Charge pumping in one-dimensional Kronig-Penney models
V. Gasparian
Department of Physics, California State University, Bakersfield, California 93311, USA
B. Altshuler
NEC Research Institute, 4 Independence Way, Princeton, New Jersey 08540, USA
M. Ortuño
Departamento de Física, Universidad de Murcia, Spain
Received 12 August 2005; published 7 November 2005
We consider adiabatic charge transport through one-dimensional open chain for two -like pumping sources.
We obtain explicitly the charge Q pumped within a period. This charge turns out to be proportional to the
conductance G
0
. For weak pumping perturbation , the charge Q is proportional to the area of the contour in
parametric space
2
sin , where is the phase difference of the oscillations of the two parameters. There is
an intermediate regime, where Q is proportional to the length of the contour and to sin /2. For large
pumping strength and not too small the charge decreases as sin
-3
.
DOI: 10.1103/PhysRevB.72.195309 PACS numbers: 73.23.-b, 73.63.Rt, 72.10.-d
I. INTRODUCTION
The phenomenon of adiabatic charge pumping, discussed
by Thouless,
1
has attracted attention of both experiment-
alists
2–4
and theorists.
5–11
By pumping we usually mean the
dc current or flow of a fluid which takes place due to some
periodic ac perturbations of the system. Such a dc current is
not a persistent current—not an equilibrium response to an
external perturbation. Nevertheless it may be entirely adia-
batic. It means that the charge transferred within the period
of the pumping t
0
=2 / is independent of this period and
remains finite, when the period tends to infinity → 0. The
pumping provides a new way of generating dc currents,
which is quite different from the usual application of a dc
voltage, and thus can have important practical advantages.
An experimental realization of an adiabatic electron
pumping through a quantum dot was reported by Swiktes et
al.
4
A similar phenomenon, drag of electrons by a traveling
acoustic wave in a semiconductor device, was demonstrated
by Talyanskii et al.
3
Until now, two approaches to the theoretical description
of adiabatic charge transport have been proposed. One of
them
7,9
is based on the conventional Green’s function for-
malism. It allows to evaluate explicitly the pumping current
for small amplitudes of the pumps and only for weakly dis-
ordered systems. The other approach
5
is based on the scat-
tering theory. In spite of these recent developments, many
details of the theory, namely the magnitude of the pumped
current, its dependence on external tunable parameters e.g.,
magnetic field, the relation between ensembles averaged
current and its mesoscopic fluctuations, the conditions under
which the current is quantized, etc., are not completely un-
derstood. There are few theoretical predictions to compare
with existing experiments. It is also unclear how this effect
changes in the crossover from weak to strong localization.
In this paper we study analytically the pumped current in
one-dimensional 1D Kronig-Penney models, where elec-
trons are subject to a potential that can be represented as a
sum of arbitrarily located delta functions with arbitrary
weights V
l
V
˜
x =
l=1
N
V
l
x - x
l
. 1
Time dependence of the factors V
l
can serve as pumping
perturbations. Our goal is to evaluate explicitly the scattering
matrix elements and their parametric derivatives for a given
set of the amplitudes by the method of the characteristic
determinant.
12
This approach seems to provide a natural
model for the study of both weak and strong disorder on the
pumping current.
II. PUMPED CURRENT IN TERMS OF THE
CHARACTERISTIC DETERMINANT
Consider the pumping charge Q transferred during a
single period trough a 1D chain of an arbitrary potential
shape Vx at zero temperature. Let two parameters of the
system be modulated periodically. The transferred charge is
independent of the frequency and, as shown in Ref. 5, is
given by
Q =
e
A
X, Y dXdY , 2
where X , Y is defined as
X, Y =
Im
s
1
*
X
s
1
Y
, 3
s
1
=1,2 are the elements of the scattering matrix s and
X = Xt and Y = Y t are two external parameters adiabati-
cally varying with time.
A
denotes integration within the
area encompassed by the contour A and the asterisk indicates
complex conjugation. According to Eq. 3 X , Y has a
meaning of adiabatic curvature.
PHYSICAL REVIEW B 72, 195309 2005
1098-0121/2005/7219/1953096/$23.00 ©2005 The American Physical Society 195309-1