EE 529 Semiconductor Optoelectronics
– Optical Processes and LED
EE529 Semiconductor
Optoelectronics
Optical Processes and Light Emitting Diodes
1.
2.
3.
4.
5.
Band-to-band optical transitions
Absorption spectrum and mechanisms
Spontaneous emission
LED principles and efficiency
Frequency response and modulation bandwidth
Reading: Liu, Sec. 13.2, 13.4-13.5, 13.7
Reference: Bhattacharya, Sec. 3.1-3.4, 5.4-5.5, 5.8
Lih Y. Lin
EE 529 Semiconductor Optoelectronics
– Optical Processes and LED
Schrödinger’s Equation and Probability
Schrödinger’s Equation:
∂Ψ
2 ∂ 2Ψ
i
=
−
+ V
Ψ

2
∂t 
2m
∂x Potential energy


Total energy
Ψ ( x, t )
Kinetic energy
Position operator: x
Momentum operator:
Energy operator: i
∂
∂t
 ∂
i ∂x
: Wave function of a matter wave
2
∫a | Ψ ( x, t ) | dx : Probability of finding the particle between a and b at time t
b
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EE 529 Semiconductor Optoelectronics
– Optical Processes and LED
Solving Schrödinger Equation
Assume time-independent potential V(x).
1. Solve the time-independent Schrödinger Equation.
 2 d 2ψ
−
+ V ψ = Eψ
2
2m dx
Obtain ψn(x) with associated energy En.
2. Initial wave function:
∞
Ψ ( x,0) =∑ cn ψ n ( x)
n =1
Cn can be obtained by matching initial conditions.
3. Wave function at subsequent time t:
∞
Ψ ( x, t ) =∑ cn ψ n ( x)e
− iEn t / 
∞
=∑ cn Ψ n ( x, t )
=
n 1=
n 1
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EE 529 Semiconductor Optoelectronics
– Optical Processes and LED
Infinite Square Well Potential
V(x)
Eigenfunction:
2
 nπ 
ψ ( x) = sin  x 
a
 a 
E3
E2
E1
0
a
Quantized energy:
x
 2 kn 2 n 2π 2  2
En =
=
2m
2ma 2
Wave function:
∞
Ψ ( x, t ) =
∑ cn
n =1
2
 nπ
sin 
a
 a

x  e − i ( n π  / 2 ma ) t

2 2
2
Expectation value of the energy:
∞
H = ∑ | cn |2 En
n =1
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EE 529 Semiconductor Optoelectronics
– Optical Processes and LED
Direct Bandgap vs. Indirect Bandgap
Direct bandgap
E
• Recall that photon momentum is very small,
compared to electron momentum.
CB
Direct Bandgap
Ec
Eg
( p = k )
Photon
• Momentum conservation can be satisfied in
direct bandgap semiconductors.
• Can be good photon absorbers and emitters.
Ev
VB
–k
k
(a) GaAs
Indirect bandgap
E
E
Indirect Bandgap, Eg
CB
kcb
–k
VB kvb
(b) Si
Ec
CB
Er
Ev
Ec
Phonon
Ev
VB
k
–k
(c) Si with a recombination center
• Momentum conservation
cannot be satisfied with
photon-only process in
indirect bandgap
semiconductors.
• Can be good photon
k absorbers, but not good
photon emitters.
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EE 529 Semiconductor Optoelectronics
– Optical Processes and LED
Band-to-Band Absorption and Recombination
Momentum of photons and electrons p = h/λ →
Photon momentum << Electron Momentum
Absorption and emission
in direct bandgap
Emission in
indirect bandgap
Absorption in
indirect bandgap
Defect center
ε ph = ω
εg + ε p
ω=
e
ω=
εg − ε p
a
e.g. GaAs, InP
Involves emission of a phonon
e.g. Si, Ge
Involves absorption or
emission of a phonon
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EE 529 Semiconductor Optoelectronics
– Optical Processes and LED
Exercise: Determine bandgap and phonon
energies from absorption experiment
The figure below shows the absorption for Ge
at 300K and 77K. Analyze the 300K data to
obtain the value of the direct bandgap, the
indirect bandgap, and the phonon energy
participating in the indirect transitions.
Note: The band structure of Ge shows possibilities of
indirect and direct transitions. The values we obtain from
this exercise will be different from the calculated data at
0K.
(Source: Kittel, “Introduction to Solid State Physics”)
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EE 529 Semiconductor Optoelectronics
– Optical Processes and LED
Absorption Spectrum of a Typical
Semiconductor
(Source: Wolfe, Holonyak, and Stillman, “Physical Properties of Semiconductors)
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EE 529 Semiconductor Optoelectronics
– Optical Processes and LED
Exciton Absorption
Q: Exciton absorption peaks are normally seen in very pure semiconductors at
low temperatures. Higher degree of confinement in the semiconductors also
greatly helps observing these, e.g., quantum wells. Why?
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9
EE 529 Semiconductor Optoelectronics
– Optical Processes and LED
Calculated Absorption Spectrum due to
Franz-Keldysh Effect
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EE 529 Semiconductor Optoelectronics
– Optical Processes and LED
Quantum-Confined Stark Effect (QCSE)
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EE 529 Semiconductor Optoelectronics
– Optical Processes and LED
Relation between Absorption and
Spontaneous Emission
8πn 2 ν 2 α 0 (ν)
R (ν ) = 2
c
e hν / kBT − 1
0
sp
Roosbroeck-Shockley relation
Spontaneous emission spectrum
of GaAs at 300K
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EE 529 Semiconductor Optoelectronics
– Optical Processes and LED
Light Emitting Diodes
A p-n junction diode typically made from a direct bandgap semiconductor.
Electron-hole pair recombination results in the emission of a photon.
Spontaneous emission; emitted photons in random direction.
Principles
Electron energy
n+
p
Ec
eVo
(a)
EF
Ec
EF
Eg
n+
p
Eg
(b)
hυ - Eg
Ev
eVo
Ev
Distance into device
V
Electron in CB
Hole in VB
Device Structures
Light output
Light output
p
n+
Epitaxial layers
n+
n+
Substrate
(a)
Metal electrode
Insulator (oxide)
p
n+
Epitaxial layer
Substrate
(b)
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EE 529 Semiconductor Optoelectronics
– Optical Processes and LED
Heterojunction LEDs
n+
(a)
AlGaAs
p
p
GaAs
Avoid re-absorption of photons along
the emission path.
AlGaAs
~ 0.2 µm
Electrons in CB
eVo
EF
Ec
(b)
Ec
∆Ec
(a) A double
No bias heterostructure diode has
2 eV
1.4 eV
EF
Ev
2 eV
Holes in VB
Ev
(b) A simplified energy
band diagram with
exaggerated features. EF
must be uniform.
With
forward
bias
(c)
n+
p
p
two junctions which are
between two different
bandgap semiconductors
(GaAs and AlGaAs)
(c) Forward biased
simplified energy band
diagram.
(d) Forward biased LED.
Schematic illustration of
photons escaping
reabsorption in the
AlGaAs layer and being
emitted from the device.
(d)
AlGaAs
GaAs
AlGaAs
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EE 529 Semiconductor Optoelectronics
– Optical Processes and LED
Exercise: LED Efficiency
An AlGaInP/GaP LED with peak emission
wavelength of 636 nm has an external quantum
efficiency of 25%. (a) To achieve 5 mW of output
optical power, what should be the injected current?
(b) If this corresponds to forward-biasing the
device at 2 V, find its power conversion efficiency.
(c) Find its luminous efficiency and luminous flux.
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EE 529 Semiconductor Optoelectronics
– Optical Processes and LED
Photon Escape Efficiency
The intensity distribution of the LED radiation is
Lambertian. At the semiconductor-air interface, if the
incident angle is greater than the critical angle
𝜃𝑐 = sin−1 (𝑛𝑎𝑎𝑎 ⁄𝑛𝑟 ), total internal reflection occurs
and the light is trapped inside the semiconductor.
Possible solutions:
(a) Shape the semiconductor
surface as a hemisphere
(expensive).
(b) Encapsulate the LED in a
transparent dome with
higher refractive index than
air to increase 𝜃𝐶 .
Lih Y. Lin
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