We perform a semiclassical calculation of the magnetoresistance of spinless
two-dimensional fermions in a long-range correlated random magnetic field. In
the regime relevant for the problem of the half filled Landau level the
perturbative Born approximation fails and we develop a new method of solving
the Boltzmann equation beyond the relaxation time approximation. In absence of
interactions, electron density modulations, in-plane fields, and Fermi surface
anisotropy we obtain a quadratic negative magnetoresistance in the weak field
limit.Comment: 12 pages, Latex, no figures, Nordita repor

A. Smith

A.D. Mirlin

A.G. Aronov

A.K. Geim

B.I. Halperin

B.L. Altshuler

D. V. Khveshchenko

D.K.K. Lee

D.V. Khveshchenko

F.D.M. Haldane

J. Miller

L.B. Ioffe

L.D. Landau

L.P. Rokhinson

N. Nagaosa

P. Hedegard

P.A. Lee

P.B. Mancoff

T. Sugiyama

V. Kalmeyer

W. Kang

Y.B. Kim

English

arXiv

We perform a semiclassical calculation of the magnetoresistance of spinless two-dimensional fermions in a long-range correlated random magnetic field. In the regime relevant for the problem of the half filled Landau level the perturbative Born approximation fails and we develop a new method of solving the Boltzmann equation beyond the relaxation time approximation. In absence of interactions, electron density modulations, in-plane fields, and Fermi surface anisotropy we obtain a quadratic negative magnetoresistance in the weak field limit

Khveshchenko, D.V.

Carolina Digital Repository

VOLUME 77, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 26 AUGUST 1996onalofofons,raticMagnetoresistance of Two-Dimensional Fermions in a Random Magnetic FieldD. V. KhveshchenkoNordisk Institut for Teoretisk Atomfysik, Blegdamsvej 17, Copenhagen DK-2100, Denmark(Received 29 April 1996)We perform a semiclassical calculation of the magnetoresistance of spinless two-dimensifermions in a long-range correlated random magnetic field. In the regime relevant for the problemthe half filled Landau level the perturbative Born approximation fails and we develop a new methodsolving the Boltzmann equation beyond the relaxation time approximation. In absence of interactielectron density modulations, in-plane fields, and Fermi surface anisotropy, we obtain a quadnegative magnetoresistance in the weak field limit. [S0031-9007(96)01015-0]PACS numbers: 72.10.Bgllyofr oF-IItotinaufatemilegyarylenaartralytimti-caistoofatdeseetnialar-ts ofs.asendHis in.ionceen)as[1]rgeoftheionFltu-llarThe problem of two-dimensional transport in spatiarandom static magnetic fields (RMF) attracted a lotattention over the last few years. There exists a numbeexperimental realizations of two-dimensional (2D) RMcreated by randomly pinned flux vortices in a typesuperconducting gate, grains of a type-I superconducor a demagnetized permanent magnet placed nearby2D electron gas [1]. A formally similar problem arisesthe contexts of the gauge theory of highTc cuprates [2]and the composite fermion theory of the half filled Landlevel (HFLL) [3,4].Various analytical and numerical results obtained soseem to indicate that in the case of a continuum syswith unbound spectrum all states are localized [5], whin the lattice case there might be an extended statethe center of the band [6]. The effective low-enerdescription in terms of the unitarys model constructed in[7] suggests that the RMF problem belongs to the unitrandom ensemble. Compared to the conventional probof potential scattering corresponding to the orthogocase, the effects of quantum interference in the RMFsuppressed as a result of broken time-reversal symmeIn particular, the logarithmic temperature dependent welocalization corrections to the conductivity appear onin the next order in the metallicity parameterkFl ¿ 1:dwls ­ 2s1yp2kFld lnstwyttr d [where the temperaturedependence comes from the inelastic phase breakingtwsT d in the regimetwsTd ¿ ttr ­ lyyF ]. It suggests alarger localization lengthLloc , l expfsp2y4d skFld2g thanin the orthogonal case.It was shown in [8] that quantum fluctuations of parcle’s positions in the direction transverse to their classitrajectories are strongly reduced by the RMF. And sothe contribution of quantum backscattering which leadsa negative magnetoresistance (MR) in the presenceweak uniform external magnetic fieldB.Therefore, in the RMF problem one might expect ththe semiclassical transport theory is applicable in a wirange of length scales and temperatures than in the catime-reversal invariant random potential scattering. Ya semiclassical treatment of the RMF scattering beyothe relaxation time approximation may lead to a nontriv0031-9007y96y77(9)y1817(4)$10.00fr,heratmley.kelarof,dMR. This classical contribution due to the bending of pticle’s trajectories dominates over the suppressed effecweak localization, at least, at not very low temperatureThe semiclassical approach to the RMF problem wundertaken in a number of publications. In [9] thSubnikov–de Haas (SdH) oscillations ofrxxsBd in thestrong field limit Vc ­ Bym ¿ 1yttr were studied bysumming over classical cyclotron orbits. However, to firxxsBd in the weak field limit before the onset of the Sdoscillations one has to develop a semiclassical analysterms of particle-hole pairs rather than single fermionsThe authors of [10] used the linear Boltzmann equatwhere the RMF played a role of a random driving forinstead of including it into the collision integral. Thcondition which makes it possible iskFj ¿ 1, wherejis a correlation length (scale of a typical spatial variatioof the RMF. However, the analysis performed in [10] wrestricted onto the case relevant for the experimentswhere, first, strong potential scattering provides a labare value ofrxx , and, second, the spatial correlationthe RMF has a finite range,j21.In the present Letter we address the case whenrxx issolely due to the scattering by the RMF described by(not necessarily short-range) correlatorkbqb2ql ­ fsqdindependent of the external fieldB. We will not restrictour consideration onto the lowest order of the perturbattheory and be able to discuss the casefsqd , e22qjrelevant for the problem of HFLL where the spatial RMcorrelations decay asr23 and the lowest order resusimply diverges.We start with the Boltzmann equation for the distribtion functiongst, $r , f, ed ­ g0sed 1 dgst, $r, f, ed whichreads as∑i≠≠t1 i $yf ? $= 1 iµVc 1bs$rdm∂≠≠f∏3dgst, $r , f, ed ­ ie $yf ? $E≠g0≠e,(1)where $yf ­ yF $nf is a vector of Fermi velocity normato the circular Fermi surface parametrized by the angu© 1996 The American Physical Society 1817VOLUME 77, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 26 AUGUST 1996icctiaofn ibearnt1],izainnnAoningsvenorelvevariablef, $E stands for an infinitesimal external electrfield, andg0 ­ us´F 2 ed is the unperturbed Fermi-Diradistribution. By contrast to the case of ordinary potenscattering (which can also be treated atkFj ¿ 1 as a ran-dom electric field) Eq. (1) implies a trivial dependencedg one, namely,dgst, $r , f, ed ­ 2dgst, $r, fdds´F 2 ed.We note, in passing, that one can use Eq. (1) evethe presence of long-range retarded gauge interactionstween fermions which spoil the Fermi liquid coherencEven though there are no well-defined fermionic quasipticles, the kinetic equation in terms of local displacemeof the fluctuating Fermi surface can still be derived [1thereby making a close contact with the idea of bosontion of 2D non-Fermi-liquids [12].The perturbed distribution function can be foundterms of the (retarded) Green function of the Boltzmaoperator for a given RMF configurationdgst, $r, f, ed ­ ieIdf0Zd $r 03 Gst; $r, $r 0; f, f0d $yf0 $EdseF 2 ed (2)and then averaged over all possible configurations.usual, the averaging over disorder restores the translatiinvariance in both$r andf spaces:kkGst; $r , $r 0; f, f0dll ­leto1818lne-.-s-salG st; $r 2 $r 0; f 2 f0d. Then the calculation of thefrequency-dependent conductivity amounts to computthe p-wave harmonics of the$q ­ 0 component of theFourier transform ofGsv; $q; f 2 f0d ­Xleilsf2f0d1iRc $q3s $nf2 $nf0 d Glsvd2p,where Rc ­ yFyVc is the cyclotron radius of fermionwith densityne,sxxsvd ­ie2ne2mfG1svd 1 G21svdg ,sxysvd ­e2ne2mfG1svd 2 G21svdg .In the absence of the RMF the bare Green function is giby its harmonicsGs0dl svd ­ 1yv 2 lVc 1 id.In our case of no potential disordersd ­ 01d the lowestorder (bosonic) self-energy correction toG21l svd ­ v 2lVc 1 Slsvd found in [10] makes it only possible tostudy theB-dependent correction to the barerxx at highfrequencies or magnetic fields. To proceed with a mcomplete account of the effects of the RMF we first sothe equation for the Green function∑i≠≠t1 i $yf $= 1 iµVc 1bs$rdm∂≠≠f∏Gst 2 t0; $r , $r 0; f, f0d ­ dst 2 t0dds$r 2 $r 0ddsf 2 f0d (3)for an arbitrary RMF configuration by Fourier transforming with respect to2 t0 and $r 2 $r 0. The solution of theresulting equation ∑v 2 $yf $q 1 i $yf $= 1 iµVc 1bs$rdm∂≠≠f∏Gsv; $q, $r; f, f0d ­ dsf 2 f0d (4)ghonionr-tronnentb-can be searched in the formGs $r , $q; f, f0d ­ 2 iZ `0dt3Idf00 eitG210 s $q;f,f00deiCst;$r ;f00 ,f0d.(5)HereC can be viewed as an “eikonal” phase of a partichole pair propagating in the RMF. Substituting (5) in(4) and integrating by parts we obtain the equation forC,µi≠≠t1 i $yf $=∂C ­ 2e2iCe2itG210bs$rdm≠≠feitG210 eiC .(6)-Provided the RMF correlation length is large enouskFj ¿ 1d one can linearize Eq. (6) (we will commentthis point below) and then end up with an explicit solutCst; $r; f, f0d ­1mZ t0dt0 bs$r 2 $Rt0d≠≠fdsf 2 f0d ,(7)where $Rt ­ RceVcts≠y≠fd $nf is a classical trajectory coresponding to the Larmor precession along the cycloorbit. Now taking the Gaussian average of the expoeiC in (7) over different realizations of the RFM we otain the averaged Green functionGsv, $0; f 2 f0d ­1iZ `0dt eitfv1iVcs≠y≠fdg exp"12Z t0dt1Z t0dt2 F√2Rc sinVcst1 2 t2d2!≠2≠f2#dsf 2 f0d ,(8)whereFsrd is a Fourier transform of the RMF correlatorfsqd. The use of Eq. (8) yields the conductivitysxxsvd ­e2nemZ `0dt cossVc 1 vdt exp"2Z t0dt0st 2 t0dF√2Rc sinVct02!#1 sv ! 2vd . (9)The formula forsxysvd differs from (9) by the additional factor tansVc 6 vdt in the integrand.VOLUME 77, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 26 AUGUST 1996n-x-alesonionxi-0)enermehathileteltorlnveAngwoal,r oo[13tio1)ndAbeticheheyreyofeestem-ain-st,antry.stionattereoft insisFtosThe exponential factor in the integrand in (9) is cotrolled by the amplitude of the RMF correlatorFsrd. Tocheck the validity of (9) in the perturbative regime we epand the exponent up to the first order inFsrd and performthet integration first [it has to be done with an infinitesimexponente2dt inserted into the integrand which specifithe retarded nature of the Green function (8)]. Thencan readily obtainsxxsvd ­ 1yttr sVc 2 vd22 1 sv ! 2vd ,where the RMF scattering rate is given by the express1yttr sBd ­Z `0dt cosVctFµ2Rc sinVct2∂, (10)which is nothing but the result of the first Born appromation (BA) [10]. One can also obtain the formula (1by estimating the,t term in the exponent in (9) in tht ! ` limit which would be equivalent to the relaxatiotime approximation.In what follows we will concentrate on the casefsqd ­8p2anee22qj relevant for the problem of HFLL wherthe dimensionless coupling constanta ­ 12 F2sniyned isproportional to the densityni of ionized Coulomb impu-rities separated by the spacer of the widthj from the 2Delectron gas and the amount of gauge flux quantaF t-tached to every electron in the case of a filling factonwith an even denominatorF [3]. In the picture of com-posite fermions (CF) each charged impurity also becoa source of a gauge magnetic flux [3,4]. It is believed tthe scattering by randomly distributed fluxes providesmain mechanism of the CF momentum relaxation whthe potential scattering is negligible.At B ­ 0 Eq. (10) gives the CF elastic transport ra1ytCFtr ­ syFy2jda which coincides with the BA resufound in [3]. Although this estimate certainly holds fa ø 1, it is no longer valid at largea when the CFmean free path (MFP)lCF ­ 2ja21 gets shorter thanj. The case of HFLLsn ­ 1y2d appears to be marginasa ­ 2d, andl determined this way just equalsj. Sincea is proportional to F2, the situation becomes eveworse for compressible states at fractions with higher edenominators.This observation signals about a failure of the Bat large couplinga when the matrix elementM $p, $p0 ,pa skFymjd describing a single event of CF scatteriby a typical magnetic impurity satisfies neither of the tconditionsM $p, $p0 ø 1ymj2 nor skFjdymj2 required forthe validity of the BA [13]. This is the semiclassicregime of a smallsDf , 1ykFj ø 1d angle scatteringwhich, however, cannot be treated in the lowest ordethe perturbation theory.Nevertheless, if the parameterkFj is large enough, sthatkFj ¿pa , (11)one can resort to the so-called eikonal approachwhich was essentially implemented by the above soluof the Boltzmann equation. In fact, it is the condition (1which allows one to solve Eq. (6) in the first order inbs$rd.estenf]nThe use of (9) then leads tosxx ­e22hskFld ­e2hskFjdeaK1sad , (12)whereK1sxd is the modified Bessel function of the secokind. At small a Eq. (12) reproduces the above Bresult while at strong coupling it predicts the MFP tol ø jp2pya.Thus the condition (11) actually provides thatkFl ¿ 1which is necessary to verify the very use of the kineequation (1) and to obtain the metallic value of tconductivity.In the intermediate coupling regime relevant for tcase of HFLL the condition (11) is fairly well met (a ­ 2and kFj ø 15) and Eq. (12) gives the CF conductivitsCFxx ­ se2yhd skFjde2K1s2d which is 2.06 times greatethan the BA result. It is worthwhile mentioning that thexperimentally measured resistivity atn ­ 1y2 is about3 times smaller than the BA estimate [3].The strong coupling behavior of the MFP (l , jypaas opposed tol , jya at weak coupling) can alreadbe seen in the self-consistent BA which in the casea finite-range potential scattering is applicable atj øB21y2. Adapting this method to the RMF problem oncan achieve a self-consistent improvement of the loworder result by inserting a factore2tyttr into the integrandin (10) and solving the resulting nonlinear equation forttr .For the case of HFLL this equation reads as1yttr ­2jayFZ `0dt e2tyttr∑t2 1µ2jyF∂2∏23y2(13)and gives the MFPlsad in agreement with the aboveikonal calculation. In terms of the conventional diagramatics Eq. (13) corresponds to the sum of uncrossed “rbow” diagrams for the bosonic self-energy. By contrathe eikonal result (8) includes contributions of the relevcrossed diagrams in all orders of the perturbation theoIn [10] the MR of the RMF problem was identified athe B dependence ofttr . It is obvious, however, that inthe absence of other mechanisms of momentum relaxaone cannot use Eq. (10) in order to obtain the MRVcttr , 1.Moreover, in the HFLL case the long-range characof Fsrd , r23 leads to a logarithmic divergency of thsecond derivative ofrxxsBd at B ! 0, which is, of course,an artifact of the above expansion. In the frameworkthe more accurate self-consistent BA one obtains thathe weak field limitttr sBd is an increasing function ofBwhich implies a negative MR. Our self-consistent analyshows that the decreasing behavior ofttr sBd (positive MR)in the case of sqd , e2s1y2dq2j2 and smalla reported in[10], in fact, holds only for large enough fields,Vc .1yttr s1y ln 1ya2d.The more complete eikonal treatment of the RMproblem confirms this prediction. Expanding (9) upthe second order inVc we obtain the MR defined aDrxxsBdyrxx ­ 2fDsxxsBdysxxg 2 fsxysBdysxxg2 inthe form1819VOLUME 77, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 26 AUGUST 19961820DrxxsBdrxx­V2c2k1l√kt2l 2 2ktl2k1l1112*Z t0dt0 s3t02t 2 4t03dFst0d+!, (14)nlycto-mdengie]ontotoRnud-[14ndisoadM.lairdthuc,ntr18ingobedtheon-re-lar,duc-ntheper-andiveticd,ga-nablett.ys.),ys.wherektnl ­R`0 dt tn expf2Rt0 dt0st 2 t0dFst0dg.Notice that in the relaxation time approximatiosktnl ­R`0 dt tne2tyttr d expression (14) contains ontwo terms proportional tokt2l andktl2 which exactly can-cel out and result in zero MR. Since the exponential faRt0 dt0st 2 t0dFst0d appearing in all our calculations instead oftyttr behaves as,t2 at smallt, the combinedeffect of these terms on the MR is negative and can donate over the remaining (strictly positive) contribution.The MR remains negative for all couplings in a wiclass of realistic RMF correlation functionsfsqd includingthe HFLL case ata ­ 2 when the Eq. (14) yieldsDrxxsBdrxx­ 20.06sVcttr d2, (15)while in the strong coupling limit of largea the MRbecomesDrxxsBdyrxx ­ sVcttr d2spy4 2 1d.Equation (9) also suggests that the MR may chasign atVcttr , 1 and remain positive up toVcttr ø 8although our semiclassical analysis becomes insufficat such high fields.In the existing artificial realizations of the RMF [1the random field depends on the applied externalNamely, the RMF correlation functionfsqd appropriatefor the references (a) and (c) from [1] is proportionalB while the one relevant for the reference (b) is,B2. Ineither case this trivialB dependence necessarily leadsa positive MR and does not allow one to explore the M(14) which is the only one present in the case offsqdindependent ofB.The quadratic positive MR was also reported in americal simulation of the lattice version of the RMF moeled as an assembly of uncorrelated random fluxesAgain, the anisotropy of the lattice fermion dispersion athe obvious lack of the large parameterkFj in the short-range case preclude us from making a direct comparwith our results.In the case of HFLL one may also think of the brominimum ofrxxsBd at n ­ 1y2 suggesting a positive MRof CF as resulting from the interference between the Rscattering of CF and their residual gauge interactionswas recently shown [15] that this interference can expthe strong nonuniversal lnT correction torxxsT d observedat n ­ 1y2 and3y2 [16].Indeed, it turns out that by contrast to the case of onary Coulomb interacting spin-polarized electrons [17]analysis of the interference correction to the CF condtivity at finite Beff ­ B 2 2pFne reveals a positive MRwhich appears to be greater than the negative RMF cobution (15). The details will be presented elsewhere [To summarize, we develop a new method of solvthe semiclassical Boltzmann equation for the RMF prlem beyond the relaxation time approximation providkFl ¿ 1. The obtained solution is used to calculateri-ente.-].nFItni-e-i-].-RMF transport time, mean free path, and zero-field cductivity which all appear to be greater than the corsponding results of the Born approximation. In particuwe propose a new estimate of the semiclassical contivity at HFLL which is about twice the value found i[3] and agrees better with the experimental data. Inabsence of interactions, Fermi surface and fermion dission anisotropy, periodic electron density modulations,in-plane fields (all these factors are known to yield positcontributions to the MR) we obtain the negative quadraMR in the weak field limit. It remains to be understoohowever, whether or not the semiclassically found netive MR is related to the quantum localization phenomein the RMF.The author is grateful to Per Hedegard for a valuadiscussion of the results of this work.[1] (a) A. K. Geim et al., Phys. Rev. B49, 5749 (1994);(b) A. Smith et al., Phys. Rev. B50, 14 726 (1994);(c) P. B. Mancoffet al., Phys. Rev. B51, 13 269 (1995).[2] L. B. Ioffe and A. I. Larkin, Phys. Rev. B39, 8988 (1989);N. Nagaosa and P. A. Lee, Phys. Rev. Lett.64, 2450(1990); Phys. Rev. B45, 966 (1992).[3] B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B47,7312 (1993).[4] V. Kalmeyer and S.-C. Zhang, Phys. Rev. B46, 9889(1992).[5] T. Sugiyama and N. Nagaosa, Phys. Rev. Lett.70, 1980(1993); D. K. K. Lee and J. T. Chalker, Phys. Rev. Le72, 1510 (1994).[6] J. Miller and J. Wang, Phys. Rev. Lett.76, 1461 (1996).[7] A. G. Aronov, A. D. Mirlin, and P. Wolfle, Phys. Rev. B49, 16 609 (1994).[8] B. L. Altshuler and L. B. Ioffe, Phys. Rev. Lett.69, 2979(1992); D. V. Khveshchenko and S. V. Meshkov, PhRev. B 47, 12 051 (1993).[9] A. D. Mirlin, E. Altshuler, and P. Wolfle, Ann. Phys.5,281 (1996).[10] P. Hedegard and A. Smith, Phys. Rev. B51, 10 869(1995).[11] Y. B. Kim, P. A. Lee, and X.-G. Wen, Phys. Rev. B52,17 275 (1995).[12] F. D. M. Haldane,Varenna Lectures,edited by R. Schri-effer and P. A. Broglia (North-Holland, New York, 1994Vol. 121.[13] L. D. Landau and E. M. Lifshitz,Quantum Mechanics(Pergamon, New York, 1977).[14] V. Kalmeyer, D. Wei, D. Arovas, and S. C. Zhang, PhRev. B 48, 11 095 (1993).[15] D. V. Khveshchenko, Phys. Rev. Lett.77, 362 (1996).[16] L. P. Rokhinsonet al., Phys. Rev. B52, R11 588 (1995);W. Kanget al., Phys. Rev. Lett.75, 4106 (1995); R. Hauget al. (unpublished).[17] P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys.57,287 (1985).[18] D. V. Khveshchenko, Report No. cond-maty9605051.

Magnetoresistance of Two-Dimensional Fermions in a Random Magnetic Field

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