Table Of ContentIntrinsic Gap of the ν = 5/2 Fractional Quantum Hall State
C.R. Dean1,⋆, B.A. Piot1,⋆, P. Hayden2, S. Das Sarma3, G. Gervais1, L.N. Pfeiffer4 and K.W. West4
1Department of Physics, McGill University, Montreal, H3A 2T8, CANADA
2School of Computer Science, McGill University, Montreal, H3A 2A7, CANADA
3Condensed Matter Theory Center, Department of Physics,
University of Maryland, College Park, MD 20742 USA and
4Bell Laboratories, Lucent Technology, Murray Hill, NJ 07974 USA
8 The fractional quantum Hall effect is observed at low magnetic field, in a regime where the
0 cyclotron energy is smaller than the Coulomb interaction energy. The ν = 52 excitation gap is
0 measured to be 262±15 mK at ∼2.6 T, in good agreement with previous measurements performed
2 on samples with similar mobility, but with electronic density larger by a factor of two. The role
of disorder on the ν = 5 gap is examined. Comparison between experiment and theory indicates
n 2
that a large discrepancy remains for the intrinsic gap extrapolated from the infinite mobility (zero
a disorder) limit. In contrast, no such large discrepancy is found for the ν = 1 Laughlin state. The
J 3
observationofthe 5 stateinthelow-fieldregimeimpliesthatinclusion ofnon-perturbativeLandau
0 2
level mixing may be necessary to betterunderstand the energetics of half-filled fractional quantum
3
Hall liquids.
]
l PACSnumbers: 73.43.-f,73.63.Hs,03.67.-a
l
a
h
- Since the discovery of the fractional quantum Hall ef- FQHE in a regime where the cyclotron energy is smaller
s
e fect (FQHE), understanding the role played by electron- than the Coulomb interaction energy. We compare the
m electroninteractionshasbeenthesourceofmajorbreak- measured energy gap with neighbouring FQH states in
. throughs in our understanding of strongly interacting the SLL, and discuss these results in the context of pre-
t
a two-dimensional electron gases (2DEGs). Chief among vious studies allowing us to deduce the intrinsic gap in
m theseisthecompositefermionpictureoftheincompress- the zero-disorder limit. Our analysis shows that large
- ible FQHliquid[1, 2], extremely successfulatexplaining discrepancies remain between theory based on a Moore-
d 5
both the complete series of observed FQH states in the ReadPfaffianstateandexperimentatν = thatcannot
n 2
first Landau level (FLL), and the absence of such a liq- be attributed to disorder alone. In contrast, a similar
o
1
c uid at precisely half-filling. In the second Landau level analysis for the ν = 3 Laughlin state shows much better
[ (SLL), however,the situationis morecomplex where ex- agreement with current models.
periments have shown, unambiguously, exact quantiza-
2 The sample used in this study was a 40 nm wide,
5
v tion of the Hall resistance at filling factor ν = 2 [3, 4] modulation-doped, GaAs/AlGaAs quantum well, with a
2 and ν = 27. In 1991, Moore and Read [5] proposed an measured density of 1.6(1) 1011 cm−2 and mobility of
31 elegant many-body wave fun5ction to explain this phe- 14(2) 106cm2/Vs. Thesa×mplewascooledinadilution
nomenonthatdescribedthe FQHstateasa‘condensa- × ·
3 2 fridge enclosed inside a shielded room, with a base tem-
tion process’of composite fermions. In recentyears,this
. perature of 16 mK in continuous mode, and equipped
1
Moore-Read‘Pfaffian’statehasreceivedconsiderablein- ∼
0 with a 9 Tesla magnet. Treatment with a red LED was
terest owing to built-in quantum statistics that are now
8 usedduringthe cooldown. In situ powderfiltersandRC
0 predicted to be non-abelian. The non-abelian composite filters were used on the sample leads to ensure efficient
5
: particles that comprise the ν = FQH state underlie a
v 2 coolingofthe2DEG.Temperaturesweremonitoredwith
paradigm for fault-tolerant topological quantum compu-
i aRuOresistivethermometer,andaCMNmagnetization
X tation first proposed by Kitaev[6] and recently exploited
thermometer,bothcalibratedwithsuperconductingfixed
r by Das Sarma, Freedman and Nayak[7]. Yet, in spite of
a points. Transportmeasurementswereperformedusing a
these many recent theoretical advances, an unequivocal
standard lock-in technique at 6.5 Hz.
experimental verification of the Moore-Read description ∼
isstillmissing. Furthermore,continueddiscrepanciesbe- Fig. 1 shows the magnetoresistance (Rxx) and corre-
5
tweenexperimentandtheory,suchasthelargedifference sponding Hall resistance (Rxy), taken around ν = 2 in
between the measured and calculated activation energy the SLL at 20 mK. A vanishingly small magnetoresis-
∼ 5
gap, remain problematic. tance is observed at ν = 2, which, together with a wide
5
plateau in the corresponding Hall trace, indicates the
2
5
In an effort to better understand electron-electron in- state is exceptionally well formed. The unambiguous
2
teractionathalffilling,wepresentinthisworkadetailed state observed here, occurring at 2.63 T, represents to
5 ∼
analysis of the ν = state for a sample with, to our our knowledge the lowest magnetic field observation of
2
5
knowledge, the lowest electron density reported to date the to date [3, 4, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
2
(by nearly a factor of two). This allows the study of the 18, 19, 20]. Strong FQHE minima are also observed at
2
13 0.51 100 60
n =5/2 0.402=1] (a) n =14/5 (b) n =8/3
WR [k] xy1112 2.60 2B.6 2[T]2.6400..349080nR [Rxyxy 5/2 71/311/56/7 00..4437n=1R [R]xyxy 10 D1 4 =9– 5 mK 2400 D64 =– 8 mK
10 0.39 0 20 40 60 0 20 40 60
8/3 100 100
14/5 (c) n =5/2 (d) n =7/3
9 0.35
(a)
] D = D =
1.0 2.4 2.6 2.8 3.0 WR [xx10 262– 15 mK 10 225– 13 mK
(b)
0.8
0 20 40 60 0 20 40 60
100 1
Wk] 0.6 16/7 ( e) n =11/5 0.8 (f ) n =7/2
R [xx0.4 3 14/5 8/3 5/2 12/57/3 11/5 2 D18 =3– 6 mK R/R xxpeak0.6 2D 5Qu–a6si =mK
0.2 10
0.4
0 20 40 60 0 20 40 60
0.0 2.4 2.6 2.8 3.0 1/T [K-1]
B [T]
14/5 19/78/3 5/2 7/3 16/7 11/5
0.006
FIG.1: (a)Hallresistanceand(b)correspondingmagnetore- Choi "high m" [17]
sistance in the second Landau level of our low density, high 0.005 Couhro di a"ltoaw m" [17] (g)
mobility 2DEG (T∼20 mK).
0.004
]B
l
2ee/ 0.003
[
D 0.002
14 8 7 16 11
ν = , , , , and , each of which exhibit plateaus
5 3 3 7 5
in R . A hint of an emerging R minimum can also 0.001
xy xx
12
be seen at ν = 5 . The four reentrant phases observed 0.000
5 0.36 0.40 0.44
in the Hall trace on either side of the plateau (two
2 1/n
peaks tending towards R =h/3e2 and two tending to-
xy
2
wards Rxy=h/2e ) together with the observation of the FIG. 2: (a)-(e) Arrhenius plot showing activated tempera-
16 12
ν = 7 state, and the emerging minimum at ν = 5 , ture behaviour in the second Landau level. (f) Quasi-gap
are all signatures of an extremely high quality sample measurement for ν = 7 (see text). (g) Energy gaps for the
2
[4, 14, 15, 17]. The deep R minima appearing in the FQH states, plotted in Coulomb energy units. Open circles
xx
are our data. Solid squares and circles, respectively, refer to
reentrantinsulatingphaseat 2.55T(Fig.1b)issimilar
∼ the “high mobility” and “low mobility” samples in reference
to that observed elsewhere upon lowering the electronic
[17].
temperaturetoaregimewherethereentrantstateisfully
formed [15, 19, 20].
ThetemperaturedependenceoftheFQHEminimaare
shown in Fig. 2a-e, with all data acquired at fixed mag- energy dependent Landau level broadening due to disor-
7 5 8
netic field in order to avoid heating effects caused by der[22]. Inthe , ,and states(Figs.2b-d),thereis a
3 2 3
varying fields. The corresponding energy gaps were de- deviation from activated behaviour at high temperature
termined by linear fits to the thermally activated trans- whose onset temperature scales with the corresponding
portregion,wheretheresistanceisgivenbytheequation gapvalue. Likely,this resultsfromk T approachingthe
B
R e−∆/2kBT. Thegaperrorquotedoneachplotwas gap value. However, the same deviation is not observed
xx
∝ 11 14
estimated from the goodness of the linear fit. Examina- inthe and FQHstates,whichhavelowermeasured
5 5
7 5
tion of weakly formed FQHE states under single shot of gaps than the and states. Interestingly, recent work
3 2
11 14
thedilutionfridgedownto 9mKindicatedtheelectrons has suggestedthe and to be Laughlinstates,while
5 5
∼ 7 8
continued to cool, suggesting that the low temperature the and areproposedtobenon-Laughlin[17,24,25].
3 3
7
tail-off observed in the data of Fig. 2 does not reflect We also observed a FQH state at ν = , the electron-
2
5
a saturation in the electronic temperature. Instead, it hole conjugate of ν = , appearing at a magnetic field
2
may indicate a transition from activated conduction to less than 2 T. However, due to a competition between
7
hopping conduction [21], and/or could result from the the weakly formed and rapidly emergentneighbouring
2
3
reentrant states, the R minimum did not fall signif-
xx 0.025 Theoretical D i [28]
icantly with temperature near base. The “quasi-gap” 5/2 (a) n = 5/2
was therefore determined by measuring the depth of the
0.020
7 Gammel (1998) [9]
t2wmo inneiimghabwouitrhinrgesppeeacktst(oRtpheaek)a[v8e,r9a,ge13r]e.siTsthaencreesouflttinhge 2ee/l]B0.015 wLoitwh eLsLtM ca alcnudl afitneidte D i5/2 EPPEiaaissnnee nn((12ss90ttee90iinn91 ))(( 12[[11902390]]02)) [[1114]]
A(Frirgh.e2nfi)ugsivpelosta,nwehsitcihmaclteeafrolrytshheow72sgaapctvivaaluteedofbeh2a5vmioKur. D [ 0.010 width corrections [33] MCPahilnole ir( ( 2(2020000807)7) )[ 2[ 1[017]9]]
∼ Our Data
0.005
In Fig. 2g, the gap values are plotted in Coulomb en-
ergy units, e2/ǫlB, where lB = p~/eB is the magnetic 0.000
length, and ǫ = 12.9 is the dielectric constant. Results 0.000 0.005 0.010 0.015 0.020
from recent gap measurements in the SLL by Choi et
0.10 Theoretical D i [28]
al. are also shown for comparison [17]. The excellent 1/3
(b) n = 1/3
agreement between our data set and that of the Choi
0.08
eittsrhtoesaniurl.srdpe(‘rln2oi.ssw8iintimges1og0bbiv1eil1eticwntmye’te−hns2eaomafuanprcdlteso3a(r.mµ2oBpfle=t1w(0111o.106c.d5m×iff×−1e20r1e10fn1o6ccrcmemt−hi2n2e/)V“eall·eonscwd)- 2De [e/l]B00..0046 Tfohre foinrieteti cwaild Dthi1 /[33 c1o,3rr3e]cted BHBWDDooaueieeltu lh(ebbg1etii 9tnnl( f91gg(s139eee9)8nrr 87 [(((3811)2 1)99[0 ]288[0295763])))] [[[323210]]]
× ×
mobility” and “high mobility” respectively). Simple di-
0.02
mensional considerations imply that the interaction en-
ergy,andhencetheFQHgap,shouldscaleas√B,which 0.00
0.00 0.05 0.10 0.15 0.20 0.25
would predict a 40% enhancement in the gap between 1/t [ps-1]
the low density (∼ours) and the high density (Choi et al.) tr
samples. Our finding that the gap is almost the same FIG. 3: (a) Comparison of the measured ν = 5 gap energy
2
for the two samples with similar mobility (independent withvaluesfoundinliterature(opensymbols). Solidtriangles
of density), while significantly enhanced in samples with representourdata. Solidsquareisthetheoreticallycalculated
higher mobility (Choi et al. “high mobility”) indicates intrinsic gap energy [27, 28] and the solid circle includes cor-
that disorder more strongly affects the gap than the ap- rections for Landau level mixing, and finite width. (b) Same
plied magnetic field. Furthermore, the similar gap value plot as in (a), but for ν = 13 energy gap values reported in
theliterature.
measured in a low magnetic field where the cyclotron
energy is reduced compared to the Coulomb interaction
suggestsnon-perturbativeLandaulevelcouplingmayaf-
5
fect the ν = FQH gap in a way not yet understood
2 For comparison, a similar plot for the measured gap
theoretically. 1
values of the Laughlin state is shown in Fig. 3b,
3
5 5
In Fig. 3a, we show a plot of all the gap val- [21, 23, 29, 30, 31, 32]. In contrast to the ν = FQH
2 2
1
ues found in the literature versus the inverse trans- state,theintrinsicgapdeterminedatν = withourpro-
3
port lifetime, τ−1, deduced from the reported mobili- cedure (∆i 0.045 e2/ǫl ) is in good agreement with
tr 1/3 ∼ B
ties [4, 9, 11, 13, 14, 17, 19, 20]. In spite of the large theory ( 0.055 e2/ǫl ) [28, 31]. Morf et al. proposed
spread in the 25 data, owing to wide ranging differences that sinc∼e the disorderB-induced Landau level broadening
in sample parameters, i.e. dopant, well width, etc., a is expected to be roughly equal for FQH states corre-
clearlydiscernibletrend(indicatedbythesolidcurveasa sponding to particle-hole conjugate pairs, then plotting
guide-to-the-eye)isobservedpointingtowardsadisorder- thecorrespondiggapvaluesasafunctionofCoulomben-
free intrinsic gap value in the range of ∆i5/2 ∼ 0.005- ergy directly gives a measure for the intrinsic gap (slope
2 5
0.010 e /ǫl . This estimate is in good agreement with of a fitted line to this data) [33]. Fig. 4 shows the and
B 2
7
a similar extrapolation reported very recently by Pan gap values obtained in our low electron density sam-
2
et al. [20, 26]. Moreover, examination of the low field ple (open squares) together with those from Ref. [14]
Shubnikov de Haas oscillations gave the level broaden- (open triangles). The dashed line shows the predicted
ing, Γ, in our sample to be Γ = 0.168 0.040 K. This trend for a disorder free gap. The slope extracted from
±
givesadirectexperimentalestimatefortheintrinsicgap, a linear fit gives the intrinsic gap for our sample to be
∆i = ∆exp +Γ, of 0.005 e2/ǫl , also in good agree- 0.018 e2/ǫl , which is in remarkable agreement with
B B
∼ ∼
mentwiththeextrapolatedintrinsicgapvalueinFig.3a. thedatafromref. [14]( 0.014)andthetheoreticalvalue
∼
Importantly, the experimentally measured intrinsic gap ( 0.016)correctedforthesampleparametersspecifiedin
∼
inferred from this data remains well below (by a fac- ref. [14, 33]. This however disagrees with the intrinsic
tor of three to five) the theoretically estimated intrinsic gap estimated both from our sample, and from the ex-
gapvalue for a Moore-Readtype Pfaffianwavefunction, trapolation towards the infinite mobility limit. Further-
( 0.025 e2/ǫl ) [27, 28]. more, the Landau Level broadening deduced from Fig. 4
B
∼
4
implies aratherlargevalue( 1.25K)thatisanorderof 3.5
magnitude larger than the va∼lue determined experimen- 3.0 Theoretical gap (no disorder) corrected
2.5 for LLM and finite width [33]
tally from the SdH oscillations ( 0.17 K).
∼ 2.0 our data
It is instructive to consider the energetics of our low- Eisenstein (2002) [14]
1.5
density 25 FQH state. At the observed field of 2.62 T [K]1.0
the cyclotron energy is 52 K, the interaction energy is D 0.5
81 K, and the Zeeman energy (assuming the GaAs band 0.0
g-factor) is 0.75 K. The level broadening in our sample -0.5
was measured to be 0.17 K and the mobility broaden- -1.0
∼ -1.5
ing 0.006 K [34]. Also important is the suppression of
0 20 40 60 80 100 120 140
∼
theidealtwo-dimensionalFQHexcitationgapduetothe e2/e l [K]
finite width d=40 nm of our quasi-2D square well sam- B
5
ple. For our 2 FQH gap, this is only about 15% (using FIG. 4: Intrinsic gap energy at ν = 5 from particle-hole
d/l =2.5in [28],where l =16nm at 2.6T).Taking 2
B B ∼ paired states [33] (see text). Dashed line indicates a theo-
alloftheseenergiesintoaccountweconclude: i)ourmea- retical gap with no disorder. Open squares are the gaps for
sured gap value of 0.262 K is at least a factor of 5 lower the 7 and 5 FQHstates. Opentrianglesaredatatakenfrom
2 2
5
than the ideal 2D theoretical excitation gap ( 2 K Ref. [14],forcomparison. Solidanddottedlinesarelinearfit
2 ∼
at 2.6 T), even if the theoretical gap is corrected for fi- to thedata.
nite width and level broadening suppression ( 1.5 K);
∼
ii)the cyclotrongap,i.e. theLandaulevelseparation,is
smaller than the interaction energy in our system, sug-
may play an important role in the understanding of the
gesting considerablenon-perturbativeinter-Landaulevel
5
enigmatic FQH state.
coupling, which has not so far been included in the the- 2
ory,maybeimportantinunderstandingthe 5 FQHstate; ThisworkhasbeensupportedbytheNaturalSciences
2
iii) the Zeeman energy at 2.6 T is extremely small com- and Engineering ResearchCouncil of Canada (NSERC),
pared with the Coulomb energy, so the observation of a the Canada Fund for Innovation (CFI), the Canadian
strong 5 gapatthisfieldmightsuggestthatthe 5 FQHE Institute for Advanced Research (CIFAR), the Canada
2 2
is spin-unpolarized. However,the 5 FQHE has been ob- ResearchChairsprogram,MITACS,QuantumWorksand
2
served in magnetic fields as large as 12T[13], i.e. with FQRNT (Qu´ebec). Two of the authors (G.G. and P.H.)
an increase in Zeeman energy by a factor of five, where acknowledge receipt of support from the Alfred P. Sloan
the system is most likely spin-polarized, without affect- Foundation through their fellowship program.
5
ing much the gap. Therefore, unless a quantum phase Special Note: Both authors marked with a star (*)
2
transition occurs between a low-field ( 2.6 T) spin- un- contributed evenly to this work.
∼
polarized state, and a high-field ( 12 T) spin-polarized
5 ∼
FQH state, our experiment rather points towards a
2
5
spin-polarizedstate at ν = even in the zero-fieldlimit,
2
consistentwith a Moore-Readpfaffian wavefunction,the
leading candidate for the 5 FQHE. [1] J.K. Jain, Phys.Rev.Lett. 63, 199 (1989).
5 2 [2] B.I. Halperin, P.A.Lee, and N.Read,Phys.Rev.B. 47,
In conclusion, the energy gap was measured for a
2 7312 (1993).
samplewithanelectrondensitynearlytwicesmallerthan
[3] R. Willett, J. P. Eisenstein, H. L. Stormer, D. C. Tsui,
previously observed,and was found to be comparable to A. C. Gossard, and J. H. English, Phys. Rev. Lett. 59,
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1 [5] G. Moore and N. Read,Nucl. Phys. B360, 362 (1991).
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3 [6] A. Yu.Kitaev, Ann.Phys.(N.Y.) 303, 2 (2003).
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5
oryandexperimentatν = cannotsimplybeattributed [8] J.P. Eisenstein, R. Willett, H.L. Stormer, D.C. Tsui,
2
to disorder, but rather may indicate that our knowledge A.C Gossard, and J.H. English, Phys.Rev.Lett.61, 997
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2
[9] P.L. Gammel, D.J. Bishop, J.P. Eisenstein, J.H. En-
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5 glish, A.C. Gossard, R. Ruel and H.L. Stormer, Phys.
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2 Rev.B38, 10128 (1988).
islargerthanthecyclotronenergy,wespeculatethatthe
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5
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