Explorations in quantum transport – phenomena and methods

Sokrates T. Pantelides Department of Physics and astronomy, Vanderbilt University, Nashville, TN

andOak Ridge National Laboratory, Oak Ridge, TN

Collaborators: Yoshihiro Gohda Zhong-yi Lu

Kalman Varga

Supported in part by Department of Energy

MOORE’S LAW

• Phenomena (using the Lippmann-Schwinger method)

• Charging of molecules during transport (Gohda)

• Transport through ultra-thin films (Lu)

• New method (Varga)

The Lippmann-Schwinger method

• Norton Lang, 1981 –

t

r

• Di Ventra, Lang, and Pantelides, 2000-2002

0 ,

ik z ik zr r

ik zl

e re z

te z

FR

FL

E

E

rrdErJ )]()(Im[2)( *

Experiment: Reed et al (2000)

T=190 K T=300 K

90°

0°

0°

90°

Theory

Nature 417, 72 (2002)

“The current is strongly suppressed up to a threshold V, then it increases in steps”

Coulomb blockade in a quantum dot

GaAs-AlGaAs-InGaAs-AlGaAs-GaAs

Barner and Ruggiero, 1987

LUMO LUMO

V=2.4V

V=1.2[V] V=3.6[V]

LUMO LUMO

AFTER SELF-CONSISTENCY,

MOLECULE IS NEUTRAL!

ELECTRODES ARE NEUTRAL!

EXCITED STATE?

C6H5S

ELIMINATE CONTACT ON LEFT

C6H4(NO2)S

-6 -4 -2 0 2

Energy (eV)

C6H5-S C6H4(NO2)-S

Energy (eV)

-3 -2 -1 0 1

0.6V0 e

1.8V1 e

4.2V1 e

Vsd = 0.1 V

Using a gate voltage

Q=0

0.3

0.8 1.2

Fowler-Nordheim tunneling

JE

J/E2 = Aexp(-B/E)

M O S

n-Si

MetalSiO2

EF

Ec

Ev

I

V

ln(J/E2)

1/E

Ohmic

Fowler-Nordheim

J/E2 = Aexp(-B/E)

I=V/R

8-layer Si(001)

Ohmic

Effective potential

EF

J

The dash-dot lines are boundary

EF

8 layers Si(001)

V=5.0v

V=1.0v

V=0.1v

Current vs thickness [Si(001)]

Bias=1.0V

I-V curve through SiO2 nano-film

Three regions:(1) 0.0 to 0.5V quasi-linear;(2) 0.5 to 4.0V non-linear;(3) Over 4.0V quasi-linear

Fowler-Nordheim I-V plot

Effective potential

J

The dash-dot lines are boundary

EF

SiO2

V= 4.0v

nano-film

V=0.5v

1.2 n m (SiO 2)

1.5 n m (SiO2)

0.9 n m (vacuum)

1.2 n m (vacuum)

1.5 n m (vacuum)

0 1 2 3 4 5

G. Timp et al (Bell Lab) 1998 calculation

The Lippmann-Schwinger method

t

r

0 ,

ik z ik zr r

ik zl

e re z

te z

FR

FL

E

E

rrdErJ )]()(Im[2)( *

0 J EVERYWHERE

DENSITY FUNCTIONAL THEORYFOR STEADY-STATE TRANSPORT

(CURRENT-DENSITY FUNCTIONAL)

[ ] [ ] 0 0E J E J A 21

2{ ( ) }xc ext H xcH i V V V A A

Static external potential ( )extV x + B.C.

( )xc J j A A *Im ( )

j

2HV 2 ( ) 0 A A J J

*

[ , ]xc

EV

J [ , ]

xc

E

J

AJ

MAP TRANSPORT ONTO AN EIGENVALUE PROBLEM

2W ( ) ))2

IW x L x R

J

( )H iW

Schrödinger equation with imaginary potential:

Source Sink

Battery!

Na wire

Real-space DFT calculationJellium electrodesBias Voltage

Experiment

(Reed et al.)

Benzene ring -- IV characteristics