 , f (x1 ), and f (x2 ). Numerical calculation, in general, also introduces error
because only a finite number of significant digits are kept during the numerical calculation. This type of error is called truncation error. Both types of error result
in an error in (y2 − y1 ), the numerator in Eq. (4.1), and an error in (x2 − x1 ), the
denominator. The errors in the numerator and the denominator get propagated to
the final result upon division. In the following, δy 0 represents the errors in y 0 due to
63
δ(x2 − x1 ) and δ(y2 − y1 ),
y2 − y1
δy = δ
x2 − x1
1
y2 − y1
≈
δ(y2 − y1 ) −
δ(x2 − x1 )
x2 − x1
(x2 − x1 )2
δ(y2 − y1 )
y2 − y1 δ(x2 − x1 )
≈
−
x 2 − x1
x2 − x 1 x2 − x1
δ(x2 − x1 )
δ(y2 − y1 )
− y0
,
≈
x 2 − x1
x2 − x1
0
(4.2)
where the first term is due to errors in y (ordinate), and the second term is due to
the errors in x (abscissa). In general, both δ(x2 − x1 ) and δ(y2 − y1 ) decrease with
decreasing (x2 − x1 ) until (x2 − x1 ) is small enough when the error is dominated
by instrument resolution,
roundoff error,
or noise in the measurement. When
δ(x2 − x1 ) and δ(y2 − y1 ) scale in proportion to (x2 − x1 ), the error δy 0 does not
depend on (x2 − x1 ). When either δ(x2 − x1 ) or δ(y2 − y1 ) or both stop decreasing
with decreasing (x2 − x1 ) when (x2 − x1 ) is small enough, δy 0 starts to increase with
decreasing (x2 − x1 ).
This is an important property of numerical differentiation. In order to control the
error term in Eq. (4.1), which is first order in (x2 − x1 ), one would wish to use a small
(x2 −x1 ), i.e. a small difference or step size. However, Eq. (4.2) indicates that the error
in the numerical derivative due to the errors in the raw data and numerical truncation
error increases with decreasing (x2 − x1 ). The smaller the (x2 − x1 ) is, the larger the
error in the derivative y 0 due to the errors in the raw data and truncation error is.
The total error in y 0 is the sum of the error due to the approximation formula and the
error due to the errors in the raw data and the truncation error. Because the former
decreases with decreasing step size, while the latter increases with decreasing step size,
there exists a lowest total error, beyond which a lower error is impossible. The lowest
64
total error corresponds to an optimal step size that is neither too big nor too small.
Depending on the application, this lowest error (from numerical differentiation) may
meet the error requirement, or it may not. When the lowest error from numerical
differentiation does not meet the error requirement, one needs to try smoothing if it
helps solve the problem, or find methods other than numerical differentiation. This is
the reason that numerical differentiation should in general be avoided if possible [114].
This is especially true when a measurement requires high resolution in abscissa, in
which case the error due to numerical differentiation can become exceedingly large.
Other approximation formulas exist for the first order derivative, with different
error terms. For example, when the data points are equally spaced in x, i.e. with a
step size of, e.g. h, then one has the following 3-point formula with an error term to
the second order in step size,
f 0 (x) =
1
[f (x + h) − f (x − h)] + O(h2 ).
2h
(4.3)
Numerical differentiation formulas using more points are discussed later in a separate
section. These formulas have error terms in higher orders of h, thus relaxing the
requirement for a smaller step size. But the analysis above about the error due to
numerical differentiation still applies. That is, a lowest error exists for numerical
differentiation, and the error in numerical derivative does not always decrease with
decreasing step size.
4.2.2
Second-Order Derivative
The second-order numerical derivative can be obtained by twice first-order numerical differentiation. The error in the second-order derivative obtained using this
65
method will have a similar step-size-dependent characteristic as inherited from firstorder numerical differentiation.
The second-order derivative can also be obtained in one step by using various
second-order numerical derivative formulas. For equally spaced data points, the simplest approximation formula for the second-order derivative is:
f 00 (x) =
1
[f (x + h) − 2f (x) + f (x − h)] + O(h2 ).
h2
(4.4)
The above analysis about the error due to numerical differentiation still applies, i.e. a
minimum error exists for numerical differentiation, and the error in numerical derivative does not always decrease with decreasing step size.
4.3
Derivatives by Harmonics Detection Method
As a good approximation, the first- and second-order derivatives can also be directly measured by the harmonics detection method [106, 107, 108, 109, 110]. The
main advantages of a harmonics detection system are high signal-to-noise ratio and
high sensitivity, as compared to the numerical differentiation method.
The harmonics detection method utilizes the fact that, if one applies a small
single-frequency AC voltage on top of a DC bias to a two-terminal (voltage-controlled)
device, the current response of the device can, to a good approximation, be expressed
as a sum of a DC component and a series of AC components at various harmonic
frequencies. One can expand the current around the DC bias V0 , with a small ideal
single-frequency AC voltage cos ωt, where the ω (= 2πf ) and are, respectively, the
66
angular frequency and amplitude of the AC voltage,
dI 1 d2 I 2
3
I(V0 + cos ωt) = I(V0 ) +
(
cos
ωt)
+
(
cos
ωt)
+
O
(
cos
ωt)
dV V0
2 dV 2 V0
dI 1 d2 I 2 1 + cos 2ωt
3
+
O
(
cos
ωt)
= I(V0 ) +
(
cos
ωt)
+
dV V0
2 dV 2 V0
2
!
1 d2 I 2
dI 1 d2 I 2
= I(V0 ) +
+
cos ωt +
cos 2ωt
4 dV 2 V0
dV V0
4 dV 2 V0
+ O ( cos ωt)3
(4.5)
Therefore, in the approximation up to the second order in , the second derivative
d2 I/dV 2 produces a contribution to the DC component of the current. But since the
correction is second order in , and thus, for non-zero I(V0 ), can be neglected. The
AC component at frequency ω is directly proportional to the first derivative dI/dV ,
and is first order in . The AC component at twice the fundamental frequency,
i.e. 2ω, is directly proportional to the second derivative d2 I/dV 2 , and is in . For
small excitation signals, this method gives a good approximation of the first- and the
second-order derivatives .
A circuit can be build to measure the amplitude of the first and second harmonics
in the current. The block diagram is shown in Fig. 4.2. A signal generator generates
a high quality sine wave at angular frequency ω, which is added to a high quality DC
voltage source which acts as the DC bias. It is critical that the signal from the generator generates a high-quality sine wave which have only the fundamental frequency,
with extremely low level higher order harmonics, as the higher order harmonics other
than the fundamental frequency directly affect the derivative measurement. The combined voltage is applied to the two-terminal device. The current flowing through the
device is measured. Specifically, the DC component, the AC component at ω, and the
AC component at 2ω are separately measured. Current measurement can be achieved
67
DC voltmeter
lock−in amplifier at ω
lock−in amplifier at 2ω
V
DC
AC, ω
DUT
Figure 4.2: Block diagram of a first and second harmonics measurement system using
the source-voltage-measure-current (SVMC) method.
by a sample resistor and a DC or AC voltmeter, or some other more advanced methods for higher performance. Upon proper calibration, the absolute values of dI/dV
and d2 I/dV 2 can be directly measured.
Because the input signal at fundamental frequency is a very small signal added
on top of a DC bias, the AC components in the current will also be weak. Specifically, the AC components at ω and 2ω in current are, respectively, first and second
order corrections to the DC current. Measuring such weak signals with acceptable
signal-to-noise ration is a challenge. Lock-in amplifiers are required to measure the
fundamental and second harmonics with good signal-to-noise ratio. Even though the
second harmonic is weaker than the first harmonic and therefore more difficult to
measure, it is still possible with a lock-in amplifier.
68
The circuit shown in Fig. 4.2 is based on the source-voltage-measure-current
(SVMC) approach, which measures dI/dV and d2 I/dV 2 in a straightforward way.
The implementation of this scheme has one potential complication. That is, the actual voltage drop across the device needs to be measured, which requires additional
circuitry because of the insertion of the current-sensing resistor between the voltage
source and the device under test. Although in principle the voltage drop across the
device can also be obtained by subtracting the voltage drop across the current-sensing
resistor (known from current measurement), this approach is less reliable and likely
less accurate than directly measuring the voltage across the device.
One way to avoid this complication is to use the source-current-measure-voltage
(SCMV) scheme, as shown in Fig. 4.3. If the SCMV scheme is used, two other
derivatives, dV /dI and d2 V /dI 2 , instead of dI/dV and d2 I/dV 2 , are measured. If a
DC
DUT
AC, ω
DC voltmeter
lock−in amplifier at ω
lock−in amplifier at 2ω
V
Figure 4.3: First and second harmonics detection circuit using the source-currentmeasure-voltage (SCMV) method.
69
SCMV system is used and therefore dV /dI and d2 V /dI 2 are measured, one can still
obtain dI/dV and d2 I/dV 2 , provided that dI/dV 6= 0. These derivatives are related
by
1
dI
= dV ,
dV
dI
−1 d2 V
d2 I
=
.
dV 3 dI 2
dV 2
(4.6)
(4.7)
dI
The conversion is done after the measurement by calculation, and the calculation
requires that all of the three quantities, d2 V /dI 2 , dV /dI, and V , are simultaneously
known at each DC current bias. Therefore, this approach requires that d2 V /dI 2 ,
dV /dI, and V are measured simultaneously at each DC current bias point.
The high signal-to-noise ratio of a harmonics detection system is mainly due to the
lock-in amplifiers used in the system, which are capable of measuring weak signals with
good signal-to-noise ratio, or in the cases of low-noise signals, achieving high precision
measurements. Overall, harmonic detection method yields a direct measurement of
the first and second order derivatives with signal-to-noise ratios better than achievable
by numerical differentiation.
For implementations and reviews of harmonic detection systems, see, e.g., [108,
107].
With digital measurement systems, the first- and second-order derivatives over a
range of biasing voltage is obtained by a sweep using a step-and-repeat approach. A
multichannel analyzer (MCA) can aid in this process by repeated sweeps of a short
time constant, which provides a fast measurement with a signal-to-noise ratio simply
improving over time with increased number of sweeps [106]. This method has the
advantage over the conventional method of a single sweep with a large time constant
70
in which the result of a measurement, which depends sensitively on the sweep rate,
modulation amplitude and lock-in amplifier time constant, is not available until the
sweep is substantially complete which usually takes a long time. The MCA approach
was not used in this thesis.
4.3.1
Resolution
The peaks in the experimentally measured phonon spectrum are broadened by
thermal broadening due to finite temperature and modulation broadening due to a
finite modulation voltage. The thermal broadening of the phonon spectrum d2 I/dV 2
due to the broadening of the Fermi-Dirac distribution function around the Fermi
energy at finite temperatures was studied by Lambe et al., and a line width at half
maximum of 5.4kB T was obtained with a line shape of ev {(v−2)ev +(v+2)]/(ev −1)3 }
where v = e(V − V0 )/(kT ) [112]. A modulation broadening of 1.22 eVω , where e is
the electron charge, and Vω is the modulation amplitude, was obtained by Klein et
al. [113]. The total full-width-at-half-maximum (FWHM) due to thermal broadening
and modulation broadening is therefore given by
FWHM =
p
(5.4 kB T )2 + (1.2 eVω )2 .
(4.8)
Therefore, in order to achieve a high resolution, the thermal broadening and modulation broadening need to be reduced by using a low temperature and low modulation
voltage respectively. At 4.2 K, 5.4 kB T = 1.95 meV. To achieve a prescribed signalto-noise ratio, the measurement time increases rapidly with decreasing modulation
amplitude. This is especially true for the second harmonic signal for the second-order
derivative measurement, which scales as the square of the modulation amplitude. The
71
modulation amplitude is typically chosen so that the modulation broadening is comparable to thermal broadening, to allow a fast measurement by using a (correspondingly) large modulation amplitude without significantly losing resolution. If a higher
resolution is desired, a lower temperature and correspondingly a lower modulation
amplitude must be used, which usually means (significantly) increased measurement
time.
4.3.2
Low Temperature
Due to thermal broadening effect, the phonon signals cannot be obtained at room
temperature. Tunneling experiments for phonon spectra are typically performed at
4.2 K or even lower temperatures. As discussed above, the resolution is mainly
determined by the temperature at which the phonon spectrum is measured, assuming
a correspondingly low-modulation-amplitude measurement can be made, which is
generally true for this type of measurements. If the measurement is to be performed
at 4.2 K, the measurement can be done simply by dunking the tunneling junction into
the liquid helium in the storage dewar. If the measurement requires a temperature
lower than 4.2 K, an appropriate cryostat is needed.
72
CHAPTER 5
SECOND-ORDER DERIVATIVE FROM NUMERICAL
DIFFERENTIATION WITH OPTIONAL SMOOTHING
The purpose of this chapter is to use real experimental data to show that, it is
virtually impossible to obtain high resolution second-order derivative concurrently
with high sensitivity by numerical differentiation of raw I-V data measured with
commercially available instruments, even with the usage of smoothing filters. This
was an approach investigated as part of this thesis, but later abandoned due to its
limitations.
5.1
5.1.1
Obtaining Numerical Derivatives
Numerical Derivatives from Interpolating Polynomials
The first-order and second-order numerical differentiation formulas used in this
thesis are listed below [114, 115]. These formulas are obtained by approximating
the underlying function by an interpolating polynomial in the region defined by the
extremes of the abscissas, and using the derivatives of the interpolating polynomial
to approximate those of the underlying function. The error terms in these numerical
differentiation formulas come from the error in the approximation of the underlying
function by a polynomial of a finite degree (in the region defined by the extremes of
73
the abscissas). These formulas can be verified by and are consistent with Taylor expansion. The formulas listed here apply to data points whose abscissas are uniformly
spaced. Including the numerical derivative formulas listed below, Appendix III of
Ref [115] lists the coefficients of the numerical derivative formulas for a wide range
of derivative order number and number of points used in the calculation.
Note these formulas are based on interpolating polynomials, i.e. polynomials that
pass the given sets of points or abscissa-ordinate pairs exactly. Equivalently, suppose
that the data points are from an underlying function, and Taylor expansion of the
underlying function is used to interpret these numerical differential formulas, it is
assumed that these formulas create no error in the ordinates and the abscissas, and
that the only error is the truncation error. In practice, the error term has to be
discarded because it is unobtainable. Any error in the ordinates or abscissas results
in additional error in the derivatives calculated using these formulas.
First-Order Numerical Derivative Using 2 Points:
f (x + h) − f (x)
+ O(h)
h
f (x) − f (x − h)
f 0 (x) =
+ O(h)
h
f 0 (x) =
(5.1)
(5.2)
These two formulas are known as the forward and backward finite difference approximations, respectively, with an error term to the first order in step size h.
74
First-Order Numerical Derivative Using 3 Points:
f (x + h) − f (x − h)
+ O(h2 )
2h
−f (x + 2h) + 4f (x + h) − 3f (x)
+ O(h2 )
f 0 (x) =
2h
f 0 (x) =
(5.3)
(5.4)
The first formula above is used for calculation of the derivative at all points except
the two end points. The second formula is used to calculate the derivative at the
first point. The formula for calculating the derivative at the last point is obtained by
changing h to −h in the second formula.
Notice that the 3-point formulas have a higher order error term of O(h2 ) than the
2-point formulas, which have a first-order error term O(h). In general, this results
in a lower error in the evaluation of the derivative, provided that h is small enough.
Note that the error term is also proportional to a certain higher-order derivative,
not displayed in the formulas listed above, but evaluated at some point in the region
defined by the extremes of the abscissas.
However, recall that these formulas assume
no error in the ordinates and the abscissas. When there is error in the ordinates or the
abscissas, for example, due to error of observation in experimentally measured data,
the error in the ordinates or abscissas results in additional error in the numerical
derivatives calculated using these formulas. In such cases, a differentiation formula
with a higher order error term (which requires the use of more points) does not
necessarily lead to a lower total error in the numerical derivative.
75
First-Order Numerical Derivative Using 5 Points:
−f (x + 2h) + 8f (x + h) − 8f (x − h) + f (x − 2h)
+ O(h4 )
(5.5)
12h
f (x + 3h) − 6f (x + 2h) + 18f (x + h) − 10f (x) − 3f (x − h)
f 0 (x) =
+ O(h4 ) (5.6)
12h
−3f (x + 4h) + 16f (x + 3h) − 36f (x + 2h) + 48f (x + h) − 25f (x)
f 0 (x) =
+ O(h4 )
12h
(5.7)
f 0 (x) =
The first formula is used to calculate the derivative at all points except the two end
points and their two adjacent points. The second formula is used to calculate the
derivative at the second point, and the third formula for the first point. Similarly, by
changing h to −h, the second and the third formulas are used for the calculation of
the derivative at the second-to-last and the last points.
Second-Order Numerical Derivative Using 3 Points:
f (x + h) − 2f (x) + f (x − h)
+ O(h2 )
h2
f (x + 2h) − 2f (x + h) + f (x)
f 00 (x) =
+ O(h)
h2
f 00 (x) =
(5.8)
(5.9)
Note that the center point formula has a higher order error term, and thus in general
lower error, than the off-center formula.
Second-Order Numerical Derivative Using 5 Points:
−f (x + 2h) + 16f (x + h) − 30f (x) + 16f (x − h) − f (x − 2h)
+ O(h4 )
12h2
−f (x + 3h) + 4f (x + 2h) + 6f (x + h) − 20f (x) + 11f (x − h)
f 00 (x) =
+ O(h3 )
2
12h
11f
(x
+
4h)
−
56f
(x
+
3h)
+
114f (x + 2h) − 104f (x + h) + 35f (x)
f 00 (x) =
+ O(h3 )
12h2
f 00 (x) =
76
Finally, note again that these formulas assume no error in the ordinates and abscissas. When there are errors in the ordinates or abscissas, for example, due to error
of observation and/or roundoff error, additional error will result in the derivatives
calculated using these formulas. In fact, the presence of error in the ordinates or
abscissas results in a new type of error that has a special dependence on step size h,
so that a lowest total error exists for a certain optimal step size. Either increase or
decrease from this optimal step size will result in a larger total error. This is discussed
previously and will be demonstrated in a separate section below.
Thus, due to the error in the ordinates or abscissas, in practice, a numerical
derivative formula using a larger number of points, i.e. usually with a higher order
error term, does not necessarily lead to a lower total error in the derivative. In general,
it is difficult to predict beforehand, for derivative of a certain order, which numerical
derivative formula (using a certain number of points) gives the lowest total error in
the derivative. A practical approach is trial and error, i.e. one attempts the numerical
differentiation formulas using various number of points, and compares the derivatives
obtained with any further knowledge about the data or derivative, in an attempt to
judge which formula gives the lowest or low-enough error.
5.1.2
Numerical Derivatives from Fitting Polynomials
When the points have error in the ordinates or abscissas, additional error will result
in the numerical derivatives calculated using the numerical differentiation formulas
by interpolating polynomials, as discussed above. The numerical derivatives obtained
in this way are likely to have error levels that are much higher than desired.
77
When the error in the ordinates or abscissas is random, i.e. with approximately
equal probability of being positive and negative, another approach of obtaining approximate derivatives exists. In this approach, one uses an appropriate function with
parameters to be determined to fit the data points, and use the derivatives of the
fitted function to approximate the derivatives of the underlying function. The error
in the derivatives obtained using this method depends on not only the quality of the
fit, but also the error or uncertainties in the raw data. Even a perfect fit does not
mean zero error in the derivatives, because the derivative still have errors originating
from the errors in the data points. Therefore, analysis of the error in the derivative
is more difficult than in the previous case in which the interpolating polynomials are
used.
Because of the difficulty in the analysis of the error in the numerical derivatives
obtained using this method, in general, this method should be avoided if possible.
However, this fitting method has the potential of achieving a smoothing effect, in
that, if one assumes or knows that the underlying function is slow varying, then
a non-perfect fit using a polynomial of a low degree can reduce the noise without
significantly altering the characteristics of the underlying function. In this case, the
fitted function is a better approximation to the underlying function than the raw data.
Situations often arise in which such a smoothing procedure does appear to reduce the
noise without significantly changing the characteristics of the underlying function,
and therefore desirable. However, one needs to keep in mind that error analysis is
difficult, and one has to use further information (from other sources) to determine
whether a certain smoothing procedure is justified or not.
78
In general, it is impossible to use a polynomial with a low degree to satisfactorily fit
the data over the entire range. Instead, one can carry out a local fit of a polynomial
of a low degree to a small number of points around a point to be smoothed (in a
narrow region), and use the value of the fitted polynomial at the point to replace the
original value.
For numerical differentiation purpose, the derivatives of the fitted
polynomial at the point are used to approximate the “true” derivatives. This process
is repeated for each point that is to be smoothed [114, 116]. The implementation of
this procedure is discussed in a later section.
5.2
Errors in Numerical Derivatives
When the tabulated values are exact (still subject to roundoff error in practice), for
example, in the numerical evaluation of a given function and its numerical derivatives
by the numerical derivative formulas from interpolating polynomials, the only error
in the numerical derivatives comes from the error term in the numerical derivative
formulas. Usually, a numerical derivative formula that use more points has an error
term in a higher order of step size h (assuming uniformly spaced data points) than
one that use fewer points. In general, numerical derivative formulas using more points
result in lower error than those using fewer points, provided that h is small enough.
However, in practice, the tabulate values contain error, either from error of observation (for experimentally measured data) or roundoff error or both. This creates
a new problem in error control for numerical derivatives using interpolating polynomials. This is discussed in the previous chapter. The result is that the errors in
the ordinates or abscissas will result in an error in the numerical derivatives using
79
interpolating polynomial formulas that, in general, increases with decreasing step size
h.
The total error in the numerical derivative is the sum of the error due to the
error term in the interpolating polynomial formulas and the error due to error in the
ordinates and abscissas. The final result is that a minimum error exists for a certain
step size which is neither too big nor too small. Both increasing or decreasing from
this optimal step size will result in increased error in the derivative. In practice, this
optimal step size is usually found by trial and error. One issue is, such optimal
step size for the lowest error in a certain derivative may not satisfy other abscissa
resolution requirements.
One way to reduce the error in the numerical derivatives caused by the errors
in the ordinates and abscissas is to smooth the data by fitting the data to a slowvarying function, typically a polynomial of a certain degree, assuming the errors in the
ordinates and abscissas are random. This is discussed in a previous section. However,
error analysis for numerical derivatives with a fitting function is more difficult as it
also depends on the fitting, such as whether an appropriate fitting function is used
and the quality of the fitting.
5.3
5.3.1
Smoothing
Least-Squares Fit with A Polynomial
Given n points defined by abscissa-ordinate pairs (xi , yi ), i = 1, · · · , n, the task
is to fit a polynomial of degree m (n ≤ m + 1)
m
f (x) = a0 + a1 x + a2 x + · · · + am x =
2
m
X
i=0
80
ai x i ,
(5.10)
where the m + 1 coefficients a0 , a1 , · · · , am are to be determined, to the points so that
the fitted polynomial is as close as possible to all points. When m + 1 = n, fitting
becomes exact, and the fitted (n − 1)th-degree polynomial passes all of the n points
exactly. In this case, fitting actually becomes interpolating.
The sum of squared error is defined by
n
X
σ=
yj − f (xj )
2
=
j=1
n X
yj −
j=1
m
X
ai xij
2
.
(5.11)
i=0
In the method of least-squares, the best fit is defined to be the one that corresponds
to the lowest sum of squared error(s) σ [114]. Note that least squares fit assumes
that all of the error is in the ordinates, but not in the abscissas, i.e. the abscissas are
assumed to be accurate.
Minimization of the sum of squared error with respect to ak , (k = 0, · · · , m),
requires that the corresponding partial derivatives to be zero,
n
m
X
X
∂σ
i
=0=
2 yj −
ai xj (−)xkj ,
∂ak
j=1
i=0
k = 0, · · · , m.
(5.12)
Rewriting, and changing order of summation yields
m
X
i=0
ai
n
X
j=1
xi+k
j
=
n
X
xkj yj ,
k = 0, · · · , m.
(5.13)
j=1
This is a set of m + 1 simultaneous (inhomogeneous) linear algebraic equations for
m + 1 unknowns a0 , a1 , · · · , am . Therefore the coefficients ak , (k = 0, · · · , m), can
be found by solving the above linear equations using standard methods such as
Gauss elimination, Gauss-Jordan elimination, LU decomposition, but perhaps most
appropriately, Singular Value Decomposition [117, 118, 114]. Therefore, he leastsquares fit with a polynomial can be found by solving a set of simultaneous linear
equations.
81
The above results can be expressed using the vector notation. Define a n×(m+1)
matrix X,


1 x1 x21 · · · xm
1
 1 x2 x2 · · · xm 
2 
2

X =  .. ..
..
..
..  ,
. .
.
.
. 
2
1 xn xn · · · xm
n
or Xij = xji ,
i = 1, · · · , n; j = 0, · · · , m. (5.14)
Define a column vector a of length m + 1 from the m + 1 coefficients a0 , a1 , · · · , am ,
and a column vector y of length n from the y values of the data points y1 , y2 , · · · , yn ,
 
 
a0
y1
 a1 
 y2 
 
 
 
 
a =  a2  ,
y =  y3  .
(5.15)
 .. 
 .. 
 . 
.
am
yn
Then Eq. (5.13) can be expressed in the matrix form as
XT X a = XT y.
(5.16)
The solution is
a = XT X
5.3.2
−1
XT y.
(5.17)
Smoothing by Local Least-Squares Fit and SavitzkyGolay Method
Suppose, as a smoothing method, one carries out a local least-squares mth degree
polynomial fit to nL points to the left of the point to be smoothed, the point to be
smoothed itself, and nR points to the right (hence a total of nL + nR + 1 points are
used in the fit), and then use the fitted polynomial evaluated at the abscissa of the
point to be smoothed to replace the original ordinate. Next, a point at the left end is
discarded, and a new point next to the old right end point is added, a new “window”
82
is obtained, and one repeats the same procedure for the new point to be smoothed
which is to the right of the old one, and the process is repeated [114, 116].
After the fitting, the values of the fitted polynomial evaluated at each of the
abscissas can be obtained by
f = Xa = X XT X
−1
XT y = Hy,
(5.18)
where
H = X XT X
−1
XT .
(5.19)
Therefore, the values of the fitted polynomial evaluated at each of the abscissas are
linear combinations of the original ordinates. In the smoothing procedure described
above, only one point, typically at or near the middle of the points, is smoothed
at a time by replacing the original ordinate with the value of the fitted polynomial
evaluated at this abscissa.
In general, the fitting and evaluating process are repeated, once for every point.
The reasons is that, for each point and therefore the associated moving window, the
set of abscissas is different, and so is matrix X. Therefore the fitting and evaluating
calculation need to be carried out once for every point. This can be computationally
intensive, especially for large data sets. Fortunately, for data sets that are uniformly
spaced in abscissa, one can utilize the translational symmetry in abscissa so that the
fitting needs to be done only once (for a give group of values of nL , nR , and m), and
the linear combination coefficients for the ordinates, obtained for one point applies
to all points except those near and including the two end points for which nL or nR
has to vary. This makes use of the property of H that, for points whose abscissas
are uniformly spaced, H is independent of the abscissas and the step size or the
83
separation between two neighboring abscissas. In other words, H is invariant with
respect to the shifting and rescaling of the abscissa. Without giving a complete proof
of this property (of H), some qualitative arguments can be made to help understand
this property.
Suppose one has done the fitting and evaluation at one point, and moves on to
the next point. The new moving window is shifted to the right (assuming one works
from left to right although this is not required) from the old window by one point.
One obtains a new matrix X because of the new set of abscissas. In general, this
requires a new fitting and evaluation process. However, if the abscissas of the points
are equally spaced with a step size of h, the abscissas of the new window is simply
those of the old window shifted to the right by a fixed h.
If one now shifts each abscissa to the left by h, the new set of abscissas becomes
the previous one, and the new window becomes the previous window. Then the
fitting and hence the linear combination coefficients for the ordinates (the matrix H)
becomes identical to those of the previous window, because the matrix X is the same.
Because the point to be smoothed is also shifted to the left by h, coinciding with the
point to be smoothed in the previous window, the coefficients for calculation of the
smoothed value are also the same. Therefore, for smoothing of the new point, all
one needs to do is a new linear combination with coefficients already obtained before.
Equivalently, this is simply a change of coordinate system in which the abscissas shift
by −h while keeping the ordinates the same. The smoothing of a certain point, or the
value of the fitted polynomial at a certain point, in the (moving) windows is invariant
with respect to this coordinate transformation. Therefore, for fixed nL , nR , and m,
the calculation of the coefficients, i.e. the matrix H from X according to Eq. (5.19),
84
needs to be carried out only once, and the coefficients apply to all points except for
those at or near the two ends where nL or nR has to vary.
In fact, the abscissa of the point to be smoothed can be further shifted to the
origin x = 0. Due to the same argument above, this does not change the smoothed
value.
In the same vein, it can be understood that, if a new set of (equally spaced with a
new spacing) abscissas are obtained by scaling the original abscissas with a constant
factor, the value of the polynomial fitted using scaled abscissas and evaluated at the
scaled abscissa of a certain point is the same as that of the polynomial fitted in the
original abscissas and evaluated at the original abscissa of the same point. Therefore,
the value of the fitted polynomial at a certain point is invariant with respect to the
scaling of the abscissa.
The special property of the matrix H is the basis of the Savitzky-Golay smoothing
method [116]. The choice of x = 0, h = 1 is an especially convenient choice which
simplifies the calculation: they make a simple X matrix, and hence making the
calculation of H simple. The coefficients in [116] are calculated with this choice.
Making the abscissa of the point to be smoothed at the origin especially simplifies
the evaluation of the derivatives. The kth order derivative of the fitted polynomial,
Eq. (5.10), is given by
m−k
m−k
X
X (i + k)!
dk f
i
=
(i
+
k)(i
+
k
−
1)
·
·
·
(i
+
1)a
x
=
ai+k xi ,
i+k
dxk
i!
i=0
i=0
k = 0, 1, · · · , m.
(5.20)
The evaluation of the derivatives is made easy if the point at which the derivatives
are evaluated is located at origin, i.e. with an abscissa of zero,
dk f = k!ak ,
k = 0, 1, · · · , m.
dxk x=0
85
(5.21)
This is can be achieved by shifting the point at which the derivatives are to be
evaluated to the origin, which, as analyzed above, changes neither the smoothed
value nor the (smoothed) derivatives.
Although the scaling of the abscissa to a unit step size h = 1 does not affect the
smoothed value, it does affect the derivatives by introducing a scaling factor. The
transformation x∗ = x/h scales a data set with an arbitrary step size h to h = 1. One
has,
dk f
1 dk f
dk f
=
=
,
dxk
d(hx∗ )k
hk dx∗k
k = 0, 1, · · · , m.
(5.22)
Therefore, after smoothing with the Savitzky-Golay method with h = 1, the derivatives need to be divided by hk in order to translate the derivatives back to that of
the original data set. The fact that the smoothed value is not affected by the scaling
is also manifested in the above equation, because a function can be considered the
zeroth order derivative of itself, and h0 = 1.
Typically, when Savitzky-Golay method is used, the coefficients are obtained with
the x = 0, h = 1 convention. This does not affect the smoothed value. However, if one
wishes to obtain the smoothed derivatives, the derivatives from the Savitzky-Golay
method should be divided by hk where k is the order number of the derivative in
order to obtain the (smoothed) derivative of the original data, as discussed above.
In the smoothing method described above (hence the Savitzky-Golay method),
nL , nR , and m are user-selectable parameters. For example, the special case of nL =
nR = n with m = 1 is called a Moving Average of 2n + 1 points, in which all of the
2n + 1 coefficients are equal to 1/(2n + 1), i.e. the smoothing is done by replacing an
ordinate with the average of 2n + 1 neighboring ordinates. The parameters nL , nR ,
and m control the amount of smoothing, or to what extent the noise is reduced and
86
to what extend the characteristics of the underlying function is altered. Generally,
the larger the nL + nR + 1 is, the stronger the smoothing effect is, because more
points are used in a fitting to a given order of polynomial. In general, the smaller
the m is, the stronger the smoothing effect is, because a lower order polynomial is
inherently smoother than a higher order polynomial. While the objective of smoothing
is to reduce the level of random error, the smoothing process alters the true signal
at the same time, resulting in an additional error, because it uses a polynomial to
approximate the underlying function of a generally unknown form (even though over
a narrow region only). One caveat is that excessive smoothing in the pursuit of
smoothness will results in poorly modeled data which eventually increase the total
error. Such effects include, among others, appreciable lowering of peak height and
increasing of peak width. The appropriate choice of the parameters nL , nR , and m,
in the objective of achieving the lowest total error, depends on the nature of the data
under consideration and whether the smoothed value or a derivative of a certain order
is output. As the error analysis is difficult, in practice, it usually takes trial and error
and experience, combined with further information from other sources, to determine
what are likely to be the most appropriate parameters.
Finally, it must be pointed out that, as discussed above, the random noise reduction capability of smoothing is limited. For example, smoothing does not replace an
accurate measurement with low noise. Due to the difficulty in error analysis, smoothing in general should be avoided if possible. Smoothing should only be used as a last
resort, in an effort to recover the last bit of information in, for example, noisy data,
which is otherwise impossible.
87
The savgol() function in [119] is used in this thesis for calculation using SavitzkyGolay method. In addition to the data to be smoothed, savgol() takes a few parameters: nL , nR , m, and k, where m = 0, 1, · · · , nL + nR , is the order of the fitting
polynomial, and k = 0, 1, 2, · · · , m, for the smoothed data, the smoothed first-order
derivative, smoothed second-order derivative, etc. When nL + nR + 1 = m + 1, fitting
becomes interpolating, and no smoothing effect is achieved.
5.4
Obtaining Measurement with the Least Error
The error in the result of an electrical measurement comes from, among others,
two sources: (1) the noise that is included in, and therefore cannot be separated from,
the signal to be measured; (2) the error due to the measuring instrument
Some noise will enter the signal before the signal reaches the measuring instrument. In an electrical measurement, noise can come from, among others, improper
shielding from stray electromagnetic fields, improper grounding in the measuring circuit. Therefore, general guidelines in performing electrical measurements should be
followed, which includes proper shielding (of signal wires by coaxial cables etc.) and
proper grounding. This part of the noise is not related to the measuring instrument.
The measuring instrument introduces additional error when it measures the signal
being fed to it. Quite generally, in a measurement procedure, there are user-selectable
parameters that affect the measuring process. The choice of these parameters has a
direct impact on the error in the measured result. For the electrical measurements
needed in this thesis, most of the user-selectable parameters are related to timing in
digital electrical measuring systems.
88
The digital readout in a measuring system is provided by analog-to-digital converters (ADCs) which convert analog signals to digital values. A parameter that is
directly related to a typical ADC is the integration time. For direct-current (DC)
measurements, usually the longer the integration time is, the more accurate the measurement result is. But the trade-off is the measurement time, or equivalently, the
measurement speed.
In a sweep measurement which uses the step-and-repeat approach, a transient
state is established at each step point which will last for a period of time. Enough
time should be given to allow the transient signal to “die out” before a proper measurement of the steady state signal can be made. This results in additional measurement time. Another issue involved in a step-and-repeat sweep measurement is, for
the abscissa, whether one records the programmed value or the actually applied value
obtained by (an additional) measurement. In principle, it makes sense to measure
the actually applied value for higher accuracy. However, this involves an additional
measurement which, as usual, suffers from measurement error. At least for the measurements required in this thesis, it is not obvious that use of actually measured
value instead of the programmed value will result in reduced error in the results of
the measurements. In addition, due to the fluctuation in the applied value and the
measurement error, the measured abscissas in general are not uniformly spaced for
uniformly spaced programmed values. Non-uniformly spaced abscissas may pose a
problem in certain mathematical operations which require a uniform step size, for
example, the numerical differentiation and smoothing methods discussed earlier in
this chapter. From these two considerations, programmed values were used for the
abscissas of the measurements presented in this thesis.
89
Finally it is worth noting that extra care is needed in measuring weak signals, as
the noise will be relatively larger as compared to the case of stronger signals, resulting
in lower signal-to-noise (S/N) ratio.
In the rest of this thesis, the two terms “error” and “noise” are not distinguished
from each other and are used interchangeably.
5.5
Numerical Derivatives of Experimental Data: Example
In this section, real experimentally collected data are used to show that, due
to the limit of the error (or noise) in the commercially available instruments, it is
impossible to obtain low-noise high-resolution second-order derivative by numerical
differentiation of measured I-V data.
Fig. 5.1 shows the I-V of an actual tunnel diode (a Si/SiGe RITD with -1/4/4
structure and 60% Ge) measured by a Keithley 4200 SCS system in the 0 to 0.1 V bias
range. For a step size of 0.5 mV, the measurement timing settings on the Keithley
4200 SCS were: DT, A/D... This timing setting was believed to give the lowest noise
that is close to the machine limit while providing a reasonable measurement speed.
Overall, this curve appears smooth and seems to indicate a high-quality measurement.
However, upon magnification, the error or the noise in the becomes revealed, which
is exacerbated when numerical derivatives are taken.
Fig. 5.2 shows the first order numerical derivative dI/dV using the 3-point numerical derivative formula. The first order numerical derivative appears smooth overall,
but a careful inspection of the details reveals that it is not.
It is evident in this
example that the “smoothness” level deteriorates going from the original measured
data to its numerical derivative, as discussed before. (The structure near zero bias
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031119.1 825C 18υm 4.2 K
Current (mA)
0.3
0.2
0.1
0
0
20
40
60
Voltage (mV)
80
100
Figure 5.1: I-V characteristics of a representative Si/SiGe RITD at 4.2 K at small
positive biases. Two inflections are visible.
is most likely due to a measurement artifact caused by the instrument, therefore can
be ignored for the current discussion.) The numerical derivatives from 2-point forward formula (with an error term of O(h)) and 5-point formula (with an error term
of O(h4 )) are plotted in Fig. 5.3, and compared with the 3-point numerical derivative (with an error term O(h2 )). It is observed that the 2-point (forward) formula
produces the noisiest derivative, while the 3-point and 5-point formulas produce very
similar result for this I-V data set. The reason that the 5-point formula, which has a
higher order error term, does not produce a derivative decisively smoother than that
by the 3-point formula is primarily due to the measurement error in the I-V data.
If the first order derivative is all one needs, this degree of smoothness or roughness is perhaps bearable, and the numerical derivative as shown in the figure is, for
practical purpose, perhaps usable. However, if one takes numerical derivative of the
numerical derivative shown above (effectively a second-order numerical derivative of
91
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031119.1 825C 18υm 4.2 K
dI/dV (1/Ω)
0.004
0.003
0.002
0.001
0
20
40
60
Voltage (mV)
80
100
Figure 5.2: The first order numerical derivative dI/dV using 3-point numerical derivative formula.
0.005
031119.1 825C 18υm 4.2 K
2pt forward
3pt
5pt
dI/dV (1/Ω)
0.004
0.003
0.002
0.001
0
20
40
60
Voltage (mV)
80
100
Figure 5.3: First-order numerical derivatives dI/dB from different numerical derivative formulas: 2-point forward, vs 3-point, vs 5-point.
92
the original data), one would expect the noise to grow in magnitude. This is indeed
the case, as is shown in Fig. 5.4. As this figure shows, the noise becomes much more
noticeable in the second order derivative. Depending on the purpose the second-order
derivative is used for, this may or may not be a problem. But for accurate extraction
of the phonon spectrum, this is unsatisfactory. The second-order numerical derivative
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031119.1 825C 18υm 4.2 K
2
2
d I/dV (1/ΩV)
0.3
0.2
0.1
0
-0.1
-0.2
0
20
40
60
Voltage (mV)
80
100
Figure 5.4: Second-order numerical derivative by first-order numerical differentiation
of first-order numerical derivative, both using the 3-point numerical derivative formula. It is observed that the error or noise gets bigger/worse in the second-order
numerical derivative.
can also be obtained using second-order numerical derivative formulas. The second
order derivative from 3-point second-order numerical derivative formula is shown in
Fig. 5.5, and is compared to the second order derivative by two successive 3-point
(first-order) numerical differentiation. It is observed that the 3-point second-order
numerical differentiation yields noisier result than twice 3-point first-order numerical
differentiation. As a comparison, the second order numerical derivative from 5-point
formula (with an error term of O(h3 ) is compared with that from 3-point formula
93
(with an error term of O(h)) in Fig. 5.6. The 5-pt result is slightly noisier than the
3-pt result. This should be mainly due to the measurement error in the I-V data.
0.4
031119.1 825C 18υm 4.2 K
2d3pt
3pt+3pt
0.3
d I/dV (1/ΩV)
0.2
2
0.1
2
0
-0.1
-0.2
0
20
40
60
Voltage (mV)
80
100
Figure 5.5: The second-order numerical derivatives from different methods: 3-point
second-order numerical differentiation vs twice 3-point first-order numerical differentiation.
The tests so far indicate that, with the current step size h that is required by
resolution requirement, the measurement noise dominates the noise in the numerical
derivatives, and the level of measurement noise in the I-V data is so excessive that
the noise in the second-order derivative is too overwhelming to extract the phonon
spectrum properly. In a later section, attempts to salvage the data will be attempted.
5.5.1
Step Size
When the step size is small enough, the error in the numerical derivatives increases
rapidly with decreasing step size, as can be seen from the following comparison. The
measurement was taken with a 0.5 mV step size. By sampling the measured I-V data
every two data points, one effectively obtains an I-V data set with a 1.0 mV step size,
94
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031119.1 825C 18υm 4.2 K
2d5pt
2d3pt
2
2
d I/dV (1/ΩV)
0.3
0.2
0.1
0
-0.1
-0.2
0
20
40
60
Voltage (mV)
80
100
Figure 5.6: The second-order numerical derivatives using different number of points:
5-point vs 3-point.
with the same measurement error. The second order numerical derivative by twice 3point (first-order) differentiation is taken and compared with that of the original data
set with 0.5 mV step size, see Fig. 5.7. Further, the comparison with a 2.0 mV step
size second order derivative by sampling the original I-V data every four points is also
included in the plot. The decrease of the noise in the numerical derivative with the
step size increasing from 0.5 mV to 1.0 mV is significant. The 1.0 mV step size curve
still suffers from appreciable noise. The loss of voltage resolution with increasing step
size is obvious in the plot. While a 2.0 mV step size produces the smoothest second
order derivative which appears good, it suffers from the lowest voltage resolution. For
example, it cannot be used to reliably resolve phonon energies with a resolution of
about 2 meV and below. For the purpose of measuring phonon energies using IETS,
the voltage resolution needs to be about 1 mV (corresponding to an energy resolution
of 1 meV) or even smaller.
95
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031119.1 825C 18υm 4.2 K 3pt+3pt
0.5 mV
1.0 mV
2.0 mV
0.2
0.1
0
2
2
d I/dV (1/ΩV)
0.3
-0.1
-0.2
0
20
40
60
Voltage (mV)
80
100
Figure 5.7: The dependence of the error in the (second-order) derivative on the step
size.
5.5.2
Averaging
In order to check if averaging of multiple measurements helps reduce the error,
ten measurements were taken successively with the same measurement settings and
the second-order numerical derivative of the average was taken. Fig. 5.8 compares
the second-order numerical derivative of one measurement and that of the average
of the ten measurements using the three-point second-order numerical differentiation
formula. Fig. 5.9 shows the second-order derivatives from twice three-point first-order
numerical differentiation of one measurement and the average of ten measurements.
In both cases, slight reduction in error is observed. But in general, this type of
averaging is not very effective in reducing the noise in the measured I-V data. This
means that the error in the measured I-V data is not entirely random noise which
can be reduced by repeating the measurement and taking the average.
96
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031119.1 825C 18υm 4.2 K 3pt
Average of 10
1 measurement
2
2
d I/dV (1/ΩV)
0.3
0.2
0.1
0
-0.1
-0.2
0
20
40
60
Voltage (mV)
80
100
Figure 5.8: For the data considered, averaging does not appreciably reduce the error
or noise in the second-order numerical derivative using 3-point second-order numerical
derivative formula.
0.4
031119.1 825C 18υm 4.2 K 3pt+3pt
Average of 10
1 measurement
2
2
d I/dV (1/ΩV)
0.3
0.2
0.1
0
-0.1
-0.2
0
20
40
60
Voltage (mV)
80
100
Figure 5.9: For the data considered, averaging does not appreciably reduce the error
or noise in the second-order numerical derivative obtained by twice 3-point first-order
numerical differentiation.
97
5.5.3
Smoothing
The approaches attempted so far have all resulted in second-order numerical
derivatives that contain too much error. As a last resort, one might consider applying a certain data smoothing procedure in an attempt to reduce the error to a
level low enough for a certain application.
Data smoothing is based on the assumption that the data is slowly varying (with
respect to the sampling interval or step size) and contains random noise. The purpose
of smoothing is to reduce the level of random noise without significantly altering its
value from the true value. A quite general smoothing method, in which a local leastsquares polynomial is used to “smooth” each data point, is described in an early
section.
For the purposes of obtaining smoothed second-order derivative, the derivative
can be obtained in a number of ways, each resulting in a different amount of error. Data smoothing can be used together with numerical derivative formulas to
obtained smoothed derivatives, or alternatively, derivatives of the least-squares polynomial obtained during the smoothing process can be used an an approximation to
the derivatives of the underlying function. For the purpose of obtaining second-order
derivative, one might even carry out a first-order numerical differentiation using one
of the two methods above, followed by another first-order derivative by the other
method, if it is desired.
Second-order derivative by smoothing and numerical differentiation
This method involves smoothing and numerical differentiation operations carried
out in series in a specific order. Smoothing can be performed before or after numerical
98
differentiation. Considering that the second-order derivative can also be obtained by
twice first-order differentiation, one might perform smoothing after the first first-order
differentiation. In order to achieve a stronger smoothing effect, one might wish to
apply multiple smoothing operations instead of only one. The multiple smoothing
steps can, then, be combined with the numerical differentiation steps, in a number
of ways. Finally, in each smoothing step, the number of points used for smoothing
and the polynomial degree number can be varied. Therefore, the number of possible
approaches to obtain second-order derivative with smoothing is significant.
It is impractical and unnecessary to list the results from all possible approaches
in order to find out the one that produces a second-order derivative with the lowest or
one with a low-enough error. But, two observations can be made, which considerably
limit the number of the most promising approaches. First, smoothing of data first
prior to differentiation is likely more justified than smoothing after differentiation.
Second, for practical purposes, two successive smoothing steps (without intervening
differentiation steps ) for a stronger smoothing effect can be approximated by one
smoothing step with appropriate parameters.
Previously, twice 3-point first-order numerical differentiation has been shown to
give the second-order derivative with the lowest noise, for the I-V data considered in
this section. Fig. 5.10 shows the second order numerical derivative (by twice 3-point
first-order numerical differentiation) of the smoothed I-V data using 3-point moving
averaging (SG(1,1,1,0)), and compares it with that without smoothing. The fact that
the second order numerical derivative of the smoothed I-V is much smoother than
that of the original I-V indicates that the error in the I-V data is dominated by the
99
measurement error and not by numerical calculation truncation error at 0.5 mV step
size.
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031119.1 825C 18υm 4.2 K 3pt+3pt
No Smoothing
3-point Moving Average
2
2
d I/dV (1/ΩV)
0.3
0.2
0.1
0
-0.1
-0.2
0
20
40
60
Voltage (mV)
80
100
Figure 5.10: Effect of 3-point moving averaging smoothing on the second-order derivative obtained by twice 3-point first-order numerical differentiation.
Increasing the number of points used in moving averaging results in higher amount
of smoothing, as observed in Fig. 5.11. However, even after an 11-point moving
average smoothing, the resultant second order derivative still has appreciable noise.
Moreover, the peak height lowering and peak width widening effects start to become
noticeable, as the number of points used in smoothing increases, as observed in the
plot.
Given enough number of points, using a higher order fitting polynomial should, in
general, result in better approximation of data. For example, it will have a stronger
tendency to maintain peak height and width even after smoothing with a relatively
100
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031119.1 825C 18υm 4.2 K 3pt+3pt
3-point Moving Average
7-point Moving Average
11-point Moving Average
2
2
d I/dV (1/ΩV)
0.3
0.2
0.1
0
-0.1
-0.2
0
20
40
60
Voltage (mV)
80
100
Figure 5.11: Moving averaging smoothing: 3-point, vs 7-point, vs 11-point.
larger number of points. But note that increasing the fitting polynomial order number effectively reduces the smoothing effect. Fig. 5.12 compares 11-point SavitzkyGolay smoothing with first-, second-, and forth-order polynomial. It it observed that,
compared to the second- and fourth-order polynomial smoothing, the first-order polynomial smoothing (moving average) results in appreciable reduction in peak height
and sizeable increase in peak width. While the peaks are reasonably close (but not
identical), in regions other than the peaks, the noise from the fourth-order polynomial
smoothing results is higher than that from the second-order polynomial smoothing.
However, it is not justified to use an unreasonably high order number. Even though
a higher order polynomial has a stronger potential to preserve weak features in the
data, it is not obvious whether the weak features after smoothing are real signals
or just noise. As an extreme example, it is in general not recommended to use a
polynomial with the order number equal to the number of points, in which case the
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031119.1 825C 18υm 4.2 K 3pt+3pt 11pt
11-point 4th order polynomial
11-point 2nd order polynomial
11-point 1st order polynomial
2
2
d I/dV (1/ΩV)
0.3
0.2
0.1
0
-0.1
-0.2
0
20
40
60
Voltage (mV)
80
100
Figure 5.12: Smoothing by local polynomial fit (Savitzky-Golay method) using 11
points using polynomials of: first-degree, second-degree, and fourth-degree.
polynomial fit effectively becomes a polynomial interpolation. The polynomial interpolation always exists, which passes all given points. As a result, no smoothing effect
is achieved. Furthermore, interpolating polynomial in general is not a good modeling
of the data because the data usually include measurement error. Usually m = 2 and
4 are used.
Keeping using the second-order fitting polynomial, Fig. 5.13 compares the SavitzkyGolay smoothing results with 11, 15, and 19 points used in the smoothing. Even when
19 points are used, the resulting second-order derivative is still appreciably noisy.
Second-order derivative by derivative of least-squares polynomial
With this method, the tasks of smoothing and numerical differentiation are effectively accomplished in one single step.
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SG(5,5,2,0): 11 point
SG(7,7,2,0): 15-point
SG(9,9,2,0): 19-point
2
2
d I/dV (1/ΩV)
0.3
0.2
0.1
0
-0.1
-0.2
0
20
40
60
Voltage (mV)
80
100
Figure 5.13: Savitzky-Golay smoothing with a second-order degree polynomial using:
11 points, 15 points, and 19 points.
Fig. 5.14 compares the second order derivative obtained using Savitzky-Golay
method with a second-order fitting polynomial with 7, 11, and 15 points, respectively. It is observed that the SG(7,7,2,2) appears to over-smooth the data, with appreciable peak height lowering and peak width increasing. SG(5,5,2,2) second-order
derivative appears to be an acceptable compromise between smoothness/noise and
peak height/width maintaining.noise reduction and peaking height and width maintaining. In passing, it is noticed that using the same 11-point second-order fitting
polynomial, the second-order derivative (directly) from the Savitzky-Golay method
is smoother than that obtained by twice differentiating the smoothed I-V using the
3-point formula.
The fitting polynomial order number m can be further varied to study its effect
on smoothing. However, as noted earlier, increasing m while keep other parameters
the same will only result in weaker smoothing. Although this effect may be to some
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031119.1 825C 18υm 4.2 K SG
SG(3,3,2,2)
SG(5,5,2,2)
SG(7,7,2,2)
2
2
d I/dV (1/ΩV)
0.3
0.2
0.1
0
-0.1
-0.2
0
20
40
60
Voltage (mV)
80
100
Figure 5.14: Second-order numerical differentiation by the second-order derivative of
the fitted local second-degree polynomial (differentiation by Savitzky-Golay method)
using: 7 points, 11 points, and 15 points.
extent compensated by increasing the number of points at the same time, one needs
to evaluate which is a better approach, a larger number of points with a higher
order fitting polynomial, or a smaller number of points with a smaller (but not too
small) order fitting polynomial.
In general, a simpler (less complicated) method
is preferred over a more complicated method unless the more complicated method
carries some extra advantage over the simpler method. Here, the advantage, if any, of
the more-points-higher-order is not obvious. Therefore, a few-points-with-resonableorder method is preferred.
Second-order derivative by mixed numerical differentiation with optional
smoothing and derivative of least-squares polynomial
In principle, one could obtain a smoothed second-order derivative by obtaining
first-order derivative by numerical differentiation (with optional smoothing) followed
by a first-order derivative by derivative of the least-squares polynomial, or vice versa.
104
But there is no obvious reason why this hybrid approach is better than the other two
types of approaches discussed above.
Summary
Of the various numerical differentiation and smoothing approaches attempted
above, the SG(5,5,2,2) appears to produce the best compromise between reducing
the noise level in the data and maintaining signal integrity (e.g. peak height and
width) for the example I-V data set. Is this “best” second-order numerical derivative
(among the derivatives from the various numerical differentiation methods attempted
above) good enough for the purpose of detecting phonon signals? The answer depends on whether one is looking for strong prominent phonon signals only or for
weak phonons also. The bottom line is, in general, the signal-to-noise ratio in the
second-order numerical derivative is limited, so that detection of weak signals with
magnitudes comparable to the noise level is impossible.
5.6
Conclusion and Discussion
Various reasonable numerical differentiation methods were attempted together
with optional smoothing while preserving signal characteristics as much as possible,
it was found that, with step sizes h up to about 0.5 to 1 mV (upper limit as required
in a typical measurement of phonon energy by tunneling spectroscopy), the signalto-noise ratios of the obtained second-order numerical derivatives were rather low,
which makes identification of relatively weak phonon signals difficult if not impossible.
Increasing the step size reduces the noise (error) in the second-order derivative, but
results in loss of energy resolution and therefore is not desired.
105
The fact that averaging and smoothing helped reducing the noise level but not to
a significant extent without losing signal features indicates that the noise was mainly
due to the measurement error from the measuring instrument, assuming a proper
measurement is made (by appropriate wiring, shielding and grounding etc.).
It is concluded that due to the measurement error from the Keithley 4200 Semiconductor Characterization System, the error in the measured I-V data is at a level
so that, even with a step size as big as 0.5 or 1 mV, it is still impossible to obtain
second-order derivative with a high signal-to-noise ratio as required by the detection
of relatively weak phonon signals. The same observation is expect to apply to another commercially available instrument, the Agilent 4156C Precision Semiconductor
Parameter Analyzer, also, although it has not been tested yet.
It might be possible to construct a custom ultra-low-noise I-V measurement system
so that the noise in the measured I-V does not lead to excessive noise in the secondorder numerical derivative. But even if it is feasible, it is likely to be non-trivial.
On the other hand, the method of harmonics detection is able to measure the firstand the second-order derivatives with a high signal-to-noise ratio and has been used
successfully in many published experiments. Therefore, harmonics detection method
is still the method of choice for tunneling spectroscopy.
106
CHAPTER 6
SECOND-ORDER DERIVATIVE FROM HARMONICS
DETECTION SYSTEM
6.1
Harmonics Detection System
The harmonics detection system used in this thesis is based on Feng et al. [120].
This system was primarily designed for cryogenic resistance thermometry. A calibrated ac current is driven through the resistor (or any two-terminal device to be
measured) and the voltage across the resistor is measured, i.e. SCMV method. This
instrument realized a high signal-to-noise ratio and an absolute accuracy better than
10−4 , and an overall thermal stability better than 10 ppm/◦ C, with ac modulation
voltages as low as 50 µV. This was realized by the use of extremely low-noise amplifiers, resistors with 0.1% tolerance and temperature coefficients as low as 2 ppm/◦ C,
combined with various design innovations. An ac modulation frequency of about
17 Hz was used to reduce the effect of stray capacitance in the measurement circuit
(including wiring in the low-temperature cryostat) while maintaining the 1/f noise
control. The 60 Hz line frequency and its harmonics pickup is reduced with an amplitude locked feedback loop. The optimal resistance measurement range ranges from
10 Ω to 1 MΩ by factors of 10. The details of the implementation is described in the
original paper.
107
The original system essentially measures the first-order derivative of V with respect to I, i.e. dV /dI, by measuring the first harmonic at ∼17 Hz. By measuring
the second harmonic voltage at ∼34 Hz, the second-order derivative d2 V /dI 2 can be
measured.
One of the advantages of this system is that by design, it can measure a wide
range of resistance (dV /dI). For devices that require a bias, a wide range of biasing
current can be realized by simply changing the biasing resistor (see below). These two
features make this system suitable for studying devices, especially Si/SiGe RITDs,
with a wide range of junction resistance or current levels.
For measurement of the bias-dependent first-order and second-order derivatives
of a tunnel diode, the tunnel diode needs to be biased. The natural choice is to bias
the tunnel diode with a voltage source, i.e. the SVMC scheme, for a tunnel diode
is fundamentally a voltage-contrlled device. But for a simpler implementation, the
SCMV scheme was used to bias the tunnel diode. The SCMV method allows the
measurement in the range of zero bias to the peak voltage of a tunnel diode, within
which the I-V is monotonous. This range also covers the range of the Si, SiGe, and Ge
phonon spectra that are usually measured in electron tunneling experiments. Special
techniques may be needed if measurement is to performed in other regions of the I-V
curve of a tunnel diode. For example, if the measurement is to be done in the negative
differential resistance (NDR) region, one may need to start the sweep from a current
higher than the peak current and ramp down.
Using the SCMV scheme, the biasing current was provided by a voltage source
and a high-precision biasing resistor.The voltage source was a 16-bit digital-to-analog
converter (DAC) controlled by a computer via an optical link (to prevent the electric
108
noise from the computer from entering the measuring circuit). The DAC output
a maximum of 10 V DC voltage. The biasing current range could be adjusted by
choosing an appropriate biasing resistor. For the measurement, the programmed
biasing current, instead of the actually measured current, was recorded. At each DC
current bias, with the measurement of the derivatives dV /dI and d2 V /dI 2 , the DC
voltage across the tunnel diode was also measured simultaneously. For a more precise
measurement of the DC voltage, a calibrated 9.97× DC precision amplifier was used
before it was measured by a digital voltmeter.
For the measurement of dV /dI and d2 V /dI 2 in a SCMV system, a constanntamplitude ac modulating current is provided. This results in a device (differential)
resistance dependent ac modulation voltage across the device. In the measurements
presented in this thesis, settings were used so that the ac modulation voltage across
the tunnel diodes normally did not exceed 0.5 mV, being always smaller than the
voltage resolution.
In this thesis, measurements were taking in the range of 0 to about 80 mV with
about 200 data points. Each data point took about 30 seconds. Therefore, it took
about 100 minutes to measure one curve.
6.2
Second-Order Derivative from Harmonics Detection System
Because the harmonic detection system used in this thesis was based on the SCMV
method, dV /dI and d2 V /dI 2 as functions of I were measured. V , dV /dI, and d2 V /dI 2
were measured in the same run as functions of I. The data point are uniformed
spaced in current I. The second order derivative with respective to V , d2 I/dV 2 ,
can be obtained in two ways: (1) by converting the directly measured d2 V /dI 2 to
109
d2 I/dV 2 using Eq. (4.7); and (2) by (first-order) numerical differentiation of the
inverse of the measured dV /dI with respect to V . As a third method, (3) the second
order derivative can also be obtained by numerical differentiation of the V -I (or IV after approximate “uniformization” in V ) data, but this method does not take
advantage of the harmonic detection system at all. The d2 I/dV 2 obtained using the
three different methods are compared below. It is observed that, with the harmonic
detection system used, the method of (first-order) numerical differentiation of the
measured conductance (or resistance) produces the most reliable and trustworthy
second-order derivative.
As discussed before, it is important to always keep in mind with respect to which
variable the derivative in question is. As tunnel diodes are fundamentally voltagecontrolled devices, the first and second derivative of our interest are all with respect
to V . Using the SVMC method is the straightforward way. However, an existing
SCMV system was used in this thesis in order to avoid the effort and time required to
construct a new SVMC system. Even though in general a SVMC system is desired,
the simple calculation needed for converting the derivatives with respect to I from a
SCMV system to derivatives with respect to V does not pose a problem by introducing
appreciable additional error, as will be demonstrated later.
For simplicity, in the following, I will use “first-order derivative” to refer to either
dI/dV or dV /dI or both, and “second-order derivative” for either d2 I/dV 2 or d2 V /dI 2
or both, unless otherwise explicitly stated.
110
6.2.1
d2 I/dV 2 : Converted from Measured d2 V /dI 2 vs Numerical Derivative of Measured dV /dI
As discussed before, a SCMV system measures d2 V /dI 2 . Eq. (4.7) is used to convert d2 V /dI 2 to d2 I/dV 2 . In the following, for simplicity, at times I will use “directly
measured second-order derivative” to loosely refer to either d2 V /dI 2 or d2 I/dV 2 or
both that is either directly measured or converted from the other derivative which is
directly measured, i.e. from the second harmonic (2ω) signal.
Fig .6.1 compares the the directly measured second derivative and the numerical derivative of the measured first derivative of the I-V characteristics of a typical
Si/SiGe RITD.
This plot shows that overall they agree with each other reasonably
well. However, some difference in shape is observed.
120
d0313d.dht.res
measured signal
d(1/R)/dV
2
80
60
2
Scaled d I/dV (1/ΩV)
100
40
20
0
-20
-40
0
10
20
30
40
50
60
Voltage (mV)
70
80
90
Figure 6.1: Comparison between the second harmonic signal from the harmonics
detection system used and the numerical derivative of the first harmonic signal using
3-point numerical derivative formula. Some difference in shape is observed. It is
believed that the second-order derivative from the (first-order) numerical derivative
of the first harmonic signal is more reliable and trustworthy.
111
Which one to trust? The second-order derivative from numerical differentiation
of the measured first-order derivative should be more reliable and trustworthy for the
following reasons: (1) The measurement of the second-order derivative requires measuring the weaker second harmonic signal that is second order in excitation amplitude,
and therefore is more difficult and challenging than measuring the first-order derivative; (2) The 3-point numerical differentiation process is well understood and reliable.
If we assume that the measured first order derivative is reliable, then the numerical
derivative of it is reliable up to the error introduced by the numerical differentiation
process.
This comparison shows that the observed difference is in (the details of) the shape,
but not in peak location. On the other hand, for the current purposes of identifying
phonons and measuring phonon energies, the general shape of the curve and peak
location are all what is needed. Therefore, both methods of obtaining the secondorder derivative serve the current purpose equally well.
Note that for the current purposes, the peak height, which is related to the strength
or magnitude of the phonon signal, is not measured. For this reason, the second harmonics signal was not calibrated in the harmonics detection system, although it could
be. The first harmonic signal was indeed calibrated, though, for (accurate) resistance
measurement purpose. In the comparison shown in Fig. 6.1, the uncalibrated second
harmonic signal is scaled (“calibrated”) to bring the “flat part” in the range of about
0 to 10 mV to overlap that in the numerical derivative of measured first derivative
curve. In other words, the second harmonics signal is “calibrated” in the plot using
the flat part in the about 0 to 10 mV range as the calibration standard. This flat
part (at the “base” of the curve) serves better as a calibration standard as it is stable
112
against perturbations in experimental conditions. For example, the peak height in
general is not a suitable calibration standard, as it can sensitively depends on the
experimental parameters. With the “calibrated” directly measured second harmonics
signal shown in Fig. 6.1, the comparison indicates that the second-order derivative
from numerical differentiation of the inverse of the measured first-order derivative
has higher peaks than the directly measured second harmonics signal. This may be
another evidence that the second-order derivative from numerical differentiation of
the measured first-order derivative is a betterquality second-order derivative.
A careful inspection of Fig. 6.1 indicates that the directly measured second derivative by the second harmonics is slightly noisier. It is possible that this is because inherently it is more challenging to measure the weaker higher-order 2ω harmonic signal.
But overall, the second-order derivatives from both methods are smooth enough, and
the amount of noise or random error present does not pose a problem.
Overall, the second order derivative from the numerical derivative of the measured
first derivative is believed to be more accurate. In the following, the second-order
derivative obtained using this mixed harmonics detection and numerical differentiation will be used, unless otherwise stated.
6.2.2
d2 I/dV 2 : First-Order Numerical Derivative of Measured First-Order Derivative vs Second-Order Numerical Derivative of Measured I-V (after Uniformization
in V )
In this section, the second-order derivative from first-order numerical differentiation of the measured first-order derivative is compared to the second-order numerical
derivative of the measured I-V data.
113
The usual numerical differentiation formulas require that the data points are
equally spaced in abscissas. Therefore, one needs to “uniformize” the data to covert
the raw V -I data points uniformly spaced in I measured using the SCMV method,
to I-V data points uniformly spaced in V as if they were measured using the SVMC
method, in order to take numerical derivatives with respect to V . This “uniformization” operation has to be an approximation. Therefore one needs to carefully evaluate the error introduced in a certain uniformization process to determine whether the
particular uniformization process should be accepted or rejected. In this thesis, uniformization by second-degree polynomial interpolation with three nearest measured
data points is used. For each desired voltage, three nearest measured data points
are first located, then the second-degree polynomial interpolation through the three
points is determined, and evaluated at the uniformized voltage point to approximate
the current at that voltage. For the two end-points, and possibly a few close-to-end
points if the I-V is highly nonlinear near the ends, a straight line (i.e. first-order
polynomial) fit to the nearest three points is used. The amount of error introduced
by this uniformization process remains to be determined.
The measured I-V data is first uniformized in V , and then the second-order numerical derivative with respective to V is taken with various amount of smoothing.
The measured dV /dI vs I is also uniformized with the help of V vs I data to obtain
dV /dI and hence dI/dV (= 1/(dV /dI)) vs V , of which the numerical derivative with
respect to V is taken to obtain d2 I/dV 2 . A representative comparison is shown in
Fig. 6.2.
In this comparison, the 3-point formula is used for first-order numerical differentiation (SG(1,1,2,1)), and the 7-point second-degree polynomial fit is used for
114
0.14
’d03_13d.dht.res.m’ u 1:4
’< savgol 3 3 2 2 d03_13d.dht.res.uniform’
0.12
"< awk ’!/^#/ {print $1, 1/$2}’ d03_13d.dht.res.uniform_r | savgol 1 1 2 1 -"
0.1
0.08
0.06
0.04
0.02
0
-0.02
-0.04
0
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
Figure 6.2: Numerical experiment of converting V-I data to I-V data for numerical
differentiation with respect to voltage.
second-order numerical differentiation (SG(3,3,2,2)). With the uniformization of the
measured first-order derivative (dV /dI), the second-order derivative from numerical
derivative of measured first-order derivative is still observed to be smooth. With
the uniformization of the V -I data, the second-order numerical differentiation with
a 7-point second-degree polynomial fit results in a second-derivative that maintains
the peak height and width reasonably well, as compared to that obtained use the
other method, but with a noticeable amount of noise. The amount of noise from
second-order numerical differentiation with smoothing appears to be similar to those
without the uniformization of data, according to experience. The noise level in the
second-order numerical derivative from uniformized I-V data, and that in the numerical derivative of the measured first-order derivative with uniformization, indicate no
significant amount of error introduced by the uniformization process.
115
Except for the noise, the trend of the two curves agree very well with each other.
The excellent overall agreement indicates that the (first-order) numerical derivative of
the measured first-order derivative is indeed reliable high-quality second-order derivative, which, in turn, indirectly indicate that the directly measured first-order derivative (by detection of the first harmonic) is high-quality first-order derivative.
The level of noise that is present in the second-order numerical derivative with
smoothing as shown in Fig. 6.2 is typical of that in the second-order numerical
derivative. This high level of noise makes identification of any possible weak signal
peaks of comparable height difficult, if not impossible.
The comparison clearly demonstrates that the first-order numerical derivative of
the measured first derivative is of a much higher quality than the second-order numerical derivative of the measure I-V . This shows that even just the direct measurement
of the first derivative helps enormously in obtaining the second order derivative. In
fact, the second-order numerical derivative of the measured I-V data may simply be
unusable depending on the resolution and/or sensitivity (peak height) requirement,
while the first-order numerical derivative of the measured first-order derivative data
is in general of a much higher quality.
6.2.3
d2 I/dV 2 : Numerical Differentiation With Respect To
V vs Numerical Differentiation With Respect To I
As yet another check, and also as a check for the error introduced in the voltage
uniformization process, the second-order numerical derivative with respect to I is
also calculated from the measured I-V data, and converted to d2 I/dV 2 using Eq. 4.7
(with the help of dV /dI). (Note this numerical experiment uses only the measured
I-V data and nothing else; it does not used the measured first-order or second-order
116
derivative.)
A typical comparison with the second-order numerical derivative of
the measured I-V after voltage uniformization is shown in Fig. 6.3. This comparison
indicates that the second-order numerical derivative with respect to I suffers from a
similarly high amount of noise as that with respect to V .
It also shows, at least
in this test, that the error introduced by the approximate uniformization process is
insignificant compared to other errors.
0.14
’< savgol 3 3 2 2 d03_13d.dht.res.uniform’
’d03_13d.dht.res.dvdi2didv.sg332x’ using 2:9
0.12
0.1
0.08
0.06
0.04
0.02
0
-0.02
-0.04
0
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
Figure 6.3: Second-order derivative with respect to V from conversion of numerical
differentiation with respect to current (of V-I data) compared to numerical differentiation with respect to voltage after uniformization in V.
6.2.4
dI/dV : Measured vs Calculated
The first-order derivative dI/dV from the first-order derivative dV /dI measured
by a SCMV system is compared with the first-order numerical derivative of the measured I-V in Fig. 6.4. Good agreement is observed except that the numerical derivative is slightly noisier. If the first order derivative is all that is needed, the slight noise
117
as shown in the first-order numerical derivative may not be much of a concern. However, if the numerical derivative is to be taken on the first-order derivative in order to
obtain the second-order derivative, the noise in the first-order derivative usually gets
“enlarged.”
For first-order numerical derivative which already has a certain larger
amount of noise, the amount of noise can get enlarged to an extent that eventually
render the second-order derivative unusable. This has been demonstrated earlier.
0.0012
’d03_13d.dht.res.dvdi2didv.sg332x’ u 2:(1/$3)
’’ u 2:8
0.0011
0.001
0.0009
0.0008
0.0007
0.0006
0.0005
0.0004
0
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
Figure 6.4: First-order derivative dI/dV : measured (first harmonic) vs calculation
(numerical differentiation).
6.2.5
Summary
The d2 I/dV 2 from various numerical differentiation methods are compared, and
are found to agree with each other overall except in terms of noise. The secondorder numerical differentiation of the measured I-V data produces the highest level
of noise or error so far, as compared to the that from the directly measured first-order
118
derivative or directly measured second-order derivative using a harmonics detection
system.
With the harmonics detection system used in this thesis, it was found that the firstorder numerical differentiation of the measured dV /dI produced the highest-quality
second-order derivative so far. The harmonics detection system used in this thesis
measured the first-order derivative with a very high precision and very low noise,
which formed the basis for the high quality second-order derivative from first-order
differentiation of the measured first-order derivative. As long as the measured firstorder derivative (by detection of first harmonic) is reliable, the numerical derivative of
the measured first-order derivative is reliable within the error introduced by numerical
differentiation.
The directly measured d2 I/dV 2 (or d2 V /dI 2 ) is generally smooth, and in general agrees with numerical differentiation results. But there is observable difference.
The slightly lesser-quality (but still in general better than second-order numerical
differentiation) directly measured second-order derivative was likely the result of the
challenge of measuring a weaker, higher-order, second harmonic signal.
In the following, unless otherwise stated, the second-order derivative was obtained
by first-order numerical differentiation of the first-order derivative measured by harmonic detection.
119
CHAPTER 7
OHMIC CONTACTS TO SILICON USING DELTA
DOPING AND NICKEL SILICIDE
7.1
Motivation
The voltage at which a peak occurs in the d2 I/dV 2 vs V phonon spectrum in
the electron tunneling spectroscopy is related to the energy of the corresponding
phonon. The voltage drop due to contact resistance could affect the phonon energy
measurement. The need for low resistance ohmic contacts to Si also arose during the
optimization of the high frequency performance of Si/SiGe RITDs by former members
of this research group. The topic is also relevant to the continued scaling down of Si
CMOS devices which demands ever decreasing specific contact resistivity for ultra-low
contact resistance of contacts of reduced size.
7.2
7.2.1
Metal-Semiconductor Contacts
Schottky Contacts and Ohmic Contacts
There are two type of metal-semiconductor contacts, Schottky contacts and ohmic
contacts, depending on the work function of the metal and the Fermi level in the
semiconductor [30]. In the band diagram of a Schottky contact, a barrier, called a
Schottky barrier, exists. As a result, the I-V characteristic of a Schottky barrier is,
120
in general, non-linear. A barrier does not exist in the band diagram of an ohmic
contact. Therefore, an ideal ohmic contact has a linear I-V characteristic.
The current flowing through a Schottky contact consists of a thermionic component due to electrons with enough thermal energy surmounting the Schottky barrier,
and a tunneling component due to electron tunneling through the barrier which is significant only when the barrier is thin. Therefore, the I-V characteristic of a Schottky
barrier depends on the barrier height and the doping level of the semiconductor. The
thermionic component is facilitated by a low Schottky barrier, while the tunneling
component is enhanced by high doping levels which create thin tunneling barriers.
When either the thermionic current or the tunneling current, or both, are promoted,
by using a low Schottky barrier and/or a high doping level, the I-V characteristic of
the Schottky contact can be approximately linear, approximating that of an ohmic
contact. Therefore, a Schottky contact may be used to approximate an ohmic contact, provided that the Schottky barrier is low and/or the semiconductor doping level
is high.
Most metals form Schottky contacts to both n-type and p-type Si. In practice,
ohmic contacts to Si are approximated by Schottky contacts with low Schottky barriers and high doping levels. The Schottky barrier height is mainly determined by
the given metal-semiconductor pair. Therefore, in practice, the ohmic contacts to Si
are realized by choosing an appropriate metal that is compatible with the fabrication
process, and using a doping level as high as allowed by the fabrication process.
For a given metal-semiconductor pair, the Schottky barrier height also depends
on the type of doping. The Schottky barrier height to the n-type semiconductor
ΦBn and that to the p-type semiconductor ΦBp are related to the band gap of the
121
semiconductor Eg by
e(ΦBn + ΦBp ) = Eg ,
(7.1)
where e = 1.6 × 10−19 C is the magnitude of the electron charge. This means that
a metal cannot have a low barrier height to both n-type Si and p-type Si. A metal
that has a low barrier height to n-type Si will have a high barrier to p-type Si, and
vice versa. For the purpose of simplifying the fabrication process, a compromise may
be made by using a metal with a barrier height of about half of the band gap of the
semiconductor, so that a single metal can be used for both the n-type and p-type
ohmic contacts.
For ohmic contact purposes, the metal-semiconductor contacts are usually annealed to form metal silicides. The advantages of metal silicides include, among
others, low resistivity, appropriate Schottky barrier heights for Schottky contacts or
ohmic contacts depending on the type of metal silicide, and high thermal stability [121, 122]. For ohmic contact purposes, the self-aligned silicide process in the
source and drain regions in the fabrication of Si metal-oxide-semiconductor field effect transistors (MOSFETs) reduces the contact resistance by increased contact area
(as compared to the contact windows), and the sheet resistance by the reduced sheet
resistivity of silicides (as compared to Si) [122]. Titanium silicide (the TiSi2 phase)
and cobalt silicide (the CoSi2 phase) have been extensively used in the production of
large scale Si integrated circuits. Nickel silicide (the NiSi phase) is currently being
actively investigated. One of the advantages of NiSi is the lower formation temperature of as low as 400–500◦ C as compared to the formation temperatures of TiSi2
(600–800◦ C) and CoSi2 (450–700◦ C) [122].
122
7.2.2
Characterization of Contact Resistance: TLM and Cross
Kelvin Test Structures
The two basic quantities characterizing an ohmic contact are: the contact resistance RC in Ω which depends on the area of the contact, and the specific contact
resistivity ρC in Ω cm2 which is independent of the area. The concept of an areaindependent specific contact resistivity is useful in comparing contacts of different
sizes.
It must be pointed out that, in general, the current flow or distribution is different,
and hence the intrinsic contact resistance is different, in different contact resistance
test structures. In the same vein, the contact resistance of a certain test structure in
general is different from that in a real device. But the concept of contact resistance is
still a good approximation and hence useful, provided that it is used within its range
of validity. In addition, all models of contact resistance structures suffer from certain
inherent limitations or inaccuracies. Therefore, the contact resistances characterized
by different experimental methods do not necessarily agree with each other.
The usefulness of the concept of specific contact resistivity is also limited. For
example, in the case of transfer length method (TLM), for electrically long contacts,
the contact resistance is not equal to the specific contact resistivity times the geometrical area of the contact. More generally, due to the two-dimensional or even
three-dimensional distribution of the current flow, and the non-uniformity of the real
world contacts, the contact resistance is not necessarily equal to the specific contact
resistivity times the apparent geometrical area of the contact. The specific contact
resistivity is useful as a good approximation only when comparing to, or predicting
123
contact resistance of, contacts of similar sizes. Comparing the specific contact resistivities of contacts of vastly differently sizes does not make much sense. Using the
specific contact resistivity obtained from a contact of one size to predict the contact
resistance of a contact of a vastly different size is highly unreliable and not recommended.
Transfer Length Method (TLM) Test Structure
Figure 7.1: The transfer length method (TLM) structure, and the plot of the resistance between two neighboring contacts as a function of the spacing between the
contacts. W − Z = δ should be as small as possible. In this plot, the electrically
long contact (L LT ) approximation is used to obtain the x intercept. The general
expression for the x intercept is −2LT coth(L/LT ). For electrically long contacts,
L LT , coth(L/LT ) ≈ 1, −2LT coth(L/LT ) ≈ −2LT . From [123].
In the transfer length method (TLM) test structure, Fig. 7.1, the resistances between two neighboring contacts are measured (using a four-wire connection) as a
function of the spacing between the contacts [123]. The contact resistances of all contacts are assumed to be equal. In the simple one-dimensional model, the resistance
124
of the semiconductor is assumed to be directly proportional to the length along the
direction of current flow. therefore the resistance measured between two neighboring
contact is equal to twice contact resistance plus the resistance of the semiconductor
between the contacts,
R = 2RC +
Rsh
d,
Z
(7.2)
where, R is the measured resistance between two neighboring contacts, RC is the
contact resistance of each contact, Rsh is the sheet resistance of the semiconductor, Z
is the width of the contacts, and d is the spacing between the two contacts, as shown
in the plot in Fig. 7.1. When the measured resistance R between two neighboring
contacts is plotted as a function of spacing d between the contacts, a least-squares fit
is performed to obtain a straight line. The contact resistance is then obtained from
the y intercept, and the semiconductor sheet resistance is obtained form the slope.
In addition, the transfer length LT can be obtained from either the y intercept or the
x intercept by solving the equations below, and the specific contact resistivity ρC is
obtained from the semiconductor sheet resistance and the transfer length,
slope =
Rsh
,
Z
(7.3)
y intercept = 2RC
=
hLi
2Rsh LT
coth
,
Z
LT
hLi
x intercept = −2LT coth
,
LT
ρC = Rsh L2T .
(7.4)
(7.5)
(7.6)
Two limiting cases are electrically long contacts and electrically short contacts,

ρ

 C
for L LT , coth(L/LT ) ≈ LT /L, electrically short contacts;
RC ≈ LZ
(7.7)
ρ

 C
for L LT , coth(L/LT ) ≈ 1,
electrically long contacts,
LT Z
corresponding to an effective contact area of LZ and LT Z respectively.
125
Cross Kelvin Test Structure
Figure 7.2: Cross Kelvin contact resistance test structure. (a) Cross section through
section A-A; (b) top view. From [123].
The Cross Kelvin test structure is shown in Fig. 7.2. Essentially, a Cross Kelvin
test structure is a structure for four-wire measurement of the resistance of contact
2/3 (shown above). A current source I is connected to contacts 1 and 2 (or 3) to
establish a current flow, and a voltage meter is connected to contacts 3 (or 2) and
4. Because of the high imput impedance of the voltmeter, a negligible current flows
through contact 4 and the semiconductor between contacts 2/3 and 4, therefore the
voltage drop across contacts 2/3 and 4 and the semiconductor in between due to
voltage measurement is negligible. As a result, the measured voltage across contacts
2/3 and 4 is, to a good approximation, the voltage drop across contacts 2/3 due to
126
current I. The (contact) resistance of contact 2/3 is simply the measured voltage
divided by the current,
RC =
V34
.
I
(7.8)
The specific contact resistivity is give by
ρC = RC AC ,
(7.9)
where AC is the contact area.
The above equation for the calculation of the contact resistance ignores the effect
of the non-zero separation δ between the edge of the contact 2/3 and the edge of
the n-type semiconductor around the corner, which is due to alignment error in most
fabrication processes. A nonzero δ causes current flow, hence additional voltage drop
in the semiconductor, around the contact, resulting in a measured V34 higher than
the voltage drop across the contact. This, according to Eq. (7.8), leads to a higher
measured contact resistance. Therefore, δ should be made as small as possible to
reduce the error.
One advantage of the Cross Kelvin structure is that usually the contact can be
made to have the same size as that used in actual devices, therefore the contact
resistance determined from the Cross Kelvin structure is directly applicable to real
devices. This is especially useful in characterizing small contacts. On the other hand,
the structure of the Cross Kelvin structure is slightly more complicated than the TLM
structure, as it requires a dielectric layer which may not be necessary in the TLM
structure.
127
7.3
Experimental Design
The idea of using nickel silicide (NiSi) and delta doping technique was proposed
in our Si/SiGe RITD research group. My role was to define the contribution by a
formal Si ohmic contact set of experiments.
NiSi was proposed for the ohmic contact to Si, mainly because of its low resistivity
and relatively low silicidation temperature. NiSi can be formed at temperatures as low
as 400◦ C. In addition, NiSi has the potential of forming low resistance ohmic contacts
to both p- and n-Si, thus simplifying the sample fabrication process. The Schottky
barrier height of NiSi to n-Si is 0.7 eV [121]. According to Eq. (7.1), the Schottky
barrier height of NiSi to p-Si is about 0.4 eV. Although NiSi is better suited to make
ohmic contact to p-Si, it may be possible that, with optimized contact structure and
processing, low resistance ohmic contact to n-Si can be obtained with NiSi. Both p-Si
and n-Si samples were designed for the ohmic contact experiments.
The use of delta doping layers in ohmic contacts was first reported by Schubert
et al. for making low resistance nonalloyed ohmic contact to n-GaAs [124]. The
extremely high effective three-dimentional doping density of delta doping layers was
utilized for the purpose of forming low resistance ohmic contacts. The use of delta
doping layers effectively increases the doping level of the semiconductor near the
metal-semiconductor interface, which increases the tunneling current by reducing the
barrier thickness. Delta doping in Si is expected to facilitate obtaining low resistance
ohmic contacts to Si.
The latest design of the structures is shown in Fig. 7.3. The Si layers are grown
on lightly doped Si(100) substrates of the opposite polarity, to allow the formation of
patterned Si conduction channels for contact resistance characterization using a mesa
128
structure. Three delta-doping layers, separated by two thin spacer layers, are used
to achieve an effectively highly doped Si region of a finite thickness. The number of
layers can be varied for structure optimization. A surface cap layer is used to provide
the Si needed for the formation of silicides, and protect the delta doping layers.
129
130
(b)
P δ-doping (1×1014 cm−2 )
P δ-doping (1×1014 cm−2 )
P δ-doping (1×1014 cm−2 )
Figure 7.3: Structures for ohmic contact experiments: (a) p+ -p-n structure; (b) n+ -n-p structure.
(a)
p-Si(100) substrate (10-20 Ω cm)
n-Si(100) substrate (1-10 Ω cm)
P-doped Si (275Å, 3×1018 to 5-6×1019 cm−3 )
P-doped Si spacer (25Å, 5-6×1019 cm−3 )
P-doped Si spacer (25Å, 5-6×1019 cm−3 )
P-doped Si cap layer (175Å, 5-6×1019 cm−3 )
P-doped Si (4000Å, 2.5×1018 cm−3 )
B δ-doping (1×1014 cm−2 )
B δ-doping (1×1014 cm−2 )
B δ-doping (1×1014 cm−2 )
B-doped Si (4000Å, 3×1018 cm−3 )
B-doped Si (275Å, 3×1018 to 5×1019 cm−3 )
B-doped Si spacer (25Å, 5×1019 cm−3 )
B-doped Si spacer (25Å, 5×1019 cm−3 )
B-doped Si cap layer (175Å, 5×1019 cm−3 )
The delta doping layers here are for the sole purpose to reduce the contact resistance only. Outside the contact region, the delta doping layers need to be removed
before contact resistance is measured, else a parallel conduction pathway along the
surface will occur in parallel. This requires the etch of a certain thickness of Si, at
least about 250Å in this case, below the surface.
The relatively thick layer just above the substrate provides the conduction channel
for contact resistance characterization. Due to the use of a substrate of the opposite
polarity, a (doping level dependent) depletion region is created around the layersubstrate interface. A minimum thickness is needed in order to ensure this layer is
not completely depleted.
7.4
7.4.1
Preliminary Results Using Large Contacts
Attempt to Anneal Composite Metal Stacks for Silicidation
Conventional process of forming metal silicides for ohmic contact purposes is a
series process of deposition of metal, rapid thermal anneal of deposited metal for
silicidation, followed by acid etch to removed unreacted metal, which is then followed
by normal metallization process for bonding pads or interconnects through the contact
window. In the case of nickel silicide, a cap layer (typically TiN) is needed on top of
deposited Ni in order to avoid oxidation of Ni.
A simpler process was proposed in an attempt to simplify the fabrication process,
by depositing all required metals before the silicidation anneal. For the application to
Si/SiGe RITDs, gold (Au) is often used as the last metal for its stability and easiness
to probe. However, these structures were found to be unable to withstand a 30 sec
anneal at temperatures as low as 400◦ C, which was the lowest temperature required
131
for the formation of NiSi. A representative example is the anneal of a patterned 150Å
Ni/800Å Mo/1000Å Au metal stack at 500◦ C for 30 seconds. “Burns” were observed
after anneal under an optical microscope, see Fig. 7.4. The surface profile obtained
with a Dektak surface profiler indicates that the top Au layer was melted. This
means that, with a programmed temperature of 500◦ C, and the actual temperature
reading never exceeding about 520◦ C on the RTA system, the top surface layer had
reached temperatures exceeding the melting point of Au, 1064◦ C, which was higher
than the nominal RTA temperature by over 500◦ C. The high transient temperature
puts a large stress on the surface layer. In the case of a multi-layer metal stack, the
top layer could be thermally damaged, in addition to other issues such as possible
interdiffusion or reactions between the different metal layers. In contrast, single Ni
layers of thicknesses of 150Å and 1000Å were able to withstand at least 750◦ C anneal
for 30 seconds with only a slight surface morphology change.
Figure 7.4: An optical microscope image of 150Å Ni/800Å Mo/1000Å Au dots after
rapid thermal anneal at 500◦ C for 30 seconds in forming gas (5% H2 /95% N2 ). The
top Au layer was melted during the anneal, as determined by the surface profile
obtained with a Dektak surface profiler. The diameter of the dots was 100 µm.
132
7.4.2
TLM Results with Ni as Inductively Coupled Plasma
Reactive Ion Etch Mask
For initial characterization of the samples, a TLM structure with 80µm×80µm
contacts and contact spacing from 10 µm to 50 µm in 5 µm increments was used. As
discussed before, the specific contact resistivity determined from such huge contacts
is not expected to be accurate for small contacts. Two TLM samples, one from the
p+ -p-n wafer and one from the n+ -n-p wafer, with Ni as the contact metal were
fabricated. The TLM samples were then cleaved into smaller pieces. Inductively
coupled plasma reactive ion etching (ICP-RIE) was used to etch the Si for delta
doping layer removal outside the contact regions. The small pieces were etched in an
Oxford Instruments Plasmalab System 100 ICP 180 system for different times, and
the etched thicknesses were measured by a Dektak surface profiler to calibrate the
etch rate. The resistances between neighboring contacts in the TLM structures were
then measured with an Agilent 4156C Semiconductor Parameter Analyzer using a
four-wire connection. The pieces were then annealed at 500◦ C for 30 seconds. Slight
roughening of Ni surface was observed, indicating that the reaction between Ni and
the underlying Si might have happened (to form silicides). The TLM structures were
measured again to determine the contact resistance after anneal.
Although not required in this initial experiment (with large contacts), RIE was
used for the etch of Si, in anticipation of later fabrication of small contacts which
requires dry etch. As a byproduct, the RIE etch in this experiment also served to
calibrate the etch rates for future experiments. The results of ICP-RIE of Si with
various etching chemistry are listed in Table 7.1. For the p+ -p-n sample, none of the
four RIE processes produced a clean and easy-to-probe Ni surface after etch. For the
133
n+ -n-p sample, the SF6 process produced a clean Ni surface which was easy to probe.
The residue after etching made it difficult to make electrical contact to Si, indicating
that Ni should not be used as the RIE etch mask for the p+ -p-n sample. Therefore,
a different material, e.g. photoresist, should be used as the etch mask, which makes
the fabrication process more complicated by adding one additional mask level.
Chemistry
HBr
SF6
SF6 + O2
BCl3 + Cl2
ICP-RIE etch of Si with Ni
p+ -p-n
Visible residue on Ni after etch
Difficult/unable to probe
Difficult/unable to probe
Difficult/unable to probe
No data
as Etch Mask
n+ -n-p
Visible residue on Ni after etch
Clean Ni surface, easy to probe
–
–
Data
Table 7.1: Results of inductively coupled plasma reactive ion etch (ICP-RIE) of Si
with Ni as etch mask.
TLM data were only obtained from the n+ -n-p samples etched with the SF6 process. As expected, the measured resistance between contacts increased with increasing
etch time, i.e. decreasing semiconductor thickness. The measured resistance also decreased after the anneal, which indicated reduced contact resistance after anneal. A
small negative y intercept was obtained from the linear fit of the measured resistance
vs contact spacing for about half of the TLM structures measured. Some representative data are shown in Fig. 7.5.
Two conclusions can be drawn from a negative y intercept. First, the contact
resistance must be very low, within the measurement error. Therefore, this preliminary result indicates that this sample is promising for low contact resistance ohmic
contacts, and further experiments are warranted to fully characterize this sample to
134
200
1.8386*x + 0.6500
1.9593*x + 2.5036
1.8371*x - 0.1857
2.3264*x + 6.6250
2.4136*x + 2.8607
3.8314*x + 0.3429
3.6771*x - 0.1143
4.3307*x - 0.4536
4.2436*x - 1.3893
4.1943*x - 1.4714
4.0036*x - 2.5321
4.8100*x - 1.3786
4.3250*x - 1.1821
Resistance (Ω)
150
100
50
n+-n-p Sample
ICP RIE, Ni Mask
0
0
5
10
15
20
25
Separation (µm)
30
35
40
Figure 7.5: Representative resistance vs separation data from the TLM structures of
the n+ -n-p sample, and the least-squares fit with a straight line. The equations of
the fitted straight lines are also displayed. About half of the fitted straight lines have
a negative y intercept. While a negative y intercept means invalid TLM data, it also
indicates low contact resistance.
produce an accurate assessment of its contact resistance. A negative y intercept is
probably due to the error in the measure resistance data, or due to the assumption
of the TLM model that the resistance below the contact does not change. The TLM
method may still be used to characterize the contact resistance of this sample, but
the results obtained so far indicate that a different geometry is needed. Alternatively,
the Cross Kelvin structure can be used for greater accuracy.
135
CHAPTER 8
PHONONS IN SI/SIGE RESONANT INTERBAND
TUNNEL DIODES
8.1
Motivation
The phonon spectra of Si, Ge, and Si1−x Gex bulk alloys have been documented
using Esaki tunnel diodes by Logan et al. [28]. With the development of the Si/SiGe
resonant interband tunnel diodes, it is a natural extension to measure the phonon
spectra in this new type of devices with a different structure with nanometer layering.
By measuring the phonon spectra using electron tunneling spectroscopy, information on the phonons that participate in the electron tunneling process, on the
structure of the tunneling junction, and on the materials that constitute the tunneling junction can be obtained. The purpose of this study is to obtain the phonon
spectra of Si/SiGe resonant interband tunneling diodes, which may help in the understanding of the fundamental electron tunneling processes in these diodes which
incorporate composite tunneling barriers.
136
8.2
Experimental Design
Because electron tunneling spectroscopy requires low temperature, the sample
needs to be packaged appropriately in order to be compatible with the low temperature Packaging requires wire bonding. Therefore, a structure with two bonding pads
required by wire bonding is needed. A cross section of the structure with two bonding
pads used in this thesis is shown previously in Fig. 3.14.
For measurement of the phonon energies, low resistance ohmic contacts are needed
to minimize the voltage drop across the contacts. Carrier freeze-out can also occur
at cryogenic temperatures. Both considerations require that the Si semiconductor in
the ohmic contact region be highly doped. Both B and P doping levels were set at
5×1019 cm−3 .
For low temperature experiments, care must be taken to limit the current flowing
through the junction, because Joule heating (P = IV ) can raise the local junction
temperature to the extent that it becomes significantly higher than the ambient temperature (the voltage drop across the junction, being in general determined by device
structure, cannot be easily or significantly varied). The current can be limited by:
(1) limiting the peak current density by diode design (tunneling barrier thickness and
composition, etc.); and (2) limiting diode area size.
Two groups of samples were designed and studied. Note that the contact schemes
studied in the previous chapter (nickel silicide and delta doping) were not employed
in these two groups of samples.
137
0/3/2 Structures with Zero to 44% Ge Content
The first group of sample used a 0/3/2 structure, but with varying Ge content: 0,
10%, 20%, 30%, 40%, and 50%.
The structure is shown in Fig. 8.1. Note that the zero percent Ge sample is actually
100 nm n+ Si
2 nm intrinsic Si
3 nm intrinsic Si1−x Gex
P δ-doping plane
B δ-doping plane
+
80 nm p Si
p+ Si(100) substrate
Figure 8.1: Si/SiGe RITDs with a 0/3/2 structure with varying Ge content: x = 0,
10%, 20%, 30%, 40%, and 50%.
a 5-nm spacer Si RITD. For this group of samples, the structure was maintained
constant, but with varying Ge content. The purpose is to study the dependence of
the phonon spectrum on the Ge percentage.
-1/4/n Structure with 60% Ge Content/Concentration
The second group of samples examined the -1/4/n structure, where n = 4, 6, 8,
10, 11, and 12 were thicknesses of Si layers in nm in the spacer region, with a 4-nm
thick Si1−x Gex layer with 60% Ge concentration in the spacer region. The general
structure is shown in Fig. 8.2. The purpose of this group of samples was to study
the effect of current magnitude on the voltage shift due to contact resistance. These
samples were designed with thicker spacers for lower currents.
138
100 nm n+ Si
n nm intrinsic Si
4 nm intrinsic Si0.40 Ge0.60 1 nm p+ Si0.40 Ge0.60
P δ-doping plane
B δ-doping plane
250 to 80 nm p+ Si
p+ Si(100) substrate
Figure 8.2: Si/SiGe RITDs with the -1/4/n structure with 60% Ge, where n =
4,6,8,10,11,12 in nm for the the thickness of SiGe layer.
8.3
Sample Fabrication
The samples were grown using low-temperature MBE by Dr. Phillip E. Thompson
at the Naval Research Laboratory in Washington, DC. For the designed 0/3/2 structure with 50% Ge content, it was discovered after growth that the actual structure
was a 0/2.7/2 configuration with 44% Ge content.
Each 3-inch MBE wafer was cleaved into smaller pieces, and each piece was used
separately for device fabrication. Using this approach, a single 3-inch wafer can be
used for making a number of different devices using different fabrication procedures,
minimizing wafer waste.
The 0/3/2 Structures with zero to 44% Ge content were annealed at 750◦ C for 1
min in a Modular Process Technology RTP-600S Rapid Thermal Annealing System.
The -1/4/n structures with 60% Ge content were annealed at 825◦ C for 1 minute.
The simplified 3-mask level process described previously was used to fabricate the
RITDs with two bonding pads. The fabricated RITDs had various mesa diameters
of 10, 18, 50, and 75 µm.
139
After testing, a small piece containing one or more working devices was cleaved
from the finished pieces. The small piece was then mounted in the cavity of a ceramic
DIP-16 package header, followed by wire bonding.
Then the packaged devices were mounted on the dipper stick by inserting the DIP16 package header containing samples into the DIP-16 socket on the dipper stick, and
dipped in liquid helium for measurement.
Fig. 8.3 shows some SEM images of a fabricated 10 µm diameter diode.
(a)
(b)
Figure 8.3: SEM images of a 10 µm diameter Si/SiGe RITD, top view: (a) overview;
(b) close-up of the circular mesa with the top contact and the bottom contact semienclosing the bottom of the mesa.
Fig. 8.4 shows a picture of packaged diodes in a ceramic DIP 16 package, and the
dipper stick on which the package is mounted.
140
(a)
(b)
Figure 8.4: (a) Packaged diodes; (b) Package mounted on the dipper stick.
8.4
Measurement
Two methods, described in earlier chapters, were used to obtain the second-order
derivative d2 I/dV 2 (vs. V ) phonon spectra in Si/SiGe RITDs, i.e. the second-order
numerical derivative (with smoothing) of the measured I-V data, and first-order numerical derivative of the dV /dI vs. I measured by a harmonic detection system.
In the first method, the I-V characteristics at room temperature and 4.2 K were
collected using the source-voltage-measure-current (SVMV) method with a Keithley
4200 Semiconductor Characterization System. The second-order numerical derivative d2 I/dV 2 (vs. V ) was then taken using Savitzky-Golay method with appropriate
smoothing parameters.
In the second “hybrid” method, the first-order derivative dV /dI vs. I was measured by detecting the first harmonic using the harmonics detection system described
earlier with the source-current-measure-voltage approach. The first-order numerical
141
derivative with respect to I was taken to obtain d2 V /dI 2 (vs. I), which was subsequently converted into d2 I/dV 2 (vs. I) using Eq. (4.7). The final d2 I/dV 2 vs. V
data is obtained by a look-up of the V -I data.
8.5
8.5.1
Results
Commercial Si Esaki Diodes: 1N2927 and 1N2930A
As a reference, two commercial Si Esaki tunnel diodes, 1N2927 and 1N2930A from
Advanced Semiconductor, Inc., were measured. The resistance dV /dI vs. current was
measured using the first harmonic detection system described earlier (and the voltage
vs. current data were measured simultaneously), and the numerical derivative of the
resistance with respect to current was taken to obtain d2 V /dI 2 , and finally converted
into d2 I/dV 2 using Eq. (4.7). The measurement of commercial Si Esaki tunnel diodes
also served as a check of the harmonic detection system and the approach used in
this thesis.
Fig. 8.5 shows their I-Vs at 300 K and 4.2 K, and their phonon spectra. The
phonon spectra agree with the Si phonon spectrum in [28], except that the voltage
resolution of this measurement was slightly lower. Specifically, even though the presence of the ∼56 meV LO phonon was evident (the left shoulder of the stronger TO
phonon peak), it was not clearly resolved from the nearby ∼59 meV TO phonon. The
better resolution of [28] was mainly due to a lower temperature, ∼1 K, being used.
The quality of the phonon spectra of 1N2927 and 1N2930A as demonstrated in
Fig. 8.5 indicates that the method of first-order numerical differentiation of the firstorder derivative measured by harmonic detection is a valid method which is capable
of producing low noise phonon spectra. The limited resolution in Fig. 8.5 was mainly
142
due to the temperature being used, i.e. 4.2 K. The resolution of phonon spectra can
be improved by using a temperature lower than 4.2 K and a correspondingly lower
ac excitation voltage.
The shape of the phonon spectrum of 1N2927 is compared to that of 1N2930A in
Fig. 8.6.
8.5.2
0/3/2 Structures with Zero to 44% Ge Content
5-nm Spacer Si RITDs
The phonon spectra of 5 nm spacer Si RITDs (no Ge content) of various sizes are
shown in Fig. 8.7.
The two strong ∼19 meV TA and ∼59 meV TO phonon peaks
in the phonon spectrum of Si Esaki tunnel diodes are still prominent in that of the Si
RITDs. But the positions of the phonon peaks shifts to higher voltages, as compared
to those of Esaki tunnel diodes, e.g. Fig. 8.5(e) and (f). As discussed before, nonignorable contact resistance in Si RITDs is the main reason for the apparent shift
in the measured d2 I/dV 2 vs. V curve. The structure of the ohmic contacts in
Si RITDs makes it difficult to obtain comparably low contact resistance as in Esaki
tunnel diodes. Contact resistance is one major issue in the study of phonon spectra in
Si/SiGe RITDs (when measurement of phonon energies is desired), as was examined
in Chapter 6.
The weak ∼47 meV LA phonon signal was still observable in the smaller (10 and
18 µm diameter) Si RITDs, but less obvious in the larger sized (50 and 75 µm) Si
RITDs.
The weak ∼56 meV LO phonon signal was not resolved from the nearby strong
∼59 meV TO phonon, because of the wide width of the ∼59 meV TO phonon as
observed.
143
0.2
6
1N2927 130 300K
0.15
4
Current (mA)
Current (mA)
1N2930A 300K
5
0.1
0.05
3
2
1
0
0
0
0.1
0.2
0.3
0.4
0.5
Voltage (V)
0.6
0.7
-1
0.8
0
0.1
0.2
0.3
0.4
0.5
Voltage (V)
(a)
0.007
0.016
0.012
Current (mA)
Current (mA)
0.004
0.003
0.002
0.01
0.008
0.006
0.004
0.001
0.002
0
10
20
30
40
50
Voltage (mV)
60
70
0
80
0
10
20
(c)
0.7
30
60
70
80
1N2930A 4.2K d(1/R)/dV
1N2930A
25
d I/dV (1/ΩV)
0.5
2
0.3
2
0.2
2
0.4
2
d I/dV (1/ΩV)
30
40
50
Voltage (mV)
(d)
1N2927 130 4.2K d(1/R)/dV
1N2927, -6.4 mV shift
0.6
0.1
20
15
10
5
0
0
-0.1
0.8
1N2930A 4.2K
0.014
0.005
0
0.7
(b)
1N2927 130 4.2K
-6.4 mV shift corrected
0.006
0.6
0
10
20
30
40
50
Voltage (mV)
60
70
-5
80
(e)
0
10
20
30
40
50
Voltage (mV)
60
70
80
(f)
Figure 8.5: The I-V characteristics at 300 K, at 4.2 K, and the d2 I/dV 2 from firstorder numerical differentiation of the measured conductance (or resistance)by first
harmonic detection for two commercial Si Esaki tunnel diodes: 1N2927 (left column)
and 1N2930A (right column).
144
35
4.2K d(1/R)/dV
1N2927, ×50, -6.4 mV shift
1N2930A
30
2
20
15
2
d I/dV (1/ΩV)
25
10
5
0
-5
0
10
20
30
40
50
Voltage (mV)
60
70
80
Figure 8.6: The shape of the phonon spectrum of 1N2927 is compare to that of
1N2930A.
10
5 nm All Si 4.2K d(1/R)/dV
10 µm, ×70, +6.2 mV shift
18 µm, ×20
50 µm, ×2.2
75 µm
2
2
d I/dV (1/ΩV)
8
6
4
2
0
-2
0
10
20
30
40
50
Voltage (mV)
60
70
80
Figure 8.7: Phonon spectra of 5-nm spacer Si RITDs of various sizes.
145
Notice that the widths of the TA and TO phonons in Si RITDs as shown in Fig. 8.7
are appreciably wider than those in Si Esaki diodes. Because the same measurement
system has been used to measure commercial Si Esaki diodes to obtain narrower
phonon peaks, the observed wider TA and TO phonon peaks in Si RITDs should be
due to the properties of the device being measured instead of a measurement artifact
(due to e.g. various broadening effects). It is likely that the weaker LA phonon peak
and the unresolved LO phonon peak also have greater widths than those in Si Esaki
diodes. Wide peak widths make it difficult, if not impossible, to fully resolve different
phonons signatures with small energy differences between them. Possible reasons for
the increased phonon peak include: (1) The layered structure of RITDs, especially
of thin spacers, as opposed to the bulk-like structure of Esaki diodes; (2) Increased
thermal broadening effect due to a higher local junction temperature as the result
of Joule heating of a higher current; and (3) Possible lower crystallinity of MBE Si
(layers) as compared to the (bulk) Si in Esaki diodes.
Compared to their respective TA phonon peaks, the TO phonon peaks are appreciably lower and wider in the 50 µm and 75 µm RITDs than those in the 10 µm
and 18 µm RITDs. One possible reason is that at 50 µm, the current, and hence the
Joule heating, are high enough to thermally broaden the TO peak appreciably.
0/3/2 Structures with 10%, 20%, 30%, 40%, and 44% Ge Content
At the time of the first generation low-temperature measurements, RITDs using
the 0/3/2 structure with 10%, 20%, 30%, 40%, and 44% Ge content were made with
polyimide (HD8001) as the dielectric layer which made wire bonding difficult. For a
very small number of fabricated RITDs, the bonding was successful.
For uniform
MBE growth, these bonded RITDs might be located outside the preferred area. The
146
0.05
0/3/2 10% Ge 750C 10µm
300 K
4.2 K
0.2
0.03
0.02
2
0.01
0
2
0.1
0/3/2 10% Ge 750C 10µm 4.2K SG(15,15,2,2)
0.04
d I/dV (1/ΩV)
Current (mA)
0.3
-0.01
-0.02
-0.03
-0.04
0
0
20
40
60
-0.05
80 100 120 140 160 180
Voltage (mV)
0
20
40
60
(a)
(b)
0.08
0/3/2 20% Ge 750C 18µm
300 K
4.2 K
0.2
2
0.04
0.02
0
2
0.1
0/3/2 20% Ge 750C 18µm 4.2K SG(15,15,2,2)
0.06
d I/dV (1/ΩV)
Current (mA)
0.3
80 100 120 140 160 180
Voltage (mV)
-0.02
-0.04
0
0
20
40
60
80
100
Voltage (mV)
120
-0.06
140
0
20
40
(c)
1.1
0.9
0.02
0.7
d I/dV (1/ΩV)
0.6
2
0.5
0.4
2
Current (mA)
140
0.3
0/3/2 40% Ge 750C 10µm 4.2K SG(15,15,2,2)
0.015
0.8
0.2
0.01
0.005
0
-0.005
0.1
0
120
(d)
0/3/2 40% Ge 750C 10µm
300 K
4.2 K
1
60
80
100
Voltage (mV)
0
50
100
150 200 250
Voltage (mV)
300
350
-0.01
400
(e)
0
50
100
150 200 250
Voltage (mV)
300
350
400
(f)
Figure 8.8: 0/3/2 structure with 10% (top row), 20% (middle row), and 40%
(bottom row) Ge content: I-V characteristics at 300 K and 4.2 K, and secondorder numerical derivative d2 I/dV 2 obtained from measured 4.2 K I-V data using
savgol(15,15,2,2).
147
low-temperature data from these RITDs are still included in this thesis. It is hoped
that the phonon spectra for these RITDs are still representative of those of the
structures.
In addition, this batch of RITDs used Pt/Al instead of Ti/Al for contacts to both
n-type and p-type Si, which formed a Schottky contact instead of an ohmic contact
with low contact resistance to n-type Si. This results in a voltage shift in the same way
as a series resistance does, except that in this case the equivalent series resistance,
in general, is nonlinear. The voltage shift due to poor ohmic contact and/or high
series resistance in the measuring circuit is, in one way, manifested as a peak voltage
significantly higher than about 0.1 V. Provided that the contact resistance is not too
high and is not too nonlinear, the shape of the I-V curve is mostly retained, even
though the voltage is shifted. Some qualitative conclusion can be drawn from the
shape of the I-V curve and its derivatives I-V curve and its derivatives, even though
quantitative measurement in voltage cannot be made.
The second-order derivative d2 I/dV 2 is obtained by numerical differentiation of
the measured I-V data, using Savitzky-Golay method with a second-order polynomial
and nL = nR = 15, i.e. savgol(15,15,2,2). Even though numerical differentiation
with Savitzky-Golay method still suffers from noise so that weak signals cannot be
reliably detected, strong signals are reasonably well preserved with appropriate choice
of smoothing parameters. This makes it possible to study strong signals, if not weak
signals, using numerical differentiation with appropriate smoothing.
Due to the difficulty in wire bonding on bonding pads on a polyimide dielectric
layer, no RITDs from 0/3/2 structure with 30% and 44% Ge content were successfully
bonded for low-temperature measurement. Fig. 8.8 shows, for 0/3/2 structure with
148
10%, 20% and 40% Ge content, the I-V characteristics at 300 K and 4.2 K, and
the phonon spectra obtained using numerical differentiation (with smoothing) of the
measured I-V data.
Despite the noise that is still present after smoothing, two peaks are prominent.
Comparing these phonon spectra (limited by the relatively high noise and inaccurate
voltage information due to non-ignorable contact resistance) obtained from Si/SiGe
RITDs with the phonon spectra of Si, Ge, and Si1−x Gex alloys obtained from Esaki
tunnel diodes in [28], the two peaks appear to be the TA and TO phonons of Si,
and no other apparent peaks are present in these spectra. Even though it will remain
inconclusive until the voltages at which these two peaks occur can be accurately
measured which requires good ohmic contacts with low contact resistance, these two
prominent peaks appear to be the TA and TO phonons of Si. Further, there appears
to be an absence of Ge influenced phonons. This may be explained by noting that the
tunneling barrier of the Si/SiGe RITDs consists of a SiGe layer and a pure Si layer.
The phonon spectrum of such a tunneling junction can therefore possibly contain
features of phonon spectra of pure Si, pure Ge, and possibly new features of the SiGe
alloys. It is also possible that a subset of thesis dominate the phonon spectrum of
the tunneling junction.
Subject to confirmation that these two prominent peaks are indeed the TA and
TO phonons of Si, the measured phonon spectra of the 0/3/2 structures with 10%,
20%, and 40% Ge content indicate that the TA and TO phonons of Si dominate the
phonon spectra, and any other possible phonon peaks (including Ge-like or Si1−x Gex like) are either too low in peak height to be detected or too close in peak position to
an observed peak to be resolved or both.
149
8.5.3
-1/4/n Structures with 60% Ge Content
Fig. 8.9 shows the I-V characteristics of the 18 µm RITDs from the -1/4/n structures with 60% Ge content.
The measured phonon spectra of the −1/4/n, n = 4, 6, 8, 10, structures with 60%
Ge content are shown in Fig. 8.10. The two major peaks located at about 20 mV
and 60 mV can be identified as the TA and TO phonons of Si, by the fact that the
tunneling barrier region includes a Si layer. The voltages of the highest points of
the peaks do not exactly correspond to the known Si TA phonon and TO phonon
energies, mainly because of the non-ignorable voltage shift due to contact resistance.
It could also come from an asymmetrical peak shape in which the peak “center” does
not coincide with the highest point of the peak.
-1/4/4 and -1/4/6 Structures
In the phonon spectra of the -1/4/4 and -1/4/6 structures, a shoulder on the left
of the major peak at about 20 mV is observed. The left shoulder may indicated the
presence of a new peak located to the left of the Si TA phonon peak, being not fully
resolved from the Si TA phonon peak because it is too close to the Si TA phonon
peak, with the finite measured peak widths.
Assuming the observed peak at about 20 mV indeed consists of two phonon peaks,
the individual peak locations, heights and widths can be extracted by a fitting procedure. As the primary objective of this thesis is to obtain the phonon energies, i.e. the
peak position, a simple symmetrical Gaussian shape is assumed in this thesis (care
must be taken in interpreting the fitted width). The following expression is used to
150
4
0.45
-1/4/4, 18 µm, 300K
3.5
0.35
3
0.3
2.5
Current (mA)
Current (mA)
-1/4/6, 18 µm, 300K
0.4
2
1.5
1
0.25
0.2
0.15
0.1
0.05
0.5
0
0
-0.05
-0.5
0
100
200
300 400 500
Voltage (mV)
600
700
-0.1
800
0
100
200
(a)
0.05
0.02
-1/4/8, 18 µm, 300K
600
700
-1/4/10, 18 µm, 300K
0.015
0.03
Current (mA)
Current (mA)
500
(b)
0.04
0.02
0.01
0.01
0.005
0
0
-0.01
0
100
200
300
400
Voltage (mV)
500
-0.005
600
0
100
(c)
0.002
200
300
400
Voltage (mV)
500
600
(d)
0.0016
-1/4/11, 18 µm, 300K
-1/4/12, 18 µm, 300K
0.0014
0.0015
0.0012
Current (mA)
Current (mA)
300
400
Voltage (mV)
0.001
0.0005
0.001
0.0008
0.0006
0.0004
0
0.0002
-0.0005
-0.0002
0
0
100
200
300
Voltage (mV)
400
500
(e)
0
100
200
300
Voltage (mV)
400
500
(f)
Figure 8.9: I-V characteristics of the 18 µm RITDs of the -1/4/n 60% Ge structures
at 300 K: (a) -1/4/4; (b) -1/4/6; (c) -1/4/8; (d) -1/4/10; (e) -1/4/11; (f) -1/4/12.
151
5
4
2.5
1
0
1.5
1
0.5
0
-1
-2
2
2
2
2
d I/dV (1/ΩV)
3
-1/4/6 60% Ge 4.2K d(1/R)/dV
10 µm, ×70, +1.9 mV shift
18 µm, ×20
50 µm, ×2.2
75 µm
3
2
2
d I/dV (1/ΩV)
3.5
-1/4/4 60% Ge 4.2K d(1/R)/dV
10 µm, ×60
18 µm, ×15
50 µm, ×1.8
75 µm, +6.2 mV shift
-0.5
0
20
40
60
Voltage (mV)
80
-1
100
0
10
20
(a)
0.8
0.12
d I/dV (1/ΩV)
0.5
2
0.3
2
0.2
2
0.4
0.1
0
80
0.08
0.06
0.04
0.02
0
-0.1
-0.2
70
-1/4/10 60% Ge 4.2K d(1/R)/dV
75 µm
0.1
2
d I/dV (1/ΩV)
0.6
60
(b)
-1/4/8 60% Ge 4.2K d(1/R)/dV
18 µm, ×22, +5.3 mV shift
18 µm, ×22
50 µm, ×2.4
75 µm
0.7
30
40
50
Voltage (mV)
0
10
20
30
40
50
Voltage (mV)
60
70
-0.02
80
(c)
0
10
20
30
40
50
Voltage (mV)
60
70
80
(d)
Figure 8.10: d2 I/dV 2 of the -1/4/n structures with 60% Ge content obtained by
numerical differentiation of the resistance (data) measured by detection of the first
harmonic. (a) -1/4/4; (b) -1/4/6; (c) -1/4/8; (d) -1/4/10.
152
fit the peak located at about 20 mV in the spectra,
f (x) = a1 e
−(
x−b1 2
)
c1
+ a2 e
−(
x−b2 2
)
c2
+ offset,
(8.1)
where the as, bs and cs correspond to the heights, centers, and widths of the two
peaks respectively. The results of the fit to the -1/4/4 and -1/4/6 structures are
plotted in Figs. 8.11 and 8.12, respectively. As can be seen, the quality of fit is very
good, indicating that the observed peak can indeed be interpreted as consisting of
two individual peaks. The fitted parameters are listed in Table 8.1.
2
0
0
20
40
Voltage (mV)
2
0.1
0.1
0
-0.1
80
60
0.2
-0.1
0
0
20
(a)
0
0
0
20
40
60
Voltage (mV)
80
4
2
2
0
0
2
1
4
2
1
-1/4/4 60% Ge, 75 µm, 4.2K; +6.2 mV shift
d I/dV (1/ΩV)
2
Current (mA)
3
-1/4/4 60% Ge, 50 µm, 4.2K
2
-1
-0.1
80
60
(b)
2
2
d I/dV (1/ΩV)
3
40
Voltage (mV)
-1
100
-2
(c)
0
20
40
60
Voltage (mV)
80
Current (mA)
-0.03
0
0.2
2
0.1
d I/dV (1/ΩV)
0.03
Current (mA)
0.2
2
d I/dV (1/ΩV)
0.06
0.3
-1/4/4 60% Ge, 18 µm, 4.2K
Current (mA)
0.3
-1/4/4 60% Ge, 10 µm, 4.2K
-2
100
(d)
Figure 8.11: Fit of the observed ∼20 mV peak with two peaks of Gaussian shape,
-1/4/4 structure with 60% Ge: (a) 10 µm; (b) 18 µm; (c) 50 µm; (d) 75 µm.
153
0.03
0.018
-1/4/6 60% Ge, 10 µm, 4.2K; +1.9mV shift
0
0.04
0.04
0.02
0
0
20
40
Voltage (mV)
60
-0.006
80
-0.04
0
0
20
(a)
2
0.6
1
0.3
0
0
0
0
20
40
Voltage (mV)
60
-0.2
80
-1
2
2
0.2
d I/dV (1/ΩV)
0.4
0
-0.02
80
-1/4/6 60% Ge, 75 µm, 4.2K
0.9
3
Current (mA)
0.4
2
2
d I/dV (1/ΩV)
0.8
-0.4
60
(b)
-1/4/6 60% Ge, 50 µm, 4.2K
0.6
1.2
40
Voltage (mV)
(c)
0
20
40
Voltage (mV)
60
Current (mA)
-0.01
0
Current (mA)
2
0.08
2
0.006
d I/dV (1/ΩV)
0.01
Current (mA)
0.012
2
2
d I/dV (1/ΩV)
0.02
-1/4/6 60% Ge, 18 µm, 4.2K
0.06
0.12
-0.3
80
(d)
Figure 8.12: Fit of the observed ∼20 mV peak with two peaks of Gaussian shape,
-1/4/6 structure with 60% Ge: (a) 10 µm; (b) 18 µm; (c) 50 µm; (d) 75 µm.
154
155
Offset
(1/ΩV)
−0.01920
−0.07164
−0.5979
−1.078
–
−0.02531
−0.2069
−0.4815
0.1253
1.177
2.918
19.47
19.59
19.42
5.828
5.500
5.354
0.04319
0.4132
0.9482
16.22
16.76
16.63
c1
a2
b2
(mV) (1/ΩV) (mV)
6.968 0.05693 20.37
6.585 0.1900 21.52
6.903
1.787
21.90
7.690
2.583
22.97
a1
b1
(1/ΩV) (mV)
0.04192 17.28
0.1676 18.49
1.470
19.58
3.042
21.75
2.638
2.085
1.972
c2
(mV)
3.005
3.257
2.743
2.880
0.00745
0.0695
0.157
b1
(mA)
0.0460
0.0493
0.500
1.05
Si0.4 Ge0.6 TA
0.00961
0.0845
0.192
b2
(mA)
0.0534
0.0593
0.575
1.13
Si TA
80.1
10.5
3.75
RC
(Ω)
31.3
47.6
5.57
3.78
15.6
16.0
16.0
TA
(mV)
15.8
16.1
16.8
17.8
Si0.4 Ge0.6
Cont. Res. Corrected
Table 8.1: Fit results of the major peak around 20 mV. a: +6.2 mV shift; b: voltage shift unknown.
RITD
(µm)
-1/4/4 10
18
50
75a
-1/4/6 10b
18
50
75
Current At
For the -1/4/4 structure, the fitted peak at lower voltage is slightly lower than
the fitted peak at higher voltage, except for the 75 µm RITD which might be due
to sample variation. The shoulder is weaker in the -1/4/6 structure than in the 1/4/4 structure, corresponding to a lower fitted left peak height to right peak height
ratio. This -1/4/4 vs -1/4/6 comparison is consistent with the interpretation that
the fitted peak at lower voltage is due to the TA phonon of the Si0.4 Ge0.6 layer in the
tunneling barrier region, and the one at higher voltage is due to the TA phonon of
Si in the tunneling barrier region, for two reasons: (1) the energies of the phonons in
(relaxed and strained) Si1−x Gex are lower than those of the corresponding phonons
in Si; (2) the Si layer in the tunneling barrier region is thicker in -1/4/6 structure,
or alternatively, the relative thickness of the Si0.4 Ge0.6 layer over the total tunneling
barrier thickness is smaller, hence a weaker Si0.4 Ge0.6 signal is expected in the -1/4/6
structures.
Assuming that the fitted left peak is indeed the TA phonon of the Si0.4 Ge0.6 layer
and the one on the right is the TA phonon of Si, the energies of these two phonons
can be measured from the obtained phonon spectra, as the electron charge e times
the voltage locations of the centers of the fitted peaks. However, it is noticed that
in the -1/4/4 structure the centers of the higher-voltage peaks, b2 , after multiplied
by the electron charge e, are higher than the Si TA phonon energy of 18.7±0.2 meV
as reported by Logan et al. in [28] by about 1 to 4 meV. The shifts of the phonon
peak positions to higher voltages were due to non-ignorable series resistance, mainly
the resistance of the two metal-to-semiconductor ohmic contacts, in the RITDs. In
the -1/4/6 structure, the fitted higher-voltage peak locations, after multiplication
by e, are within 1 meV of the Si TA phonon energy. These small voltage shifts
156
due to contact resistance were primarily due to the low currents in RITDs of this
structure (as compared to the -1/4/4 structure), instead of lower contact resistance,
since the RITDs of different structures were fabricated in the same batch and similar
contact resistance was expected. This is one example that a low current alleviates the
problem of voltage shift due to contact resistance. The intrinsic peak location can
be obtained if the voltage shift due to contact resistance can be corrected, provided
that the contact resistance, or more generally, the I-V characteristics of the metalsemiconductor contacts, are known. The two ohmic contacts in the RITDs have not
yet been characterized independently. However, the fact that the measured voltage
shift was not excessive means that the contact resistance was not too high. If all
contacts have linear I-V characteristics, i.e. ohmic contacts, or if the overall I-V
characteristic of all contacts is linear, the total contact resistance can be determined
by comparing the measured energy of a phonon with its known energy (when the
phonon energy is determined by the peak location in the second-order derivative
d2 I/dV 2 , it is still a good approximation as the I-V of the tunneling junction is
still largely linear, and the non-liearity is weak). If the overall I-V characteristic of
all contacts is nonlinear, then a reasonably good approximate voltage shift correction
can be obtained if a nearby point can be used as a reference or “calibration standard,”
as if the contacts have an overall linear contact resistance. Since the Si0.4 Ge0.6 TA
phonon has the Si TA phonon nearby in energy, an approximate contact resistance
correction can be obtained by using the Si TA phonon energy as the reference.
The contact resistance is estimated by comparing the measured Si TA phonon peak
position and the known Si TA phonon energy and dividing the voltage shift difference
by the current at which the peak is obtained. Then, using the current corresponding to
157
the observed Si0.4 Ge0.6 TA phonon peak, the voltage shift is obtained, and subtracted
from the measured Si0.4 Ge0.6 TA phonon peak position. The relevant experimental
data and the results are listed in Table 8.1.
The fact that the measured (and fitted) peak voltage shift increases with increasing
RITD size, i.e. increasing current, indicates that it is preferable to use smaller, and
thus lower-current, RITDs for phonon energy measurement, in order to reduce the
effect of contact resistance. A strong dependence of the Si0.4 Ge0.6 TA phonon energy
on current remains after the estimated contact resistance correction, over the large
current level span in the -1/4/4 structure, which has the highest current density
among the -1/4/n structures measured. This was most likely due to the nonlinear
I-V characteristics of the contacts, being manifested as voltage shift more clearly at
higher current.
For the -1/4/4 RITD with a diameter of 10 µm RITD, it is noticed that even
though it produced a phonon spectrum whose shape is consistent with those from
other RITDs, the I-V characteristics at 4.2 K are not consistent with that at room
temperature (the peak current at 4.2 K being about half of that at room temperature
for most Si/SiGe RITDs measured) or the 4.2 K I-V of diodes of other sizes but of
the same structure (current scaling with area of diode assuming a constant current
density). With a 4.2 K I-V characteristic that shows noticeably weaker inflections
than those from other RITDs, the second-order derivative has a weak magnitude
with regard to the current level of the I-V characteristic, and is noticeably noisier
than those from other RITDs. This could be due to an unknown abnormality of this
device. The data from this diode are still included in this thesis for completeness,
but is not used for further analysis.
158
The Si0.4 Ge0.6 TA phonon energy after estimated contact resistance correction
vs current is plotted in Fig. 8.13. In both the -1/4/4 and -1/4/6 structures, the
extrapolation of the trends to zero current yields the same improved estimate of the
Si0.4 Ge0.6 TA phonon energy of about 15.8 meV, with an estimated total measurement
error of about ±1 meV. This is consistent with that the additional 2 nm Si grown
on top of the Si0.4 Ge0.6 layer for the -1/4/6i RITD structures, which is expected to
have negligible effect on the residue strain in the Si1−x Gex , and therefore the phonon
Estimated Si0.4Ge0.6 TA Phonon Energy (meV)
energies of, the SiGe layer (in the n-on-p structures measured).
18
-1/4/4
-1/4/6
17
16
15
0
0.2
0.4
0.6
0.8
Current (mA)
1
1.2
Figure 8.13: Energy of the Si0.4 Ge0.6 TA phonon after estimated contact resistance
correction vs. current. In the case of the -1/4/4 structure, the dependence of the estimated Si0.4 Ge0.6 TA phonon on current indicates that the contact resistance correction
is incomplete, especially at higher currents. This was most likely due to non-linear
I-V characteristics of the contacts, manifested especially at higher currents. In both
-1/4/4 and -1/4/6 structures, the extrapolation of the trends to zero current yields
the same improved estimate of the Si0.4 Ge0.6 TA phonon energy of about 15.8 meV,
with an estimated total measurement error of about ±1 meV. Note that data from
the -1/4/4 10 µm RITD is not included in this plot.
159
There might be a Si0.4 Ge0.6 TO phonon in the other broad peak, located around
60 mV, in the measured spectra. If this is the case, then there are at least three
individual peaks, Si0.4 Ge0.6 TO, Si LO and Si TO, that make up the observed broad
peak as indicated by peak fit analysis. But, as the accurate peak shapes of the
various phonons involved are unknown, fitting with three and more peaks is difficult
and unreliable. Therefore, no effort was made to fit the observed broad peak at about
60 mV with three or more peaks in an attempt to extract information of additional
phonons.
-1/4/8 and -1/4/10 Structures
A similar shoulder at about 20 mV may or may not be visible in the -1/4/8
structure, but it is completely absent in the -1/4/10 RITD structure.
Actually, the measured phonon spectrum of the -1/4/10 structure is very similar
to the phonon spectrum of the Si Esaki diode, and the two spectra are indistinguishable from each other. This may be explained by the fact that, of the -1/4/n
structures measured, the -1/4/10 structure had the highest “weighted average” of Si
in the tunneling barrier layer and thus was expected to have the highest “weighted
average” of Si phonon spectral features in its phonon spectrum. It is determined that
the Si phonons completely dominated the phonon spectrum of the -1/4/10 structure
with 60% Ge content, and no phonons of SiGe were observed within measurement
limitations. In addition, the measured phonon spectrum was not appreciably affected
by voltage shift due to contact resistance. This is the result of the extremely low
current density of the -1/4/10 RITD structure, and not from better ohmic contacts
than other samples, as the -1/4/n samples were fabricated in the same batch. This
160
is another example that a low current alleviates the problem of voltage shift due to
contact resistance.
-1/4/11 and -1/4/12 Structures
Because of the increased Si layer thickness, the -1/4/11 and -1/4/12 structures
should have even lower current density and exhibit an even stronger Si phonon dominance in their phonon spectra. Therefore the -1/4/11 and -1/4/12 structures were
not measured yet at the time of this thesis.
8.5.4
Discussion
The energy of the TA phonon of Si0.4 Ge0.6 was estimated to be ∼12 meV by Logan
et al. [28]. Noticing that the SiGe layers in the Si/SiGe RITDs grown on Si substrates
were compressively strained, while the SiGe used in the measurement in [28], being
bulk, was not strained, the about 4 ± 1 meV increase in the TA phonon energy should
be due to the residue strain in the Si0.4 Ge0.6 layer in the Si/SiGe RITDs. These
results show that electron tunneling spectroscopy is an effective way of characterizing strain in the Si1−x Gex layer in the Si/SiGe RITDs and other similar structures.
Electron tunneling spectroscopy provides additional information by measuring the
phonon energies, supplementing the characterization of strain by Raman scattering
and the characterization of the crystal structure and measuring the lattice constant
by X ray diffraction (XRD). In addition, in contrast to Raman scattering and XRD
which are fundamentally surface analysis techniques, electron tunneling spectroscopy
is able to characterize layers buried below the surface. If the phonon spectrum can
be calibrated, quantitative measurement of strain by electron tunneling spectroscopy
is possible.
161
It is reasonable to expect that the “weighted average” of Si1−x Gex phonons in the
phonon spectrum of a Si/SiGe RITD depends on the “weighted average” of Ge content in the spacer region of the RITD. The spacer region of a typical Si/SiGe RITD
is a composite of a Si layer and a Si1−x Gex layer. Therefore, the “weighted average”
of Ge depends on: (1) the ratio of the thickness of the Si1−x Gex to the total spacer
thickness (i.e. the sum of the Si layer thickness and the Si1−x Gex layer thickness), and
(2) the Ge content x in the Si1−x Gex alloy. The 0/3/2 structures considered in the
previous section had thicker SiGe layers (60% total spacer thickness) as compared to
the -1/4/4 structure (50% total spacer thickness). However, of the 0/3/2 structures
for which the phonon spectra have been measured, the highest Ge content was only
40%, lower than that of the -1/4/4 structure with 60% Ge content. A relatively
thicker Si1−x Gex layer is expected to increase the Si1−x Gex phonon peak height to
Si phonon peak height ratio, making the Si1−x Gex phonon peaks easier to identify.
However, a lower Ge content in the Si1−x Gex layer results in SiGe phonons closer to
those of the corresponding Si phonons in energy, making it difficult to resolve the
SiGe phonons from the Si phonons. The intrinsic phonon peak width, (temperaturedependent) thermal broadening, and in the case of harmonics detection system the
(modulation voltage-dependent) modulation voltage broadening will ultimately determine the energy resolution limit.
8.5.5
Summary
For typical Si/SiGe RITDs using a composite spacer consisting of a Si1−x Gex layer
and a Si layer, the Si phonons were found to dominate the measured phonon spectra.
162
The TA phonon of Si0.4 Ge0.6 was identified in the phonon spectra of Si/SiGe
RITDs of -1/4/4 and -1/4/6 structures, and the energy was estimated to be ∼
15.8 ± 1 meV after approximate contact resistance correction assuming the I-V characteristics of the ohmic contacts were linear. The measured Si0.4 Ge0.6 TA phonon
energy is higher than that of relaxed Si0.4 Ge0.6 of ∼12 meV reported in literature.
The increase in the TA phonon energy of about 4±1 meV is attributed to the compressive strain in the Si0.4 Ge0.6 layers in the Si/SiGe RITDs grown on Si substrates.
The TO phonon of Si0.4 Ge0.6 might be present in the measured phonon spectra, but
could not be unambiguously identified. The weaker LA and LO phonons of Si0.4 Ge0.6
were not observed.
No phonons of Si0.4 Ge0.6 were observed in the RITDs of the -1/4/8 and -1/4/10
structures. This should be the result of the higher weighted average of Si component
in the tunneling barrier region.
The TA phonons of Si1−x Gex with x = 0.10, 0.20, and 0.44 were not found in
the phonon spectra of Si/SiGe RITDs with the 0/3/2 structure, where the weighted
average of Ge was lower.
Contact resistance in Si/SiGe RITDs is an important issue if accurate measurements of the phonon energy is desired. It was also observed that, using RITDs of
lower current (dependent on current density of the structure and the size of the diode),
the problem of voltage shift due to contact resistance was alleviated. A current of
30 µA at a bias of about 80 mV was low enough so that the contact resistance of the
current (n+,p+)Si/Ti/Au ohmic contact scheme did not result in appreciable voltage
shifts. However, low current is just a workaround (with limitations), not a solution,
to the problem of voltage shift in the measurement of phonon energies by electron
163
tunneling spectroscopy. Ohmic contacts with lower contact resistance need to be employed in the Si/SiGe RITDs for more accurate measurement of the phonon energies.
The contact schemes studied in the previous chapter are potential candidates for this
purpose.
164
CHAPTER 9
CONCLUSIONS AND FUTURE WORK
Preliminary results of the TA phonon of compressively strained Si1−x Gex with
x = 0.6 were obtained in this thesis. Still, it is worthwhile to obtain a more complete
data set of the phonon energies of strained Si1−x Gex as a function of x, and possibly
as a function of strain also. The methods and procedures outlined here may serve as
a guide for the eventual completion of these studies. Barriers still remain, however.
The phonon spectra in Si1−x Gex alloys measured in Esaki diodes provided some of
the important starting information in this field. The development of Si/SiGe resonant
interband tunnel diodes provides a new platform on which new information can be
obtained. In Si/SiGe RITDs grown on Si substrates, the Si1−x Gex layer in the spacer
region is compressively strained. By using tunneling spectroscopy, some information
on the phonon modes in the strained Si1−x Gex layer can be obtained. However, there
are many challenges along the way.
For the purpose of low-temperature measurement and the study of the phonon
spectrum, the 0/3/2 structure suffers from high current density. The problems associated with a high current are: (1) local Joule heating leading to possible excessive
thermal broadening of phonon peaks, and (2) the voltage shift due to contact resistance becoming exacerbated.
165
In order to study the effect of Ge content on the phonon spectra of Si/SiGe RITDs,
it may be desirable to use an all Si1−x Gex spacer, but this could have catastrophic
effects on dopant interdiffusion. In order to avoid excessive Joule heating which leads
to thermal broadening of the phonon peaks, and to alleviate the problem of voltage
shift due to contact resistance, it is desirable to have a very low current, preferably by
reducing the current density by increasing the spacer thickness. However, the growth
of a thick Si1−x Gex on a Si substrate is limited by the critical thickness at which a
misfit dislocation network is created to relieve the strain energy. Adding a pure Si
layer in the spacer region reduces the current density, but it also adds a Si component
to the phonon spectrum of the Si/SiGe RITDs, which makes studying the effects of
the Ge content difficult and intertwined. The alternative way of reducing the current
flowing through the RITDs is to reduce the diode size. 5 µm diameter RITDs can be
reliably fabricated using contact mode photolithography. 2 or even 1 µm RITDs may
be fabricated using contact mode photolithography, though it is quite challenging.
RITDs of sizes below 1 µm require an optical stepper or direct-write electron beam
lithography. It should be noted that submicron Si/SiGe RITDs have not been fully
fabricated and characterized yet, and ohmic contact to submicron RITDs may pose a
problem in the quest for ultra-low contact resistances. These studies are continuing.
For the purpose of studying the effect of Ge content on the phonon spectrum, it
may be difficult to reduce the current density to a desired level, as discussed above. If
the contact resistance issue is solved, it will greatly help the study of phonon spectra
of strained Si1−x Gex layers.
166
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