Lecture 11: Light-emitting diodes
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Internal quantum efficiency
Double heterostructures
Extraction efficiency
External quantum efficiency
Power conversion efficiency
Responsivity
Spectral distribution
Modulation
Reading: Senior Chapter 7
Keiser 4.2
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LED, SOA, LD
n
p
Light-emitting diode
n
p
Semiconductor
Optical Amplifier
n
p
Laser Diode
2
Light-emitting diodes (LEDs)
• The light output of an LED is the spontaneous emission generated by
radiative recombination of electrons and holes in the active region of
the diode under forward bias.
• The semiconductor material is direct-bandgap to ensure high
quantum efficiency, often III-V semiconductors.
• An LED emits incoherent, non-directional, and unpolarized
spontaneous photons that are not amplified by stimulated emission.
• An LED does not have a threshold current. It starts emitting light as
soon as an injection current flows across the junction.
3
Electron energy
A heavily doped p-n junction under forward-bias
injection
e(V0-V)
EFc
Eg
hu
hu
Eg
eV > Eg
EFv
injection
• The internal photon flux: F = hint i/e
position
hint: int. quantum efficiency
(injection electroluminescence)
4
Internal quantum efficiency
• The internal quantum efficiency hint of a semiconductor material:
the ratio of the radiative electron-hole recombination coefficient to the
total (radiative and nonradiative) recombination coefficient.
• This parameter is significant because it determines the efficiency of
light generation in a semiconductor material.
• Recall that the total rate of recombination = r n p [cm-3 s-1]
• If the recombination coefficient r is split into a sum of radiative and
nonradiative parts, r = rr + rnr, the internal quantum efficiency is
hint = rr / r = rr / (rr + rnr)
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• The internal quantum efficiency may also be written in terms of the
recombination lifetimes as t is inversely proportional to r.
• Define the radiative and nonradiative recombination lifetimes tr and tnr
1/t = 1/tr + 1/tnr
• The internal quantum efficiency is then given by rr/r = (1/tr)/(1/t)
hint = t / tr = tnr / (tr + tnr)
*Semiconductor optical sources require hint to be large
(in typical direct bandgap materials tr ≈ tnr).
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Order-of-magnitude values for recombination
coefficients and lifetimes
material
rr(cm3 s-1)
tr
tnr
t
hint
Si
10-15
10 ms
100 ns
100 ns
10-5
GaAs
10-10
100 ns
100 ns
50 ns
0.5
*assuming a lightly doped n-type material with a carrier concentration no = 1017 cm-3 and a defect
concentration of 1015 cm-3 at T = 300 K
• The radiative lifetime for bulk Si is orders of magnitude longer than its
overall lifetime because of its indirect bandgap. This results in a small
internal quantum efficiency.
• For GaAs, the radiative transitions are sufficiently fast because of its
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direct bandgap, and the internal quantum efficiency is large.
Electroluminescence in the presence of carrier injection
• The internal photon flux F (photons per second), generated within a
volume V of the semiconductor, is directly proportional to the carrierpair injection rate R (pairs/cm3-s).
• The steady-state excess-carrier concentration Dn = Rt (recombination
rate = injection rate), where t is the total recombination lifetime
(1/t = 1/tr + 1/tnr).
• The injection of RV carrier pairs per second therefore leads to the
generation of a photon flux F = hint RV photons/s.
F = hint RV = hint V Dn/t = V Dn/tr
Or
F = hint V (i/e)/V = hint i/e
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Enhancing the internal photon flux
F  Dn/tr
=> Need higher Dn, shorter tr
1. The excess carriers in a LED with homojunction (same materials on
the p and n sides) are neither confined nor concentrated but are spread
by carrier diffusion.
 The thickness of the active layer in a homojunction is normally
on the order of one to a few micrometers, depending on the diffusion
lengths of electrons and holes.
2. There is no waveguiding mechanism in the structure for optical
confinement. It is therefore difficult to control the spatial mode
characteristics (essential for laser diode).
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A p+-n+ homojunction under forward-bias
Eg
hu
Eg
EFv
Optical
Refractive
Excess carrier
field
index
distribution
distribution profile
EFc
~ 1 - few mm
Photons generated
can be absorbed
outside the active
region
carriers diffuse
x
homostructure
x
no waveguiding
x
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Double heterostructures
• Very effective carrier and optical confinement can be simultaneously
accomplished with double heterostructures. A basic configuration can
be either P-p-N or P-n-N (the capital P, N represents wide-gap materials,
p, n represents narrow-gap materials). The middle layer is a narrow-gap
material. (e.g. Ga1-yAlyAs-GaAs-Ga1-xAlxAs)
• Almost all of the excess carriers created by current injection are
injected into the narrow-gap active layer and are confined within this
layer by the energy barriers of the heterojunctions on both sides of the
active layer.
• Because the narrow-gap active layer has a higher refractive index than
the wide-gap outer layers on both sides, an optical waveguide with the
active layer being the waveguide core is built into the double
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heterostructure.
A P+-p-N+ double heterostructure under forwardbias (GaAlAs/GaAs/GaAlAs)
EFc
DEc
Optical
Refractive
Excess carrier
field
index
distribution
distribution profile
hu
EFv
DEv
wide-gap outer
layers are
transparent to the
optical wave
~ 0.1 mm
carriers confined
x
~few %
Double
heterostructure
x
waveguiding
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x
Double heterostructures
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e.g. The radiative and nonradiative recombination lifetimes of
the minority carriers in the active region of a LED are 60 ns and
100 ns. Determine the total carrier recombination lifetime and
the power internally generated within the device when the peak
emission wavelength is 870 nm at a driving current of 40 mA.
• The total carrier recombination lifetime is given by
t = trtnr / (tr +tnr) = 37.5 ns
• The internal quantum efficiency
hint = t / tr = 0.625
=> Pint = hint i/e • (1240 eV-nm / 870 nm)  36 mW!
(However, this power level is not readily out-coupled from the device! )
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Output photon flux and efficiency
• The photon flux spontaneously generated in the junction active region
is radiated uniformly in all directions. However, the flux that emerges
from the device (output photon flux) depends on the direction of
emission.
A
B
n
qc
C
l1
active region
p
e.g. Ray A at normal incidence is partially reflected. Ray B at oblique
incidence suffers more reflection. Ray C lies outside the critical angle
and thus is trapped in the structure by total internal reflection.
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• The photon flux (optical power) traveling in the direction of ray A
(normal incidence) is attenuated by the factor
h1 = exp(-al1)
where a is the absorption coefficient (cm-1) of the n-type material, and l1
is the distance from the junction to the surface of the device.
• For normal incidence, reflection at the semiconductor-air boundary
permits only a fraction of the light to escape (recall Fresnel reflection)
reflectance
h2 = 1 – [(n-1)2/(n+1)2] = 4n / (n+1)2
where n is the refractive index of the semiconductor material.
(For GaAs, n = 3.6, h2 = 0.68. The overall transmittance for the
photon flux (power) traveling in the direction of ray A is hA=h1 h2)
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• The photon flux traveling in the direction of ray B has farther to travel
suffers a larger absorption;
a larger incident angle at the semiconductor-air interface
=> a greater Fresnel reflection loss
=> hB < hA
• The photon flux emitted along directions lying outside a cone of
critical angle qc = sin-1(1/n) (ray C) suffer total internal reflection.
hC = 0
(for n = 3.6, qc = 16o)
Only rays that lie inside the cone of critical angle can escape – so
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called “escape cone”
Escape cone
A
r
qc
An arbitrary point source
with spherical emission
in the active junction
• The fraction of light lies within the escape cone from a point
source:
A / 4pr2 = (1 – cos qc)/2 ≈ 1/4n2
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• Here we estimate the fraction of the total generated photon flux that
lies within the escape cone. The area of the circular disk cap atop this
cone is (assuming a spherical emission distribution radius r)
qc
A = ∫ 2pr sinq r dq = 2pr2 (1 - cos qc)
0
• The fraction of the emitted light that lies within the solid angle
subtended by this escape cone is A/4pr2
=> h3 = ½ (1 – cos qc) = ½ (1 – (1 – 1/n2)1/2) ≈ 1/4n2
e.g. For a material with refractive index n = 3.6, only 1.9% of the
total generated photon flux lies within the escape cone.
• The efficiency with which the internal photons can be extracted
from the LED structure is known as the extraction efficiency he.
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Output photon flux and external quantum efficiency
• The output photon flux Fo is related to the internal photon flux F
Fo = he F = he (hint i/e)
where the extraction efficiency he specifies how much of the internal
photon flux is transmitted out of the structure.
• A single quantum efficiency that accommodates both he and hint
is the external quantum efficiency hext
hext ≡ he hint
=> The output photon flux Fo = hext i/e
=> hext is simply the ratio of the output photon flux Fo to the injected
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electron flux i/e.
Output optical power
• The LED output optical power Po:
Po = hu Fo = hext hu i/e
• The internal efficiency hint for LEDs ranges between 50% and just
about 100%, while the extraction efficiency he can be rather low
(may stretch up to 50%).
The external quantum efficiency hext of LEDs is thus typically below
50%.
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Responsivity
• The responsivity R of an LED is defined as the ratio of the emitted
optical power Po to injected current i, i.e. R = Po/i
R = Po/i = hu Fo/i = hext hu/e
• The responsivity in W/A, when lo is expressed in mm,
R = hext 1.24/lo
• The linear dependence of the LED output power Po on the injected
current i is valid only when the current is less than a certain value
(say few tens of mA on a typical LED). For larger currents, saturation
causes the proportionality to fail (known as “current droop”).
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Output optical power Po (mW)
Optical power at the output of an LED vs. injection current
*saturation at high injection
current (known as
“current droop”)
Slope = responsivity R = hext 1.24/l0
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Current i (mA)
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Power-conversion efficiency
• Another measure of performance is the powerconversion efficiency (or wall-plug efficiency),
defined as the ratio of the emitted optical power Po to the
applied electrical power.
hc ≡ Po / iV = hext hu/eV
where V is the voltage drop across the device
• Note that hc  hext because hu  eV, where eV = EFc –
EFv in a degenerate (heavily doped) junction.
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Surface-emitting and edge-emitting
• Surface-emitting diodes radiate from the face parallel to the p-n
junction plane.
(The light emitted in the opposite direction can be reflected by a
metallic contact.)
• Edge-emitting diodes radiate from the edge of the junction region.
Multimode
fiber
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Spatial pattern of LED emitted light
• The far-field radiation pattern for light emitted into air from a planar
surface-emitting LED is given by a Lambertian distribution:
I(q) = Io cos q
=> The intensity decreases to half its value at q = 60o
I(q)
Planar
surface
q
Io
Lambertian spatial pattern in the
absence of a lens
• In contrast, the radiation pattern from edge-emitting LEDs (and laser
diodes) is usually quite narrow and can often be empirically described
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by the function (cos q)s, with s > 1.
MM fiber
SM fiber
surface-emitting
edge-emitting
• The coupling efficiency hcouple (assuming Lambertian spatial pattern):
qa
p/2
hcouple =  I(q) sin q dq /  I(q) sin q dq
0
= sin2 qa = NA2
0
Qa: Fiber acceptance angle
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Epoxy-encapsulated LEDs
• Transparent epoxy lenses of different shapes alter the emission pattern
in different ways (e.g. hemispherical vs. parabolic lenses)
LED chip
• Epoxy lenses can also enhance the extraction efficiency he – a lens
with a refractive index close to that of the semiconductor reduces index
mismatch, and thus optimizes the extraction of light from the
semiconductor into the epoxy. (epoxy : semiconductor ~ 1.5 : 3.5)
In practice, epoxy lenses can yield a factor of 2-3 enhancement in
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light extraction.
Enhancing the extraction efficiency
• LED die geometry designs vs. simple planar-surface-emitting LEDs
(limited by Fresnel reflection)
• Roughen the planar surface - permitting rays beyond the critical
angle to escape via scattering (commonly adopted by industry)
• Contact geometry designs - Top-emitting LEDs make use of currentspreading layers, which are transparent conductive semiconductor layers
(typically indium-tin-oxide (ITO)) that spread the region of light
emission beyond that surrounding the electrical contact.
• Also include the use of reflective and transparent contacts,
transparent substrates (flip-chip packaging allows light to be extracted
through the substrate), distributed Bragg reflectors, 2D photonic crystals,
etc.
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Summary: LED efficiencies
• Internal quantum efficiency hint - only a fraction of the electron-hole
recombinations are radiative in nature
• Extraction efficiency he – only a small fraction of the light generated
in the junction region can escape from the high-index medium
• External quantum efficiency hext = he hint (can be measured from
the responsivity R = Po/i)
• Power-conversion (wall-plug) efficiency hc – efficiency of converting
electrical power to optical power
• Coupling efficiency hcouple – only a fraction of the light emitted from
the LED can be coupled to an optical fiber
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LED spectra
• The spectral intensity Rsp(u) of light spontaneously emitted from a
semiconductor in quasi-equilibrium (upon injection) can be determined
as a function of the concentration of injected carriers Dn.
• The spectral intensity of the direct band-to-band injectionelectroluminescence has precisely the same shape as the thermalequilibrium spectral intensity, but its magnitude is increased by the
factor exp [eV/kBT] = exp [(EFc – EFv)/kBT], which can be very large in
the presence of injection upon a forward-bias voltage V.
(assuming EFc, EFv within the bandgap for this enhancement factor)
Eg
1.2
1.3
1.4
2kBT
1.5
1.6
1.7
hu
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Spectral intensities vs. wavelength for LEDs
*LEDs are
broadband
incoherent
sources.
0.2
0.3
0.4
0.5
Wavelength mm
0.6
0.7
AlN: the largest III-nitride bandgap, emitting at 210 nm
AlGaN: mid and near UV
InGaN: violet, blue, and green
AlInGaP: yellow, orange, and red
InGaAsP: near IR (1.3 – 1.55 mm)
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Modulation
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Direct current modulation
• An LED can be directly modulated by applying the modulation signal
to the injection current, an approach known as direct current modulation.
• There are two factors that limit the modulation bandwidth of an LED:
the junction capacitance and the diffusion capacitance.
• Because an LED is operated under a forward bias, the diffusion
capacitance is the dominating factor for its frequency response.
• The diffusion capacitance is a function of the carrier lifetime t, which
is the total carrier recombination lifetime (1/t = 1/tr + 1/tnr),
because it is associated with the injection and removal of carriers in the
diffusion region in response to the modulation on the injection current.
The intrinsic speed of an LED is primarily determined by the lifetime
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of the injected carriers in the active region.
• For an LED that is biased at a DC injection current level io and is
modulated at an angular frequency W = 2pf with a modulation index m,
the total time-dependent current that is injected to the LED is
i(t) = io + i1(t) = io (1 + m cos Wt)
• In the linear response regime under the condition that m << 1 (i.e.
small-signal modulation), the output optical power of the LED in
response to this modulation can be expressed as
P(t) = Po + P1(t) = Po [1 + |r| cos (Wt – j)]
where Po is the constant optical output power at the bias current level
of io, |r| is the magnitude of the response to the modulation, and j is the
phase delay of the response to the modulated signal (due to the carrier
lifetime t).
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• For an LED modulated in the linear response regime, the complex
response as a function of modulation frequency W is (following a RC
low-pass filter analysis)
r(W) = |r| eij = m/(1 – iWt)
• The frequency response and modulation bandwidth of an LED are
usually measured in terms of the electrical power spectrum of a
broadband, high-speed photodetector.
• In the linear operating regime of the photodetector, the photodetector
current is linearly proportional to the optical power of the LED.
The electrical power spectrum of the detector output is proportional
to |r|2:
R(f) = |r(f)|2 = m2/(1 + 4p2f2t2)
A 3-dB bandwidth of
f3dB = 1/2pt
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Normalized modulation response
R(f)/R(0) (dB)
Normalized current-modulation frequency response of an
LED measured in terms of the electrical power spectrum of
a photodetector
0
t = 10 ns
-1
*in electronics,
f3dB ≈ 0.35/rise time
-2
=> rise time ≈ 2.2 t
-3
-4
f3dB = 15.9 MHz
-5
0
5
10
15
Modulation frequency, f (MHz)
20
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Modulation bandwidth
• The spontaneous carrier lifetime t is normally on the order of a
few hundred to 1 ns for an LED.
The modulation bandwidth of an LED is typically in the range
of a few megahertz to a few hundred megahertz.
• A modulation bandwidth up to 1 GHz can be obtained with a
reduction in the internal quantum efficiency (hint = t/tr) of the LED by
reducing the carrier lifetime to the sub-nanosecond range.
• Aside from this intrinsic response speed determined by the carrier
lifetime, the modulation bandwidth of an LED can be further limited
by parasitic effects from its electrical contacts and packaging, as well
as from its driving circuitry.
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Power-bandwidth product
A 3-dB bandwidth
f3dB = 1/2pt
=> hint f3dB = (t/tr) (1/2pt) = 1/2ptr
• One can obtain a certain internal-quantum-efficiency-bandwidth
product by choosing the semiconductor with a certain radiative lifetime.
At an injection current i, the output optical power and the smallsignal modulation bandwidth of an LED have the following powerbandwidth product (i.e. a tradeoff between power and bandwidth):
Pof3dB = he hint (i/e) hu (1/2pt) = he (i/e) hu (1/2ptr)
At a given injection level, the modulation bandwidth of an LED is
inversely proportional to its output power. A high-power LED tends to
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have a low speed, and vice versa. (P0f3dB  i)