URL:http://link.aps.org/doi/10.1103/PhysRevLett.94.086801
DOI:10.1103/PhysRevLett.94.086801Recent experiments have studied the tunneling current between the edges of a fractional quantum Hall liquid as a function of temperature and voltage. The results of the experiment are puzzling because at “high” temperature (600-900 mK) the behavior of the tunneling conductance is consistent with the theory of tunneling between chiral Luttinger liquids, but at low temperature it strongly deviates from that prediction dropping to zero with decreasing temperature. In this Letter we suggest a possible explanation of this behavior in terms of the strong temperature dependence of the tunneling amplitude.We are grateful to S. Roddaro, V. Pellegrini, and F. Beltram for useful discussions and the use of their experimental
data. We kindly acknowledge the hospitality of the Max Planck Institute for the Physics of Complex Systems in Dresden where part of this work was completed. This research was supported by NEST-INFM PRA-Mesodyf and NSF DMR-0313681. R. D'A. acknowledges the financial support by NEST-INFM PRA-Mesodyf

D'Agosta, Roberto

Raimondi, Roberto

Vignale, Giovanni, 1957-

Physical Review Letters

arXiv

Roberto D’Agosta

Giovanni Vignale

Roberto Raimondi

Crossref

Temperature Dependence of the Tunneling Amplitude between Quantum Hall Edges

Recent experiments have studied the tunneling current between the edges of a fractional quantum Hall liquid as a function of temperature and voltage. The results of the experiment are puzzling because at "high" temperature (600–900 mK) the behavior of the tunneling conductance is consistent with the theory of tunneling between chiral Luttinger liquids, but at low temperature it strongly deviates from that prediction dropping to zero with decreasing temperature. In this Letter we suggest a possible explanation of this behavior in terms of the strong temperature dependence of the tunneling amplitude

Vignale, Giovanni

Archivio Aperto di Ateneo

University of Missouri: MOspace

PRL 94, 086801 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending4 MARCH 2005Temperature Dependence of the Tunneling Amplitude between Quantum Hall EdgesRoberto D’Agosta,1,* Giovanni Vignale,1 and Roberto Raimondi21NEST-INFM and Department of Physics and Astronomy, University of Missouri - Columbia, 65211, Columbia, Missouri, USA2NEST-INFM and Dipartimento di Fisica, Universita` di Roma Tre, via della Vasca Navale 84, 00146, Roma, Italy(Received 2 July 2004; published 1 March 2005)0031-9007=Recent experiments have studied the tunneling current between the edges of a fractional quantum Hallliquid as a function of temperature and voltage. The results of the experiment are puzzling because at‘‘high’’ temperature (600–900 mK) the behavior of the tunneling conductance is consistent with thetheory of tunneling between chiral Luttinger liquids, but at low temperature it strongly deviates from thatprediction dropping to zero with decreasing temperature. In this Letter we suggest a possible explanationof this behavior in terms of the strong temperature dependence of the tunneling amplitude.DOI: 10.1103/PhysRevLett.94.086801 PACS numbers: 73.43.Cd, 71.10.Pm, 73.43.Lp0FIG. 1. Simple scheme of the experimental setup. The currentis carried by the quasiparticle in the edge states that are forced tostay close by the presence of the geometrical constriction ofwidth D. Note that, inside the constriction, the edges are at thedistance dwith d D. We assumed as the 0 for the y coordinatethe position where the edge distance is minimal and the tunnel-ing takes place (see Ref. [9] for further details on the choice ofthe reference frame and Ref. [12] for the details about the actualdevice).In the last 20 years quantum Hall systems have been arich source of information about the physics of correlatedelectron systems. For example, the edge of a fractionalquantum Hall system represents one of the best realizationsof a strictly one-dimensional interacting system. Indeed,Wen showed that the low-energy density excitations local-ized along the edges of a fractional quantum Hall liquid areeffectively described by a chiral Luttinger liquid (LL)model [1], with the effective interaction parameter givenby the bulk filling factor b.Tunneling experiments offer an effective way to probe indetail the predictions of the LL model [2,3]. For example,measurements of the tunneling current from an externalmetallic gate into the edge of a two-dimensional electrongas (2DEG) [4–6] have confirmed the theoretical predic-tion of a tunneling current proportional to VT with  1=b at the incompressible filling factor b  1=3 (VTbeing the potential difference between the edge and thegate) but have also revealed a smooth variation of theexponent  with filling factor in the range 1=4< b < 1[7]—in conflict with the original theory of Ref. [2].The tunneling of fractionally charged quasiparticles be-tween the edges of a fractional quantum Hall liquid hasalso been studied experimentally by several groups [10–12]. This Letter focuses on the recent measurements per-formed in the weak tunneling regime at b  1=3 byRoddaro et al. [12]. According to the theory, one expectsthat, in this experiment, the tunneling current must scale asV2b2T for kBT  eVT and be linear in VT for eVT  kBT.The zero-bias tunneling conductance, dITdVT jVT0, further-more, should grow as T2b2 with decreasing temperature[2,3,9]. Contrary to this expectation, below 600 mK onesees a dramatic drop in the tunneling conductance. Weemphasize that this is in glaring contrast not only withthe prediction of the weak tunneling theory, but also withan exact solvable model [13] valid in both weak- andstrong-tunneling regimes.In this Letter we argue that these puzzling data may beexplained by a strong temperature dependence of the in-05=94(8)=086801(4)$23.00 08680teredge tunneling amplitude. More precisely, we will showthat the spatial separation between the edges of a fractionalquantum Hall liquid increases with decreasing tempera-ture, resulting in a rapid loss of overlap between the edgesand a consequent collapse of the tunneling amplitude on atemperature scale T0 quite comparable to the 600 mKobserved in the experiment.The common starting point for calculating the differen-tial tunneling conductance is a model consisting of twoLLs (the two edges) coupled by the tunnelingHamiltonianHT  	^yT0	^B0  		^yB0	^T0; (1)where  is a phenomenological tunneling amplitude andthe operators 	^TB0 destroy a quasiparticle of fractionalcharge e	  be at a point ‘‘0’’ in the top or bottom edge,respectively, (see Fig. 1). A standard perturbative calcula-tion leads to the following expression for the differential1-1  2005 The American Physical SocietyPRL 94, 086801 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending4 MARCH 2005tunneling conductance G  dITdVT at VT  0[2,9]:G  e2hhv2kBTh!02b2Bb; b; (2)where v is the velocity of the edge modes, !0 is anultraviolet frequency cutoff related to the microscopic cut-off length a by !0  va , and Bx; y is the Euler betafunction [14].It is normally assumed that the tunneling amplitude isindependent of temperature: if this were true it wouldimply G / T2b2, increasing with decreasing tempera-ture. However, an analysis of the experimental data ofRef. [12], shows that the situation is quite different. Weextract the value of  from the measured values of theconductance simply by inverting Eq. (2), using v ’ 4105 m=s for the edge wave velocity [15] and a  100 A[2] for the ultraviolet length cutoff. The values of  ob-tained in this manner are shown as solid dots in Fig. 2.Notice that  increases rapidly with temperature belowabout 600 mK and more slowly for T > 600 mK.To understand this unexpected behavior, we begin byrecalling that the tunneling amplitude arises from the over-lap of single particle states localized in front of each otheron the top and bottom edges. For two coherent states in thelowest Landau level centered, respectively, at 0; T and 0; B(see Fig. 1), the matrix element of the noninteractingHamiltonian is (up to an irrelevant phase factor)  h22m	‘ed2=4‘2; (3)where d is the distance between the edges at the center ofthe constriction, ‘ is the magnetic length, and m	 is theeffective mass. It is important to realize that d is typicallymuch smaller than the geometric separation, D, betweenthe split gates (in the experiments of Ref. [12], with m	 ’0:067 m for GaAs, and ‘ ’ 100 A, one has d 3–5‘ [16],FIG. 2 (color online). The variation of jj2 with temperature.The points (red) are the experimental results [12] obtained fromthe evaluation of Eq. (2). The lines are two fits with the functiong expT=T0. The solid line (black) is a fit with all the experi-mental data and gives g  2:6 meV A2 and T0  400 mKwhile for the dashed line (blue) we have considered onlytemperatures below 400 mK and gives the estimates g 0:5 meV A2 and T0  140 mK.08680while D 30‘) and that its value is determined by equi-librium considerations discussed in detail below. Becauseof the exponential dependence of  on d even a relativelysmall variation of d with temperature can have a largeeffect on . Moreover, we will show that, at low tempera-tures, d varies linearly with the temperature.Our picture of the system is shown in the inset of Fig. 3.The center of the Hall bar is occupied by an incompressiblequantum Hall strip of width d, sandwiched between twocompressible regions of smoothly varying density. Sincethe density tapers off from the uniform value in the incom-pressible strip to zero over a distance of several magneticlengths, what we are showing here is essentially the situ-ation depicted by Chklovskii et al. in their classical electro-static theory of edge channels [17–19]. The density profileis determined, at T  0, by minimizing the sum of theelectrostatic energy and the confinement energy, subject tothe constraint of having an incompressible strip at thecenter of the system. In order to arrive at an analyticallytractable model we assume that the system is translation-ally invariant in the y direction (i.e., the density profiledepends only on x) [20] and that the electron-electroninteraction is screened, due to the presence of the splitgates, beyond a characteristic screening length , also ofthe order of several magnetic lengths. We also assume thatthe system is symmetric with respect to x  0 and studybelow only the part with x > 0: thus we neglect anyinteraction between the top and the bottom part of thesystem. None of these simplifications alter the qualitativefeatures of the solution.The total energy associated with a given density profilenx can be written asE  e2L!bZnx2dx LZVxnxdx; (4)where !b is the dielectric constant, Vx is the externalconfining potential (from gates, etc.), and L is the length ofthe system in the y direction. The integral runs over the topinhomogeneous region. At finite temperature, we must alsoFIG. 3. The solution of Eq. (6) for various temperatureskBT=U  0:01, 0.51, 0.81, 1.01. Inset: plot of the local fillingfactor profile at T  0 (solid line) and of the confining potential(dashed line).1-2PRL 94, 086801 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending4 MARCH 2005include the electronic entropy. In the inhomogeneous re-gion noninteracting electrons would give rise to a largezero-temperature entropyS   kBL2‘2Zfx lnx 1 x ln1 xgdx; (5)which follows from the interpretation of x  2‘2nxas the probability of a single particle state centered at x inthe lowest Landau level to be occupied. Undoubtedly, thisfree-electron entropy overestimates the real entropy of thesystem. However, we expect a large degeneracy to persistin the correlated state. For example, introducing compositefermions (CF) [8] to mimic the interactions between theoriginal fermions, we find a zero-temperature entropy thatis similar in form to Eq. (5), but smaller. The use of Eq. (5)has the advantage of allowing a completely analytic solu-tion of the model, while different forms of the entropy, e.g.,the form appropriate for noninteracting CF, give qualita-tively similar results [21].The edge density profile is computed from the require-ment that the free energy F  E TS is stationary withrespect to small variations of the density, subject to theconstraint of global particle number conservation [20] andwith the further condition x  b at the edge of theincompressible strip (notice that the position of this edge isitself to be determined). These requirements easily lead tothe equationUx  Vx  kBT lnx1 x &; (6)which must be satisfied in the compressible region deter-mined by the conditions 0< x< b. Here U e2=!b‘2 represents a typical interaction energy, & is thechemical potential, which fixes the total particle number,and the edge of the incompressible strip occurs at theposition for which x  dT=2  b (cf. Fig. 3). Forthe sake of simplicity we take the position of the edge atT  0 as the origin of the coordinate, x! x d0=2. Toproceed, we assume that around this point the externalpotential can be linearly expanded, Vx  eEx[22], whereE is the electric field. Equation (6) admits an elegantsolution in this case. However, we expect that nonlinearterms yield no qualitative differences as long as one con-siders not too high temperatures. To begin with, by settingT  0, we easily find the zero-temperature solution0x 8>>><>>>:b1 x; 0< x< b; x < 00; x > (7)where   Ub=eE is the width of the compressible re-gion and &  Ub.At finite temperature, the chemical potential must bechosen in such a way that the total particle number remains08680the same as at T  0. Therefore, we must haveZ 1x0xdx Z x00xdx  bx0 b2 ; (8)where the position of the edge, x0, is determined by thecondition x0  b. Because of the linearity of the ex-ternal potential, the integral on the left-hand side of Eq. (8)can be evaluated analytically by a change of variable fromx to  after an integration by parts. This yieldsZ 1x0xdx  bx0 b2 1eE f&b  kBTb lnb 1 b ln1 bg: (9)By comparing Eq. (8) and Eq. (9) we get the temperatureshift of the chemical potential&T&0  kBTbb lnb1b ln1b (10)and by evaluating Eq. (6) for x  x0, we havex0  kBTUln1 b2b; (11)which yields the effective edge separation by recalling thatdT  d0  2x0. Figure 3 shows the numerical solutionof Eq. (6) for x obtained for different temperatures.Notice that the edge of the incompressible strip shiftsinward as predicted by Eq. (11).Putting Eq. (11) in Eq. (3) we finally arrive atjTj2  j0j2eT=T0 ; (12)wherekBT0  bln1 b 2be2!bd: (13)From the experiments [12] we estimate that  ’ 3 [23]and d0 ’ 4‘: thus we obtain T0 ’ 600 mK, which iscomparable with the value obtained from the fits shownin Fig. 2 [21].Since   Ub=eE, one expects that the characteristictemperature scale T0 increases by making the confiningpotential steeper. This prediction appears to be qualita-tively in agreement with recent experiments [24] wherethe behavior of the tunneling conductance has been inves-tigated as a function of the gate voltage controlling thequantum point contact. For sufficiently negative gate volt-age the tunneling conductance is consistent with the pre-diction of the LL model with a constant . This in turn isconsistent with a large characteristic temperature scale T0as predicted by Eq. (13).We believe that our electrostatic model, in spite of itssimplicity, captures the essential aspects of the observedtemperature dependence of the tunneling amplitude. Themain effect of the temperature is to remove particles fromthe incompressible strip transferring them into the zonethat was depleted at T  0. This causes a linear increase in1-3PRL 94, 086801 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending4 MARCH 2005entropy, coming primarily from the population of statesthat were initially empty. Let us emphasize that Eq. (5)takes into account only the entropy of the compressiblestrip and that the electrons in the incompressible strip arelocked in a collective state of essentially zero entropy fortemperatures below the fractional quantum Hall gap. Thuswe do not expect that the general scenario presented herewill be significantly affected by introducing more realisticfeatures in the calculation of the energy and of the confin-ing potential.On the other hand, our analysis of the experiment as-sumes the validity of Eq. (2), itself a consequence of theweak tunneling theory of Wen and its extensions. Recently,there have been suggestions that Eq. (2) might be invali-dated by additional interactions between electrons on thesame edge, since these interactions appear to change thescaling dimension of the tunneling [25]. For that mecha-nism to be effective the long range intraedge interactionmust be stronger than the interedge interaction: this con-dition is unlikely to be satisfied in the present experimentalsetup.As a final point, we note that the dependence of theinteredge separation on temperature is not expected totranslate into a dependence of this quantity on the appliedvoltage. Indeed, in the present experiment this voltage isjust the Hall voltage created by the dc current injected inthe Hall bar [12]. The effect of this current is to createdifferent quasiparticle populations on the two edges.However, in our model, this will cause a rigid shift ofboth edges in the same direction, thus leaving the distancebetween them and hence the tunneling amplitudeunaffected.In conclusion, in this Letter we have addressed theproblem of determining the temperature dependence ofthe tunneling amplitude in the tunneling process betweenthe edges of a fractional quantum Hall liquid. We haveshown that the temperature modifies in a nontrivial way theequilibrium distance between the edges, and therefore thetunneling amplitude which is a very sensitive function ofthe temperature.We are grateful to S. Roddaro, V. Pellegrini, and F.Beltram for useful discussions and the use of their experi-mental data. We kindly acknowledge the hospitality of theMax Planck Institute for the Physics of Complex Systemsin Dresden where part of this work was completed. Thisresearch was supported by NEST-INFM PRA-Mesodyfand NSF DMR-0313681. R. D’A. acknowledges the finan-cial support by NEST-INFM PRA-Mesodyf.*Electronic address: dagostar@missouri.edu[1] X. G. Wen, Phys. Rev. B 41, 12838 (1990).[2] X. G. Wen, Phys. Rev. B 44, 5708 (1991).[3] C. L. Kane and M. P. A. Fisher, Phys. Rev. Lett. 68, 1220(1992).08680[4] A. M. Chang, L. N. Pfeiffer, and K. W. West, Phys. Rev.Lett. 77, 2538 (1996).[5] M. Grayson, D. C. Tsui, L. N. Pfeiffer, K. W. West, andA. M. Chang, Phys. Rev. Lett. 80, 1062 (1998).[6] A. M. Chang, M. K. Wu, C. C. Chi, L. N. Pfeiffer, andK. W. West, Phys. Rev. Lett. 86, 143 (2001).[7] A recent theory by S. S. Mandal and J. K. Jain, Phys. Rev.Lett. 89, 096801 (2002) based on the idea of interactingCF [8] results in good agreement with the experimentaldata at b  13 , 35 , and 37 , whereas the noninteracting CFtheory, as well as the original Wen’s theory, yields   3at these three filling factors. See also [9] for a simpleexplanation of the approximate relation  1b .[8] J. K. Jain, Phys. Rev. Lett. 63, 199 (1989).[9] R. D’Agosta, R. Raimondi, and G. Vignale, Phys. Rev. B68, 035314 (2003).[10] Y. C. Chung, M. Heiblum, Y. Oreg, V. Umansky, and D.Mahalu, Phys. Rev. B 67, 201104(R) (2003).[11] Y. C. Chung, M. Heiblum, and V. Umansky, Phys. Rev.Lett. 91, 216804 (2003b).[12] S. Roddaro, V. Pellegrini, F. Beltram, G. Biasiol, L. Sorba,R. Raimondi, and G. Vignale, Phys. Rev. Lett. 90, 046805(2003).[13] P. Fendley, A. W. W. Ludwig, and H. Saleur, Phys. Rev. B52, 8934 (1995).[14] Handbook of Mathematical Functions, edited by M.Abramowitz and I. A. Stegun (National Bureau ofStandards, Washington, D.C., 1964).[15] I. L. Aleiner and L. I. Glazman, Phys. Rev. Lett. 72, 2935(1994).[16] This estimate follows from Eq. (3) after the observationthat  ’ 5–25 meV A.[17] D. B. Chklovskii, B. I. Shklovskii, and L. I. Glazman,Phys. Rev. B 46, 4026 (1992).[18] D. B. Chklovskii, K. A. Matveev, and B. I. Shklovskii,Phys. Rev. B 47, 12605 (1993).[19] J. H. Oh and R. R. Gerhardts, Phys. Rev. B 56, 13519(1997).[20] To ensure translational invariance we impose periodicboundary conditions in the y direction; i.e., the electrongas is enclosed in a ring of given radius. This guaranteesthe conservation of the total particle number.[21] Using the CF entropy we obtain again x0  T with <0 as in Eq. (11) but the temperature scale T0 is smallerthan the one estimated from Eq. (13). Because of the lowerentropy of CF the edge position becomes more sensitive toa change in temperature.[22] The constant term V0 represents merely a shift in thechemical potential.[23] We estimate that the screening length  must be smallerthan or of the same order of magnitude of the distance ofthe electron gas from the surface where the gates wereetched. In the experiment reported in [12] this distance is100 nm ’ 10‘ while  is half the distance between thesplit gates which is 300 nm.[24] S. Roddaro, V. Pellegrini, F. Beltram, G. Biasiol, and L.Sorba, Phys. Rev. Lett. 93, 046801 (2004).[25] E. Papa and A. H. MacDonald, Phys. Rev. Lett. 93, 126801(2004).1-4

D'Agosta R

Vignale G

RAIMONDI, Roberto

Archivio della Ricerca - Università di Roma 3

Temperature dependence of the tunneling amplitude between Quantum Hall edges

Recent experiments have studied the tunneling current between the edges of a fractional quantum Hall liquid as a function of temperature and voltage. The results of the experiment are puzzling because at “high” temperature (600–900 mK) the behavior of the tunneling conductance is consistent with the theory of tunneling between chiral Luttinger liquids, but at low temperature it strongly deviates from that prediction dropping to zero with decreasing temperature. In this Letter we suggest a possible explanation of this behavior in terms of the strong temperature dependence of the tunneling amplitude

Temperature Dependence of the Tunneling Amplitude between Quantum Hall Edges

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Archivio Aperto di Ateneo

University of Missouri: MOspace

Archivio della Ricerca - Università di Roma 3

Archivio della Ricerca - Università di Roma 3