Effect of density on quantum Hall stripe orientation in tilted magnetic fields
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Effect of density on quantum Hall stripe orientation in tilted magnetic fields
Q. Shi, M. A. Zudov, Q. Qian, J. D. Watson ∗ , and M. J. Manfra
2, 3, 4 School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA Department of Physics and Astronomy and Birck Nanotechnology Center,Purdue University, West Lafayette, Indiana 47907, USA Station Q Purdue, Purdue University, West Lafayette, Indiana 47907, USA School of Materials Engineering and School of Electrical and Computer Engineering,Purdue University, West Lafayette, Indiana 47907, USA (Received September 25, 2018)We investigate quantum Hall stripes under in-plane magnetic field B k in a variable-density two-dimensional electron gas. At filling factor ν = 9 /
2, we observe one, two, and zero B k -inducedreorientations at low, intermediate, and high densities, respectively. The appearance of these distinctregimes is due to a strong density dependence of the B k -induced orienting mechanism which triggersthe second reorientation, rendering stripes parallel to B k . In contrast, the mechanism which reorientsstripes perpendicular to B k showed no noticeable dependence on density. Measurements at ν = 9 / / B k -induced symmetry-breaking fields. Quantum Hall stripe phases [1–21] represent one classof exotic states that appear in a two-dimensional elec-tron gas (2DEG) subjected to perpendicular magneticfields and low temperatures. These phases manifestcharge clustering originating from a box-like interac-tion potential [1, 2] owing to ring-shaped wavefuctionsin higher Landau levels (LLs). A built-in symmetry-breaking potential in the GaAs quantum well, hostinga two-dimensional electron gas (2DEG), macroscopicallyorients stripes along h i crystal direction, with veryfew exceptions [8, 15, 20]. Despite continuing efforts[13, 14, 20, 22], the origin of such preferred native orien-tation remains a mystery. It is known, however, that dueto a finite thickness of the 2DEG, an in-plane magneticfield B k modifies both the wavefunction and interactionswhich, in turn, can change stripe orientation [23, 24].While early experiments [5–7] and theories [23, 24] con-sistently showed that B k favors stripes perpendicular toit [25], subsequent studies revealed limitations of this“standard picture”. For example, in a tunable-densityheterostructure insulated gate field effect transistor [8]native stripes along h i crystal direction did not re-orient by B k . In other experiments, however, reorien-tation occurred even when B k was applied perpendicu-lar to the native stripes [11, 19, 26]. Finally, it was re-cently reported that B k applied along native stripes caninduce two successive reorientations, first perpendicularand then parallel to B k [19].Together, these experiments indicate that the impactof B k on stripe orientation remains poorly understoodand is far more complex than suggested by a “standardpicture” [23, 24]. In particular, all examples mentionedabove revealed that B k can, in fact, favor parallel (to B k ) stripe alignment. It was also found that the B k -induced mechanism which favors such alignment is highly sensitive to both spin and orbital quantum numbers [19].To shed light onto the nature of this mechanism, it isvery desirable to identify a tuning parameter that wouldenable one to control stripe orientation under B k .In this Rapid Communication we study the effect ofthe carrier density n e on stripe orientation in a single-subband 2DEG under B k applied along native stripes( || h i ). At filling factor ν = 9 /
2, we demonstrate threedistinct classes of behavior. At low n e , we observe asingle reorientation (at B k = B ) which renders stripesperpendicular to B k , in agreement with the “standardpicture” [5, 6, 23, 24]. At intermediate n e , we also de-tect the second reorientation (at B k = B ) which revertsstripes back to their native direction, parallel to B k . Fi-nally, at higher n e we find that B k cannot alter stripeorientation. We further construct a phase diagram of thestripe orientation which reveals that B is independentof n e , whereas B decreases rapidly with n e and even-tually merges with B . The appearance of the robustregime of stripes parallel to B k at higher n e can be at-tributed to a reduced screening due to increased inter-LLspacing. At the same time, a density sweep at ν = 9 / ν = 11 / B k and identifying the nativesymmetry-breaking field.Our 2DEG resides in a 30-nm GaAs/AlGaAs quan-tum well (about 200 nm below the sample surface) thatis doped in a 2 nm GaAs quantum well at a setback of63 nm. The in situ gate consists of an n + GaAs layersituated 850 nm below the bottom of the quantum well[27]. Eight Ohmic contacts were fabricated at the corners (cid:1)(cid:2)(cid:3)(cid:4) (cid:1) (cid:2) (cid:5) (cid:6) (cid:7) (cid:8) (cid:1)(cid:9)(cid:10)(cid:1)(cid:9)(cid:11)(cid:1)(cid:9)(cid:2)(cid:1)(cid:9)(cid:4)(cid:2)(cid:9)(cid:12)(cid:2)(cid:9)(cid:10)(cid:2)(cid:9)(cid:11) (cid:3) (cid:4) (cid:5)(cid:6)(cid:3)(cid:4) (cid:3)(cid:3) (cid:5)(cid:13)(cid:14) (cid:15)(cid:2)(cid:5) (cid:8)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21)(cid:16)(cid:5)(cid:22)(cid:22) (cid:5)(cid:6) (cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21)(cid:16)(cid:5)(cid:22)(cid:22)(cid:5) (cid:2)(cid:1) (cid:3) (cid:1) (cid:2)
FIG. 1. (Color online) Stripe orientation as a function of n e and B y at ν = 9/2. Triangles (circles) mark stripe orienta-tion perpendicular (parallel) to B k = B y . The big circle wasobtained from the density sweep in Fig. 2. Squares mark theisotropic state. The lower (upper) phase boundary (dashedline) is a guide to the eyes marking B k = B ( B k = B ), seethe text. and midsides of the lithographically-defined 1 × Van der Pauw mesa. The electron density n e was variedcontinuously from 2.2 to 3 . × cm − . The peak mo-bility was about µ ≈ . × cm /Vs at n e ≈ . × cm − . Resistances R xx (ˆ x ≡ h i ) and R yy (ˆ y ≡ h i )were measured by a standard low-frequency lock-in tech-nique at temperature of about 0.1 K to avoid possiblemetastable orientations [8, 9]. An in-plane magnetic field B k ≡ B y was introduced by tilting the sample.In Fig. 1 we summarize our experimental findings at ν = 9/2, namely the phase diagram of stripe orientationin the ( n e , B y )-plane. The diagram contains two distinctphases, “stripes k ˆ x ” and “stripes k ˆ y ”. While the na-tive stripes are along the ˆ y -axis at all densities studied,one easily identifies three distinct evolutions of stripe ori-entation with B k . At low densities we observe a single reorientation (ˆ y → ˆ x ), in accord with the “standard pic-ture” [5–7, 23, 24, 28]. At intermediate densities, thestripes undergo two successive reorientations (ˆ y → ˆ x andˆ x → ˆ y ), ultimately aligning along B k [19]. Finally, thehigh density regime reveals no reorientations whatsoever,and the native direction of the stripes ( k y ) is preservedat all B k . While each of these regimes was previouslyrealized in individual samples [5, 6, 8, 11, 19], to ourknowledge, it is the first observation of all three classesof behavior in a single device.Further examination of the phase diagram (Fig. 1) shows that the characteristic in-plane field B y = B , de-scribing the first (ˆ y → ˆ x ) reorientation, is virtually inde-pendent of n e , as revealed by essentially horizontal lowerboundary at B ≈ .
25 T of the “stripes k ˆ x ” phase.On the other hand, the in-plane field B y = B , corre-sponding to the second (ˆ x → ˆ y ) reorientation (the upperboundary of the “stripes k ˆ x ” phase) decreases sharplywith n e until it merges with B at n e ≈ . × cm − .Indeed, B drops by an order of magnitude over a densityvariation of less than 20 %. It is this steep dependenceof B on n e that is responsible for the appearance of thethree distinct regimes discussed above.As pointed out in Ref. 19, which investigated both B and B at fixed n e , B depends strongly on spin andorbital indices, in sharp contrast to B ; at ν = 11 / B is significantly higher than at ν = 9 /
2. This ob-servation, together with theoretical considerations [23]predicting similar B k -induced anisotropy energies favor-ing perpendicular stripes at these filling factors, has leadto the conclusion that the second reorientation is of adifferent origin [19]. The observation of strong (weak) n e -dependence of B ( B ) lends further support to thisnotion.The B k -induced anisotropy energy E A evaluated at B k = B is routinely used as a measure of the nativeanisotropy energy E N >
0, which aligns stripes alongthe h i direction at B k = 0. More specifically, thepositive (negative) sign of the total anisotropy energy E = E N − E A [29] is reflected in the parallel (perpendic-ular) stripe alignment with respect to B k . Within thispicture, n e -independent B suggests that E is not af-fected by n e at B k ≈ B . However, E A depends on theperpendicular magnetic field B z and on the separationbetween subbands ∆, both of which change appreciably[30] within the density range of Fig. 1. While the exacteffect of n e on E N is not known, two experiments [8, 9]revealed that E N vanishes and becomes negative above acertain n e . In light of all these effects it is indeed surpris-ing that B [defined by E N ( n e ) = E A ( B , n e )] does notdepend on n e reflecting either that none of these effectsis significant or that the respective changes in E A and E N compensate each other.The rest of the phase diagram in Fig. 1 clearly showsthat stripe orientation is determined not by B k alone,but also by n e . In particular, the rapid decay of B and its merger with B indicate that at higher n e (andhigher B k ) stripes are more likely to be oriented parallelto B k . The decrease of B with n e , in principle, can bedue to either increasing E N [8, 9] and/or decreasing E A .However, in the regime of large B k ≫ B , any change of E N is unlikely to play a big role and, as we show below,it is indeed not the driving force for the n e -induced stripereorientation observed at B k > B in Fig. 1.As discussed above, E A is governed by B z and by theinter-subband splitting ∆, both of which vary with n e atfixed ν = 9 /
2, complicating the interpretation of Fig. 1. (cid:1)(cid:2)(cid:3)(cid:1)(cid:2)(cid:4)(cid:1)(cid:2)(cid:5)(cid:1)(cid:2)(cid:1) (cid:1) (cid:2)(cid:2) (cid:3) (cid:6)(cid:7) (cid:1) (cid:4)(cid:4) (cid:3) (cid:7) (cid:8) (cid:9) W ) (cid:3)(cid:2)(cid:10)(cid:3)(cid:2)(cid:11)(cid:3)(cid:2)(cid:12)(cid:3)(cid:2)(cid:4)(cid:3)(cid:2)(cid:1)(cid:4)(cid:2)(cid:10) (cid:5) (cid:6) (cid:7)(cid:8)(cid:5)(cid:1) (cid:5)(cid:5) (cid:7)(cid:13)(cid:14) (cid:15)(cid:4)(cid:7) (cid:16) (cid:7) (cid:8) (cid:7)(cid:17)(cid:7)(cid:4)(cid:2)(cid:10)(cid:7)(cid:18) (cid:7) (cid:4) (cid:7)(cid:17)(cid:7)(cid:5)(cid:2)(cid:10)(cid:7)(cid:18)(cid:19)(cid:20)(cid:4) (cid:5)(cid:5)(cid:20)(cid:4) FIG. 2. (Color online) R xx and R yy vs. n e measured at B z = 2.8 T and B y = 1.8 T. Additional information can be obtained if one fixes B z and B k and compares ν = 9 / / n e [31]. To this end we have measured R xx and R yy at afixed B z = 2 . B y = 1 . R xx (solid line) and R yy (dotted line) as a function of n e . At ν = 9/2, which occurs at a lower n e , R xx > R yy and stripes are parallel to ˆ y , as a result of the secondreorientation which has just occurred (cf. open circle inFig. 1). In contrast, at ν = 11 /
2, which is at a higher n e , we find R xx < R yy implying that stripes are stillperpendicular to B k . This finding might appear puzzlingas it indicates that the overall trend in Fig. 1, namelythat higher n e favors stripes parallel to B k , is completelyreversed by simply changing the spin index.Before discussing E A , we first examine if any possi-ble density dependence of E N can explain opposite reori-entation behaviors in Fig. 1 and Fig. 2. Increasing gatevoltage (at either fixed ν or fixed B z ) modifies quantumconfinement which can affect E N , e.g. by changing thespin-orbit coupling [22] and the strength of the interfacepotential experienced by electrons [8]. However, sinceall the effects associated with quantum confinement areincluded Fig. 1 and Fig. 2 on an equal footing and theycannot be the reason for the contrasting behaviors [32].Having concluded that the change of E N is of minorimportance, we can now focus on E A alone. Since theeffects associated with the change of B z are absent inFig. 2, but present in Fig. 1, it follows that they shouldplay a dominant role in triggering n e -induced parallelstripe alignment observed at B k = B in Fig. 1. One such effect is screening from other LLs which gets weakerwith B z due to increased inter-LL spacing. While theoryalways yields E A > E A < B k if screening in realistic sam-ples is weaker than calculations suggest [33–35]. Whilethere are other B z -related effects that might affect E A ,(ˆ x → ˆ y ) stripe reorientation with increasing n e in Fig. 1can be explained qualitatively by decreasing screeningwhich favors stripes parallel to B k .On the other hand, what exactly drives the reorienta-tion in Fig. 2 is not clear. Since E A relies on the finitethickness of the 2DEG, the initial decrease of ∆ enhances E A [23, 24], in agreement with recent measurements of B [20]. However, when the valence LL is sufficiently closeto the second subband (i.e., when ~ ω c / ∆ is slightly below0.5 at ν = 9 / E A [7, 23, 36].The n e -driven (ˆ y → ˆ x ) reorientation of stripes in Fig. 2implies that E increases with decreasing ∆. However,judging what happens to E A based on theoretical calcu-lations [23] is not possible because a decrease of E A with B k , observed at both ν = 9 / E N mightalso change in the density sweep.It remains to be understood why stripes parallel to B k in a single-subband quantum well are never predictedby theories [23] that calculate the dielectric function us-ing the random phase approximation (RPA) [33]. Whilewe cannot point out the exact reason, it appears plausi-ble that such calculations might not accurately capturea real experimental situation. For example, the period ofthe stripe phase might be different from what Hartree-Fock calculations suggest. Experiments employing sur-face acoustic waves have obtained about a 30% largerstripe period than suggested by theory [1, 2]. In addition,LL mixing effects beyond the RPA or disorder-inducedLL broadening [22] were not taken into account.While the phase diagram of stripe orientation shown inFig. 1 clearly identifies a robust regime of stripes parallelto B k , it would indeed be interesting to extend studies toeven higher carrier densities without populating the sec-ond subband. In particular, it might allow observationof all three distinct regimes at other filling factors, e.g. ν = 11 / , / , /
2. In addition, higher n e might re-veal a regime of native stripe orientation along the h i crystal direction which might allow us to establish a con-nection, if any, with the findings of Ref. 8.In summary, we have studied the effect of the carrierdensity n e on stripe orientation in a single-subband 30nm-wide GaAs quantum well under B k applied along na-tive stripes ( k h i ). At filling factor ν = 9 /
2, we haveobserved one, two, and zero B k -induced stripe reorienta-tions at low, intermediate, and high density, respectively.The in-plane magnetic field B k = B , which reorientsstripes perpendicular to it in accord with the “standardpicture” [5, 6, 23, 24], changes only slightly, if at all, overa wide range of densities. In contrast, the second char-acteristic field B k = B , which renders stripes parallelto B k , rapidly decays with density eventually mergingwith B . The observation that increasing carrier den-sity promotes stripes parallel to B k can be qualitativelyascribed to a weaker screening due to increased inter-LL spacing which can reduce B k -induced anisotropy en-ergy and even change its sign [23]. At the same time,our data suggest that the density dependence of the na-tive symmetry-breaking field, if any, is not an importantfactor in determining stripe orientation. Our findingsprovide guidance to future theories attempting to ex-plain parallel stripe alignment with respect to B k and toidentify the native symmetry-breaking field. These theo-ries should also take into account experimental evidence[8, 11, 19] for anisotropic nature of E A .We thank I. Dmitriev, A. Kamenev, B. Shklovskii,and I. Sodemann for discussions and H. Baek, G. Jones,S. Hannas, T. Murphy, and J. Park for technical as-sistance. 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.[1] A. A. Koulakov, M. M. Fogler, and B. I. Shklovskii, Phys.Rev. Lett. , 499 (1996).[2] M. M. Fogler, A. A. Koulakov, and B. I. Shklovskii, Phys.Rev. B , 1853 (1996).[3] M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer,and K. W. West, Phys. Rev. Lett. , 394 (1999).[4] R. R. Du, D. C. Tsui, H. L. Stormer, L. N. Pfeiffer, K. W.Baldwin, and K. W. West, Solid State Commun. , 389(1999).[5] M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer,and K. W. West, Phys. Rev. Lett. , 824 (1999).[6] 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).[7] K. B. Cooper, M. P. Lilly, J. P. Eisenstein, T. Jungwirth,L. N. Pfeiffer, and K. W. West, Solid State Commun. , 89 (2001).[8] J. Zhu, W. Pan, H. L. Stormer, L. N. Pfeiffer, and K. W.West, Phys. Rev. Lett. , 116803 (2002).[9] K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W.West, Phys. Rev. Lett. , 026806 (2004).[10] G. Sambandamurthy, R. M. Lewis, H. Zhu, Y. P. Chen, L. W. Engel, D. C. Tsui, L. N. Pfeiffer, and K. W. West,Phys. Rev. Lett. , 256801 (2008).[11] H. Zhu, G. Sambandamurthy, L. W. Engel, D. C. Tsui,L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. ,136804 (2009).[12] I. V. Kukushkin, V. Umansky, K. von Klitzing, and J. H.Smet, Phys. Rev. Lett. , 206804 (2011).[13] 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).[14] Y. Liu, D. Kamburov, M. Shayegan, L. N. Pfeiffer, K. W.West, and K. W. Baldwin, Phys. Rev. B , 075314(2013).[15] Y. Liu, S. Hasdemir, M. Shayegan, L. N. Pfeiffer, K. W.West, and K. W. Baldwin, Phys. Rev. B , 035307(2013).[16] B. Friess, V. Umansky, L. Tiemann, K. von Klitzing, andJ. H. Smet, Phys. Rev. Lett. , 076803 (2014).[17] N. Samkharadze, K. A. Schreiber, G. C. Gardner, M. J.Manfra, E. Fradkin, and G. A. Csathy, Nat. Phys. ,191 (2016).[18] Q. Shi, S. A. Studenikin, M. A. Zudov, K. W. Baldwin,L. N. Pfeiffer, and K. W. West, Phys. Rev. B , 121305(2016).[19] Q. Shi, M. A. Zudov, J. D. Watson, G. C. Gardner, andM. J. Manfra, Phys. Rev. B , 121411 (2016).[20] J. Pollanen, K. B. Cooper, S. Brandsen, J. P. Eisenstein,L. N. Pfeiffer, and K. W. West, Phys. Rev. B , 115410(2015).[21] M. A. Mueed, M. S. Hossain, L. N. Pfeiffer, K. W. West,K. W. Baldwin, and M. Shayegan, Phys. Rev. Lett. ,076803 (2016).[22] I. Sodemann and A. H. MacDonald, arXiv:1307.5489(2013).[23] T. Jungwirth, A. H. MacDonald, L. Smrˇcka, and S. M.Girvin, Phys. Rev. B , 15574 (1999).[24] T. D. Stanescu, I. Martin, and P. Phillips, Phys. Rev.Lett. , 1288 (2000).[25] We limit our discussion to single-subband 2DEG.[26] Even early experiments [5] have shown that B k appliedperpendicular to the native stripes can reduce or eveneliminate the anisotropy at ν = 9 / ν = 13 /
2, incontrast to theoretical predictions [23, 24].[27] J. D. Watson, G. A. Cs´athy, and M. J. Manfra, Phys.Rev. Applied , 064004 (2015).[28] We limit the range of B y to avoid complications fromLL-mixing effects.[29] At B k = 0, E N = E and E A = 0.[30] In our sample ∆ decreases by about 15% over the densityrange studied [27].[31] ν = 9 / / ν , larger n e also translates to a reduced stripeperiod which can also affect E N . However, Ref. 8 foundthat E N vanishes at both ν = 9 / n e , indicating that this effect is of minor importance.[33] I. L. Aleiner and L. I. Glazman, Phys. Rev. B , 11296(1995).[34] I. V. Kukushkin, S. V. Meshkov, and V. B. Timofeev,Phys. Usp. , 511 (1988).[35] Screening at a wavevector of stripe modulation is charac-terized by ε (1 + a/πa B ), where ε ≈ . a ≈ . R c is the stripe period, R c is the cyclotron radius, and a B ≈
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