Computational Solid State Physics

計算物性学特論　第９回

9. Transport properties I:

Diffusive transport

Electron transport properties

le: mean free path 　 of electrons

lφ: phase coherence length

λF: Fermi wavelength

Examples of quantum transport

le : mean free path of electrons lφ: phase coherence lengthλF: Fermi wavelength single electron charging

key quantities

Point contact: ballistic

h

eGo

22

quantum conductance

Aharanov Bohm effect: phase coherent

eh /0

quantum magnetic flux

Quantum dot: single electron charging

Shubnikov-de Haas oscillations and quantum Hall effect

Diffusive transport

'

2

scat

g

1

)()()()(V)(2

P

)(1

dt

ddt

d

kkkk'

kkkk

P

kkrrr

kk

Vr

Fk

Equation of motion for electrons

Scattering Rate

k: wave vecot of Bloch electron

How to solve equations of motion for electrons with scattering?

Relaxation time approximation for scattering Direct numerical solution:

Monte Carlo simulation Boltzmann equation for distribution function

of electrons

Relaxation time approximation

m

kk c 2

)(22

Fmdk

kd

dt

dk

dt

dvg 1)(12

2

)( BvEeF

vmF

dt

dvm

g

gg

equation of motion

genvj current density

m: effective mass

n: electron concentration

E: electric field

B: magnetic field

Drude model: B=0

gg v

meEdt

dvm )exp(0 tiEE

Eim

neenvj

Eim

ev

g

g

1

1

1

1

2

Ej

im

ne

1

1)(

2

conductivity

: drift velocity

Drude model: steady state solution in magnetic field

)( BvEeF

vmF

dt

dvm

g

gg

),,( zxyyx EBvEBvEeF

z

y

x

c

c

c

cgz

gy

gx

E

E

E

m

e

v

v

v

2222

100

01

01

1

1

: B is assumed parallel to z.

m

eBc : cyclotron

frequency

drift velocity

Conductivity tensor in magnetic field

2222

2

100

01

01

1

1

c

c

c

cm

ne

m

eBc Eenvj g

zB //

xI //

z

y

x

c

c

c

c E

E

E

m

nej

2222

2

100

01

01

1

1

xx

y

Em

nej

j

20

no transverse magneto-resistance

Bne

jEE xxcy Hall effect

Hall effect

m

eBc

Monte Carlo simulation for electron motion

)()(

)()(V)(2

P

)(1

dt

ddt

d

2

scat

g

kk

rrr

kk

Vr

Fk

kkkk

)()(

)()(V)(2

P

)(1

dt

ddt

d

2

scat

g

kk

rrr

kk

Vr

Fk

kkkk

Drift

Scattering

Drift

Scattering

tdtVt

Vt

xx )(1

0 xx venj : current

Drift velocity as a function of time

Boltzmann equation

r

k

tF

k

tvr g

Motion of electrons in r-k space during infinitesimal time Interval Δt

Equation of motion for distribution function

r

fv

dt

trftdtvrfdt

trfdttrff

kg

kgk

kkdiffk

),(),(

),(),(|

equation of motion for electron distribution function fk(r,t).

k

fFdt

trftrfdt

trfdttrff

k

kFdtk

kkforcek

),(),(

),(),(|

Boltzmann equation

scattkforcekdiffkk ffff |||

scattkfieldkdiffk fff |||

Steady state

scattkkrgkk ffvfF

|

Boltzmann equation

'][)2(

1

'])1()1([)2(

1|

''3

''''3

dkPff

dkPffPfff

kkkk

kkkkkkkkscattk

Electron scattering

detailed balance condition for transition probability

kkkk PP ''

Scattering term

)(1

')]()[()2(

1

'][)2(

1

0

'0''

0

3

''3

kkk

kkkkkk

kkkk

ff

dkPffff

dkPff

assume: elastic scattering, spherical symmetry

')cos1()2(

11''3dkP kkkk

k

),( '' kkkk kPP

Transport scattering time

')cos1()2(

11''3dkP kkkk

k

),( '' kkkk kPP k

k’

Θkk’

Contribution of forward scattering is not efficient.Contribution of backward scattering is efficient.

Linearized Boltzmann equation

)(1 0

00

kkk

kk

kk ffF

fvT

T

fv

')cos1()2(

11

)(

''3

000

dkP

Ff

TT

fvff

kkkkk

kkk

kkk

),( '' kkkk kPP

Fermi sphere is shifted by electric field.

Current density and conductivity

kdeEf

vevj kk

kk3

0

3)(

)2(

2

kdevfj k3

3)2(

2

kdf

ve

Ej

kkxxx

xxxx

30

2

3

2

4

Electron mobility in GaAs

Energy flux and thermal conductivity

TkdT

fvvk

kdTT

fvvkU

kk

kk

kk

kk

3

0

3

30

3

)()2(

2

)()()2(

2

kdT

fvvkK k

kkk

30

3)(

)2(

2

TKU

thermal conductivity

Problems 9

Calculate both the conductivity and the resistivity tensors in the static magnetic fields, by solving the equation of motion in the relaxation time approximation.

Study the temperature dependence of electron mobility in n-type Si.

Calculate the electron mobility in n-type silicon for both impurity scattering and acoustic phonon scattering mechanisms, by using the linearized Boltzmann equation.