We develop a theory for describing frictional drag in bilayer systems with
in-plane periodic potential modulations, and use it to investigate the drag
between bilayer systems in which one of the layers is modulated in one
direction. At low temperatures, as the density of carriers in the modulated
layer is changed, we show that the transresistivity component in the direction
of modulation can change its sign. We also give a physical explanation for this
behavior.Comment: 4 pages, 4 figure

A. Kamenev

A.-P. Jauho

Antti-Pekka Jauho

Audrius Alkauskas

B.N. Narozhny

Ben Yu-Kuang Hu

C. Albrecht

C. Jörger

D. Weiss

F. von Oppen

H. Rubel

J.G.S. Lok

K. Flensberg

Karsten Flensberg

L. Zheng

L. Świerkowski

M.C. Bønsager

M.P. Lilly

M.S. Gudiksen

N.A. Mortensen

N.P.R. Hill

R. Pillarisetty

R.A. Deutschmann

T.J. Gramila

U. Sivan

English

arXiv

Alkauskas, Audrius

Flensberg, Karsten

Hu, Ben Yu Kuang

Jauho, Antti Pekka

Copenhagen University Research Information System

Sign reversal of drag in bilayer systems with in-plane periodic potential modulation

Crossref

We develop a theory for describing frictional drag in bilayer systems with in-plane periodic potential modulations, and use it to investigate the drag between bilayer systems in which one of the layers is modulated in one direction. At low temperatures, as the density of carriers in the modulated layer is changed, we show that it is possible for the transresistivity component in the direction of modulation to change its sign. We also give a physical explanation for this behavior

Alkauskas, A.

Hu, Ben Yu-Kuang

Jauho, Antti-Pekka

Online Research Database In Technology

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Downloaded from orbit.dtu.dk on: Dec 17, 2017Sign reversal of drag in bilayer systems with in-plane periodic potential modulationAlkauskas, A.; Flensberg, Karsten; Hu, Ben Yu-Kuang; Jauho, Antti-PekkaPublished in:Physical Review B (Condensed Matter and Materials Physics)Link to article, DOI:10.1103/PhysRevB.66.201304Publication date:2002Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Alkauskas, A., Flensberg, K., Hu, B. Y-K., & Jauho, A-P. (2002). Sign reversal of drag in bilayer systems with in-plane periodic potential modulation. Physical Review B (Condensed Matter and Materials Physics), 66(20),201304. DOI: 10.1103/PhysRevB.66.201304Sign reversal of drag in bilayer systems with in-plane periodic potential modulationAudrius Alkauskas, 1,2,4 Karsten Flensberg,1 Ben Yu-Kuang Hu,3 and Antti-Pekka Jauho,41O rsted Laboratoriet, Universitetsparken 5, Kobenhavns Universitet, DK-2100 Kobenhavn O , Denmark2Physics Faculty, Vilnius University, Saule˙tekio 9, Vilnius 2040, Lithuania3Department of Physics, University of Akron, Akron, Ohio 44325-40014Mikroelektronik Centret, Danmarks Tekniske Universitet, DK-2800 Lyngby, Denmark~Received 12 June 2002; revised manuscript received 28 August 2002; published 26 November 2002!We develop a theory for describing frictional drag in bilayer systems with in-plane periodic potentialmodulations, and use it to investigate the drag between bilayer systems in which one of the layers is modulatedin one direction. At low temperatures, as the density of carriers in the modulated layer is changed, we show thatit is possible for the transresistivity component in the direction of modulation to change its sign. We also givea physical explanation for this behavior.DOI: 10.1103/PhysRevB.66.201304 PACS number~s!: 73.23.2b, 73.50.2h, 73.61.2rThroughout the past decade there has been a great deal ofexperimental and theoretical activity in frictional drag in bi-layer systems, following the seminal experiments by Gramilaet al.1 These drag experiments involved a double quantum-well system where the layers are individually contacted byingenious fabrication techniques. The barrier between thewells is made thick enough to suppress tunneling but thinenough to allow significant interlayer interactions. An aver-age current density j1 is driven through layer 1 and circuit iskept open in layer 2, so that j250. The interlayer interactioncauses the electrons in layer 1 to drag along the electrons inlayer 2, and hence a counterbalancing electric field E2 formsin layer 2 to maintain a zero net j2. The transresistivity ten-sor rJ21 , defined by E25rJ21j1, can be extracted experimen-tally and can reveal important information about the proper-ties of the effective interlayer interactions, the individuallayers and the coupled bilayer system.Since the original work of Gramila et al.,1 which wasdone on a closely spaced electron-electron system at lowtemperatures without an applied magnetic field, many varia-tions on the theme of the original experiments have beenperformed. For instance, drag has been measured inelectron-hole2 and hole-hole3 systems, in widely separatedlayers,4,5 and in the presence of a perpendicular magneticfield.3,6–8 Very recently, low-density systems have been stud-ied to probe the suggested metal-insulator phase transition instrongly correlated disordered two-dimensional systems.9,10Another modern trend is to examine mesoscopic effects inCoulomb drag.11–13 In general, drag without an applied Bfield is reasonably well understood within the framework ofa standard weak-interlayer coupling theory.14–16 The theorysuccessfully accounts for several unusual features such aslarge enhancements in the transresistivity ~up to an order ofthe magnitude; some intriguing discrepancies, however, dopersist for the most dilute systems studied10! due to intra-layer correlations17 and plasmon mediated scattering.18,19 Onthe other hand, the understanding of magnetodrag ~i.e., dragin the presence of a perpendicular B field! in bilayer systemsis less complete, and several puzzling experimental resultsremain unexplained. For instance, under certain circum-stances, the diagonal terms in the magnetotransresistivity(r21xx and r21yy) has been observed to reverse sign when thechemical potential is changed in one layer while being keptfixed in the other.20 This sign reversal with changing chemi-cal potential ~which incidentally has not been observed atB50) cannot be obtained from magnetodrag calculations us-ing the self-consistent Born approximation,21,22 and despiterecent theoretical progress,23 a fully satisfactory explanationof this phenomenon is not yet available.In this paper, we suggest that a reversal of the sign of thetransresistivity is possible at B50 in bilayer systems thathave periodic potential modulations in the plane of the lay-ers. The periodic potential modulation creates minibands,and the charge carriers can evolve from electronlike to hole-like behavior with a relatively small change in the density.Furthermore, for systems that are modulated in one direction,it is possible to observe anisotropic drag, implying that theelectric-field response in the drag layer is in a different di-rection from that of the driving current. This demonstratesthe important role band structure plays in determining thetransresistivity of the system.24 We note that experiments ontwo-dimensional electron gases with strong potential modu-lations in one direction have already been reported in theliterature,25 and hence we believe that the theory describedbelow is amenable to experimental tests in near future.To investigate drag in these modulated systems, we usethe Kubo formalism15,16 to calculate the transconductivitytensor sJ21 , which is related to the transresistivity by rJ2152rJ22sJ21rJ11 in the weak-interlayer coupling limit. In thismethod, the transconductivity is expressed as a current-current correlation function, which can be calculated withstandard perturbation-theory techniques.The Hamiltonian of the system is Hˆ 5( i51,2Hˆ i1Hˆ 12 ,where Hˆ i is the Hamiltonian of layer i and Hˆ 12 is the inter-layer interaction term. We assume Hˆ 21 is due to Coulombinteractions, so that Hˆ 125A 21(nˆ 1(q)nˆ 2(2q)V12(q),where nˆ i(q) and V12(q) are the Fourier transforms of thedensity operator and the interlayer Coulomb interaction, re-spectively.We define, within the Matsubara formalism, DW to be thecorrelation function26DW ~q,q8;ivn ,ivn8!52E0bdtE0bdt8eivnte2ivn8t83^TtJˆ~0 !nˆ ~q,t!nˆ ~2q8,t8!&0 . ~1!RAPID COMMUNICATIONSPHYSICAL REVIEW B 66, 201304~R! ~2002!0163-1829/2002/66~20!/201304~4!/$20.00 ©2002 The American Physical Society66 201304-1For systems that have a periodic potential modulation withreciprocal vectors G, only q2q85G terms are nonzero. Ex-panding in powers of V12 , the first nonvanishing term fors21 in the dc limit is the second-order term. We obtains21dg5e2hA (q (G1G2V12~q!V12~2q1G1!dG1G23E0‘dv2p D2d~q,q1G2 ;v1i01,v2i01!3D1g~q,q1G1 ;v1i01,v2i01!@2]vnB~v!# .~2!Evaluation of DW (q,q1G;v1i01,v2i01) is analogousto Ref. 16. In this paper, we assume the system is in the weakscattering limit, ,@d ,a (, is the impurity mean-free path, dis the layer separation, and a is the modulation period!27allowing us to ignore vertex corrections at the charge verti-ces. Then, one obtainsDW ~q,q1G,v1i01,v2i01!54pA (knn8@vn8k1qt tr,n8~k1q!2vnkt tr,n~k!#3@~nF~«nk!2nF~«nk2v!#d~«nk2«n8k1q2v!3h~k1qn8,kn;q!h~kn ,k1qn8;2q2G!. ~3!Here, vnk is the band velocity, « is the energy, nF(«)5$exp@b(«2m)#11%21 (m is the chemical potential!, t tr isthe transport time, h(k8n8,kn;q)5^k8n8uexp(2iqr)ukn&(n ,n8 are the band indices!. The sJ21 obtained using Eq. ~2!and the weak scattering result Eq. ~3! can, with the approxi-mation h51, also be derived from the semiclassical Boltz-mann equation.19A complete calculation of drag, using Eq. ~3! in Eq. ~2!, isan arduous task, requiring a numerical evaluation of the bandstructure~s! of the layers, calculation of the matrix elementsh(k8n8,kn;q), and summation of different bands n, and re-ciprocal lattice vectors G. For incommensurate lattices onealways has G1505G2 , and in the remaining part of thepaper we assume this to be the case. The other technicalsteps do not pose conceptual difficulties, and in the presentcontext we find it appropriate to consider simplified systemswhere to a certain extent analytic progress can be made, andfor which the physics is transparent.The central issue of this paper is the possible sign reversalof the drag. We demonstrate this first for a one-dimensionalmodel,28 neglecting interband processes and the momentumdependence of the transport relaxation time. For this case,correlation function ~3! becomes D5t trF(q ,v), whereF~q ,v!522pL (k ~vk2vk1q!3@nF~«k!2nF~«k1q!#d~«k2«k1q2v!5(kisign~vki2vki1q!@nF~«ki!2nF~«ki1q!# ,~4!where ki are the solutions of «ki2«ki1q2v50. For illustra-tive purposes, we consider a cosine band, «k52\2/(ma2)cos ka, for which there are two ~or no! solutions,and one findsF~q ,v!5sign~v22v1!@nF~2«2!2nF~«1!2nF~2«1!2nF~«2!# . ~5!At half-filling the chemical potential m vanishes, and makinguse of nF ,m(2«)512nF ,2m(«), it is easy to see that theresult ~5! is an odd function of m. Thus, in an experimentwhere one of the subsystems is kept unchanged while in theother the chemical potential is moved through half-filling,the drag will change sign. While the above discussion is animportant demonstration of principle, it is necessary to alsoconsider periodically modulated two-dimensional electrongases, which are the most commonly studied systems in thiscontext.A system which has an identical periodic modulation inboth x and y directions is characterized by particle-hole sym-metry, and it seems natural that the drag passes through zerowhen the two carrier species are matched. The experimen-tally most relevant systems are those, however, where themodulation is only in one direction25 ~the strongest modula-tions have been achieved for these systems!, and hencewe choose the model system as follows: ~1! There is a singleband ~the dispersion law and corresponding density ofstates are illustrated in Fig. 1! with a tight-binding disper-sion relation29 «(kx ,ky)5\2@12cos(kxa)#/(mxa2)1\2ky2/(2my), and hence the velocity components are vx(kx)5\sin(kxa)/(mxa) and vy(ky)5\ky /my . ~2! t tr is k indepen-dent. ~3! The interlayer interaction V12(q) is significant onlyfor small q.30At low temperatures, it would appear permissible to ex-pand in v , because ]nB(v)/]v in the integrand cuts off thehigher v contributions. Following this procedure yields ananalytic expression for F, and the resulting drag resistivityFIG. 1. The density of states for a two-dimensional system withperiodical modulation in one direction. The inset shows the constantenergy surfaces for the dispersion relation used in this work. Theenergy is in units of kBT052\2/(mxa2).RAPID COMMUNICATIONSALKAUSKAS et al. PHYSICAL REVIEW B 66, 201304~R! ~2002!201304-2obeys the familiar T2 dependence known from unmodulatedtwo-dimensional systems.1,14 In this scheme, the transresisi-tivity diverges when m52\2/(mxa2) with the opposite signfrom the low density rxx. This divergence, which is relatedto the divergence in the density of states at this energy ~seeFig. 1!, is unphysical because it only occurs in the experi-mentally unreachable T50 limit. Nevertheless, it is interest-ing to observe that this approximation leads to a change ofsign in the transconductance, and we should expect this be-havior to be most prominent when m52\2/(mxa2).To cure this spurious divergence one must avoid expan-sions, and perform a numerical evaluation. As a startingpoint we have found it convenient to useFx~q,v!5my~2p!2\qymxaE2p/a1qx/2p/a2qx/2dkx@sin~kx2qx/2!a2sin~kx1qx/2!a#$nF@kx2qx/2,ky0~kx!2qy/2#2nF@kx1qx/2,ky0~kx!1qy/2#%, ~6!whereky0~kx!5my\2qyF\v1 \2mxa2 @cos~kx1qx/2!a2cos~kx2qx/2!a#G . ~7!While the Kubo formula gives the transconductivity, it isoften most convenient to express the results in terms of tran-sresisistivity ~this is the object usually recorded in experi-ments! rJ21 , whose components are given byr21xx52s21xxs11xxs22xx2s12xxs21xx.2s21xxs11xxs22xx, ~8!and analogously for the yy component. The transresisivitytensor has the additional advantage that does not involve thetransport relaxation times for the individual layers, as long asthese are momentum independent. The computed transresisi-tivity is shown in Fig. 2 for four different temperatures. Themost important feature is that the drag indeed changes sign;the effect is most prominent for low temperatures, and den-sities close to a fully occupied band.31 In Figs. 3 and 4 wecompare the xx and yy components of the computedtransconductances ~the xx component was used in calculat-ing the results of Fig. 2!. We observe that the sign changedoes not take place for s12yy, nevertheless an interestingdouble-peak structure emerges.An analysis of the several assumptions made in our cal-culations is now in place. We have assumed that the systemonly has one band. Clearly, this assumption breaks downwhen the density so large that Fermi energy significantlyexceeds 2\2/(2mxa2), because the carriers will start to oc-cupy higher bands. We also have assumed that temperature islow enough that the inelastic mean-free path , in is muchlonger than period of the potential modulation, a. For finite, in the system is roughly divided into coherent regions oforder , in2. If , in&a , the electrons do not coherently feel theFIG. 2. The calculated transresistivity rDxx, as a function of thedensity for four different temperatures. Superlattice perioda5140 Å, and the layer separation is 280 Å.FIG. 3. The normalized transconductivity sDxx, as a function ofthe density for the same temperatures as in Fig. 1.FIG. 4. The normalized transconductivity sDyy, as a function ofthe density for four different temperatures.RAPID COMMUNICATIONSSIGN REVERSAL OF DRAG IN BILAYER SYSTEMS . . . PHYSICAL REVIEW B 66, 201304~R! ~2002!201304-3periodic potential, and the drag characteristics will be givenby an average of the drag over the density fluctuation causedby the potential modulation. Since the system as a whole actslike a ~nearly! uniform system in this case, effects describedin this paper will not be observable at temperatures for which, in&a .To summarize, we have developed a theory for drag inbilayer systems where there is a periodic potential modula-tion. We have calculated the drag for the case where there ispotential modulation in one direction in one of the two lay-ers. For the studied system, at low temperatures, the transre-sistivity changes sign as the density is increased.31 Experi-mentally, it may be possible to fabricate the systeminvestigated here by overgrowing a pair of quantum wellsover a cleaved edge,25 or using lithographic techniques.32A.A. acknowledges a NORFA grant, which was crucialfor this study. B.Y.K.H. was supported by DOE Grant No.DE-G02001ER45948, Research Corporation Grant No.CC5168, and by the University of Akron. We are pleased toacknowledge collaboration with Niels Asger Mortensen andHenrik Smith during the early stages of this project.1 T.J. Gramila, J.P. Eisenstein, A.H. MacDonald, L.N. Pfeiffer, andK.W. West, Phys. Rev. Lett. 66, 1216 ~1991!.2 U. Sivan, P.M. Solomon, and H. Shtrikman, Phys. Rev. Lett. 68,1196 ~1992!.3 C. Jo¨rger, S.J. Cheng, H. Rubel, W. Dietsche, R. Gerhardts, P.Specht, K. Eberl, and K. von Klitzing, Phys. Rev. B 62, 1572~2000!.4 T.J. Gramila, J.P. Eisenstein, A.H. MacDonald, L.N. Pfeiffer, andK.W. West, Phys. Rev. B 47, 12 957 ~1993!.5 T.J. Gramila, J.P. Eisenstein, A.H. MacDonald, L.N. Pfeiffer, andK.W. West, Physica B 197, 442 ~1994!.6 N.P.R. Hill, J.T. Nicholls, E.H. Linfield, M. Pepper, D.A. Ritchie,A.R. Hamilton, G.A.C. Jones, J. Phys.: Condens. Matter 8, L557~1996!.7 H. Rubel, A. Fisher, W. Dietsche, K. von Klitzing, and K. Eberl,Phys. Rev. Lett. 78, 1763 ~1997!.8 M.P. Lilly, J.P. Eisenstein, L.N. Preiffer, and K.W. West, Phys.Rev. Lett. 80, 1714 ~1998!.9 R. Pillarisetty, Hwayong Noh, D.C. Tsui, E.P. De Poortere, E.Tutuc, and M. Shayegan, Phys. Rev. Lett. 89, 016805 ~2002!.10 E.H. Hwang and S. Das Sarma, cond-mat/0202249 ~unpublished!.11 B.N. Narozhny and I.L. Aleiner, Phys. Rev. Lett. 84, 5383 ~2000!.12 N.A. Mortensen, K. Flensberg, and A.P. Jauho, Phys. Rev. Lett.86, 1841 ~2001!.13 N.A. Mortensen, K. Flensberg, and A.P. Jauho, Phys. Rev. B 65,085317 ~2002!.14 A.-P. Jauho and H. Smith, Phys. Rev. B 47, 4420 ~1993!.15 A. Kamenev and Y. Oreg, Phys. Rev. B 52, 7516 ~1995!.16 K. Flensberg, B.Y.-K. Hu, A.-P. Jauho, and J. Kinaret, Phys. Rev.B 52, 14 761 ~1995!.17 L. S´wierkowski, J. Szyman´ski, and Z.W. Gortel, Phys. Rev. Lett.74, 3245 ~1995!.18 K. Flensberg and B.Y.-K. Hu, Phys. Rev. Lett. 73, 3572 ~1994!.19 K. Flensberg and B.Y.-K. Hu, Phys. Rev. B 52, 14 796 ~1995!.20 J.G.S. Lok, S. Kraus, M. Pohlt, W. Dietsche, K. von Klitzing, W.Wegscheider, and M. Bichler, Phys. Rev. B 63, 041305~R!~2001!.21 M.C. Bonsager, K. Flensberg, B.Y.-K. Hu, and A.-P. Jauho, Phys.Rev. Lett. 77, 1366 ~1996!.22 M.C. Bonsager, K. Flensberg, B.Y.-K. Hu, and A.-P. Jauho, Phys.Rev. B 56, 10 314 ~1997!.23 F. von Oppen, S.H. Simon, and A. Stern, Phys. Rev. Lett. 87,106803 ~2001!.24 Miniband effects have recently been observed by C. Albrecht,J.H. Smet, D. Weiss, K. von Klitzing, R. Hennig, M. Langen-buch, M. Suhrke, U. Rossler, V. Umansky, and H. Schweizer,Phys. Rev. Lett. 83, 2234 ~1999!.25 R.A. Deutschmann, W. Wegscheider, M. Rother, M. Bichler, G.Abstreiter, C. Albrecht, and J.H. Smet, Phys. Rev. Lett. 86, 1857~2001!.26 Note that this definition is different from the D used in Ref. 16.The D defined in Ref. 16 is appropiate only for homogenoussystems.27 Strong scattering ~diffusive! effects are negligible at B50, see,e. g., L. Zheng and A.H. MacDonald, Phys. Rev. B 48, 8203~1993!.28 Importantly, significant progress towards fabricating modulatedone-dimensional nanowires has recently been achieved by M.S.Gudiksen, L.J. Lauhon, J. Wang, D.C. Smith, and C.M. Lieber,Nature ~London! 415, 617 ~2002!.29 This energy spectrum underlies the magnetoresistance oscillationsdiscovered by D. Weiss, K. von Klitzing, K. Ploog, and G. We-imann, Europhys. Lett. 8, 179 ~1989!.30 In our calculations we use the statically screened Coulombinteraction, V(q)5(2pe2/k)q/@2qTF,1qTF,2sinhqd1q(q1qTF,11qTF,2)expqd#, where qTF,i are the Thomas-Fermi screeningwave numbers for the two layers. For the modulated layer thescreening wave number has a slight temperature and densitydependence, which, however, has a rather small effect for thecalculated drag.31 It can be shown that a necessary condition for the sign change ofs21 ii is that ]@t tr(k)v i(k)#/]ki vanishes when integrated overthe Fermi surface. The sign-change is thus not a special propertyof the cosine-band.32 T. J. Gramila ~private communication!.RAPID COMMUNICATIONSALKAUSKAS et al. PHYSICAL REVIEW B 66, 201304~R! ~2002!201304-4

Sign-reversal of drag in bilayer systems with in-plane periodic
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Sign-reversal of drag in bilayer systems with in-plane periodic potential modulation

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