Stability of multielectron bubbles in high Landau levels
Dohyung Ro, S.A. Myers, N. Deng, J.D. Watson, M.J. Manfra, L.N Pfeiffer, K.W. West, G.A. Csáthy
SStability of multielectron bubbles in high Landau levels
Dohyung Ro , S.A. Myers , N. Deng , J.D. Watson , M.J. Manfra , , , L.N Pfeiffer , K.W. West , and G.A. Cs´athy , Department of Physics and Astronomy, Purdue University, West Lafayette, IN 47907, USA School of Materials Engineering and School of Electrical and Computer Engineering,Purdue University, West Lafayette, IN 47907, USA Birck Nanotechnology Center Purdue University, West Lafayette, IN 47907, USA Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA (Dated: February 24, 2021)We study multielectron bubble phases in the N = 2 and N = 3 Landau levels in a high mobilityGaAs/AlGaAs sample. We found that the longitudinal magnetoresistance versus temperature curvesin the multielectron bubble region exhibit sharp peaks, irrespective of the Landau level index. Weassociate these peaks with an enhanced scattering caused by thermally fluctuating domains of abubble phase and a uniform uncorrelated electron liquid at the onset of the bubble phases. Withinthe N = 3 Landau level, onset temperatures of three-electron and two-electron bubbles exhibitlinear trends with respect to the filling factor; the onset temperatures of three-electron bubbles aresystematically higher than those of two-electron bubbles. Furthermore, onset temperatures of thetwo-electron bubble phases across N = 2 and N = 3 Landau levels are similar, but exhibit an offset.This offset and the dominant nature of the three-electron bubbles in the N = 3 Landau level revealsthe role of the short-range part of the electron-electron interaction in the formation of the bubbles. The two-dimensional electron gas (2DEG) exposed toperpendicular magnetic fields is a rich model system thathosts a variety of electronic phases. Perhaps the mostwell known of these phases are the fractional quantumHall states which harbor topological order. Electronsolids possessing charge order form yet another distinctgroup of phases. Examples of electronic solids are theWigner solid , electronic bubble phases, and quantumHall nematic or stripe phases .Bubble phases are among the most recently discov-ered phases of 2DEGs which have not yet revealed alltheir properties. They were predicted by a Hartree-Focktheory and confirmed by exact diagonalization anddensity matrix renormalization group studies to be aperiodic arrangement of clusters or bubbles of electrons.In linear transport bubble phases are identified by reen-trant integer quantum Hall behavior . In additionmicrowave absorption , non-linear transport ,surface acoustic wave propagation , temperaturedependence , and thermopower measurements also support the formation of bubbles. However, we stilllack direct probes of the morphology of the bubbles.Bubble phases are commonly observed in 2DEGs inGaAs/AlGaAs and have also been recently seenin graphene . In the former system, bubbles form inhigh Landau levels, at orbital Landau level index N greater or equal to 1. Here we used the customary la-beling of quantum numbers of energy levels associatedwith cyclotron motion, N = 0 being the lowest Lan-dau level. Theories allow for different types of bubblephases within a given Landau level . The differenttypes of bubble phases are distinguished by the numberof electrons per bubble M ; a modest change in the Lan-dau level filling factor was predicted to result in a phasetransition between two different types of bubble phases.Measurements for nearly two decades did not resolve suchdistinct bubble phases. Only recently were distinct bub- ble phases observed in the N = 3 Landau level ; theLandau level filling factors of these bubble phases werein excellent agreement with calculations. These observa-tions allowed the assignment of the number of electronsper bubble for each bubble phase and cemented the bub-ble interpretation of the reentrant integer quantum Hallstates.Recent observations of distinct multielectron bubblephases within one Landau level , that at N = 3,opened up the possibility for their qualitative and quan-titative analysis both within one Landau level and alsoacross different Landau levels. We found that, in ourhigh mobility GaAs/AlGaAs sample bubble phases in the N = 3 Landau level exhibit sharp peaks in the longitu-dinal magnetoresistance versus temperature curves, asmeasured at fixed magnetic fields. Similar peaks weredetected in the N = 1 and N = 2 Landau levels in highmobility GaAs/AlGaAs and also in a graphenesample , but such peaks appear to be absent in a lowmobility GaAs sample containing alloy disorder . Wethink these peaks are due to scattering through the bulkof the sample when the bulk consists of interpenetratingand fluctuating domains of a bubble phase and a uni-form uncorrelated liquid. Within this interpretation, thetemperature of the peak is identified with the onset tem-perature of the bubble phase. We found that the onsettemperatures of the bubble phases determined this wayhave a linear trend with the filling factor and a particu-lar dependence on the number of electrons per bubble.Within the N = 3 Landau level, the onset tempera-tures of M = 3 bubbles are higher than those of M = 2bubbles and exhibit different trends with the filling fac-tor. Furthermore, when comparing the M = 2 bubblephases across N = 2 and N = 3 Landau levels, we findthat onset temperatures are similar but exhibit an offset.These measurements offer information on bubble ener-getics that may be used for a qualitative comparison to a r X i v : . [ c ond - m a t . s t r- e l ] F e b
01 0 0
R6aR6b R6cR6d R7aR7b R7cR7d
N = 2
R4a R4d R5a R5d
M = 2 M = 3 qH nematic or stripe qH nematic or stripeqH nematic or stripe
03 5
N = 3 n Ryy [ W ] qH nematic or stripe FIG. 1. The dependence of the longitudinal magnetoresistance R yy on filling factor ν in the N = 2 (top panel) and N = 3(bottom panel) Landau levels. The two-electron bubble phases ( M = 2) are shaded in yellow, whereas the three-electron bubblephases ( M = 3) are shaded in blue. Vanishing R yy near integer filling factors indicate integer quantum Hall states, whereasareas shaded in green near half-integer filling factors are quantum Hall nematics. Data collected at T = 59 mK. theories and reveal details of the short-range part of theeffective electron-electron interaction.We measured a 2DEG confined to a 30 nm wideGaAs/AlGaAs quantum well. This sample has an elec-tron density n = 2 . × cm − and mobility µ =15 × cm /Vs and it is the same as the one reportedon in Ref. . In order to stabilize the temperature ofthe sample, we took advantage of the large heat capacityof liquid He-3 by mounting our sample in a He-3 im-mersion cell . The temperature in this experiment ismeasured by a common resistive ruthenium oxide ther-mometer. The sample is grown on the (100) face of GaAsand it is cleaved into a 4 × square shape.In Fig.1 we show magnetotransport against the Lan-dau level filling factor ν in the N = 2 and N = 3 Lan-dau level. Here ν = hn/eB , h is the Planck constant, e the elementary charge, and B the applied magnetic field.Regions of vanishing longitudinal resistance R yy in thisfigure are associated with a variety of phases. At integervalues of the Landau level filling factor ν = i , R yy = 0and the Hall resistance is quantized to h/ie , indicatinginteger quantum Hall states . Here i = 4 , , , and 7. Athalf integer values ν = i +1 / . Finally, at other non-integer valuesof ν bubble phases form. In the N = 2 Landau level onlyone type of multielectron bubble phase develops . Weextensively report on R yy , the longitudinal magnetoresis-tance along the [110] crystallographic axis of our sample.In the region of bubble phases, the magnetoresistance isnearly isotropic .In contrast to the N = 2 Landau level, as recently dis-covered, in the N = 3 Landau level there are two differ- ent types of multielectron bubble phases . Based onan excellent agreement of the measured filling factors ofthese phases with those predicted by the theory, the num-ber of electrons per bubble was identified for each bubblephase. In Fig.1 we shaded and labeled the two-electron( M = 2) and three-electron ( M = 3) bubble phases.Multielectron bubble phases of the N = 3 Landau levelare separated by a small magnetoresistive feature .The Hall resistance of bubble phases was found to bequantized to integer values of the nearest integer quan-tum Hall plateau (not shown in Fig.1). Usingtechniques other than transport, in these Landau levels M = 1 bubbles also form as part of plateaus of integerquantum Hall states . However, our transport ex-periments cannot distinguish them from other localizedstates and thus in this Article the M = 1 bubble phaseswill not be further discussed.Similarly shaded bubble phases in Fig.1 appear toform at particle-hole conjugated filling factors . Inthe following we examine this apparent symmetry to agreater detail. Bubble phases in high mobility samples,such as ours, form in a range of filling factors. We de-fine ν c , the central filling factor of a bubble phase, asthe filling factor of its highest stability. Thus the cen-tral filling factor is the filling factor of the local mini-mum in the longitudinal magnetoresistance in the bub-ble phase region that may confidently be detected atthe highest possible temperature . For example, inFig.2a we observe that for the R a bubble phase thereis a local minimum in R yy isotherms that persists totemperatures as high as T = 97 mK. This local min-imum is observed at ν R ac = 7 .
30. Similarly, for the
Ryy [ W ] T [ m K ]
Ryy [ W ] a b R 7 a
R 7 b
R7b R7a n R 7 a c T R 7 a c T R 7 b c W n R 7 b c n FIG. 2. (a) Isothermal evolution of R yy versus B -field for the R a and R b bubble phases. Numbers on each trace are thetemperature in units of mK. Arrows mark the central fillingfactors ν c of these phases. (b) Evolution of R yy versus T of the R a and R b bubble phases at their respective central fillingfactors ν R ac = 7 .
30, and ν R bc = 7 .
22. Arrows indicate theonset temperatures T c of these phases near the peak region ofthe R yy versus T curves. more fragile R b phase there is a local resistance min-imum developed at ν R bc = 7 .
22 at temperatures as highas T = 75 mK. These and central filling factors of othermultielectron bubble phases of the N = 3 Landau levelare shown in Table.I. Errors for filling factors are ± . M = 3 bubble phases can be written in the form ν c = 6 + 0 . , − . , . , − .
30 for R a , R d , R a , and R d , respectively. Furthermore, the filling fac-tors of the family of M = 2 bubble phases can be writtenin the form ν c = 6 + 0 . , − . , . , − .
22 for R b , R c , R b , and R c , respectively. We thus foundthat, similarly to the bubble phases of the N = 1 and2 Landau levels , those of the N = 3 Landau levelalso form at central filling factors related by particle-holeconjugation .While the isotherm at T = 97 mK in Fig.2a exhibits alocal minimum near ν R ac = 7 .
30, that at T = 104 mK ex- TABLE I. Central filling factors ν c and onset temperatures T c of the bubble phases of the N = 3 Landau level. R a R b R c R d R a R b R c R dν c T c [mK] 117 100 91 117 101 80 71 100
03 06 0
R 6 a
Ryy [ W ] T [ m K ]
R 6 d R 7 a R 7 d
R 6 b
R 6 c
R 7 b
R 7 c
FIG. 3. Temperature dependence of the longitudinal magne-toresistance measured at fixed filling factor ν = ν C for theeight bubble phases in the N = 3 Landau level. Curves ex-hibit a sharp peak near the onset temperatures T c of the bub-ble phases. hibits a local maximum. We define T R ac , the onset tem-perature of R a , as the average of highest temperature atwhich there is a local minimum in R yy and the next high-est temperature of measurement. The difference betweenthese two temperatures signifies the error in determining T c . Values obtained from such an analysis of this andother multielectron bubble phases of the N = 3 Landaulevel are found in Table.I. Errors for T c are ± T = 104 mK R yy isotherm measured near ν R ac = 7 .
30, shown in Fig.2a,may still be associated with the bubble phase R a ; thislocal maximum indicates a precursor of the bubble phase R a .In Fig.2b we plot the evolution of R yy with T as mea-sured at the central filling factor ν c for the bubble phases R a and R b . We denote such curves as R yy ( T ) | ν = ν c .These R yy ( T ) | ν = ν c curves may be thought of as cutsalong a constant filling factor ν = ν c in the R yy ( ν, T )manifold having two independent variables ν and T . Asexpected, R yy ( T ) | ν = ν c is vanishingly small at the lowestmeasured temperatures, indicating well developed bubblephases. In addition R yy ( T ) | ν = ν c has a finite and nearly T -independent value at T >
200 mK. However, near T = T R ac = 101 mK, R yy ( T ) | ν = ν c for the R a phase ex-hibits a sharp peak. Similar sharp peaks in R yy ( T ) | ν = ν c curves were measured at the onset temperatures of bub-ble phases in the N = 1 and N = 2 Landau levels .As seen in Fig.3, we now detect such peaks for all mul-tielectron bubble phases of the N = 3 Landau level. Weconclude that, in high mobility samples there is a sharppeak present in the R yy ( T ) | ν = ν c curves near the onset ofmultielectron bubble phases, irrespective of the Landaulevel they develop in.Data available for bubble phases in the N = 3 Landaulevel in an alloy sample offer a chance for comparison.Because of the deliberately introduced Al into the GaAschannel during the sample growth process, thereby form-ing a dilute AlGaAs alloy, the alloy sample in Ref. hada mobility of µ = 3 . × cm /Vs. This number isabout a factor of 4 times less than that of our sample.Quite interestingly, in the alloy sample bubble phasesof the N = 3 Landau level develop at the same fillingfactors and also in a similar temperature range as thosein our work . A consequence of the reduced mobility,which can be seen at temperature much above bubbleonset, is the enhanced longitudinal magnetoresistance of ≈
80 Ω in the alloy sample , compared to ≈
18 Ω inour sample. Another consequence is the conspicuous ab-sence of the sharp peak in the R yy ( T ) | ν = ν c curves . In-deed, as the temperature increases in the sample withadded disorder , the longitudinal resistance of the bub-ble phase increases and saturates past 135 mK, withoutthe development of a sharp peak. Transport in bubblephases is currently understood as follows: at T << T c the bubble phase is pinned by the disorder present in thesample, whereas at T >> T c electrons form a uniformuncorrelated liquid. In this interpretation, near T = T c these two phases compete by forming an interpenetratingnetwork of domains throughout the bulk of the sample.The presence of a peak in R yy ( T ) | ν = ν c in our high mo-bility GaAs sample and also in graphene in a narrowrange of temperatures near T = T c indicates excess scat-tering due to enhanced thermal fluctuations between thedomains of the two competing phases. We think thatsuch thermal fluctuations and the associated sharp re-sistance peak are suppressed in the alloy sample by thedisorder present .We now examine the onset temperatures of bubblephases, quantities related to the corresponding cohesiveenergies calculated in Hartree-Fock theories . Wefound that onset temperatures of the M = 2 and M = 3bubble phases in the N = 3 Landau level are close toeach other. This property is consistent with the Hartee-Fock predictions . Quantitative comparisons withcalculated cohesive energies are, however, tenuous. Thisis partly because cohesive energies are calculated underidealized conditions, such as no disorder and no Landaulevel mixing. Discrepancies of more than two orders ofmagnitude between the onset temperatures and cal-culated cohesive energies in the N = 1 and N = 2 Landaulevels were indeed attributed to these idealizedconditions. We found that these discrepancies persist inthe N = 3 Landau level .Nonetheless, comparisons of onset temperatures andcohesive energies provides useful insight to the na-ture of electronic interactions. It is well-knownthat the clustering of electrons into bubbles is pro-moted by competing short-range and long-range elec-tronic interactions . The long-range interactionis Coulombic in nature, while the short-range interac-tion is a softened Coulomb potential. At the root ofsuch a potential softening we find overlapping single Tc [mK] n c N = 2 N = 3
R 4 a R 4 d R 5 a R 5 d R 6 bR 6 a R 6 cR 6 d R 7 bR 7 a R 7 cR 7 d o f f s e t
FIG. 4. Dependence of the onset temperature T c on the cen-tral factor of the M = 2 and M = 3 bubble phases in the N = 2 and N = 3 Landau levels. Shaded bands illustratetrends of onset temperatures for phases with the same num-ber of electrons per bubble. Near ν = 6, the dimensionlessonset temperatures of the M = 2 bubble phases exhibit anoffset marked by the double arrow. particle wavefunctions and finite layer thicknesseffects .At first sight, the onset temperatures in the N = 3Landau level listed in Table.I. do not seem to follow aparticular trend. However, a closer inspection revealssome interesting properties. Within one Landau level,onset temperatures of a given type of bubble phase forman approximately linear trend. In Fig.4 we show onsettemperatures T c for multielectron bubble phases in the N = 2 and N = 3 Landau levels. The three coloredbands in Fig.4 indicate these linear trends for the M = 2bubble phases of the N = 2 Landau level, for the M = 2bubble phases of the N = 3 Landau level, and for the M = 3 bubble phases of the N = 3 Landau level. Sincedata for bubble phases forming in different spin branchesof a given orbital Landau level lie on the same line, weconclude that onset temperatures are not influenced bythe spin quantum number.Identifying the dominant bubble phase in the N = 3Landau level reveals details on the short-range electron-electron interaction that drive bubble formation. Wenote, that Hartree-Fock calculations do not provide con-sistent results for the dominant, i.e. the most stable, bub-ble phase. Indeed, in the N = 3 Landau level Refs. predict the M = 3 bubbles to be dominant, whereasRefs. find the M = 2 bubbles to be stronger. Theformer results agree, but the latter ones are contrary toour findings. A likely cause of different dominant bub-ble phase may be different effective electron-electron in-teraction. To see this, the work of Ettouhami et al. isparticularly useful. In this work authors tuned the short-range part of the electron-electron interaction through tc=kBTc/EC · n c N = 2 N = 3
R 6 a R 6 dR 4 a R 4 d R 6 b R 6 c R 7 a R 7 dR 5 dR 5 a R 7 b R 7 c o f f s e t
FIG. 5. Dependence of the dimensionless onset temperature t c = k B T c /E C on the central factor of the M = 2 and M = 3bubble phases in the N = 2 and N = 3 Landau levels. Shadedbands illustrate trends of onset temperatures for phases withthe same number of electrons per bubble. Near ν = 6, thedimensionless onset temperatures of the M = 2 bubble phasesexhibit an offset marked by the double arrow. the layer thickness parameter λ , while keeping the long-range Coulombic potential unchanged . It was foundthat in the N = 3 Landau level the energy balance canbe significantly tilted: the M = 3 bubbles are dominantfor λ = 0, whereas the M = 3 bubbles have nearly thesame energy with M = 2 bubbles at λ = 1, i.e. whenthe electron-electron interaction was softened at shortdistances . We then surmise that a further softeningof the potential may reverse the energy balance of the M = 3 and M = 2 bubble phases and therefore may yieldthe experimentally observed dominant bubble phase.A comparison of the energetics of M = 2 bubble phasesin the N = 2 and N = 3 Landau levels reveals thatthe electronic short-range interaction is dependent on theLandau index N . We discussed earlier the linear trendof both T c versus ν c for the M = 2 bubble phases. Theselinear trends exhibit a vertical offset when N changesfrom 2 to 3 in the vicinity of ν = 6. Indeed, as seenin Fig.4, the two colored bands associated with M = 2bubble phases in the N = 3 Landau level acquired anoffset when compared to that for M = 2 bubble phasein the N = 2 Landau level. We attribute this offset toa variation of the effective electron-electron interaction,specifically its short-range part, with the Landau index N . While finite layer thickness effects soften the electron-electron interaction, they are not expected to depend onthe Landau level index. In contrast, a short-range po-tential that results from the overlapping single electronwavefunctions is Landau index dependent . Thisis because the number of nodes in these wavefunctions directly influences bubble energetics. The comparison ofthe energetics of the M = 2 bubble phases in the N = 2and N = 3 Landau levels thus provides direct evidencethat the overlapping electronic wavefunctions play a rolein shaping the short-range part of the electron-electroninteraction.In the following, we examine properties of the dimen-sionless onset temperatures t c = k B T c /E C , a quantityclosely related to the dimensionless cohesive energy ofHartree-Fock calculations. Here k B is the Boltzmannconstant, E C = e / π(cid:15)l B the Coulomb energy, and l B is the magnetic length. Dimensionless onset tempera-tures t c for the multielectron bubble phases in the N = 2and N = 3 Landau levels are plotted in Fig.5. Trends al-ready discussed for T c are also observed for t c : within oneLandau level, both of these quantities have a linear trendwith ν c and these linear trends exhibit a vertical offsetwhen N changes from 2 to 3 in the vicinity of ν = 6. Inaddition, we find that across the different Landau levels,the linear trend of t c versus ν c for the M = 2 bubblephases have a similar slope, ∂t c /∂ν c ≈ − . × − inboth the N = 2 and N = 3 Landau levels. We thusfound that bubble phases with the same number of elec-trons forming in different Landau levels share a simi-lar ∂t c /∂ν c . In contrast, the M = 3 bubble phases inthe N = 3 Landau level have a significantly diminished ∂t c /∂ν c slope, reduced by about a factor 5 as comparedto that of the M = 2 bubble phases.In conclusion, we observed qualitative and quantitativeaspects of bubble formation in the N = 2 and N = 3 Lan-dau levels. We found that in our high mobility samplethe longitudinal magnetoresistance versus temperaturecurves exhibit sharp peaks in the multielectron bubbleregions both in the N = 2 and N = 3 Landau levels. Weused these peaks to extract the onset temperatures forthe bubble phases. The recent assignment of the num-ber of electrons per bubble to these phases allowed ananalysis of the measured onset temperatures. We foundthat within the N = 3 Landau level, onset temperaturesof different bubble phases exhibit linear trands with thefilling factor. 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