Lundstrom ECE 305 S15

ECE-305: Spring 2016

The Semiconductor Equations

Professor Mark Lundstrom Electrical and Computer Engineering

Purdue University, West Lafayette, IN USA lundstro@purdue.edu

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

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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

✓

✓

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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

✓

✓

✓ ✓

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