We develop a method to calculate electronic transport properties through a
mesoscopic scattering region in the presence of a series of time-periodic
potentials. Using the method, the quantum charge pumping driven by
time-periodic potentials is studied. Jumps in the pumped current are observed
at the peak positions of the Wigner delay time. Our main results in both the
weak pumping and strong pumping regimes are consistent with experimental
results. More interestingly, we also observed the nonzero pumping at the phase
difference phi=0 and addressed its relevance to the experimental result.Comment: 5 page

B. L. Altshuler

B. Wang

D. J. Thouless

E. P. Wigner

F. T. Smith

F. Zhou

G. Burmeister

I. L. Aleiner

M. Büttiker

M. Switkes

M. Wagner

P. W. Brouwer

Q. Niu

R. Landauer

S. L. Zhu

Shi-Liang Zhu

T. A. Shutenko

W. Li

Y. Levinson

Y. Wei

Z. D. Wang

English

arXiv

We develop a method to calculate electronic transport properties through a mesoscopic scattering region in the presence of a series of time-periodic potentials. Using the method, the quantum charge pumping driven by time-periodic potentials is studied. Jumps in the pumped current are observed at the peak positions of the Wigner delay time. Our main results in both the weak pumping and strong pumping regimes are consistent with experimental results. More interestingly, we also observed the nonzero pumping at the phase difference φ=0 and addressed its relevance to the experimental result.published_or_final_versio

Zhu, SL

Wang, ZD

HKU Scholars Hub

Title Charge pumping in a quantum wire driven by a series of localtime-periodic potentialsAuthor(s) Zhu, SL; Wang, ZDCitation Physical Review B - Condensed Matter And Materials Physics,2002, v. 65 n. 15, p. 1553131-1553135Issued Date 2002URL http://hdl.handle.net/10722/43350Rights Creative Commons: Attribution 3.0 Hong Kong LicensePHYSICAL REVIEW B, VOLUME 65, 155313Charge pumping in a quantum wire driven by a series of local time-periodic potentialsShi-Liang Zhu1,2 and Z. D. Wang1,3,*1Department of Physics, University of Hong Kong, Pokfulam Road, Hong Kong, China2Department of Physics, South China Normal University, Guangzhou 510631, China3Department of Material Science and Engineering, University of Science and Technology of China, Hefei, China~Received 14 November 2000; revised manuscript received 8 March 2001; published 28 March 2002!We develop a method to calculate electronic transport properties through a mesoscopic scattering region inthe presence of a series of time-periodic potentials. Using the method, the quantum charge pumping driven bytime-periodic potentials is studied. Jumps in the pumped current are observed at the peak positions of theWigner delay time. Our main results in both the weak pumping and strong pumping regimes are consistent withexperimental results. More interestingly, we also observed the nonzero pumping at the phase difference f50 and addressed its relevance to the experimental result.DOI: 10.1103/PhysRevB.65.155313 PACS number~s!: 73.23.2b, 72.10.Bg, 73.50.Pz, 03.65.TaA parametric electron pump has attracted considerable at-tention in recent years.1–12 It is a device that generates a dccurrent at zero bias by cyclic deformations of systemparameters.1–3 The quantum pumping mechanism was origi-nally proposed by Thouless,1 who studied the integrated par-ticle current produced by a slow periodic variation of thepotential, and showed that in a finite torus the integral of thecurrent over a period can vary continuously, but it must havean integer value in an infinite periodic system with fullbands. Such quantized charge transport was proposed to be-come an electric current standard.4Quite recently, the charge pumping was observedexperimentally.5 For technical reasons, instead of measuringcharge currents, the pumped dc voltage Vdot is measured in aquantum dot where two gates with oscillating voltages con-trol the deformation of the shape of the dot. For weak pump-ing, the observed charge pumping has a sinusoidal depen-dence on the phase difference f between the two shape-distorting ac voltages applied to the gates, and isproportional to the square of pumping strength V. For strongpumping, the pumped current deviates from the square de-pendence on V and becomes nonsinusoidal, being alwaysantisymmetric about f5p . The charge pumping may have aclose relation to the adiabatic Berry’s phase since the evolu-tion of the system is cyclic and is controlled by several sys-tem parameters, referred to as the parametric pumping.Based on this understanding, the total charge pumped percycle is proportional to the area enclosed by the path in theparameter space, and nonzero pumping current requires atleast two parameters.5,6 The pumped charge drived by twoparameters should be zero if two parameters are in phase(f50) since the area enclosed by the path is zero. However,it is in contradiction with the observed current I(f50)Þ0.5One of possible mechanisms of nonzero currents for f50 isphotovoltaic effects introduced in Ref. 2, where a surprisingresult, nonzero dc current generated by a single pumpinggate voltage, is also reported. The general physics of a quan-tum pump has been the subject of several theoreticalanalyses.2,3 Zhou et al. demonstrated that at low tempera-tures both the magnitude and the sign of the pumped chargesare sample specific quantities, and the typical value in disor-dered ~chaotic! systems turns out to be determined by quan-0163-1829/2002/65~15!/155313~5!/$20.00 65 1553tum interference effects. Another general expression for theaverage transmitted charge current was derived by Brouwer3under the adiabatic condition and based on the time-dependent S-matrix method,7 which appears to be quite suc-cessful for ~adiabatic! weak pumping. Adiabaticity heremeans that the oscillating period t of the system is muchlarger than the Wigner delay time tw .3,8 Note that the adia-batic condition does not simply imply that the pumpingstrength V should be very small. In fact, the adiabatic condi-tion requires that t must be larger as V increases. On theother hand, the pumping was not weak in the experiments.5The main purpose of the paper is to develop a theory, whichis also applicable in the case of strong pumping. By using theFloquet theorem, the photon-assisted transport has beentaken into account.11 We calculate the pumped currentthrough a mesoscopic region in the presence of time-periodicpotentials. Our main results in the weak pumping regime, aswell as those in the strong pumping regime are consistentwith the experiment reported in Ref. 5.Consider electrons transmitting through a one-dimensional scattering region ranging from x0 to x01d . Thepotential is given byV~x ,t !5H 0, x0,0, x.x01d ,Vs~x ,t !, x0<x<x01d ~1!with Vs(x ,t)5V01Vscos(vt1fs). The Schro¨dinger equa-tion can be written asi\]C~x ,t !]t52\22m*]2C~x ,t !]x21V~x ,t !C~x ,t !, ~2!with m* as the electron effective mass. Equation 2 can besolved by using the Floquet theorem.13 By setting CFl(x ,t)5e2iEFlt/\c(x ,t), where EFl is the Floquet eigenenergy andc(x ,t) is a periodic function c(x ,t)5c(x ,t1t) with periodt52p/v , the Schro¨dinger equation takes the formEFlc~x ,t !52\22m*]2c~x ,t !]x22i\]c~x ,t !]t1V~x ,t !c~x ,t !.©2002 The American Physical Society13-1SHI-LIANG ZHU AND Z. D. WANG PHYSICAL REVIEW B 65 155313Substituting c(x ,t)5g(x) f (t), we have two separated equa-tions with an introduced constant E,2\22m*]2]x2g~x !1V0g~x !5Eg~x !, ~3!i\] f ~ t !]t2Vs cos~vt1fs! f ~ t !5~E2EFl! f ~ t !. ~4!Integrating Eq. ~4! givesf ~ t !5eiVssin fs /\v2i(E2EFl)t/\ (n52‘‘e2nfsJnS Vs\v D e2invt,~5!where Jn(x) is the Bessel function of the first kind of ordern . Since f (t) is periodic in time with period t , it followsfrom Eq. ~5! that E2EFl5mv with m as an integer. Theequation for g(x) has a solutiong~x !5e6ikmsx, ~kms !252m*~EFl1m\v2V0!/\2. ~6!Thus c(x ,t) becomescm~x ,t !5ei(Vs /\v)sin fs6ikmsx(nFn2me2invt, ~7!with Fn2m5exp@2i(n2m)fs#Jn2m(Vs /\v).We consider an incoming wave from the left with theenergy E05\2k02/2m*, then the outgoing waves should bedivided into different modes En , which satisfies En5E01n\v with n50,61,62, . . . . The propagating modesmean that En.0, while the evanescent modes mean thatEn<0. The latter exists only in the neighborhood of the os-cillating barrier and do not propagate. Denote kn5A2m*En/\ , the solutions of the Schro¨dinger equation canbe written asC l5 (n52‘‘~Ani eiknx1Anoe2iknx!e2iEnt/\,~x,x0!,Cs5e2iEFlt\ (m ,n52‘‘~ameikmsx1bme2ikmsx!Fn2me2invt,~x0<x<x01d!,Cr5 (n52‘‘~Bni e2iknx1Bnoeiknx!e2iEnt/\,~x.x01d!,where Ani and Bni are the probability amplitudes of the in-coming waves from the left and right, respectively, while Anoand Bno are those of the outgoing waves. We can characterizethe barrier by a scattering matrix S which is a matrix con-necting the incoming and outgoing channels15531S AoBo D 5SS AiBi D ,where the S matrix can be derived by the matching condi-tions for the wave function C(x ,t) and its derivative]xC(x ,t) at x5x0 and x5x01d . After eliminating am andbm , we have13S5S R→ T←T→ R←D , ~8!where T←5LLTLR21, R←5LR21RLR21, T→5LR21TLL , andR→5LLRLL . Here the left ~right! arrow indicates incomingwaves from right ~left!, the matrices LL and LR are defined as(LL)mn5exp@ iknx0#dmn and (LR)mn5exp@ ikn(x01d)#dmn . T and R are given byT5~C121D11C221D2!/2, ~9!R5~C121D12C221D2!/2, ~10!whereC15~Ls2 I˜ !KsF†2~Ls1 I˜ !F†K ,D152~Ls2 I˜ !KsF†2~Ls1 I˜ !F†K ,C25~Ls1 I˜ !KsF†2~Ls2 I˜ !F†K ,D25~Ls1 I˜ !KsF†1~Ls2 I˜ !F†K ,with the matrices (Ls)mn5exp@iknsd#dmn , (Ks)mn5kns dmn ,Kmn5kndmn , I˜ as the unit matrix and F† as the Hermitianconjugate of F. The electronic transport properties of thescattering region may be obtained straightforward from Eq.~8!.The above method may be generalized to l time-periodicbarriers described byV~x ,t !550, x,0, x.al ,V1~x ,t !, 0<x,a1 ,V2~x ,t !, a1<x,a2 , . . . ,Vl~x ,t !, al21<x<al ,~11!where V1(x ,t)5V101V1cos(v1t1f1), V2(x ,t)5V201V2cos(v2t1f2), . . . , and Vl(x ,t)5Vl01Vlcos(vlt1fl).This potential may be more a realistic model for experi-ments. Obviously the transport properties for each barriercan be characterized by an S matrix given bySa5S R→a T←aT→a R←a D ,where a51,2, . . . ,l , T→a, T←a, R→a, and R←a can be derivedby the same method presented above. Now the propagatingmode En should be replaced by3-2CHARGE PUMPING IN A QUANTUM WIRE DRIVEN BY . . . PHYSICAL REVIEW B 65 155313E~n j!5E01 (n j52‘‘n j\v j , ~ j51, . . . ,l !. ~12!The associated transfer matrix M a for the ath barrier may bederived directly from the Sa matrixM a5S ~T←a !21 2~T←a !21R→aR←a ~T←a !21 T→a 2R←a ~T←a !21R→a D .The total transfer-matrix M t for all those barriers is deter-mined byM t5S M 11t M 12tM 21t M 22t D 5M lM n2lM 1,where M i jt (i , j51,2) are the partitioned matrices with thesame size as T→a. The total scattering matrix St can be de-rived from M t asSt5S 2~M 11t !21M 12t ~M 11t !21M 22t 2M 21t ~M 11t !21M 12t M 21t ~M 11t !21D .In each cycle a net charge current may pass through thescattering region in the direction determined from the de-tailed form of St matrix. We define a net transmission coef-ficient ~for an incoming wave in mode E05E) byFIG. 1. The pumped currents I(f5p/2) versus the barrierheight V for different pumping frequencies. Dotted lines fit the re-lation I (V)}V2.15531Tnet5 (E(n j).0A2E~n j!m*@ uT→ ,n j0t ~E0!u22uT← ,n j0t ~E0!u2# .The average net current per period t ~for E0) through thosebarriers is j(E0)5Tnet(E0). If the system is connectedthrough two ideal leads to two electron reservoirs with thesame chemical potential m , the average pumped current perperiod t is given by13,14I~m!5eE0‘dEg~E ! f ~E2m!Tnet~E !, ~13!where g(E)5A2m*/E/h is the density of electrons contrib-uting to the current in one direction, and f (E2m) is theFermi-Dirac distribution. At zero temperature, it becomesI~m!52eh E0mdEAm*2E Tnet~E !. ~14!Another important quantity is the Wigner delay timewhich gives the time delay of the scattered electron due to itsinteraction with the scattering field ~here the oscillating po-tential!. It relates to the S matrix by15tw~E !52i\NcTrF ~St!† dStdE G52 i\Nc ddE ln~det St!, ~15!where Nc is the number of open channels. Physically, theWigner time represents the time spent by a wave packetFIG. 2. The pumped currents versus the phase difference f forthree different V and v53.0.3-3SHI-LIANG ZHU AND Z. D. WANG PHYSICAL REVIEW B 65 155313passing through the scattering region. The charge pumping issupposed to be adiabatic when t is much greater than theWigner delay time tw .It is obvious that the net charge transfer in one cycle iszero for a single time-periodic barrier since T→5T← . Thenthe simplest system which may induce the nontrivial chargepumping should include at least two barriers. As an examplewe consider a mesoscopic system with two time-periodicbarriers connected through ideal leads to two electron reser-voirs with the same chemical potentials m . The potentials aredescribed by V1(x ,t)5V101V1cos(vt), V2(x ,t)5V20, andV3(x ,t)5V301V3cos(vt1f). This appears to be a simplifiedmodel for the Switkes et al. experiment, nevertheless it turnsout that some essential characteristics can be exhibited, as wewill address below. In the following numerical calculations,m575 mev , m*50.067me ~with me as the mass of the freeelectron!, V205230 mev , V250, and Nc is determined by anatural condition: uTu21uRu221.0<ce with ce (51.031024 in this paper! as a defined error.The general characteristics of quantum pumping areshown in Figs. 1 and 2. The parameters in Figs. 1, 2, and 3are chosen as V105V30550 mev , and V15V35V . Figure 1shows that the pumped current I(V) is proportional to V2 forsmall pumping amplitude V, with the proportional factor de-pending on the driving frequency (\51). But it deviatesfrom V2 dependence for the strong pumping case. On theother hand, the pumped current is sinusoidal dependence onf for weak pumping, and becomes nonsinusoidal depen-dence on f when V increases, as seen in Fig. 2. Anotherimportant characteristic shown in Fig. 2 is that I(p1f ,V)FIG. 3. The pumped current and the Wigner delay time versusthe insert energy E0 for V57.0 and v56.0.1553152I(f,V) for all amplitude strengths, and uI(f ,V)u is maxi-mum at f5p/2 or f53p/2 for weak pumping. Remark-ably, these results for I(f ,V) are in agreement with the ex-perimental observation in Ref. 3.Figure 3 shows that sharp peaks in the Wigner time occurat the resonance insert energies E05n\v . In addition, jumpsin the pumped current as a function of E0 appear at the peakpositions of the Wigner time. The direction of the currentdepends crucially on the insert energy. It is interesting tonote that the adiabatic condition is not necessary in our cal-culations. Figure 3 indicates that the maximum value of tw isabout 5.5 ns for \v56.0 mev ~corresponding to t;0.7 ps), which is much greater than the pumping cyclictime t . Then we may say that the method described here isbeyond adiabaticity. Actually, the nonadiabatic effects areonly important for the strongly photon-assisted transportsince tw is greater than t only if the energy of the incomingwave is approximately equal to the resonance energy forphoton-assisted tunneling. Physically, by emitting or absorb-ing photons, the outgoing waves may be at the quantumstates different from that of the incoming wave. Conse-quently, the adiabatic condition, which requires that thequantum state is at the same instant state in the whole evo-lution, is not satisfied. Note that the formula derived byBrouwer3 may be valid merely under the adiabatic condition,and thus the method developed here may be quite useful.It is quilt intriguing to note from Fig. 4 that I(f50) isnonzero for V1ÞV3, while the corresponding areas enclosedby the path in the parameter space $V1(x ,t),V3(x ,t)% arezero. Although the pumped currents in the above case wereFIG. 4. The pumped currents versus the phase difference f forv53.0 for different V1 and V3.3-4CHARGE PUMPING IN A QUANTUM WIRE DRIVEN BY . . . PHYSICAL REVIEW B 65 155313predicated to be zero under the adiabatic approximation, thedeviation from zero is reported experimentally at strongpumping,5 just as we observed here in terms of a rigoroustheoretical analysis which is also valid for strong pumping.Moreover, from comparison with that I(f50)50 for V15V3, it is now clear that the present nonzero pumped cur-rents stem from the spatial asymmetry of potentials V1ÞV3,which is coincident with the result obtained by Wagner inRef. 11: the nonzero currents may be observed in a singleosscillating potential but with asymmetric static potential.Actually, to observe a pump current at zero applied bias, itseems that the inversion symmetry should be broken, eitherin real or in k space.The fact that I(f50) is nonzero at strong pumping maybe understood based on a scenario of the nonadiabatic geo-metric phase.16 Pumped currents are determined by geomet-ric phase accumulated in the evolution.1,5,6 Under the adia-batic approximation, I(f50)50 is predicted theoreticallybecause the corresponding adiabatic geometric phase is zero.While it is now clear that the nonadiabatic geometric phasemay be nonzero even in the case where the area enclosed bythe path in the parameter space is zero ~thus the adiabaticphase is zero!.16 Therefore, the nonadiabatic correction to thecurrents should be taken into account for strong pumpingwhenever the adiabatic condition is not well satisfied. Physi-15531cally, it is reasonable to believe that the observed nonzeropumping at phase f50 for the strong pumping stems fromthe nonadiabatic correction when the inversion symmetry isbroken. Practically, the asymmetric spatial potential might bepresent in the experiment, which may originate from eitherthe shape-distorting ac voltages, or from the internal poten-tial established during transport.7 Since the current calculatedin this approach is conserved since uTu21uRu251.0, nointernal potential appears explicitly in the present formulism.It is worth pointing out that nonzero pumped currents forf5p are also seen in Fig. 4, which seems to contradict withthat in Ref. 5. Also note that a nonzero I(f5p) was alsopredicted by another totally different theoretical study,17 sothis contradication is still an interesting open question atpresent.In summary, we developed a method to calculate thepumped current and Wigner delay time in a mesoscopic sys-tem with a series of time-periodic barriers connected to twoelectron reservoirs, which appears to be applicable for strongpumping cases.We thank the support from a RGC grant of Hong Kong~Grant No. HKU7118/00P! and a CRCG grant at the Univer-sity of Hong Kong.*To whom correspondence should be addressed. Email address:zwang@hkucc.hku.hk1 D. J. Thouless, Phys. Rev. B 27, 6083 ~1983!.2 F. Zhou, B. Spivak, and B. Altshuler, Phys. Rev. Lett. 82, 608~1999!.3 P. W. Brouwer, Phys. Rev. B 58, R10 135 ~1998!.4 Q. Niu, Phys. Rev. Lett. 64, 1812 ~1990!.5 M. Switkes, C. M. Marcus, K. Campman, and A. C. Gossard,Science 283, 1905 ~1999!.6 B. L. Altshuler and L. I. Glazman, Science 283, 1864 ~1999!.7 M. Bu¨ttiker, H. Thomas, and A. Preˆtre, Z. Phys. B: Condens.Matter 94, 133 ~1994!.8 A. Andreev and A. Kamenev, cond-mat/0001460 ~unpublished!.9 I. L. Aleiner and A. V. Andreev, Phys. Rev. Lett. 81, 1286 ~1998!;T. A. Shutenko, I. L. Aleiner, and B. L. Altshuler, Phys. Rev. B61, 10 366 ~2000!.10 Y. Levinson, O. Entin-Wohlman, and P. Wo¨lfle, Phys. Rev. Lett.85, 634 ~2000!.11 M. Wagner, Phys. Rev. Lett. 85, 174 ~2000!; M. Wagner and F.Sols, ibid. 83, 4377 ~1999!; M. Wagner, ibid. 76, 4010 ~1996!.12 Y. Wei, J. Wang, and H. Guo, Phys. Rev. B 62, 9947 ~2000!.13 G. Burmeister and K. Maschke, Phys. Rev. B 57, 13 050 ~1998!;W. Li and L. E. Reichl, ibid. 60, 15 732 ~1999!.14 R. Landauer, Philos. Mag. 21, 863 ~1970!; M. Bu¨ttiker, Phys.Rev. Lett. 57, 1761 ~1986!.15 E. P. Wigner, Phys. Rev. 98, 145 ~1955!; F. T. Smith, ibid. 118,349 ~1960!.16 S. L. Zhu and Z. D. Wang, Phys. Rev. Lett. 85, 1076 ~2000!; S. L.Zhu, Z. D. Wang, and Y. D. Zhang, Phys. Rev. B 61, 1142~2000!.17 B. Wang, J. Wang, and H. Guo, Phys. Rev. B 65, 073306 ~2002!.3-5

Charge pumping in a quantum wire driven by a series of local time-periodic potentials

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Charge pumping in a quantum wire driven by a series of local
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Charge pumping in a quantum wire driven by a series of local time-periodic potentials

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