We investigate the parametric electron pumping of a double barrier structure
in the presence of a superconducting lead. The parametric pumping is
facilitated by cyclic variation of the barrier heights $x_1$ and $x_2$ of the
barriers. In the weak coupling regime, there exists a resonance line in the
parameter space $(x_1,x_2)$ so that the energy of the quasi-bound state is in
line with the incoming Fermi energy. Levinson et al found recently that the
pumped charge for each pumping cycle is quantized with $Q=2e$ for normal
structure when the pumping contour encircles the resonance line. In the
presence of a superconducting lead, we find that the pumped charge is quantized
with the value $2e$

A.F. Andreev

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.K. Yip

M.L. Polianski

P. Sharma

P.W. Brouwer

Q. Niu

Q.F. Sun

T. Gramespacher

T.A. Shutenko

V. Gasparian

X.B. Wang

Y. Levinson

Y.D. Wei

English

arXiv

We investigate the parametric electron pumping of a double barrier structure in the presence of a superconducting lead. The parametric pumping is facilitated by cyclic variation of the barrier heights x1 and x2 of the barriers. In the weak-coupling regime, there exists a resonance line in the parameter space (x1,x2) so that the energy of the quasibound state is in line with the incoming Fermi energy. Levinson et al. found recently that the pumped charge for each pumping cycle is quantized with Q=2e for normal structure when the pumping contour encircles the resonance line. In the presence of a superconducting lead, we find that the pumped charge is quantized with the value 2e.published_or_final_versio

Wang, J

Wang, B

HKU Scholars Hub

Title Quantization of adiabatic pumped charge in the presence ofsuperconducting leadAuthor(s) Wang, J; Wang, BCitation Physical Review B (Condensed Matter and Materials Physics),2002, v. 65 n. 15, p. 153311:1-4Issued Date 2002URL http://hdl.handle.net/10722/43352Rights Creative Commons: Attribution 3.0 Hong Kong LicensePHYSICAL REVIEW B, VOLUME 65, 153311Quantization of adiabatic pumped charge in the presence of superconducting leadJian Wang and Baigeng WangDepartment of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China~Received 19 October 2001; published 3 April 2002!We investigate the parametric electron pumping of a double barrier structure in the presence of a supercon-ducting lead. The parametric pumping is facilitated by cyclic variation of the barrier heights x1 and x2 of thebarriers. In the weak-coupling regime, there exists a resonance line in the parameter space (x1 ,x2) so that theenergy of the quasibound state is in line with the incoming Fermi energy. Levinson et al. found recently thatthe pumped charge for each pumping cycle is quantized with Q52e for normal structure when the pumpingcontour encircles the resonance line. In the presence of a superconducting lead, we find that the pumped chargeis quantized with the value 2e .DOI: 10.1103/PhysRevB.65.153311 PACS number~s!: 73.23.Ad, 73.40.Gk, 72.10.Bg, 74.50.1rPhysics of parametric electron pump has attracted greatattention recently.1–12 A classical example of electron pumpis the Thouless pump facilitated by a traveling wavepotential.13 The pumped charge is quantized13 and can beused as a quantum standard for electric charge.14 The quan-tization of pumped charge has also been studied for a large,almost open quantum dot15,16 and a small, strongly pinchedquantum dot.17 In the latter case, there exists a resonance linealong which the transmission through the quantum dot is atresonance. The pumped charge is quantized if the pumpingcontour in parameter space is properly chosen to encircle theresonance line.17 Recently, we have studied the parametricpumping in presence of a superconducting lead.18 At thenormal-conductor–superconductor ~NS! interface, an incom-ing electronlike excitation can be Andreev reflected as aholelike excitation.19 In contrast to the current doublingeffect,20 we found that due to the quantum interference ofdirect reflection and the multiple Andreev reflection, thepumped current is four times of the value when the leads arenormal in the weak pumping regime. In this paper, we ex-plore the effect of superconducting lead on electron pumpingin the opposite limit, i.e., we study the pumped charge duringthe pumping cycle in the the strong pumping regime. Herethe pumped charge is equal to the pumped current multipliedby the period of pumping cycle. Similar to the Ref. 17, weexamine the behavior of pumped charge near the resonanceline. We find that the pumped charge in one pumping cycle isquantized with the value of Q52e when one of the leads issuperconducting.21We consider a parametric pump, which consists of adouble barrier tunneling structure attached to a normal leftlead and a superconducting right lead. Due to the cyclicvariation of external parameters x1 and x2, the adiabaticcharge transfer in the presence of a superconducting leadis1,22,23QNS52eE0tdtFdNLdx1 dx1dt 1 dNLdx2 dx2dt G , ~1!where t is the period of cyclic variation and the quantitydNL /dx is the injectivity25,26 given, at zero temperature, by0163-1829/2002/65~15!/153311~4!/$20.00 65 1533dNLdx j512pImFSee* ]See]x j 2She* ]She]x j G , ~2!where the first term is the injectivity of the electron due tothe variation of the external parameter,25,26 i.e., the partialdensity of states for an electron coming from the left leadand exiting the system as an electron, and the second term isthe injectivity of a hole, i.e., the DOS for a hole coming fromthe left lead and exiting the system as an electron. Using theGreen’s theorem, the pumped charge can be expressed assurface integral over area A enclosed by the pathx1(t),x2(t) in the parameter space1QNS52ep EAdx1dx2PNS~x1 ,x2! ~3!withPNS~x1 ,x2!5ImF]See*]x1 ]See]x2 2 ]She*]x1 ]She]x2 G . ~4!Note that the area A is a measure of variation of pumpingparameters x1 and x2 . A is very small in the weak pumpinglimit while it remains finite in the strong pumping regime.For the NS structures, the scattering matrix is describedby 232 matrix Sˆ when the Fermi energy is within the su-perconducting gap DSˆ 5S See SehShe ShhD , ~5!where See ~or She) is the scattering amplitude of the incidentelectron reflected as an electron ~or a hole!. Using Andreevapproximation,19 we have20,27Sˆ 5Sˆ 111Sˆ 12~12Rˆ ISˆ 22!21Rˆ ISˆ 21 , ~6!where Sˆ bg(E) (b ,g51,2) is a diagonal 232 scattering ma-trix for the double barrier structure with matrix elementSbg(E) and Sbg* (2E). For instance, we haveSˆ 115S S11~E ! 00 S11* ~2E ! D . ~7!©2002 The American Physical Society11-1BRIEF REPORTS PHYSICAL REVIEW B 65 153311In Eq. ~6! Rˆ I5asx is the 232 scattering matrix at NS in-terface due to the Andreev reflection with off-diagonal ma-trix element a . Here a5(E2inAD22E2)/D with n51when E.2D and n521 when E,2D . In Eq. ~6!, theenergy E is measured relative to the chemical potential m ofthe superconducting lead. Equation ~6! has a clear physicalmeaning.27 The first term is the direct reflection from thenormal scattering structure and the second term can be ex-panded as Sˆ 12Rˆ ISˆ 211Sˆ 12Rˆ ISˆ 22Rˆ ISˆ 211 , which is clearlythe sum of the multiple Andreev reflection in the hybridstructure. It is the quantum interference of these two termswhich gives rise the enhancement of pumped current in theweak pumping regime for NS system.18 From Eq. ~6! weobtain the well-known expressions for the scattering matrixSee and She ~Ref. 20!See~E !5S11~E !1a2S12~E !S22* ~2E !M eS21~E !, ~8!She~E !5aS12* ~2E !M eS21~E !, ~9!andM e5@12a2S22~E !S22* ~2E !#21. ~10!The double barrier structure, which we consider, is modeledby potential U(y)5V1d(y1a/2)1V2d(y2a/2), where V1and V2 are barrier heights that vary in a cyclic fashion toallow the charge pumping. For this system the retardedGreen’s function Gr(y ,y8) can be calculated exactly.30 Thisis done by applying the Dyson’s equation regarding the factthat any one of the d barrier is just a perturbation of theremaining system. This way Gr(y ,y8) is obtained by apply-ing Dyson’s equation twice starting from the Green’s func-tion of the one-dimensional free space. With Gr(y ,y8) wecan calculate scattering matrix exactly from the Fisher-Leerelation28Sbg52dbg1ivGbgr, ~11!where Gbgr 5Gr(yb ,yg) and v52k is the electron velocityin the normal lead. For normal structure, we have17S115@12ix22~11ix1!s2#/D , ~12!S225@12ix12~11ix2!s2#/D , ~13!andS125S215x1x2s/D , ~14!where D52(12ix1)(12ix2)1s2, x1,252kV1,2 , and s5exp(ika). For the double barrier structure, the resonant tun-neling is mediated by the quasibound state. When the energyof the incident electron is in line with the energy of thequasibound state the transmission coefficient reaches itsmaximum. The energy of quasibound states can be deter-mined either by looking at the pole of the scattering matrix,17which works well in one dimension, or by calculating thedwell time of the incident electron for two- or three-dimensional systems.29 In the case of double d barriers struc-ture, the energy of quasibound state is given by17 E5Er1DE with DE52(kr /a)(x11x2), where Er5kr2153315(np/a)2 is the energy of the bound state when the system isisolated. This defines a resonance line x11x252d in pa-rameter space (x1 ,x2) along which the transmission is atresonance.17 Here d,0 is the detuning of the Fermi energyfrom the bound state.To show the quantization of charge transfer in the NSsystem, it is useful to recall the calculation of the normalcase and make the comparison. In the normal case the chargetransfer is given by1,21QN52ep EAdx1dx2PN~x1 ,x2!, ~15!PN~x1 ,x2!5ImF]S11*]x1 ]S11]x2 1 ]S12*]x1 ]S12]x2 G . ~16!The pumped charge in this case has been calculated in Ref.17. In the weak pumping limit, it is easy to show that only]xS11 contributes to the pumped charge. In the strong pump-ing regime, we will show in the following that the contribu-tion from ]xS12 to the pumped charge in normal structure iszero. As discussed in detail in Ref. 17, we neglect the smoothenergy dependence of x1 and x2. From Eq. ~16!, we obtainthe contribution due to ]xS12P1N~x1 ,x2!5F1~x1 ,x2!/F22~x1 ,x2! ~17!withF1~x1 ,x2!522x1x2~x12x2!sin2~d/2!, ~18!F2~x1 ,x2!5x12x221~x11x2!212~x11x2!sin d12~12x1x2!~12cos d!. ~19!To compute the surface integral of P1N in Eq. ~17!, it isconvenient to change the variables from x1,2 to p and z,x152pd~11z !/2 ~20!andx252pd~12z !/2 ~21!with 0,p,‘ and 21,z,1. Substituting Eqs. ~20! and~21! into Eqs. ~18! and ~19! and expanding Eqs. ~18! and~19! in terms of small d , we haveF15z~12z2!d5p3/8 ~22!F25d2@~12p !21d2g~p ,z !# ~23!where g(p ,z) @an even function of z# is given in Eq. ~8! ofRef. 17. Since F1 is an odd function of z, the contributiondue to ]xS12 to the pumped charge is zero.Now we follow the same procedure to calculate thepumped charge for the NS system. For the parametric pump-ing at zero temperature, we only need the scattering matrix atthe Fermi level, i.e., at E50. From Eqs. ~9! and ~10!, we see1-2BRIEF REPORTS PHYSICAL REVIEW B 65 153311FIG. 1. ~a!. The integrand P of Eqs. ~3! and~15! as a function of x1 and x2 for d520.2.For illustration, the origin of PN(x1 ,x2) hasbeen shifted by (0.1,0.1). ~b!. The cross sectionof P along the resonance line x11x252d .Solid line, PNS; dotted line, PN. Insets show thecross section of P along the direction x12x25c0, which is perpendicular to the resonanceline. Left inset, c050.01; right inset, c0520.042.that She is a real quantity and hence makes no contribution tothe pumped charge in Eq. ~3!. It is straightforward to calcu-late PNS using Eq. ~8!, from which we obtain,PNS(x1 ,x2)5F3(x1 , x2)/F43(x1 ,x2), where F354x14x23(222 cos d1x2sin d) and F45x12x2212(x11x2)214(x11x2)sin d14(12x1x2)(12cos d).In Fig. 1 we plot both PNS and PN as well as their crosssections along and perpendicular to the resonance line. Wesee that PNS and PN are peaked around the resonance line.Two features are worth noticing. First of all, the peak of PNSis much sharper than that of PN. This is understandable andis due to the resonance nature of NS structures near the reso-nance line. In the Breit-Wigner form, the transmission coef-ficients for normal and NS structures are, respectively,uS21u25G1G2 /@(E2Er)21G2/4# and ~see Ref. 20! uS heu254G12G22/@4(E2Er)21G121G22#2, where Er is the resonantlevel, G1 and G2 are the decay widths into the left and rightleads. Hence uS heu2 decays much faster away from Er thanuS21u2. The scattering matrix S21 and She will appear, respec-tively, in Eqs. ~3! and ~15! implicitly as can be seen fromFisher-Lee relation Eq. ~11! and the Dyson equation ]X2G11r5G12r G21r.31 Second, the peak height of PNS is four timeslarger than that of PN. This is precisely due to the construc-tive interference of direct reflection and multiple Andreevreflection.18 Now the physics of pumping at resonance isclear. For the resonance pumping in the weak pumping re-gime, we are looking at the small neighborhood of the peak.The area of the neighborhood has to be small since it is theweak pumping. The neighborhood has to be around the peakwith x1;x2, since only around the peak the transmissioncoefficient is approximately 1. As a result, we obtain imme-diately that the pumped charge or pumped current of the NSstructure near the resonance is four times that of the corre-15331sponding normal structure. In the other extreme, for strongpumping, we take a large contour enclosing entire resonanceline. Since PNS decreases much faster than PN away fromthe peak, it is understandable that the pumped charges ~theintegral of P over the area enclosed by the contour! for bothnormal and NS structures are equal, which will be shownanalytically below.After the expansion in powers of d in Eqs. ~18! and ~19!and keeping the leading orders of d , we haveF35p7@21p~211z !#~211z !3~11z !4d964 , ~24!F452~12p !2d21F2 16 1 2p3 1 12 p2~211z2!1116 p4~211z2!2Gd4. ~25!So Eq. ~3! becomesQNS5 epE0‘pdpE211dzF3F43 d2, ~26!using the fact that limd2.0d5/(x21d2)35(3/8)pd(x), Eq.~26! becomesQNS53A2eE211dz~12z2!3~11z !2~116z21z4!5/252e . ~27!Hence the pumped charge for NS system is quantized at thesame value as that of the normal structure.Now we have a better physical picture for the transportproperties of the NS structure. For the conductance or the1-3BRIEF REPORTS PHYSICAL REVIEW B 65 153311I-V curve, we need S21 or She . For normal structure, thecurrent is given by IN52e/h*dE@ f (E2eV1)2 f (E2eV2)#uS12u2 and hence at resonance and at zero tempera-ture GN5IN/(V12V2)52e2/h . For NS structure, wehave20,32 INS52e/h*dE@ f (E2eV1)2 f (E1eV1)#uS heu2and at resonance GNS5INS/V154e2/h , which is the well-known doubling of the conductance. For pumped charge orpumped current at resonance, however, it depends only on]xiS11 or ]xiSee (E50 is assumed!. Due to the constructiveinterference between direct reflection and multiple Andreevreflection in the weak pumping regime, the charge transferincreases by a factor of 4 when one of the leads becomessuperconducting. In the strong pumping regime, however,the charge transfer is quantized at the value equal to that ofnormal structure, if the pumping contour is chosen such thatthe resonance line is enclosed. The physics behind this canbe understood as follows. In the normal case, the contourenclosing the resonance line in the parameter space passesthrough the resonance line at two points (x1 ,x2)5(0,2d)when the left contact is almost closed and (2d ,0) when theright contact is almost closed. When passing through thesetwo points, the resonance level of the dot crosses the Fermienergy. At each crossing, the occupation of the level changes,and two electrons with opposite spins enter or exit the regionbetween the barriers. Since one of the tunnel barriers haszero conductance at these points, it is clear that the electronsmust have tunneled through the other contact upon entering15331or leaving the quantum dot. Hence, in the pumping cycle,electrons are shuttled pairwise through the dot. In the pres-ence of the superconducting lead, the resonance level ~boththe energy and the width! is exactly the same as that ofnormal case since the scattering matrix is given by She5iuS12u2/(11uS22u2) when E50. Therefore, the same argu-ment applies to the superconducting case and the quantiza-tion unit is 2e . Note that our statement is only valid when theelectron interaction is neglected. For the case of two normal-metal contacts, if interactions are included the quantizationwill remain, but now the quantum is only e: Only one elec-tron at a time can enter the region between the barriers; ad-dition of a second electron is forbidden by the Coulombblockade. In the presence of the superconducting lead, sincethe Andreev reflection requires two electrons with oppositespins in order to produce the supercurrent, it seems that thepumping is not allowed in the strong pumping regime due tothe Coulomb blockade. In this paper, we have also neglectedeffects of the temperature and inelastic scattering. As dis-cussed in Ref. 17 the temperature will destroy the quantiza-tion of the pumped charge. When inelastic channel is presentan additional physical mechanism for an incoherent pumpeffect will show up.33We gratefully acknowledge support by a RGC grant fromthe SAR Government of Hong Kong under Grant No. HKU7215/99P.1 P.W. Brouwer, Phys. Rev. B 58, R10 135 ~1998!.2 M. Switkes et al., Science 283, 1905 ~1999!.3 F. Zhou, B. Spivak, and B.L. Altshuler, Phys. Rev. Lett. 82, 608~1999!.4 M. Wagner, Phys. Rev. Lett. 85, 174 ~2000!.5 J.E. Avron et al., Phys. Rev. B 62, R10 618 ~2000!.6 I.L. Aleiner, B.L. Altshuler, and A. Kamenev, Phys. Rev. B 62,10 373 ~2000!.7 Y.D. Wei, J. Wang, and H. Guo, Phys. Rev. B 62, 9947 ~2000!.8 M.G. Vavilov, V. Ambegaokar, and I.L. Aleiner, Phys. Rev. B 63,195313 ~2001!.9 P.W. Brouwer, Phys. Rev. B 63, 121303 ~2001!; M.L. Polianskiand P.W. Brouwer, ibid. 64, 075304 ~2001!.10 P. Sharma and C. Chamon, Phys. Rev. Lett. 87, 096401 ~2001!.11 X.B. Wang and V.E. Kravtsov, Phys. Rev. B 64, 033313 ~2001!.12 Y.D. Wei et al., Phys. Rev. B 64, 115321 ~2001!.13 D.J. Thouless, Phys. Rev. B 27, 6083 ~1983!.14 Q. Niu, Phys. Rev. Lett. 64, 1812 ~1990!.15 I.L. Aleiner and A.V. Andreev, Phys. Rev. Lett. 81, 1286 ~1998!.16 T.A. Shutenko, I.L. Aleiner, and B.L. Altshuler, Phys. Rev. B 61,10 366 ~2000!.17 Y. Levinson, O. Entin-Wohlman, and P. Wolfle, Physica A 302,335 ~2001!.18 J. Wang et al., Appl. Phys. Lett. 79, 3977 ~2001!.19 A.F. Andreev, Zh. E´ ksp. Teor. Fiz. 46, 1823 ~1964! @Sov. Phys.JETP 19, 1228 ~1964!#.20 C.W.J. Beenakker, Rev. Mod. Phys. 69, 731 ~1997!.21 A factor of 2 has been included for spin.22 The units are fixed by setting \52m51 in the following analy-sis. For the GaAs system with a51000 Å, the energy uint isE556 meV.23 This formula can be derived using the Keldysh nonequilibriumGreen’s function method ~see Ref. 24! and we have assumedthat the Coulomb blockade effect can be neglected.24 B.G. Wang, J. Wang, and H. Guo, Phys. Rev. B 65, 073306~2002!.25 T. Gramespacher and M. Buttiker, Phys. Rev. B 61, 8125 ~2000!.26 J. Wang et al., Phys. 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Quantization of adiabatic pumped charge in the presence of superconducting lead

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Quantization of adiabatic pumped charge in the presence of
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Quantization of adiabatic pumped charge in the presence of superconducting lead

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