Quasiparticle Tunneling in the Fractional Quantum Hall State at ν= 5/2
Iuliana P. Radu, J. B. Miller, C. M. Marcus, M. A. Kastner, L. N. Pfeiffer, K. W. West
QQuasiparticle Tunneling in the Fractional Quantum Hall State at ν = 5 / Iuliana P. Radu, J. B. Miller, C. M. Marcus, M. A. Kastner, L. N. Pfeiffer, and K. W. West Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Department of Physics, Harvard University, Cambridge, Massachusetts 02138 Bell Labs, Lucent Technologies, Murray Hill, New Jersey 07974 (Dated: 7 March 2008)Theory predicts that quasiparticle tunneling between the counter-propagating edges in a fractionalquantum Hall state can be used to measure the effective quasiparticle charge e ∗ and dimensionlessinteraction parameter g , and thereby characterize the many-body wavefunction describing the state.We report measurements of quasiparticle tunneling in a high mobility GaAs two-dimensional electronsystem in the fractional quantum Hall state at ν = 5 / ν = 5 / g . Among these models, thenon-abelian states with e ∗ = 1 / g = 1 / The fractional quantum Hall (FQH) effect [1] resultsfrom the formation of novel states of a two-dimensionalelectron system (2DES) at high magnetic field and lowtemperature, in which electron-electron interactions leadto gaps in the bulk excitation spectra. Because of thesegaps, current can only flow via extended states that prop-agate around the edges of the 2DES [2]. At a constrictionin the 2DES, such as that formed by a quantum pointcontact (QPC), counter-propagating edge states comeclose enough together that quasiparticles can tunnel be-tween them. According to theory [3], weak quasiparticletunneling depends strongly on the voltage difference be-tween the edges (or, because of the Hall effect, the currentthrough the QPC), and should scale with temperaturein a way that provides a measurement of the effectivecharge, e ∗ , of the quasiparticles and the strength of theCoulomb interaction, g . Since both e ∗ and g are specificto the FQH state, such measurements provide a discrim-inating probe of FQH wavefunctions.The FQH state at ν = 5 / ν = 5 /
2, alter-natives with abelian properties have also been proposed[14, 15, 16]. All candidate wavefunctions for ν = 5 / e ∗ = 1 /
4, but theydiffer in the predicted values of g [8, 9, 17, 18, 19].Weak tunneling theory, developed originally for Laugh-lin FQH states [3], has also been extended to non-abelian states [17, 18, 19, 20, 21]. Tunneling measurements ona single constriction can distinguish among candidatewavefunctions for ν = 5 /
2; existing proposals to find di-rect evidence for non-abelian statistics, however, requiremultiple constrictions to create interference among tun-
Device 1Device 21µmDevice 1 G3Device 2 R D , R xy ( h / e ) D R xy R D , R xy ( h / e ) D not annealedR D annealedR xy A2A1 G4G1G2
FIG. 1: Magnetic field dependence of the diagonal ( R D ) andHall ( R xy ) resistance for device 2 at fixed gate voltage from ν = 2 to ν = 4 illustrating that both the QPC and the bulkare at the same filling fraction. The upper inset shows low-field data from the same device (device 2) emphasizing thatthe carrier density in the annealed QPC is nearly the same asthat of the bulk (red and black traces with almost matchingslopes), while in the non-annealed QPC (green trace) the den-sity shifts significantly. For clarity, the non-annealed data hasbeen offset vertically by 0 .
003 h/e . Lower insets are scanningelectron micrographs of devices with similar gate geometry tothose used in these experiments. In device 2, grounded gatesheld are artificially colored gray. a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r I d c ( n A ) T=13 mK A / R xy ( e ² / h ) T=20 mK B I d c ( n A ) T=30 mK C / R xy ( e ² / h ) T=50 mK D T (e²/h) FIG. 2: Differential tunneling conductance g T (device 2) as a function of magnetic field and dc bias current at several tem-peratures (marked on figure). On each graph, the zero-dc-bias R xy trace from the same temperature is superimposed (rightaxis). The field range encompasses the FQH states 7 /
3, 5 / / ∼
30 mK. neling paths [11, 22, 23, 24, 25, 26].Experimentally, the quasiparticle charge, e ∗ , has beeninvestigated for FQH states at ν < ν = 5 / e ∗ = 1 / g , has been measured in studies oftunneling of ν = 1 / ν = 5 /
2, in theregime where the filling fraction (and carrier density) in the QPC and the bulk 2DES are the same. We findthat tunneling conductance across the QPC exhibits astrong zero-bias peak that scales with temperature inquantitative agreement with the theory for weak tunnel-ing [3, 18, 19]. From these measurements, we extract e ∗ and g . We observe that among the candidate states for ν = 5 /
2, the anti-Pfaffian [8, 9] and the U (1) × SU (2)[7] are most consistent with the data. Sample and experimental setup.
The sample is aGaAs/AlGaAs heterostructure with the 2DES 200 nmbelow the surface and two Si δ -doping layers 100 nmabove and below the 2DES. Hall bars with a width of150 µ m are patterned on this heterostructure. The mo-bility (before the gates are energized) is 2000 m /Vs, thecarrier density is 2.6 × m − , and the ν = 5 / ∼
130 mK in the bulk [34]. The QPCs are formedby Cr/Au top gates, which are patterned on the Hallbar using e-beam lithography. By applying a negativegate voltage V g to these gates, the electrons underneaththem are depleted, creating a constriction tunable with V g . We report measurements on devices with two differ-ent gate geometries (lower insets of Fig. 1). Device 1 isa simple QPC with gate separation of 800 nm. Device 2is a channel ∼ ≥
20 mK, the mixing chamber andelectron temperatures have been measured to be equalusing resonant electron tunneling in a lateral quantumdot. Temperatures below 20 mK have been estimatedusing both resonant tunneling and by tracking severalstrongly temperature-dependent quantum Hall featuresin the bulk, with consistent results. (See Supporting On-line Material.) The magnetic field is oriented perpendic-ular to the plane of the 2DES.Measurements are performed using standard 4-probelock-in techniques with an ac current excitation between100 - 400 pA and in some cases a dc bias current of up to20 nA. To determine the tunneling conductance g T , wesimultaneously measure the Hall resistance R xy (voltageprobes on opposite sides of the Hall bar away from theQPC) and the diagonal resistance R D (voltage probes onopposite sides of the Hall bar and also opposite sides ofthe QPC)[34, 36]. In the weak tunneling regime [3], whenthe bulk of the sample is at a quantum Hall plateau, thetunneling voltage is the same as the Hall voltage, while R D reflects the differential tunneling conductance via: g T = R D − R xy R xy (1)Note that R xy is independent of dc bias when the bulkis at a FQH plateau. If one assumes that the under-lying edge has a filling fraction ν under , then the reflec-tion of the 5 / R = g T R xy / [(1 /ν under ) h/e − R xy ]. Same filling fraction in QPC and bulk.
A key differ-ence between this work and previous tunneling exper-iments [31, 32, 33, 34] is that we are able to depletethe electrons under the gates and induce tunneling with-out significantly changing the filling fraction in the QPC.This is achieved by applying a gate voltage of -3 V whileat 4 K and allowing the system to relax for several hours,which we refer to as annealing. We then cool the sampleand limit the voltage to the range -2 to -3 V when atdilution refrigerator temperatures. After annealing, R D and R xy are measured over several integer plateaus andthe fields marking the ends of the plateaus are found tocoincide for QPC and the bulk (see Fig. 1), indicatingthat the filling factors are the same. The extra resis-tance in R D at FQH states is consistent with tunneling,as discussed below. Additional evidence that the fillingfraction changes little once the QPC is annealed i! sshown in the inset of Fig. 1: the slopes of R xy and R D at I d c ( n A ) -3 -2.5 -2V g (V) 0.20.1g T (e²/h)T=13mK -3 -2.5 -2V g (V)-3 -2.5 -2V g (V) T=20mK T=40mK
A B C
FIG. 3: Differential tunneling conductance g T (device 1) asa function of V g and dc bias current at several temperatures: A. T = 13 mK, B. T = 20 mK, C. T = 40 mK. A peakin both dc bias and V g becomes visible at T=40 mK. Thevertical dashed line marks the center of this resonance. low magnetic field, inversely proportional to carrier den-sity, differ by 2% or less. For comparison, we show datafrom a non-annealed QPC where the density decreasesby ∼ Bias and temperature dependence
In the following, wefocus on the dependence of R D on the dc current bias I dc through the QPC and Hall bar. Figure 2 shows acolor-scale plot of the dependence of R D on both I dc andmagnetic field B at four temperatures; a measurement of R xy is shown for comparison. As seen most clearly at thehighest temperatures, these field sweeps reveal a series ofFQH states [37] around ν = 5 /
2, including the 7/3 and8/3. At the lowest temperatures strong re-entrant integerquantum Hall (RIQH) features are also visible on eitherside of 5/2, both in the bulk and in the QPC (see Fig. 2).The dc bias behavior at FQH plateaus is quite differentfrom that of the RIQH features: At FQH plateaus, zero-bias peaks in g T persist up to at least 50 mK (Fig. 2D).By contrast, RIQH states have more complex dc biassignatures, which decrease rapidly with temperature, dis-appearing by 30 mK both in the bulk ( R xy ) and in theQPC ( g T ). Qualitatively similar ! results are observedfor device 1. To study the FQH state at ν = 5 /
2, we setthe magnetic field to the center of a bulk FQH plateau( B = 4 .
31 T for device 2, vertical line in Fig. 2C, and B = 4 . V g on the zero-bias peak at several temperatures is pre-sented in Fig. 3. At the lowest temperatures (Fig. 3A),the zero-bias peak persists throughout the V g range. Athigher temperatures, a peak in both dc bias and V g isobserved, centered near V g = − . V g to the center of thispeak, the feature that persists to the highest temper-ature, because theory predicts that tunneling decreasesslowly, as a power law, with temperature.Having chosen the magnetic field and gate voltage inthis way, we measure the dc bias dependence in device 1at various temperatures (Fig. 4). The traces in Fig. 4A R D ( h / e ²) -10 -5 0 5 10I dc (nA) 0.200.150.10 g T ( e ² / h ) dc (µV) = I dc x R xy
13 mK 16 mK 20 mK 40 mK 60 mK m K m K m K m K m K ( R D - R ¥ ) / T - . ( a . u . ) dc /T (nA/mK)-50 0 50eV dc /kT 13 mK 16 mK 20 mK 40 mK 60 mK EAD BC F W H M ( n A ) H e i gh t ( h / e ²)
20 30 40 50 6010 T (mK) R D ( h / e ²) -5 0 5 I dc (nA)-5 0 5 -5 0 5 -5 0 5 -5 0 5 g =0.35e*=0.177 H (cid:181) T -1.3 FIG. 4: A. R D (device 1) as a function of dc bias at fixed magnetic field ( B = 4 . ν = 5 /
2) and fixed gate voltage( V g = − . R xy is independent of dc bias over the range of I dc (not shown), which makes thebias dependence of R D proportional to that of g T (right axis) up to a constant. B. Zero dc bias peak height as a function oftemperature. The red line is the best fit with a power law where the exponent is -1.3. C. The peak full width at half maximum(FWHM) as a function of temperature. The red line is the best fit with a line going through zero. D. Data collapsed onto asingle curve using an exponent -1.3. E. Best fit of all the data in A with the weak tunneling formula (eq. 2) returns e ∗ = 0 . g = 0 . are slices along the dashed lines in Fig. 3. Since thevoltage drop between the two counter-propagating edgestates in the QPC is the dc current multiplied by the Hallresistance, we have labeled the horizontal axis with boththe current and the dc voltage, using R xy = 0 . [3].All these traces saturate at the same value R ∞ at highdc bias, higher than the expected value 0.40 h/e . Theheight of the peak, measured from R ∞ , decreases with in-creasing temperature, following a power law in tempera-ture with exponent − . T and the verti-cal axis is scaled by T − . (after subtracting a common background R ∞ ). Extracting g and e*.
The observed temperature de-pendence of the peak height and FWHM is consistentwith the theoretical predictions of weak quasiparticletunneling between fractional edge states [3, 18, 19]. Inthat picture, the zero-bias peak height is expected to varywith temperature as T g − , which gives g = 0 .
35 for thedata in Fig. 4B. The weak-tunneling expression, whichincludes the effects of dc bias [3] has the form g T = AT (2 g − F ( g, e ∗ I dc R xy kT ) , (2)(see Supporting Online Material for details). This func-tional form fits the experimental data very well, as seenin Fig. 4E (Note that R D and g T differ only by an offset g N o r m a li z e d f i t e rr o r (1/4,1/8)(1/4,1/4)(1/4,3/8)(1/4,1/2) FIG. 5: Map of the fit quality. Normalized fit error is theresidual from the least-squares fit, divided by the number ofpoints and by the noise of the measurement. Also marked onthe map are proposed theoretical pairs ( e ∗ , g ). and scale factor.) All five temperatures are fit simulta-neously with four free parameters: a single vertical off-set corresponding to R ∞ , an amplitude A , and the twoquantities g and e ∗ . A least-squares fit over the full dataset gives best-fit values g = 0 . e ∗ = 0 .
17. Uncertainties in these values will be discussedbelow. Similar analysis performed on data from a dif-ferent device (device 2 but energizing only gates G1 andG4) yields quantitatively similar results.To characterize the uncertainty of these measured val-ues, a matrix of fits to the weak-tunneling form, Eq. (2),with g , e ∗ fixed and A , R ∞ as fit parameters is shown inFig. 5. The color scale in Fig. 5 represents the normal-ized fit error, defined as the residual of the fit per pointdivided by 0.0005 h/e , the noise of the measurement.A fit error (cid:46) ν = 5 / e ∗ = 1 /
4, but g can differ.States with abelian quasiparticle statistics include the(331) state [14, 15], which has a predicted g = 3 / K = 8 state with g = 1 / g = 1 / g = 1 / U (1) × SU (2) state [7], also with g = 1 /
2. Parameter pairs( e ∗ , g ) representing these candidate states are marked inFig. 5. Evidently, the states with e ∗ = 1 / g = 1 / e ∗ = 1 / g = 3 / e ∗ = 1 / Strong tunneling.
In contrast to device 1, the dc biasdata from device 2 show evidence for strong tunneling.Device 2 has a long, channel-like geometry, which couldincrease the number of tunneling sites and hence the tun-neling strength. Diagonal resistance, R D as a function ofdc bias at several temperatures is shown in Fig. 6A. Com-paring these data to those from the short QPC (Fig. 4A),shows qualitative differences at lower temperatures. Athigher temperatures, the zero-bias peak height can bedescribed by a power law in temperature with an expo-nent similar to that in the QPC (Fig. 6B), and a FWHMthat is proportional to temperature (Fig. 6C). At lowertemperatures, the peak height deviates from a power lawand saturates at the lowest temperatures at a value ofresistance consistent with the resistance at ν = 7 / / by the background R ∞ − . R D ( h / e ²) -10 -5 0 5 10I dc (nA) 0.20.10 g T ( e ² / h ) -100 -50 0 50 100V dc (µV) = I dc x R xy
13 mK 18 mK 24 mK 30 mK 60 mK
A BC H e i gh t ( h / e ²)
20 30 40506010 T (mK)1086420 F W H M ( n A ) FIG. 6: A. R D (device 2) as a function of dc bias at fixedmagnetic field ( B = 4 .
31 T, middle of ν = 5 /
2) and fixedgate voltage ( V g = − . R xy isindependent of dc bias over this range of I dc (not shown).At the lowest temperature, the peak develops a flat top at avalue of resistance consistent with the resistance at ν = 7 / B. Zero-bias peak height as a function of temperature. Thepeak height saturates at the lowest temperatures. C. Peakwidth as a function of temperature. The red line is best fitof the high temperature data with a line going through zero.Note that below ∼
30 mK the peak width no longer followsthis line. strong tunneling regime for ν = 5 /
2. However, qualita-tive comparisons with strong tunneling theory [38] andexperiment [31, 32, 33] at other FQH states ( ν <
1) canbe made. For strong tunneling, the edge states associatedwith the topmost fractional state ( ν = 5 / ν < R D at the peak is consistent with full backscat-tering of the 5 / ν = 7 / Summary and outlook.
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