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Spin-polarized parametric pumping: Theory and numerical
results
Wu, J; Wang, B; Wang, J
Physical Review B (Condensed Matter and Materials Physics),
2002, v. 66 n. 20, p. 205327:1-9
2002
http://hdl.handle.net/10722/43369
Physical Review B (Condensed Matter and Materials Physics).
Copyright © American Physical Society.; This work is licensed
under a Creative Commons Attribution-NonCommercialNoDerivatives 4.0 International License.
PHYSICAL REVIEW B 66, 205327 共2002兲
Spin-polarized parametric pumping: Theory and numerical results
Junling Wu,1 Baigeng Wang,1 and Jian Wang1,2,*
1
Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China
2
Institute of Solid State Physics, Chinese Academy of Sciences, Hefei, Anhui, China
共Received 29 April 2002; revised manuscript received 12 July 2002; published 27 November 2002兲
We have extended the previous parametric electron pumping theory to include the spin-polarized pumping
effect. Specifically, we consider a parametric pump consisting of a nonmagnetic system with two ferromagnetic
leads whose magnetic moments orient at an angle ␪ with respect to each other. In our theory, the leads can be
maintained at different chemical potentials. As a result, the current is driven due to both the external bias and
the pumping potentials. When both ␪ and the external bias are zero, our theory recovers the known theory. In
particular, two cases are considered: 共i兲 in the adiabatic regime, we have derived the pumped current for an
arbitrary pumping amplitude and external bias and 共ii兲 at finite frequency, the system is away from equilibrium,
and we have derived the pumped current up to quadratic order in pumping amplitude. From our numerical
results we found that the pumped current can be modulated by the angle ␪ , showing interesting spin-valve
effects.
DOI: 10.1103/PhysRevB.66.205327
PACS number共s兲: 73.23.Ad, 73.40.Gk, 72.10.Bg
I. INTRODUCTION
Recently, there has been considerable interest in parametric pumping.1–29 The parametric pump is facilitated by cyclic
variations of pumping potentials inside the scattering system
and has been realized experimentally by Switkes et al.3 On
the theoretical side, much progress has been made towards
understanding various features related to the parametric
pump. This includes quantization of the pumped
charge,2,9,20,25 the influence of discrete spatial symmetries
and magnetic field,5,7 the rectification of displacement
current,10 as well as inelastic scattering11 to the pumped current. The concept of an optimal pump has been proposed
with the lower bound for the dissipation derived.12 Within
the formalism of time-dependent scattering matrix theory, the
heat current and shot noise in the pumping process17,23,24,27
has also been discussed. Recently, the original adiabatic
pumping theory has been extended to account for the effect
due to finite frequency,14,19 Andreev reflection in the presence of superconducting leads,13,28 and strong electron interaction in the Kondo regime.26 This gives us more physical
insight into parametric pumping. For instance, the experimentally observed anomaly of pumped current at ␾ ⫽0 and
␾ ⫽ ␲ can be explained using finite frequency theory19 as
due to the quantum interference of different photon-assisted
processes. When a superconducting lead is present, the interference between the direct reflection and multiple Andreev
reflections gives rise to an enhancement of pumped current
which is four times that of the normal system.13 It will be
interesting to further extend the parametric theory to the case
where the ferromagnetic leads are present. With the theory
extended, much different physics is foreseen30 which may
lead to operational paradigms for future spintronic devices.31
In this paper, we have extended the previous parametric electron pumping theory to include the spin-polarized pumping
effect. Specifically, we consider a parametric pump consisting of a nonmagnetic system with two ferromagnetic leads
whose magnetic moments orient at an angle ␪ with respect to
each other. Our theory is based on a nonequilibrium Green’s0163-1829/2002/66共20兲/205327共9兲/$20.00
function approach and focused on current perpendicular to
plane geometry. The parametric pump generates current at
zero external bias. It would be interesting to see the interplay
of the role played by the pumping potential and external bias
if the leads are maintained at different chemical potentials.22
Hence in our theory, the external bias is also included. In the
adiabatic regime, the pumped current is proportional to
pumping frequency. In this regime, we have derived the
parametric pumping theory for finite pumping amplitude. At
the finite pumping frequency, the system is away from equilibrium, and we have performed perturbation up to the second order in pumping amplitude and obtained the pumped
current at finite frequencies. Our theory allows one to study
the pumped current for a variety of parameters, such as the
pumping amplitude, pumping frequency, phase difference
between two pumping potentials, the angular dependence between the magnetization of two leads, as well as the external
bias. We have applied our theory to a tunneling magnetoresistance 共TMR兲 junction.32 Due to the reported roomtemperature operation of TMR, the fundamental principle
and transport properties of TMR devices have attracted increasing attention.33 From our numerical results we found
that the pumped current can be modulated by the angle ␪
showing interesting spin-valve effects. The paper is organized as follows. In Sec. II, we derive the general theory of a
parametric pump in the presence of ferromagnetic leads. The
numerical results and summary are presented in Sec. III.
II. THEORY
The system we examine consists of a nonmagnetic system
connected by two ferromagnetic electrodes to the reservoir.
The magnetic moment M of the left electrode is pointing in
the z direction, the electric current is flowing in the y direction, while the moment of the right electrode is at an angle ␪
to the z axis in the x-z plane 共see the inset of Fig. 1 for a
schematic picture兲. The Hamiltonian of the system is of the
following form:
66 205327-1
H⫽H L ⫹H R ⫹H 0 ⫹V p ⫹H T ,
共1兲
©2002 The American Physical Society
PHYSICAL REVIEW B 66, 205327 共2002兲
WU, WANG, AND WANG
not in the Coulomb blockade regime. In addition, for the
nonmagnetic regions in which we are interested, the Kondo
effect can be neglected.
To proceed, we first apply the following Bogoliubov
transformation34 to diagonalize the Hamiltonian of the right
electrode,35
c kR ␴ →cos共 ␪ /2兲 C kR ␴ ⫺ ␴ sin共 ␪ /2兲 C kR ¯␴ ,
⫹
⫹
⫹
c kR
␴ →cos共 ␪ /2 兲 C kR ␴ ⫺ ␴ sin共 ␪ /2 兲 C kR ¯␴ ,
共6兲
共7兲
from which we obtain the effective Hamiltonian
H R⫽
FIG. 1. The transmission coefficient as a function of Fermi energy at different angles ␪ between the magnetizations of two leads:
␪ ⫽0 共dashed line兲, ␪ ⫽ ␲ /2 共solid line兲, and ␪ ⫽ ␲ 共dotted line兲.
Inset: schematic picture of the system.
H T⫽
H R⫽
兺
k␴
⫹
†
共 ⑀ kL ⫹ ␴ M 兲 c kL
兺
␴ c kL ␴ ,
k␴
冋
共2兲
⑀ n d n† ␴ d n ␴ .
兺
n␴
共3兲
兺
k␣n␴
关 T k ␣ n c k†␣ ␴ d n ␴ ⫹c.c.兴 .
冊
共9兲
In this subsection, we examine the pumping current at the
low-frequency limit while maintaining the pumping amplitude finite. In this limit, the system is nearly in equilibrium
and we will use the equilibrium Green’s function36 –38 to
characterize the pumping process. Using the distribution
function, the total charge in the system during the pumping is
given by
共4兲
Q 共 x,t 兲 ⫽⫺iq
V p is the time-dependent pumping potential and H T describes the coupling between electrodes and the NM scattering region with hopping matrix T k ␣ n . To simplify the analysis, we assume the hopping matrix to be independent of spin
index, hence
H T⫽
␪ †
†
T kLn c kL
␴ d n ␴ ⫹T kRn cos C kR ␴
2
A. Pumping in the low-frequency limit
In Eq. 共1兲, H 0 describes the nonmagnetic 共NM兲 scattering
region,
H 0⫽
冉
册
共8兲
In the following subsections, we will consider two cases: 共i兲
parametric pumping in the low-frequency limit with finite
pumping amplitude and 共ii兲 pumping in the weak pumping
limit with finite pumping frequency.
†
共 ⑀ kR ⫹ ␴ M cos ␪ 兲 c kR
␴ c kR ␴
†
M sin ␪ 关 c kR
兺
␴ c kR ¯␴ 兴 .
k␴
冋
␪ †
⫺ ␴ sin C kR ¯␴ d n ␴ ⫹c.c. .
2
where H L and H R describe the left and right electrodes,
H L⫽
兺
kn ␴
†
关共 ⑀ kR ⫹ ␴ M 兲 C kR
兺
␴ C kR ␴ 兴 ,
k␴
共 dE/2␲ 兲关 G⬍ 共 E, 兵 V 共 t 兲 其 兲兴 xx ,
共10兲
where G⬍ is the lesser Green’s function in real space, x
labels the position, and 兵 V(t) 其 describes a set of external
parameters which facilitates the pumping process. Within a
Hartree approximation, G⬍ is related to the retarded and advanced Green’s functions Gr and Ga ,
共5兲
In these expressions ⑀ k ␣ ⫽ ⑀ 0k ⫹qV ␣ with ␣ ⫽L,R; c k†␣ ␴ 共with
␴ ⫽↑,↓ or ⫾1, and ¯␴ ⫽⫺ ␴ ) is the creation operator of electrons with spin index ␴ inside the ␣ electrode. Similarly d n† ␴
is the creation operator of electrons with spin ␴ at energy
level n for the NM scattering region. In writing Eqs. 共2兲 and
共3兲, we have made a simplification that the value of the molecular field M is the same for the two electrodes, thus the
spin-valve effect is obtained32 by varying the angle ␪ . Essentially, M mimics the difference of the density of states between spin-up and -down electrons32 in the electrodes. In this
paper, we only consider the single electron behavior. The
charge quantization is not considered so that our system is
冕
G⬍ 共 E, 兵 V 其 兲 ⫽Gr 共 E, 兵 V 其 兲 i
兺␣ ⌫␣ f ␣共 E 兲 Ga共 E, 兵 V 其 兲 ,
共11兲
where f ␣ (E)⫽ f (E⫺qV ␣ ). In the low-frequency limit, the
retarded Green’s function in real space is given by
Gr 共 E, 兵 X 其 兲 ⫽
1
E⫺H 0 ⫺Vp ⫺⌺r
,
共12兲
where ⌺r ⬅ 兺 ␣ ⌺␣r is the self-energy, and ⌫␣ ⫽⫺2 Im关 ⌺␣r 兴 is
the linewidth function. In the above equations, Gr,a,⬍ der,a⬍
notes a 2⫻2 matrix with matrix elements G ␴ , ␴ ⬘ and ␴
⫽↑,↑. Vp ⫽V p I where I is a 2⫻2 unit matrix in the spin
space. In a real-space representation, V p is a diagonal matrix
205327-2
PHYSICAL REVIEW B 66, 205327 共2002兲
SPIN-POLARIZED PARAMETRIC PUMPING: THEORY . . .
describing the variation of the potential landscape due to the
external pumping parameter V. The self-energies are given
by35
⌺␣r 共 E 兲 ⫽R̂ ␣
冉
⌺ ␣r ↑
0
0
⌺ ␣r ↓
冊
冉
⌫ ␣0 ↑
0
0
⌫ ␣0 ↓
R̂ ␣†
共13兲
冉
冊
cos ␪ ␣ /2
sin ␪ ␣ /2
⫺sin ␪ ␣ /2
cos ␪ ␣ /2
R̂ ␣†
冊
T * T k␣n
兺k E⫺k⑀␣0m
.
⫹
共15兲
兺j
冕
dQ p ␣ 共 t 兲 ⫽⫺q
k ␣ ␴ ⫹i ␦
,
共16兲
冕
共19兲
兺i ⳵ V Tr关 Q 共 x,t 兲兴 ␦ V i共 t 兲 .
i
1
Tr关 iGr ⌫␣ Ga ⌫␤ Gr ⌬ j ⫹c.c.兴
4␲
共20兲
冕
dE
dN ␣␤
␦ V j共 t 兲.
dV j
兺␤ 共 ⫺ ⳵ E f ␤ 兲 兺j
共21兲
If we include the charge passing through contact ␣ due the
external bias, then
dQ ␣ 共 t 兲 ⫽⫺q
⫺q
冕
冕
dN ␣␤ dV j 共 t 兲
dt
dV j
dt
dE
兺␤ 共 ⫺ ⳵ E f ␤ 兲 兺j
dE
兺␤ Tr关 ⌫␣ Gr ⌫␤ Ga 兴共 f ␣ ⫺ f ␤ 兲 dt.
共22兲
Furthermore, the total current flowing through contact ␣ due
to both the variation of parameters V j and external bias, in
one period, is given by
J ␣⫽
1
␶
冕
␶
0
共23兲
dtdQ ␣ /dt,
where ␶ ⫽2 ␲ / ␻ is the period of cyclic variation. If there are
two pumping parameters, Eq. 共23兲 can be written as when
␣ ⫽1,
共17兲
J (1)
1 ⫽
冕 冕
q
2␶
dt
dE ⳵ E 共 f 1 ⫺ f 2 兲
册
⫺
dN 12 dV j 共 t 兲
dt
dV j
dt
⫺
q
␶
冕 冕
dt
兺j
冋
dN 11 dV j 共 t 兲
dV j
dt
dE Tr关 ⌫1 Gr ⌫2 Ga 兴共 f 1 ⫺ f 2 兲
共24兲
and
dE
2␲
兺␤ Tr关 Gr ⌬j Gr ⌫␤ Ga
J (2)
1 ⫽
⫹Gr ⌫␤ Ga ⌬ j Ga 兴 f ␤ 共 E 兲 ␦ V j 共 t 兲
⫽⫺q
共18兲
with 兺 ␣ dN ␣␤ /dV j ⫽dN ␤ /dV j , we obtain
It is easily seen that the total charge in the system in a period
is zero which is required for the charge conservation. To
calculate the pumped current, we have to find the charge
dQ p ␣ passing through contact ␣ due to the change of the
system parameters. Using the Dyson equation ⳵ V i Gr
⫽Gr ⌬ i Gr , the above equation becomes
dQ p 共 t 兲 ⫽q
dN ␤
␦ V j共 t 兲,
dV j
dN ␣␤
1
⫽
Tr关 Gr ⌫␣ Gr ⌬ j ⫹c.c.兴 ␦ ␣␤
dV j
4␲
共14兲
and ⌫ ␣0 ␴ ⫽⫺2 Im(⌺ ␣r ␴ ) is the linewidth function when ␪
⫽0.
In order for a parametric electron pump to function at low
frequency, we need simultaneous variation of two or more
system parameters controlled by gate voltages: V i (t)⫽V i0
⫹V ip cos(␻t⫹␾i). Hence, in our case, the potential due to the
gates can be written as V p ⫽ 兺 i V i ⌬i , where ⌬i is the potential profile due to each pumping potential. For simplicity we
assume a constant gate potential such that (⌬1 ) xx is one for x
in the first gate region and zero otherwise. If the time variation of these parameters is slow, i.e., for V(t)⫽V 0
⫹ ␦ V cos(␻t), then the charge of the system coming from all
contacts due to the infinitesimal change of the system parameter ( ␦ V→0) is
dQ p 共 t 兲 ⫽
兺␤ 共 ⳵ E f ␤ 兲 兺j
Using the partial density of states dN ␣␤ /dV j defined as41
Here angle ␪ ␣ is defined as ␪ L ⫽0 and ␪ R ⫽ ␪ ,⌺ ␣r ␴ is given
by
⌺ ␣r ␴ mn ⫽
dE
dN ␤
1
⫽
Tr关 Ga ⌫␤ Gr ⌬ j 兴 .
dV j 2 ␲
with the rotational matrix R̂ ␣ for electrode ␣ defined as
R̂⫽
冕
where we have used the injectivity40
and
⌺␣⬍ 共 E 兲 ⫽i f ␣ R̂ ␣
dQ p 共 t 兲 ⫽q
dE
2␲
兺j 兺␤
⫹
f ␤ Tr共 ⳵ E 关 Ga ⌫␤ Gr ⌬ j 兴 兲 ␦ V j 共 t 兲 ,
where the wideband limit has been taken. Integrating by part,
we obtain
q
2␶
冕 冕
dt
dE ⳵ E 共 f 1 ⫹ f 2 兲
册
dN 12 dV j 共 t 兲
dt,
dV j
dt
兺j
冋
dN 11 dV j 共 t 兲
dV j
dt
共25兲
(2)
(1)
where J 1 ⫽J (1)
1 ⫹J 2 . In Ref. 22, J 1 has been identified as
the current due to the external bias and J (2) as pumping
current. For ␪ ⫽0,␲ , all of the 2⫻2 matrix is diagonal. In
205327-3
PHYSICAL REVIEW B 66, 205327 共2002兲
WU, WANG, AND WANG
this case, it is easy to show that Eq. 共23兲 agrees with the
result of Ref. 22. If the external bias is zero, Eq. 共23兲 reduces
to the familiar formula1 when there are two pumping potentials,
J ␣⫽
q␻
2␲
冕 冋
␶
dt
0
册
dN ␣ dX 1 dN ␣ dX 2
⫹
.
dX 1 dt
dX 2 dt
共26兲
In this subsection, we will calculate the pumping current
at finite frequency. The Keldysh nonequilibrium Green’sfunction approach used here is in the standard tight-binding
representation.36 We could not use the momentum space version because the time-dependent perturbation 共pumping potential兲 inside the scattering region is position dependent.
Hence it is most suitable to use a tight-binding real-space
technique. In contrast, in previous investigations39 on
photon-assisted processes the time-dependent potential is
uniform throughout the dot and therefore a momentum space
method is easier to apply.
Assuming the time-dependent perturbations located at the
different sites, j⫽i 0 , j 0 , and k 0 , etc. with
冕
t
⫺⬁
(2)r
G11
共 t,t 1 兲 ⫽
兺jk
⬅
共29兲
where the scattering Green’s functions and self-energy are
defined in the usual manner:
冕冕
0r
dxdyG0r
1 j 共 t⫺x 兲 V j 共 x 兲 G jk 共 x⫺y 兲
兺jk G0r1 j Vj G0rjk Vk Gk10r .
共36兲
Substituting Eq. 共36兲 into Eq. 共33兲 and completing the
integration over time x,y,t 1 ,t, it is not difficult to calculate
the average current 具 J L1 典 due to the first term in Eq. 共33兲
共see the Appendix for details兲,
具 J L1 共 t 兲 典 ⫽⫺ 兺
jk
T 共 E 兲 ⫽Tr关 ⌫Lr 共 E 兲 Gr 共 E 兲 ⌫Rr 共 E 兲 Ga 共 E 兲兴 ,
共35兲
0r
⫻Vk 共 y 兲 Gk1
共 y⫺t 1 兲
共28兲
and a transmission coefficient
共34兲
Now we make use of the time-dependent pumping potentials
V j (t) as the perturbations to calculate all kinds of Green’s
functions up to the second order, and corresponding average
current.
First, we calculate the current corresponding to the term
r
(t,t 1 )⌺L⬍ (t 1 ,t) in Eq. 共33兲. The Dyson equation for
G11
r
G11(t,t 1 ) gives the second-order contribution
dt 1 Tr关 Gr 共 t,t 1 兲 ⌺␣⬍ 共 t 1 ,t 兲 ⫹G⬍ 共 t,t 1 兲 ⌺␣a 共 t 1 ,t 兲
⫹c.c.兴
E⫺H 0 ⫺⌺r
and G0⬍ is related to the retarded and advanced Green’s
functions G0r and G0a by
共27兲
When there is no interaction between electrons in the ideal
leads L and R, the standard nonequilibrium Green’s-function
theory gives the following expression for the time-dependent
current,37
J ␣ 共 t 兲 ⫽⫺q
1
G0⬍ 共 E 兲 ⫽G0r 共 E 兲 ⌺⬍ 共 E 兲 G0a 共 E 兲 .
B. Finite frequency pumping in the weak pumping limit
V j 共 t 兲 ⫽V j cos共 ␻ t⫹ ␾ j 兲 .
G0r 共 E 兲 ⫽
qV j V k
4
冕
dE
0r
i⌬ k j
Tr兵 ⌺L⬍ G0r
1 j 关 G jk 共 E ⫺ 兲 e
2␲
0r
⫺i⌬ k j
⫹G0r
兴 Gk1
其,
jk 共 E ⫹ 兲 e
共37兲
where ⌬ k j ⫽ ␾ k ⫺ ␾ j is the phase difference, E ⫾ ⫽E⫾ ␻ , and
G r,a,⬍ ⬅G r,a,⬍ (E).
⬍
(t,t 1 )⌺La (t 1 ,t) in
Now we calculate the second term G11
⬍
Eq. 共33兲. Using the Keldysh equation, G ⫽Gr ⌺⬍ Ga , we
have
⫹
r,a
Gi j ␴␴ ⬘ 共 t 1 ,t 2 兲 ⫽⫿i ␪ 共 ⫾t 1 ⫿t 2 兲 具 兵 d i, ␴ 共 t 1 兲 ,d j, ␴ ⬘ 共 t 2 兲 其 典 ,
⬍
⫹
Gi j, ␴ , ␴ ⬘ 共 t 1 ,t 2 兲 ⫽i 具 d j, ␴ ⬘ 共 t 2 兲 d i, ␴ , 共 t 1 兲 典 ,
⌺r,a,⬍
␣ i j 共 t 1 ,t 2 兲 ⫽
兺k T ␣*ki T ␣ k j g␣r,a,⬍共 t 1 ,t 2 兲 .
共30兲
r
⬍ a
r
⬍ a
G⬍
jk ⫽G j1 ⌺L G1k ⫹G jN ⌺R GNk ,
共31兲
where ⌺␣⬍ ⫽i⌫␣ f ␣ . Expanding G r,a up to the second order
in pumping parameters, we obtain the second-order contribu⬍
,
tion from G11
共32兲
(2)⬍
r
a
r
a
⫽G11
⌺L⬍ G11
⫹G1N
⌺R⬍ GN1
G11
The average current J L (t) from the left lead can be written as
q
具 J L 共 t 兲 典 ⫽⫺
␶
冕 冕
␶
dt
0
t
⫺⬁
(2)r ⬍ 0a
(1)r ⬍ (1)a
0r ⬍ (2)a
⫽G11
⌺L G11 ⫹G11
⌺L G11 ⫹G11
⌺L G11
(2)r ⬍ 0a
(1)r ⬍ (1)a
0r ⬍ (2)a
⫹G1N
⌺R GN1 ⫹G1N
⌺R GN1 ⫹G1N
⌺R GN1
r
dt 1 Tr关 G11
共 t,t 1 兲 ⌺L⬍ 共 t 1 ,t 兲
⬍
⫹G11
共 t,t 1 兲 ⌺La 共 t 1 ,t 兲 ⫹c.c.兴 .
共38兲
⫽
共33兲
In the absence of pumping, the retarded Green’s function
is defined in terms of the Hamiltonian H 0 ,
205327-4
0a
兺jk 关 G0r1 j Vj G0rjk Vk Gk10⬍ ⫹G0r1 j Vj G0⬍
jk Vk Gk1
0a
0a
⫹G0⬍
1 j V j G jk Vk Gk1 兴
⫽g 2 ⫹g 3 ⫹g 4 ,
共39兲
PHYSICAL REVIEW B 66, 205327 共2002兲
SPIN-POLARIZED PARAMETRIC PUMPING: THEORY . . .
where we have used Eq. 共38兲 to simplify the expression.
Plugging Eq. 共39兲 into Eq. 共33兲 and after some algebra, we
have the following three expressions corresponding to each
term in Eq. 共39兲,
J L ⫽i
兺jk
qV j V k
4
冕
具 J L3 典 ⫽⫺ 兺
jk
⫹e
qV j V k
4
⫺i⌬ k j
具 J L4 典 ⫽⫺ 兺
jk
冕
冕
0r
G0r
ji Gik ⫽
共41兲
J L⫽
共42兲
共 E ⫾ 兲 ⫽⫺ f 共 E ⫾ 兲关 G 共 E ⫾ 兲 ⫺G 共 E ⫾ 兲兴 .
0r
0a
共44兲
Eqs. 共37兲, 共40兲, and 共42兲* lead to
兺jk
qV j V k
4
冕
dE
0r
i⌬ k j
Tr兵 ⌫L G0r
1 j f 共 E 兲关 G jk 共 E ⫺ 兲 e
2␲
0a
⫹e ⫺i⌬ k j G0r
k j 共 E ⫹ 兲兴 Gk1 其 ,
共45兲
while Eqs. 共37兲, 共40兲* , and 共42兲 give
i
兺jk
qV j V k
4
冕
dE
0a
i⌬ k j
Tr兵 ⌫L G0r
1 j f 共 E 兲关 G jk 共 E ⫺ 兲 e
2␲
0a
⫹e ⫺i⌬ k j G0a
jk 共 E ⫹ 兲兴 Gk1 其 .
共46兲
Furthermore, Eq. 共41兲 plus the complex conjugate becomes
i
兺jk
qV j V k
4
冕
dE
0r
0a
i⌬ k j
Tr„⌫L G0r
1 j 兵 f ⫺ 关 G jk 共 E ⫺ 兲 ⫺G jk 共 E ⫺ 兲兴 e
2␲
0a
0a
⫹e ⫺i⌬ k j f ⫹ 关 G0r
jk 共 E ⫹ 兲 ⫺G jk 共 E ⫹ 兲兴 其 Gk1 …,
where f ⫾ ⫽ f (E ⫾ ).
Combining Eqs. 共45兲–共47兲, we finally obtain
共47兲
共50兲
兺jk
q ␻ V j V k sin共 ⌬ k j 兲
2
冕
dE
⳵ f 共 E 兲 Tr兵 ⌫L G0r
1j共E兲
2␲ E
0a
0a
⫻关 G0r
jk 共 E 兲 ⫺G jk 共 E 兲兴 Gk1 共 E 兲 其
⫽⫺
兺jk
冋
iq ␻ V j V k sin共 ⌬ jk 兲
2
⫻Tr ⌫L
冕
dE
⳵ f 共E兲
2␲ E
册
0r
0a
0r
0a
⳵ G11
⳵ G11
⳵ G12
⳵ G21
⌫L
⫹⌫L
⌫R
,
⳵ Vj
⳵ Vk
⳵ Vj
⳵ Vk
共51兲
which is the same as Ref. 1 when ␪ ⫽0.
共43兲
and
⳵ G0r
jk
.
⳵ Vi
We obtain
dE
0a
i⌬ k j
Tr兵 ⌺La G0⬍
1 j 关 G jk 共 E ⫺ 兲 e
2␲
G0⬍ 共 E 兲 ⫽⫺ f 共 E 兲关 G0r 共 E 兲 ⫺G0a 共 E 兲兴
⫺i
共49兲
and the Dyson equation
The final pumped current is the sum of Eqs. 共37兲, 共40兲–共42兲,
and their complex conjugates, in addition to the current directly due to the external bias 关see second term of Eq. 共24兲兴.
If the external bias is zero, the expression of the pumped
current can be simplified significantly. Note that in the equilibrium, the lesser Green’s function satisfies the fluctuationdissipation theorem,
G
G0r ⫺G0a ⫽⫺iG0r ⌫G0a
dE
0⬍
i⌬ k j
Tr兵 ⌺La G0r
1 j 关 G jk 共 E ⫺ 兲 e
2␲
0a
⫹e ⫺i⌬ k j G0a
jk 共 E ⫹ 兲兴 Gk1 其 .
0⬍
共48兲
In the limit of small frequency, we expand Eq. 共48兲 up to the
first order in frequency and use the fact that
共40兲
0a
G0⬍
jk 共 E ⫹ 兲兴 Gk1 其 ,
qV j V k
4
dE
0r
Tr„⌫L G0r
1 j 兵 共 f ⫺ ⫺ f 兲关 G jk 共 E ⫺ 兲
2␲
0a
⫺G0a
jk 共 E ⫹ 兲兴 其 Gk1 ….
dE
0r
i⌬ k j
Tr兵 ⌺La G0r
1 j 关 G jk 共 E ⫺ 兲 e
2␲
0⬍
⫹e ⫺i⌬ k j G0r
jk 共 E ⫹ 兲兴 Gk1 其 ,
冕
i⌬ k j
⫺G0a
⫹e ⫺i⌬ k j 共 f ⫹ ⫺ f 兲关 G0r
jk 共 E ⫺ 兲兴 e
jk 共 E ⫹ 兲
具 J L2 典 ⫽the term corresponding tog 2 in Eq. 共 39兲
⫽⫺
qV j V k
4
兺jk
III. RESULTS
We now apply our formula, Eqs. 共24兲, 共25兲, and 共48兲, to a
TMR junction. For current perpendicular to the plane geometry, the TMR junction can be modeled by a one-dimensional
quantum structure with a double barrier potential U(x)
⫽X 1 ␦ (x⫹a)⫹X 2 ␦ (x⫺a) where 2a is the well width. For
this system the Green’s function G(x,x ⬘ ) can be calculated
exactly.43 The adiabatic pump that we consider is operated
by changing barrier heights adiabatically and periodically:
X 1 ⫽V 0 ⫹V p sin(␻t) and X 2 ⫽V 0 ⫹V p sin(␻t⫹␾). In the following, we will study zero-temperature behavior of the
pumped current. In the calculation, we have chosen M
⫽37.0 and V 0 ⫽79.2. Finally the unit is set by ប⫽2m⫽1. 42
We first study the pumped current with two pumping potentials in the adiabatic regime. Figure 1 depicts the transmission coefficient T versus Fermi energy at several angles ␪
between magnetizations of ferromagnetic leads. In general,
all the transmission coefficients display the resonant feature.
At ␪ ⫽0 the resonance is much sharper. As expected, among
different angles, T is the largest at ␪ ⫽0 and smallest at ␪
⫽ ␲ with the ratio T max (0)/T max ( ␲ )⬃4. This gives the
usual spin-valve effect.32 Figure 2 plots the pumped current
versus Fermi energy at different angles ␪ . Here we have set
the phase difference of two pumping potentials to be ␲ /2.
Similar to the transmission coefficient, we obtain the largest
pumped current at ␪ ⫽0 and the smallest current at ␪ ⫽ ␲ .
We found that the ratio I max (0)/I max ( ␲ ) is about the same as
that of the transmission coefficient 共see also the left inset of
205327-5
PHYSICAL REVIEW B 66, 205327 共2002兲
WU, WANG, AND WANG
FIG. 2. The pumped current as a function of Fermi energy at
different ␪ : ␪ ⫽0 共solid line兲, ␪ ⫽ ␲ /2 共dot-dashed line兲, and ␪ ⫽ ␲
共dotted line兲. Other parameters are ␾ ⫽ ␲ /2 and V p ⫽0.05V 0 . Left
inset: the same as the main figure except V p ⫽0.1V 0 . Right inset:
the pumped current as a function of ␪ . Here ␾ ⫽ ␲ /2, E F ⫽37.55,
and V p ⫽0.05V 0 . For the unit for the pumped current, see Ref. 42.
Fig. 2兲. This demonstrates the spin-valve effect for the
pumped current. To understand this, we note that in the presence of the ferromagnetic leads, the electrons with spin up
and down experience different potentials and hence generate
different currents for different spin. Hence, because of the
difference in density of states for spin-up and -down electrons, the pumped current is spin polarized. In addition as
one varies the angle between magnetization in the ferromagnetic lead, the pumped current can be modulated by the angle
␪ . As the pumping amplitude doubles, the peak of pumped
current is broadened and the maximum pumped current is
nearly doubled 共see the left inset of Fig. 2兲. This broadening
is understandable since at large pumping amplitude the instantaneous resonant level oscillates with a large amplitude
and hence can generate heat current in a broad range of energy. The doubling effect of the pumped current, as pumping
amplitude doubles, persists for larger pumping amplitude.
The physics may be different from that of Ref. 5. For an
adiabatic pump, the pumped current is sensitive to the configuration of the system. After the random average over different configurations, Ref. 5 found that the pumped current
scales as the square root of the pumping amplitude. For a
chaotic system, Brouwer10 found similar results. However,
our system is not chaotic. It is in the ballistic regime and
does not require the random average over a different configuration. However, since our pumping amplitude is not
large enough, we cannot rule out that the range of our pumping amplitude is in the intermediate regime. The spin-valve
effect of pumped current is illustrated in the right inset of
Fig. 2 where the pumped current versus the angles ␪ is
shown when the system is at resonance. We see that the
pumped current is maximum at ␪ ⫽0 and decreases quickly
as one increases ␪ from 0 to ␲ . For larger pumping amplitude, we have similar behavior 共not shown兲. In Fig. 3, we
plot the pumped current, as a function of phase difference ␾
between two pumping potentials. We see that the pumped
current is antisymmetric about the ␾ ⫽ ␲ . In the weak pumping regime, the dependence of pumped current as a function
FIG. 3. The pumped current as a function of phase difference ␾
at different ␪ . ␪ ⫽0 共dashed line兲, ␪ ⫽ ␲ /2 共dotted line兲, and ␪ ⫽ ␲
共solid line兲. Other parameters are E F ⫽37.55 and V p ⫽0.05V 0 .
of the phase difference is sinusoidal.1 In Fig. 3, the nonlinear
behavior is clearly seen which deviates from the sinusoidal
behavior at small pumping amplitude, indicating the onset of
the strong pumping regime. We also notice that the peak of
the pumped current shifts to the larger ␾ . In Fig. 4, we show
the pumped current in the presence of external bias. In the
calculation we assume that V L ⫽⫺ ␻ V/2 and V R ⫽ ␻ V/2 so
that the external bias is against the pumped current when V is
positive. Due to the external bias, the total pumped current
共dashed line兲 decreases near resonant energy and reverses the
direction at other energies. Now we turn to the case of finite
frequency pumping. We first present our results 共Fig. 5–Fig.
7兲 at small pumping frequency ␻ ⫽0.002. In Fig. 5 we plot
the pumped current as a function of phase difference ␾ near
the resonant energy. At ␪ ⫽0, the magnitude of pumped current is much larger than that at ␪ ⫽ ␲ /2 or ␲ . This again
demonstrates the spin-valve effect for the pumped current at
finite frequency. We notice that at ␾ ⫽0 and ␾ ⫽ ␲ , the
pumped current is nonzero, similar to the experimental
anomaly observed experimentally for a nonmagnetic
system.3 The pumped current away from resonant energy is
FIG. 4. The pumped current as a function of Fermi energy with
an external bias V L ⫺V R ⫽⫺0.02␻ at ␪ ⫽ ␲ . Solid line: pumped
current, dotted line: current due to external bias, dashed line: total
current. Here ␾ ⫽ ␲ /2 and V p ⫽0.05V 0 . For ␪ ⫽ ␲ /2 and ␪ ⫽0 similar behavior is observed.
205327-6
PHYSICAL REVIEW B 66, 205327 共2002兲
SPIN-POLARIZED PARAMETRIC PUMPING: THEORY . . .
FIG. 5. The pumped current as a function of ␾ at finite frequency. Dashed line: ␪ ⫽0, dotted line: ␪ ⫽ ␲ /2, and solid line: ␪
⫽ ␲ . Here E F ⫽37.55 and ␻ ⫽0.002.
FIG. 7. The pumped current as a function of ␪ at finite frequency. Here dashed line: ␾ ⫽0, dotted line: ␾ ⫽ ␲ /2, and solid
line: ␾ ⫽ ␲ . Other parameters are E F ⫽37.55 and ␻ ⫽0.002.
shown in Fig. 6. At ␪ ⫽0, we see that the pumped current is
sharply peaked at resonant energy. The pumped current is
positive for ␾ ⫽ ␲ /2 and negative for ␾ ⫽0 and ␲ . At ␪
⫽ ␲ /2 or ␲ , the pumped current at ␾ ⫽ ␲ /2 共dashed line兲 is
much larger than that at other angles 共not shown兲. Figure 7
displays the pumped current as a function of ␪ near resonant
energy. For ␾ ⫽ ␲ /2 共dotted line兲, we see the usual behavior
that large pumped current occurs at ␪ ⫽0 and it decreases to
the minimum at ␪ ⫽ ␲ . For ␾ ⫽0 or ␲ , however, we see
completely different behavior. The pumped current is the still
the largest at ␪ ⫽0 but the direction of the pumped current is
reversed. As one increases ␪ , the pumped current decreases
and reaches a flat region with almost zero pumped current.
Now we study the effect of frequency to the pumped current
versus ␪ 关see Fig. 8共a兲兴. We will fix the phase difference to
be ␾ ⫽ ␲ /2 with energy near resonance. At small frequency
␻ ⫽0.002 共dashed line兲, the pumped current versus ␪ shows
usual behavior. When the frequency is increased to ␻
⫽0.004 共dot-dashed line兲, two peaks show up symmetrically
near ␪ ⫽⫾ ␲ /4 while the minimum is still at ␪ ⫽ ␲ . As frequency is increased further to ␻ ⫽0.006 共dotted line兲, the
pumped current near ␪ ⫽0 reverses the direction and the new
peak position shifts to ␪ ⬃0.35␲ . Upon further increasing ␻ ,
the curve of pumped current versus ␪ develops a flat region
between ␪ ⫽⫾0.35␲ with positive current, while the magnitude of the negative pumped current at ␪ ⫽0 becomes larger
FIG. 6. The pumped current as a function of Fermi energy at
finite frequency at ␪ ⫽0: Dotted line: ␾ ⫽0, dashed line: ␾ ⫽ ␲ /2,
and solid line: ␾ ⫽ ␲ . Here ␻ ⫽0.002.
FIG. 8. The pumped current as a function of ␪ at different frequencies. 共a兲 ␻ ⫽0.002 共short dashed line兲, 0.004 共dot-dashed line兲,
0.006 共dotted line兲, and 0.008 共solid line兲. 共b兲 ␻ ⫽0.01 共solid line兲,
0.02 共dot-dashed line兲, 0.05 共short dashed line兲, and 0.1 共dotted
line兲. Other parameters are E F ⫽37.55 and ␾ ⫽ ␲ /2. Inset: the
pumped current as a function of frequency. Here ␪ ⫽0, ␾ ⫽ ␲ /2,
and E F ⫽37.55.
205327-7
PHYSICAL REVIEW B 66, 205327 共2002兲
WU, WANG, AND WANG
关see Fig. 8共b兲兴. Finally, at even larger frequency ␻ ⫽0.1, all
the pumped currents are negative. This behavior can be understood from the photon-assisted process.19 The quantum
interference between contributions due to photon emission
共or absorption兲 near two pumping potentials is essential to
understand the nature of pumped current. In addition to the
interference effect, the pumped current is also affected by a
competition between the photon emission and absorption
processes which tend to cancel each other. It is the interplay
between this competition and interference that gives rise to
the interesting spin-valve effect for the pumped current. In
the inset of Fig. 8共b兲, we show the pumped current versus
pumping frequency at ␪ ⫽0. We see that at small frequency
the current is positive and small, and at large frequency the
current is much larger and is negative.
In summary, we have extended the previous parametric
electron pumping theory to include the spin-polarized pumping effect. Our theory is based on the nonequilibrium
Green’s-function method, is valid for multimodes 共in two or
three dimensions兲, and can be easily extended to the case of
multiprobes 共although most of calculations are for two
probes兲. In the parametric pumping, two kinds of driving
forces are present: multiple pumping potentials inside the
scattering system as well as the external bias in the multiprobes. Two cases are considered. In the adiabatic regime,
the system is near equilibrium. In this case our theory is for
general pumping amplitude. At finite frequency, the system is
away from equilibrium. Our theory is up the quadratic order
in pumping amplitude. This theory allows us to examine the
pumped current in broader parameter space including pumping amplitude, pumping frequency, phase difference between
two pumping potentials, and the angle between magnetization of two leads. From our numerical calculation, the spinvalve effect is clearly seen and the pumped current can be
modulated by the angle ␪ .
F共 t 兲 ⫽
B⫽
冕 冕
冕␲
␶
t
dt
0
V ␣V ␤
⫽
4
⫺⬁
dt 1 Tr关 F0 共 t 1 ,t 兲 G共 t,t 1 兲兴
⫽
and F i ’s (i⫽0,1,2,3) satisfy F i (t 1 ,t 2 )⫽F i (t 1 ⫺t 2 ).
Taking the Fourier transform
冕 冕 冕
␶
t
dt
⫺⬁
0
dt 1
dE
F 共 E 兲 e ⫺iE(t 1 ⫺t)
2␲ 0
冕
dxdy
1
␶
冕 冕 兿 ␲ 冕 ␲冕 冕
␶
dt
i⫽1,5
0
dE i
2
dE
2
t
⫺⬁
dt 1
dxdy
⫻e i(E⫺E 1 )t e i(E 5 ⫺E)t 1 e i(E 1 ⫺E 2 ⫺E 3 )x e i(E 3 ⫺E 4 ⫺E 5 )y ,
where
V␣ 共 E 兲 ⫽ ␲ V ␣ 关 e i ␾ ␣ ␦ 共 E ⫹ 兲 ⫹e ⫺i ␾ ␣ ␦ 共 E ⫺ 兲兴 .
共A4兲
Integrating over x and y yields
B⫽
1
␶
冕 冕 兿 ␲ 冕 ␲冕
␶
dt
i⫽1,5
0
dE i
2
dE
2
t
⫺⬁
dt 1 F0 共 E 兲
⫻F1 共 E 1 兲 V␣ 共 E 2 兲 F2 共 E 3 兲 V␤ 共 E 4 兲 F3 共 E 5 兲
⫻e i(E⫺E 1 )t e i(E 5 ⫺E)t 1 共 2 ␲ 兲 2 ␦ 共 E 1 ⫺E 2 ⫺E 3 兲
⫻ ␦ 共 E 5 ⫺E 4 ⫺E 5 兲 .
Integrating over t 1 ,t and using Eq. 共A4兲, we have
B⫽
冕
dE 2 dE 4 dE 5
2␲ 2␲ 2␲
冕
dE
F0 共 E 兲
F 共 E ⫹E 4
2 ␲ i 关 E 5 ⫺E⫺i ␦ 兴 1 2
⫹E 5 兲 V␣ 共 E 2 兲 F2 共 E 4 ⫹E 5 兲 V␤ 共 E 4 兲 F3 共 E 5 兲 ␦ 共 E 2 ⫹E 4 兲
⫽
V ␣V ␤
4
冕
dE 5 dE
F0 共 E 兲
F 共 E 兲关 F 共 E
2 ␲ 2 ␲ i 关 E 5 ⫺E⫺i ␦ 兴 1 5 2 5
⫹ ␻ 兲 e i⌬ ␤␣ ⫹e ⫺i⌬ ␤␣ F2 共 E 5 ⫺ ␻ 兲兴 F3 共 E 5 兲 .
冕
共A1兲
dE
F0 共 E 兲
⫽2 ␲ F0 共 E 5 兲
i 关 E 5 ⫺E⫺i ␦ 兴
共A5兲
and we thus obtain Eq. 共A1兲. For case 共ii兲 G is analytic in the
upper half plane, we have
where
G⬅F1 V␣ F2 V␤ F3
1
␶
ÃF0 共 E 兲 F1 共 E 1 兲 V␣ 共 E 2 兲 F2 共 E 3 兲 V␤ 共 E 4 兲 F3 共 E 5 兲
dE
Tr兵 F0 共 E 兲 F1 共 E 兲关 F2 共 E ⫹ 兲 e i⌬ ␤␣
2
⫹e ⫺i⌬ ␤␣ F2 共 E ⫺ 兲兴 F3 共 E 兲 其 ,
共A3兲
Now we look at Eq. 共33兲, where there are two cases: 共i兲 F0
(2)⬍
⫽⌺a and G⫽G11
关see Eqs. 共A2兲 and 共36兲兴 and 共ii兲 F0
(2)r
⬍
⫽⌺ and G⫽G11
. In case 共i兲 since F0 is analytic in the
lower half plane, we use the theorem of residue to obtain
Now we show that
1
␶
dEe ⫺iEt F共 E 兲
⫻ 关 F1 共 t⫺x 兲 V ␣ 共 x 兲 F2 共 x⫺y 兲 V ␤ 共 y 兲 F3 共 y⫺t 1 兲兴
We gratefully acknowledge support of a RGC grant from
the SAR Government of Hong Kong under Grant No. HKU
7091/01P and a CRCG grant from the University of Hong
Kong.
B⬅
冕
we obtain
ACKNOWLEDGMENTS
APPENDIX
1
2␲
冕
共A2兲
dE 5
so Eq. 共A1兲 remains.
205327-8
G共 E 5 兲
⫽2 ␲ G共 E 兲 ,
i 关 E 5 ⫺E⫺i ␦ 兴
共A6兲
PHYSICAL REVIEW B 66, 205327 共2002兲
SPIN-POLARIZED PARAMETRIC PUMPING: THEORY . . .
*Electronic address: [email protected]
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For the system of Fe/Ge/Fe with a⫽100A, the energy unit is E
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on the pumping frequency ␻ . If ␻ ⫽100 MHz, then the pumped
current is 1.6⫻10⫺11 A. For pumped current at finite frequency,
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