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Measurements of quasiparticle tunneling in the =5/2

fractional quantum Hall state

Citation

Lin, X. et al. “Measurements of Quasiparticle Tunneling in the

=5/2 Fractional Quantum Hall State.” Physical Review B 85.16

(2012). ©2012 American Physical Society

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http://dx.doi.org/10.1103/PhysRevB.85.165321

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Page 2

PHYSICAL REVIEW B 85, 165321 (2012)

Measurements of quasiparticle tunneling in the υ =5

X. Lin,1,*C. Dillard,2M. A. Kastner,2L. N. Pfeiffer,3and K. W. West3

1International Center for Quantum Materials, Peking University, Beijing, People’s Republic of China 100871

2Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

3Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA

(Received 17 January 2012; published 25 April 2012)

2fractional quantum Hall state

Some models of the 5/2 fractional quantum Hall state predict that the quasiparticles, which carry the charge,

have non-Abelian statistics: exchange of two quasiparticles changes the wave function more dramatically than

just the usual change of phase factor. Such non-Abelian statistics would make the system less sensitive to

decoherence, making it a candidate for implementation of topological quantum computation. We measure

quasiparticle tunneling as a function of temperature and dc bias between counterpropagating edge states. Fits

to theory give e∗, the quasiparticle effective charge, close to the expected value of e/4 and g, the strength of

the interaction between quasiparticles, close to 3/8. Fits corresponding to the various proposed wave functions,

along with qualitative features of the data, strongly favor the Abelian 331 state.

DOI: 10.1103/PhysRevB.85.165321 PACS number(s): 73.43.Jn, 05.30.Pr, 03.67.Lx

I. INTRODUCTION

The collective interactions of a two-dimensional electron

gas (2DEG) in a strong perpendicular magnetic field B give

rise to the fractional quantum Hall effect (FQHE).1Because

of the energy gap in the bulk, motion of the quasiparticles that

arise in the FQHE is generally constrained to one-dimensional

chiral edge channels.2However, if two opposite channels are

brought close together, quasiparticles may tunnel between

them. Studies of such tunneling have led to measurements

of the quasiparticle charge3,4and creation of quasiparticle

interferometers.5,6

The states comprising the FQHE are determined by the

filling factor υ = n/(B/?0), where n is the electron sheet

density and ?0= h/e is the quantum of magnetic flux. The

υ = 5/2 state7is of particular interest because it is one of only

afewphysically realizable systemsthought topossiblyexhibit

non-Abelian particle statistics.8–13A number of different

ground-state wave functions have been proposed for the

5/2 state, some with non-Abelian statistics and some with

prosaic Abelian statistics. Were the existence of non-Abelian

statistics confirmed, it would be an exciting discovery of a

new state of matter and would possibly enable topological

quantum computation.14A great deal of theoretical and

experimentalworkhasbeendoneonthe5/2staterecently.15–32

Experimentally, the quasiparticle charge e∗has been found to

be consistent with the predicted value e/4.24,25,31Numerical

simulations indicate a preference for the non-Abelian Pfaffian

and anti-Pfaffian wave functions over various Abelian wave

functions.18,21,23,32,33The degree of electron spin polarization

also provides valuable information about the wave function,

but experimental results are contradictory.29,30Recent exper-

imental results from an interferometer have been interpreted

as evidence for non-Abelian statistics at υ = 5/2.26,27The

observation of a counterpropagating neutral mode is also most

easily explained by the existence of a non-Abelian state.28

Wehavestudiedtheυ = 5/2stateintwodifferentquantum

point contact (QPC) geometries and present temperature and

dc bias dependence of quasiparticle tunneling conductance

across each QPC in the weak tunneling regime. We have

improved the signal-to-noise ratio by a factor ∼2 compared

to previous similar measurements.24By fitting these results to

the theoretical form,34we extract the quasiparticle charge e∗

and interaction parameter g. The resulting e∗is in agreement

with the predicted value of e/4, and the value of g best agrees

with that predicted for the Abelian 331 state. Fixing g at the

values predicted by other proposed states produces fits that are

qualitatively and quantitatively worse. In addition, qualitative

features of the dc bias dependence also favor the 331 state.

II. EXPERIMENTAL DETAILS

ThedeviceusedisthesameasoneofthosestudiedbyRadu

et al.,24and we briefly summarize its characteristics here. The

GaAs/AlGaAs heterostructure has a measured mobility of 1 ×

107cm2/V s and electron density n = 2.6 × 1011cm−2. The

mobility is only half that previously reported, but the cause

of this degradation is unknown. The sample still exhibits a

strong 5/2 fractional quantum Hall effect, with a quantized

Hall plateau and vanishing longitudinal resistance.35Metallic

topgatesarebiasednegativelytodepletetheunderlying2DEG

and induce tunneling between edge channels. The gate pattern

isshowninFig.1(a).TwodifferentQPCgeometriesarecreated

by applying negative voltages to some of the gates while

keeping the remaining gates grounded. Geometry A is a short

QPC of nominal width ∼0.6 μm, and Geometry B is a long

channel of nominal width ∼1.2 μm and length ∼2.2 μm. For

convenience we refer to both geometries as QPCs. The gates

biased to form the two geometries are listed in the caption of

Fig. 1. The measurement setup is illustrated in Fig. 1(b). A dc

current Idcof up to ±10 nA with a 0.4 nA ac modulation is

applied to the source at one end of the Hall bar, with the drain

at the other end. Using standard lock-in techniques at 17 Hz,

measurements are made of the differential Hall resistance

(RXY)andlongitudinalresistance(RXX)atpointsremotefrom

theQPCandofthedifferentialdiagonalresistance(RD)across

the QPC. Experiments are made in a dilution refrigerator

with a mixing chamber base temperature of ∼5 mK. The

temperatures quoted below are electron temperatures, which

track the mixing chamber temperature well down to ∼20 mK.

Lower electron temperatures are calibrated against thermally

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1098-0121/2012/85(16)/165321(5)©2012 American Physical Society

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LIN, DILLARD, KASTNER, PFEIFFER, AND WESTPHYSICAL REVIEW B 85, 165321 (2012)

(a)

ac

dc

(b)

FIG. 1. Device image and measurement setup. (a) Scanning

electron micrograph of a device fabricated similarly to the one used

in this experiment. Gates A1, G3, and G4 are biased negatively for

GeometryA,withtheothergatesgrounded.ForGeometryB:G1,G2,

G3, and G4 are energized, with A1 and A2 grounded. (b) Simplified

diagram of the Hall bar mesa and measurement setup. The mesa is

outlined, and top gates are shaded grey.

broadened Coulomb blockade peaks in a quantum dot and

against quantum Hall features showing linear temperature

dependence.35

In order to preserve the same electron density, and hence

filling factor, both inside and outside the QPC, we anneal

the device with bias voltage applied to the gates at 4 K for

approximately60hbeforecoolingtobasetemperature.24Each

geometryisannealedbyapplying−2400mVtothegateslisted

inthecaptionofFig.1,whileleavingtheothergatesgrounded.

Afterannealingandloweringthetemperature,thegatevoltage

VGis constrained to the range of −2400 to −1800 mV. We

find that after annealing the filling factor in the QPC matches

that of the bulk of the Hall bar. This is important in order to be

confident that the measurements reflect tunneling of υ = 5/2

quasiparticles from one chiral edge state to the other. The

success of annealing is confirmed by the observation that RXY

(sensitive to the 2DEG far from the QPC) and RD(sensitive to

the QPC) exhibit the same dependence on magnetic field.35In

particular, the integer QHE plateaus begin and end at the same

values of magnetic field, and the low-field slopes of RXYvs B

andRDvsBarethesame.ThisidenticaldeviceinGeometryB

hasbeenpreviouslymeasuredinthestrongtunnelingregime,24

andwehaverepeatedthesemeasurementswithsimilarresults.

Here, we anneal at a less negative gate voltage enabling us to

access the weak tunneling regime to much lower temperature.

III. RESULTS

For eachgeometrywemeasurethedcbiasandtemperature

dependence of RD at υ = 5/2, as shown in Figs. 2(a) and

2(b). The peak positions deviate from Idc= 0 by ∼0.2 nA

because of hysteresis in the sweep direction; all measurements

shown in this paper are taken with increasing dc bias, and the

small offset is subtracted when fitting. The magnetic field and

gate voltage are chosen, following the measurement technique

described in Radu et al.,24to maximize the temperature range

exhibiting a zero-dc-bias peak and to minimize variations

in the background resistance.35The magnetic field is set to

4.31 T, which is the center of the RXY plateau for υ = 5/2.

Geometry A is measured at VG= −2100 mV and Geometry

B at −2148 mV.

In the limit of weak quasiparticle tunneling, RDis linearly

related to the differential tunneling conductance gTby

gT = (RD− RXY)/R2

XY,

(1)

(a)

(b)

(c)

dc

dc

dc

FIG. 2. (Color) dc bias and temperature dependence of RD, with

best fit. (a) RDmeasured in Geometry A with an applied gate voltage

of VG= −2100 mV. (b) RDmeasured in Geometry B with VG=

−2148 mV. For both, the magnetic field is set to the center of the

υ = 5/2 RXY plateau (B = 4.31 T), and the sample is annealed at

VG= −2400 mV as discussed in the text. (c) Least-squares best fit

of RDto Eq. (2) for Geometry B, with all temperatures in the range

27–80 mK fit simultaneously. Tunneling conductance peaks at each

temperature (labeled on graph) are concatenated to produce a single

curve. Experimentalresultsarered,and thefitisblack. Ticksindicate

0 and ±5 nA for each temperature.

provided that the sample is in an integer or fractional quantum

Hall state. This allows us to fit to the theoretical form for weak

quasiparticle tunneling in an arbitrary FQHE state:34

?

gT = AT(2g−2)F g,e∗IDCRXY

kBT

?

.

(2)

The quasiparticle charge e∗and interaction parameter g

are physical constants characterizing the FQHE state, while

the fit parameter A accounts for the tunneling amplitude. F

has a complicated functional form, peaked at zero dc bias

(see supplementary material35). Fapproaches zero at infinite

dc bias and for g < 1/2 exhibits minima on the sides of

the zero-bias peak; these minima are absent for g ? 1/2.

Equation (2) predicts that the zero-bias peak height follows

a power-law temperature dependence and that the full-width

165321-2

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MEASUREMENTS OF QUASIPARTICLE TUNNELING IN ...

PHYSICAL REVIEW B 85, 165321 (2012)

at half maximum in dc bias is linear in temperature. Like Radu

etal.24andasseeninFigs.2(a)and2(b),wefindabackground

resistance R∞,which is larger than the expected quantized

value 0.4 h/e2. However, this background is independent of dc

biasandtemperaturewithintheresolutionofthemeasurement,

under the measurement conditions chosen.

A least-squares fit to Eq. (2) of the RDmeasurements for

Geometry B for many temperatures is shown in Fig. 2(c). We

limit the temperature range for fitting to 20–80 mK for Geom-

etryAandto27–80mKforGeometryBtoensurethatonlythe

weak tunneling regime is included. At lower temperatures the

dc bias curves start to exhibit features associated with strong

tunneling.24,36,37However,wenotethatthefittingresultsdonot

change significantly if a different temperature range is chosen.

Within the chosen range, measurements for all temperatures

are fit simultaneously with the same fitting parameters. The

best fit for Geometry A returns e∗= 0.25e and g = 0.42,

and for Geometry B returns e∗= 0.22e and g = 0.34. These

values are similar to those obtained in a previous weak

tunneling measurement24(e∗= 0.17e, g = 0.35), but we find

e∗much closer to the predicted value of e/4.38

In order to understand the level of confidence we can place

in these fit results, we fix (e∗, g) pairs over a range of values

and fit to the weak tunneling form, allowing A and R∞to vary.

The residual of each fit is divided by the measurement noise of

∼2 × 10−4h/e2and plotted against the (e∗, g) pairs in Fig. 3.

(a)

(b)

FIG. 3. (Color) Matrix of fit residual for fixed pairs of (e∗, g)

divided by the experimental noise. (a) Results for Geometry A.

(b) Results for Geometry B. For both, pairs of (e∗, g) corresponding

to proposed non-Abelian states (green squares) and Abelian states

(white circles) are plotted; see the text for more details. Contours of

the normalized fit error are included as guides to the eye.

Pairsof (e∗,g)values withafitresiduenolarger thanthenoise

fall within the “1” contour. Also indicated are the pairs of (e∗,

g) values corresponding to the proposed wave functions listed

below.

IV. INTERPRETATION AND DISCUSSION

The simplest interpretation of our data is that the value

of g derived from the fits directly reflects the nature of the

quasiparticles,andthereforethewavefunction,ofthe5/2state

inoursystem.Thisallowsustodistinguishbetweencompeting

proposalsbytheirexpectedvalueofg.ProposedAbelianstates

include 33139,40with g = 3/8 and K = 841with g = 1/8.

Non-Abelian states include the Pfaffian9with g = 1/4, the

particle-hole conjugate anti-Pfaffian11,12with g = 1/2, and

U(1) × SU2(2)10with g = 1/2. All these states support

quasiparticles both of charge e/4 and charge e/2. However,

tunneling measurements are expected to be dominated by

quasiparticles of charge e/4.42Hence, we examine the fits

to Eq. (2) with e∗fixed at e/4 and g fixed at a value predicted

by one of the proposed states: 1/8, 1/4, 3/8, or 1/2. Of these

fouroptions,fitswithg = 3/8andg = 1/2producethelowest

residuals, as can be seen in Fig. 3; fits for g = 1/8 and 1/4 are

very poor, both quantitatively and qualitatively.35

Fits for Geometry A with g = 1/2 and 3/8 are shown in

Figs. 4(a) and 4(b), respectively, and similarly for Geometry B

in Figs. 4(c) and 4(d). The fits with g = 3/8, corresponding to

the Abelian 331 state, are clearly the best, following the data

closely, including the temperature dependence of the peak

height and the obvious minima on either side of the main

peak. For g = 1/2, the minima on either side of the peak are

absent, and peak heights are not well described. Our ability to

better discriminate between g = 1/2 and g = 3/8 than with

previous weak tunneling measurements24results from a factor

of about two reduction in the noise. Reaching a lower level

of noise is particularly important because it makes the dc bias

minima, which are important qualitative features of Eq. (2),

clearly distinguishable from the noise. The minima are even

more prominent at base temperature, as can be seen in Fig. 2.

In fact, the presence of these minima, which have also been

observed in previous strong tunneling measurements,24is an

indication that g is strictly less than 1/2. Expanding the exact

form of the current transmitted through a QPC in the FQHE

around the high dc bias limit,36one finds

1

RD

=

dI

dVD

= g + C

?TB

VD

?2(1−g)

(2g − 1) + ···.

(3)

Here, C is a negative constant, TBis a temperature scale

reflecting the strength of the edge channel interaction in the

QPC, and VD is the diagonal voltage. As TB/VD increases

(correspondingto|Idc|decreasingfrominfinity),RDdecreases

for g < 1/2, is constant for g = 1/2, and increases for g >

1/2. Because RDeventually increases as the bias approaches

zero, this produces a minimum in RD only for g < 1/2.

This behavior is also reflected in the weak tunneling formula,

Eq.(2).Thisanalysisneglectshigher-orderterms,whichcould

conceivably produce a minimum in RD at g = 1/2, but to

lowest order, the presence of minima requires g < 1/2.

165321-3

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LIN, DILLARD, KASTNER, PFEIFFER, AND WESTPHYSICAL REVIEW B 85, 165321 (2012)

(a)

(b)

(c)

(d)

dc

dc

dc

dc

FIG. 4. (Color) Fits of RDto the theoretical form of quasiparticle

weaktunnelingwithfixede∗andg.(a)and(b)FitsofRDinGeometry

A to Eq. (2) with fixed g = 1/2 [anti-Pfaffian and U(1) × SU2(2)

wave functions] and g = 3/8 (331 wave function), respectively.

(c) and (d) Fits of RD in Geometry B to Eq. (2) with fixed

g = 1/2 and g = 3/8, respectively. In all cases e∗is fixed to e/4,

and all temperatures shown are fit simultaneously with the same

fit parameters. Tunneling conductance peaks at each temperature

[labeled on graph] are concatenated to produce a single curve.

Experimental results are red, and fits are black. Ticks indicate 0

and ±5 nA for each temperature.

It may be that the value of g extracted from Eq. (2)

does not directly reflect the nature of the 5/2 state in our

system. Reconstruction of edge channels43and interactions

between edge channels may cause the value of g observed

by tunneling experiments to change.44,45The presence of

striped phases, which have been found to be energetically

favorable for some confinement potentials17or random do-

mains of different ground-state wave functions,15may also

complicate matters. Even the Luttinger liquid theory of

quasiparticle tunneling may be an incomplete description

of experimental systems. Experimental results of electron

tunneling in cleaved-edge-overgrowth samples are not fully

described by the predictions of chiral Luttinger liquid theory,

indicating that perhaps a more nuanced picture is needed to

accurately describe tunneling experiments.46Measurements

of transmission through QPC devices in the lowest Landau

level provide some confidence that Luttinger liquid theory

can be applied to quasiparticle tunneling across a QPC.37,47,48

Ideally, one would like to repeat the measurement technique

of Radu et al.24at a simple filling fraction, such as υ = 1/3,

and fit the results to Eq. (2). Unfortunately, the sample

studied here is not suitable for such measurements, because

of the relatively wide width of the QPC and the low electron

density.

V. CONCLUSION

In conclusion, fits based on quasiparticle weak tunneling

theory34favor the presence of the 331 Abelian wave function

in our sample, while excluding other states, including the

non-Abelian Pfaffian and anti-Pfaffian. Particularly telling is

the presence of minima in the dc bias dependence, which

requires g < 1/2. Given previous studies favoring the Pfaffian

andanti-Pfaffianwavefunctions18,21,23,29,32,33oranon-Abelian

wave function in general,27,28it seems possible that different

states may be physically realizable at υ = 5/2. The device

geometryandheterostructurecharacteristicsmaybeimportant

factors in determining which state is favored. For example,

thereisnumericalevidencethatthestrengthoftheconfinement

potential influences the wave function exhibited in a FQHE

system at υ = 5/2.17Better understanding of the effects of

these factors on the 5/2 state will likely be vital for any

efforts to further explore non-Abelian particle statistics or

realize a topological quantum computer. We recommend that

similar measurements of e∗and g be performed on other

heterostructures and device geometries, especially the ones

in which evidence of a non-Abelian wave function has been

observed.27,28

ACKNOWLEDGMENTS

We are grateful to Claudio Chamon, Dmitri Feldman,

Paul Fendley, Charles Marcus, Chetan Nayak, and Xiao-Gang

Wen for helpful discussions. We thank Jeff Miller from the

Marcus Lab for sample fabrication at Harvard’s Center for

Nanoscale Systems, with support from Microsoft Project Q.

The work at MIT was funded by National Science Foundation

under Grant No. DMR-1104394. The work at Princeton was

partially funded by the Gordon and Betty Moore Foundation

as well as the National Science Foundation MRSEC Pro-

gram through the Princeton Center for Complex Materials

(Grant No. DMR-0819860).

165321-4