IOP PUBLISHING NANOTECHNOLOGY
Nanotechnology 18 (2007) 424014 (6pp) doi:10.1088/0957-4484/18/42/424014
The quantum interference effect transistor
Charles A Stafford
1
, David M Cardamone
2
and Sumit Mazumdar
1
1
Department of Physics, University of Arizona, 1118 E. 4th Street, Tucson, AZ 85721, USA
2
Department of Physics, Simon Fraser University, 8888 University Drive, Burnaby, BC,
V5A 1S6, Canada
E-mail: stafford@physics.arizona.edu
Received 3 May 2007, in final form 25 July 2007
Published 13 September 2007
Online at stacks.iop.org/Nano/18/424014
Abstract
We give a detailed discussion of the quantum interference effect transistor
(QuIET), a proposed device which exploits the interference between electron
paths through aromatic molecules to modulate the current flow. In the off
state, perfect destructive interference stemming from the molecular
symmetry blocks the current, while in the on state, the current is allowed to
flow by locally introducing either decoherence or elastic scattering. Details
of a model calculation demonstrating the efficacy of the QuIET are presented,
and various fabrication scenarios are proposed, including the possibility of
using conducting polymers to connect the QuIET with multiple leads.
1. Introduction
Despite their low cost and extreme versatility, modern
semiconductor transistors face fundamental obstacles to
continued miniaturization. First, top-down fabrication gives
them microscopic variability from device to device, which,
while acceptable at today’s length scales, renders them
unscalable in the nanometre regime. Second, these devices,
like all field-effect devices, function by raising and lowering
an energy barrier to charge transport of at least k
B
T ; each
device therefore dissipates energy of this magnitude into the
environment with every switching cycle
3
. At device densities
greater than the current state of the art, the cost and engineering
challenges associated with removing the resultant heat are
daunting [2]. While the first challenge can be met by utilizing
the bottom-up chemical fabrication of single-molecule devices
(e.g. [3]), this approach in itself does nothing to address the
need for a cooler switching mechanism.
An alternative paradigm to raising and lowering an energy
barrier is to exploit the wave nature of the electron to control
the current flow [4–9]. Traditionally, such interference-based
devices are modulated via the Aharanov–Bohm effect [10].
This, however, is incompatible with the small size of molecular
devices [4]: through a 1 nm
2
device, a magnetic field of over
600 T would be required to generate a phase shift of order
1 rad. Similarly, a device based on an electrostatic phase
shift [7] would require voltages incompatible with structural
stability. Previously [11], we have proposed a solution,
3
This is to be distinguished from the heating due to irreversible computation
of [1].
Figure 1. Artist’s conception of a quantum interference effect
transistor based on 1,3-benzenedithiol. The coloured spheres
represent individual carbon (green), hydrogen (purple), sulfur
(yellow), and gold (gold) atoms. In the ‘off’ state of the device,
destructive interference blocks the flow of current between the source
(bottom) and drain (right) electrodes. Decoherence introduced by the
scanning transmission microscope (STM) tip (upper left) suppresses
interference, allowing current flow. Image by Helen M Giesel.
called the quantum interference effect transistor (QuIET) (see
figure 1), which exploits a perfect destructive interference due
to molecular symmetry and controls quantum transport by
introducing decoherence or elastic scattering.
The purpose of this article is to communicate the details of
this proposal, including several potential chemical structures,
to facilitate fabrication and testing of this device. In section 2,
we describe the theoretical framework used to model the
device. Section 3 explains the QuIET’s operating mechanism.
0957-4484/07/424014+06$30.00 1 ? 2007 IOP Publishing Ltd Printed in the UK
Nanotechnology 18 (2007) 424014 C A Stafford et al
Section 4 discusses practical implementations of the device.
We conclude in section 5.
2. Theoretical model
The QuIET consists of a central molecular element, two leads
chemically bonded to the molecule, and a third lead, which can
be coupled to the molecule either capacitively or via tunnelling.
The Hamiltonian of this system can be written as the sum of
three terms:
H = H
mol
+ H
leads
+ H
tun
. (1)
The first is the π-electron molecular Hamiltonian
H
mol
=
summationdisplay
nσ
ε
n
d
?
nσ
d
nσ
?
summationdisplay
nmσ
parenleftbig
t
nm
d
?
nσ
d
mσ
+ H.c.
parenrightbig
+
summationdisplay
nm
U
nm
2
Q
n
Q
m
, (2)
where d
?
nσ
creates an electron of spin σ =↑,↓ in the π-orbital
of the nth carbon atom, and ε
n
are the orbital energies. We use
a tight-binding model for the hopping matrix elements with
t
nm
= 2.2, 2.6 or 2.4 eV for orbitals connected by a single
bond, double bond, or within an aromatic ring, respectively,
and zero otherwise. The final term of equation (2) contains
intrasite and intersite Coulomb interactions, as well as the
electrostatic coupling to the leads. The interaction energies are
given by the Ohno parameterization [15, 16]:
U
nm
=
11.13 eV
radicalBig
1 + .6117
parenleftbig
R
nm
/
?
A
parenrightbig
2
, (3)
where R
nm
is the distance between orbitals n and m.
Q
n
=
summationdisplay
σ
d
?
nσ
d
nσ
?
summationdisplay
α
C
nα
V
α
/e ? 1(4)
is an effective charge operator [17] for orbital n,wherethe
second term represents a polarization charge. Here C
nα
is the
capacitance between orbital n and lead α, chosen consistent
with the interaction energies of equation (3) and the geometry
of the device, and V
α
is the voltage on lead α. e is the
magnitude of the electron charge.
Each metal lead α possesses a continuum of states, and
their total Hamiltonian is
H
leads
=
3
summationdisplay
α=1
summationdisplay
k∈α
σ
epsilon1
k
c
?
kσ
c
kσ
, (5)
where the epsilon1
k
are the energies of the single-particle levels in the
leads, and c
?
kσ
is an electron creation operator. Here leads 1
and 2 are the source and drain, respectively, and lead 3 is the
control, or gate electrode.
Tunnelling between molecule and leads is provided by the
final term of the Hamiltonian,
H
tun
=
summationdisplay
〈nα〉
summationdisplay
k∈α
σ
parenleftbig
V
nk
d
?
nσ
c
kσ
+ H.c.
parenrightbig
, (6)
where the V
nk
are the tunnelling matrix elements from a level
k within lead α to the nearby π-orbital n of the molecule.
Coupling of the leads to the molecule via molecular chains,
as may be desirable for fabrication purposes, can be included
in the effective V
nk
, as can the effect of substituents (e.g., thiol
groups) used to bond the leads to the molecule [18, 19].
We use the non-equilibrium Green function (NEGF)
approach [20, 21] to describe transport in this open quantum
system. The retarded Green function of the full system is
G(E) = [E ? H
mol
? Sigma1(E)]
?1
, (7)
where Sigma1 is an operator, known as the retarded self-energy,
describing the coupling of the molecule to the leads. The
QuIET is intended for use at room temperature, and it operates
in a voltage regime where there are no unpaired electrons in
the molecule. Thus lead–lead and lead–molecule correlations,
such as the Kondo effect, do not play an important role.
Electron–electron interactions may therefore be included via
the self-consistent Hartree–Fock method. H
mol
is replaced
by the corresponding mean-field Hartree–Fock Hamiltonian
H
HF
mol
, which is quadratic in electron creation and annihilation
operators, and contains long-range hopping. Within mean-field
theory, the self-energy is a diagonal matrix
Sigma1
nσ,mσ
prime (E) = δ
nm
δ
σσ
prime
summationdisplay
〈aα〉
δ
na
Sigma1
α
(E), (8)
with nonzero entries on the π-orbitals adjacent to each lead α:
Sigma1
α
(E) =
summationdisplay
k∈α
〈nα〉
|V
nk
|
2
E ? epsilon1
k
+ i0
+
. (9)
The imaginary parts of the self-energy matrix elements
determine the Fermi’s Golden Rule tunnelling widths
Gamma1
α
(E) ≡?2ImSigma1
α
(E) = 2π
summationdisplay
k∈α
|V
nk
|
2
δ (E ? epsilon1
k
) . (10)
As a consequence, the molecular density of states changes from
a discrete spectrum of delta functions to a continuous, width-
broadened distribution. We take the broad-band limit [20],
treating the Gamma1
α
as constants characterizing the coupling of the
leads to the molecule. Typical estimates [19] using the method
of [22] yield Gamma1
α
lessorsimilar 0.5 eV, but values as large as 1 eV have
been suggested [18].
The effective hopping and orbital energies in H
HF
mol
depend
on the equal-time correlation functions, which are found in the
NEGF approach to be
〈d
?
nσ
d
mσ
〉=
summationdisplay
〈aα〉
Gamma1
α
2π
integraldisplay
∞
?∞
dEG
nσ,aσ
(E)G
?
aσ,mσ
(E) f
α
(E),
(11)
where f
α
(E) ={1 + exp[(E ? μ
α
)/k
B
T ]}
?1
is the Fermi
function for lead α. Finally, the Green function is determined
by iterating the self-consistent loop, equations (7)–(11).
The current in lead α is given by the multi-terminal current
formula [23]
I
α
=
2e
h
3
summationdisplay
β=1
integraldisplay
∞
?∞
dET
βα
(E)
bracketleftbig
f
β
(E) ? f
α
(E)
bracketrightbig
, (12)
where
T
βα
(E) = Gamma1
β
Gamma1
α
|G
ba
(E)|
2
(13)
is the transmission probability [21] from lead α to lead β,and
a (b) is the orbital coupled to lead α(β). Similar mean-field
NEGF calculations have been widely used to treat two-terminal
transport through single molecules [13].
2
Nanotechnology 18 (2007) 424014 C A Stafford et al
(a)
(b)
(c)
Figure 2. Effective transmission probability
?
T
12
of the device shown
in figure 1, at room temperature, with Gamma1
1
= 1.2eVandGamma1
2
= .48 eV.
Here ε
F
is the Fermi level of the molecule. (a) Sigma1
3
= 0; (b)
Sigma1
3
=?iGamma1
3
/2, where Gamma1
3
= 0 in the lowest curve, and increases by
.24 eV in each successive one; (c) Sigma1
3
is given by equation (15) with
a single resonance at ε
ν
= ε
F
+ 4eV.Heret
ν
= 0inthelowest
curve, and increases by 0.5 eV in each successive curve. Inset: full
vertical scale for t
ν
= 1 eV. Reprinted with permission from [11].
? 2006 American Chemical Society.
3. Switching mechanism
The QuIET exploits quantum interference stemming from the
symmetry of monocyclic aromatic annulenes such as benzene.
Quantum transport through single benzene molecules with two
metallic leads connected at para positions has been the subject
of extensive experimental and theoretical investigation [13];
however, a QuIET based on benzene requires the source (1)
and drain (2) to be connected at meta positions, as illustrated
in figure 1. The transmission probability T
12
of this device,
for Sigma1
3
= 0, is shown in figure 2. Due to the molecular
symmetry [8], there is a node in T
12
(E), located midway
between the highest occupied molecular orbital (HOMO) and
lowest unoccupied molecular orbital (LUMO) energy levels
(see figure 2(b), lowest curve). This mid-gap node, at the Fermi
level of the molecule, plays an essential role in the operation of
the QuIET.
The existence of a transmission node for the meta
connection can be understood in terms of the Feynman path
integral formulation of quantum mechanics [24], according
to which an electron moving from lead 1 to lead 2 takes all
possible paths within the molecule; observables relate only
to the complex sum over paths. In the absence of a third
lead (Sigma1
3
= 0), these paths all lie within the benzene ring.
An electron entering the molecule at the Fermi level has de
Broglie wavevector k
F
= π/2d,whered = 1.397
?
Ais
the intersite spacing of benzene (note that k
F
is a purely
geometrical quantity, which is unaltered by electron–electron
interactions [25]). The two most direct paths through the ring
have lengths 2d and 4d, with a phase difference k
F
2d = π,so
they interfere destructively. Similarly, all of the paths through
the ring cancel exactly in a pairwise fashion, leading to a node
in the transmission probability at E = ε
F
.
This transmission node can be lifted by introducing
decoherence or elastic scattering that breaks the molecular
symmetry. Figures 2(b) and (c) illustrate the effect of coupling
a third lead to the molecule, introducing a complex self-energy
Sigma1
3
(E) on the π-orbital adjacent to that connected to lead 1
or 2. An imaginary self-energy Sigma1
3
=?iGamma1
3
/2 corresponds to
coupling a third metallic lead directly to the benzene molecule,
as shown in figure 1. If the third lead functions as an
infinite-impedance voltage probe, the effective two-terminal
transmission is [12]
?
T
12
= T
12
+
T
13
T
32
T
13
+ T
32
. (14)
The third lead introduces decoherence [12] and additional
paths that are not cancelled, thus allowing current to flow, as
shown in figure 2(b). This quantum-mechanical effect of the
third lead is a fundamentally different switching mechanism
from other proposed molecular transistors, such as [3], which
rely on electrostatic gating to control the current. As a proof
of principle, a QuIET could be constructed using a scanning
tunnelling microscope (STM) tip as the third lead (cf figure 1),
with tunnelling coupling Gamma1
3
(x) to the appropriate π-orbital of
the benzene ring, the control variable x being the piezo-voltage
controlling the tip–molecule distance.
By contrast, a real self-energy Sigma1
3
introduces elastic
scattering, which can also break the molecular symmetry. This
can be achieved by attaching a second molecule to the benzene
ring, for example an alkene chain. The retarded self-energy
due to the presence of a second molecule is
Sigma1
3
(E) =
summationdisplay
ν
|t
ν
|
2
E ? ε
ν
+ i0
+
, (15)
where ε
ν
is the energy of the νth molecular orbital of the
second molecule, and t
ν
is the hopping integral coupling this
orbital with the neighbouring π-orbital of the benzene ring.
Figure 2(c) shows the transmission probability T
12
(E) in the
vicinity of the Fermi energy of the molecule, for the case of
a single side orbital at ε
ν
= ε
F
+ 4 eV. As the coupling
t
ν
is increased, the node in transmission at E = ε
F
is
lifted due to scattering from the side orbital. The side group
introduces Fano antiresonances [5, 26], which suppress current
through one arm of the annulene, thus lifting the destructive
interference. Put another way, the second molecule’s orbitals
hybridize with those of the annulene, and a state that connects
leads 1 and 2 is created in the gap (see figure 2(c) (inset)). In
practice, either t
ν
or ε
ν
might be varied to control the strength
of Fano scattering.
Tunable current suppression occurs over a broad energy
range, as shown in figure 2(b); the QuIET functions with any
metallic leads whose work function lies within the annulene
gap. Fortunately, this is the case for many bulk metals,
among them palladium, iridium, platinum, and gold [27].
Appropriately doped semiconductor electrodes [14] could also
be used.
We show in figure 3 the I–V characteristic of a QuIET
based on sulfonated vinylbenzene. The three metallic
3
Nanotechnology 18 (2007) 424014 C A Stafford et al
Figure 3. Room-temperature I –V characteristic of a QuIET based on
sulfonated vinylbenzene. The current in lead 1 is shown, where
V
αβ
= V
α
? V
β
. Here, Gamma1
1
= Gamma1
2
= 1eV.Gamma1
3
is taken as 0.0024 eV,
which allows a small current in the third lead, so that the device
amplifies current. A field-effect device with almost identical I –V can
be achieved by taking Gamma1
3
= 0. The curve for I
3
is for the case of
1.00 V bias voltage; I
3
for other biases look similar. Reprinted with
permission from [11]. ? 2006 American Chemical Society.
electrodes were taken as bulk gold, with Gamma1
1
= Gamma1
2
= 1eV,
while Gamma1
3
= 0.0024 eV, so that the coupling of the third
electrode to the alkene side group is primarily electrostatic.
The device characteristic resembles that of a macroscopic
transistor. As the voltage on lead 3 is increased, scattering from
the antibonding orbital of the alkene side group increases as it
approaches the Fermi energies of leads 1 and 2, leading to a
broad peak in the current. For Gamma1
1,2
greatermuch Gamma1
3
negationslash= 0, the device
amplifies the current in the third lead (dotted curve), emulating
a bipolar junction transistor. Alkene chains containing four
and six carbon atoms were also studied, yielding devices
with characteristics similar to that shown in figure 3, with
the maximum current I
1
shifting to smaller values of V
32
with increasing chain length. As evidence that the transistor
behaviour shown in figure 3 is due to the tunable interference
mechanism discussed above, we point out that if hopping
between the benzene ring and the alkene side group is set to
zero, so that the coupling of the side group to benzene is purely
electrostatic, almost no current flows between leads 1 and 2.
For Gamma1
3
= 0, I
3
= 0 and the QuIET behaves as a field-
effect transistor. The transconductance dI/dV
32
of such a
device is shown in figure 4. For comparison, we note that an
ideal single-electron transistor [28] with Gamma1
1
= Gamma1
2
= 1eVhas
peak transconductance (1/17)G
0
at bias voltage .25 V, and
(1/2)G
0
at bias 1 V, where G
0
= 2e
2
/h is the conductance
quantum. For low biases, the proposed QuIET thus has
a higher transconductance than the prototypical nanoscale
amplifier, while even for large biases its peak transconductance
is comparable. Likewise, the load resistances required for a
QuIET to have gain (load times transconductance) greater than
one while in its ‘on’ state are comparable to other nanoscale
devices, ～10/G
0
.
Operation of the QuIET does not depend sensitively on the
magnitude of the lead–molecule coupling
ˉ
Gamma1 = Gamma1
1
Gamma1
2
/(Gamma1
1
+
Gamma1
2
). The current through the device decreases with decreasing
ˉ
Gamma1, but aside from that, the device characteristic was found
to be qualitatively similar when
ˉ
Gamma1 was varied over one order
of magnitude. The QuIET is also insensitive to molecular
vibrations: only vibrational modes that simultaneously
alter the carbon–carbon bond lengths and break the six-
fold symmetry within the benzene component can cause
Figure 4. Transconductance dI/dV
32
of a QuIET based on
sulfonated vinylbenzene with Gamma1
3
= 0. The characteristic is similar to
that of a field-effect transistor, i.e. I
3
= 0 while I
1
=?I
2
= I .Asin
figure 3, Gamma1
1
= Gamma1
2
= 1 eV, and the calculation was done for room
temperature. Reprinted with permission from [11]. ? 2006
American Chemical Society.
decoherence in a benzene ‘interferometer’. Such modes are
only excited at temperatures greater than about 500 K.
The position of the third lead affects the degree to which
destructive interference is suppressed. For benzene, the most
effective location for the third lead is shown in figure 1.Itmay
also be placed at the site immediately between leads 1 and 2,
but the transistor effect is somewhat reduced, since coupling
to the charge carriers is less. The third, three-fold symmetric
configuration of leads completely decouples the third lead from
electrons travelling between the first two leads. For each
monocyclic aromatic annulene, one three-fold symmetric lead
configuration exists, yielding no transistor behaviour.
While H
mol
of equation (2) is well known to reproduce
the basic experimental features of conjugated molecules [29],
the QuIET’s characteristics, based on general principles
of quantum mechanics and symmetry, are qualitatively
independent of the particulars of the quantum chemical
method. Inclusion of σ electrons, for example, has little effect:
they form a separate system of localized bonds, and so cannot
contribute strongly to transport. We have verified via an all-
valence extended H¨uckel theory [30] that the QuIET’s tunable
coherent current suppression persists for such extensions of the
basis set.
The QuIET’s operating mechanism, tunable coherent
current suppression, occurs over a broad energy range within
the gap of each monocyclic aromatic annulene; it is thus a
very robust effect, insensitive to moderate fluctuations of the
electrical environment of the molecule. Although based on an
entirely different, quantum-mechanical, switching mechanism,
the QuIET nonetheless reproduces the functionality of
macroscopic transistors on the scale of a single molecule.
Furthermore, since the current flow is not blocked by an energy
barrier, which must be raised and lowered with each switching
cycle, heating of the environment is greatly reduced.
4. Implementations
As daunting as the fundamental problem of the switching
mechanism is the practical one of nanofabrication. The QuIET
requires a third lead coupled locally to the central molecule,
and, while there has recently been significant progress in
4
Nanotechnology 18 (2007) 424014 C A Stafford et al
(a) (b) (c)
(d)
Figure 5. Source–drain lead configurations possible in a QuIET based on [18]annulene. The bold lines represent the positioning of the two
leads. Each of the four arrangements has a different phase difference associated with it: (a) π;(b)3π;(c)5π;and(d)7π. The slight
deviations of [18]annulene’s hydrogen atoms from the molecular plane do not significantly affect the QuIET’s switching mechanism.
Reprinted with permission from [11]. ? 2006 American Chemical Society.
Figure 6. Schematic of various QuIETs based on a benzene ring. A, B, and C represent the various substituents which may be placed in series
between the ring and each lead. In particular, the conducting polymers like polyaniline and polythiophene may be useful in overcoming the
‘third lead’ problem.
V
a
V
b
R
Figure 7. Possible embodiment of a QuIET integrated with conventional circuitry on a chip. The source (1) and drain (2) electrodes are
connected via conducting polymers (in this case, polythiophene) to the central aromatic ring, while the gate electrode (3) is coupled
electrostatically to an alkene side group.
(This figure is in colour only in the electronic version)
that direction [14, 31, 32], to date, only two-lead single-
molecular devices, sometimes with global gating, have been
achieved [13]. With this in mind, we turn to potential practical
realizations of the device.
Using novel fabrication techniques, such as ultra-sharp
STM tips [31] or substrate pitting [32], it may soon be possible
to attach multiple leads to large molecules. Fortunately, the
QuIET mechanism applies not only to benzene, but to any
5
Nanotechnology 18 (2007) 424014 C A Stafford et al
monocyclic aromatic annulene with leads 1 and 2 positioned
so the two most direct paths have a phase difference of
π. Furthermore, larger molecules have other possible lead
configurations, based on phase differences of 3π,5π,etc;
as an example, figure 5 shows the lead configurations for a
QuIET based on [18]annulene. Other large ring-like molecules
that possess an aromatic conjugated electron system, such as
[14]annulene and the divalent metal-phthalocyanines, would
also serve well.
Another method of increasing the effective size of the
molecule is to introduce molecular wires linking the central
ring and leads (see figures 6 and 7). Conducting polymers,
such as polythiophene or polyaniline, are ideal for this task.
Such changes can be absorbed into the diagonal elements of
the self-energy Sigma1(E), and so only modify G(E) locally. As
such, while they can significantly modify the on-resonance
behaviour of a molecular device, the off-resonance function is
largely unaltered. In particular, the transmission node at the
centre of the gap is unaffected. An example of such a QuIET
integrated with conventional circuitry on a chip is shown in
figure 7.
5. Conclusions
The quantum interference effect transistor represents one way
to simultaneously overcome the problems of scalability and
power dissipation which face the next generation of transistors.
Because of the exact symmetry possible in molecular devices,
it possesses a perfect mid-gap transmission node, which serves
as the off state for the device. Tunably introduced decoherence
or elastic scattering can lift this quantum interference effect,
with the result of current modulation. Furthermore, a vast
variety of potential chemical structures possess the requisite
symmetry, easing fabrication difficulties. In particular,
molecular wires, such as conducting polymers, can be used to
extend the molecule to arbitrary size.
Acknowledgments
This work was supported in part by National Science
Foundation Grant Nos PHY0210750, DMR0312028, and
DMR0705163.
References
[1] Landauer R 1961 IBMJ.Res.Dev.5 183
[2] International Technology Roadmap for Semiconductors: 2006
Update http://public.itrs.net
[3] Di Ventra M, Pantelides S T and Lang N D 2000 Appl. Phys.
Lett. 76 3448
[4] Sautet P and Joachim C 1988 Chem. Phys. Lett. 153 511
[5] Sols F, Macucci M, Ravaioli U and Hess K 1989 Appl. Phys.
Lett. 54 350
[6] Joachim C and Gimzewski J K 1997 Chem. Phys. Lett. 265 353
Joachim C, Gimzewski J K and Tang H 1998 Phys. Rev. B
58 16407
[7] Baer R and Neuhauser D 2002 J. Am. Chem. Soc. 124 4200
[8] Yaliraki S N and Ratner M A 2002 Ann. New York Acad. Sci.
960 153
[9] Stadler R, Forshaw M and Joachim C 2003 Nanotechnology
14 138
Stadler R, Ami S, Forshaw M and Joachim C 2003
Nanotechnology 14 722
[10] Washburn S and Webb R A 1986 Adv. Phys. 35 375
[11] Cardamone D M, Stafford C A and Mazumdar S 2006 Nano
Lett. 6 2422
[12] B¨uttiker M 1988 IBMJ.Res.Dev.32 63
[13] Nitzan A and Ratner M A 2003 Science 300 1384 and
references therein
[14] Piva P G, DiLabio G A, Pitters J L, Zikovsky J, Rezeq M,
Dogel S, Hofer W A and Wolkow R A 2005 Nature 435 658
[15] Ohno K 1964 Theor. Chim. Acta 2 219
[16] Chandross M, Mazumdar S, Liess M, Lane P A, Vardeny Z V,
Hamaguchi M and Yoshino K 1997 Phys. Rev. B 55 1486
[17] Stafford C A, Kotlyar R and Das Sarma S 1998 Phys. Rev. B
58 7091
[18] Tian W, Datta S, Hong S, Reifenberger R, Henderson J I and
Kubiak C P 1998 J. Chem. Phys. 109 2874
[19] Nitzan A 2001 Annu. Rev. Phys. Chem. 52 681
[20] Jauho A-P, Wingreen N S and Meir Y 1994 Phys. Rev. B
50 5528
[21] Datta S 1995 Electronic Transport in Mesoscopic Systems
(Cambridge: Cambridge University Press) pp 293–342
[22] Mujica V, Kemp M and Ratner M A 1994 J. Chem. Phys.
101 6849
[23] B¨uttiker M 1986 Phys.Rev.Lett.57 1761
[24] Feynman R P and Hibbs A R 1965 Quantum Mechanics and
Path Integrals (New York: McGraw-Hill)
[25] Luttinger J M 1960 Phys. Rev. 119 1153
[26] Clerk A A, Waintal X and Brouwer P W 2001 Phys.Rev.Lett.
86 4636
[27] Marder M P 2000 Condensed Matter Physics (New York:
Wiley)
[28] Kastner M A 1992 Rev. Mod. Phys. 64 849
[29] Salem L 1966 The Molecular Orbital Theory of Conjugated
Systems (New York: Benjamin)
[30] Hoffman R 1963 J. Chem. Phys. 39 1397
[31] Rezeq M, Pitters J and Wolkow R 2006 J. Chem. Phys.
124 204716
[32] Mativetsky J M, Burke S A, Fostner S and Grutter P 2007 Small
3 818
6
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