Probing the Microscopic Structure of the Stripe Phase at Filling Factor 5=2

Benedikt Friess,1 Vladimir Umansky,2 Lars Tiemann,1 Klaus von Klitzing,1 and Jurgen H. Smet11Max Planck Institute for Solid State Research, Heisenbergstraße 1, D-70569 Stuttgart, Germany2Braun Center for Submicron Research, Weizmann Institute of Science, Rehovot 76100, Israel

(Received 27 March 2014; published 13 August 2014)

A prominent manifestation of the competition between repulsive and attractive interactions acting ondifferent length scales is the self-organized ordering of electrons in a stripelike fashion in material systemssuch as high-Tc superconductors. Such stripe phases are also believed to occur in two-dimensional electronsystems exposed to a perpendicular magnetic field, where they cause a strong anisotropy in transport. Theaddition of an in-plane field even enables us to expel fractional quantum Hall states, to the benefit of suchanisotropic phases. An important example represents the disappearance of the 5=2 fractional state. Here, wereport the use of nuclear magnetic resonance spectroscopy to probe the electron density distribution of thisemergent anisotropic phase. A surprisingly strong spatial density modulation was found. The observedbehavior suggests a stripe pattern with a period of 2.6� 0.6 magnetic lengths and an amplitude as large as20% relative to the total density.

DOI: 10.1103/PhysRevLett.113.076803 PACS numbers: 73.43.-f, 76.60.Cq

When a two-dimensional electron system (2DES) issubject to a perpendicular magnetic field B⊥, a largevariety of competing, interaction-driven phases emergesbetween integer quantum Hall states (IQHSs) [1,2]. At highB⊥, when the first Landau level is partially filled, fractionalquantum Hall states (FQHSs) are prevalent. In contrast,starting from the third Landau level towards higher fillingfactors, density modulated phases take over. For thesephases, the electron density is believed to become non-uniform and arrange in geometrical patterns. Two distinctphases are known to exist. They are classified according tothe underlying symmetry as “bubble” and “stripe” phases.Both phases can be identified in transport experiments,either by a reentrance of the IQH effect or a strong transportanisotropy [3–5]. Their dominance over the FQHSs inhigher Landau levels is rooted in the altered exchange anddirect Coulomb interaction as a result of the modified shapeand extent of the wave function when populating a Landaulevel with higher index. The competition between the FQHeffect and density modulated phases culminates in thesecond Landau level. Here, reentrant phases as well asFQHSs coexist and minute changes of B⊥ or the electrondensity can lead to transitions between them. This rivalryfurther manifests itself in the observation that the quantizedHall states at filling factor ν ¼ 5=2 and 7=2 give way to ananisotropic phase if a sufficiently strong in-plane magneticfield B∥ is applied [3,6–8]. According to Hartree-Fockcalculations, these anisotropic phases may be understood asunidirectional charge density waves (CDWs) [9–11]. In thismodel, the local filling factor is modulated in parallelstripes of nearby integer quantum Hall states. The under-standing of stripe phases has progressed further by drawingon the analogy to liquid crystal behavior. These anisotropicelectron liquid phases can be categorized according to the

strength and symmetry of shape fluctuations [12]: thesmectic, the nematic, as well as the stripe crystal phases.Experimentally, little is known about the microscopic

structure of anisotropic phases in the quantum Hall regime.The temperature dependence of the resistance anisotropywas found to be consistent with a nematic phase [13,14].Yet, with this technique, microscopic details remain elu-sive. Scanning probe methods would serve as an ideal tool[15]. Their implementation, however, is impeded by therequired temperature (∼100 mK) and the necessity to burythe 2DES under insulating layers. The latter is problematicbecause the depth of the 2DES inherently sets a lowerbound to the achievable spatial resolution. Unfortunately,the stripe pattern is predicted to occur on a length scale of afew magnetic lengths only (lB ≈ 26 nm=

ffiffiffiffiffiffi

B⊥p

with B⊥ inunits of Tesla) [9–11]—much smaller than the typical depthof the 2DES in a GaAs-AlGaAs heterostructure.Here, we used nuclear spins as local detectors to study

the spatial electron distribution of the anisotropic phaseemerging around ν ¼ 5=2 in tilted magnetic fields. Thistechnique relies on the characteristic shift of the nuclearmagnetic resonance (NMR) frequency experienced bynuclei in contact with a spin polarized electron system(Knight shift) due to the hyperfine interaction [16].Conventional NMR measurements have been used beforeto study density modulated phases in different materialsystems [17–20]. In the present case, the ultralow temper-atures require a modified, more sensitive detection scheme.Here, the resonance condition was detected via the longi-tudinal resistance of the sample (Rxx), again, mediated bythe hyperfine coupling. This technique has been establishedas a powerful tool to investigate the electron spin polari-zation on the one hand and its spatial distribution on theother hand [21–23]. Even though recording Rxx is a

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macroscopic measurement, the NMR is local in the sensethat the Knight shift of each individual nucleus dependsonly on those electrons whose wave functions have anoverlap with the nucleus (contact hyperfine interaction).The overall NMR line shape is given by the integral overthe individual nuclear resonance spectra. As a consequence,a nonuniform distribution of the electron spin polarizationwithin the 2DES modifies the overall line shape due to thelocal variation of the Knight shift. Taking advantage of thisspatial sensitivity, we found a remarkably strong in-planemodulation of the charge density of more than 20% for theanisotropic phase at ν ¼ 5=2.Sample and setup.—The sample under study was a 2DES

residing in a 30 nm GaAs quantum well flanked by AlGaAslayers. An in situ grown backgate provided fast control overthe electron density and allowed tuning ν at a constantmagnetic field. The sample was patterned in a 400 μmwidevan der Pauw geometry. To perform NMR experiments, acoil was wound around the sample as depicted in Fig. 1(a).Magnetotransport measurements.—Transport measure-

ments were performed with standard low frequency lock-intechniques.Figure1(b) showsRxx betweenν ¼ 2and3alongperpendicularcurrentdirectionsfordifferent tilt angles. In thecase of perpendicular field orientation, both current direc-tions exhibit similar behavior, revealing well developedFQHSs at ν ¼ 7=3, 8=3, and 5=2. Upon tilting the sample,the5=2 state becomesweaker and eventually vanishes,whileat the same time a strong transport anisotropy develops.Thehardaxis isorientedalongB∥. Incontrast, the7=3and8=3

states get strengthened by B∥. These observations are con-sistent with previous publications [3,6,24,25].Resistively detected NMR.—Local information about the

anisotropic phase around ν ¼ 5=2 was obtained by usingresistively detected NMR. The measurement sequence isdepicted in Fig. 1(c). It is based on a two-step process. Inthe first step, a radio frequency (rf) close to the expectedresonance frequency of the nuclei is applied to the coil whilethe electron system rests at filling νprobe. It is the electron spindistribution at this filling factor which is reflected in thenuclear resonance frequency. The rf-induced change in thenuclear spin polarization is detected by measuring Rxx at adifferent filling νdetect which is chosen such that Rxx issensitive to changes in the nuclear spin polarization (seedetails in the Supplemental Material [26]).NMR response and interpretation.—Using the above

technique, we measured several nuclear resonance spectraat a rotation angle of 60° and constant total magnetic fieldBtot ¼ 6.9 T for different values of νprobe between 2 and 3.The recorded spectra are displayed in Fig. 2(b). Thecorresponding transport behavior under these conditions isshown in Fig. 2(a). For a proper comparison between thetransport data and the NMR response, the transport mea-surements were done under off-resonant rf exposure in orderto ensure equal temperature conditions (T ∼ 70 mK). Therf-induced heating is responsible for the disappearance ofthe 7=3 and 8=3 states in Fig. 2(a) [compared to Fig. 1(b)].The strong transport anisotropy, however, is mostly unaf-fected by the increase in temperature.

FIG. 1 (color). (a) Schematic of the sample configuration. (b) Rxx as a function of B⊥, measured for different tilt angles (offset) at basetemperature (T ∼ 20 mK). Shown in blue (red) is the current flow along (perpendicular to) B∥. (c) Measurement scheme for resistivelydetected NMR experiments (see the text for details).

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Looking at the NMR spectra in Fig. 2(b), severalobservations can be made. The steady trend to lowerresonance frequencies while approaching ν ¼ 3 reflectsthe increasing degree of electron spin polarization. Startingfrom the unpolarized quantum Hall state at ν ¼ 2, thepopulation of the spin-up branch of the second Landaulevel rises with increasing filling. This change of polari-zation results in a down-shift of the nuclear resonancefrequency (Knight shift). At ν ¼ 3, the electron spinpolarization reaches its highest value. The substantialbroadening of the resonance line shape with increasingspin polarization is a consequence of the spatial extent ofthe electron wave function perpendicular to the quantumwell. The changing electron density along the z directionleads to a variation of the Knight shift, which ultimatelybroadens the NMR line shape [22]. For an unpolarizedelectron system, the Knight shift is 0 irrespective of theelectron distribution. This is reflected in the narrow lineshape at ν ¼ 2.Of main interest is the appearance of a second resonance

peak. It is attributed to an in-plane modulation of theelectron spin density. Its presence is inherently linked to theanisotropic transport behavior. Other mechanisms known tocause a splitting of the resonance line are addressed inthe Supplemental Material but can be excluded [26]. Theappearance of an anisotropic in-plane phase as a plausibleexplanation for the double resonance feature is furthercorroborated by the disappearance of the second resonancepeak under purely perpendicular fields as well as at

elevated temperatures of roughly 800 mK (not shown),both of which strongly reduced the transport anisotropy.Discussion and modeling.—From the above considera-

tions, we conclude that the anisotropic phase exhibits aspatial modulation of the electron spin density. Assumingthat the electron Zeeman energy is large compared to theLandau level broadening, it is safe to infer that this spindensity modulation goes along with a spatial variation of thecharge density. This deduction is consistent with theoreticalcalculations, according to which the anisotropic phases athalf filled higher Landau levels are formed by either ananisotropic charge density wave or an electron liquid crystalphase such as a smectic or nematic phase [9–12].In an attempt to model the observed NMR response,

we assumed an equally spaced, unidirectional CDW in thesecond Landau level [Fig. 3(a), left]. A crystalline chargeorder as in the bubble phase can be excluded because of thestrongly anisotropic transport behavior. For the unidirec-tional CDW, we first calculated the 2D electron distributionρ. The spatial extent of the electron wave function leads to asmooth variation of the electron density [Fig. 3(a), right].Given the density distribution, we then simulated the NMRresponse by integrating over the local Knight shifts (seedetails in the Supplemental Material [26]).Figure 3(b) displays the calculated spectra for different

stripe periods at ν ¼ 5=2, where the relative stripe width is1∶1. Figure 3(c) illustrates the density distribution in thesecond Landau level for prominent values of the stripeperiod in Fig. 3(b). In the limiting cases of high and low λ

FIG. 2 (color). (a) Rxx between ν ¼ 2 and 3 for a fixed rotation angle of 60° and Btot ¼ 6.9 T, measured at T ∼ 70 mK. (b) Resistivelydetected NMR spectra of 75As nuclei taken at various filling factors as indicated by colored bars in (a). Resistance changes ΔR wereinverted, normalized, and offset for clarity. (c) Color plot of simulated NMR response with stripe period 2.6lB.

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values, the simulation behaves as expected: If λ is largecompared to the in-plane spatial extent of the electron wavefunction, the electron distribution closely resembles thesteplike pattern imposed by the filling factor. In this case,the spin density alternates between unpolarized and com-pletely spin polarized. As a consequence, two resonancesappear: the unshifted resonance and a resonance withmaximal Knight shift. In contrast, for small values ofλ=lB, the filling factor modulation is smeared out.Therefore, only a single resonance is observed, shiftedby Kmax=2 due to the half filled spin split Landau level atν ¼ 5=2. For intermediate values of λ=lB, the calculatedspectra depend sensitively on the CDW period. Around4.4lB, the NMR response contracts to a single resonanceline indicative of a uniform density distribution, as shownin Fig. 3(c). This behavior stems from the characteristicnode in the electronic wave function of the secondLandau level.At a phenomenological level, the best agreement with the

observations in Fig. 2(b) is achieved for λ ¼ ð2.6� 0.6ÞlB,which corresponds to λ ¼ ð1.5� 0.4ÞRc. Here, Rc is thecyclotron radius. In this range, two resonances appear, bothshifted in frequency with respect to the unpolarized case.Interestingly, the density distribution for this stripe periodorders antiphase to the modulation of the filling factor. Theexperimentally extracted value for the stripe period induced

by an in-plane field is lower than the theoretically predictedvalue of 2.8Rc, calculated for the case of perpendicularmagnetic fields [2]. Other experimentally determined valuesof the stripe period are scarce. Kukushkin et al. [29] probedthe anisotropic phase in the third Landau level at ν ¼ 9=2using surface acoustic waves and found a stripe period of3.6Rc. Inour case, probing the stripephases inhigherLandaulevels was hindered by the lower B⊥, which significantlyreduced the NMR sensitivity. Figure 3(c) reveals that aremarkably strong densitymodulation of almost 60%withina spin split Landau level (20% total density) is necessary toproduce the significant splitting of the nuclear resonanceobserved in experiment. This needs to be contrasted withtheoretical calculationspredictingmodulationsofabout20%within one Landau level spin branch in the case ofperpendicular magnetic fields [9,10]. While one mayintuitively expect that the electron system acquires one-dimensional character in an in-plane field,we are unaware oftheoretical studies addressing the influence of an in-planemagnetic field on the modulation strength.For λ ¼ 2.6lB, the calculated filling factor dependence of

the NMR response is depicted in Fig. 2(c). Even though ourmodel is relatively straightforward and does not take shapefluctuations of the stripe pattern into account, it resemblesqualitatively the experimental results in Fig. 2(b). Itproperly captures the existence of two resonance peaksover a broad filling factor range as well as the evolutionback to a single peak at ν ¼ 3. Apparent discrepanciesbetween experiment and simulation may largely be attrib-uted to the fact that our model does not consider the densitydistribution in the z direction. This would not only lead to abroadening of the resonance line shape, as mentionedearlier, but may also slightly alter the filling factordependence of the NMR response due to the modifiedshape of the quantum well when tuning the filling factorelectrostatically. A full 3D calculation, however, is chal-lenging and beyond the scope of this work because of thecomplicated influence of B∥ on the wave function in thez direction.The clear twofold resonance observed in Fig. 2(b)

suggests a high degree of stripe order. Shape fluctuationswould cause a spatially varying distance between thestripes. In view of the strong dependence of the line shapeon the stripe periodicity, this would smear out the NMRresponse and a twofold resonance as observed in experi-ment would no longer be visible. The pronounced stripeorder is presumably a consequence of the tilted field sincethe B∥ component increases the anisotropy energy [30].This also manifests itself in the persistence of transportanisotropy up to higher temperatures under tilt [14].In conclusion, we have used resistively detected NMR to

probe the local nature of the anisotropic phase emergingaround ν ¼ 5=2 in tilted magnetic fields. A twofold reso-nance was observed in the filling factor region where trans-port reveals a large anisotropy. A detailed analysis of the line

FIG. 3 (color). (a) Illustration of the stripe phase model used tosimulate the NMR response at ν ¼ 5=2. The local filling factoralternates between νloc ¼ 2 and 3 in a stripe pattern with 1Dperiodicity (left). The finite width of the electron wave functionleads to a smooth variation of the electron density (right).(b) Color plot of the simulated NMR response for differentstripe periods λ in units of the magnetic length at ν ¼ 5=2.(c) One-dimensional electron distribution in the second Landaulevel for different stripe periods, as indicated by arrows in (b).

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shape indicates a strong stripelike spatial modulation of theelectron density with an estimated period of 2.6lB. Theseresultshavetobeseeninthecontextof therichvarietyofstripephases occurring in other material systems [31].

We are grateful to B. Rosenow, I. Kukushkin, B.Shklovskii, and R. Morf for helpful discussions. We thankJ. Falson for comments on the manuscript. This work hasbeen supported financially by the German Ministry ofScience and Education (BMBF) as well as the German-Israeli Foundation (GIF).

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