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Spin and charge pumping in a quantum wire: the role of spin-flip scattering and Zeeman
splitting
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2011 J. Phys.: Condens. Matter 23 405301
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IOP PUBLISHING
JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 23 (2011) 405301 (11pp)
doi:10.1088/0953-8984/23/40/405301
Spin and charge pumping in a quantum
wire: the role of spin-flip scattering and
Zeeman splitting
T Kwapiński and R Taranko
Institute of Physics, M Curie-Skłodowska University, PL-20-031 Lublin, Poland
E-mail: tomasz.kwapinski@umcs.lublin.pl
Received 3 June 2011, in final form 17 August 2011
Published 19 September 2011
Online at stacks.iop.org/JPhysCM/23/405301
Abstract
We investigate theoretically charge and spin pumps based on a linear configuration of quantum
dots (quantum wire) which are disturbed by an external time-dependent perturbation. This
perturbation forms an impulse which moves as a train pulse through the wire. It is found that
the charge pumped through the system depends non-monotonically on the wire length, N. In
the presence of the Zeeman splitting pure spin current flowing through the wire can be
generated in the absence of charge current. Moreover, we observe electron pumping in a
direction which does not coincide with the propagation direction of the pulse and the spin
pumping direction (spin–charge separation). Additionally, on-site spin-flip processes
significantly influence electron transport through the system and can also reverse the charge
current direction.
1. Introduction
the ac signal. The easiest way to break the time-reversal
symmetry is to add the second harmonic to the driving
periodic field or use time-dependent dipole forces, [17].
The functional principle of spin pumps is closely related to
charge pumps. However, a charge–spin system often reveals a
fascinating phenomenon, i.e. it can generate pure spin current
in the absence of charge (electron) current. Such spin pumps
were studied, for example, for a two-dimensional electron
gas [18], a single molecule (quantum dot, QD) [6, 19, 20]
or a double QD system [11, 21]. In the last case in the
presence of ac-driven perturbations the system behaves as a
spin filter. Pumping effects in one-dimensional systems were
reported for an oscillating potential [22, 23] or time-periodic
(harmonic, bi-harmonic, pulsed) delta-like potentials applied
to two sites of a wire [10, 24, 25]. Such time-dependent
two-site perturbations act as electron pumping sources in
these systems. It was found that the pumped charge depends
monotonically (increases) with the wire length or shows
a step-like behaviour, [23]. Also by applying two slowly
varying orthogonal gate electric fields on different sections
of the wire one can generate spin pumping, [26]. Moreover,
time-dependent magnetic fields acting on two quantum wire
sites can also produce the spin current [27].
Charge–spin pumps have attracted a great deal of attention
from scientists and engineers due to their significant
advantages and potential applications in spintronics and
quantum computing. Subtle experimental techniques were
applied to investigate new phenomena like conductance
oscillations [1], conductance quantization [2], spin–charge
separation [3] and others. Many interesting effects were
found in low-dimensional systems driven by external timedependent fields like photon-assisted tunnelling (PAT) [4, 5]
or quantum pumps [6–11].
A charge or spin pump is an electronic device which
generates a net current flowing between unbiased leads.
A quantum pump involves periodic changes of two or
more device-control parameters [12, 13], which break the
spatial symmetry of the system. Such electron pumps
were realized experimentally, e.g. [8, 14]. Recently it was
shown, both theoretically and experimentally, that an electron
pump can exist also for a single time-dependent parameter
(monoparametric pumping), [15, 16]. Generally a pump
current can be generated by an external time-dependent signal
(i) due to the absence of spatial symmetry in the system
or (ii) in the case of a lack of time-reversal symmetry in
0953-8984/11/405301+11$33.00
1
c 2011 IOP Publishing Ltd Printed in the UK & the USA
J. Phys.: Condens. Matter 23 (2011) 405301
T Kwapiński and R Taranko
In our studies for a row of coupled QDs (or atomic sites
which form a quantum wire, QW) the source of pumping is
the time-dependent potential applied to the gate electrodes
forming the tunnel barriers between QDs. This potential
influences the couplings in the wire (hopping integrals).
The couplings vary non-homogeneously in time and form
an impulse/perturbation which moves between leads, along
the wire (train impulse). Because no source–drain voltage is
applied to the left and right electrodes the system behaves as
a charge pump or, in the presence of magnetic fields, as a
spin or spin–charge pump. The system considered here reveals
new physical possibilities for spin and charge pumping which
are often quite different in comparison with two-site pumping
sources in a wire (studied, for example, in [10, 24, 25]). Our
goal is to investigate the role of the Zeeman splitting and
the intradot spin-flip scattering on charge and spin pumping
through a QW in the presence of a train impulse. These studies
allow us to answer the following intriguing questions. Is it
possible to observe the spin–charge separation effect with no
Coulomb interactions in the wire? Then, next, do the pumping
currents (charge and spin) change monotonically as a function
of the wire length or do they oscillate? Especially interesting
for us are spin pumps in the presence of both the Zeeman
splitting and spin-flip scattering. It is known that coupling of
a small quantum system (like a QD or QW) to its surrounding
environment is usually responsible for spin decoherence or
spin dissipation and can lead to spin-flip processes. A possible
mechanism of spin relaxation in QDs is based on coupling
of nuclear spins to those of QD electrons where the rate of
spin relaxation depends inversely on the number of nuclei
in the dot [28]. Other possibilities for spin-flip scattering are
spin–orbit coupling, phonon interaction or magnetic impurity.
Spin-flip transitions give rise to additional steps in the
electric current through a quantum system which leads to
additional peaks in the conductance [29–31]. Moreover, for
magnetic tunnel junctions the spin-flip processes suppress
the tunnel magnetoresistance [32, 33], but in general the
spin-polarized current depends on the magnetic configuration
of both leads [34]. Also the conductance of an interacting
QD in the presence of spin-flip tunnelling is suppressed [35].
For a magnetic QD with time-dependent pulsed bias voltage
and attached to normal and ferromagnetic leads the spin-flip
scattering influences the amplitude of the quantum beats in
the total charge and spin currents, [36]—both currents are
suppressed in that case.
In our calculations we use a tight-binding Hamiltonian
and the evolution operator method. This method, which
can be applied for arbitrary time dependence of external
perturbations (and does not require the Floquet theory), was
successfully used to describe time-dependent phenomena
like scattering of atoms at the surface or photon-assisted
tunnelling in QD systems [37, 38]. A possible experimental
realization of the considered system is based on oscillating
molecules or QDs in a harmonic potential, like in a quantum
shuttle [39–41]. The moving QD affects its nearest-neighbour
hoppings in time and drastically influences the electron
transport through the system. Alternatively, one can use
coherently coupled QDs, e.g. [5, 9], with additional external
Figure 1. Schematic view of a wire consisting of N sites and
coupled with the left (L) and right (R) electron reservoirs. fi (t)
functions stand for the time-dependent hybridization matrix
elements between sites.
(This figure is in colour only in the electronic version)
electrodes between them. Using these electrodes one can
arbitrary change in time all hopping integrals between dots.
It is also possible to use mechanical vibrations or acoustic
waves [42], which can influence nearest-neighbour hoppings
in a wire.
The organization of this paper is as follows. The
model Hamiltonian of a wire and theoretical calculations are
presented in section 2. Electron pumping effects through a
single QD and a QW are analysed in section 3. In section 4
spin pumps are studied in the presence of the Zeeman splitting
and in section 5 the role of the spin-flip scattering is discussed.
Section 6 is devoted to conclusions.
2. Model and theoretical description
Our model consists of a monatomic QW or coupled QDs
connected with the left electrode (via the first wire site) and
the right one (via the last wire site), figure 1. The hopping
integrals between the first/last wire site and the left/right
electrode as well as inside the wire can be changed in time.
We assume that the wire remains straight during this process
and the next-neighbour couplings are neglected.
The time-dependent Hamiltonian written in the second
quantization notation is H = H0 + V(t), where
H0 =
X
+
+
εkασ
E
E aE akασ
kασ
E
kα,σ
V(t) =
X
kLσ
N−1
XX
i=1
i=1 σ
+
VkL
E (t)aE a1σ +
E
kL,σ
+
N X
X
σ
X
εiσ a+
iσ aiσ ,
+
VkR
E (t)aE aNσ
kRσ
E
kR,σ
Vi (t)a+
iσ ai+1σ +
(1)
N X
X
i=1 σ
Vsf a+
iσ ai−σ + h.c.,
(2)
+
+
and α = L, R, operators akασ
E (akασ
E ), aiσ (aiσ ) are the electron
annihilation (creation) operators in the α electrode or at the ith
wire site, respectively. Vsf stands for the spin-flip scattering
strength (the same for all QW sites) and Vkα
E (t), Vi (t) are the
hybridization matrix elements (hopping integrals) between the
electron states in the left/right electrode and the first/last wire
site or between the nearest-neighbour sites in the wire and in
general one can assume
VkL/R
(t) = VkL/R
f1/N+1 (t),
E
E
Vi (t) = Vi fi+1 (t), (3)
i.e. fi (t) functions are responsible for the time dependence
of all hopping integrals. In our calculations we neglect the
electron–electron correlations as we do not consider here the
2
J. Phys.: Condens. Matter 23 (2011) 405301
T Kwapiński and R Taranko
many-body effects (e.g. the Kondo effect). In the case of
strong Coulomb correlation one can consider the tunnelling in
the neighbourhood of a single Coulomb oscillation peak [43].
In general, the Coulomb interactions play no role when the
broadening of the molecular levels due to the coupling to the
leads is larger than any broadening due to the interactions
within the wire, [44]. Note that one-dimensional charge and
spin pumps exist also for an interacting wire (Luttinger
liquid) [45, 46] and spin currents can be achieved without the
presence of a magnetic field [47].
In the Hamiltonian the spin index, σ , is written explicitly
and is responsible for magnetic properties of the system. An
external magnetic field which is applied to the wire leads to
the Zeeman splitting of the wire on-site energy levels (e.g. [11,
21, 48, 49]) and thus in general εiσ 6= εi−σ . In that case the
magnetic field cancels the spin degeneracy of these levels.
The strength of the Zeeman splitting is expressed by the h
parameter, h = εi−σ − εiσ , and its value (in our calculations)
corresponds to magnetic fields which are usually used in many
experiments, [48, 49]. Moreover, we assume that both leads
are unpolarized.
To describe the spin and charge pumping effects through
the wire in the presence of time-dependent couplings we use
the evolution operator method, [37, 38]. The total charge or
spin pumped through the system can be obtained from the
time-dependent current flowing, e.g. from the left electrode,
jLσ (t). We calculate this current from the time derivative of
the total number of electrons in the left reservoir, [50]:
jLσ (t) = −ednLσ (t)/dt,
elements can be written:
∂
i(εE − ε0σ )t
0 (t) = VE (t)e kLσ
U1σ,EqLσ 0 (t),
(8)
UE
kL
∂t kLσ,EqLσ
∂
i Uiσ,kLσ
E 0 (t) = Vi (t)Ui+1σ,kLσ
E 0 (t)
∂t
i(ε0σ −ε0−σ )t
Ui−σ,kLσ
+ Vi−1 (t)Ui−1σ,kLσ
E 0 (t)
E 0 (t) + Vsf e
X
ei(ε0σ −εqELσ )t UqELσ,kLσ
+ δi,1 δσ,σ 0 VkL
E (t)
E (t)
i
qELσ
+ δi,N δσ,σ 0 VkR
E (t)
E kRσ
E
kL,
2
nkRσ
E 0 (t0 )|UkLσ,
E
E 0 (t, t0 )| .
kRσ
∂
U E 0 (t) = −iVi (t)Ui+1σ,kLσ
E 0 (t) − iVi−1 (t)
∂t iσ,kLσ
i(ε0σ −ε0−σ )t
Ui−σ,kLσ
× Ui−1σ,kLσ
E 0 (t)
E 0 (t) − iVsf e
i(ε0σ −εkLσ
E )t
− δi,1 δσ,σ 0 iVkL
E f1 (t)e
0L 2
R
2
+
f (t)U1σ,kLσ
E (t) − δi,N δσ,σ 0 0 /2fN+1 (t)
2 1
× UNσ,kLσ
E (t).
(4)
jLσ (t) = −e0f12 (t)n1σ (t)


X

i(εE −ε0σ )t
+2eRe
nkLσ
U1σ,kLσ
E (0)iVkL
E f1 (t)e kLσ
E (t) ,


(5)
E
kL
(11)
Here the time evolution operator, U(t, t0 ) (in the interaction
representation), satisfies the equation of motion (h̄ = 1):
∂
U(t, t0 ) = Ṽ(t) U(t, t0 ),
∂t
(10)
A similar equation can be written for Uiσ,kRσ
E 0 (t). Note that the
Vsf term mixes the set of differential equations with spin σ and
spin −σ via the matrix elements Ui−σ,kLσ
E 0 (t). Assuming the
L
same strengths of the 0 parameters, 0 = 0 R = 0, the current
flowing through the system can be expressed as follows:
0
i
(9)
0
and similar equations for Uiσ,kRσ
E 0 (t) and UkRσ,E
E
qRσ (t). Note
that within
Pthe wide∗ band limit approximation [50], i.e. for
0 α = 2π kα
E ), α = L, R, and using the
E Vkα
E Vkα
E δ(ε − εkασ
0
solutions of the corresponding equations for Ukασ,E
E
qα σ (t),
equation (9) takes the form:
E qLσ 0
kL,E
X
E )t U
ei(ε0σ −εkRσ
E
E (t),
kRσ,
kLσ
E
kRσ
where the left electrode occupation, nLσ (t), can be expressed
in terms of the evolution operator matrix elements, [37, 38]:
X
2
0
nLσ (t) =
nqELσ 0 (t0 )|UkLσ,E
E
qLσ (t, t0 )|
+
X
where the charge, n1σ (t), is calculated from equation (7) and
Uiσ,kLσ
elements are obtained from the set of differential
E
equations, equation (10). Here the current, equation (11),
can be written
coefficients,
P in terms of the transmittance
P
i.e. j(t) = kL
E nkL
E (0)TLR (t) −
EkR nkR
E (0)TRL (t), where
TLR/RL is expressed using the evolution operator elements,
0 (t). Note that, in general, TLR (t) 6= TRL (t), and one
Uiσ,kL/Rσ
E
cannot write the current in the form of the Landauer-like
formula. Thus electrons can be pumped between electrodes
with no source–drain voltage [17]. The expression for the
current, equation (11), together with the set of differential
equations for the evolution operator elements, equation (10),
are the main analytical formulae of this paper.
The total charge pumped during one time-dependent
cycle of a train impulse (period) through the system is
obtained from
Z
NL/Rσ = jL/Rσ (t) dt,
(12)
(6)
where Ṽ(t) = U0 (t, t0 ) V(t) U0+ (t, t0 ) and U0 (t, t0 ) = T
Rt 0
0
exp(i t0 dt H0 (t )). In equation (5) nkασ
E (t0 ) represents the
initial filling of the corresponding single-particle state in the
left or right leads (we assume the initial wire site occupations,
niσ (t0 ) = 0). For t > t0 the charge localized at the ith wire site
can be expressed in terms of the appropriate matrix elements
of the evolution operator, i.e.
X
2
niσ (t) =
nkασ
(7)
E 0 (t0 )|Uiσ,kασ
E 0 (t, t0 )| .
E 0
kασ
Assuming the same positions of all on-site energies in the
wire, εiσ = ε0σ (which is reasonable for a regular wire
consisting of almost identical sites) and t0 = 0, the following
set of differential equations for the evolution operator matrix
3
J. Phys.: Condens. Matter 23 (2011) 405301
T Kwapiński and R Taranko
and the relation NLσ = −NRσ is satisfied in our system.
We assume that the external perturbation does not spread
immediately through the wire but there is a kind of a wire
inertia which leads to different couplings between QW sites
for a given time. It means that for our wire the coupling (first
QD)–(L electrode) is different in comparison with those for
(first QD)–(second QD) or (last QD)–(R electrode). Here we
consider two possibilities for time-dependent perturbations.
The first one assumes that the couplings QD–QD and
QD–electrode change in time like the Gaussian distribution
function, i.e.
!
[t − (t00 + (j − 1)tx )]2
fj (t) = 1 + exp −
,
(13)
σ0
where j = 1, 2, . . . , N + 1 corresponds to the following
connection: (L electrode)–(first QD) for j = 1, (first
QD)–(second QD) for j = 2, . . ., (last QD)–(R electrode) for
j = N +1, respectively. The parameter tx stands for a time shift
of the external signal between the nearest-neighbour hoppings
and for tx = 0 all hoppings are driven homogeneously (the
coupling strengths change in time in the same way). σ0
corresponds to the half-width of the Gaussian signal and
t00 stands for the time for the first Gaussian maximum,
i.e. the maximum of the f1 function. The couplings between
appropriate sites in the system are responsible for the distance
between these sites (the weaker the V parameter the larger
the distance between sites). Thus the perturbation described
above (one-Gaussian function) leads to decrease the distance
between sites (in the first stage of perturbation) and then
the system returns to its initial arrangement. The second
time-dependent perturbation considered in this paper has a
structure of two, positive and negative, Gaussian functions
which can be written as follows:
!
[t − (t00 + (j − 1)tx )]2
fj (t) = 1 + exp −
σ0
!
[t − t1 − (t00 + (j − 1)tx )]2
,
(14)
− exp −
σ0
Figure 2. The case of a single QD coupled with two electrodes.
Upper panels: time-dependent perturbations between the left
electrode and the QD site (solid lines) and between the right
electrode and the QD site (broken lines). Charge occupations of the
QD site: the current flowing from the left electrode and from the
right one are shown in the middle and bottom panels, respectively.
The left (right) panels correspond to the one-Gaussian
(two-Gaussian) perturbation. The broken and solid lines correspond
to ε0 = 4 and ε0 = −4, respectively. The other parameters are
tx = 5, σ = 10, VL = VR = 4, t00 = 50 and 0 = 1.
where t1 is a time shift between the maxima of two-Gaussian
functions. In the absence of a time shift between these
functions, t1 = 0, the system is not disturbed. The examples
of both kinds of perturbations are described in section 3
and depicted in figure 2, upper panels. Throughout the paper
we consider no source–drain voltage (µL = µR = 0), which
implies that the Fermi energy is zero, EF = 0. The current
is expressed in units of 2e0/h̄ and time in h̄/ 0. In our
calculations we assume large values of the t00 parameter in
order to avoid the initial switch effects, which appear for t ≥
t0 = 0. The positive value of the left/right current means that
electrons tunnel from the left/right electrode to the first/last
QW site.
we consider a wire in the absence of Zeeman splitting,
i.e. ε0σ = ε0−σ = ε0 , (h = 0). Thus the current flowing
through the system, charge occupations and the pumped
charge do not depend on σ , i.e. jLσ (t) = jL−σ (t) = jL (t),
niσ (t) = ni−σ (t) = ni (t) and NLσ = NL−σ = NL and similar
relations hold for the R index. Also the spin-flip scattering is
not taken into consideration in this section.
3.1. Single QD pump
First we study the pumping effect for a system consisting
of one site (single QD) coupled with two electrodes. The
couplings change in time but there is a time shift between
the left and right couplings, figure 2, upper panels—the left
(right) panel for one-Gaussian (two-Gaussian) perturbations.
A similar system was also considered in [8], but the couplings
were switched on and off alternately from zero to maximal
3. Pure charge pumps
To find the basic reactions of the one-dimensional system to
the Gaussian-like time-dependent perturbations in this section
4
J. Phys.: Condens. Matter 23 (2011) 405301
T Kwapiński and R Taranko
pump which will be discussed further. This asymmetry is very
visible for the two-Gaussian perturbation, right panels. To find
the total charge pumped from the left electrode to the right
one, NL , one should integrate the left current jL (t) over time.
In figure 3 the charge pumped through a single QD
system is shown as a function of the time shift parameter,
tx , for different values of ε0 as well as the one-Gaussian
perturbation (upper panel) and the two-Gaussian perturbation
(bottom panel). If there is no time shift between f1 and f2
functions, tx = 0, the charge is not pumped—in this case
f1 and f2 functions are the same and the system is driven
homogeneously. For a very large value of the tx parameter,
the net charge also equals zero as both functions f1 and f2 are
quite separate (do not overlap each other).
√The maximal value
of the net charge is observed for tx = σ . It is interesting
that, for ε0 = 0 (thick lines), the charge does not flow
through the system although tx 6= 0. In this case the system
is symmetrical in the energy scale and the total occupation
of the QD site remains constant. For ε0 6= 0 and in the
presence of a time-dependent perturbation, the QD occupation
changes in time which leads to the charge pumping effect.
Note that for the one-Gaussian perturbation and positive ε0
the pumped charge is positive with the same absolute values
as for negative ε0 (not shown here). For the two-Gaussian
perturbation and negative ε0 , the charge flows from the right
to the left electrode (for small tx ), or in the reverse direction
(for larger tx ). The opposite conclusion is valid for positive ε0 .
Figure 3. Charge pumped through a single QD as a function of tx
for ε0 = 0, −2 and −4 (thick solid, broken and thin solid lines,
respectively). The upper (lower) panel corresponds to the
one-Gaussian (two-Gaussian) perturbation. The other parameters
are as in figure 2.
values (quantum ratchets). The QD occupation and the current
flowing from the left and right electrodes in the presence
of these perturbations are shown in the middle and bottom
panels in figure 2. The broken and solid lines correspond
to different positions of the on-site QD energy, ε0 = 4 and
ε0 = −4, respectively. The charge occupations of these sites
are characterized by two local minima (for negative ε0 ) or
maxima (for positive ε0 ) which appear for maximal values
of the left and right couplings, i.e. for t = 50 and 55, and
are related to the extremes of the f1 and f2 functions. This
conclusion is also valid for the two-Gaussian perturbation,
right panels. It is interesting that for negative ε0 , the QD
occupation, n1 (t), decreases for larger couplings VL/R (t) and
increases for smaller VL/R (t). The opposite relation holds for
positive ε0 .
If both functions f1 and f2 do not change in time the
current does not flow through the system as we consider here
the same chemical potentials of both leads. In the presence
of time-dependent perturbations the current oscillates in time.
Depending on the position of ε0 , for a given time, it can
be positive or negative. It is interesting that the left (right)
current is equal to zero for the extremal values of the f1
(f2 ) function. Moreover, the right and left currents seem
to be strictly time-shifted by the tx parameter. However,
there are small differences between both currents—compare,
for example, the maximal values of jL (t) and jR (t) (solid
lines, right panels). Additionally, the current curves look
like an ‘odd’ function for the one-Gaussian perturbation (left
panels) but the precise calculations show that these curves
are non-symmetrical and the system behaves as an electron
3.2. Quantum wire pump
In this subsection we consider pumping effects through a short
QW subjected to external time-dependent perturbations (train
impulse). First, we analyse the case of a QW consisting of
three sites and in figure 4 we show the net charge pumped
through the system as a function of the on-site energy, ε0 ,
and the time shift parameter, tx . The upper (bottom) panel
corresponds to the one-Gaussian (two-Gaussian) perturbation.
Before the external impulse starts the system is characterized
√
by three molecular states localized at ε0 and ε0 ± V1 2 '
ε0 ± 5.6. In the presence of the time-dependent perturbations
the parameter V1 changes, which moves these peaks in the
energy scale. Thus the maximal pumped charge is observed
for larger ε0 , i.e. ε0 ' ±6.8. For negative values of ε0 the
charge at the first QW site flows out to the left electrode (if
the coupling of the left electrode with the first QD increases)
and, due to nonzero coupling between the first and second
QW sites, electrons flow from the second site to the first
one in order to compensate the charge which escapes from
the first site. This is the reason that, for negative ε0 , the
pumped charge flows from the right electrode to the left one.
The opposite conclusion is valid for positive values of ε0 .
This result is crucial as it concerns spin current devices and
will be discussed in section 4. It is interesting that electron
charge is pumped through the system also for small absolute
values of ε0 (but for ε0 = 0 the current is zero, cf figure 3).
It is related to a nonzero value of the wire density of states
(DOS) at the Fermi level for odd N. The pumped charge is
rather small in this case (ε0 ≈ 0, upper panel) because the
5
J. Phys.: Condens. Matter 23 (2011) 405301
T Kwapiński and R Taranko
Figure 5. Charge pumped from the left electrode as a function of
the wire length, N, for ε0 = 0.5, 4.0, 5.56 and 9.0. The other
parameters are V1 = 4, tx = 3, σ0 = 10 and 0 = 1. The lines serve
as a guide to the eyes.
figures 3 and 4. Unfortunately, in general, this conclusion is
not valid and the pumping current strongly depends on the
system parameters. To corroborate this effect in figure 5 we
study the charge pumped along a wire as a function of the
wire length, N. Note that for ε0 = 0 the net charge is not
pumped through the system independently of the wire length
(not shown in figure 5) but for small nonzero ε0 even–odd
pumped charge oscillations are visible (thin solid line). These
oscillations are characterized by the maximal values of NL for
every odd-length wire but the oscillation amplitude decreases
with N. The other positions of ε0 were chosen in such a way
that they satisfy the condition of conductance oscillations for
periods 3 and 4, [51]. Thus for ε0 = 4 (thick solid line) the
oscillation period of the pumped charge is 3 and for ε0 = 5.56
(broken line) the period is 4. As one can see the pumped
charge changes non-monotonically as a function of N and
oscillates with the wire length (similarly to the conductance
oscillation effect). These oscillations are very visible for short
wires. For longer N there are many molecular states of the
system which are localized near (below and above) the Fermi
energy and they all play a role in pumping in both directions.
Thus the oscillation amplitude of NL decreases with N. We
have found that the charge current oscillates as a function of N
for |ε0 | < 2V1 —molecular states which lie between ε0 − 2V1
and ε0 + 2V1 (which corresponds to the wire bandwidth) are
responsible for these oscillations. For the case of |ε0 | > 2V1
the pumped current increases with N, see the dotted line in
figure 5 for ε0 = 9.0. It results from the fact that most external
molecular states of the wire are shifted with N and tend to
ε0 ± 2V1 . Thus for the large positive ε0 one of these states,
which lies close to ε0 − 2V1 , moves towards the Fermi energy.
For longer wires there are no significant differences in the
energy positions of these edge states and the pumped current
does not change with N.
Note that in the presence of an oscillating potential the
charge pumped per one cycle is proportional to the frequency
of an external perturbation [18, 22], and the pumping current
vanishes in the adiabatic limit (ω → 0). In our studies we
consider only one pumping cycle, so the pumped charge does
Figure 4. The absolute value of the pumped charge through the
three-site wire with the one-Gaussian (upper panel) and
two-Gaussian (lower panel) perturbations versus ε0 and tx . The
other parameters are as in figure 2.
system is almost symmetrical in the energy scale and the
occupation of QW sites is about 0.5. In comparison with
the one-Gaussian impulse (upper panel) for the two-Gaussian
perturbation (bottom panel)√a large value of the net charge is
pumped also for |ε0 | < V1 2. In this case the V1 parameter
decreases to a very small value for a moment (cf figure 2,
upper right panel) and in that case the wire DOS peaks
are shifted towards the Fermi energy (the current increases).
Moreover, for the two-Gaussian perturbation the net charge is
also pumped through the system for larger tx .
It is worth noting that for short QWs the pumped charge
decreases very fast for tx > σ . However, for longer wires
the net charge can be nonzero even for larger tx . This effect
is related to the wire capacity—the excess charge must be
transported through all QW sites in order to obtain a new
equilibrium state. For such a system it takes some time to get
rid of the excess charge and it also leads to the pumping effect
for larger tx . Thus one expects that the pumped charge should
increase with the wire length, [23]. It seems true as the charge
which is pumped per cycle through a short wire is rather large
(about ten times) in comparison with a single QD system, see
6
J. Phys.: Condens. Matter 23 (2011) 405301
T Kwapiński and R Taranko
Figure 7. Spin current pumped through a wire consisting of
N = 2–5 QDs as a function of the h parameter for ε0−σ = ε0σ = 0
at h = 0 and Vi = 4, i = 1–4. The other parameters are the same as
in figure 6.
(e.g. in figure 3). For nonzero h electrons with spins −σ and
σ are pumped in the opposite direction (there are spin-up
and spin-down currents). As a consequence, the total charge
flowing through the system, NLσ + NL−σ (thick solid line),
is equal to zero independently of the strength of the Zeeman
splitting. However, the spin current which is proportional to
NLσ − NL−σ (thin solid line) is nonzero, i.e. spins are pumped
through the system although there is no charge current.
In the lower panel the initial positions of the QD spin-up
and spin-down energy levels are not localized at the Fermi
level—in the absence of Zeeman splitting we have ε0−σ =
ε0σ = 1. Thus for h = 0 the electron charge is pumped from
the left electrode through both QD spin levels and the total
electron current is maximal, thick solid line, bottom panel.
In this case, however, the total spin current (thin solid line)
does not flow because spin-up and spin-down currents are
equal to each other (broken curves). In the presence of the
Zeeman splitting spin-up and spin-down charges (which are
pumped through the system) are characterized by different
values, which leads to nonzero total spin–charge currents.
However, for some values of h, the total electron current does
not flow but the spin current exists, e.g. for h = 2.9. Moreover,
the charge current can be positive (for small Zeeman splitting)
or negative for larger h which results directly from the relative
positions of ε0σ , ε0−σ and the Fermi energy. It is worth noting
that the charge current is symmetrical in relation to h (for
positive and negative values of h the charge current is the
same) whereas the spin current curves are antisymmetrical
versus h = 0 (thin solid line).
The next interesting question is whether the spin and
charge pumping effects exist also for a QW (chain of QDs).
Therefore, we have calculated the corresponding currents as a
function of the h parameter for the symmetrical case ε0−σ =
−ε0σ (i.e. ε0−σ = ε0σ = 0 at h = 0), and for a wire composed
of N = 2–5 QDs, see figure 7. Note that, in this case, for
arbitrary N, the total charge current does not flow through the
system, independently of h. Thus, in figure 7 we show only
the total spin current, NLσ − NL−σ . As one can see, for h = 0
(no Zeeman splitting) the spin current does not flow, which
is in accordance with the previous results. For h 6= 0 the spin
Figure 6. Charge and spin currents flowing through a single QD
system, N = 1, as a function of the level separation h (Zeeman
splitting parameter) for different positions of ε0σ for the vanishing
magnetic field (ε0−σ = ε0σ = 0–upper panel–and
εσ = ε−σ = 1–bottom panel). Thick broken (thin broken) lines
correspond to NL−σ (NLσ ) and the thick solid (thin solid) lines
represent charge (spin) currents. The other parameters are tx = 5,
σ = 10 (one-Gaussian perturbation), VL = VR = 4 and 0 = 1.
not depend on the perturbation frequency and is nonzero for a
wide range of the system parameters, ε0 , tx and N.
4. Spin and charge pumps in the presence of
magnetic field
If a magnetic field is applied to the wire all on-site energies
are split (Zeeman splitting) which leads to different values of
ε0σ and ε0−σ states. Therefore, in the presence of external
time-dependent perturbations, fi , electrons with spin σ can
be pumped through the system in one direction and, at the
same time, electrons with spin −σ can flow in the opposite
direction. In that case for particular values of ε0±σ the spin
current could appear although the charge current is zero.
(Note that no source–drain voltage is applied to the system.)
To confirm this effect in figure 6 we show electron charges
which are pumped through a single-site system (QD), as a
function of the Zeeman splitting parameter, h (thin and thick
broken curves correspond to spins σ and −σ , respectively).
In the upper panel, for no Zeeman splitting, h = 0, the
QD spin levels satisfy the relation ε0−σ = ε0σ = 0 and
in the presence of magnetic field both levels are separated
symmetrically versus the Fermi energy, i.e. ε0−σ = −ε0σ .
As one can see no charge is pumped through the system for
h = 0 which is the same effect as discussed in section 4,
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J. Phys.: Condens. Matter 23 (2011) 405301
T Kwapiński and R Taranko
current appears and, for example, for N = 2 it is visible at only
one current extreme (the second one appears for negative h,
not shown in figure 7). Generally, for the even or odd number
of N there are N or N + 1 extrema on the spin current curve.
For odd N the spin current possesses a small minimum near
h = 0.5 (slightly visible for N = 3 and 5) while for even N,
the spin current does not flow for the same h. It results from
the structure of the wire’s molecular states—only odd-length
wires are characterized by a state at the Fermi level (if N is
even the wire DOS is flat at the Fermi level) [51]. Thus, for
example, for N = 2 and h = 0 the system is characterized
by two molecular states at ±V1 (not at the Fermi level). In
the presence of a magnetic field each molecular state is split
and four spin states are observed (two with spins σ and two
with −σ ). For small h the spin current does not flow through
the system because all molecular states lie far from the Fermi
level. However, for larger h the spin molecular states cross
the Fermi level and the spin current appears (spin-up and
-down molecular states are localized symmetrically versus EF
in this case and instead of four spin current peaks we observe
only two of them). For stronger and stronger magnetic fields
there are no molecular states near the Fermi level and the spin
current does not flow again.
As the most prominent feature obtained in figure 7 we
find that the total spin current pumped during one cycle of
time-dependent perturbation strongly depends on the number
of QW sites, N. In particular, for N = 1 only about 0.15
electron spin is pumped (figure 6, upper panel) whereas for
N = 5 this value exceeds 1.6 per cycle (it is ten times larger),
figure 7. This effect is related to the structure of the wire
molecular states. Namely, for longer wires there are many
molecular states localized symmetrically below and above
the Fermi energy. For a nonzero magnetic field each state
is split and (depending on the h parameter) can cross the
Fermi energy. Thus even a small change in QD–QD coupling
(during the time-dependent perturbation) strongly influences
the site occupations. As a consequence, a high-value spin
current flows through the system in the absence of the charge
current. For N = 1 a single molecular state lies at the Fermi
level and is very wide in comparison with the case of larger N.
Thus the same time-dependent disturbance does not influence
the spin current in the same way as for other N. However, all
the above-mentioned values of the spin current depend on the
magnetic field, h, and in general the current does not change
monotonically with the wire length, see also figure 5. Only
for h > ε0σ + 2V1 , which corresponds to h > 16, does the
spin current increase with N, figure 7, but for the other h the
sequence is different. For small h (h ' 1) the spin current
oscillates with N and this effect will be discussed later.
Next it is interesting to study a similar problem in the
presence of nonzero charge current. Thus we consider an
N-site wire with the Zeeman splitting and asymmetrically
localized ε0σ and ε0−σ levels, figure 8. In contrast to figure 7
here we have chosen ε0−σ = ε0σ = 1 for h = 0 and therefore
the initial positions of the QD spin-up and spin-down energy
levels do not correspond to the Fermi level. As one can see,
for h = 0 electrons are pumped through the system but the
net pumped charge is rather small for all N. In that case the
Figure 8. Charge current (upper panel) and spin current (bottom
panel) through a wire consisting of N = 2–5 sites as a function of
the Zeeman splitting parameter, h, for the asymmetrical case, i.e.
ε0σ = ε0−σ = 1 for h = 0. Letters A and B indicate extreme points
on the thick curve which are discussed in the text. The other
parameters are the same as in figure 7.
spin current does not flow because ε0σ = ε0−σ (bottom panel).
For larger h the charge current increases and, for example,
for N = 5 about 0.8 of an electron can be pumped during
one time-dependent cycle (upper panel). Moreover, the charge
current direction strongly depends on the magnetic field and
may be positive or negative (each current peak coexists with
its neighbouring dip). To explain this effect we concentrate on
the two-sites wire N = 2 (thick line) which is characterized
by two molecular state structures for every spin (there are two
peaks in the wire DOS localized at ε ± = ε0σ ± V1 ). Thus for
the small magnetic field the system is characterized by very
low values of DOS near the Fermi level which leads to no
charge and no spin pumping. For the larger magnetic field
the molecular states ε ± cross the Fermi level at different h
and the total charge pumped through the system depends on
a relative distance between ε + , ε − and the Fermi level. (For
a state which lies below/above the Fermi level, the pumping
current is negative/positive.) Thus for h = 8 the charge current
is positive (peak indicated by point A) and for a stronger
magnetic field the pumping current is characterized by a local
dip (point B). For larger h the charge current vanishes because
both +σ and −σ molecular states are situated far from the
Fermi level.
It is interesting that some inflection points and extrema
are also visible on the spin current curves NLσ − NL−σ ,
figure 8, bottom panel. For all N and for h > 0 the spin
current is always negative (the current flows in one direction)
8
J. Phys.: Condens. Matter 23 (2011) 405301
T Kwapiński and R Taranko
but at the same time the charge current changes its value
from positive to negative, depending on the Zeeman splitting
parameter. It leads to a kind of spin–charge separation,
i.e. spin and charge move in opposite directions. It is
surprising that this effect holds for no electron–electron
interactions in the wire. The structure of NLσ − NL−σ for
other N (maxima and inflection points) is related to the wire
molecular states and their relative distance from the Fermi
level.
To conclude, by changing the Zeeman splitting parameter
one can change the pumping current from the positive
value to the negative one (and vice versa). This effect is
very interesting because the time-dependent Gaussian-like
perturbation (train impulse) moves along the wire only in
one direction, from the left electrode to the right one. Thus
one expects that electrons should follow this impulse, but in
the considered system they can flow in the opposite direction
which leads to the spin–charge separation effect. Note that in
that case the direction of the spin current remains unchanged
(the curves are asymmetrical versus h = 0).
Figure 9. QD charge occupations nσ (solid lines) and the
corresponding currents flowing from the left electrode (broken
lines) as a function of time and spin-flip coupling Vsf = 0, 1 and 4
(thin, normal and thick lines) in the case of the single QD coupled
with two leads. The one-Gaussian perturbation is considered with
the parameters tx = 3, σ = 10, t00 = 25 and VL = VR = 4, 0 = 1,
h = 1 and ε0±σ = 1 for the vanishing magnetic field.
5. Pumping effects with the spin-flip scattering
the charge current, NLσ + NL−σ , pumped through the system.
Thus in figure 10 we show the charge and spin currents
for a single-site wire and for a different Vsf parameter as
a function of the Zeeman splitting strength, h. The upper
panel corresponds to the symmetrical case ε0−σ = ε0σ (the
charge current does not flow in this case, see also figure 6)
and the middle and bottom panels show the results for the
non-symmetrical case, ε0σ = ε0−σ = 1 for h = 0. As one
can see for the symmetrical case (upper panel) the absolute
value of the spin current decreases with Vsf and its direction
is the same for all positive (or negative) h. However, for
the non-symmetrical case and for a certain range of the h
parameter the absolute value of the spin current pumped in
the system increases with the spin-flip scattering Vsf , (middle
panel for −2 < h < +2). This interesting inversion can be
explained in terms of molecular states of the system. For
|h| < 2 and Vsf = 0 both spin states, ε0−σ and ε0σ , lie above
the Fermi level. In the presence of the spin-flip scattering the
hybridized spin states move with Vsf and one of them crosses
the Fermi energy. In that case the spin current increases with
Vsf , e.g. for h = 1, the lines for Vsf = 1 and Vsf = 2, middle
panel. For a larger spin-flip parameter, Vsf > 2, the absolute
value of the spin current decreases as the hybridized spin
states move away from the Fermi level. Note that the spin
current does not change its direction at different values of
Vsf (for a given h). However, the charge current pumped
through the system (bottom panel) can be positive or negative
depending on the Vsf value and the Zeeman splitting term,
h. It is worth noting that the charge current is maximal
for no magnetic field and for no spin-flip scattering (thick
solid line, bottom panel). For nonzero Vsf the current rapidly
decreases and can be negative (changes its direction). Similar
conclusions hold for a system consisting of N QD sites. We
have checked that the spin and charge currents observed,
e.g. in figure 8, are somewhat modified only for a small
Zeeman splitting parameter. In this regime new maxima in
both pumping currents can appear which lead to an increase
In this section we concentrate on charge and spin pumping
through a QW in the presence of the intradot spin-flip
coupling, Vsf . In figure 9 we study the role of the spin-flip
scattering on time-dependent occupations (solid lines) and
corresponding currents flowing from the left electrode (broken
lines) for different values of the Vsf parameter. A single-site
wire, N = 1, and one-Gaussian time-dependent perturbation
are considered. For the stationary case (before the perturbation
starts, t 25) the current does not flow through the system
and the charge nσ localized on the QD is constant but depends
on the spin-flip strength—the occupation increases with Vsf .
Such a behaviour of nσ results from the fact that in the
presence of the spin-flip scattering spin-degenerate energy
levels, ε0σ and ε0−σ , are replaced by new energies of the
system and two peaks are
q observed in the local DOS for
ε ± = (ε0σ + ε0−σ )/2 ± (ε0σ − ε0−σ )2 +4Vsf2 /2. Thus for
larger Vsf one peak is shifted below the Fermi energy, the
occupation of this state increases and simultaneously the
probability thatqthe electron in this state has a spin σ , (1 −
(ε0−σ − ε0σ )/ (ε0σ − ε0−σ )2 +4Vsf2 )/2, also increases. In
the presence of the time-dependent perturbation the charge
occupation and the pumping current change in time. It is
interesting that, depending on the spin-flip strength, the charge
nσ can increase, decrease or remain almost unchanged. This
time dependence of nσ for different values of Vsf is similar
to the dependence of the QD charge occupation, see figure 2
for ε0 > EF . For larger Vsf the curve nσ (t) has a minimum for
the same time interval, as in that case one peak of the DOS
is localized below the Fermi energy which was also observed
for ε0 = −4 in figure 2. Note that the corresponding currents
flowing from the left electrode can also change direction in
the presence of the spin-flip scattering (broken lines).
The next interesting point is whether the spin-flip
coupling influences the net spin current, NLσ − NL−σ , and
9
J. Phys.: Condens. Matter 23 (2011) 405301
T Kwapiński and R Taranko
Figure 11. Charge pumped from the left electrode, NLσ and NL−σ
(two upper panels) and charge and spin currents (two bottom
panels) as a function of the wire length, N, for Vsf = 0–2. The other
parameters are ε0σ = −2, ε0−σ = 4, h = 6, V1 = 4, tx = 3, σ0 = 10
and 0 = 1. The lines serve as a guide to the eyes.
Figure 10. Spin current, NLσ − NL−σ , and charge current,
NLσ + NL−σ , pumped through the one-site wire as a function of the
Zeeman splitting parameter h, for the symmetrical case ε0σ/−σ = 0
for h = 0 (upper panel) and the asymmetrical one ε0σ/−σ = 1 for
h = 0 (middle and bottom panels). The spin-flip couplings are
Vsf = 0–4 (thick solid, thin solid, dotted, solid and broken lines,
respectively). The other parameters are the same as in figure 9.
values of charge for N = 2, 5, . . .. At the same time the
charge NLσ does not reveal such oscillations because for
ε0σ = −2 the condition for the conductance oscillations is
not satisfied, [51]. Oscillations with the period of three sites
are also reflected on the charge and spin currents (two lower
panels). For longer wires these oscillations decrease as in
this case there are many appropriate states near the Fermi
level which influence the oscillation effect. It is interesting
that for short wires, e.g. N = 2 or 5, the charge current, as
well as the absolute values of the spin current, NLσ − NL−σ ,
increase with the spin-flip parameter. However, the charge
current and the spin current flow in opposite directions in
this case (spin–charge separation). Moreover, the spin current
does not change its sign for all N while the charge current
can be positive or negative depending on the wire length
and the spin-flip parameter. The above results confirm that
the pumped charge does not change monotonically with the
wire length but it oscillates depending on the ε0±σ positions.
Moreover, we have found that for a given N the pumped
charge increases with the spin-flip scattering strength.
of these currents as a function of Vsf (similar to the case of
N = 1).
The final studies in this paper are devoted to the
wire-length-dependent spin and charge pumping through a
QW. In figure 11 we show the charge pumped from the left
electrode, NLσ , and NL−σ (the first and second panels), the
charge current, (NLσ + NL−σ , the third panel), and the spin
current (NLσ − NL−σ , the bottom panel) as a function of
the wire length, N, for a different spin-flip parameter Vsf .
A one-Gaussian time-dependent impulse acts on the wire
and the asymmetrical case with the Zeeman magnetic field
is considered. For Vsf = 0 and N = 1 the spin-degenerate
energy levels are localized at ε0−σ = 4 and ε0σ = −2. The
value of ε0−σ corresponds to the case of the conductance
oscillations with the period of 3, [51]. As one can see, the
pumping charge, NL−σ , oscillates with N with the maximal
10
J. Phys.: Condens. Matter 23 (2011) 405301
T Kwapiński and R Taranko
6. Conclusions
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Using the evolution operator method and the tight-binding
Hamiltonian the on-site wire occupations and the spin and
charge currents flowing through a linear configuration of
quantum dots (quantum wire) at the vanishing source–drain
voltage have been calculated. The system has been affected
by a Gaussian-like train impulse moving along the wire. This
perturbation has influenced all hopping integrals between the
nearest-neighbour sites in the system.
We have found that in the presence of full spatial
symmetry (and the same on-site energies which lie at the
Fermi level) the current flowing from the left and right
electrodes oscillates in time but no charge and no spin are
pumped along the system. In other cases, the net charge
can be pumped but its value strongly depends on the wire
length. In the presence of a magnetic field which leads to the
spin-level separation (Zeeman splitting) pure spin pumps exist
in the absence of the charge current. Moreover, the spin-flip
scattering, Vsf , can significantly modify the charge current
and can change its direction. It leads to the spin–charge
separation which manifests in opposite directions of both spin
and charge currents. It is interesting that we have observed this
effect in the system with no electron–electron correlations.
We have also found that for the asymmetrical case the spin
current increases with the strength of the spin-flip scattering.
Additionally, the pumping currents (charge and spin) oscillate
as a function of the wire length.
Acknowledgments
This work has been partially supported by grant nos N N202
263 138 and N N202 330 939 of the Polish Ministry of
Science and Higher Education.
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