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 GohdaZhong-yi LuKalman 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

Ψ+Ψ=Ψ VG00

{0 ,

ik z ik zr r

ik zl

e re z

te z

−

−+ →+∞

→−∞Ψ ∝

� Ψ∇Ψ−=FR

FL

E

E

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

Experiment: Reed et al (2000)

T=190 K T=300 K

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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

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V=2.4V

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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

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n-Si

MetalSiO2

EF

Ec

Ev

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I=V/R

8-layer Si(001)

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EF

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The dash-dot lines are boundary

EF

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V=5.0v

V=1.0v

V=0.1v

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

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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

Ψ+Ψ=Ψ VG00

{0 ,

ik z ik zr r

ik zl

e re z

te z

−

−+ →+∞

→−∞Ψ ∝

� Ψ∇Ψ−=FR

FL

E

E

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

0∇ ⋅ =J EVERYWHERE

DENSITY FUNCTIONAL THEORYFOR STEADY-STATE TRANSPORT

(CURRENT-DENSITY FUNCTIONAL)

[ ] [ ] 0 0E J E J Aαα α αδ = ∂ =21

2{ ( ) }xc ext H xcH i V V Vψ ψ εψ= − ∇ + + + + + =A A

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

( )xcρ= + +J j A A *Im ( )ψ

ψ ψ= − ∇�j

2HV ρ∇ = − 2 ( ) 0∇ − ∇ ∇ ⋅ = − ∇ ⋅ =A A J J

*

ψρ ψ ψ=�

[ , ]xc

EV

δ ρδρ

= J [ , ]xc

Eδ ρδ

= JA

J

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2W ( ) ))2

IW x L x Rρ δ δ

ρ∇ ⋅ = = ( − − ( −J

( )H iW ψ εψ+ =

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Source Sink

Battery!

Na wire

0.0 0.5 1.0 1.5 2.0Bias Voltage (V)

0.00

0.02

0.04

0.06

0.08

0.10

Cur

rent

(m

A)

Real-space DFT calculationJellium electrodesBias Voltage

Experiment

(Reed et al.)

Benzene ring -- IV characteristics