ece606: solid state devices lecture 8 - college of engineeringee606/downloads/... · direct...

Klimeck – ECE606 Fall 2012 – notes adopted from Alam

ECE606: Solid State DevicesLecture 8

Gerhard [email protected]


Reference: Vol. 6, Ch. 4

Presentation Outline

•Reminder: »Basic concepts of donors and acceptors »Statistics of donors and acceptor levels»Intrinsic carrier concentration

•Temperature dependence of carrier concentration

•Multiple doping, co-doping, and heavy-doping•Conclusion


Donor Atoms in H2-analogy

= +r0

= r0

3


Donor Atoms in Real and Energy Space

( ) 20

20

40

2

02 4

13

1

16

πε= −

= − ×

ℏ

*host

s ,hos

*host

s ,host

tm

m

m K

q m

K

m

.

r0

ET=E1

( )4

1 2

02 4πε= −

ℏ

*host

s ,host

q

KE

m

~10s meV

1/β~kBT~25meV at T=300K4


How to Read the Table …


Summary …

0ADNp n dVN −+ − + + = ∫

0AD Nn Np + −− + + =

10

12 4F V B C F B

F D A F BB

( E E ) / k T ( E E ) / k T A( E E ) /V A

D( E E ) k/ k TT

Ne e

eN N

N

e− − −

−−

−+ +− + − =

A bulk material must be charge neutral over all …

Further if the material is spatially homogenous




•Reminder»Basic concepts of donors and acceptors »Statistics of donors and acceptor levels»Intrinsic carrier concentration




Carrier-density with Uniform Doping

0D Ap n N N dV+ − − + + = ∫

0D Ap n N N+ −− + + =

01 2 1 4

− − − −− −− + − =

+ +V B C B

D B A

F F

F BF

( E ) / k T ( E ) / k TV A ( E ) / k T ( E

E EE E )

D/ k T

AN e N ee e

N N

A bulk material must be charge neutral over all …

Further if the doping is spatially homogenous

Once you know EF, you can calculate n, p, ND+, NA

-.

( ) ( )1 2 1 2

2 20

1 2 1 4β ββ βπ π − − − − − + − = + +F FD AF F E EV V A C ( E )

D)

A( E

NN F E E

NN F

e eE E

(approx.)

FD integral vs. FD function ?


Intrinsic Concentration

0D Ap n N N+ −− + + =

01 2 1 4

− − − −− −− + − =

+ +F V B C F B

F D B A F B

( E E ) / k T ( E E ) / k T( E E ) / k T ( E E )V C

Ak T

D/

N

eNN e

Ne

e

( ) ( )0 β β− − + −− = ⇒ =c F v FE E EV

ECn p e eN N

1

2 2β≡ = + VG

F iC

EE E l

N

Nn








Carrier Density with Donors

0D Ap n N N+ −− + + =

01 2 1 4

− − − −− −− + − =

+ +F V B C F B

F D B A F B

( E E ) / k T ( E E ) / k T A( E E ) / k T ( E E ) / k T

DCVN

Ne e

e eN

N

In spatially homogenous field-free region …

Assume N-type doping …

n(will plot in next slide)


Temperature-dependent Concentration

D

n

N

Temperature

1

Freezeout Extrinsic

Intrinsic

i

D

n

N


Physical Interpretation

D

n

N

Temperature

1

Freezeout Extrinsic

Intrinsic

i

D

n

N


Electron Concentration with Donors

1 2 β+

−=+ DFD E

D( E )

Ne

N

β β β− −= ⇒ =FC FC( E )E E

C

EC

n

Nn e e eN

1 2 β −

=

+

c D

D

( E E )

C

eN

N

n 1ξ

≡+

DNn

N


Electron concentration with Donors

0+− + =Dp n N

01 2

F V B C F B

F D B

( E E ) / k T ( E E ) / k T DV C ( E E ) / k T

NN e N e

e− − − −

−− + =+

2

01

ξ

− + =+

i Dnn

N

N

nn

No approximation so far ….

2× = ip n n


High Donor density/Freeze out T

2

01

ξ

− + =+

i Dnn

N

N

nn

D iN n≫

2

01

0

ξ

ξ ξ

⇒ − + ≈+

⇒ + − =

D

D

N

N

N N

n

N

n

n n1 2

41 1

2ξ

ξ

= + −

DN

nN

N

D

n

N

Temperature

1

Freezeout Extrinsic

Intrinsic

i

D

n

N

( )2ξ

β− −≡ C DE ECNN e


Extrinsic T

( )2

C DE E / kTCD

NN e Nξ

− −≡ ≫

1 2

41 1

2ξ

ξ

= + −

DN

nN

N

Electron concentration equals donor densityhole concentration by nxp=ni2

D

n

N

Temperature

1

Freezeout Extrinsic

Intrinsic

i

D

n

N

41

21 1

2ξ

ξ

≈ + −

DNN

N

≈ DN


Intrinsic T

2

01

ξ

− + =+

i Dnn

N

Nn

n

1

2

22

2 4

= + +

i

D D nnN N

2 for

1+

−= ≈ <+ F D B( E E ) /

DDk TD F D

NN E

eN E

D

n

N

Temperature

1

Freezeout Extrinsic

Intrinsic

i

D

n

N

0

2

0− + ≈iD

nn N

n

2 2 0⇒ − + − =i Dn n N n


Extrinsic/Intrinsic T

For i Dn N≫

1 222

2 4

= + + ≈

i i

D Dn nN N

n

2 1 2

2

2 4

= + + ≈

i

D DDn n

N NN

For D iN n≫

D

n

N

Temperature

1

Freezeout Extrinsic

Intrinsic

i

D

n

N

What will happen if you use silicon circuits at very high temperatures ?

Bandgap determines the intrinsic carrier density.


Determination of Fermi-level

( ) 1β

β− −

= ⇒ = +

FcEC C

EF

C

N e EN

nlnn E

0Dp n N +− + =

01 2

− − − −−− + =

+F FV B C B

D BF

( E ) / k T ( E ) / k T DV

E E( kEC E ) / T

NN e N e

e








Multiple Donor Levels

210

1 2 1 2 1 4− − −− + + − =+ + +DF B F B A F BD

D D A( E ) / k T ( E ) / k T ( E EE kE ) / T

N N Np n

e e e

Multiple levels of same donor …

1 2

1 2 01 2 1 2 1 4− − −− + + − =

+ + +DF B F B AD F B

D A( E ) / k T ( E ) / k T ( E

DE E ) / k TE

N Np n

e e

N

e

Codoping…


Heavy Doping Effects: Bandtail States

E

k

k ?

E


Heavy Doping Effects: Hopping Conduction

E

k

Bandgap narrowing

β−× =*

GC V

Ep n N N e

Band transport vs.

hopping-transport

e.g. Base of HBTsK?

E

e.g. a-silicon, OLED


Arrangement of Atoms

Poly-crystallineThin Film Transistors

Crystalline

AmorphousOxides


Poly-crystalline material

Isotropic bandgap and increase in scattering


Band-structure and Periodicity

Periodicity is sufficient, but not necessary for bandgap. Many amorphous material show full isotropic bandgap

PR

B, 4

, 250

8, 1

971

Eda

gaw

a, P

RL,

100

,013

901,

200

8


Conclusions

1. Charge neutrality condition and law of mass-action allows calculation of Fermi-level and all carrier concentration.

2. For semiconductors with field, charge neutrality will not hold and we will need to use Poisson equation.

3. Heaving doping effects play an important role in carrier transport.


Outline

29

1) Non-equilibrium systems

2) Recombination generation events

3) Steady-state and transient response

4) Derivation of R-G formula

5) Conclusion

Ref. Chapter 5, pp. 134-146


Current Flow Through Semiconductors

30

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 Statisti cal Mechanics ⇒ Encapsulated into drift-diffusion equation with

recombination-generation (Ch. 5 & 6)

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

and occupation factors (Ch. 1-4)


Non-equilibrium Systems

31

vs.

Chapter 6 Chapter 5

I

V

How does the systemgo BACK to equilibrium?


Direct Band-to-band Recombination

32

Photon

GaAs, InP, InSb (3D)

Lasers, LEDs, etc.

In real space … In energy space …

PhotonDirect transistion –direct gap material

e and h must have same wavelength

1 in 1,000,000 encounters


Direct Excitonic Recombination

33

Photon (wavelength reduced from bulk)

CNT, InP, ID-systems

Transistors, Lasers, Solar cells, etc.

In energy space …

In real space …

Mostly in 1D systemsRequires strong coulomb interactions


Indirect Recombination (Trap-assisted)

34

Phonon

Ge, Si, ….

Transistors, Solar cells, etc.

Trap needs to be mid-gap to be effective. Cu or Au in Si


Auger Recombination

35

Phonon (heat)

InP, GaAs, …

Lasers, etc.

12

3

1 2

4

3

4

Requires very high electron density


Impact Ionization – A Generation Mechanism

36

Si, Ge, InP

Lasers, Transistors, etc.

4

3

1

2


Indirect vs. Direct Bandgap

37

The top & bottom of bands do not align atsame wavevector k for indirect bandgap material


Photon Energy and Wavevector

38

2

1 21 in eV

π=photonE. / 4

2 2

5 10 m

π πµ−<< =

×a

photon

photonk

E

=

=

+

+ℏ ℏ ℏV

V phot

photon

n C

C

o

k k

E E

k

E

Photon has large energy for excitation through bandgap,but its wavevector is negligible compared to size of BZ

2πa

2

in m

πλ µ

=photonk


Phonon Energy and Wavevector

39

2 2

phonphon

souo

nn

d on

kE/

= =ℏ

π πλ υ

=

=

+

+ℏ ℏ ℏV

V phon

phonon

n C

C

o

k k

E E

k

E

4

2 2

5 10 m

π πµ−≈ =

×a

Phonon has large wavevector comparable to BZ,but negligible energy compared to bandgap

vsound ~ 103 m/s << vlight=c ~ 106m/s

λsound >> λlight


Localized Traps and Wavevector

40

4

2 2

5 10trap ~ka m

π πµ−≈

×

Trap provides the wavevectornecessary for indirect transition

a


Outline

41





5) Conclusion



Equilibrium, Steady state, Transient

42

Device

Steady state

Transient

Equilibrium

time

(n,p

)

time

(n,p

)Environment


Detailed Balance: Simple Explanation

43

Mexico

China

India

USA

3 3

22

44

The rates of exchange of people (particles) between every pair of countries (energy levels) is balanced. Hence the name “Detailed Balance”.

Detailed balance is the property of equilibrium

The population of each of the countries (energy levels) remains constant under detailed balance.

The concept of detailed balance is powerful, because it can be used for many things (e.g. reduce the number of unknown rate constants by half, and derive particle distributions like Fermi-Dirac, Bose-Einstein distributions, etc.)

•9 in & 9 out •All numbers are people/unit time.

Equilibrium is a very active place

Fermi-Dirac distribution demands exploration of allowed states


Steady-state Response

44

Mexico

China

India

USA

4 6

32

45

Disturbing the detailed balance requires non-equilibrium conditions (needs energy). Unidirectional forces (red lines) can create such Non-equilibrium conditions.

The rates of exchange of people (particles) between every pair of countries (energy levels) is NOT balanced, but the sum of all arrival and departures to all countries is zero.

The flux at steady state is balanced overall, but the flux is NOT the same as in detailed balance(e.g. 12 in and 12 out in SS vs. 9 in and 9 out for Detailed Balance, for example).

The population of a country (energy level) remains constant with time after steady state is reached.

One can use the requirement that net flux at steady state be zero to calculate steady state population of a country (Eq. 5.21)

12 in and 12 out from USA

1

1


Detailed Balance, Transient, Steady-state

45

Mexico

China

India

USA

3 3

22

44

9 in 9 out Population conserved

Equilibrium =Detailed balance

Mexico

China

India

USA

3 5

3

45

1

1

2

Forced unidirectional connections(red lines) disturbs equilibrium (e.g. 10 in/12 out at time t1local populations not conserved, but global population is ….

Transient populations

Mexico

China

India

USA

4 6

32

45

1

1

12 in 12 out Population stabilized

Steady State But NOT Equilibrium


Outline

46





5) Conclusion



Indirect Recombination (Trap-assisted)

47

Phonon

Ge, Si, ….

Transistors, Solar cells, etc.


Physical view of Carrier Capture/Recombination

48

(2) After electron capture(3) After hole capture

TT TN pn= +

(1) Before a capture

electron-filledtraps

emptytraps

totaltraps

electronhole

Crystal / atomswith vibrations

Traps have destroyedone electron-hole pair

No change in nT and pT


Carrier Capture Coefficients

49

21 3

2 2*

thm kTυ =

υtht

σ υ≡ − ≡ n tT n hnc cnp

υ σ×× × = − × ×

T nthAdnn

dt A t

t p

710th

cm

sυ ≈

Tp RelAreaVolumednn

dt TotalArea t

× = − × ×

×


Capture Cross-section

50

20n rσ π=

e1

e2

e3

h1

h1

16 -22 10 cm−×

156 10−×185 10−×

147 10−×

Zn capture model …

Cascade model for capture


Conclusions

51

1) There are wide variety of generation-recombination

events that allow restoration of equilibrium once the

stimulus is removed.

2) Direct recombination is photon-assisted, indirect

recombination phonon assisted.

3) Concepts of equilibrium, steady state, and transient

dynamics should be clearly understood.


Outline

52

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

53

(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

54

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

55

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

56

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


Expressions for (n1) and (p1)

57

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)

58

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!

( )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

59

Tn

t

∂ =∂ 3 4,

p

t

∂+∂

(1)

(3)

(2)

(4)

n Tc n p= n Te n− p Tc p n− p Te p+

1 2,

n

t

∂−∂

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


Steady-state Trap Population

60

0Tn

t

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

( ) ( ) ( )11

1 1

n T p TT n T T

n p

c N n c N pn c np n n

c n n c p p

+= = −

+ + +

(1)

(3)

(2)

(4)


Net Rate of Recombination-Generation

61

( ) ( )

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

62




4) Conclusion


Case 1: Low-level Injection in p-type

63

( )( )( ) ( )

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

64

( )( )( ) ( )

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 …

65

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!

66


Outline

67




4) Conclusion


Band-to-band Recombination

68

( )( ) 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

69

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

70

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

71

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

72

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.

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