Quantum transport of the semiconductor pump: Due to an axialexternal field

Yun-Chang Xiao a,n, Ri-Xing Wang a, Wei-Ying Deng bQ1

a College of Electrical and Information Engineering, Hunan University of Arts and Science, Changde 415000, Chinab Department of Physics, South China University of Technology, Guangzhou 510640, China

a r t i c l e i n f o

Article history:Received 25 October 2013Received in revised form6 December 2013Accepted 29 April 2014

Keywords:Floquet transportExternal field modulationRashba effectPumped current

a b s t r a c t

Parametric semiconductor pump modulated by the external field is investigated. The pump center attachingto two normal leads is driven by the potentials formed in the interfaces. With the Floquet scattering matrixmethod, the pumped currents modulated by the parameters are studied. Results reveal that the charge andspin currents pumped from the system can be strengthen by the external field besides the potentials.Directed spin currents can be pumped more strongly than the charge currents, and even the pure spincurrents can be achieved in some external field couplings to the pump parameters.

& 2014 Published by Elsevier B.V.

1. Introduction

In solid state physics, extensive interests have been attracted tothe quantum pumping mechanism [1–16]. As a way of generatingdirected currents by the cyclic adiabatic change of two or moresystemic parameters in the absence of any external bias voltage,the mechanism was originally proposed by Thouless [1]. Expan-sion theories and experiments were widely processed and theseinvestigations in the latest two decades, which make the quantumpumping realization sound be not ever impossible [3,10,13–15].

As a precise tool in control of the reversible flow via time-dependent potentials, the quantum pump is an interesting andworthy physical system to be realized. One of the most importantis the spin quantum pump, which comes from the gate voltagemodulation in semiconductors, has been focused particularly[17,18]. This typical pump can carry out oscillation currents indifferent energy traversal paths, and has very sensitive dependenceon parameters control, such as the incoming spins, the spin–orbitcoupling (SOC) and the magnetic fields [3,6]. Pumped currentsalways are period proportional to the pump parameters in differentregimes. With appropriate parameter regulations, some wonderfuluse can be achieved, such as pumping pure spin currents withoutaccompanying charge current, spin-charge separations, spin flippingand phase dependent electron transport [7,8,16].

Actually, for the quantum pumping mechanism, besides the gatevoltage driving, the time potential driving named parametric pumpsare practical [19–21]. To be exact, the time potentials are not alwaysweak in related experiments and investigations [8,22,23]. It is a viablemethod to take into account the Floquet theorem [24,25], a photon-assisted transport to solve problems both in weak and strong regime.The developed theoretical work was first applied to the quantummechanical pump in mesoscopic conductors by Moskalets et al. [24].While due to an external field, consider the double delta potentialsdriving semiconductor pump, the spin distribution will be modulatedand lots of interesting phenomena appear, i.e., the spin currents can beseparated from the system easily, and pure spin currents are sig-nificant achieved in some parameter couplings. In this paper, we aregoing to investigate these phenomena by the external field and theSOC modulating pump. And the charge and spin pumped currentvariations are mainly studied in related parameter modulations.

The paper is organized as follows. In Section 2 the system isintroduced and related theoretical formalisms are provided. Section 3gives some analytic and numerical results, and Section 4 presents abrief conclusion of this work.

2. Model and formulation

The quantum pumping, which consists of three different regions,can be realized by two time dependent delta potentials driving, asshown in Fig. 1. The potentials are formed by the controllable laserbeams placing between the interfaces of the leads and the SOC part.The left region L and right region R are the source and drain, they are

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Physica B

http://dx.doi.org/10.1016/j.physb.2014.04.0820921-4526/& 2014 Published by Elsevier B.V.

n Corresponding author. Tel.: þ86 20 134 30276056; fax: þ86 20 28658632.E-mail addresses: phyxiaofan@163.com (Y.-C. Xiao),

wangrixing@sina.com (R.-X. Wang), weiyindeng@gmail.com (W.-Y. Deng).

Please cite this article as: Y.-C. Xiao, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.04.082i

Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎

connected to the central region as usual leads, in which no SOC ispresent. While dynamic direction of the electron is chosen along the xaxis, the system can be described as a quasi-one-dimensional quan-tum wire transport, and the Hamiltonian can be written as [24,26]

HLead ðx; tÞ ¼ p2x2mf

þu ðx; tÞ; ð1Þ

where px; mf is the momentum operator and the effective mass of anelectron, respectively. In the central Rashba SOC region, due to theaxial external field, the Hamiltonian takes the form as [27,28]

Hsoc ðxÞ ¼p2x2ms

þℏksoms

szpxþℏ2B0

mssx: ð2Þ

wherems is the effective mass of an electron in the semiconductor, B0

is the external field strength, kso is the Rashba term and siði¼ x; y; zÞare the Pauli spin operators. Time dependent potentials in theinterfaces are expressed by

u ðx; tÞ ¼ uLðtÞδðxÞþuRðtÞδðx�ℓÞ ð3Þ

at x¼ 0;ℓ, and δðxÞ is the delta function. We set the identical static (ac)driving potentials us udð Þ of uαðtÞ ¼ usþud cos ωtþϕα

� �;α¼ L;R,

where ω is the oscillation frequency and Δϕ¼ ϕR�ϕL is the phasedifference.

As affected by the time dependent potentials, the Floqueteigenstates are given by series position wave functions withdifferent propagating energy [24,29]

Ψεðx; tÞ ¼ ∑1

n ¼ �1ψαnðxÞ exp � i ðεþnℏωÞt

ℏ

� �: ð4Þ

here ε is the Floquet energy and n are the propagating modes, ℏωrepresents an energy quantum with ω the oscillation frequency.With the spin eigenstates in sx presentation, an electron movingin the normal lead of the nth propagating mode generally have thex-component spin wave function [30,31]

ψLnðxÞ ¼

δ0neiknxþr↑ne� iknxffiffiffi2

p 11

� �þ r↓nffiffiffi

2p e� iknx

1�1

� �; ð5Þ

for xo0, and for x4ℓ are

ψRnðxÞ ¼

t↑nffiffiffi2

p eiknx11

� �þ t↓nffiffiffi

2p eiknx

1�1

� �; ð6Þ

where term δ0n are the left incoming spin up electron parts and

kn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ms

ℏ2ðεþnℏωÞ

qdenote the wave vectors. We neglect the

evanescent part of the propagating modes, i.e. En ¼ εþnℏω40.In the nth eigenstate, for linear combination of the moment andthe spin operator, electron wave functions in the middle semi-conductor region can be written as

ψSnðxÞ ¼ ∑

4

j ¼ 1cnj

�B0

k2nj �k2n2 þksoknj

0@

1Aeiknjx ð7Þ

for 0rxrℓ, where cnj are the parameters to be determined, andthe wave vectors knj are deduced from the eigen equations

k4nj�2 k2nþ2k2so

k2njþk4n�4B20 ¼ 0: ð8Þ

In every propagating mode, wave functions ψLn;ψ

Rn and ψS

n

in three different regions are connected by the Griffith boun-dary conditions [32–34] (as is shown in Appendix A). The reflec-tion and transmission coefficients rsn and tsn ðs¼ ↑; ↓Þ in the

lead wave functions can be represented as χnj ¼ χ*nj; χ+nj

T¼ �B0;

k2nj �k2n2 þksoknj

� �T

by the components of Eq. (7), where

ϑ¼*;+ are the sz spin states in the central part. The matrix formof any incoming spin electron can be written as

Un

χnLjχnRj

!þ2τχ0ne

� ik0ℓδRα ¼Dnþ1

χðnþ1ÞLjχðnþ1ÞRj

!þ In�1

χðn�1ÞLjχðn�1ÞRj

!;

ð9Þwhere χnαj ¼ χnjeiknjℓδRα and χ0n;Un; Dnþ1; In�1 are the matricesgiven in Appendix B. As refer to the Floquet technology, usingthe recurrent equations in corresponding propagating modes, onecan get the solutions of the reflection and transmission coefficientsas spin elements of the Floquet scattering matrix [35,36]

bsLn

bsRn

!¼

rs0LsLn ts0RsL

n

ts0LsRn rs0RsR

n

!as0Ln

as0Rn

!: ð10Þ

In adiabatic pumping regime, i.e. ℏω{kBT , the current flowingin the Floquet transport can be obtained from the transmissioncoefficients [24,29,37,38], in the right lead it is expressed as

IsR ¼ eω2π

∑nn jts0LsR

n j2�jts0RsLn j2� �

: ð11Þ

This formula can be understood easily, all propagating modesmake up the current flowing. Spin electrons in the right lead havecontribution to the current and in the other lead should be got ridof, thus the pumped charge current is defined as Ic ¼ I↑Rþ I↓R.However, for the pumped spin current Is ¼ I↑R� I↓R, it means thatonly the spin up electrons in the right and the spin down electronsin the left have contributions to the current, the others should bededucted.

3. Numerical results and discussions

In this work, we take ms ¼ 0:036mf in the typical InAs-basedsemiconductors [39], where mf ¼me is the free electron mass. Thesystem center is fixed as ℓ¼ 100 nm of the semiconductor. Theenergy quantum is ℏω¼ 6:0 meV and the insert energy isε¼ 0:3 eV, correspondingly the energy wave is kF ¼ 3� 109 m�1

and a reference energy wave is chosen as κF ¼ 0:01kF. The staticand the ac driving potential strength in following calculations areset as ud ¼ 3us ¼ 60 meV.

Numerical results of the charge and spin currents pumped fromthe system are shown in the following figures. In these figures, theinitial left potential phase is kept as ϕL ¼ 0 and the pumpedcurrent variations versus the phase difference are plotted. Thedash dot lines are additional to get a more straightforward revealof the pure spin pumped currents. In Fig. 2, when we set B¼0,which means that no external field is added to the system and theparameter pump is only modulated by the Rashba SOC [29]. Itshows that there exists only small charge current (solid line) andno pumped spin currents (dash line) are observed, just as theresults of Ref. [6]. While taking into account the external field, i.e.

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Fig. 1. Schematic diagram of the quantum pump. The left and right regions are thenormal leads. The middle region is the Rashba SOC part, which is due to an axialexternal field. Interfaces of the SOC part and the leads are subjected to the timedependent potentials.

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B0 ¼ κF, the pumped charge currents are strengthen apparentlyand the spin currents turn up, as shown in Fig. 2(b). The spin currentshave a larger amplitude oscillation than that of the charge currents.These phenomena originate from that the axial magnetic field makesa remarkable breaking to the time reversal of the spin precession inthe system. Meanwhile, Fig. 2(b) shows that the value of the pumpedspin and charge currents are still keeping the sinusoidal modulationswith same period, which makes the pure spin current be unachie-vable. However, form the figure we can get that the pumped spin andcharge currents have different directions, these easily make the spincurrents separated from the system.

Since the spin precessions have so obvious affects to the pumpingtransport, that the viable ways of modulating the Rashba SOC and theexternal field to get a stronger pumped current can be easily realized.In Fig. 3, we plotted the pumped currents versus the Rashba SOCstrength. The axial external field is chosen as B0 ¼ κF and the phasedifference isΔϕ¼ 0:5π. It shows that the pumped currents can bemodulated by the SOC strength. A larger spin current pumped fromthe system can be achieved for certain SOC strength. Furthermore,the value difference between the charge and spin currents are verysmall, which can be explained as follows: the strong SOC makes thetime reversal of the system in charge and spin be primary and thesymmetry of the system is hard to broken. In this figure only a smallpure spin current can be observed in some special Rashba SOC, i.e.only around Is � 0:05 nA, as shown by the dot dash lines of Fig. 3.This tells us that although the SOC effects have a good precession tothe current pumping, it is still hard to separate the pumped spin andcharge currents perfectly in value. However, interestingly in someSOC strength, i.e. 1okso=κFo2 or 3:5okso=κFo4:5, direction of thetwo currents are totally different. This can make the spin and chargecurrents be separated easily, which is another good way of gettingpure spin current.

Then we take account the external field effect into the pumpedcurrents, the phenomena are much more obvious and the realiza-tion is also very convenient [5,40,41]. Compared with the situationof SOC modulations, the pump current oscillations versus theexternal field strength are described in Fig. 4. Here we choose theSOC strength as kso ¼ 5κF and the phase difference is Δϕ¼ 0:5π.When the external field strength is in the small value, from thefigure we get that both the pumped spin and charge currents aresmall and almost be zero for B0r0:6κF. Increase the external fieldstrength, we find that the currents pumped from the systemapparently, as shown in Fig. 4. The charge and spin currents arecorresponding to the solid and the dash lines. It demonstrates thatboth the spin and charge pumped currents oscillate with theexternal field. From the figure, it can be seen that there aremaxima and minima pumped currents for some external fields.

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Fig. 2. Charge (solid lines) and spin (dash lines) currents versus the phase difference. Here we set the Rashba SOC strength is kso ¼ 5κF, and different external fields areconsidered in the panels (a) and (b), respectively.

Fig. 3. The pumped currents versus the Rashba SOC strength. The solid, dashedlines correspond to the charge and spin currents, the external field strength isB0 ¼ κF and the phase difference is Δϕ¼ 0:5π.

Fig. 4. Pumped currents versus the external field strength, the solid, dash linescorrespond to the charge and spin currents. The strength of the Rashba SOC iskso ¼ 5κF and the phase difference is Δϕ¼ 0:5π.

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Excitingly, for many different external fields the spin currents arealways stronger than the charge currents, as shown in the dashedlines. Moreover, in some external field strengths the strongenough pure spin currents can be achieved, as shown by the dashdot lines. These phenomena show that for certain Rashba SOC, thestrong external fields make the time revision of the system brokenheavily and the pumping mechanism is very prominent. At thesame time, the spin distribution balance is much more sensitive tothe external field than that of the charge, thus a strong spincurrents come out.

In order to take a deeper investigation to the pumped currents, itis worthy of choosing some special external field to studied thepumping transport. In Fig. 4, we still plot the current variationsversus the phase difference, and the Rashba SOC is set as kso ¼ 5κF.However, the external field part is taken corresponding to theabscissa of the crossing dash dot lines as depicted in Fig. 4. Someinteresting current oscillations are presented in Fig. 5. To get a moreclear description of the currents varying, we plot a certain enlargedpumped charge currents, as shown by the solid lines in the figure. Inpanel (a), the external field is B0 ¼ 0:9605κF, a 40 times enlargedcharge current is plotted, and the zero charge currents are not only inthe integral π phase difference any more. Oscillations changedobviously and which easily states that the pure spin currents canbe pumped from the system, as shown by the crossing dash dot linesof Fig. 5(a). The pure pumped spin currents of every dash dot linescorresponding to the external fields can be achieved, and both of thecurrents variations are not extremely different in value. Increasingthe external field to a larger value, which can still get a strong purespin current pumped, as shown in Fig. 5(b), here B0 ¼ 1:015κF.However, in this external field, the spin currents become even largerthan the charge currents in another phase difference. From the figurewe can also get that the pure spin currents are symmetric in plus–minus phase differences. These mean that the external field alwayskeeping invariable in the periodical oscillation versus the phasedifference.

4. Conclusions

In summary, by taking the Rashba SOC and the axial externalfield into the parametric pump, we have studied the pumpedcurrent properties of the pump mechanism. Comparing with theprevious related works, we obtain a general and concrete descrip-tion of the external field modulating pumped current in thesemiconductor pump. Moreover, the charge and spin pumpedcurrent are calculated and discussed in detail. As affected by someexternal field couplings, oscillations of the spin current are alwaysstronger than that of the charge current. Numerical results indicate

that the external field modulations can separate the spin currents indirections, and even can get pure spin currents in pumping currentvalues. As expected, for the external field modulation to the pumpingcurrent, strong pure spin currents can be made prominent and theyare still symmetric in the phase difference. Usually, in quantumdevice control the external field modulation can be the magneticfield, but it is hard to achieve. Recently, the SOC simulation in coldatoms the Raman coupling term just have the same effect as theexternal field [42], this means the pure spin current modulating insome systems will be very convenient. The results may be helpful forunderstanding and designing specific quantum pumping devices,especially meaningful for advising the pure spin current realizationsand modulations.

Acknowledgments

Wewould like to acknowledge Dr. Z.Q. Liu and Q.H. Zhong, Prof.R. Zhu for their interesting and helpful discussions. This work wassupported by the Program for the NSF-China (grant nos. 11174088).Y.C. Xiao and R.X. Wang are supported by the Startup Fund forDoctor of Hunan University of Arts and Science (no. 10133004) and(no. 13102009).

Appendix A

In every propagating mode, the wave-functions in the threedifferent regions are connected by the Griffith boundary condi-tions, which obtained from the matching condition of the wavefunction

Ψεjx ¼ 0� ¼Ψεjx ¼ 0þ ;Ψεjx ¼ ℓ� ¼Ψεjx ¼ ℓþ ; ðA1Þ

and the current conservation

∂xΨεjx ¼ 0þ �τ∂xΨεjx ¼ 0� ¼ 2msuLℏ2 � iksosz

Ψεjx ¼ 0;

τ∂xΨεjx ¼ ℓþ �∂xΨεjx ¼ ℓ� ¼ 2msuRℏ2

þ iksosz

Ψεjx ¼ ℓ:

ðA2Þ

Appendix B

Substitute Eqs. (5)–(6) into the boundary conditions, we obtainthe solving Eq. (9). Where the spin part of any incoming electron

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Fig. 5. Pumped current versus the phase difference in some special external fields. The Rashba SOC strength is set as kso ¼ 5κF and the external fields are chosen asB0 ¼ 0:9605κF and B0 ¼ 1:015κF corresponding to the panel (a) and (b).

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can be written as

χ0n ¼ δ0nδLα

δRα

!¼ δ0nffiffiffi

2p

δLαδ↑sþδLαδ↓s

δLαδ↑s�δLαδ↓s

δRαδ↑sþδRαδ↓s

δRαδ↑s�δRαδ↓s

0BBBB@

1CCCCA: ðB1Þ

The matrices Un; Dnþ1; In�1 in according propagating modes ofEq. (9) are derived from the equations below

∑jγϑnLjχ

ϑLnjcnj�2τδ0nδLαe� ik0ℓδRα

¼ hdLn

�eiϕ1∑

jχϑLðnþ1Þjcðnþ1Þjþe� iϕ1∑

jχϑLðn�1Þjcðn�1Þj

�; ðB2Þ

∑jγϑnRjχ

ϑRnjcnj�2τδ0nδRαe� ik0ℓδRα

¼ hdRn

�eiϕ2∑

jχϑRðnþ1Þjcðnþ1Þjþe� iϕ2∑

jχϑRðn�1Þjcðn�1Þj

�: ðB3Þ

where the related potential parameters are hns ¼msus=ℏ2kn; hnd ¼�msud=ℏ

2kn, and the expressions are defined as γϑnαj ¼ τþhnsð Þþð�1ÞδRα knj7kα=kn

� �.

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