ECE606: Solid State Devices Lecture 7ee606/downloads/ECE606_f12_Lecture7.pdfLecture 7 Gerhard...

Klimeck – ECE606 Fall 2012 – notes adopted from Alam

ECE606: Solid State Devices

Lecture 7Gerhard Klimeck

[email protected]

1


Reference: Vol. 6, Ch. 3 & 4

Presentation Outline

• Intrinsic carrier concentration•Potential, field, and charge•E-k diagram vs. band-diagram•Basic concepts of donors and acceptors •Law of mass-action & intrinsic concentration •Statistics of donors and acceptor levels• Intrinsic carrier concentration•Temperature dependence of carrier concentration•Multiple doping, co-doping, and heavy-doping•Conclusions

2


Carrier Distribution

( )cg E

Eυ

FE( )g Eυ

cE

( )1 f E−

( )f E

( ) ( )cg E f E

( ) ( )1g E f Eυ −

( ) ( )top

c

E

cEn g E f E dE= ∫

( ) ( )1bot

E

Ep g E f E dE

υ

υ = − ∫

DOS F-D concentrationE E

1

3


Electron Concentration in 3D solids

( ) ( )

( )( )2 3

2 12

2 1 βπ −

=

−= ×

+

∫

∫ℏ

top

c

top

Fc

E

cE

* *E n n C

E EE

n g E f E dE

m m E EdE

e

( )3 2

2 1 2

0 1

22

*n

C

dF

e

mN

h ξ ηπ β ξ ξη

∞

−=

≡ +∫

( )( ) ( )2 3

2 1

1 β βπ − −

∞

+

−∫≃

ℏ c Fc c

*C

E

n

E EE E

*nm m

dEe e

E E

( ) ( )1 2

2 η η βπ

= ≡ −c c FC CF EN E

Assume wide bands

Include spinfactor of 2

4


Boltzmann vs. Fermi-Dirac Statistics

( ) ( )1 2

23ηη η β

π= → − ≡ − >c

C c C c C Fn N F N e if E E

( ) ( )cg E f E

( ) ( )1g E f Eυ − FE

ηce ( )1 2 ηcF

5


Effective Density of States

( ) ( )1 2

23c FE E

C c C c Fn N F N e if E Eβη βπ

− −= → − >

( ) ( )cg E f E

( ) ( )1g E f Eυ −

CN

VN

FEFE

As if all states are at a single level EC

6


Law of Mass-Action

( )

( )

β

β

− −

+ −

=

=

c F

v F

E EC

E EV

n N e

p N eFE

( )β

β

− −

−

× =

=

c v

g

E EC V

E

C V

n p N N e

N N e

Product is independent of the Fermi level!Very useful balance equation! Will use it often

7


Fermi-Level for Intrinsic Semiconductors

2β−=≡

gE

i

F

V

i

Cn e

E

N

E

NFE

( ) ( )

1

2 2

β β

β

− − + −= ⇒ =

= +

c i v iE E E E

Gi

V

C

VCn p e e

EE ln

N

N

N

N

E

k

3

2

2 β−

= =

= g

V

i

E

i CN

p n

n eN

n

8


• We discussed how electrons are distributed in electronic states

defined by the solution of Schrodinger equation.

• Since electrons are distributed according to their energy,

irrespective of their momentum states, the previously

developed concepts of constant energy surfaces, density of

states etc. turn out to be very useful.

=> will not discuss Schroedinger Eq. anymore

=> everything is captured in bandedges and effective masses

• We still do not know where EF is for general semiconductors … If

we did, we could calculate electron concentration.

Summary – DOS and Fermi Functions

9





10


E-k diagram vs. band-diagram

c refP.E. E E qV= − = −

Eυ

cEKinetic energy

PotentialEnergy

11


Position Resolved E-k Diagram

Eυ

cE

Eυ

cE

Eυ

cE

12


E-k Diagram vs. Band-diagram

( )c refP.E. E E qV x= − = −

refE

V=0 V=V1

13


Potential, Field and Charge

c refqV E E− = −

refE

2

2

d d V

dx dxρ = =E

c refP.E. E E= −

1 cdV dE

dx q dx= − =E

In most practical cases start from charge and derive potentials!=> Useful to learn “graphical” integration

14





15


Carrier Distribution

( )cg E

Eυ

FE

( )g Eυ

cE

( )1 f E−

( )f E

( ) ( )cg E f E

( ) ( )1g E f Eυ −

( ) ( )top

c

E

cEn g E f E dE= ∫

( ) ( )1bot

E

Ep g E f E dE

υ

υ = − ∫

DOS F-D concentrationE E

1

Critical items here;Intrinsic semiconductor has VERY few active electronsni is of the order of 1010/cm3. In 1022/cm3 atoms!Do not include the coulomb interactions of individual free electronsOften good enough to forget about the Distribution of carriers in energy => Replace by delta functions at band edge

CN

VN

FE

16


Effective Density of States

( ) ( ) ( )1 2

23βη β

π− −= → − >c FE E

C c C c Fn N F N e if E E

( ) ( )cg E f E

( ) ( )1g E f Eυ −

CN

VN

FEFE

As if all states are at a single level EC

Often good enough to forget about the Distribution of carriers in energy => Replace by delta functions at band edge 17


E-k Diagram vs. Band-diagram

CE

VE

V=0 V=V1

All quantum mechanics is now hidden in a single point per band!18


Potential, Field and Charge

c refP.E. E E= −

c refqV E E− = −

refE

2

2

d d V

dx dxρ = =E

1 cdV dE

dx q dx= − =E

In most practical cases start from charge and derive potentials!=> Useful to learn “graphical” integration

19





20


Insulator, Semiconductor, Metal

• Metal:»Conducts electrons even at very low temperatures

»Fermi Level crosses multiple bands even at very low temperaturs

• Semiconductor»Very weakly conducting

»Si: Eg=1.1eV ni~1010/cm3 in 1023/cm3 0.1 in a trillion

»GaAs: Eg=1.42eV ni~106/cm3 in 1023/cm3

»Ge: Eg=0.8eV ni~1013/cm3 in 1023/cm3 0.1 in a billion

• Insulator»“Not” conducting

»SiO2, Eg= 9eV, ni~10-68/cm3 � The whole earch has about 1050 atoms! If you made the whole worl out of

glass there would be not one electon conductive at room temperature!2β−=

≡

gE

i

F

V

i

Cn e

E

N

E

N

21


Simplified Planar View of Atoms

22


Donor Atoms

Even with donors, material is charge neutral

23


Donor Atoms in H2-analogy

= +r0

= r0

24


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=300K25


Assumption of Large Radius …

20

1 2

4πε=

ℏs ,host,P *

host

Kr

m qr0

1

12 90 53 12 9

0 53= × =,P

.r . A . A

.

02

02

10

0

4πε=

=

ℏ s,host*host

s ,hostH *

host,

m K

m

K

m

q

r/ m

m

a~0.5nm=5A => hundreds of Si atoms

26


Characteristics of Donor Atoms

The number of donor atoms is much smaller compared to host atoms. Therefore, the electrons from one donor atom can go to the other donor atoms only via the conduction /valence bands of the host crystal.

Just like a Hydrogen atom, it is possible to have multiple localized level for a given atom (see the blue levels).

Good donors live close to the conduction band, so that they can offer electrons easily. However, if they are below the midgap, the donor levels are marked with (D) to differentiate them from acceptor atoms (which live close to the valence band).

(D)

27


Acceptor Atoms

Even with acceptor, material is charge neutral

28


Characteristics of Acceptor Atoms

A

r0

ET

29


Amphoteric Dopants

acceptor-typeDonor-type

30


How to Read the Table …

31


Conclusion

Intrinsic carrier concentration is so small that semiconductor

must be doped to make it useful.

A doping atom behaves like a H-atom, except that the

dielectric constant and effective masses are given by by those

of the host atom.

32





33


Looking ahead: Carrier-Density w/Doping

0−+ − + − = ∫ D ANp n dVN

0+ −− + − =ADp n NN

1 10

42− − −

− −−

++ =

+− −

F D

F

A B

C

F

B F B

B

V D( E E )

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

/ k TE ) / k TC / k TVN

NN

eee

Ne

A bulk material must be charge neutral over all …

Further if the material is spatially homogenous

Let us see how the formula come about …

34


Characteristics of Donor Atoms

(D)

r0

ET


Localized vs. Delocalized States

2N states/per-band (with spin)

2N-2 states/per-band (with spin)


Statistics of Donor Levels

0 0 0 0 1

1 0 1 1

0 1 1 1

i F

B

i F

B

i i i

( E E )

k T

( E E )

k T

u / d E N P

/ / Z

/ e Z

/ e Z

−−

−−

1/1 x x x

− − − −

− −= ≡∑

i i F B F B

i i F

ii

B

( E N E ) / k T ( E ) / k T

i ( E N E ) /

N

k T

i

Ee eP

e Z

Coulomb interaction forbids this configuration


Statistics of Donor Levels

0 0 0 0 1

0 1 1 1

1 0 1 1

i F

B

i F

B

i i i

( E E )

k T

( E E )

k T

u / d E N P

/ / Z

/ e Z

/ e Z

−−

−−

0000

00 01 10 2

1 1

1 2 1− − −= = =+ + + +i F B F i B( E E ) / k T ( E E ) / k T

P / Zf

P P P / Z e / Z e

00

1 11 1

1 11

22 −

−− = − =

+ +F i B

i F B

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

fe e

Prob. that the donoris empty (charged)

Prob. that the donoris filled with at leastone electron (neutral)

Note the extra factor ….


Coulomb Exclusion for Band Electrons?

0 0 0 1

1 1 1i F

B

( E E

i

)

i

T

i

k

state E N P

/ Z

e Z−

−

i i F B i i F B

i i F B

( E N E ) / k T ( E N E ) / k T

i ( E N E ) / k T

i

e eP

e Z

− − − −

− −= ≡∑

Ei

E=0

E=2E=4


Localized vs. Band Electrons

E1 ← 2π/Lx

E2 ← 4π/Lx

E3 ← 6π/Lx

E4 ← 8π/Lx

E5 ← 10π/Lx

E6 ← 12π/Lx

E1’ ← 2p/(Lx/2)

E2’ ← 4π/(Lx/2)

E3’ ← 6π/(Lx/2)

Lx

Lx/2 Lx/2

Two electrons (even with opposite spin) can not be at the same position and same energy becauseof electrostatic repulsion

Band electrons (with opposite spin) need not be at the same position, so they can share occupy same energy level.

When we divide space by a factor of 2, the number of states (e.g. 6 here) does not change.


Acceptor Atoms

State [1] …. Hole present … N-1 charges

State [0] … Hole filled …. N charges


Statistics of Acceptor Levels in Si and Ge

1. Each atom contributes 2 states (up & down spin) to a band, therefore a band has 2N states.

2. Every time a host atom is replaced by a impurity atom, 2 states are disappear per a band and appear as localized states (sort of).

3. Therefore an acceptor atom close to hh and lh bands removes four states from those bands.

4. Because of Coulomb interaction only 1 hole can seat in these 4 states: the states are 0000, 0001, 0010, 0100, 1000.

5. One now uses P to compute the occupation

from lhfrom hh

EC

EV


Number and Energy Considerations …

4N-4 StatesIn HH/LH bands

0000 1000 0100 0010 0001

1) [0000] is the charged state as it has N electrons, but N-1 protons.

2) Single hole configuration [0001] is uncharged, as we have N-1 electrons, and N-1 protons … same is true for [0010], [0100], [1000] states.

3) Going from [0000] to [0001] states, the number of electrons goes down by 1 (Ni=-1).

4) Going from [0000] to [0001] states energy goes down by –EA, because one electron is no longer occupying the high energy level at EA.


Statistics of Acceptor Levels

0000 1000 0100 0010 00010 0

0000

1F B

i i F B

( xE ) / k T

( E N E ) / k T

i

eP

e Z

− −

− −= ≡∑

1

0001 0010 0100 1000

A F B A F B

i i F B

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

( E N E ) / k T

i

e eP P P P

e Z

− − − − −

− −= = = = ≡∑

Steps 3 & 4

00000000

0000 1000 0100 0010 0001

1

1 4 A F B( E E ) / k T

Pf

P P P P P e −= =+ + + + +


Filled and empty Donor/Acceptor Levels

00

1

1 2 −= =+ F i BD D ( E E ) / k T

N f Ne

[ ]0000−≡ =filled

A A AN N N f

2N-2 states

4N-4 StatesIn HH/LH bands

(Two holes can not seat together)

0000 1000 0100 0010 0001

00 10 01 +≡emptyD DN N

1

1 4 −=+ A F BA ( E E ) / k T

Ne


Distributions are physical ….

1 −=+ F D B

DD ( E / T

DE ) kg

Nf

e

1 1ε − −

= =+ +F D B F

'B D B

D DD ( E E ) / kk T ET ( E ) / k T

N Nf

e e e

Degeneracy factor …

Effective donor level


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− − −

−−

−+ +− + − =


Further if the material is spatially homogenous





48


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


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





51


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





62


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.

ECE606: Solid State Devices Lecture 7ee606/downloads/ECE606_f12_Lecture7.pdfLecture 7 Gerhard...

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