Apparent temperature-induced reorientation of quantum Hall stripes
Q. Shi, M. A. Zudov, B. Friess, J. Smet, J. D. Watson, G. C. Gardner, M. J. Manfra
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Apparent temperature-induced reorientation of quantum Hall stripes
Q. Shi, M. A. Zudov, ∗ B. Friess, J. Smet, J. D. Watson ,
3, 4
G. C. Gardner,
4, 5 and M. J. Manfra
3, 4, 5, 6, 7 School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA Max-Planck-Institute for Solid State Research, Heisenbergstrasse 1, D-75569 Stuttgart, Germany Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA School of Materials Engineering, Purdue University, West Lafayette, Indiana 47907, USA School of Electrical and Computer Engineering,Purdue University, West Lafayette, Indiana 47907, USA Station Q Purdue, Purdue University, West Lafayette, Indiana 47907, USA (Received September 5, 2018)Our magnetotransport measurements of quantum Hall stripes in a high-quality GaAs quantumwell in a slightly tilted magnetic field reveal that the orientation of stripes can be changed bytemperature. Field-cooling and field-warming measurements, as well as observation of hysteresis atintermediate temperatures allow us to conclude that the observed temperature-induced reorientationof stripes is owing to the existence of two distinct minima in the symmetry-breaking potential. Wealso find that the native symmetry-breaking mechanism does not depend on temperature and thatlow-temperature magnetotransport data should be treated with caution as they do not necessarilyreveal the true ground state, even in the absence of hysteresis.
Electronic nematic phases have been observed ina variety of systems [1], ranging from a clean two-dimensional(2D) electron gas (2DEG) formed in GaAsquantum wells [2–7] to ruthenates [8], high tempera-ture superconductors [9, 10], and heavy fermion systems[11]. In a 2DEG in GaAs, the appearance of nematicphases (often referred to as stripes) is marked by the re-sistance minima (maxima) along the easy (hard) trans-port direction near half-integer values of the filling factor ν = n e h/eB , where n e is the electron density and B isthe magnetic field. Stripe phases stem from the competi-tion between (long-range) repulsive and (short-range) at-tractive components of Coulomb interaction and can beviewed as a unidirectional charge density wave consistingof stripe regions with either higher or lower integer fill-ing factors. While native stripes are usually aligned alongthe h i crystal direction [12], what exactly determinessuch preferred orientation remains a mystery [13–15].One way to learn about the native symmetry-breakingis to introduce some external mechanism which wouldcause a reorientation of stripes. The oldest (and still themost popular) tool to reorient stripes is applying an in-plane magnetic field [15–27]. Other parameters whichwere manipulated to change the orientation of stripesinclude electron density [22, 28], filling factor within agiven Landau level [23, 25], electric current [29], mechan-ical strain [14], symmetry of the quantum confinementcontaining 2DEG [28], capping layer thickness [15], and,most recently, weak periodic density modulation [30].It is also known that the orientation can be history-dependent and be governed by the direction of a density[22] or a magnetic field [23] sweep. The observed hys-teresis was attributed to a bi-directionality of the nativesymmetry breaking potential [23].In this Rapid Communication we report on magneto- transport measurements in a high-quality GaAs quantumwell, subjected to a magnetic field which is tilted slightlyaway from the sample normal. The weak in-plane mag-netic field B k was introduced in order to reorient stripesonly far away from half-filling [25] which is essential forour observations discussed below. Our magnetotransportmeasurements revealed orthogonal orientations of quan-tum Hall stripes at different temperatures. At a highertemperature ( T ≈
75 mK) stripes near half-integer fillingfactors are oriented along the native direction, ˆ y ≡ h i ,as the magnitude of B k is not sufficient to induce reori-entation. However, measurements at a base tempera-ture ( T ≈
20 mK) show a strong transport anisotropyindicative of a reorientation along the ˆ x ≡ h i direc-tion. Field-cooling/warming measurements and detec-tion of hysteresis at intermediate temperatures suggestthat the observed temperature-induced reorientation ofstripes is due to the existence of two distinct minima inthe symmetry-breaking potential. One important impli-cation of our findings is that the low-temperature trans-port data do not necessarily reflect the equilibrium elec-tron configuration even if no hysteresis is observed. Ourfindings also shed new light on the recently reported fill-ing factor dependence of the native symmetry-breakingfield [25], e.g., that its observation calls for sufficientlyhigh temperatures as it can be completely hidden inthe low-temperature magnetotransport [31]. Finally, ourdata indicate a very weak, if any, temperature depen-dence of the native symmetry-breaking potential.Our sample is a 4 × n e ≈ . × cm − and µ ≈ . × cm /Vs, respectively [32]. Eightindium contacts were fabricated at the corners and mid-sides of the sample. The longitudinal resistances, R xx (cid:1)(cid:2)(cid:3) (cid:1) (cid:2)(cid:2) (cid:3) (cid:4)(cid:5) (cid:1) (cid:4)(cid:4) (cid:5) (cid:6) (cid:7) W (cid:8) (cid:9)(cid:10)(cid:11)(cid:10)(cid:10)(cid:12)(cid:11)(cid:10)(cid:12) n (cid:5) (cid:5) º (cid:1)(cid:3)(cid:5)(cid:13)(cid:14)(cid:6)(cid:15)(cid:8) (cid:6) (cid:16)(cid:16) (cid:5)(cid:17)(cid:5) (cid:6) (cid:4) (cid:4) (cid:5)(cid:6)(cid:18)(cid:2)(cid:2)(cid:3)(cid:19)(cid:8) (cid:2) (cid:5)(cid:6)(cid:18)(cid:2)(cid:20)(cid:2)(cid:3)(cid:19)(cid:8)(cid:1)(cid:2)(cid:3) (cid:1) (cid:2)(cid:2) (cid:3) (cid:4)(cid:5) (cid:1) (cid:4)(cid:4) (cid:5) (cid:6) (cid:7) W (cid:8) (cid:5) (cid:5) º (cid:21)(cid:10)(cid:5)(cid:13)(cid:14)(cid:6)(cid:22)(cid:8) cc n + n - (cid:6) (cid:16)(cid:16) (cid:5)(cid:17)(cid:5) (cid:6) (cid:4) (cid:2) (cid:5)(cid:6)(cid:18)(cid:2)(cid:20)(cid:2)(cid:3)(cid:19)(cid:8) (cid:4) (cid:5)(cid:6)(cid:18)(cid:2)(cid:2)(cid:3)(cid:19)(cid:8) FIG. 1. (Color online) R xx (solid line) and R yy (dotted line)as a function of ν measured at (a) T ≈
75 mK and (b) T ≈ ≈ ◦ . Cartoons illustrate orientation of the stripeswith respect to the direction of B k (thick arrow). Dotted linesaround ν = 9 / ν + ≈ . ν − ≈ .
43 near which the anisotropy energy changes signat this tilt angle. and R yy , were measured using standard four-terminal,low-frequency lock-in technique with the current (10 nA,7 Hz) sent through the midside contacts and the voltagedrop measured between the corner contacts. All resultswere obtained with the magnetic field tilted by ≈ ◦ [33]about the ˆ x axis with respect to the sample normal andswept at a rate of 0.1 T/min.In Fig. 1(a) we present R xx (solid line) and R yy (dot-ted line) as a function of filling factor ν measured at T ≈
75 mK. The data demonstrate strong anisotropy(with R xx ≫ R yy ) near both ν = 9 / ν = 11 / y = h i , direction. Around fractionalfillings, ν ≈ .
28 and ν ≈ .
72, both R xx and R yy re-veal insulating states owing to the formation of bubblephases [2–4, 29, 34–36]. Closer examination of the datain Fig. 1(a) around ν = 9 / R yy ≫ R xx (marked by ↓ ). This obser-vation indicates that stripes at these filling factors have,in fact, already undergone B k -induced reorientation and are now aligned along the ˆ x direction. This happens be-cause stripes away from half-filling require a smaller B k to be reoriented, and, as a result, orientation of stripeswithin a given Landau level at a fixed tilt angle dependssensitively on the filling factor [25]. As we show below,this filling factor dependence is responsible for the ap-parent reorientation of stripes seen in Fig. 1 (b) whichwe discuss next.Like Fig. 1(a), Fig. 1(b) shows R xx (solid line) and R yy (dotted line) versus ν but measured at a lower tempera-ture, T ≈
20 mK. One observes that the bubble phasesare now almost indistinguishable from the neighboringinteger quantum Hall states. The most striking change,however, is that now one finds R xx ≪ R yy near both ν = 9 / ν = 11 /
2, indicating alignment of stripesalong the ˆ x direction.Since both data sets shown in Fig. 1 are acquired at thesame tilt angle, one can argue that lowering the temper-ature is the sole cause for the observed reorientation. Ifso, one could be tempted to conclude that, surprisingly,the native symmetry-breaking potential becomes weaker at lower T as it is now easier to be overcome by the B k -induced anisotropy energy. However, as we show next,such a conclusion is premature.First, as already mentioned, stripes in our sample havealready been reoriented away from half-filling by B k . Sec-ond, we recall that the native stripes align along eitherˆ y = h i (most common scenario) or ˆ x = h i orienta-tion [15, 22, 23, 28] but never otherwise. Moreover, thereis strong evidence for the existence of two distinct min-ima in the native orientation potential along orthogonaldirections [23]. The anisotropy energy can be defined asa difference between these two minima, E = E x − E y .While the orientation of stripes is usually decided by thesign of E (i.e., E > y direction), it can also be determined by the history. Thelatter happens when (i) E changes sign at some fillingfactor within a region where stripes form and (ii) the re-laxation is slow enough for the system to relax from thehigher minimum to the ground state during the magneticfield sweep. Indeed, both Ref. 22 and Ref. 23 observedthat when ν is swept from high to low (low to high)stripes near ν = 9 / h i ( h i ) di-rection. Field-cooling [23] confirmed the existence of thefilling factor at which E changes sign. Thus, observedhysteresis demonstrates that the orientation of stripesmanifested in low-temperature measurements can be gov-erned by the sign of E preceding the degeneracy of thetwo minima. In other words, no matter which orientationis realized first, stripes can get stuck in that orientationeven though the sign of E suggests an orthogonal one.While magnetotransport at both T ≈
75 mK and 20mK showed negligible sensitivity to the field sweep direc-tion, our experimental situation bears clear similarities tothat of Refs. 22 and 23, albeit with one important differ-ence. Indeed, our higher temperature data clearly illus- (cid:1)(cid:2)(cid:3)(cid:4) (cid:1) (cid:2)(cid:2) (cid:5) (cid:6) (cid:7) W (cid:8) (cid:9)(cid:10)(cid:11)(cid:12)(cid:9)(cid:10)(cid:11)(cid:9)(cid:10)(cid:12)(cid:12)(cid:9)(cid:10)(cid:12) n (cid:6)(cid:13)(cid:13)(cid:8)(cid:6)(cid:13)(cid:14)(cid:8) (cid:6)(cid:13)(cid:13)(cid:13)(cid:8)(cid:6)(cid:13)(cid:8) (cid:15)(cid:16)(cid:16)(cid:17)(cid:13)(cid:18)(cid:19)(cid:20)(cid:21)(cid:22)(cid:23)(cid:13)(cid:18)(cid:19) (cid:3) (cid:5) º (cid:24)(cid:12)(cid:5)(cid:23)(cid:25) (cid:3) (cid:5) º (cid:2)(cid:4)(cid:5)(cid:23)(cid:25) (cid:3) (cid:5) º (cid:2)(cid:4)(cid:5)(cid:23)(cid:25) (cid:4) (cid:26)(cid:26) (cid:5)(cid:27)(cid:5) (cid:4) (cid:5) n º n + FIG. 2. (Color online) R xx vs. ν measured during (i) cooling(dotted line) from T ≈
75 mK (square) down to T ≈
20 mK(circle) at ν ≈ .
52, (ii) down-field sweep at T ≈
20 mK (topcurve), (iii) up-field sweep at T ≈
20 mK (bottom curve),and (iv) subsequent warming (dashed line) from T ≈
20 mK(triangle) to T ≈
75 mK (square) at ν ≈ .
52. Cartoonsillustrate orientation of stripes. The magnetic field was tiltedwith respect to the sample normal by ≈ ◦ . Vertical dottedline is drawn at ν = ν + ≈ . trate the existence of characteristic filling factors around ν = 9 / ν + and ν − , separating stripes of orthogonal ori-entations [cf. dotted lines in Fig. 1(a)]. The anisotropyenergy is negative when ν − < ν < ν + , vanishes at ν + and ν − , and is positive further away from half-filling. As aresult, crossing either ν − or ν + leads to a sign change of E , very much like the situation of Ref. 23. The importantdistinction, however, is the existence of two such fillingfactors in the filling factor range within which stripesform. As we explain below, this fact plays a crucial rolein our observation of the apparent T -induced reorienta-tion illustrated in Fig. 1.We now argue that, in contrast to what the data inFig. 1(b) imply, stripes parallel to ˆ x do not reflect anequilibrium configuration of the system near ν = 9 / ν ≈ .
52 [37] starting from T ≈
75 mK (square)down to T ≈
20 mK (circle) and observe an increase in R xx (dotted line), suggesting that stripes stay parallel toˆ y . We next sweep the magnetic field away from the striperegion (upper curve), return back to ν ≈ .
52 (lowercurve), and find that R xx ≈
0, i.e. that stripes are nowparallel to ˆ x . Subsequent warming (dashed line) leadsto an increase of R xx signaling reorientation of stripesback to along the ˆ y direction. We also note that neitherup-sweep nor down-sweep reveals any sensitivity of ori- entation to the filling factor, in contrast to the higher- T data of Fig. 1(a).The data in Fig. 2 suggest that when ν − < ν < ν + , theequilibrium configuration corresponds to stripes alongthe ˆ y direction, in accord with Fig. 1(a). When the sys-tem is cooled at ν within the range ν − < ν < ν + ,the orientation does not change. However, after cool-ing at ν outside this range of filling factors, magne-toresistance reflects the orthogonal orientation as shownin Fig. 1(b). This happens because before entering therange ν − < ν < ν + ( E > x direction ( E < ν < ν − and ν > ν + . At low T ,the relaxation time is long enough to preserve the initialorientation for the duration of the field sweep. Indeed,crossing the dotted line drawn at ν = ν + from either sidedoes not trigger the reorientation of stripes. As men-tioned above, the unique feature of our experiment is theexistence of two characteristic filling factors at which E changes sign. It is this feature which leads to persistentobservation of stripes of metastable orientation withouthysteresis at low temperatures, as shown in Fig. 1(b).While no hysteresis is seen at either T ≈
20 mk or T ≈
75 mK (cf. Fig. 1), we do indeed observe sensitivityto the field sweep direction at intermediate temperatures.In Fig. 3 we present R xx (solid line) and R yy (dottedline) measured in (a) up-sweep and (b) down-sweep at T ≈
45 mK. One immediately observes that the data near ν = 9 / ν , the first anisotropic phase characterized by E < x ) is marked by a stronger resistanceanisotropy [cf. → in (a), ← in (b)] than its counterpartwhich occurs on the opposite side of half-filling. In fact,the anisotropy away from half-filling can be as strong oreven stronger than the anisotropy at exactly ν = 9 /
2, insharp contrast with the data in Fig. 1(a). This behaviormanifests that a relaxation time at this T is comparableto the sweep time of about a minute [38]. At higher T ,the relaxation becomes considerably faster which resultsin the data presented in Fig. 1(a) showing the equilibriumorientation of stripes with no hysteresis. At lower T ,the relaxation is slower, and the transport data show themetastable orientation which is set by the sign of E awayfrom half-filling.Further examination of the data in Fig. 3 shows that, incontrast to ν = 9 /
2, the magnetoresistance near ν = 11 / ν = 11 / T is short enough for the trans-port to always reflect the true ground state. Faster re-laxation near ν = 11 / | E | at this filling factor compared to that at ν = 9 / B k -induced reorientation requires smaller B k at ν = 9 / ν = 11 / (cid:1)(cid:2)(cid:3) (cid:1) (cid:2)(cid:2) (cid:3) (cid:4)(cid:5) (cid:1) (cid:4)(cid:4) (cid:5) (cid:6) (cid:7) W (cid:8) (cid:9)(cid:10)(cid:11)(cid:10)(cid:10)(cid:12)(cid:11)(cid:10)(cid:12) n (cid:5) (cid:5) º (cid:12)(cid:10)(cid:5)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:17)(cid:18)(cid:5)(cid:19)(cid:17)(cid:20)(cid:21)(cid:6)(cid:22)(cid:8) (cid:6) (cid:23)(cid:23) (cid:5)(cid:24)(cid:5) (cid:6) (cid:4) y (cid:1)(cid:2)(cid:3) (cid:1) (cid:2)(cid:2) (cid:3) (cid:4)(cid:5) (cid:1) (cid:4)(cid:4) (cid:5) (cid:6) (cid:7) W (cid:8) (cid:5) (cid:5) º (cid:12)(cid:10)(cid:5)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:17)(cid:18)(cid:5)(cid:25)(cid:26)(cid:27)(cid:28)(cid:21)(cid:6)(cid:29)(cid:8) (cid:6) (cid:23)(cid:23) (cid:5)(cid:24)(cid:5) (cid:6) (cid:4) z FIG. 3. (Color online) R xx (solid line) and R yy (dotted line)as a function of ν measured in (a) down-sweep and (b) up-sweep at T ≈
45 mK. The magnetic field was tilted withrespect to the sample normal by ≈ ◦ . more, a faster relaxation near ν = 11 / T ≈
20 mK shown inFig. 1(b) compared to T ≈
45 mK data shown in Fig. 3.Here, although the majority of the domains remain in ametastable orientation during the magnetic field sweep,some of the domains might have managed to relax to theground state, thus lowering the resistance anisotropy athalf-filling.Finally, our findings allow us to revisit the conclu-sion of Ref. 25, namely, that the magnitude of the nativeanisotropy energy quickly decays as the filling factor de-viates from half-filling. Having confirmed the existenceof the two distinct minima in the native orientation po-tential in our sample, we can now conclude that stripesnear (away from) half-filling are coupled more strongly tothe minimum of the native potential which favors stripesalong the h i ( h i ) direction. This finding shouldbe taken into account by future proposals concerning thenature of the native symmetry breaking field.In summary, magnetoresistance measurements in ahigh-quality GaAs quantum well with the magnetic fieldtilted slightly away from the sample normal revealedstripes along the h i at T ≈
75 mK and along h i at T ≈
20 mK. Sample cooling at a fixed filling factor near ν = 9 /
2, followed by magnetic field sweeps and warm-ing at the same filling factor, as well as the observationof hysteresis at intermediate temperatures, let us con-clude that, in contrast to R xx ≪ R yy detected in mag-netotransport at T ≈
20 mK, stripes along h i crystaldirection represent the true ground state. These find-ings demonstrate that the low-temperature magnetore-sistance, which is routinely used to determine orientationof stripes, can be misleading as, even in the absence ofhysteresis, it does not necessarily reflect the ground state.In addition, a recently reported filling factor dependenceof reorientation of stripes [25] can be completely hid-den in the low-temperature magnetotransport. Finally,since our data were obtained on the verge of reorientation( E ≈ B k along ˆ y = h i direction to in-troduce ν + and ν − , similar observations can be expectedunder orthogonal orientation of B k , as it can also inducereorientation of stripes [26].We thank J. Geurs for assistance with the experi-mental setup and I. Dmitriev for commenting on themanuscript. The work at Minnesota (Purdue) was sup-ported by the U.S. Department of Energy, Office of Sci-ence, Basic Energy Sciences, under Award Current address: QuTech and Kavli Institute of Nano -science, Delft Technical University, 2600 GA Delft, TheNetherlands. ∗ Corresponding author: [email protected][1] E. Fradkin, S. A. Kivelson, M. J. Lawler, J. P. Eisenstein,and A. P. Mackenzie, Annu. Rev. Condens. Matter Phys. , 153 (2010).[2] M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer,and K. W. West, Phys. Rev. Lett. , 394 (1999).[3] R. R. Du, D. C. Tsui, H. L. Stormer, L. N. Pfeiffer, K. W.Baldwin, and K. W. West, Solid State Commun. , 389(1999).[4] A. A. Koulakov, M. M. Fogler, and B. I. Shklovskii, Phys.Rev. Lett. , 499 (1996).[5] M. M. Fogler, A. A. Koulakov, and B. I. Shklovskii, Phys.Rev. B , 1853 (1996).[6] E. Fradkin and S. A. Kivelson, Phys. Rev. B , 8065(1999).[7] E. Fradkin, S. A. Kivelson, E. Manousakis, and K. Nho,Phys. Rev. Lett. , 1982 (2000).[8] R. A. Borzi, S. A. Grigera, J. Farrell, R. S. Perry, S. J. S.Lister, S. L. Lee, D. A. Tennant, Y. Maeno, and A. P.Mackenzie, Science , 214 (2007).[9] R. Daou, J. Chang, D. LeBoeuf, O. Cyr-Choiniere,F. Laliberte, N. Doiron-Leyraud, B. J. Ramshaw,R. Liang, D. A. Bonn, W. N. Hardy, et al., Nature ,519 (2010). [10] J.-H. Chu, J. G. Analytis, K. De Greve, P. L. McMahon,Z. Islam, Y. Yamamoto, and I. R. Fisher, Science ,824 (2010).[11] R. Okazaki, T. Shibauchi, H. J. Shi, Y. Haga, T. D. Mat-suda, E. Yamamoto, Y. Onuki, H. Ikeda, and Y. Mat-suda, Science , 439 (2011).[12] Refs. 22 and 15 reported stripes along h i direction.[13] I. Sodemann and A. H. MacDonald, arXiv:1307.5489(2013).[14] S. P. Koduvayur, Y. Lyanda-Geller, S. Khlebnikov,G. Csathy, M. J. Manfra, L. N. Pfeiffer, K. W. West, andL. P. Rokhinson, Phys. Rev. Lett. , 016804 (2011).[15] J. Pollanen, K. B. Cooper, S. Brandsen, J. P. Eisenstein,L. N. Pfeiffer, and K. W. West, Phys. Rev. B , 115410(2015).[16] M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer,and K. W. West, Phys. Rev. Lett. , 824 (1999).[17] W. Pan, R. R. Du, H. L. Stormer, D. C. Tsui, L. N.Pfeiffer, K. W. Baldwin, and K. W. West, Phys. Rev.Lett. , 820 (1999).[18] W. Pan, T. Jungwirth, H. L. Stormer, D. C. Tsui, A. H.MacDonald, S. M. Girvin, L. Smrˇcka, L. N. Pfeiffer,K. W. Baldwin, and K. W. West, Phys. Rev. Lett. ,3257 (2000).[19] K. B. Cooper, M. P. Lilly, J. P. Eisenstein, T. Jungwirth,L. N. Pfeiffer, and K. W. West, Solid State Commun. , 89 (2001).[20] T. Jungwirth, A. H. MacDonald, L. Smrˇcka, and S. M.Girvin, Phys. Rev. B , 15574 (1999).[21] T. D. Stanescu, I. Martin, and P. Phillips, Phys. Rev.Lett. , 1288 (2000).[22] J. Zhu, W. Pan, H. L. Stormer, L. N. Pfeiffer, and K. W.West, Phys. Rev. Lett. , 116803 (2002).[23] K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W.West, Phys. Rev. Lett. , 026806 (2004).[24] H. Zhu, G. Sambandamurthy, L. W. Engel, D. C. Tsui,L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. ,136804 (2009). [25] Q. Shi, M. A. Zudov, J. D. Watson, G. C. Gardner, andM. J. Manfra, Phys. Rev. B , 121404 (2016).[26] Q. Shi, M. A. Zudov, J. D. Watson, G. C. Gardner, andM. J. Manfra, Phys. Rev. B , 121411 (2016).[27] Q. Shi, M. A. Zudov, Q. Qian, G. C. Watson, and M. J.Manfra, submitted (2016).[28] Y. Liu, D. Kamburov, M. Shayegan, L. N. Pfeiffer, K. W.West, and K. W. Baldwin, Phys. Rev. B , 075314(2013).[29] J. Gores, G. Gamez, J. H. Smet, L. Pfeiffer, K. West,A. Yacoby, V. Umansky, and K. von Klitzing, Phys. Rev.Lett. , 246402 (2007).[30] M. A. Mueed, M. S. Hossain, L. N. Pfeiffer, K. W. West,K. W. Baldwin, and M. Shayegan, Phys. Rev. Lett. ,076803 (2016).[31] T ≈
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52 in Fig. 1 (a). At low T the peak moves to evenhigher ν which might account for the initial rise of R xxxx