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Lundstrom ECE 305 S15
ECE-305: Spring 2016
The Semiconductor Equations
Professor Mark Lundstrom Electrical and Computer Engineering
Purdue University, West Lafayette, IN USA [email protected]
2/6/16
Pierret, Semiconductor Device Fundamentals (SDF) pp. 104-124
equations for n and p
2
If we know the hole density and the electric field, then we can find the hole current. How do we find the hole density?
!J p = pqµ p
!E − qDp
!∇p
p0 x( ) ≈ N A x( )
Lundstrom ECE 305 S15 3
outline
1. Drift-diffusion current
2. The continuity equation
3. Recombination-generation
4. Quasi-Fermi levels
5. The Semiconductor Equations
✓
continuity equation for holes
4
∂p∂t
= −∇i
!Jpq
+Gp − Rp
in-flow
out-flow
∂p ∂t
recombination generation
in-flow - out-flow + G - R ∂p∂t
=
Next: generation and recombination
5
∂p∂t
= −∇i
!Jpq
+Gp − Rp
photoelectric effect (optical generation)
6
EF
EVAC
ΦM
E = hf > ΦM
Einstein, in 1905, when he wrote the Annus Mirabilis papers
http://en.wikipedia.org/wiki/Photoelectric_effect
optical generation
7
EC
EV
E = hf > EG
f λ = c f = cλ
E = hf = hcλ
EGλ < hc
EG
example: N-type sample
8
EC
EV
EG
ND = 1017 cm-3
n0 = 1017 cm-3
n0p0 = ni2
p0 =ni2
n0= 103 cm-3
EF
before we turn on the light: equilibrium
Subscript “0” denotes equilibrium.
turn the light on: “excess carriers”
9
EC
EV
EGGop m-3s-1
ND = 1017 cm-3
n0 = 1017 cm-3
n0p0 = ni2
p0 =ni2
n0= 103 cm-3
Δn = 1010 cm-3
Δp = 1010 cm-3
p = Δp ≈1010 cm-3
n = n0 + Δn ≈1017 cm-3
“Low-level injection”
“majority carriers”
“minority carriers”
the np product
10
EC
EV
EGGop m-3s-1
n0p0 = ni2
Δn = 1010 cm-3
Δp = 1010 cm-3
Δn << n0
p = Δp ≈1010 cm-3
n = n0 ≈1017 cm-3
“Low-level injection”
np = 1027 cm-3
np ≠ ni2
away from equilibrium
turn the light off
11
EC
EV
EG
Δn = 1010 cm-3
Δp t = 0( ) = 1010 cm-3
Question: What happens?
Answer: The system returns to equilibrium.
How long does it take? A time known as the minority carrier lifetime.
τ p sec
n t( ) ≈ n0 = 1017 cm-3
p t( ) >> p0 ≈ Δp t( )
carrier recombination
12
EC
EV
EG
n t( ) ≈ n0
Δp t( )
Rp t( ) = ∂p∂t R−G
= −Δp t( )τ p
(low-level injection)
R-G processes
13
Fig. 3.15a from R.F. Pierret, Semiconductor Device Fundamentals
Shockley-Read-Hall (SRH)
Next: generation and recombination
14
∂p∂t
= −∇i
!Jpq
+Gp − Rp
Lundstrom ECE 305 S15 15
outline
1. Drift-diffusion current
2. The continuity equation
3. Recombination-generation
4. Quasi-Fermi levels
5. The Semiconductor Equations
✓
✓
✓
where is the Fermi level?
16
EC
EV
EG
n0 = 1017 cm-3
n0 = nieEF−Ei( ) kBT
p0 =ni2
n0= 103 cm-3
EF
Before we created the excess holes
p0 = nieEi−EF( ) kBT
Where is the Fermi level?
17
EC
EV
EGGop m-3s-1
n = 1017 cm-3
p = Δp = 1010 cm-3
a) Where it was in equilibrium b) Closer to the conduction band c) Closer to the valence band d) Near the middle of the band e) None of the above
After the light has been on for a long time….
Same # of electrons, more hole -> need 2 Fermi levels!
quasi-Fermi levels
18
EC
EV
n = 1017 cm-3
n0 = nieEF−Ei( ) kBT
p = 1010 cm-3
p0 = nieEi−EF( ) kBT
Fnn = nie
Fn−Ei( ) kBT
Fn = EF
p = nieEi−Fp( ) kBT
Fp < EF
Fp
The QFL’s are split In equilibrium: Fn = Fp = EF
equilibrium vs. non-equilibrium
19
n0 = nieEF−Ei( ) kBT
p0 = nieEi−EF( ) kBT
n = nieFn−Ei( ) kBT
p = nieEi−Fp( ) kBT
n0p0 = ni2 np ≠ ni
2
equilibrium non-equilibrium
f0 =1
1+ e E−EF( ) kBTfc =
11+ e E−Fn( ) kBT
1− fv = 1−1
1+ e E−Fp( ) kBT
Lundstrom ECE 305 S15 20
outline
1. Drift-diffusion current
2. The continuity equation
3. Recombination-generation
4. Quasi-Fermi levels
5. The Semiconductor Equations
✓
✓
✓ ✓
“the semiconductor equations”
21
∂ p∂t
= −∇i
!Jpq
⎛
⎝⎜⎞
⎠⎟+Gp − Rp
∂n∂t
= −∇i!Jn−q
⎛⎝⎜
⎞⎠⎟+Gn − Rn
0 = −∇i ε
!E( ) + ρ
Three equations in three unknowns:
p!r( ), n !r( ), V !r( )
!J p = pqµ p
!E − qDp
!∇p
!Jn = nqµn
!E + qDn
!∇n
ρ = q p − n + N D
+ − N A−( )
!
E !r( ) = ∇V !r( )
Lundstrom ECE 305 S15 22
outline 1. Drift-diffusion current
2. The continuity equation
3. Recombination-generation
4. Quasi-Fermi levels
5. The Semiconductor Equations
✓
✓
✓ ✓
✓
