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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
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MOORE’S LAW
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• Phenomena (using the Lippmann-Schwinger method)
• Charging of molecules during transport (Gohda)
• Transport through ultra-thin films (Lu)
• New method (Varga)
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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)( *
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Experiment: Reed et al (2000)
T=190 K T=300 K
90°
0°
0°
90°
Theory
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Nature 417, 72 (2002)
“The current is strongly suppressed up to a threshold V, then it increases in steps”
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Coulomb blockade in a quantum dot
GaAs-AlGaAs-InGaAs-AlGaAs-GaAs
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Barner and Ruggiero, 1987
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LUMO LUMO
V=2.4V
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V=1.2[V] V=3.6[V]
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LUMO LUMO
AFTER SELF-CONSISTENCY,
MOLECULE IS NEUTRAL!
ELECTRODES ARE NEUTRAL!
EXCITED STATE?
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C6H5S
ELIMINATE CONTACT ON LEFT
C6H4(NO2)S
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-6 -4 -2 0 2
Energy (eV)
C6H5-S C6H4(NO2)-S
Energy (eV)
-3 -2 -1 0 1
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0.6V0 e
1.8V1 e
4.2V1 e
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Vsd = 0.1 V
Using a gate voltage
Q=0
0.3
0.8 1.2
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Fowler-Nordheim tunneling
JE
J/E2 = Aexp(-B/E)
M O S
n-Si
MetalSiO2
EF
Ec
Ev
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I
V
ln(J/E2)
1/E
Ohmic
Fowler-Nordheim
J/E2 = Aexp(-B/E)
I=V/R
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8-layer Si(001)
Ohmic
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Effective potential
EF
J
The dash-dot lines are boundary
EF
8 layers Si(001)
V=5.0v
V=1.0v
V=0.1v
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Current vs thickness [Si(001)]
Bias=1.0V
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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
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Fowler-Nordheim I-V plot
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Effective potential
J
The dash-dot lines are boundary
EF
SiO2
V= 4.0v
nano-film
V=0.5v
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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
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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
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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
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MAP TRANSPORT ONTO AN EIGENVALUE PROBLEM
2W ( ) ))2
IW x L x R
J
( )H iW
Schrödinger equation with imaginary potential:
Source Sink
Battery!
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Na wire
Real-space DFT calculationJellium electrodesBias Voltage
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Experiment
(Reed et al.)
Benzene ring -- IV characteristics