Semiconductor statistics Transportsoktyabr/NNSE508/NNSE508...Summary of carrier statistics in...

NNSE 508 EM Lecture #11

1

Lecture contents

• Semiconductor statistics

• Transport


2 Statistics of carriers: General

Electron concentration in the energy range

E to E+dE close to the conduction band

minimum:

Total electron concentration in the

conduction band

1exp

2)(

21

32

23*

Tk

EE

dEEEmdEEn

B

F

Ce

cE

B

F

Ce

Tk

EE

dEEEmn

1exp

221

32

23*

Tk

EEx

B

C

Tk

EE

B

CFc

0

2123

32

23*

1exp

2

cB

e

x

dxxTk

m

0

2123

2

*

1exp

2

22

c

Be

x

dxxTkm

General equation for carrier concentration

(effective density of states) x (Fermi integral of ½ order): )(2/1 cCNn

)()()( EfENEn cElectron concentration at the energy E

(Density of states) x (distribution function):


3 Statistics of carriers: General

The same is true for holes in

the valence band:

Effective density of states of electrons (or holes)

0

2123

2

*

1exp

2

22

V

Bh

x

dxxTkmp

)(2/1 VVNp

)(2/1

Concentration of

mobile (band) carriers

Fermi level

position

One-to-one correspondence


4 Statistics of carriers: Non-degenerate system

If all the C.B. energies are far

from Fermi level:

EC – EF >> kBT (> 3 kBT) :

)(2/1 VVNp

21exp0

21

0

21)(

0

21ccc edxxeedxxe

x

dxx xx

c

1exp cx

0

21

1exp

2

cC

x

dxxNnGeneral equation:

Tk

EE

B

CFc

Tk

EENn

B

CFC exp

Tk

EENp

B

FVV exp

Tk

EEx

B

C

Concentration of band carriers

Non-generate system: General case:

)(2/1 cCNn

2exp iB

g

VC nTk

ENNnp

Definition of intrinsic carrier concentration


5

Carriers in intrinsic semiconductors

Fermi level position in

intrinsic semiconductor


6 Effective mass approximation

)()()(2 0

2

rErrVrVm

pii

One-electron Schrödinger equation with

weak and slow varying perturbation Vi

(Effective mass approximation):

Small perturbation of periodicity: shallow impurities,

most of “hand-made” structures,

external forces

And as usual build a solution as a wave

packet of Bloch wavefunctions : )()()(

,

ruekcr nkikr

kn

n

Bloch wave packet:

With dimensions in real

and k-space 0

1a

kr

)()()( 0 rurFr

iV

Large perturbation of periodicity other bands need to

be considered: deep impurities

Depending on sign of the perturbation, the top-

most or bottom-most state splits from the band :


7

Example of EMA: Hydrogen-like impurity (donor)

Solution for energy :

r

erVi

2

)(

Hydrogen-like impurity = shallow impurity

Schrödinger equation for Hydrogen atom with

effective mass and screened Coulomb potential:

)()()(*2

22

rFEErFr

e

m

pC

22222

4 11*1

2

*

nm

mRy

n

meEd

BB

a

r

a

rF exp1

)(213

Envelope function of the ground state :

*

0

20

2

m

m

emaB

For donors in GaAs (m*=0.07m: and = 12.6 ):

Effective Ry* :

Ry* =6.6 meV, aB = 91 A

From Yu and Cordona, 2003


8

Donors

Shallow donors: - usually group V elements in Si and Ge (P, As, Sb)

- group IV elements on group III sublattice in III-V’s (Si, Sn in GaAs)

- group VI elements on group V sublattice in III-V’s (S, Te, Se in GaAs)


9

Acceptors

+

B

Shallow acceptors: - usually group III elements in Si and Ge (B, Al, Ga, In)

- group II elements on group III sublattice in III-V’s (Be, Mg, Zn in GaAs)

- group IV elements on group V sublattice in III-V’s (C, Si, Ge in GaAs)

Group IV impurities in III-V’s are often amphoteric.


10 Adding impurities: Extrinsic semiconductors

Simple impurity with two charge

states, e.g. simple donor:

d0 d+ + e,

Total donor concentration:

Concentration of neutral (filled with

electron) and ionized donors:

d0 has a degeneracy factor g

Ratio of neutral to charged donors:

g=2 for simple donors and

g=4 for simple acceptors

Ionization ratio for donors

and acceptors:

1

1)(

Tk

EEFD

B

F

e

Ef)(0

dFDdd EfgNN

FDdd gfNN 1

Tk

EE

FD

FD

d

d B

dF

gegf

gf

N

N

1

0

0ddd NNN

Tk

EE

d

d

d

B

dF

egN

N

1

1

Tk

EE

a

a

a

B

Fa

egN

N

1

1


11

Extrinsic semiconductors: no compensation

da NpNnIn extrinsic semiconductors charge neutrality

condition includes ionized impurities ( instead of n

= p in intrinsic semiconductors):

When impurity of one type (say donors) are present:

Then general equation for Fermi level (needs to be

solved for degenerate semiconductors) :

And in non-degenerate case ( – ionization energy):

or

Fermi level position

(non-degenerate) using

What happens with Fermi level if semiconductors contains impurities?

npifNNpn dd ;

Tk

EEN

eg

N

B

cFc

Tk

EE

d

d

B

dF2/1

1

n

eN

ng

N

Tk

cd

d

B

d

1

114

2 1

1

n

Nnn d Tk

d

c B

d

eg

Nn1with

Tk

EENn

B

CFC exp 11

4

2ln

1

1

n

N

N

nTkEE d

cBCF


12

Extrinsic semiconductors: no compensation

At high temperatures (kBT > d ) , for

At low temperatures, for

Fermi level

and concentration

Carriers are “freezing out”

Fermi level position in n-Ge

(uncompensated)

Cd

dBdCF

Ng

NTkEE ln

22

dd Nnor

n

N,1

4

1

Tk

d

Cd B

d

eg

NNn

2

14

1n

Nd

dNn


13 Extrinsic semiconductors with compensation

(results for a non-degenerate case)

At high temperatures, for

What is the accuracy of assumption ?

For n-type material:

For p-type material:

At low temperatures, for

Fermi level

and concentration

Add

ABdF

NNg

NTkEE ln

AdA NNNn ,

Tk

d

C

A

Ad B

d

eg

N

N

NNn

ANn1

Ad NNn

14

2

1

1

nN

nNN

A

Ad and

np


14

Doped semiconductors: Temperature dependence

Carrier concentration vs. temperature

curve has 3 distinct regions (4 regions in

compensated semiconductor)

Typical dependence for Si


15

Strong degeneracy, i.e. Fermi level lies in the

conduction (or valence) band:

Carrier concentration:

Substituting Fermi function by

step function (good for )

Finally:

Which is similar to simple metal

Strong non-degeneracy: metals again

CF

B

FC EEorTk

EE1exp

)(2/1 cCNnTk

EE

B

CFc

Tk

EE

C

c

C

B

CF

dxxNx

dxxNn

0

21

0

21 2

1exp

2

TkEE BCF 3

23

3

4

Tk

EENn

B

CFC


16 Summary of carrier statistics in semiconductors

Importance of doping:

Intrinsic

n-type

p-type

Si Resistivity

Undoped 2 x 105 -cm

Doped w/ 1015

As atoms/cm35 -cm

1015 As atoms/cm3 in 5x1022 Si atoms/cm3

5 order of magnitude resistivity change

due to 1 in 50 million impurities !

• Electronic properties are extremely

sensitive to impurities, defects, fields,

stresses …

• Fermi level determines static carrier

concentrations

• General equations can be simplified in

non-degenerate and strongly degenerate

cases

link to Java applets http://jas.eng.buffalo.edu

http://jas.eng.buffalo.edu/


17 Electron transport: General considerations

• Drude model again: Motion in real space = thermal motion + drift + scattering

• Motion in wavevector space: inside a band valley (unless intervalley scattering is involved)

• Current density is proportional to drift velocity of carriers

and gives Ohm’s law:

• For semiconductor containing both

electron and holes:

denvJ

dJ env en

*

e

m

m

en

m

een

e

m

*

2

he pne

–drift mobility


18

Scattering mechanisms

In low electric fields

• ionized impurities

• acoustic phonons

– Deformation potential

– Piezoelectric

In high electric fields

• optical phonons

• intervalley scattering

At high concentrations

• carrier-carrier scattering

To understand how these mechanisms affect mobility one needs to

consider dependence (E)

Mathiessen’s rule for relaxation time:

...

11111

niiiopacm


19 Conductivity effective mass

• For non-degenerate conduction band minimum ( -

point):

• Indirect conduction band minimum: effective mass

will depend on direction!

– For example: Si along [100] direction:

– Conductivity effective mass of Si [100] electrons::

• Valence band maximum: need to add up

contributions of light and heavy holes:

• Conductivity effective mass of holes

tlc mmm

21

3

11*

mme

*

2 2

2 2 26 6

m mn ex ey ez

l t

e en nJ J J J

m m

mhhh

lh

lhhhlhp e

m

p

m

pJJJ 2

2323

2121

*

11

hhlh

hhlh

hh

hh

lh

lh

v mm

mm

m

p

m

p

pm

Effective masses and low field mobilities at room temperature:


20 Temperature dependence of mobility

Mobility in n-Si

From Yu and Cordona, 2003,

and Shur, 2003

Mobility in p-Si

Mobility in n-GaAs Relaxation time m depends on energy!

LO phonon scattering

Intravalley acoustic phonon

scattering + (2TA+LO)

intervaley phonon scattering


21

High field electron transport

Hot electrons transfer energy into thermal

vibrations of the crystal lattice (phonons).

Such vibrations can be modeled as harmonic

oscillations with a certain frequency, ph .

The energy levels of a harmonic oscillator

are equidistant with the energy difference

between the levels equal to

Two types of phonons scatter electrons

differently: acoustical (slow) and optical (fast):

Hence, process for a hot electron can be

represented as follows. The electron

accelerates in the electric field until it

gains enough energy to excite optical phonon:

phphE

meVE optopt 40

kTE acac


22

Intervalley scattering in high electric field

GaAs has an L valley just 0.29 eV higher than

-valley

At high fields electrons can be transferred

into L-valleys. The conductivity will

depend on concentrations and mobilities

of electrons in both valleys:

Model for dependence of drift velocity on

electric field in GaAs

From Shur, 2003

LLNNe


23 Diffusion and drift of carriers

• If we increase or decrease carrier

concentration in some region of a sample,

the carriers will tend to diffuse to return to

equilibrium

• Diffusion is described by the first Fick’s

law:

Electron diffusion current

Hole diffusion current

• If electric field (weak) is applied:

Total electron current

Total hole current

neDJ nn

peDJ pp

(Flux of particles)= -(Diffusion coef.) x (gradient of concentration)

neDneJ nnn

peDpeJ pop


24 Einstein relation

At equilibrium condition (no total current)

diffusion current is equal to drift current:

For nondegenerate semiconductor:

Carrier concentration is changing in the

electric field (r):

Gradient of electron concentration is :

And substituting, obtain Einstein relation:

0neDne nn

Tk

EENn

B

CFC exp

Tk

xenxn

B

)(exp)( 0

Tk

enx

Tk

enn

BB

)(

EF

EC

Drift flux

Diffusion flux

e

TkD B

*

e

m

m

e

Semiconductor statistics Transportsoktyabr/NNSE508/NNSE508...Summary of carrier statistics in...

Documents

Transcript of Semiconductor statistics Transportsoktyabr/NNSE508/NNSE508...Summary of carrier statistics in...