ECE606: Solid State Devices Lecture...

Klimeck – ECE606 Fall 2012 – notes adopted from Alam

ECE606: Solid State DevicesLecture 10

Gerhard [email protected]


Outline

2

1) Derivation of SRH formula

2) Application of SRH formula for special cases

3) Direct and Auger recombination

4) Conclusion

Ref. ADF, Chapter 5, pp. 141-154


Sub-processes of SRH Recombination

3

(1)

(3)

(2)

(4)

(1)+(3): one electron reduced from Conduction-band & one-hole reduced from valence-band

(2)+(4): one hole created in valence band and one electron created in conduction band


SRH Recombination

4

Physical picture

(1)

(3)

(2)

(4)

(1)+(3): one electron reduced from C-band & one-hole reduced from valence-band

Equivalent picture

(1)

(3)

(2)

(4)

(2)+(4): one hole created in valence band & one electron created in conduction band


Changes in electron and hole Densities

5

1 2,

n

t

∂ =∂

− n Tc n p ( )1n T ce n f+ −

p Tc p n− υ+ p Te p f3 4,

p

t

∂ =∂

(1) (2)

(3) (4)


Detailed Balance in Equilibrium

6

1 2,

n

t

∂ =∂ 0 0− n Tc n p 0+ Tne n

(1) (2)

(3) (4)

0 0 00 = − +T n Tnn p e nc

0 01

0

= ≡T

Tn nn

n pn

nce c

( )0 0 0 10 = − −T Tn n p n nc

p Tc p n−p Te p+

3 4,

p

t

∂ =∂

0 0 00 p T T pc p n p e= − +0 0

10

p Tp p

T

c p ne c p

p≡ =

( )0 0 0 10 = − −T Tp p n p pc

( )1 1cf− ≈ Assume non-degenerate

Fundamental concept of detailed balance,

connecting 2 counteracting

processes.

Subscripts 0 indicate equilibrium


Expressions for (n1) and (p1)

7

01

0 0= T

T

n p

nn

(1) (2)

(3) (4)

01

0 0= T

T

p n

pp

0 0 0 0

0 01 1 = ×T T

T T

n p nn

p

n pp 2

0 0= = in p n


Expressions for (n1) and (p1)

8

001

0

=T

Tpnn

n

( ) ( )

( )1

F i T F

T i

E E E Ei D

E Ei D

n n e g e

n g e

β β

β

− −

−

=

=

21 1 = ip n n

( )

21 1

1 i T

i

E Ei D

p n n

n g eβ −−

=

=

( )( )

00

000 1

=−

T

T

Nn

N f

f

( ) ( )0001

1 β −= =+

−T F

TT T E E

D

Nn

g efN

ET

Trap is like a donor! f00=empty trap prob.

( )00

11

1 expf

g=

+− 00

11

1 xf

g e p= −

+

00 1

g expf

g exp=

+00

00 111

1

g exp/ g exp

g exp g exf

f p= =

+ +−


Dynamics of Trap Population

9

Tn

t

∂ =∂ 3 4,

p

t

∂+∂

(1)

(3)

(2)

(4)

n Tc n p= n Te n− p Tc pn− p Te p+

1 2,

n

t

∂−∂

( )1n T Tc np n n= − ( )1p T Tc p n p p− −


Steady-state Trap Population

10

0Tn

t

∂ =∂

( )1n T Tc np n n= − ( )1p T Tc p n p p− −

( ) ( )1

1 1

n T p TT

n p

c N n c N pn

c n n c p p

+=

+ + +

(1)

(3)

(2)

(4)

T T TN p n= +

T T Tp N n= −

( )( ) ( )( )1 10 n T T T p T T Tc n N n n n c pn N n p= − − − − −

( )1 1 1T n n p p n T p Tn c n c n c p c p c nN c p N+ + + = +


Net Rate of Recombination-Generation

11

( ) ( )

2

1 1

1 1

−=

+ + +

i

p T n T

np n

n n p pc N c N

(1)

(3)

(2)

(4)

( )1= − = −p T T

dpR c p n p p

dt

τ n τ p

( ) ( )1

1 1

n T p TT

n p

c N n c N pn

c n n c p p

+=

+ + +

T T Tp N n= −

( )1 1p T T Tc p n N p n p= − +

( )1 1p T p Tc n p p c N p= + −

1n T p Tc N n c N p

A

+=

( ) 11 1

n T p Tp p T

c N n c N p Ac p p c N p

A A

+= + −

( )1 1 1 1 1 1 1 1 1 1p T

n n p p n n p p

c Npc n p c n pc p p c p p c n p c n p c p p c p

A= + + + − − − −

( )1 1p T nc N c

pn p nA

= −

21 1 ip n n=


Net Rate of Recombination-Generation

12

( ) ( )

2

1 1

1 1

−=

+ + +

i

p T n T

np n

n n p pc N c N

(1)

(3)

(2)

(4)

( )1= − = −p T T

dpR c p n p p

dt

τ n τ p


Outline

13




4) Conclusion


Case 1: Low-level Injection in p-type

14

( )( )( ) ( )

20 0

0 1 0 1τ τ∆

∆∆+ + −

=+ ∆+ + + +

i

p n

n p n

n n pn

n

p p

n

( ) ( )2

1 1

i

p n

np nR

n n p pτ τ−=

+ + +

( )( ) ( )

20 0

0 1 0 1τ τ+ +

=+ ∆+ +

∆ ∆∆ + +p n

nn n

n

p

n pn p p

2

0 0

0n

p n n

∆ ≈∆≫ ≫

( )( )

0

0τ τ∆=

∆=

n n

p

p

n n

0 ∆+n n

0 ∆+p n

Lots of holes, few electrons => independent of holes


Case 2: High-level Injection

15

( )( )( ) ( )

20 0

0 1 0 1τ τ∆

∆∆+ + −

=+ ∆+ + + +

i

p n

n p n

n n pn

p

p p

n

( ) ( )2

1 1

i

p n

np nR

n n p pτ τ−=

+ + +

( )( ) ( )

0 0

0 1 1

2

0τ τ+ +

=+ ∆+ +

∆ ∆∆ + +p n

nn n

n

p

n nn p p

0 0n p n∆ ≫ ≫( ) ( )2

τ τ τ τ∆ ∆

∆= =

+ +n p n p

n n

n

0 ∆+n n

0 ∆+p n

e.g. organic solar cells

Lots of holes, lots of electrons => dependent on both relaxations


High/Low Level Injection …

16

0 0n p n∆ ≫ ≫( )high

n p

lowp

nR

nR

τ τ

τ

∆=+

∆= 0 0p n n∆≫ ≫

which one is larger and why?


Case 3: Generation in Depletion Region

( ) ( )2

1 1

i

p n

np nR

n n p pτ τ−=

+ + +

( ) ( )2

1 1

i

p n

n

n pτ τ−=+

1 1n n p p≪ ≪

Depletion region – in PN diode: n=p=0

NEGATIVE Recombination => Generation

n=p=0 << ni => generation to create n,pEquilibrium restoration!


Outline

18




4) Conclusion


Band-to-band Recombination

19

( )( ) 20 0 0∆ = + + −∆ × ∆≈iR B n p n Bppn n

( )2 2i iR B np n Bn= − ≈ −

( )2iR B np n= −

( )0 0n n p p∆ = ∆≪ ≪

0n, p∼

Direct generation in depletion region

Direct recombination at low-level injection

B is a material property


Auger Recombination

20

22

1ττ

≈ =∆ =∆p A auger

auger p A

R c Nc N

nn

( ) ( )2 2

29 6

2 2

10 cm /sec−

= + −−n p i

n

i

p

R c c np n p

c ,c ~

n p n n

( ) ( )0 0 An n p p N∆ = ∆ =≪ ≪

Auger recombination at low-level injection

2 electron & 1 hole


Effective Carrier Lifetime

21

SRH direct AugerR R R R= + +

Light

( ) ( )0 τ−

∆ = ∆ = eff

t

n t n t e∆n 1 1 1

SRH direct Auger

nτ τ τ

= ∆ + +

( )2n T D n,auger Dn c N BN c N= ∆ + +

( ) 12τ−

= + +eff n T D n,auger Dc N BN c N


Effective Carrier Lifetime with all Processes

22

ND

( ) 12τ−

= + +eff n T D n,auger Dc N BN c N

2τ −≈eff n,auger Dc N

Elec. Dev. Lett., 12(8), 1991.


Conclusion

23

SRH is an important recombination mechanism in important semiconductors like Si and Ge.

SRH formula is complicated, therefore simplification for special cases are often desired.

Direct band-to-band and Auger recombination can also be described with simple phenomenological formula.

These expressions for recombination events have been widely validated by measurements.


Outline

24

1) Nature of interface states

2) SRH formula adapted to interface states

3) Surface recombination in depletion region

4) Conclusion

REF: ADF, Chapter 5, pp. 154-167


Surface states: Little bit of history

25

Emitter Base

Collector

Hoerni’s diagram of Mesa and planar transistors

One of the fundamental advances in semiconductor history

Emitter Base

Collector

Defect Induced Leakage Path

Silicon oxide


Atomic configuration of Surface States

26

Single bonds


Surface States

27

x

E

k


Multiple Levels of Surface States

28

x

E E

k


Multiple Levels of Surface States

29

xwrongokay

x


Outline

30




4) Conclusion


Surface Recombination Current

31

( ) ( )

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

32

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

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 …

33

( )

( )

( )

( )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 …

34

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 …

35

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

36

( ) ( )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

37

( )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

38




4) Conclusion


Case 2: Recombination in Depletion

39

( ) ( ) ( )( ) ( )

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

40

( )

( )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?

41

( ) ( )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

42

( )

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...

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