Excess carrier, recombination rate and continuity equation

1. Course: Electronic Devices
paper code: EC301
Course Coordinator: Arpan Deyasi
Department of Electronics and Communication Engineering
RCC Institute of Information Technology
Kolkata, India
8/14/2020 1Arpan Deyasi, RCCIIT
Topic: Generation and Recombination

2. 8/14/2020 Arpan Deyasi, RCCIIT 2
Quasi-Fermi Level
EC
EV
EFI
(EC+EV)/2
(EC+EV)/2
EF
EFn
For n-type semiconductor

3. 8/14/2020 Arpan Deyasi, RCCIIT 3
Quasi-Fermi Level
EC
EV
EFI
(EC+EV)/2
(EC+EV)/2
EF
EFn
For p-type semiconductor

4. 8/14/2020 Arpan Deyasi, RCCIIT 4
Quasi-Fermi Level
1
( )
1 exp
n
Fn
f E
E E
kT
=
− 
+  
 
1
( )
1 exp
p
Fp
f E
E E
kT
=
− 
+  
 

5. 8/14/2020 Arpan Deyasi, RCCIIT 5
Generation and Recombination
In presence of excess carrier
0 ( )n n n tδ= +
0 ( )p p p tδ= +
Net rate of change of electron concentration
2
( ( )) ( ( ) ( ))r i
d
n t n n t p t
dt
α= −

6. 8/14/2020 Arpan Deyasi, RCCIIT 6
Generation and Recombination
0 0 0 0( ( )) ( ( ( ))( ( )))r
d
n t n p n n t p p t
dt
α δ δ= − + +
0 0( ( )) ( ( ) ( ))r
d
n t n p n t p t
dt
α= −
0 0( ( )) ( ( ) ( ) ( ) ( ))r
d
n t n p t p n t n t p t
dt
α δ δ δ δ=− + +

7. 8/14/2020 Arpan Deyasi, RCCIIT 7
Generation and Recombination
For excess carrier
( ) ( )n t p tδ δ=
0 0 0( ( )) ( ( ) ( ) ( ) ( ))r
d
n n t n n t p n t n t n t
dt
δ α δ δ δ δ+ =− + +
0 0 0( ( )) ( )( ( ))r
d
n n t n t n p n t
dt
δ α δ δ+ =− + +

8. 8/14/2020 Arpan Deyasi, RCCIIT 8
Generation and Recombination
0 0( ( )) ( )( ( ))r
d
n t n t n p n t
dt
δ α δ δ=− + +
Under low loss injection
0 ( )n n tδ>>
0 0( ( )) ( )( )r
d
n t n t n p
dt
δ α δ=− +

9. 8/14/2020 Arpan Deyasi, RCCIIT 9
Generation and Recombination
For n-type material
0 0n p>>
0( ( )) ( )r
d
n t n n t
dt
δ α δ= −
Solution gives
( ) (0)exp
n
t
n t nδ δ
τ
 
= − 
 
excess minority
carrier lifetime

10. 8/14/2020 Arpan Deyasi, RCCIIT 10
Generation and Recombination
Generation rate
0
( )
( ( )) ( )r
n
d n t
n t n n t
dt
δ
δ α δ
τ
=− =−
Recombination rate
0
( )
( ( )) ( )r
n
d n t
n t n n t
dt
δ
δ α δ
τ
− = =

11. 8/14/2020 Arpan Deyasi, RCCIIT 11
Continuity Equation
Assume: non-uniformly doped semiconductor
thermal equilibrium
Carrier transport mechanisms
1. Drift due to external electric field
2. Diffusion due to concentration gradient
3. Generation of excess carriers
4. Recombination of excess carriers

12. 8/14/2020 Arpan Deyasi, RCCIIT 12
Continuity Equation
Recombination rate
( )
.r
n
n t
R A z
δ
τ
= ∆
Rate of particle flow
( ) ( )n n
p
J z J z z
R A
q q
 + ∆
= − 
− − 
( )1
. .n
p
J z
R A z
q z
∂
= ∆
∂

13. 8/14/2020 Arpan Deyasi, RCCIIT 13
Continuity Equation
Total rate of electron build-up
( , )
.b
n z t
R A z
t
∂
= ∆
∂
Under thermal equilibrium
b r p GR R R R=− + +

14. 8/14/2020 Arpan Deyasi, RCCIIT 14
Continuity Equation
( )( , ) ( ) 1
. . . . . .n
n
J zn z t n t
A z A z A z G A z
t q z
δ
τ
∂∂
∆ =− ∆ + ∆ + ∆
∂ ∂
( )( , ) ( ) 1 n
n
J zn z t n t
G
t q z
δ
τ
∂∂
=− + +
∂ ∂
For p-type
( )( , ) ( ) 1 p
p
J zp z t p t
G
t q z
δ
τ
∂∂
=− + +
∂ ∂

15. 8/14/2020 Arpan Deyasi, RCCIIT 15
Continuity Equation
n n
dn
J qD
dz
=
2
2
( , ) ( )
n
n
n z t n t n
D G
t z
δ
τ
∂ ∂
=− + +
∂ ∂
p p
dp
J qD
dz
= −
2
2
( , ) ( )
p
p
p z t p t p
D G
t z
δ
τ
∂ ∂
=− + +
∂ ∂

16. 8/14/2020 Arpan Deyasi, RCCIIT 16
Continuity Equation
Excluding generation rate
2
2
( , ) ( )
n
n
n z t n t n
D
t z
δ
τ
∂ ∂
=− +
∂ ∂
2
2
( , ) ( )
p
p
p z t p t p
D
t z
δ
τ
∂ ∂
=− +
∂ ∂

17. 8/14/2020 Arpan Deyasi, RCCIIT 17
Continuity Equation
At steady state
2
22
n
n n
z L
δ∂
=
∂
2
22
p
p p
z L
δ∂
=
∂
Ln, Lp: diffusion lengths
, , ,n p n p n pL D τ=