ece606: solid state devices lecture 11 interface states...

Klimeck – ECE606 Fall 2012 – notes adopted from Alam

ECE606: Solid State Devices

Lecture 11

Interface States Recombination

Carrier Transport

Gerhard [email protected]


Outline

2

1) SRH formula adapted to interface states

2) Surface recombination in depletion region

3) Conclusion


Surface Recombination Current

3

( ) ( )

2

1 1

1 1i

bulk

p nT TN

np nR

n n p pc c N

−=+ + +

( ) ( ) ( )( ) ( )

2

1 1

1 1

−=

+ + +

s s i

s s s sp

T

s ns

Dn p nR E

n n pc c

E

p

E d

( )C

V

E

E

R R E dE= ∫

For single level bulk traps ….

( )( ) ( )

2

1 1

1 1i

p n

Tnp n

n n p pc c

N−=

+ + +

For single level interface trap at E …


Case 1: Minority Carrier Recombination

4

( ) ( )( ) ( )( ) ( )

20 0

0 1 0 0

0 0

0 1

1 1

+ ∆

∆

+ ∆ − =+ + ∆+ + +

s s i

s s

s

s

s IT

s s sps ns

n p n

n

n p D E d

p

ER

pE

n n pc c

( )0

1

0

0

10

0

1

∆=

+ +

s IT

s ss

ps ps s ns s

s p D E dE

n pn

c c n c n

n

( )0

1 1

0 0

1

∆=

+ +

ps s IT

pss s

s ns s

c p D E dE

cn p

n c n

Donor doped

ns0+∆ns0

ps0+∆ps0


Consider the Denominator …

5

( )

( )

( )

( )1i

F i

i

F i

ps

ns

E Ei

E E E Ei i

E Ei n ec

n

n

n

e

e ec− −

− − −

= + +β

β

β

β

1 1

0

1

0

11 1s s

ps ps

ns ns

ss s

D

s

D

n n

n n

c cD

c c

p p

N N= + + = + +

( ) ( )1 F FE E E Eps

ns

ce e

c− −= + +β β

( )1 x xFe ae x E E−= + + ≡ −β

ns0+∆ns0

ps0+∆ps0


Consider the Denominator …

6

At 0 1 1 2psF i

ns

cE E E , x D small

c= > = = + + × ≈

FE 'FE

At 1 1psi ii

D ns D

cn nE E D

N c N= ⇒ = + + ≈

At 1 1 2F iE E ' E , D small= < = + + =

( ) ( )1

i iE E E Epsi i

D ns D

cn e n eD

N c N

β β− − −

= + +

iE

1

2

iEFE

D

FE '( ) ( )1 F FE E E Eps

ns

ce e

cβ β− −= + +


Approximate the Denominator …

7

iE

FE 'FE

( ) ( )1

i iE E E Epsi i

D ns D

cn e n eD

N c N

− − −

= + +β β

iE

1

2

FE

D

FE '

1 for

otherwise

'F FE E E

D ≤ ≤

≈ ∞


Integrated Recombination

8

( ) ( )0

1 1

0 0

1

∆= =

+ +

∫ ∫C C

V V

E Eps s IT

psE E s s

s ns s

c p D E dER R E

cn p

n c n

( )0≈ ∆∫F

'F

E

ps s

E

c p D E dE

FE 'FEiE

1

2D


Surface Recombination Velocity

9

( )0≈ ∆∫F

F

E

ps s IT

E

R c p D E dE

Surface recombination velocity

( ) 0'

ps IT F F sc D E E p= − ∆

0sgs p= ∆


Outline

10

1) Nature of interface states

2) SRH formula adapted to interface states

3) Surface recombination in depletion region

4) Conclusion


Case 2: Recombination in Depletion

11

( ) ( ) ( )( ) ( )

2

1 1

1 1

−=

+ + +

i IT

s sp

s s

s nss s

n p n D E

n

dER E

n pc c

p

( )

( )2 1

i

i

E E

ns IT iE Ens

ps

e dEc D n

ce

c

−

−= −

+

β

β

( )

( )2 1

C i

iV

E E E

ns IT iE EnsE

ps

e dER c D n

ce

c

−

−= −

+∫

β

β

ps~0

ns~0( ) ( ) ( )β β− − −= −+

i i

ii ITE E E E

i i

ps ns

nn D E dE

n e n e

c c


Case 2: Recombination in Depletion

12

( )

( )2 1

i

i

E E

ns IT iE Ens

ps

e dEc D n

ce

c

β

β

+∞ −

−−∞

= −+

∫

( )

( )2 1

C i

iV

E E E

ns IT iE EnsE

ps

e dER c D n

ce

c

−

−= −

+∫

β

β

Ps~0

ns~0

20 1

psns IT i

ns

c dxc D n

c xφ

+∞

= −+∫

2ns ps IT ic c D nπβ= −


Why do donors/acceptors not act as R-G

Centers?

13

( ) ( )0

1 1

0 0

1

ps sD

pss s

s ns s

c p D E dER E

cn p

n c n

∆=

+ +

FE 'FE iE

1

2D

( )0 0Dps

D

s Nc p

D E

∆= →

( ) ( )0

1 1

0 0

1

ps sA

pss s

s ns s

c p D E dER E

cn p

n c n

∆=

+ +

( )0 0Aps

A

s Nc p

D E

∆= →


Summary

14

( )

Interface (depletion)2

Interface (minority)

Bulk (minority)

ns ps IT i

'ps IT F F s

p T

R c c D n

R c D E E p

R c N p

πβ= − ×

= − ∆

= ∆


ECE606: Solid State Devices

Lecture 12

Carrier Transport

Gerhard [email protected]


Outline

16

1) Overview

2) Drift Current

3) Physics of Mobility

4) High field effects

5) Conclusion

REF: Advanced Device Fundmentals, Pages 175- 192


Current Flow Through Semiconductors

17

I

V

Depends on chemical composition, crystal structure, temperature, doping, etc.

Carrier Density

velocity

I G V

q n Av

= ×= × × ×

Transport with scattering, non-equilibrium Statistical Mechanics ⇒ Encapsulated into drift-diffusion equation with

recombination-generation (Ch. 5 & 6)

Quantum Mechanics + Equilibrium Statistical Mechanics ⇒ Encapsulated into concepts of effective masses

and occupation factors (Ch. 1-4)


Non-equilibrium Systems

1

vs.

Chapter 6Chapter 5

I

V


Summary of Transport Equations …

19

1∂ = − ∇ • − +∂

JP P P

pr g

t q

( )+ −∇ • = − + −D AD q p n N N

P P Pqp E qD pµ= − ∇J

1N N N

nr g

t q

∂ = ∇ • − +∂

J

J µ= + ∇N N Nqn E qD n

I

V


Outline

20

1) Overview

2) Drift Current



5) Conclusion


Meaning of Effective Mass …

21

0m

*

nm

2 2

20

( ) (2

) ψ ψ

− + =

+ℏ

ecry xtsU Ud

xx Em dx

2 2

*0

2)

2( φ φ

− + =

ℏ

extU xd

Edxm


Drift by Electric field ….

22

n nJ qnµ= E

* *( )υ υτ

= − −n n

n

d m mq

dtE

*( ) 1 ττυ

− = − −

n

t

n

n

qt e

mE x x x x


Drift by Electric field ….

23

*( ) 1 ττυ

− = − −

n

t

n

n

qt e

mE

*( , 1-2 ps)

τ= − → ∞n

n

tq

mE

µ≡ nE

n nJ qnµ= Eυ

x x x x

time(Theory valid once t > 1-2 ps)

friction

1( )υ E

2( )υ E


Outline

24

1) Overview

2) Drift Current



5) Conclusion


Mobility and Physics of Scattering Time

25

*

τµ =n

nn

q

m0m

*

nm

Fermi’s Golden rule …

2

1 *2( ) ( ) ( )

π ψ ψτ∞

−

−∞∫∼

ℏn x U x x dx

x x x x


Phonon and Ionized Impurity Scattering

26

0m

*

nm

3 2~n Tτ −

Higher temperature, more phonon scattering

Ionized impurity

3 2

~nD

T

Nτ


Multiple Scattering Events

27

1 1 1 1

phIIn s

= + + +⋯τ τττ

*

nm� Ionized impurity

� Phonon scattering� others ….

1 1 1

p IIn hµµ µ= +

*1 n

n n

m

qµ τ=

Matthession Rule ….

IIn

II

ph

ph

µµ

µµ

µ⇒ =

+

min minp

I

h

h

II

Ip

µµ

µ µµ

µ

= + − +

( )0

min

01 IN Nα

µµ

= + +


Model for Ionized impurity Scattering

28

*

nm

( ),mi0,

0,

n1 nn

I n

nn

N Nαµ

µµ

= + +

,minnµ


Temperature-dependent Mobility

29

3 2~n n Tµ τ −∼


Outline

30

1) Overview

2) Drift Current


4) High Field Effects

5) Conclusion


Mobility at High Fields?

31

01*

1

τ µυ µ= ≡ ⇒

+

NN

nN n

C

q

mE E E

E

E

E

What causes velocity saturation at high fields?

Where does all the mobility formulain device simulator come from?

x x x x

( )υ E

cE


Velocity Saturation in Si/Ge

3

10 0JJ J −+= = − =E

2 1c J JJ J−+= − >≪E E

3 2c JJ JJ −+≈ = − >E E

4 3c J JJ J−+= − ≈≫E E


Velocity Overshoot & Inter-valley Transfer

33

What type of scattering would you need for inter-valley transfer?

Smaller m*

Larger m*


Doping dependent Resistivity

34

ρ= JE

( )µ µ= +n pq n pJ E

1

( )n pq n pρ

µ µ=

+1

for n-typen Dq Nµ

= …

1= for p-type

p Aq Nµ…

J

V


Conclusion

35

1) Poisson and drift-diffusion equations form a complete semi-classical transport model that can explain wide variety of device phenomena.

2) Drift current results from response of electrons/holes to electric field. The physics of mobility is complex and material dependent.

3) Constancy of low-field mobility can be checked by experiments.

ece606: solid state devices lecture 11 interface states...

Documents