Charge and spin pumping through a double quantum dot

Roman-Pascal Riwar and Janine SplettstoesserInstitut für Theorie der Statistischen Physik, RWTH Aachen University, D-52056 Aachen, Germany

and JARA-Fundamentals of Future Information Technology , Germany�Received 28 May 2010; revised manuscript received 24 September 2010; published 8 November 2010�

We calculate adiabatic charge and spin pumping through a serial double quantum dot with strong Coulombinteraction, coupled to normal metal or ferromagnetic contacts. We use a real-time diagrammatic approach inthe regime of weak coupling to the reservoirs. In the case of weak interdot tunnel coupling we investigate theinfluence of tunnel-induced renormalization effects due to charge fluctuations on the pumped charge and spin.We show that tunneling through thermally excited states can play an important role in the strong interdotcoupling regime. In particular, for ferromagnetic contacts, both effects enable the generation of pure spincurrents. Furthermore they can lead to an inverted spin-valve effect or even to the inversion of the transportdirection going along with a diverging tunneling magnetoresistance.

DOI: 10.1103/PhysRevB.82.205308 PACS number�s�: 72.25.�b, 73.23.Hk

I. INTRODUCTION

Pumping through mesoscopic devices is realized in theabsence of an external bias by the periodic variation of cer-tain system parameters in time, thus creating directed trans-port of electrons from one contact to the other. When thisvariation is slow compared to the characteristic lifetime ofthe electronic states on the device, one speaks of adiabaticpumping.1–6 In recent years, experiments on quantizedpumping were performed, aiming at the realization of aquantum standard for the current or for the repeated initial-ization of coherent quantum states, useful for quantumoperations.7–9 Fewer experimental realizations dealt with therole of quantum interference effects giving rise to a directedcurrent due to time-dependent periodic fields.10,11 Further-more, when the system through which the charge is pumpedis small, as say, quantum dots, the Coulomb interaction can-not be neglected because of the small capacitance of thesystem. This impact of electron-electron interactions on adia-batic pumping through quantum dots, quantum wires, andmetallic islands has lately been considered in theory.12–21

Particularly, charge pumping can be uniquely due to Cou-lomb interactions in the nonlinear regime22 or due tointeraction-induced renormalization effects.23

Adiabatic pumping requires the time-dependent modula-tion of at least two of the system’s parameters. In the workpresented here we are interested in charge and spin pumpingthrough a serially coupled double dot, where the energy lev-els of both quantum dots can be independently varied bytime-dependent gate voltages. This was so far examined fornoninteracting systems24,25 with normal conducting leads,and further studies were performed in the high-frequencyregime.26–28 The study of double dots is particularly interest-ing due to their complex internal spectrum: the eigenstates ofthe double dot are differently coupled to the leads, and thesecouplings are effectively energy-level dependent. This resultsin new effects for pumping, as demonstrated in this paper,which are controllable through the mere modulation of thebare dot energy spectrum. Importantly here, this choice oftwo independently tunable parameters, i.e., the spectra of thetwo dots, has an easier experimental access than any avail-

able choice of parameters in a single dot. The fact that adia-batic pumping acts as a useful spectroscopy tool,22 revealingsystem specific features which are not accessible through astatic measurement, establishes one of our motivations tostudy adiabatic pumping through interacting double dots invarious regimes and setups.

Previous studies on transport through a double-dot systemwith Coulomb interaction in the static regime revealedcharge fluctuation effects, related to the interplay of thedouble dot’s internal structure and the reservoirs,29,30 anddouble dots were studied in an interferometer setup,31 toname a few examples. Lately double dots have been exten-sively proposed and used as measurement devices: the read-out of spin properties is discussed in Ref. 32 and referencestherein, the Pauli spin blockade was used to investigate re-laxation times33 where the influence of spin-orbit couplingand the role of nuclear spins have been treated. Furthermore,a double quantum dot device can act as a noise detector.34 InRef. 35 transitions in the double-dot spectrum due to abosonic environment were studied.

In addition to the charge degree of freedom also the spindegree of freedom of the electrons plays an important role intransport, in particular, in the context of spintronics36 andapplications in quantum computation.37 The spin-valveeffect38–40 in systems containing differently polarized ferro-magnets and its tunability, e.g., by a gate control41 are there-fore of interest. In the study of quantum dot spin valves42–46

the effect of Coulomb interaction and the properties of thedot spectrum were taken into account.

In the field of pumping, spin-dependent charge transportand spin transport have been studied extensively in the high-frequency regime.27,47 Adiabatic spin pumping has been ex-plored in both the presence of spin-orbit coupling,48 and fi-nite magnetic fields,49 as well as for a quantum dot attachedto ferromagnetic contacts.50 Recently adiabatic spin pumpingin a magnetic wire with domain walls was investigated.51

Adiabatic pumping through double dots in the presence ofspin-orbit coupling can be used to study the spin dynamics ina Pauli-blockade configuration.25 Also charge and spin in fer-romagnetic hybrid structures pumped by magnetization dy-namics have been investigated in detail.52 An experimentalrealization of a spin pump used the modulation of a single

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quantum dot in the presence of a Zeeman field.11

In this paper we study a double-dot system with onsiteCoulomb interaction contacted weakly to electronic reser-voirs. We consider the situations where both leads are normalconducting �N-DD-N�, see Fig. 1, where only one of thereservoirs is normal conducting while the other is replacedby a ferromagnetic contact �N-DD-F�, and finally the casewhere both contacts are ferromagnetic �F-DD-F�. We inves-tigate the effects of quantum charge fluctuations and the im-pact of different effective coupling to the hybridized double-dot states on the pumped charge. Based on these effects thepumped charge shows characteristic sign changes.

Also in the presence of spin-polarized leads, these twoeffects matter. With respect to an earlier work on pumpingthrough a single interacting quantum dot,50 the present studyof a double dot reveals a number of strikingly different ef-fects, such as pure spin currents and the inversion of thetransport direction due to polarized leads only.

We use a real-time diagrammatic approach,53 extended tothe adiabatic regime,23 to calculate the pumped charge andspin, taking into account Coulomb interactions. For the in-terdot hopping we treat both the case of weak and strongcouplings.

This paper is organized as follows: we introduce themodel and the formalism used for our theoretical investiga-tions in Sec. II. In the following we present our results forcharge pumping through a double dot coupled to normal con-ductor leads �Sec. III�. We discuss spin pumping when one ofthe leads is spin polarized in Sec. IV. In the last part weconsider the transport behavior in the presence of two ferro-magnetic leads with arbitrary polarization angle �Sec. V�. Weset �=c=1 for the rest of this paper.

II. MODEL AND FORMALISM

A. Hamiltonian

We consider two single-level spin-degenerate quantumdots coupled in series to each other and tunnel coupled to aleft �L� and a right �R� lead, as depicted in Fig. 1. The Hamil-tonian of this system is given by

H = Hdd�t� + ��=L,R

H� + Htunnel. �1�

The Hamiltonian of the double dot is explicitly time depen-dent,

Hdd�t� = ��=L,R

���t�n� + UnLnR + U��nL↑nL↓ + nR↑nR↓�

−�

2 ��=↑,↓

�dL�† dR� + dR�

† dL�� . �2�

The single level ���t� of each dot �=L,R is modulated byperiodically time-dependent gate voltages, with a frequency� and a respective phase shift. We define the number opera-tor for spin �= ↑ ,↓ on dot � as n��=d��

† d�� and the totalnumber operator of dot � as n�=��n��, where d��

† �d���creates �annihilates� an electron with spin � on the dot �.Electrons on each of the dots are subject to on-site Coulombinteraction U� and to Coulomb interaction with electrons onthe neighboring dot U. The charging energies are related tothe capacitances of the dots and are taken into account withina constant interaction model, see, for example, Ref. 54. Hop-ping from one dot to the other occurs with the interdot cou-pling amplitude −� /2, where � is taken to be real and posi-tive. It is useful to define the mean double-dot energy

E=E�t�= ��L�t�+�R�t�� /2¬ E+�E�t� as well as the level dif-ference �=��t�=�L�t�−�R�t�¬ �+���t�, separating eachquantity into a time-averaged and a time-dependent part. Wewill later apply the convention of omitting the time argumentand denoting time-averaged quantities by a bar also for func-tions containing these parameters. In the following we takeU� to be the largest energy scale, and thus every dot is atmost singly occupied, excluding exchange coupling.

The noninteracting electrons in the left and right leads aredescribed by

H� = �k�

�kc�k�† c�k� �3�

with the creation �annihilation� operators c�k�† �c�k�� of an

electron with spin � and momentum k in lead �. We willinclude both cases of either ferromagnetic or normal conduc-tor leads. We assume the leads to have equal electrochemicalpotentials, L=R, not influenced by the time-dependentgate voltages. The effect of a time-dependent bias is dis-cussed in Ref. 22. The tunnel Hamiltonian describing thedot-lead coupling is given as

Htunnel = ��=L,R

�k,�

V��c�k�† d�� + H.c.� �4�

with the tunnel matrix elements VL and VR, which do notdepend on momentum or spin. The tunnel rates are definedas �=2��V��2�� for �=L,R. We define as the sum of theleft and right tunnel rates, =L+R.We assume all baretunneling rates ,� to be time independent. We will see lateron that the tunneling rates to hybrid dot states become effec-tively time-dependent via the time-dependent dot energy lev-els. Both reservoirs are taken in the wideband limit wheretheir densities of states �� are energy independent. We willlater consider ferromagnetic leads by introducing spin-dependent densities of states in the respective lead �=L,R.The technical details for the treatment of ferromagnetic leadswill be discussed in Sec. V.

(t)Lε

(t)ε R

L R

∆ ΓRΓL

FIG. 1. Schematic picture of the system. The left and right dotsare coupled to the electron reservoirs L and R, with equal electro-chemical potentials, by barriers with tunnel coupling strength L

and R, respectively. The coupling between the single-level quan-tum dots is given by � and the two dots have time-dependent en-ergy levels �L�t� and �R�t�.

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B. Real-time diagrammatic approach

We describe the system by the reduced density matrix ofthe double dot; the degrees of freedom of the leads, taking upthe role of baths, are traced out. We write the accessiblestates on the double dot as � �. The time evolution of thedensity matrix P 1

2 = � 2��dd� 1� is captured in the generalizedmaster equation �kinetic equation�

d

dtP 1

2�t� = − i � 1�, 2�

L 1, 1� 2, 2��t�P

1� 2��t�

+ � 1�, 2�

−�

t

dt�W 1, 1� 2, 2��t,t��P

1� 2��t�� . �5�

The first part represents the interdot dynamics via aLiouville superoperator, where we define thematrix elements � 2�L�t��dd�t�� 1�= � 2��Hdd�t� ,�dd�t��� 1�=� 1�, 2�

L 1, 1� 2, 2��t�P

1� 2��t�. The kernel W

1, 1� 2, 2� in the second part of

Eq. �5� incorporates the lead-dot tunneling transitions be-tween states 1� and 2� at time t�, and states 1 and 2 at timet and is calculated using a real-time diagrammaticapproach.53 We write the diagonal and off-diagonalelements of the density matrix in vector form,

P= �P �� = � , P

�� � ��, enabling a matrix representationfor the kernel W, and likewise for the Liouvillian L. Thegeneralized master equation, Eq. �5�, contains the full timedependence of the problem, which is affected by the time-dependent dot energy levels. In the limit when the lifetime ofthe double-dot states is much larger than the time scale givenby the variation of the parameters, we can perform an adia-batic approximation along the lines of Ref. 23. One firstcarries out a Taylor expansion, P�t��=� j�t�− t� j / j ! dj

dtj P�t�,around time t of the matrix elements of the reduced densitymatrix in the integrand of Eq. �5�, accounting for a finitememory of the kernel. In addition, the internal time depen-dence of the kernel via the time-dependent parameters istaken into account up to first order in �. The adiabatic ex-pansion of the kernel contains an instantaneous part �i�,where all parameters are fixed to their value at timet, X���→X�t�. It further contains a first-order correction term�a� which is found by considering X���→X�t�+ ��− t� dX

d� ��=tin the evaluation of the kernel and by taking into accountsystematically first-order terms in the time derivatives. Thisyields the two terms of the expansion

W�t,t�� → Wt�i��t − t�� + Wt

�a��t − t�� . �6�

The resulting parametric time dependence is indicated by thesubscript t. With this, a set of equations for the instantaneousand the adiabatic part, Pt

�i� and Pt�a�, of the reduced density

matrix is constructed. The instantaneous contribution of thedensity matrix fulfills the stationary generalized master equa-tion with system parameters frozen to values at time t. Theadiabatic contribution accounts for the fact that the actualdensity matrix slightly lags behind its instantaneous value.

We here restrict ourselves to the case of weak coupling.Therefore, on top of the adiabatic expansion for small � weperform a perturbation expansion in the tunnel coupling be-

tween lead and dots. We take into account only terms up tofirst order in the tunnel coupling, which is valid if the broad-ening due to tunneling is smaller than the temperature broad-ening, �kBT, where kB is the Boltzmann constant. We per-form an expansion of the generalized master equation orderby order in , as used in Ref. 53 and subsequent works. Thissystematic expansion ensures that all contributions to a givenorder are taken into account �it may fail in certain situationswhen higher order terms in play a role,55 which is not thecase in the regime considered here�. We will also considerthe case where is not the only small parameter but theintradot coupling � and the level difference � have the samemagnitude. Then coherences become important already infirst order , and the perturbation expansion has to be per-formed taking into account consistently all three small pa-rameters, see Secs. III A and IV A, as well as Appendix A.Gathering the terms of the lowest order in � and the smallparameters of the perturbative expansion, we end up with

0 = �Wt�i,1� − iLt

�i,1��Pt�i,0�, �7a�

d

dtPt

�i,0� = �Wt�i,1� − iLt

�i,1��Pt�a,−1�. �7b�

Here we have introduced the zero-frequency Laplace trans-form Wt

�i�=�−�t dt�Wt

�i��t− t��. The order in the perturbationexpansion is indicated by numbers in the superscript of therespective quantities, i.e., first order in the transition ele-ments W�i,1�, and the superscript of L�i,1� indicates that thedouble-dot Liouvillian enters this equation in the same orderof the Kernel and that its time dependence is instantaneous.The instantaneous as well as the adiabatic reduced density-matrix elements are P�i,0� and P�a,−1�. Note that the adiabaticcorrection to the reduced density matrix starts in minus firstorder in ; this is justified as these terms are proportional to� /, which is a small parameter in the adiabatic regime.Since we do not consider higher orders in the perturbationexpansion, the corresponding superscript for the above dis-cussed quantities is unambiguous, and is suppressed in thefollowing for simplicity.

Equations �7a� and �7b� have a very similar structure andtherefore one can obtain Eq. �7a� from Eq. �7b� by setting theleft-hand side to zero and replacing the reduced density ma-trix by the instantaneous contribution. This will be usedthroughout the remainder of this paper.

A similar expansion is performed for the current throughthe system. Generally, we can give the expression for thepumping current through the double quantum dot as

IL�t� = eeT−�

t

dt�WL�t,t��P�t�� �8�

with the unity charge e. The trace of all contributingelements is performed by applying the vectoreT= �1, . . . ,1 ,0 , . . . ,0�, where the number of ones is equal tothe dimensionality of the reduced Hilbert space. The kernelWL�t , t�� with the zero-frequency Laplace transform WL

takes into account processes which involve an exchange ofcharge between the left lead and the quantum dot subsystem,see Ref. 53. In the case of zero bias, considered in this paper,

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the instantaneous contribution to the current, I�i�, vanishes atall times and the only remaining term is the adiabatic contri-bution, I�a�. In lowest �zeroth� order in the tunnel couplingwe obtain

IL�t� ª IL�a��t� = eeTWt

L�i�Pt�a�, �9�

where we drop the superscript �a� for simplicity. Expressions�8� and �9� represent the tunneling current into the left lead.Note that due to the time dependence, additionally also dis-placement currents are present. In Ref. 56 this has been stud-ied in detail for a single dot. Importantly, the displacementcurrent is a pure ac current and it averages out when inte-grating over one period of the pumping cycle T=2� /�. Inthe following we are mainly interested in the pumped chargeper cycle, hence only the tunneling current plays a role andwe discard the displacement current from now on. The num-ber of pumped electrons is then given by

N =1

e

0

T

dtIL�t� . �10�

We consider the system in a time-dependent steady state, i.e.,no transient behavior; therefore the total charge of the doubledot is conserved after one pumping cycle.57

III. NORMAL CONDUCTOR RESERVOIRS

In this section we discuss the pumped charge through thedouble-dot system in contact with two normal-conductingleads �N-DD-N�.

A. Weak interdot coupling

We first consider the situation where the two dots areweakly coupled to each other, ��kBT, and the energy dif-ference between a singly occupied left and a singly occupiedright dot is small as well, ��kBT, such that both parametersare on the order of the coupling strength, � �. We ac-count for expressions in the master equation up to first orderin these parameters. In this case a rigorous expansion in thecoupling parameters �up to lowest order in both lead-dot anddot-dot couplings� and the level difference � is performed.This system has been considered in the static case in Ref. 29.For the remainder of this section we assume that not only U�but also the charging energy U is much larger than all otherenergy scales �such as temperature, the modulation fre-quency, and the level difference ��. Therefore the double-dotsystem can only be singly occupied or empty. In this case thestates �L�� and �R��, with the electron with spin �= ↑ ,↓ inthe L or R dot, are almost-degenerate quasieigenstates of thesystem and coherent superpositions of �L�� and �R�� play animportant role, see Appendix A. These coherent superposi-tions are captured in the off-diagonal elements of the reduceddensity matrix of the double-dot system. Even in the absenceof bias, they do not vanish for an asymmetrically coupleddouble dot due to the time dependence of the system. Thevector of the reduced density-matrix elements isP= �P0 , PL↑ , PL↓ , PR↑ , PR↓ , PL↑

R↑ , PL↓R↓ , PR↑

L↑ , PR↓L↓�, where we

write the diagonal elements of the density matrix P ¬P .

We end up with a master equation for the occupation prob-abilities for an empty dot P0 and for a singly occupied dotP1= PL+ PR, where the total occupation with different spinsis PL= PL↑+ PL↓ and PR= PR↑+ PR↓. We find

d

dtP0 = − 2f+�E�P0 + e�z�f−�E�S� +

1

2f−�E�P1 �11�

and P1 is obtained via the probability conservationP0

�i�+ P1�i�=1 and P0

�a�+ P1�a�=0, for the instantaneous and the

adiabatic parts of the reduced density-matrix elements. Thetunnel coupling asymmetry is given by �= �L−R� / tak-ing values between −1 and 1, where zero is the case of sym-metric coupling to the leads. The Fermi function isf+�E�= 1

e�E+1and f−�E�=1− f+�E� with the inverse tempera-

ture �=1 /kBT. The vector e�z projects out the z component ofthe pseudospin vector which captures the off-diagonal ele-ments and the difference in the occupation of the left and theright dots,

S� =1

2� PRL + PL

R

iPRL − iPL

R

PL − PR� . �12�

The dynamics of the pseudospin is described by a Bloch-typeequation

d

dtS� = e�z�� f+�E�P0 −

1

4f−�E�P1� −

1

2f−�E�S� + B� � S� .

�13�

The time evolution of the pseudospin has a contribution dueto the accumulation of the pseudospin z component, i.e., anunbalance of the left-right occupation, cf. first line of Eq.�13�. Accumulation occurs only for asymmetric lead cou-pling. Relaxation of the pseudospin, given by the first con-tribution of the second line of Eq. �13� takes place indepen-dently of the coupling asymmetry. Furthermore, we find aprecession of the pseudospin around an effective magneticfield, via which the internal dynamics of the double dot enter.It is given by

B� = �− �

0

�ren� . �14�

We note that instead of the bare level difference �, a renor-malized one is entering the effective field, given by

�ren = � + �

2���E� . �15�

This expression is explicitly time dependent via � and E. Thefunction � entering the renormalized level difference isgiven by

��E� = Re���1

2+ i

�E

2��� − ��1

2+

��cutoff

2�� , �16�

where � denotes the digamma function. The renormalizationarises from quantum charge fluctuations. It enters the z com-ponent of the effective field, which acts as a Zeeman field for

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the pseudospin. This Zeeman field affects the pseudospin dy-namics through a precession around the z axis, thereby cou-pling to the x and y components of the pseudospin, whicharise due to coherent superpositions of �L�� and �R��. Thelevel renormalization is a pure Coulomb interaction effect,i.e., it vanishes for U=U�=0. Furthermore, the renormaliza-tion is zero for symmetric coupling �=0. Here, Coulombinteractions are large and �cutoff provides the cutoff energy.The full expression for the renormalization of � for finiteinterdot and intradot Coulomb interaction is found in Ref.29. The effective field due to the interdot tunneling lies in thex ,y plane which is spanned by the pseudospin contributiondue to the coherent superpositions of �L�� and �R��.

The instantaneous solution of the master equation, Eqs.�11� and �13�, is given by Boltzmann distributions. The av-erage instantaneous occupation number on the double dot�n��i�=0· P0+1 · P1 is given by

�n��i� =4e−�E

1 + 4e−�E . �17�

The factor four stems from the total fourfold degeneracy ofthe singly occupied state. For the same reason, we also have

S� �i��t� = 0 �18�

in lowest order in the tunnel couplings and � and the leveldifference � �this holds as long as no bias voltage is present�.As opposed to the instantaneous solution, the adiabatic cor-rection of the pseudospin S� �a,−1� does in general not vanish.This is due to an occupation difference in pseudospin space,introduced by the time dependence and asymmetric couplingto the left and right leads, ��0. Therefore, in the case ofsymmetric dot-lead coupling, the adiabatic correction for thepseudospin also vanishes. For the adiabatic current we findfrom Eq. �9�,

IL�t� =e

2

d�n��i�

dt �1 +��2

�1 − �2��1

42�f−�E��2 + �ren

2 � + �2� .

�19�

Note that the parameters � and E depend on time and that thepumping current is proportional to the time derivative of theoccupation number, depending on E. Therefore, a necessaryprerequisite for a nonzero adiabatic current is an explicitlytime-dependent E. Importantly, also the prefactor of d

dt �n��i�

is in general time dependent. This is the necessary conditionfor a nonvanishing average pumped charge.58

If �=0 we find that the time-dependent current througheach contact �=L,R is given by I��t�= e

2ddt �n��i�. Therefore

the currents injected into the left and the right leads are equalat any time and the pumped charge per cycle vanishes; it istherefore directly sensitive to the coupling asymmetry.

The pumped charge is obtained from Eq. �10�. We areinterested in the regime of bilinear response for the modula-tion, which is valid if the pumped charge per infinitesimalarea in parameter space varies little within the area enclosedby the cycle. Then the pumped charge is proportional to the

cycle area, A=�0Tdt���E. We find the number of pumped

charges N per area in parameter space as a function of the

average quantities � and E,

N

A=

− ��1 − �2��ren�2

��1 − �2���

2f−�E��2

+ �ren2 � + �2�2

d�n��i�

dE. �20�

The pumped charge has a peak when E is close to resonanceand fulfills the relations

N��ren� = − N�− �ren� , �21a�

N��� = − N�− �� . �21b�

Inverting both parameters consequently maps the function Nonto itself again. As discussed above, the pumped chargevanishes if �=0, as well as at �= �1 when one of the leadsis completely decoupled. Since the pumped charge also van-ishes if �ren=0 �see Fig. 2�, the level renormalization can bedirectly read out by means of pumping through the doubledot, when scanning through the time-averaged left-right leveldifference �. This important property occurs due to the fol-lowing reason: the prefactor of d

dt �n��i� is an even function ofthe renormalized difference of the left and right level posi-tion �ren�t�; therefore in the limit of small pumping ampli-tudes �bilinear response, i.e., only the time dependence of �matters in the prefactor� a sign change in the average �ren hasthe same effect as a shift of the modulation by �. Doing theaverage over one pumping period, the transport direction istherefore reversed.

The level renormalization due to quantum charge fluctua-tions, which can be measured in the pumped charge, is dis-tinguishable from possible level renormalization effects dueto an energy-dependent density of states by its temperaturedependence, which is logarithmic for large kBT.

B. Strong interdot coupling

A finite interdot coupling amplitude � leads to a hybrid-ization of the energy levels of the right and the left dots. Inthe limit of strong coupling between the two dots, ��, we

�1.0 �0.5 0.5 1.0 1.5Ε

�

�0.015

�0.010

�0.005

0.005

0.010

0.015

N �A��2�

εren = 0

FIG. 2. �Color online� Plot of the pumped charge N with respect

to the mean bare level difference � for E=1.5kBT and asymmetrictunnel coupling, �=1 /2. The node N=0 occurs when �ren=0. Theother parameters are �=, kBT=2, and �cutoff=50kBT.

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write the single-particle part of the double-dot Hamiltonianin terms of bonding �b� and antibonding �a� states,������t�d��

† d��, where �=b,a and �= ↑ ,↓. The energy ofthe bonding and antibonding state is given by

�b/a = E �1

2��2 + �2. �22�

The corresponding operators creating �annihilating� an elec-tron in the bonding and the antibonding states are related tothe ones creating an electron in the left or the right dot by theequality

dL/R† =

1�2�1 �

�

��2 + �2db

† �1�2�1 �

�

��2 + �2da

†,

�23�

where we omitted the spin indices for simplicity. The bond-ing �b� and antibonding �a� states have an energy differencemuch larger than the level broadening. This means that theinterdot dynamics is much faster than the dot-lead hopping.Therefore, coherent superpositions of bonding and antibond-ing states are suppressed in lowest order in the tunneling andno dynamics of the off-diagonal elements of the reduceddensity matrix have to be considered �in contrast to the pre-vious case discussed in Sec. III A�. We now take U to befinite and we therefore also consider the occupation of theleft and right dots with one electron each. We still assumelarge on-site Coulomb interaction, inhibiting double occupa-tion of each single dot. As a consequence, the doubly occu-pied �fourfold degenerate� states are �L�R���. We can nowwrite the double-dot Hamiltonian � E � �� � in the basis ofthe eigenstates � �. Here the eigenenergy E0 of the empty dot�0� equals zero, the eigenenergy Eb of the spin-degeneratebonding states �b��=db�

† �0� is �b and analogously theeigenenergy Ea of the spin-degenerate antibonding states�a��=da�

† �0� is �a. Finally the four doubly occupied states�L�R���, with �= ↑ ,↓ and ��= ↑ ,↓, have the eigenenergiesE���=2E+U. The tunnel coupling between the double dotand the leads �=L,R is captured via effective rates for tun-neling through the hybrid states �b� and �a�. These rates areexplicitly time dependent and are given by

�� =1

2�1 − ���

��2 + �2��. �24�

Tunneling through the hybrid single-particle states is denotedby the subscript �=b,a. The difference of the couplingstrengths for the two transport channels is proportional to afactor of � /��2+�2. �This factor is related to the relativeposition of the bonding and antibonding states with respectto the localized states L and R.� In order to bring out thisimportant property we use the following notation: if � isused as a variable rather than a coefficient it takes the value+1 �−1� for b �a�; equally if �=L,R is used as a variable ittakes the values +1 for L and −1 for R. The sum of theserates is �=����. With this, the master equation for theoccupation probabilities P0, P�= P�↑+ P�↓, and Pd=��,��=↑,↓PL�R�� reads

d

dtP0 = �

�=b,a�− 2�f+����P0 + �f−����P�� , �25a�

d

dtP� = 2�f+����P0 + �− �f−���� − 2�f+��� + U��P�

+ �f−��� + U�Pd, �25b�

and Pd is obtained via the probability conservationP0

�i�+ Pb�i�+ Pa

�i�+ Pd�i�=1 and P0

�a�+ Pb�a�+ Pa

�a�+ Pd�a�=0. The in-

stantaneous probabilities are again given by the Boltzmanndistribution.

As described before, in Sec. II B, the instantaneous occu-pation probabilities and their first-order corrections in theadiabatic expansion are found as solutions of this masterequation. The adiabatic current is then calculated straightfor-wardly as

IL�t� = e ��=b,a

L�

�� d

dtP�

�i� +d

dtPd

�i�� . �26�

The current expression is divided into two different contri-butions, IL�t�= IL,b�t�+ IL,a�t�. It is apparent that the dot statetransitions �0�↔ �b� and �a�↔ �d� occur with the rate b. Andlikewise, the transitions �0�↔ �a� and �b�↔ �d� are due totunneling with the rate a.

In bilinear response for small pumping amplitudes, thepumped charge through the double dot becomes

N

A= �

�=b,a

d

d�� L�

�

� d

dE�P�

�i� + Pd�i�� . �27�

The derivative with respect to E of the sum of the probabili-ties is always negative for both the bonding and the anti-bonding contributions. In contrast, the prefactor, namely, thederivative with respect to � of the relative effective coupling,has opposite signs for the different hybrid states, since thetime-dependent part of the effective coupling comes alwayswith opposite signs, respectively, see Eq. �24�.

As a direct result, one finds that the terms Nb and Na,derived from the current separated into IL,b�t� and IL,a�t�,always contribute to the pumped charge per area with oppo-site signs. This result is a central point of our paper; thisdouble-dot feature cannot be provided by a single-dot systemand is of fundamental importance for many of the effectsdiscussed in the following.

In Fig. 3 we plot the pumped charge as a function of the

mean level E in the case of spatial symmetry �=0, �=0 andfor different values of U. Importantly, the finite Coulombinteraction shifts the resonance positions and furthermore en-hances the pumped charge. The two resonant peaks haveopposite signs; they appear when the addition energy to gofrom an empty dot to the ground state of the singly occupieddot is at resonance, �b�0 and when the addition energy togo from the singly occupied ground state to a doubly occu-pied dot is at resonance, �a+U�0, except for the commontemperature-dependent shift due to different charging and de-charging rates. Since the main contribution at �b�0 comesfrom tunneling involving the hybrid state �b� and the main

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contribution at �a+U�0 comes from tunneling involving thehybrid state �a�, the pumped charge has opposite signs at thetwo resonances. The pumped charge from the two contribu-tions does not cancel out for U=0 because of the finite levelspacing. This result agrees with Ref. 24 and 59, where chargepumping through a double dot with noninteracting spinlesselectrons was considered.

If the double-dot parameters are such that the level split-ting ��b− �a� is much larger than kBT, the probability of hav-ing state �a� occupied vanishes. Thus, the transport close tothe single-electron resonance occurs solely via charging anddecharging the single-electron ground state �b�, via the pro-cesses �0�↔ �b�, contributing with rate b. Equally, at theresonance for the transition between singly and doubly occu-pied double dots, only the transitions �b�↔ �d� contributewith rate a.

In the regime when the excited state is still thermallyaccessible, ��b− �a��kBT, also the transitions with the ex-cited state, �0�↔ �a� and �a�↔ �d�, start to contribute. By in-troducing a spatial asymmetry to the double dot, choosing��0 or ��0, the two contributions �for transport involvingthe ground state and transport involving the thermally ex-cited state� are changed by different amounts. In spite of thefact that the larger time-dependent current flow stems fromtransport through the ground state, for the pumped charge,i.e., the time-averaged quantity, the contribution through theground state can become the minor one. This means that bychoosing appropriate double-dot parameters charging and de-charging of the ground state leads to smaller directed trans-port than charging and decharging of the excited state, aunique feature of time-averaged transport due to time-dependent fields. Thereby, we can achieve a sign change inthe pumped charge at one of the resonances, see Fig. 4�a�,�analogously for the other resonance at �a+U�0 a signchange appears for �→−� or �→−��.

We plot the peak height of the pumped charge close to theresonance �b�0 for the full parameter spectrum of � and �in Fig. 4�b�. There, we see that for most parameter configu-rations, the bonding channel is dominant hence the peak ispositive. The blue regions �brighter areas in the lower rightand the upper left� show the parameter regimes where a signchange appears due to the dominant transport through thethermally accessible antibonding state. These sign changes inthe peaks with changing asymmetry will however vanish,once the level splitting becomes significantly larger than kBTand the excited state is no longer thermally accessible. We

also find that the pumped charge is point symmetric withrespect to the parameter � and the bare, time-averaged �,N�� , ��=N�−� ,−��.

IV. SPIN PUMPING THROUGH N-DD-F

We now replace one of the contacts by a ferromagneticlead �we choose the right one�, thereby breaking spin-rotation invariance which enables spin in addition to chargepumping. Spin and charge pumping through a single inter-acting quantum dot in the presence of a ferromagnetic leadhas been studied before;50 spin pumping through a doubledot is particularly promising due to the following reason: thepossible sign reversal of the charge transport which in theweak-coupling regime is taking place at the renormalizedlevel difference �ren being zero, and which in the strong-coupling case is due to thermal accessibility of the excitedlevel, is expected to affect the transport for different spindifferently.

We now discuss the representative situation at the reso-nance between empty and singly occupied double dots andtherefore restrict the calculation to infinite Coulomb interac-tions U and U� for the remainder of this paper. In a spin-polarized contact � the density of states is spin dependent,��↑���↓. The spin polarization strength of lead � is definedas

p� =��↑ − ��↓

��↑ + ��↓. �28�

Due to the spin-dependent density of states also the tunnelingrates to the right lead become spin dependent

�15 �10 �5 5EkB T

�0.015

0.015N �A��2�U = 10kBT

U = 5kBT

U = 0

FIG. 3. Plot of the pumped charge N as a function of E fordifferent values for the Coulomb interaction U=0,5kBT ,10kBT�dashed, dotted, and solid�, in the symmetric case �=0, �=0, andthe interdot coupling is �=kBT and kBT=2.

�15 �10 �5 5EkB T

�0.015

0.015N �A��2�

��

�

εb ≈ 0

ε = 0, λ = 0

ε = −kBT, λ = 0.6

(a)

��

�

N[AΓ−2

]Peak height plot

(b)

FIG. 4. �Color online� �a� Plot of N as a function of E for theasymmetric case �blue dashed� and the symmetric case taken fromFig. 3 as reference �black solid�. �b� Map of the peak height of thepumped charge N at the single-electron resonance �dashed verticalline in �a�� in dependence of � and �= �L−R� /. The empty andfilled circles indicate the symmetry configurations taken in �a�. Forall plots the interdot coupling is �=kBT and the Coulomb interac-tion is U=10kBT. The thermal energy is kBT=2.

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�� = �1 + �p���, �29�

where �=+1 for �↑ and �=−1 for �↓ and �=L,R. Thetotal tunneling rate for lead � is �= 1

2 ��↑+�↓�. In thissection only pR is different from zero hence L�=L. Wechoose the polarization axis of the spin in all parts of theN-DD-F system along the axis of the majority spin of theferromagnetic contact and we can study the dynamics ofelectrons with spin up and spin down separately. Due to thespin-dependent tunneling rates, the spin-up and the spin-down channels have different pumping dynamics, leading toa generally nonzero net spin transport. Analogously to Sec.III, we discuss in the following the two limits for weak andstrong interdot couplings. We find that the equations for thespin-resolved pumped charge are formally equivalent to theresults of the unpolarized case, see Sec. III, where one re-places �→��. We will show that the two previously dis-cussed different regimes both lead to a pure pumped spincurrent, relying on different effects.

A. Weak interdot coupling

We first discuss the case of weak interdot coupling, where� �, analogous to Sec. III A. In order to calculate thetotal pumped charge N=N↑+N↓ and the total pumped spinNS=N↑−N↓, we evaluate the number of pumped electronswith spin �. We obtain an expression similar to the unpolar-ized case from Eq. �20�,

N�

A= −

1

2

d�n��i�

dE

���1 − ��2��ren,��2

��1 − ��2���L� + R�

2f−�E��2

+ �ren,�2 � + �2�2 �30�

with the spin-dependent quantities

�� =L� − R�

L� + R�

, �31a�

�ren,� = � +�L� + R����

2���E� . �31b�

Importantly, the coupling asymmetry is spin dependent here,more specifically, for the spin-dependent coupling asymme-tries we always find �↑����↓. This also makes the renor-malized energy-level distance �ren,� spin dependent. As wehave elaborated in Sec. III A, there are nodes in the pumpedcharge whenever the left and right tunnel rates are equal, andwhen the time-averaged renormalized level difference �ren iszero. In analogy, we find nodes for the spin-resolved pumpedcharge N�: zero net transfer of particles with spin � occurs if��=0 and if �ren,�=0.

The spin-resolved pumped charges N↑ and N↓ are plottedin the upper panels of Fig. 5 for different values of the cou-pling asymmetry. Clearly, the number of pumped charges

with spin up and down differs strongly from each other lead-ing to a finite spin transport. If the coupling asymmetry issuch that R↓�L�R↑, then spin-up and spin-down elec-trons are even mostly pumped in opposite directions as isshown in the upper panel of Fig. 5�b�.

The lower panels of Fig. 5 show the total pumped chargeand the total pumped spin. We see that the pumped chargeand spin have different nodes: whenever N↑=N↓ the pumpedspin vanishes, as indicated by the vertical dotted line in Fig.5�a�, while the pumped charge can still be finite. On the otherhand, when N↑=−N↓ the pumped charge vanishes, as indi-cated by the vertical dotted line in Fig. 5�b� while thepumped spin can still be finite. This results in both the pos-sibility to pump charge in the absence of net spin transport,and quite more intriguing, the possibility to pump spin with-out net charge transport by electrical control only. In theregime studied here, pure spin pumping in the absence ofcharge pumping is due to spin-dependent quantum chargefluctuations, induced by Coulomb interactions, resulting indiffering spin-dependent level renormalization effects.

�2 �1 1 2

�0.010�0.005

0.0050.010

�2 �1 1 2

�0.005

0.005

�A��2�a) λ = −0.6

NS

N

N↑

N↓

�2 �1 1 2Ε

�

�0.010�0.005

0.0050.010

�2 �1 1 2Ε

�

�0.005

0.005

�A��2�b) λ = −0.1

NS

N

N↑

N↓

FIG. 5. �Color online� Plot of the pumped charge through theup-spin state N↑ and down-spin state N↓ �upper plot�, and the totalpumped charge N=N↑+N↓ as well as the total pumped spinNS=N↑−N↓ �lower plot�, for different coupling symmetries �a��=−0.6 and �b� �=−0.1. The polarization strength of the right res-

ervoir is pR=0.5. The other parameters are �=, E=1.5kBT, andkBT=2.

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B. Strong interdot coupling

We now turn to the strong interdot coupling regime,��, which is studied in Sec. III B for unpolarized leads.Similar to Eq. �27�, we obtain for the spin-resolved pumpedcharge through the two states �=b,a,

N��

A=

1

2

d

d�� L�

� + �pRR�

�dP��i�

dE. �32�

These quantities depend explicitly on the spin and the state �through which the charge is pumped and therefore all fourquantities are in general expected to be different, leading topossible pure dc charge as well as pure dc spin transport, seeFig. 6�c�.

To explain the origin of different N↑ and N↓ we considerFigs. 6�a� and 6�b�. In Fig. 6�a� we resolve the pumpedcharge for each channel separately, namely, for a spin-upelectron through both hybrid states, Nb↑ and Na↑, and like-wise for a spin-down electron, Nb↓ and Na↓. For the plots inFig. 6 we take �b�0 in the case of strong-coupling asymme-try �=0.6 where we consider � and � such that the excitedstate �antibonding� is thermally accessible. Figure 6�a� showscontributions due to transport through the bonding state�red�, which have always an opposite sign to the chargepumped through the antibonding state �blue�. Since the spinpolarization enters Eq. �24� only as a modification of theprefactor R, the important sign dependence of the time-dependent part of the effective tunnel coupling for differenthybrid channels is not altered. Consequently, as discussed inSec. III B, also the spin-resolved pumped charge has oppo-site signs for different hybrid channels. Whether the chargepumped through the excited, antibonding level can dominate

over the contribution through the bonding state, dependsstrongly on the coupling asymmetry to the leads. In the ex-ample shown here, for the spin-down channel the increasedeffective coupling asymmetry allows for dominant transportthrough the antibonding hybrid state. This leads to a signchange in the pumped charge with spin down N↓ as a func-tion of �, in the limit of a thermally accessible excited state,see Fig. 6�b�. In contrast, there is no configuration in whichthe transport of electrons with spin up takes place prevalentlythrough the excited state since �↑�� is too small to invertthe transport direction. The important result is that N↑ and N↓can have opposite signs for certain values of �. Therefore wefind again the possibility of pure spin pumping while thepumped charge vanishes and vice versa.

V. NONCOLLINEAR FERROMAGNETIC RESERVOIRS

In this section we discuss the pumping characteristicsthrough the double-dot device coupled to two differently po-larized ferromagnetic leads, F-DD-F, as sketched in Fig. 7.We concentrate on the strong interdot coupling limit ��and start from the Hamiltonian discussed in Sec. II A; it isnow important to keep track of the different tunnel couplingfor minority and majority spin of the different reservoirs.

The polarization strength p�, Eq. �28�, and the polariza-tion direction defining the polarization axes n�� are differentfor the two leads �=L,R. We choose the spin quantizationaxis of each lead along the respective magnetization axis, n�Land n�R, where the two vectors enclose an angle �. FollowingRef. 42, we now take a coordinate system such that its basisvectors ex, ey, and ez align with n�L+n�R, n�L−n�R, andn�L�n�R, respectively. For the dots’ spin quantization axis wechoose the z axis of this coordinate system, which is perpen-dicular to the plane spanned by the leads’ polarization vec-tors, see Fig. 7. In this basis, the left and right lead magne-tization vectors, p�L= pLn�L and p�R= pRn�R, are given as

p�L = pL�cos��/2�sin��/2�

0�, p�R = pR� cos��/2�

− sin��/2�0

� . �33�

It is useful to express the tunneling Hamiltonian, Eq. �4�, interms of creation �annihilation� operators for an electron witha spin along the spin quantization axis of the respective sys-tem part. We therefore write

�4 �2 2 4Ε

kB T

0.005

0.010

0.015�4 �2 2 4

Ε

kB T

0.004

0.008

�4 �2 2 4Ε

kB T�0.005

0.005

0.010�A��2�

NS

N

(c)

N↑

N↓

(b)

Nb↑

Nb↓

Na↑

Na↓

(a)

FIG. 6. �Color online� �a� Plot of the pumped charge resolvedfor all four possible channels N��, �=b,a, and �= ↑ ,↓. �b� Spin-resolved pumped charge, N↑ and N↓. �c� The total pumped chargeN=N↑+N↓ as well as the total pumped spin NS=N↑−N↓. For allplots the coupling symmetry is �=0.6, the polarization strength ofthe right reservoir is pR=0.5, and the other parameters are �=kBT,

E=1.5kBT, and kBT=2.

(t)Lε

(t)ε R

pL Rp

∆ ΓRΓL

ϕ2

ϕ2

RppL

z

FIG. 7. A schematic picture of the Hamiltonian with ferromag-netic leads. The magnetization of the leads is given by p�L and p�R.

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Htunnel = ��=L,R

V�

�2�

k

c�k+† �ei��/4d�↑ + e−i��/4d�↓�

+ c�k−† �− ei��/4d�↑ + e−i��/4d�↓�� + H.c., �34�

where c�k�† �c�k�� creates �annihilates� an electron with mo-

mentum k in lead � with majority/minority spin of lead �.Again, we use the notation that whenever � is used as avariable rather than a coefficient, it takes the values +1 for Land −1 for R. In this representation of the tunneling Hamil-tonian it becomes apparent that the diagonal elements of thereduced density-matrix couple to the off-diagonal ones withrespect to the real spin. The reason for this coupling of di-agonal and off-diagonal elements is that tracing out the fer-romagnetic leads, we obtain dynamics for the reduced den-sity matrix of the double dot which do not conserve spin.Therefore, off-diagonal elements of the density matrix haveto be taken into account here, and we consider the density-matrix elements P= �P0 , Pb↑ , Pb↓ , Pa↑ , Pa↓ , Pb↑

b↓ , Pb↓b↑ , Pa↑

a↓ , Pa↓a↑�.

The off-diagonal elements and the difference of the occupa-tion number for different spins are contained in the vector

S�� =1

2� P�↓�↑ + P�↑

�↓

i�P�↓�↑ − P�↑

�↓�P�↑ − P�↓

� . �35�

for �=b,a. The generalized master equation for the elementsof the reduced density matrix reads

d

dtP� = 2 �

�=L,R���f+����P0 − p����f−����S��

−1

2��f−����P�� , �36a�

d

dtS�� = �

�=L,R�p����f+����P0 −

1

2p����f−����P�

− ��f−����S��� + B� � � S��. �36b�

Again P0 is obtained via the probability conservation P0�i�

+ Pb�i�+ Pa

�i�=1 and P0�a�+ Pb

�a�+ Pa�a�=0. The last line in Eq.

�36b� represents a Bloch-type equation for the spin expecta-tion value, which is affected by relaxation and accumulationof spin and by a rotation due to an effective magnetic fieldwhich reads

B� � =1

��

�=L,Rp��������� . �37�

The function � is defined by Eq. �16�. This effective mag-netic field is induced by coherent transitions between spinstates and is different for the bonding and the antibondingchannels. Due to the absence of a bias voltage, we find thatthe instantaneous density-matrix elements do not differ fromthe previous case without magnetization. In particular, wefind that the instantaneous spin expectation value is zero,S��

�i�=0. The adiabatic correction to the spin is different fromzero; nevertheless, in the absence of bias, the adiabatic spin

vector S���a,−1� turns out to be parallel to the effective magnetic

field,50 hence no precession takes place and the effectivemagnetic field does not influence the transport dynamics.

We calculate the pumping current through the double-dotsystem in the presence of arbitrarily polarized ferromagneticleads and find

IL�t� = e ��=b,a

L�

��1 + R�

�p�R − p�L��� �

�2 − 2�� �

2 � d

dtP�

�i� �38�

with the definition of

�� � =L�

p�L +

R�

p�R. �39�

The sign of the correction term to the pumping current withrespect to the nonmagnetic case �second term in Eq. �38��differs depending on the difference in the lead polarizations.This means that the pumping current can be reduced in pres-ence of differently polarized ferromagnetic leads, called thespin-valve effect,38,39 or enhanced, showing an inverted spin-valve effect. Such an observation was made for the pumpingcurrent through a single quantum dot attached to ferromag-netic leads.23 In this case of a single dot, the spin-valve effectfor the pumped charge can only be inverted, when inducing astrong spatial asymmetry regarding the tunnel coupling.Considering a static transport bias, an inverted spin-valveeffect has, for example, been found for a single quantum dotdue to a spin dependence of interfacial phase shifts45 as wellas for a double quantum dot due to charge fluctuations in thenonlinear bias regime.46

Qualitatively very different features for the spin-valve ef-fect are found when pumping through a double dot. We showin the following that this is due to effective spin-dependentasymmetries in the coupling to the different hybrid states.

For the pumped charge per area of the pumping cycle �inbilinear response� we find

Np

A= �

�=b,a

d

d��L�

� − p�L�� �

�2 − 2�� �

2 � d

dEP�

�i�. �40�

We use the abbreviation Np to indicate the pumped charge inpresence of spin polarized leads to be contrasted with thepumped charge N0=N��=0�=N�pL= pR=0�, in the presenceof normal conducting leads �see Eq. �27��. According to Sec.III B we separate the two contributions corresponding topumping through the bonding and antibonding states, con-tributing with opposite signs to the pumped charge,Nb

p /A�0 and Nap /A�0. The total pumped charge in the pres-

ence of polarized leads with respect to the pumped charge forvanishing polarization, Np /N0, depends on the strength of thepolarization of the two leads and on their respective angle �.

The quantity Np /N0 is independent of E; it is shown fordifferent parameter regimes in Fig. 8. As a function of theangle �, the relative pumped charge Np /N0 is 2� periodic,and it equals one if �=0. The stronger the polarization of theleads, the more the functional dependence on � deviatesfrom a cosinelike behavior.47

Three different parameter regimes are distinguished inpanels �a�–�c�, where we choose pL= pR= p. The upper plot,

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Fig. 8�a� shows results for a symmetric double dot, �=0 and�=0, where the usual spin-valve effect is found. The spin-valve effect is most dominant when the left and right polar-ization vectors are in antiparallel alignment, �=�. When thedouble dot is made asymmetric, the spin-valve effect can beinverted, Np�N0, ergo we find an anomalous spin-valve ef-fect, see Fig. 8�b�. In this case, the deviation from the cosinebehavior goes so far that for large polarizations, the maxi-mum effect can even be found at angles away from �. Thisimplies that in the limit p→1, the relative pumped chargeequals one for �=0 and zero elsewhere. The most surprisingresult is that even the transport direction can be inverted ascompared to the normal case, leading to Np /N0�0, see Fig.8�c�. Also here, there is a shift of the maximum effect awayfrom �=� as in Fig. 8�b�, which becomes apparent for largepolarizations and which is not shown in this figure.

TMR of the pumped charge

To classify these three cases in a more convenient way wedefine in analogy to the static case38,46 a tunneling magne-toresistance �TMR� for pumping

TMR =N0 − Np

N0 + Np . �41�

Values between 0 and 1 represent the case where transport issuppressed due to the spin-polarized leads, see the examplein Fig. 8�a�. Values between −1 and 0 on the other hand

indicate enhanced transport �reversed spin-valve effect�, seethe example in Fig. 8�b�. In the more exotic case, wheretransport is reversed, the absolute value of the TMR is biggerthan one and it diverges when Np=−N0. The TMR is plottedas a function of the double-dot parameters in Fig. 9, wherewe show the case of antiparallel alignment of the magneti-zations of the two leads, �=� and p= pL= pR. The threedifferent regimes are represented by a color scale: the redarea denotes the ordinary spin-valve effect �0�TMR�1�and in the blue area the spin-valve effect is reversed�−1�TMR�0�. The parameter area in which the chargetransport itself is inverted ��TMR��1� is drawn in yellow,and the black line dividing the yellow area is indicatingNp=−N0. For comparison, the parameter choices for thethree plots of Fig. 8 are indicated by the correspondingletters.

The inverted spin-valve effect, −1�TMR�0, can be eas-ily explained for this case of �=�. Now N↑, the number ofpumped electrons with spin up, relates, e.g., to the majorityspin in the left lead and the minority spin in the right lead.The shift between N↑ and N↓, similar to what is shown in Fig.6, depends on the polarization strength, in particular, the shiftis zero for p=0. This can lead to an increase or a decrease inthe total pumped charge as a function of � and �, implyingthat the total pumped charge in the presence of polarizedleads, Np, can be larger than for normal conducting leads, N0.This effect can occur independently of the accessibility ofthe excited �antibonding� state, as it does not rely on a signchange in N↑ or N↓.

When ��b− �a��kBT parameter regimes exist where thetransport changes direction, i.e., Np and N0 have differentsigns �as depicted in Fig. 8�c��. As we have stated before, thepolarization changes the transport through the spin-dependent bonding and antibonding channels differently.Consequently the transport through the excited antibondingstate becomes dominant at different parameter configura-tions, depending on the polarization of the leads. Thereforewe can obtain different nodes for Nb

0+Na0=0 �contours sepa-

0.4

0.6

0.8

1.0

(a) ε = 0, λ = 0

Np/N0 Normal spin valve:

0.9

1.0

1.1

1.2

(b) ε = −kBT , λ = 0

p = 0.7

p = 0.4

p = 0.2

Anomalous spin valve:

�3�2�101

(c) ε = −kBT , λ = 0.6

0 π 2π 3π 4π

ϕ

Inverted transport:

FIG. 8. Plot of the ratio of the pumped charge for polarizedleads to the pumped charge for nonmagnetic contacts Np /N0, as afunction of the magnetization angle � for increasing polarizationstrength p= pL= pR= 0.2,0.4,0.7�. �a� Plot for the symmetric case�=0 and �=0. �b� Plot for the case of �=−kBT and �=0. �c� Plot forthe strongly asymmetric case �=−kBT and �=0.6. For all threeplots we take �=kBT and kBT=2.

FIG. 9. �Color online� Overview map of the tunneling magne-toresistance of the pumping system for p=0.7, �=�, and �=kBT;

the TMR is independent of E. Red represents the spin-valve effect,blue its inversion, and the yellow area denotes the region of in-verted charge transport �TMR��1. The black contours separatingtwo yellow areas represent the divergence of TMR where Np=−N0. Any contour separating blue �red� and yellow represents N0

=0 �Np=0�. The black dots indicate the symmetry configurationschosen in Figs. 8�a�–8�c�.

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rating yellow and blue areas in Fig. 9� and Nbp+Na

p=0 �con-tours separating yellow and red areas in Fig. 9�. In betweenthese nodes, the charge transport is reversed. The regions ofinverted transport can be enlarged by increasing polarizationstrengths p.

We compare this with the tunneling magnetoresistance fora double-dot system subject to a static bias in the linear-response regime. All parameters are therefore time indepen-dent. We define the static tunneling magnetoresistance as

TMRstat =G0 − Gp

G0 + Gp , �42�

via the linear conductance in the presence of normal leads,G0, and in the presence of spin-polarized leads, Gp. Theirexplicit form is given in Appendix B. One can show that forall possible parameters in the regime of strong interdot cou-pling

0 � TMRstat � 1. �43�

This means that the spin-valve effect in the static linear-response regime in lowest order in the tunnel coupling can-not be reversed.

An inverted spin-valve effect was found even withouttime-dependent parameters for spin-dependent tunneling45

and for nonlinear bias.46 In contrast, the charge transportinversion, which we find for the pumped charge, correspond-ing to �TMRpump��1, is not expected in a static situation,because it is induced by thermal transport through the ex-cited state. In the static linear response, thermal transportthrough an excited state is always a minor correction totransport through the ground state. In contrast, we showed inthis section that due to the time dependence of the systemparameters one can realize certain configurations �tuned bythe double dot’s parameters � and �� in which the contribu-tion from transport through the ground state is suppressed inthe time-averaged pumped charge.

We finally mention results for the TMR in the weak inter-dot coupling regime, which is not discussed in detail here.Also in this case the TMR takes values in the whole range�−� ,��, which can be shown directly from a study of theN-DD-F system in this regime: already if only one lead ispolarized, we find a reversed spin-valve effect as well asreversed charge transport in the presence of polarized leads,due to polarization dependent quantum charge fluctuations.

A multiple quantum dot in a carbon nanotube in contactwith ferromagnetic leads was recently realizedexperimentally60—a system which is at the basis of our studyin Secs. IV and V. The application of time-dependent fieldsto such a setup would therefore allow to experimentallyverify our results regarding spin-dependent transport.

VI. CONCLUSION

We have investigated charge and spin pumping through aserially coupled double quantum dot in both the weak andstrong interdot coupling regimes, contacted to either normalor ferromagnetic reservoirs. We take into account Coulombinteraction with the only restriction that double occupation of

a single dot is excluded. When the two dots are weaklycoupled, interaction-induced quantum charge fluctuations oc-cur, resulting in a renormalized level difference. We showthat this level renormalization can be directly measured as anode in the pumped charge. In the case of strong interdotcoupling, we find that hybridized double-dot states contrib-ute to the transport with opposite sign. If their level spacingis within thermal reach, the time-averaged transport of theexcited channel can outmatch the ground-state one. This re-sults in a sign change in the averaged pumped charge,uniquely found in the presence of time-dependent fields.

Including one ferromagnetic contact, both mechanisms,i.e., quantum charge fluctuations in the weak interdot cou-pling and thermal accessibility of the excited state in thestrong interdot coupling, enable the possibility to pump spinin the absence of net charge transport.

We finally studied the spin-valve effect for noncollinearferromagnetic contacts. Depending on the system’s param-eters, we find a normal as well as an anomalous spin-valveeffect, and even an actual change in the transport directioncan be observed, leading to a diverging tunneling magnetore-sistance.

ACKNOWLEDGMENTS

The authors would like to thank A. Cottet, M. Governale,G. Güntherodt, J. König, T. Kontos, F. Reckermann, and M.R. Wegewijs for helpful comments and valuable discussions.We acknowledge financial support by the Ministry of Inno-vation NRW.

APPENDIX A: PERTURBATION EXPANSION FOR WEAKINTERDOT COUPLING

The single-electron eigenstates of the isolated double-dotsystem are given by the bonding and the antibonding states,see Sec. III B. If the level spacing between these two states islarge with respect to the coupling to the leads , off-diagonalelements of the reduced density matrix become fundamen-tally important only in second order in �Ref. 55� but aresuppressed in lowest order in the tunnel coupling, �kBT.The present work is limited to the case of weak coupling,where a description up to first order in is justified. In Secs.III A and IV A, we are interested in the regime where thesplitting of the two levels is small and coherent oscillationsbetween the related states play an important role already infirst order in . Therefore, the interplay of the internal dy-namics and the dynamics due to the coupling to the leads ismost interesting when all three parameters are of the sameorder, � � , and thus have to be considered on the samelevel of approximation.29

In this regime, when coupling between the left and theright dots is therefore small, ��, it is intuitive to describethe double-dot system by the states �L�� and �R��. These arethe eigenstates of the fully decoupled single dots, and thecorresponding basis does not represent an exact eigenbasis ofthe isolated double-dot system.

The localized states differ from the hybrid bonding andantibonding states only when � is taken into account in at

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least first order. On the other hand the Kernel W, entering thelast term of the generalized master equation, Eq. �5�, de-scribes tunneling to the leads, and therefore starts in firstorder in . When treating the parameters �, � and on thesame footing, for the calculation of the Kernel, the eigen-states of the double-dot system are therefore taken into ac-count in zeroth order in the small parameters and equal �L��and �R�� within the limits of the approximation. This addi-tionally justifies the description of the dynamics of the sys-tem in terms of the localized eigenstates, �L�� and �R��.

APPENDIX B: STATIC TMR

We calculate the conductance for the static case �see Eq.�42�� in the linear-response regime. We focus on strong in-terdot coupling �� and infinite Coulomb interactions Uand U� in order to compare to the results found in Sec. V.The linear conductance in the presence of polarized leads,Gp, is obtained by solving the stationary master equation, Eq.�7a� for leads with different Fermi energies and by calculat-ing the stationary current response from Eq. �9�. We find forthe conductance

Gp = 2e2� ��=b,a

f+����L�R�

��1 −

L�R�

�2 − 2�� �

2

�C�

2�f−�����2 + ��p�L − p�R�B� ��2

�2�f−�����2 + B� �

2 �P0 �B1�

with

C = �p�L − p�R�2 + �p�Lp�R�2 − pL2 pR

2 �B2�

and the static probability of having an empty dot in lowestorder in tunneling P0= �1+2e−��b +2e−��a�−1. The linearconductance for unpolarized leads G0 is found by settingthe polarizations to zero, making the correction term inthe parentheses vanish. We insert the linear conductancesfound from Eq. �B1� into Eq. �42� in order to findTMRstat. The static tunneling magnetoresistance depends

on E in contrast to the pumping TMR. Furthermore,due to the applied bias, the effective magnetic field entersin the static case, which it does not in the case ofpumping for vanishing bias. In contrast to the pumpingcase, TMRstat takes always values within in the interval�0,1�.

1 M. Büttiker, H. Thomas, and A. Prêtre, Z. Phys. B: Condens.Matter 94, 133 �1994�.

2 P. W. Brouwer, Phys. Rev. B 58, R10135 �1998�.3 F. Zhou, B. Spivak, and B. Altshuler, Phys. Rev. Lett. 82, 608

�1999�.4 Y. Makhlin and A. D. Mirlin, Phys. Rev. Lett. 87, 276803

�2001�.5 M. Moskalets and M. Büttiker, Phys. Rev. B 64, 201305�R�

�2001�; 66, 205320 �2002�.6 O. Entin-Wohlman, A. Aharony, and Y. Levinson, Phys. Rev. B

65, 195411 �2002�.7 H. Pothier, P. Lafarge, C. Urbina, D. Esteve, and M. H. Devoret,

Europhys. Lett. 17, 249 �1992�.8 N. E. Fletcher, J. Ebbecke, T. J. B. M. Janssen, F. J. Ahlers, M.

Pepper, H. E. Beere, and D. A. Ritchie, Phys. Rev. B 68, 245310�2003�; J. Ebbecke, N. E. Fletcher, T. J. B. M. Janssen, H. E.Beere, D. A. Ritchie, and M. Pepper, ibid. 72, 121311 �2005�; P.J. Leek, M. R. Buitelaar, V. I. Talyanskii, C. G. Smith, D. Ander-son, G. A. C. Jones, J. Wei, and D. H. Cobden, Phys. Rev. Lett.95, 256802 �2005�.

9 G. Fève, A. Mahé, J.-M. Berroir, T. Kontos, B. Plaçais, D. C.Glattli, A. Cavanna, B. Etienne, and Y. Jin, Science 316, 1169�2007�.

10 M. Switkes, C. M. Marcus, K. Campman, and A. C. Gossard,Science 283, 1905 �1999�.

11 S. K. Watson, R. M. Potok, C. M. Marcus, and V. Umansky,Phys. Rev. Lett. 91, 258301 �2003�.

12 R. Citro, N. Andrei, and Q. Niu, Phys. Rev. B 68, 165312�2003�.

13 T. Aono, Phys. Rev. Lett. 93, 116601 �2004�.14 J. Splettstoesser, M. Governale, J. König, and R. Fazio, Phys.

Rev. Lett. 95, 246803 �2005�.

15 P. W. Brouwer, A. Lamacraft, and K. Flensberg, Phys. Rev. B72, 075316 �2005�.

16 E. Sela and Y. Oreg, Phys. Rev. Lett. 96, 166802 �2006�.17 L. Arrachea, A. Levy Yeyati, and A. Martin-Rodero, Phys. Rev.

B 77, 165326 �2008�.18 D. Fioretto and A. Silva, Phys. Rev. Lett. 100, 236803 �2008�.19 A. R. Hernández, F. A. Pinheiro, C. H. Lewenkopf, and E. R.

Mucciolo, Phys. Rev. B 80, 115311 �2009�.20 F. Cavaliere, M. Governale, and J. König, Phys. Rev. Lett. 103,

136801 �2009�.21 N. Winkler, M. Governale, and J. König, Phys. Rev. B 79,

235309 �2009�.22 F. Reckermann, J. Splettstoesser, and M. R. Wegewijs, Phys.

Rev. Lett. 104, 226803 �2010�.23 J. Splettstoesser, M. Governale, J. König, and R. Fazio, Phys.

Rev. B 74, 085305 �2006�.24 V. Kashcheyevs, Proc. SPIE 7142, 714206 �2008�.25 F. Romeo and R. Citro, Phys. Rev. B 80, 165311 �2009�.26 B. L. Hazelzet, M. R. Wegewijs, T. H. Stoof, and Y. V. Nazarov,

Phys. Rev. B 63, 165313 �2001�.27 E. Cota, R. Aguado, and G. Platero, Phys. Rev. Lett. 94, 107202

�2005�; 94, 229901�E� �2005�; R. Sánchez, E. Cota, R. Aguado,and G. Platero, Phys. Rev. B 74, 035326 �2006�.

28 D. V. Khomitsky and E. Y. Sherman, Phys. Rev. B 79, 245321�2009�.

29 B. Wunsch, M. Braun, J. König, and D. Pfannkuche, Phys. Rev.B 72, 205319 �2005�.

30 J. König and Y. Gefen, Phys. Rev. B 71, 201308�R� �2005�.31 D. Urban and J. König, Phys. Rev. B 79, 165319 �2009�.32 R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L.

M. K. Vandersypen, Rev. Mod. Phys. 79, 1217 �2007�.33 A. B. Vorontsov and M. G. Vavilov, Phys. Rev. Lett. 101,

CHARGE AND SPIN PUMPING THROUGH A DOUBLE… PHYSICAL REVIEW B 82, 205308 �2010�

205308-13

226805 �2008�; M. R. Buitelaar, J. Fransson, A. L. Cantone, C.G. Smith, D. Anderson, G. A. C. Jones, A. Ardavan, A. N. Khlo-bystov, A. A. R. Watt, K. Porfyrakis, and G. A. D. Briggs, Phys.Rev. B 77, 245439 �2008�.

34 R. Aguado and L. P. Kouwenhoven, Phys. Rev. Lett. 84, 1986�2000�.

35 W. G. van der Wiel, T. Fujisawa, S. Tarucha, and L. P. Kouwen-hoven, Jpn. J. Appl. Phys., Part 1 40, 2100 �2001�; T. Fujisawa,T. H. Oosterkamp, W. G. van der Wiel, B. W. Broer, R. Aguado,S. Tarucha, and L. P. Kouwenhoven, Science 282, 932 �1998�.

36 G. A. Prinz, Science 282, 1660 �1998�; S. A. Wolf, D. D. Aw-schalom, R. A. Buhrman, J. M. Daughton, S. von Molnar, M. L.Roukes, A. Y. Chtchelkanova, and D. M. Treger, ibid. 294, 1488�2001�.

37 D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 �1998�.38 M. Julliere, Phys. Lett. A 54, 225 �1975�.39 J. C. Slonczewski, Phys. Rev. B 39, 6995 �1989�.40 M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F.

Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas,Phys. Rev. Lett. 61, 2472 �1988�; G. Binasch, P. Grünberg, F.Saurenbach, and W. Zinn, Phys. Rev. B 39, 4828 �1989�; J. S.Moodera and L. R. Kinder, J. Appl. Phys. 79, 4724 �1996�.

41 S. Datta and B. Das, Appl. Phys. Lett. 56, 665 �1990�; T.Schäpers, J. Nitta, H. B. Heersche, and H. Takayanagi, Phys.Rev. B 64, 125314 �2001�.

42 M. Braun, J. König, and J. Martinek, Phys. Rev. B 70, 195345�2004�.

43 W. Rudziński and J. Barnaś, Phys. Rev. B 64, 085318 �2001�; G.Usaj and H. U. Baranger, ibid. 63, 184418 �2001�; 71,179903�E� �2005�; J. König and J. Martinek, Phys. Rev. Lett.90, 166602 �2003�; M.-S. Choi, D. Sánchez, and R. López, ibid.92, 056601 �2004�.

44 S. Sahoo, T. Kontos, J. Furer, C. Hoffmann, M. Graber, A. Cot-tet, and C. Schönenberger, Nat. Phys. 1, 99 �2005�.

45 A. Cottet and M.-S. Choi, Phys. Rev. B 74, 235316 �2006�.46 P. Trocha, I. Weymann, and J. Barnaś, Phys. Rev. B 80, 165333

�2009�.47 B. Wang, J. Wang, and H. Guo, Phys. Rev. B 67, 092408 �2003�;

J. Fransson and M. Galperin, ibid. 81, 075311 �2010�.

48 M. Governale, F. Taddei, and R. Fazio, Phys. Rev. B 68, 155324�2003�; V. Brosco, M. Jerger, P. San-Jose, G. Zarand, A. Shnir-man, and G. Schön, arXiv:0912.1096 �unpublished�.

49 E. R. Mucciolo, C. Chamon, and C. M. Marcus, Phys. Rev. Lett.89, 146802 �2002�; M. Blaauboer and C. M. L. Fricot, Phys.Rev. B 71, 041303 �2005�.

50 J. Splettstoesser, M. Governale, and J. König, Phys. Rev. B 77,195320 �2008�.

51 R. Zhu and J. Berakdar, Phys. Rev. B 81, 014403 �2010�.52 Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B

66, 224403 �2002�; A. Brataas, Y. Tserkovnyak, G. E. W. Bauer,and B. I. Halperin, ibid. 66, 060404 �2002�; B. Heinrich, Y.Tserkovnyak, G. Woltersdorf, A. Brataas, R. Urban, and G. E.W. Bauer, Phys. Rev. Lett. 90, 187601 �2003�; J. Xiao, G. E. W.Bauer, and A. Brataas, Phys. Rev. B 77, 180407 �2008�.

53 J. König, H. Schoeller, and G. Schön, Phys. Rev. Lett. 76, 1715�1996�; J. König, J. Schmid, H. Schoeller, and G. Schön, Phys.Rev. B 54, 16820 �1996�.

54 G. Schön, in Quantum Transport and Dissipation, edited by T.Dittrich, P. Hänggi, G. Ingold, B. Kramer, G. Schön, and W.Zwerger �VCH Verlag, Berlin, 1997�, p. 149.

55 M. Leijnse and M. R. Wegewijs, Phys. Rev. B 78, 235424�2008�.

56 C. Bruder and H. Schoeller, Phys. Rev. Lett. 72, 1076 �1994�.57 In the following we will focus on the pumped charge for small

pumping amplitudes on account of simplicity. Note however thatthis is no fundamental limitation, and the expression for thetime-dependent current is valid also in the nonlinear regime.This regime of large pumping amplitudes becomes important ifone is interested in quantized pumping.

58 When considering the total current pumped out of the dotthrough the left and the right leads, we recover the total dis-placement current e d

dt �n��i�, as required.59 M. R. Buitelaar, V. Kashcheyevs, P. J. Leek, V. I. Talyanskii, C.

G. Smith, D. Anderson, G. A. C. Jones, J. Wei, and D. H. Cob-den, Phys. Rev. Lett. 101, 126803 �2008�.

60 C. Feuillet-Palma, T. Delattre, P. Morfin, J.-M. Berroir, G. Fève,D. C. Glattli, B. Plaçais, A. Cottet, and T. Kontos, Phys. Rev. B81, 115414 �2010�.

ROMAN-PASCAL RIWAR AND JANINE SPLETTSTOESSER PHYSICAL REVIEW B 82, 205308 �2010�

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