We investigate the parametric pumping of a hybrid structure consisting of a
normal quantum dot, a normal lead and a superconducting lead. Using the time
dependent scattering matrix theory, we have derived a general expression for
the pumped electric current and heat current. We have also derived the
relationship among the instantaneous pumped heat current, electric current, and
shot noise. This gives a lower bound for the pumped heat current in the hybrid
system similar to that of the normal case obtained by Avron et al

B.G. Wang

Baigeng Wang

C.W.J. Beenakker

D.J. Thouless

D.S. Fisher

F. Zhou

G.B. Lesovik

I.L. Aleiner

J. Wang

J.E. Avron

Jian Wang

M. Moskalets

M. Switkes

M. Wagner

M.G. Vavilov

M.L. Polianski

M.P. Anantram

O. Entin-Wohlman

P.W. Brouwer

Q. Niu

Q.F. Sun

T.A. Shutenko

V. Gasparian

X.B. Wang

Y. Levinson

Y.D. Wei

English

arXiv

We investigate the parametric pumping of a hybrid structure consisting of a normal quantum dot, a normal lead, and a superconducting lead. Using the time-dependent scattering matrix theory, we have derived a general expression for the pumped electric current and heat current. We have also derived the relationship among the instantaneous heat current generated by the pump, electric current, and the noise. This gives a lower bound for the heat current generated in the hybrid system similar to that of the normal case obtained by Avron et al. [Phys. Rev. Lett. 87, 236601 (2001)].published_or_final_versio

Wang, B

Wang, J

HKU Scholars Hub

Title Optimal quantum pump in the presence of a superconductingleadAuthor(s) Wang, B; Wang, JCitation Physical Review B (Condensed Matter and Materials Physics),2002, v. 66 n. 20, p. 201305:1-4Issued Date 2002URL http://hdl.handle.net/10722/43368Rights Creative Commons: Attribution 3.0 Hong Kong LicenseRAPID COMMUNICATIONSPHYSICAL REVIEW B 66, 201305~R! ~2002!Optimal quantum pump in the presence of a superconducting leadBaigeng Wang1 and Jian Wang1,2,*1Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China2Institute of Solid State Physics, Chinese Academy of Sciences, Hefei, Anhui, China~Received 29 August 2002; published 26 November 2002!We investigate the parametric pumping of a hybrid structure consisting of a normal quantum dot, a normallead, and a superconducting lead. Using the time-dependent scattering matrix theory, we have derived a generalexpression for the pumped electric current and heat current. We have also derived the relationship among theinstantaneous heat current generated by the pump, electric current, and the noise. This gives a lower bound forthe heat current generated in the hybrid system similar to that of the normal case obtained by Avron et al.@Phys. Rev. Lett. 87, 236601 ~2001!#.DOI: 10.1103/PhysRevB.66.201305 PACS number~s!: 73.23.Ad, 73.40.Gk, 72.10.Bg, 74.50.1rSince the seminar work of Thouless1 and Brouwer,2 thephysics of parametric pumping has attracted increasingattention.3–18 Recently, Avron et al.19 have considered theheat current generated by the pump in the adiabatic regimeand found a general lower bound for the heat current. Thisdefines an optimal pump if the heat current equals to thepower of Joule heat dissipated during the pumping process.19As a consequence, the optimal pump is noiseless and thecharge transported is quantized. The physics of heat currenthas also been investigated by Moskalets and Buttiker20 whohave derived a general formula for the heat current in theweak pumping regime and the noise generated during thepumping process. In the strong pumping regime, the heatcurrent has been studied within the time-dependent scatteringmatrix theory21 and the existence of optimal pump has beenexamined. For chaotic quantum dots, Polianski et al.22 havedeveloped a time-dependent scattering matrix theory to ac-count for the noise for parametric pumping and mesoscopicfluctuation for arbitrary temperature and beyond bilinear re-sponse. In this paper, we investigate the pumped heat currentfor a normal superconducting ~NS! hybrid system that con-sists of a normal quantum dot, a normal lead, and a super-conducting lead. In the adiabatic regime, the energy ofcharged carriers ~electron or hole! is within the supercon-ducting energy gap and hence physics of the Andreevreflection23 dominates. We have derived a general expressionfor the pumped electric current and heat current in the pres-ence of a superconducting lead which is valid at finite pump-ing amplitude and finite temperature. Our theory is based onthe time-dependent scattering matrix theory.10 Going beyondthe adiabatic regime, we can in principle obtain the pumpedelectric current and heat current to any order in frequency. Inthe adiabatic regime, we have also derived a relationshipamong the instantaneous heat current, electric current, andthe noise. This sets a lower bound for the heat current gen-erated by the pump. Similar to the normal system,19 a quan-tum pump will be optimal if the heat current reaches itslower bound. As a result, the charge transported will bequantized and the system is noiseless just like the normalsystem. We have also compared with the heat current of NSstructure with that of normal structure. For a single pumpingpotential, the total heat currents generated are the same forNS and normal systems. For two pumping potentials the heat0163-1829/2002/66~20!/201305~4!/$20.00 66 2013current for NS system can be either larger or smaller thanthat of normal system depending on the phase differencebetween two pumping potentials.For the purpose of presentation, we consider the pumpedelectric current first. We start with the general definition forthe electric current of type a ~electron or hole! in the leftnormal lead in scattering matrix theory (\5q51),20Ie ,La5 limDt>‘1DtE0Dtdt^Iˆe ,La& , ~1!where ^& denotes the quantum average and Iˆe ,La is theelectric current operator of type a in the left lead,Iˆe ,La5qa@bˆ La† ~ t !bˆ La~ t !2aˆ La† ~ t !aˆ La~ t !# . ~2!Here the operators bˆ La and aˆ La are annihilation operators forthe outgoing and incoming carriers of type a in the left leadand qa51,21 for a5e ,h . They are related by the scatteringmatrix,20,26bˆ La~ t !5(bE dt8Sab~ t ,t8!aˆ Lb~ t8!, ~3!where the time dependence of the scattering matrix S is dueto the slowly time-varying pumping potential X(t). The dis-tribution function can be obtained by taking the quantumaverage,20^aˆ La† ~E !aˆ Lb~E8!&5dabd~E2E8! f L~E !, ~4!where aˆ La(E) is the Fourier transform of aˆ La(t), and f L(E)is the Fermi distribution function of the left lead. From Eqs.~2!–~4!, the pumped electric current is given byIe ,La5 limDt>‘qaDtE0DtdtE dt1dt2(bSab~ t ,t1!3 f ~ t12t2!Sab* ~ t ,t2!2qaE dE2p f ~E !, ~5!where f (t)[*(dE/2p)exp(2iEt)f(E). After changing of thevariables t05(t11t2)/2 and t5t12t2, and using the follow-ing Wigner transform for the scattering matrix:10©2002 The American Physical Society05-1RAPID COMMUNICATIONSB. WANG AND J. WANG PHYSICAL REVIEW B 66, 201305~R! ~2002!S~ t ,t8!5E dE2p e2iE(t2t8)SS E , t1t82 D , ~6!Eq. ~5! becomesIe ,La5 limDt>‘qa4p2DtE0DtdtE dt0dtdE1dE2 f ~t!3e2iE1(t2t02t/2)eiE2(t2t01t/2)(bSabS E1 , t1t021t4 D3Sab* S E2 , t1t02 2 t4 D2qaE dE2p f ~E !. ~7!Changing the variables again to t15t2t0 and t85(t1t0)/2 and integrating over t1, we obtainIe ,La5 limDt>‘qa2pDtE2DtDtdt8dtdEeiEt f ~t!3(bSab~E ,t81t/4!Sab* ~E ,t82t/4!2qa3E dE2p f ~E !. ~8!Using the fact thatlimDt>‘E2DtDtdt8(bSab~E ,t81t/4!Sab* ~E ,t82t/4!5 limDt>‘E2DtDtdt(bSab~E ,t !e2(t/2)] tSab* ~E ,t !, ~9!Eq. ~8! becomesIe ,La5qapTpE0TpdtE dE$Sˆ ~E ,t !3@ f ~E1i] t/2!2 f ~E !#Sˆ †~E ,t !%aa , ~10!where Tp is the period of the pumping cycle and Sˆ is a 232 scattering matrix for NS structure with matrix elementSab(a ,b5e ,h). Equation ~10! is symbolic and is the centralresult of this paper. One can in principle obtain the pumpedelectric current to any order in frequency.24 For instance, toget the electric current up to v , it is enough to expand f (E1i] t/2) up to the first order in ] t , from which we obtainIe ,La5iqa2pTpE0TpdtE dE]E f 3@] tSˆ †~E ,t !Sˆ ~E ,t !#aa .~11!Note that from the unitary condition of the scattering matrixSˆ , we have(bSab* Sab51. ~12!Taking the derivative with respect to time, we obtain2520130(b] tSab* Sab1c.c.50. ~13!Hence Im@(] tSˆ †Sˆ )aa#52i(] tSˆ †Sˆ )aa . In the adiabatic re-gime, we have10 ] tSab5( i@]XiSab] tXi1]X˙ iSab] tX˙ i1], where X˙ [dX/dt . Up to the order v , we can neglectthe contribution from ]X˙ isab . At zero temperature, Eq. ~11!becomesIe ,La5iqa2pTpE0Tpdt@]XiSˆSˆ †#aa] tXi , ~14!which agrees with the theory of nonequilibrium Green’sfunction.27Now we proceed to derive the heat current generated bythe pump for NS structure. We note that the heat current isdefined as the particle current multiplied by the energy mea-sured from the Fermi level. We thus have from Eq. ~10!,Iq ,La51pTpE0TpdtE dE~E2EF!$Sˆ ~E ,t !3@ f ~E1i] t/2!2 f ~E !#Sˆ †~E ,t !%aa . ~15!Expanding the heat current up to ] t2 and after some algebra,we finally obtained the heat current up to v2,Iq ,La5218pTpE0TpdtE dE]E f ~] tSˆ ] tSˆ †!aa . ~16!Now we derive the relationship between instantaneous elec-tric current @denoted as Ie(t)52iqa(] tSˆ †Sˆ )aa] and heatcurrent @Iq(t)5(] tSˆ †] tSˆ )aa# . Now the instantaneous heatcurrent becomesIq~ t !5~] tSˆ †] tSˆ !aa5~] tSˆ †SˆSˆ †] tSˆ !aa5(b~] tSˆ †Sˆ !ab~Sˆ †] tSˆ !ba . ~17!We see that the diagonal term in Eq. ~17! is just Ie2(t) and theoff-diagonal term is the noise S0(t) generated during thepumping process.20,19,28 Therefore, we have the following re-lationship:Iq~ t !5Ie2~ t !1S0~ t ! ~18!or general lower bound for the heat currentIq~ t !>Ie2~ t !. ~19!The condition of optimal pump for NS structure is defined asS0(t)50. Following Avron et al.,19 it is straightforward toshow that the charge transported through the system percycle is quantized if the quantum pump is optimal.Now we consider the adiabatic and weak pumping limitfor a symmetric double-barrier structure in the presence ofsuperconducting lead. The double-barrier structure is mod-eled by potential V(x)5X1(t)d(x1a)1X2(t)d(x2a),where X1(t)5X01X1sin(vt) and X2(t)5X01X2sin(vt5-2RAPID COMMUNICATIONSOPTIMAL QUANTUM PUMP IN THE PRESENCE OF A . . . PHYSICAL REVIEW B 66, 201305~R! ~2002!1f). In this limit, we keep only the quadratic order in pump-ing amplitude in Eqs. ~14! and ~16!. It is easy to show thatthe pumped electric current and heat current are given, re-spectively, byIe ,La5vqasin fX1X22p Im@~]X1Sˆ†]X2Sˆ !aa# ~20!andIq ,La5v216p @X12]X1Sˆ †]X1Sˆ 1X22]X2Sˆ †]X2Sˆ12cosfX1X2Re~]X1Sˆ ]X2Sˆ †!#aa . ~21!In Eqs. ~20! and ~21!, we have set Xi50 in S after the partialderivatives. Now we will calculate the heat current Iq ,La5efor the double-barrier NS system. For the NS system, thescattering matrices See and She are given by23,29Sˆ 5Sˆ 111Sˆ 12~12Rˆ ISˆ 22!21Rˆ ISˆ 21 , ~22!whereSˆ i j~E !5S Si j~E ! 00 Si j~2E ! D , ~23!with Si j being usual scattering matrix for the normal struc-ture. Rˆ I5asx is the 232 scattering matrix at NS interfacewith off-diagonal matrix element a . Here a5(E2inAD22E2)/D with n51 when E.2D and n521when E,2D . In Eq. ~22!, the energy E is measured relativeto the chemical potential m of the superconducting lead.Equation ~22! has clear physical meaning.29 The first term isthe direct reflection from the normal scattering structure andthe second term can be expanded as Sˆ 12Rˆ ISˆ 211Sˆ 12Rˆ ISˆ 22Rˆ ISˆ 211 , which is clearly the multiple An-dreev reflection in the hybrid structure. From Eq. ~22! weobtain the well-known expressions for the scattering matricesSee and She ,23See~E !5S11~E !1a2S12~E !S22* ~2E !M eS21~E ! ~24!andShe~E !5aS12* ~2E !M eS21~E !, ~25!with M e5@12a2S22(E)S22* (2E)#21. In the case of para-metric pumping, we assume that the Fermi energy is in linewith the chemical potential of superconducting lead, so E50 and a52i . For the symmetric NS system at resonance,we have9 S1150 and S125e22ika in the absence of pumpingpotential. Therefore, from Eqs. ~24! and ~25!, we have]X1/2See5]X1/2S112S122 ]X2/1S11* ~26!and]XShe52i~]XS12* S121c.c.!, ~27!where we have used the fact that ]X1S225]X2S11 . UsingFisher-Lee relation30 Sab52dab1ivGabr and the Dyson20130equation ]X jGabr 5Ga jr G jbr,31 we have ]X1S115ivG11r G11r52i/v , ]X2S115ivG12r G21r 52iS122 /v , and ]X1/2S1252iS12 /v with the velocity v52k . Thus from Eqs. ~26!and ~27!, we have ]X jSee52]X jS11 and ]XShe50. From Eq.~21!, we obtainIq ,LeNS 5v216k2@X121X2212 cos fX1X2 cos 4ka# , ~28!which should be compared with the heat current in the nor-mal case,Iq ,LN 5Iq ,RN 5v232k2@X121X221cos fX1X2~11cos 4ka !# .~29!We note that in the NS system, the heat current flows outonly through the normal lead; while for normal system, theheat current pumps out through both leads. Comparing Eqs.~28! and ~29!, we haveIq ,LN 1Iq ,RN 2Iq ,LeNS 5v216k2cos fX13X2 ~12cos 4ka !.~30!Hence the total heat current generated in the normal systemcan be either larger or smaller than that in the NS systemdepending on the phase difference of two pumping poten-tials. For a single pump, by setting X250 in Eqs. ~28! and~29!, we see that the total heat currents are the same for bothNS and normal systems. This is different from the pumpedelectric current where in the weak pumping regime at reso-nance, the electric current for NS system is four times largerthan that of normal system.9In summary, we have derived a general expression for thepumped electric current and heat current in the presence ofsuperconducting lead using the time-dependent scatteringmatrix theory. Our theory is valid at finite pumping ampli-tude and can be applied to the multichannel systems. Usingour theory, we can expand Eq. ~15! to higher order in fre-quency and hence approach to the nonadiabatic regime. Ourtheory can also be easily extended to the case of multitermi-nal structures. Although our expression is derived for NSsystem, it is also valid the normal system as well by simplyreplacing the NS scattering matrix Sab with ab5e ,h bynormal system scattering matrix Si j with i , j51,2 in Eqs.~10! and ~15!. For the NS system, we have found the lowerbound for the heat current similar to that of Avron et al.19 forthe normal system. As a result, the optimal pump can existfor NS system as well. In the weak pumping limit, we haveexamined the heat current for NS system for a double-barrierstructure at resonance. For two-parameter pump, we foundthat the total heat current for NS structure can be larger orsmaller than that of normal structure depending on the phasedifferent between two pumping parameters.We gratefully acknowledge support by a RGC grant fromthe SAR Government of Hong Kong under Grant No. HKU7091/01P and a CRCG grant from The University of HongKong.5-3RAPID COMMUNICATIONSB. WANG AND J. WANG PHYSICAL REVIEW B 66, 201305~R! ~2002!*Electronic address: jianwang@hkusub.hku.hk1 D.J. Thouless, Phys. Rev. B 27, 6083 ~1983!; Q. Niu, Phys. Rev.Lett. 64, 1812 ~1990!.2 P.W. Brouwer, Phys. Rev. B 58, R10135 ~1998!.3 M. Switkes, C. Marcus, K. Capman, and A.C. Gossard, Science283, 1905 ~1999!.4 F. Zhou, B. Spivak, and B.L. Altshuler, Phys. Rev. Lett. 82, 608~1999!.5 M. Wagner, Phys. Rev. Lett. 85, 174 ~2000!.6 J.E. Avron, A. Elgart, G.M. Graf, and L. Sadun, Phys. Rev. B 62,R10 618 ~2000!.7 I.L. Aleiner, B.L. Altshuler, and A. Kamenev, Phys. Rev. B 62,10 373 ~2000!.8 Y.D. Wei et al., Phys. Rev. B 62, 9947 ~2000!; 64, 115321 ~2001!;cond-mat/0207473 ~unpublished!.9 J. Wang et al., Appl. Phys. Lett. 79, 3977 ~2001!.10 M.G. Vavilov et al., Phys. Rev. B 63, 195313 ~2001!.11 P.W. Brouwer, Phys. Rev. B 63, 121303 ~2001!; M.L. Polianskiand P.W. Brouwer, Ibis 64, 075304 ~2001!.12 X.B. Wang and V.E. Kravtsov, Phys. Rev. B 64, 033313 ~2001!.13 I.L. Aleiner and A.V. Andreev, Phys. 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Optimal quantum pump in the presence of a superconducting lead

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