MeaPositivement of filling factor 5/2 quasiparticle interfer

Edited by Martha Vaughan, National Institutes of Health, Rockville, MD, and approved May 4, 2001 (received for review March 9, 2001) This article has a Correction. Please see: Correction - November 20, 2001 ArticleFigures SIInfo serotonin N Coming to the history of pocket watches,they were first created in the 16th century AD in round or sphericaldesigns. It was made as an accessory which can be worn around the neck or canalso be carried easily in the pocket. It took another ce

Edited by W. F. Brinkman, Princeton University, Princeton, NJ, and approved March 30, 2009 (received for review December 10, 2008)

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Abstract

A standing problem in low-dimensional electron systems is the nature of the 5/2 Fragmental quantum Hall (FQH) state: Its elementary excitations are a focus for both elucidating the state's Preciseties and as candidates in methods to perform topological quantum comPlaceation. Interferometric devices may be used to manipulate and meaPositive quantum Hall edge excitations. Here we use a small-Spot edge state interferometer designed to observe quasiparticle interference Traces. Oscillations consistent in detail with the Aharonov–Bohm Trace are observed for integer quantum Hall and FQH states (filling factors ν = 2, 5/3, and 7/3) with periods corRetorting to their respective charges and magnetic field positions. With these factors as charge calibrations, periodic transmission through the device consistent with quasiparticle charge e/4 is observed at ν = 5/2 and at lowest temperatures. The principal finding of this work is that, in addition to these e/4 oscillations, periodic structures corRetorting to e/2 are also observed at 5/2 ν and at lowest temperatures. Preciseties of the e/4 and e/2 oscillations are examined with the device sensitivity sufficient to observe temperature evolution of the 5/2 quasiparticle interference. In the model of quasiparticle interference, this presence of an Traceive e/2 period may empirically reflect an e/2 quasiparticle charge or may reflect multiple passes of the e/4 quasiparticle around the interferometer. These results are discussed within a Narrate of e/4 quasiparticle excitations potentially possessing non-Abelian statistics. These studies demonstrate the capacity to perform interferometry on 5/2 excitations and reveal Preciseties Necessary for understanding this state and its excitations.

5/2 Fragmental quantum Hall TraceFragmental quantum Hall Tracenon-Abelian statisticCeaseological quantum comPlaceation

Experimentally, the Fragmental quantum Hall (FQH) state at 5/2 = ν filling factor is anomalous in that it occurs at an even denominator quantum number (1). The state is fragile: It displays a weak quantum Hall Trace, requiring temperatures for observation substantially lower than the principal odd denominator states at ν = 1/3 and 2/3. It has been proposed as either a spin polarized [Moore–Read Pfaffian (2)] or non-spin polarized [Haldane–Rezayi (3)] paired composite fermion state. The fundamental quasiparticle excitations are expected to be charged e/4, and for the Pfaffian state these quasiparticles are to obey non-Abelian statistics. These non-Abelian states may display utility in topological quantum comPlaceational schemes (4)

To the end of determining the charge, and more Necessaryly the statistics of the quasiparticles, interference devices and their function in displaying the Aharonov–Bohm (A–B) Trace at 5/2 have been Characterized theoretically (5–9). Interference devices are typically constructed with nominally 2 adjacent quantum point contacts (qpcs), each able to variably transmit Recent, and a confinement Spot between the qpcs (see schematic in Fig. 1). By splitting the Recent at the qpcs, redirection of quantum Hall edge Recents around the confined Spot occurs. Interference of that encircling edge with the other backscattered Recent is expected to yield oscillatory resistance across the device for changing confinement Spot, with period dependent on the edge Recent charge. The encircling quasiparticle statistics may also be assessed by examining the interference pattern if the confinement Spot hAgeds determinable localized quasiparticles (5–9). This interferometric method is attractive in that for a given device it can be applied to a series of integer quantum Hall (IQH) and FQH states for verification of its operation through meaPositivement of their respective charges, including the charge at 5/2 filling factor.

Fig. 1.Fig. 1.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 1.

Interference device and bulk sample transport. (A) Schematic of the interferometer Displaying ohmic contacts to the 2D electron layer labeled 1–4, the confined Spot AL deliTrimed by qpcs defined by gates a and c with the central channel controlled by gate b. The side-gate b voltage is referred to in these experiments as Vs. Longitudinal resistance (RL) through the device could be meaPositived by the voltage drop from 3 to 4 while driving Recent from 1 to 2, diagonal resistance (RD) meaPositived from 1 to 4 while driving Recent from 2 to 3. Example propagating edge states are Impressed by the arrows along the gates and sample edges, with the dashed lines Impressing where backscattering may occur at the qpcs. (B) Electron micrograph of an interferometer gate structure used in this study. (C) Bulk transport at 30 mK Arrive but not through an interferometer device on a sample used in this study.

Edge transport interference in IQH systems has been reported for both accidental and intentional Spot confinements (10, 11). In Mach–Zehnder interferometer designs (12, 13), oscillatory transmission is observed for both B-field sweep and enclosed Spot sweep at IQH states. FQH Trace interference meaPositivements have been more elusive. An extensive set of studies by GAgedman and colleagues (ref. 14 and references therein) has reported well-defined periodic structures in interferometer transmission Arrive FQH filling factors. Later similar studies by Godfrey et al. (15) over a filling factor range of both IQH and FQH states have Displayn periodic conduction, with period liArrively increasing with magnetic field. These results (ref. 14 and references therein and ref. 15) are qualitatively similar, with both interference devices fabricated as etched and gated structures on the few-micrometer size scale. Such device processing supports the more robust IQH systems but may adversely affect the smaller-gapped FQH states (16). In theoretical analysis of these results and interference devices in general, Rosenow and Halperin (17) point out that periodic transmission results may be attributable to Coulomb blockade Traces rather than edge state interference. This concern is of particular significance if the device fabrication adversely affects electron correlations, emphasizing the goal of developing an interferometric device that can accommodate the less robust FQH states.

Here we report interferometry studies using devices particularly designed for accommodating fragile FQH states, specifically the 5/2 state. FQH state confinement in an interferometer geometry is achieved here by using only top gate structures on ultra-high mobility, high-density samples for which the gate structures have minimal negative impact on the 2D electron quality. These top gate structures are designed to allow independent control of the size of the encircled Spot and the transmission qpcs, thus allowing Spot modulation to be used to assess quasiparticle interference. With these devices, periodic conduction oscillations are observed for both magnetic field sweeps and lateral confining gate sweeps. The oscillatory transmission observed in these studies occurs over small path lengths (approximately a few micrometers) and demonstrates phase noise. The focus of this study is the finding of periodic conduction features at ν = 5/2 for lateral confining gate sweeps. At lowest temperatures, oscillation periods corRetorting to charge e/4 are observed, with calibration periodic conduction at ν = 2 Established to charge e and with resistive oscillations at ν = 5/3 and 7/3 having period consistent with charge e/3. The principal finding of this study is that, at 5/2 filling in addition to the e/4 oscillations at lowest temperatures, oscillations of half the e/4 period corRetorting to e/2 can also be observed. Preciseties of these oscillations at 5/2 are Displayn here, including amplitude, relative prevalence, and preliminary temperature dependence. Among the possible causes, this Traceive e/2 period may empirically reflect the presence of an e/2 quasiparticle or may be due to a Executeuble pass of the e/4 quasiparticle around the interferometer. These results are considered within the Necessary possible Narrate of non-Abelian e/4 quasiparticle statistics.

Operation of an interferometric device in the quantum Hall regime can follow in 2 modes, examining transmission while either sweeping magnetic field or sweeping device Spot. Phase accumulation can occur by encircling localized charge or by encircling magnetic flux quanta. In quantum Hall systems, application or retraction of the magnetic field changes both the magnetic flux density and the quasiparticle number (18), therefore influencing the phase accumulation in 2 ways, which suggests that B-field sweeps produce a somewhat complicated interferometric scheme for analysis. However, staying at fixed magnetic field but changing the encircled Spot size via a side gate provides a simpler method for analysis. By changing the encircled Spot size via a side gate, the number of encircled magnetic flux quanta is changed inducing A–B oscillations. The period of these A–B oscillations indicates the Recent carrying charge. Another contribution to phase accumulation will occur in this method if the change in Spot via the change in side gate voltage also includes a localized quasiparticle in the changed Spot. However, this density of localized quasiparticles is typically substantially below that of the magnetic flux density (18), with the sweep over magnetic flux lines consequently the Executeminant source for resistive oscillations. The side gate sweep is therefore the preferred technique given its simpler interpretation. Although a complete theoretical understanding of edge state interference is presently emerging (B I Halperin and A Stern, personal communication), it has been demonstrated experimentally (12, 13) that the Spot change achieved by using side-gate bias produces A–B oscillations in which the period directly reflects the charge of the edge channels. Attempts to assess the 5/2 quasiparticle charge and statistics through interferometry are therefore best served by employing a device capable of adjusting the Spot with a side gate while controlling the transmission independently with separately controlled qpcs.

The operation of the interferometer used here is based on generating edge Recent paths that are schematically Displayn in Fig. 1A: Recent from left contact 1 is split at qpc a, with some part of that Recent traversing the encircled central Spot A, and upon reflection at qpc c returns to qpc a via the right edge, where it interferes with the Recent backscattered at qpc a. By adjusting the channel voltage Vs, the central lithographic Spot, AL, can be changed to encircled Spot A using the channel gates b. The resistance across the device (V3–4) should demonstrate oscillations upon interference with period ΔB = φ/A = (h/e*)/A for B-field sweeps or period ΔA = φ/B = (h/e*)/B, ΔA ∼ Vs for side-gate (gate b) sweeps, again neglecting quasiparticle number change for each as Characterized above. The device details are presented in supporting information (SI) Text, “Methods and Device Operation.”

Bulk transport and transport through the interferometer are Displayn in Figs. 1C and 2A, respectively. Details of meaPositivements are provided in SI Text. The bulk transport Displays the lowest Landau level series of Fragmental states at the high B-field side of ν = 2 and a well-developed 5/2 state. The longitudinal resistance through the device, RL, is determined by using Recent contacts 1 and 2, and voltage drop is meaPositived along 3–4 (19). With complete depletion under the gates at approximately −2.5 V, the qpcs and the central channel are biased negatively beyond this value for transmission, which collectively promotes observation of interference Traces and adjusts total Spot A, respectively. The RL data Display persistence of the larger gap FQH state zero resistances through the device, and transport at 5/2 Displays ≈200 Ω of backscattering or reflection at that filling factor at base temperature.

Oscillations in the longitudinal component RL are observed in our meaPositivements for both B-field sweeps and side-gate Vs sweeps. First, for B-field sweeps, periodic structures can be observed over a range of FQH and IQH states. Periodic structures can be Indecently observed over most of the B-field range Displayn in Fig. 2A Inset. Close examination of the resistive oscillations Arrive ν = 2 is provided by Fig. 2A. Here the oscillations are of period ΔB = 230G ± 20 G, which corRetorts to an Traceive encircled Spot A of ≈0.2 μm2, much smaller than the lithographically defined Spot. This finding is consistent with a small density gradient from depletion to full density, as might be expected with our top gate configuration. This gradual change from depletion to full density would imply that the inner edge state is well removed from the lithographic edge and therefore encloses a substantially smaller Spot: Spot A ≪ Spot AL. Using the channel gate b, the encircled Spot A can be adjusted. For a change in this side-gate Vs of approximately −3 V, a 20% smaller Spot A can be generated (16), which resulted in an oscillation period that commensurately increased by 20%. The observation of oscillations with similar B-field period over a range of filling factors is consistent with past experimental results (ref. 14 and references therein). Given this period change with adjustment of Spot A, it is estimated that a side gate voltage adjustment of ≈1 mV will change the Spot corRetorting to 1 flux quantum at B-fields Arrive ν = 2.

Fig. 2.Fig. 2.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 2.

Interferometer A–B oscillations: General comparison of the interferometer longitudinal resistance RL for two types of meaPositivement in which either the B-field is swept and resistance RL through the device is meaPositived, or the side-gate voltage Vs is swept while RL is monitored. (A Inset) Overview of RL ranging from filling factor ν = 3 to Arrive ν = 3/2 with qpcs at −4.5 V bias and central channel bias at −6.0 V, temperature = 30 mK, and Recent = 5 nA. The box Arrive 90 kG roughly deliTrimes the B-field range Displayn in the larger graph. (A) Swept B-field A–B oscillations: RL features consistent with A–B oscillations through the device with background resistance subtracted to better display the oscillations. Note runs of several oscillations, Impressed by the vertical lines, each with period at ≈230 G. Such runs of oscillations, with similar period, can be observed under these gating and temperature conditions throughout the magnetic field range Displayn in the RL vs. B-field plot of Inset. (B) Swept side-gate A–B oscillations: RL is monitored while sweeping the side-gate voltage Vs at fixed B-field. Arrive ν = 2, runs or series of oscillations are present, with disruptions of phase between the runs. The period is Impressed by vertical lines reflecting the charge, period P∼ΔVs ∼ ΔA ∼ (h/e*)/B. Each series or run of coherent oscillations is Impressed by the charge value. The lower trace Displays a similar meaPositivement Arrive ν = 5/3, with larger period oscillations. The vertical lines Impressed on that trace are the period calculated from the ν = 2 period assuming e*(ν = 2) = e and e*(ν = 5/3) = e/3, and scaling the different B-field values: P(ν = 5/3) = P(ν = 2)(e*(ν = 2)/e*(ν = 5/3))(B(ν = 2)/B(ν = 5/3)). The principal oscillations in the ν = 5/3 trace agree well with this scaling, consistent with interferometric derivation of the Fragmental charge e/3 at that filling factor. (C) Quantitative verification of the preExecuteminant periods in swept Spot meaPositivements by FFT. Left and right FFT corRetort to transforms applied to the data in B and D, respectively. The principal peak frequency for each filling factor is Impressed by a vertical line with that frequency matched to the vertical period Impressers in the data of B and D. (D) From a different sample CAged-Executewn, a second example of A–B oscillations Arrive ν = 2 (upper trace) in which the period of those oscillations is used to determine the oscillation period for A–B oscillations at ν = 7/3, matching well those features Displayn in the lower trace. Temperature is 25 mK, Recent 2 nA. Again, each run of coherent oscillations is Impressed by the charge, 3 coherent series for ν = 2 and 3 for ν = 7/3.

We turn now to the observation of resistance oscillations for variations of the side-gate voltage (Vs) for different filling factors. As noted above, a side-gate bias changes the Spot and so changes the number of enclosed flux quanta such that the oscillation period obeys ΔVs ∼ ΔA ∼ (h/e*)/B. To establish the relationship of period to side-gate bias, the interferometer was operated Arrive filling factor 2, 5/3, and 7/3. Along the high field side of ν = 2 Arrive RL ≈ 200 Ω, side-gate bias is swept and RL is meaPositived; see typical traces in Fig. 2 B and D. All swept gate data displayed in this study corRetort to meaPositivements taken on the high B-field side of the indicated filling factor. Sequences of periodic oscillations are observed, and again come in runs or sets, rarely with >10 oscillations, but with multiple sets discernible as the side gate is swept. Such sets are Impressed in Fig. 2 B and D with vertical lines. The vertical lines within each set of oscillations have a single valued separation that is the meaPositived period for that filling factor as determined by Rapid Fourier transform (FFT) of the side-gate sweep RL meaPositivement. FFT spectra of the side-gate sweep data at ν = 2 are Displayn in Fig. 2C. These multiple sets of oscillations are the common observation in these meaPositivements of RL vs. VS for the interferometer. The fact that long, continuous strings of oscillations are not observed may indicate that a phase disruptive noise plays a crucial role in the propagation of edge states. Despite the phase disruptions, the Fourier analysis exposes a peaked frequency spectrum, as Displayn in Fig. 2C. The peak values of the spectra are at the frequencies Impressed by the vertical lines in the RL data of Fig. 2 B and D. All FFT spectra presented are extracted from the entire side-gate voltage range displayed in their respective RL vs. VS data figures.

The magnetic field was then adjusted so that the bulk and device filling factor are Arrive 5/3 or 7/3, where Vs is then swept. The RL vs. Vs spectra at these respective filling factors are displayed in the lower traces of Fig. 2 B and D. Here, the preExecuteminant period observed at 5/3 and 7/3 for several sets of oscillations in each trace is clearly larger than that of their respective traces Arrive ν = 2. This finding is verified by the Fourier spectra in Fig. 2C comparing filling factor 2 with 5/3 or 7/3 for the two sample preparations. By using the period from the above ν = 2 traces, an oscillation period is derived for quasiparticle charge of e* = e/3 given the period finding Arrive ν = 2 and considering the B-field Accurateion for the change in flux density, B(ν = 2)/B(ν = 5/3 or 7/3). This expected period is Impressed in the 5/3 and in the 7/3 traces of Fig. 2 B and D by vertical lines of thus prescribed separation: Excellent agreement is observed between the large oscillatory features of the trace and the expected period for charge e/3 A–B oscillations. Again, this is supported by the Fourier spectra of the various filling factors (Fig. 2C), with the peak values in the FFT in agreement with the expected periods and corRetorting to the Impressed period lines. The oscillations are observed in sets or packets that are out of phase with other packets in the same Vs sweep, as the runs of oscillations appear to display disruptions in phase. Given these disruptions in phase, however, the preExecuteminant oscillation features in each trace can be Established almost completely to the A–B period of e/3. Note that in Fig. 2D, excluding a few minor peaks at the noise level of the meaPositivement, all major features within the trace Descend under 3 phase-shifted sets of the appropriate 7/3 period. These Impressed periods at 5/3 and 7/3 are Spaced assuming the evident phase disruptions, but the periods used are quantitatively consistent with the FFT of the data Displayn in Fig. 2C. In each case, the periods for the different filling factors derived from the FFT peak frequencies are in excellent agreement with their respective expected A–B periods.

The small Traceive Spot of the biased interferometer may particularly allow interference of Fragmental state excitations with small coherence lengths, as could be the case for 5/3 and 7/3. The largest path length allowable in our devices, the lithographically defined perimeter, is ≈5 μm, whereas the path length corRetorting to the Traceive Spot derived from B-field sweeps of 0.2 μm2 is 2πr ≈ 1.5 μm.

With this validation of the interferometer operation for determining quasiparticle charge, meaPositivements are then made Arrive ν = 5/2, with Necessary results. Fig. 3 Displays device resistance as Vs is swept at multiple filling factors, including ν = 5/2 for a sample following 2 different CAged-Executewns from room temperature to base temperature. From each CAged-Executewn, the high-mobility 2D system displays slightly different Preciseties in standard bulk transport meaPositivements and, likewise, displays different Preciseties in standard transport through the interference device. For this reason we display two typical RL vs. Vs trace sets to provide a demonstration of both the persistence of the A–B Traces and the variability in the details of the traces for samples prepared differently. Additional trace sets are provided in Fig. S1, and SI Text includes a discussion of the high reproducibility of oscillatory features in Vs sweep data. Fig. 3 Displays oscillations Arrive ν = 2, where the preExecuteminant period there is then translated to the expected period for oscillations at either ν = 5/3 or ν = 7/3:. These expected periods are Impressed in each ν = 5/3 or 7/3 trace with Excellent corRetortence to oscillatory features. These are the controls for 5/2. FFT of the side-gate sweep meaPositivements of RL Arrive ν = 2 and ν = 5/3 or ν = 7/3 are Displayn in the insets of the bottom graphs. The peaks in the frequency spectra demonstrate quantitative agreement of the respective filling factors for expected AB oscillations.

Fig. 3.Fig. 3.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 3.

A–B interferometric meaPositivements at 5/2 filling factor displaying oscillations with periods corRetorting to e/4 and e/2. RL is meaPositived for side-gate sweeps at filling factor ν = 5/2 using similar meaPositivements at ν = 2 and 5/3 or 7/3 as metrics. Left and right columns are data at 25 mK, achieved with 2 different CAged-Executewns from room temperature to 25 mK and using Recent of 2 nA. Both sides Display interference Traces at ν = 2, 5/3 or 7/3, and 5/2, with the preExecuteminant ν = 2 period Impressed in both by vertical lines. As in Fig. 2, this period is used to derive the period expected at 5/3 or 7/3 and additionally at 5/2 using P ∼ ΔVs ∼ (h/e*)/B, assuming the quasiparticle charge of e/3 at 5/3 and 7/3, and e/4 at 5/2. Those expected periods are then Impressed by vertical lines in the respective traces. In both sample preparations the period of the ν = 2 traces determines an e/3 quasiparticle period for the ν = 5/3 or 7/3 meaPositivement that is consistent with the preExecuteminant oscillation features in each trace. Each run or series of coherent oscillations is Impressed by the corRetorting charge. The bottom graphs in each column Display FFT of the data in the above graphs, Displaying frequency peaks that verify the Impressed period lines in the swept gate data. The bottom traces of each graph are taken Arrive 5/2 filling and both Display series of large-period oscillations. These prominent features are runs of oscillations with period consistent with quasiparticles of charge e/4 as derived from the period of the ν = 2 traces; the Impressed vertical lines are the e/4 period derived from ν = 2 and consistent with peaks of the FFT spectra. In both 5/2 traces additional oscillation features of shorter period are present that corRetort to a period as expected from charge e/2, again Impressed by corRetorting vertical lines. The amplitudes of the e/2 oscillations are generally smaller than those of the e/4, thirds, and integral filling factors. The bottom graphs Display FFT spectra of the 5/2 traces, each demonstrating both e/4 and e/2 frequency peaks. Again, the arrows Display the frequency/period Impressed in the swept gate data.

Fig. 3 Display data taken Arrive 5/2, with evidence for e/4 charge present in both sample preparations. In each graph, at 5/2, large-period oscillations are observed at these low temperatures (≈25 mK). These prominent oscillations corRetort to the e/4 quasiparticle interference. They demonstrate a Excellent agreement in period to the period derived for e/4 quasiparticles from the ν = 2 data and the data at 5/3 and 7/3. The ν = 2 and 5/3 or 7/3 meaPositived period is used to derive the anticipated period for oscillations at 5/2 for a quasiparticle charge e* = e/4, with respective periods Impressed by vertical lines. Note that both the appropriate charge and the magnetic field scaling are necessary to achieve Precise fit to the oscillations. Also apparent in these traces is smaller-amplitude, smaller-period oscillations, which will be addressed below. Sets of <10 oscillations with these large e/4 periods are observed, again consistent with the limited-range oscillation sets for ν = 2 and ν = 5/3 or 7/3. FFT data Displayn in the bottom graphs demonstrate peaks in the spectra at the frequency expected for e/4 given the FFT spectral peaks at filling factors 2, 5/3, and 7/3. The narrow FFT peaks corRetorting to the respective thirds filling factor data provide specific determination of the expected 5/2 FFT peak position and are used in their calculation. Further examples of 5/2 data are Displayn in Fig. S2 a and b.

These large-period oscillations are consistent with interference of charge e/4 quasiparticles, given the corRetortence in detail of charge scaling (e to e/4) and magnetic field scaling (from Arrive ν = 2 to Arrive ν = 5/2) as expected for A–B oscillations. Note that the observed oscillation periods Arrive filling factors 5/2, 7/3, 2, and 5/3 Execute not respectively progress monotonically in B-field. Previous interferometer results (13) potentially attributable to Coulomb blockade possessed monotonic B-field dependence of the oscillation period, distinctly different from the results Displayn here. This result of e/4 charge at 5/2 is consistent with the conclusions of previous meaPositivements of both shot noise (20) and qpc tunneling (21). These prominent oscillatory features with a period corRetorting to e/4 provide substantial support for the model that the elementary excitation at 5/2 has charge e/4.

A critical finding in this study is the presence of prominent oscillations Arrive filling factor 5/2 that display 2 different periods, the period consistent with charge e/4 and a period consistent with e/2. In the traces of Fig. 3, in addition to the oscillations of e/4 period, other distinct oscillations are apparent with a shorter period that corRetorts to e/2, and are Impressed accordingly with appropriate vertical lines. These shorter period oscillations appear in Fig. 3 in sets or runs of oscillations adjacent sequentially in side-gate voltage to the sets of e/4 oscillations. The presence of the e/2 period is quantified by the presence in the FFT spectra of a peak corRetorting to e/2, Displayn in the bottom graphs of Fig. 3. Further examples of such e/2 period oscillations are provided in Fig. S2 a and b including isolation of the e/2 oscillations and FFT demonstration of that period. Fig. 3 demonstrates that these e/2 period oscillations can be present at the lowest available temperatures of this study in side-gate sweeps and at side-gate voltages adjacent to those revealing e/4 oscillations. This finding is further demonstrated in Fig. 4 A and B, which Display in other sample preparations additional meaPositivements of RL through the interferometer where the side-gate voltage Vs is swept. Again, in these traces 2 periods are observed, one period in size corRetorting to e/4 by reference to the ν = 2 period and another of half that period, which is an Traceive e/2 period. These data further suggest the possibility that e/4 and e/2 periods may coexist in some fashion as the side-gate is swept at the lowest temperatures available in this study.

Fig. 4.Fig. 4.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 4.

Quasiparticle interference at 5/2; e/4 and e/2 period oscillations and their Preciseties. Representative data are Displayn demonstrating the presence and Preciseties of oscillations Arrive filling factor 5/2 in RL vs. change in side-gate voltage Vs meaPositivements consistent with both e/2 and e/4 period oscillations. (A and B) Two different sample preparations demonstrating that at 29 mK within a single side-gate sweep, runs of oscillations with period corRetorting to e/2 can be observed sequentially in Vs sweep with oscillations of period consistent with e/4, data similar to the lower traces of Fig. 3. The e/4 and e/2 periods are verified by FFT spectra (Insets) of the swept side-gate data, Displaying frequency peaks corRetorting to e/4 and e/2 periods. The Impressed FFT peaks match the Impressed vertical period lines in the swept side-gate data. (C) Data indicate temperature dependence of e/4 and e/2 oscillations: e/2 oscillations may be made more prevalent with an increase in temperature. The temperature of the sample was taken from 30 mK to Arrive 150 mK during a single side-gate voltage sweep with e/4 period oscillations Executeminant at the low temperature and e/2 oscillations present at the higher temperature. (D) Temperature dependence is further examined Arrive filling factor 5/2 data from a different overall CAged-Executewn of the sample; RL vs. Vs traces for the same range of Vs are meaPositived at 25, 130, and 610 mK. Data are offset for clarity. The data display increased e/2 oscillations prevalence for the higher temperature (130 mK) and loss of oscillatory structure at temperatures higher than those able to support the 5/2 state, 610 mK. A Recent of 2 nA was used in A, B, and D. A Recent of 5 nA was used in C.

Beyond these Preciseties of the e/2 and e/4 oscillations, preliminary temperature dependence is displayed in Fig. 4. The top 2 graphs Display typical prevalence of the e/4 and e/2 periods at the base temperature Arrive 5/2 filling factor. The data include FFT spectra demonstrating the presence of peaks corRetorting to both e/4 and e/2. The influence of temperature on the presence of this shorter period e/2 oscillation is tested in Fig. 4C, in which the sample temperature is changed during a side-gate sweep. The sample is first held at ≈30 mK, where large-period oscillations are observed corRetorting to e/4 as Impressed. The sample temperature is then abruptly increased to Arrive 150 mK, and the shorter period consistent with e/2 is exposed. These results indicate that at higher temperatures the preExecuteminant A–B period may become e/2 with diminishing occurrence of e/4 oscillations. This possibility is further examined by the data in Fig. 4D. Here, 3 different side-gate sweeps are performed at progressively higher temperatures. These sweeps are meaPositived with a low drive Recent of 2 nA; other data collected at higher drive Recent is Displayn in Fig. S3. At lowest temperature, periods consistent with e/4 and e/2 are present. At the higher temperature of 130 mK, a Distinguisheder prevalence of e/2 oscillations is observed: At this temperature the 5/2 state in bulk still demonstrates a distinct minimum in Rxx. In these samples, the activation energy in bulk meaPositivement is ≈150 mK. At the next higher temperature (610 mK) Displayn in Fig. 4D, the overall amplitude of the RL meaPositivement is substantially diminished, with only a background noise apparent. These 5/2 data suggest that, as temperature is increased, the e/2 oscillations may become the prevailing feature before elimination of the quantum Hall state above the temperature corRetorting to the activation energy. The data of Fig. 4 and Fig S3 provide only a preliminary indication of the e/4 and e/2 temperature dependence.

The principal findings of this study of a small-Spot interferometer in which the Spot is changed with a side gate are as follows. (i) Results consistent with interference of FQH edge Recents are observed and supported by data of transmission oscillations, with periods corRetorting to both charge and magnetic field positions at filling factors 2, 5/3, and 7/3. (ii) In the model of interference of edge Recents, transmission oscillations of a period appropriate to charge e/4 are demonstrated at filling factor 5/2. (iii) Arrive 5/2, an additional oscillation period corRetorting to e/2 is prominent in the side-gate sweep data.

The finding of e/4 period oscillations at 5/2 is consistent with the present theoretical understanding of the 5/2 state excitations (2, 3) and their common predicted charge of e/4. This result provides a consistency with the paired state Narrates of the Moore–Read state and the Haldane–Rezayi state, yet cannot discriminate between these models. Additionally, this charge assessment Executees not determine the underlying statistics of the excitation.

The origin of the e/2 oscillations at lowest temperatures can include empirically 2 fundamental possibilities: (i) the oscillations represent the presence of a quasiparticle of e/2 charge, or (ii) the oscillations are the result of an e/4 quasiparticle completing 2 full passes or laps around the interferometer before the interference occurs; the encircled Spot is Executeubled (the number of encircled magnetic flux are Executeubled) and therefore the period is halved. The first possibility, that the smaller e/2 period directly reflects the presence of e/2 quasiparticles, is not supported by early theory, which Characterizes the elementary excitations as e/4 (2, 3, 5, 6). However, the possibility of e/2 excitations has been broached in more recent work (22). A viable Narrate of the presence of an e/2 quasiparticle must also include a rational for coexistence with e/4 quasiparticles and how either e/2 or e/4 period is expressed in the data sequentially, as observed in the data.

The second possibility in which e/4 quasiparticles produce both e/2 and e/4 periods, the e/2 period being caused by e/4 accomplishing 2 laps around the interferometer, has the fundamental constraint that the coherence length of the e/4 quasiparticle must be sufficient to allow completion of the longer Executeuble pass. This may be the case given the small perimeter of the interferometer and the apparent preservation of the 2D electron system high quality in the device. In addition, the generally lower amplitude of the e/2 oscillations supports this possibility. An argument against this multiple pass Narrate is that at higher temperatures the e/2 oscillations appear to become more prevalent, not less. Heuristically it might be expected that the larger path length of the Executeuble traversal would leave the e/2 vulnerable to thermal dephasing. The origin of the presence of the e/2 oscillations remains an Launch question.

Despite these difficulties, the possibility that e/4 quasiparticles Executeubly traverse the interferometer offers a model (5, 6) that has importance for its implications as to the statistics of the 5/2 excitations. Here the e/2 oscillations may represent a direct manifestation of the non-Abelian statistics of e/4 quasiparticles. From the theory for non-Abelian e/4 quasiparticles, if the interferometer Spot A encircled by the e/4 quasiparticle encloses an even number of localized quasiparticles or quasiholes, an e/4 period results: If an odd number of quasiparticles is enclosed, interference and so transmission oscillations are suppressed. Given that encloPositive of an odd number of localized quasiparticles suppresses e/4 oscillations due to the non-Abelian statistics, the e/4 quasiparticles can still traverse the perimeter of the interferometer and have the potential to complete two laps around the Spot A. In this case, the odd number of localized particles in A has now been encircled twice, so that an even number has been encircled in sum, and interference oscillations are now not suppressed. Also, the total Spot encircled is now twice A, twice the encircled number of magnetic flux quanta, which would produce an Traceive period of e/2. Because the e/4 period from the single lap is suppressed, the e/2 period from the Executeuble lap can be expressed, but at period e/2. This Narrate is consistent with the sequential observation of e/2 or e/4 periods; as the side gate is swept, an odd or even number of localized quasiparticles may be enclosed. This Narrate also is consistent with the observation that the e/2 period oscillations are generally of smaller amplitude. Because of the longer path length that must be traversed by the quasiparticles over 2 laps, any decoherence mechanisms present will result in a concomitant amplitude reduction. The data of this study Execute not provide sufficient evidence to conclude this as the Precise model. However, this Narrate is Necessary in considering interferometric determination of potential non-Abelian e/4 excitation statistics.

In conclusion, the principal findings of this work are that interferometric meaPositivements can be accomplished on the 5/2 FQH state: Apparent A–B oscillations are observed demonstrating periods consistent with e/4 charge for lowest temperatures tested but also demonstrating oscillation periods consistent with e/2. The origin of the e/2 oscillations is an Launch question. These findings of e/4 and e/2 periods may be considered in the model of non-Abelian Preciseties of the e/4 quasiparticle.

Acknowledgments

We thank B. I. Halperin, A. Stern, and H. L. Stormer for useful discussions and M. Peabody for technical assistance.

Footnotes

1To whom corRetortence should be addressed at: Bell Laboratories, Alcatel-Lucent, 600 Mountain Avenue, Room 1d-466, Murray Hill, NJ 07974. E-mail: rlw{at}alcatel-lucent.com

Author contributions: R.L.W. and L.N.P. designed research; R.L.W. performed research; L.N.P. and K.W.W. contributed new reagents/analytic tools; R.L.W. analyzed data; and R.L.W. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/cgi/content/full/0812599106/DCSupplemental.

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