445.664 445.664
Light
Light - - Emitting Diodes Emitting Diodes
Chapter 3.
Chapter 3.
Theory of Radiative Recombination Theory of Radiative Recombination
Euijoon Yoon Euijoon Yoon
Theory of Radiative Recombination Theory of Radiative Recombination
Semiclassical model of radiative recombination
based on equilibrium generation and recombination
•van Roosbroeck-Shockley model
ÎCalculation of the spontaneous radiative recombination rate under equilibrium and non-equilibrium conditions
•Einstein model
ÎCalculation of the spontaneous and stimulated transitions in a two-level atom
445.664 (Intro. LED) / Euijoon Yoon 3
The van Roosbroeck
The van Roosbroeck- -Shockley Model Shockley Model
α(ν) absorption coefficient [cm-1]
α(ν)-1 mean distance that a photon travels before being absorbed
υgr group velocity of photons propagating in the semiconductor
τ(ν) time that it takes for a photon to be absorbed
= (α υgr)-1
ν frequency of a generated photon
445.664 (Intro. LED) / Euijoon Yoon 4
Photon Absorption Probability Photon Absorption Probability
) d(
d ) / 1 ( d d d d
ν ν λ
ν υ ω
c n
gr = = =
k
) n d(
) d ( ) ( ) (
1
ν ν ν
α υ ν ν α
τ = gr = c
group velocity of photons
angular frequency wave vector refractive index
( )
⎟⎠
⎜ ⎞
⎝⎛ =
⎟⎠
⎜ ⎞
⎝⎛ =
=
λν λ
π πν ω
n c k
2 2
Îphoton absorption probability per unit time inverse photon lifetime
445.664 (Intro. LED) / Euijoon Yoon 5
Density of Photons Density of Photons
λ λ λ π
λ
νd
1 1 8
d )
(
4 /= h kT − N e
ν ν ν λ ν
λ ν λν
d d ) ( d ) ( d
2
n n
c n
c n c
−
=
=
⇒
=
Density of photons per unit volume(equilibrium conditions) by Planck’s black body radiation formula
ν ν ν ν
ν π
ν
νd
1 1 d
) ( d 8
d )
(
3 /2 2
= h kT −
e n c
N n
The van Roosbroeck
The van Roosbroeck- -Shockley Equation Shockley Equation
ν ν α ν
π
ν ν ν
α ν ν
ν ν
π ν τ
ν ν ν
ν
ν
ν
1d ) ( 8
) d(
) d ( 1d
1 d
) d(
8
) (
d ) ( d ) (
/ 2
2 2
/ 3
2 2 0
= −
⎟⎟⎠
⎜⎜ ⎞
⎝
⋅⎛
⎟⎟⎠
⎜⎜ ⎞
⎝
⎛
= −
=
kT h
kT h
e c
n
c n e
n c
n R N
ν ν α ν
ν π
ν ν d
1 ) ( 8
d ) (
/
0 2
2 2
0 0
0 =
∫
∞R =∫
∞ c n eh kT −R
Phonon absorption rate per unit volume
= Absorption probability ×Photon density
Absorption rate per unit volume in the frequency interval νand ν+ dν
Absorption rate per unit volume
van Roosbroeck-Shockley Equation
445.664 (Intro. LED) / Euijoon Yoon 7
The Simplified van Roosbroeck
The Simplified van Roosbroeck- -Shockley Model Shockley Model
e x x x x h
c kT E n kT c R
xg x
g g
1 d 8
2 3 2
0 =
π α
⎜⎝⎛ ⎟⎠⎞∫
∞ −−g g
E E E−
= α0
α
Absorption coefficient
Eg band gap energy of the semiconductor α0 absorption coefficient at hν= 2Eg
Neglecting the frequency dependence of the refractive index Using the absorption coefficient at the band edge ( α= α0 )
( )
νν d / d
/
/ /
kT h x
kT E x
kT E kT h x
g g
=
=
=
=
The simplified van Roosbroeck-Shockley Equation
445.664 (Intro. LED) / Euijoon Yoon 8
Bimolecular Recombination Rate Bimolecular Recombination Rate
0
2
Bnp Bn R
R = = i =
Under equilibrium conditions,
photon absorption rate (R0) = photon emission rate
|| ||
carrier generation rate = carrier recombination rate (R) Equilibrium carrier recombination rate
B bimolecular coefficient
Direct band gap B = 10-9~ 10-11cm3/s Indirect band gap
B = 10-13~ 10-15cm3/s
445.664 (Intro. LED) / Euijoon Yoon 9
Einstein Model Einstein Model
A B ( )
2 1
1
2→ = → ρν +
W
) ( 1 2
2
1→ = B→ ρν W
A Spontaneous transition rate
B Induced (stimulated) transition rate ∝photon density, ρ(ν) Probability per unit time
downward transition (2 Æ1) upward transition (1 Æ2)
Induced emission spontaneous emission
Induced absorption
Einstein showed that
1. B = B2Æ1 = B1Æ2 ÎStimulated absorption and stimulated emission are complementary processes.
medium
isotropic
an in constant
/ 8
2. A/B = π n3hν3 c3=
