semiconductor equations: ii...bermel ece 305 f16 2 outline 1. semiconductor equations 2. equilibrium...

ECE-305: Spring 2016

Semiconductor Equations: II

Professor Peter BermelElectrical and Computer Engineering

Purdue University, West Lafayette, IN USApbermel@purdue.edu

Pierret, Semiconductor Device Fundamentals (SDF)pp. 104-124

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Bermel ECE 305 F16 2

outline

1. Semiconductor equations

2. Equilibrium versus non-equilibrium

3. Minority carrier diffusion equation

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drift- diffusion equation

3

current = drift current + diffusion current

total current = electron current + hole current

D

p

p D

n

n k

BT q

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continuity equation for holes

4

in-flow

out-flow

p t

recombinationgeneration

in-flow - out-flow + G - Rp

t

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Continuity Equations for Electron/Holes

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Continuity Equations for Electron/Holes

1n N N

nJ g r

t q

+ -

1 - + -

P P P

pJ g r

t q

( )J x

( )J x dx+

Ng

n

Nr

p

( )pJ x

( )pJ x dx+

rp

gp

5

Bermel ECE 305 F166

outline

1. Semiconductor equations

2. Equilibrium versus non-equilibrium

3. Minority carrier diffusion equation

✓

equilibrium (no G-R)

7

in-flow

out-flow

p t

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current and QFL’s

8

p nieEi-Fp kBT

dp

dx nie

Ei-Fp kBT´1

kBT

dEi

dx-

dFp

dx

p

kBT

dEi

dx-

dFp

dx

dEi

dx qE x

n nieFn-Ei kBT

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Direct Band-to-band Recombination

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Photon(light)

GaAs, InP, InSb (3D)

Lasers, LEDs, etc.

In real space … In energy space …

Photon

9

Indirect Recombination (Trap-assisted)

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Phonon (Thermal Energy)

Ge, Si, ….

Transistors, Solar cells, etc.

10

Auger Recombination

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Phonon (Thermal Energy)

InP, GaAs, …

Lasers, etc.

12

3

1 2

4

3

4

11

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outline

1. Semiconductor equations

2. Equilibrium versus non-equilibrium

3. Minority carrier diffusion equation

✓

✓

Minority Carrier Equation

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0 0 0

D A

D A

D q p n N N

q p p n p N N

+ -

+ -

- + -

® + D - - D + - ®

0

20

2

1 1p pNN N N N

n

p p p p

N p

n

n n nnr g g

t q t q x

n n n n nD g

t t x

+ D D - + ® - +

+ D D D D - +

JJ

t

t

( ~0)

N N N

N

qn qD n

nqD

x

+

®

J E

E

13

Various approximations …

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2

2

p p p

N p

n

n n nD g

t x

D D D - +

t

Time dependence

density gradient

recombination

generation

14

Summary: Equation of State

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0 0 0D A D AD q p n N N q p p n p N N+ - + - - + - ® + D - - D + - ®

2

2

1 1 p n nP P P p P p

n n

p p p pr g g D g

t q q x x

- D D D - + ® - - + ® - - +

JJ

t t

( ~0)P P P P

pqp qD p qD

x

- ® -

J E E

2

2

1 1 p pNN N N N N p

n n

n nn nr g g D g

t q q x x

D D D - + ® - + ® - - +

JJ

t t

( ~0)N N N N

nqn qD n qD

x

+ ®

J E E

15

when is the electric field zero?

17

x

n x ND x

1017 cm-3

1018 cm-3

n x » ND x

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e-band diagram

18

EF

EC x

EV x

Ei x

x

E qE x

dEC

dx

Dp

t¹ Dp

d 2Dp

dx2-Dp

t p

+GL

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example #1: N-type sample in ll injection

19

Steady-state, uniform generation, no spatial variation

Solve for Δp and for the QFL’s.

1) Simplify the MCDE2) Solve the MCDE3) Deduce Fp from Δp

Dp

t Dp

d2Dp

dx2-Dp

t p

+GL

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example #1: solution

20

x

Dp x

Dp x GLt p

x L 200 mx 0

Steady-state, uniform generation, no spatial variation9/19/2016 Bermel ECE 305 F16

Example 2A: Transient, No Illumination

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1N N N

nr g

t q

- +

J J + N N Nqn E qD n

(uniform)

0( )

n

n n nG

t t

+ D D - +

Acceptor doped

1 - - +

J p P p

pr g

t q - J p p Pqp E qD p

(uniform)

0( )

t

+ D D - +

p

p p pG

tMajority carrier

0 0 0+ - + - - + - + D- - + - DD A D AD q p n N N q p n N Npn21

Dn

time

Example 2A: Transient, No Illumination

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( )

n

n nG

t

D D - +

t

( , ) ntn x t A Be-D + tAcceptor doped

22

000,

00,

nBnxn

Axn

DDD

¥D

ntt

entxn-

DD 0,

Dn

time

Example 2B: Transient, Uniform Illumination

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1N N N

nr g

t q

- +

J

1

J + N N Nqn E qD n

(uniform)

0( )

n

n n nG

t t

+ D D - +

Acceptor doped

1 - - +

J p P p

pr g

t q - J p p Pqp E qD p

(uniform)

0( )

t

+ D D - +

p

p p pG

tMajority carrier

0 0 0+ - + - - + - + D- - + - DD A D AD q p n N N q p n N Npn23

Example 2B: Transient, Uniform Illumination

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1

( )

n

n nG

t

D D - +

t

( , ) ntn x t A Be-D + t

( , ) 1 ntnn x t G e tt -D -

Acceptor doped

0, ( ,0) 0

, ( , ) n

t n x A B

t n x G A

D -

®¥ D ¥ t

time

24

example #3

25

Solve for Δp and for the QFL’s.

1) Simplify the MCDE2) Solve the MCDE3) Deduce Fp from Δp

Dp

t Dp

d2Dp

dx2-Dp

t p

+GL

Transient, no generation, no spatial variation

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example #3

26

x

Dp x

Dp t 0 GLt p

x L 200 mx 0

transient, no generation, no spatial variation

Dp t Dp t 0 e-t /t p

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example #4

27

Steady-state, sample long compared to the diffusion length.i.e., a short diffusion length

fixedDp x 0

1) Simplify the MCDE2) Solve the MCDE3) Deduce Fp from Δp

Dp

t Dp

d2Dp

dx2-Dp

t p

+GL

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example #4

28

x

Dp x

Dp x ®¥ 0

Dp 0

Dp x Dp 0 e-x/Lp

x L 200 mx 0

Lp Dpt p << L

Steady-state, sample long compared to the diffusion length.9/19/2016 Bermel ECE 305 F16

Continuity Equations…

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1 - - +

JP P P

pr g

t q

D AD q p n N N+ - - + -

P P Pqp qD p - J E

1N N N

nr g

t q

- +

J

N N Nqn qD n + J E

1

time

Dn

time

Analytic solutions

ntt

entxn-

DD 0,

( , ) 1 ntnn x t G e tt -D -

Dn

Example 5: One sided Minority Diffusion

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1 nN N

n dJr g

t q dx

- +

N N N

dnqn E qD

dx +J

2

20 N

d nD

dx

Steady state, no generation/recombination, acceptor dopedLong diffusion length

30

1

0,' D txn

a 0x’

Metal contact

Example: One sided Minority Diffusion

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, ( ' ) 0 D -x a n x a C Da

'( , ) ( 0 ') 1

D D -

xn x t n x

a

2

20 N

d nD

dx

( , ) 'n x t C DxD +

x’

a

Metal contact

0 ', ( ' 0 ') D x n x C

0x’

31

0,' D txn

Conclusions

1) We will often be using minority carrier diffusion equation to understand the mechanics of carrier transport in electronic devices. Review the problem carefully to see if the assumption of minority carrier transport is satisfied.

2) Divide all complex problems into solvable parts, solve the parts sequentially and then put the partial solutions back by using proper boundary conditions to arrive at the complete solution.

3) Explore analytical solution whenever possible, however numerical solutions are also of great value.

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