ECE606: Solid State Devices Lecture 8 Temperature

Klimeck – ECE606 Fall 2012 – notes adopted from Alam

ECE606: Solid State Devices Lecture 8

Temperature Dependent Carrier Density Concepts of Recombination

Gerhard Klimeck [email protected]

mailto:[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

202 4

13

1

16

πε= −

= − ×

*host

s ,hos

*host

s ,host

tm

mm K

q mK

m

.

r0

ET=E1

( )4

1 202 4πε

= −

*host

s ,host

qK

E m

~10s meV

1/β~kBT~25meV at T=300K 4


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 TTNe 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 22 2 0

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 Fe e

E 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/

Ne

NN e Nee

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

ECn p e eN N

12 2β

≡ = + VGF i

C

EE E l NN

n








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

nN

Temperature

1

Freeze out Extrinsic

Intrinsic

i

D

nN


Physical Interpretation

D

nN

Temperature

1


Intrinsic

i

D

nN


Electron Concentration with Donors

1 2 β+

−=+ DFD E

D( E )N

eN

β β β− −= ⇒ =FC FC( E )E E

C

EC

nN

n e e eN

1 2 β −

=

+

c D

D

( E E )

C

eN

Nn 1

ξ

≡+

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 ee

− − − −−− + =

+

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

0ξ

ξ ξ

⇒ − + ≈+

⇒ + − =

D

D

N

N

N N

n

N

n

n n1 2

41 12ξ

ξ

= + −

DNn N

N

D

nN

Temperature

1


Intrinsic

i

D

nN

( )

2ξβ− −≡ C DE ECNN e


Extrinsic T

( )

2C DE E / kTC

DNN e Nξ

− −≡

1 241 1

2ξ

ξ

= + −

DNn

NN

Electron concentration equals donor density hole concentration by nxp=ni

2

D

nN

Temperature

1


Intrinsic

i

D

nN

412

1 12ξ

ξ

≈ + −

DNNN

≈ DN


Intrinsic T

2

01

ξ

− + =+

i Dnn

N

Nnn

12

22

2 4

= + +

iD D nn N N

2 for

1+

−= ≈ <+ F D B( E E ) /

DDk TD F D

N N Ee

N E

D

nN

Temperature

1


Intrinsic

i

D

nN

0

2

0− + ≈iD

n n Nn

2 2 0⇒ − + − =i Dn n N n


Extrinsic/Intrinsic T

For i Dn N

1 222

2 4

= + + ≈

i iD Dn nN N n

2 1 22

2 4

= + + ≈

iD D

Dn nN N N

For D iN n

D

nN

Temperature

1


Intrinsic

i

D

nN

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 ENnlnn 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 ee








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 ne 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 ne e

Ne

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 HBTs K?

E

e.g. a-silicon, OLED


Arrangement of Atoms

Poly-crystalline Thin Film Transistors

Crystalline

Amorphous Oxides


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 Vq 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

31

vs.

Chapter 6 Chapter 5

I

V

How does the system go BACK to equilibrium?


Direct Band-to-band Recombination

32

Photon

GaAs, InP, InSb (3D)

Lasers, LEDs, etc.

In real space … In energy space …

Photon Direct 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 systems Requires 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.

1 2

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 at same wavevector k for indirect bandgap material


Photon Energy and Wavevector

38

21 21 in eV

π=

photonE. / 4

2 25 10 m

π πµ−<< =

×a

photon

photonkE

=

=

+

+ V

V phot

photon

n C

C

o

k kE E

kE

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

2πa

2 in mπ

λ µ=photonk


Phonon Energy and Wavevector

39

2 2

phonphon

souo

nn

d on

kE/

= =

π πλ υ

=

=

+

+ V

V phon

phonon

n C

C

o

k kE E

kE

4

2 25 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 25 10trap ~k

a mπ π

µ−≈×

Trap provides the wavevector necessary for indirect transition

a


Outline

41

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

ECE606: Solid State Devices Lecture 8 Temperature

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