Computational Solid State Physics

計算物性学特論　第１０回

10. Transport properties II:

Ballistic transport

Electron transport properties

le: electronic mean free path

lφ: phase coherence length

λF: Fermi wavelength

Tunneling transport

I L

Current in one-dimension

L

L

L

U

L

U

LL

U

LL

dkTkfh

e

hv

dkTkvkfeI

dv

dd

dkdk

dkkTkvkfeI

)(]),([2

)()(]),([2

1

2)()(]),([2

T(k): transmission coefficient

LU

LL dTfh

eI )(],[

2

RU

RR dTfh

eI )(],[

2

LU

RLRL dTffh

eIII )()],(),([

2

Total current in one-dimension

Low bias limit

),(),(),(),(

feV

feVff RL

dTf

h

eG

VIG

dTf

h

VeI

L

L

U

U

)(2

/

)(2

2

2

)(2

)(

2

Th

eG

f

LU

RLRL dTffh

eIII )()],(),([

2

: conductance

at low temperatures

Landauer’s formula

)(

)(2

)(2

2

2

T

Th

eG

VTh

eI

： transmission coefficient

kR

μS

0 8.25

7.38

2

2

0

e

hh

eG

：Conductance

： Quantum conductance

： Quantum resistance

I: current, V: bias

Two- and four- terminal measurements

Tow- and four- terminal measurement

T

T

e

hR

Te

hR

IVR mnpqpqmn

1

2

1

2

/

243,21

212,21

,

2-terminal measurement

4-terminal measurement

Conductance of a quantum point contact

m

zkzkn z

nz 2

)()(),(

22 1)( T

Only one channel (n=1) is open.

for n=1

Conductance of a quantum point contact

Quantization of transverse motion

Nanowire of Au

Nanowire of Au

Nanowire of Au

Mechanically Controllable Break Junction

Histogram of conductance of a relay junction

Conductance through a quantum dot

dEeVEfEfETh

eI ds )]()()[(

2

22

2

]1)[exp(

)exp(1)(

)()(

kTEEkTEE

kTEf

dE

d

EEET

F

F

N

:Lorentzian broadening

of resonant tunneling through quantized energy EN of a dot:Thermal broadening

Tunneling current via quantum dot

A bound state and a resonant state

Transmission coefficient for resonant tunneling

)(2

)2/

(1)(

2

RL

pk

pk

TTa

v

EE

TET

2)(

4

RL

RLpk TT

TTT

1)( pkET

If TL=TR

Transmission coefficient of a resonant-tunneling structure

Characteristics of resonant tunneling diode

Resonant tunneling current

)]),,(()2(

2)[()(2

22),(

)()exp(

2

2

0

2222

,

Lzzzzz

L

zLz

kkk

kkfkd

kTkvdk

eI

m

k

m

kUkk

zurikzz

LU

LDL

zLL

BB

D

dEETEnh

eI

m

kU

TkTmk

n

)()(

2

))/exp(1ln()(

2

22

22

Dn2

:wave function

:energy

L

L

U

RDLD

U

LDL

zL

dEETEnEnh

eJ

dEETEnh

eJ

m

kUE

)()]()([

)()(

2

22

2

22

L

LU

L dEETEm

h

eJ

)()(2

Large bias and low temperature limit

Total resonant tunneling current

Transmission coefficient for resonant tunneling

)(2

)2/

(1)(

2

RL

pk

pk

TTa

v

EE

TET

2)(

4

RL

RLpk TT

TTT

1)( pkET

If TL=TR

Profile through a three-dimensional resonant-tunnelling diode. The bias increases from (a) to (d), giving rise to the I(V) characteristic shown in (e). The shaded areas on the left and right are the Fermi seas of electrons.

Profile through a three-dimensional resonant tunneling diode

L

Problems 10

Calculate the density of states for free electrons in one, two and three dimensions.

Calculate the ballistic current in two dimensions.

Calculate the transmission coefficient for a square barrier potential.

Calculate the transmission coefficient for a double square barrier potential.