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NMR probing of the spin polarization of the ν = 5/2 quantum Hall state M. Stern1,∗ B. A. Piot2, Y. Vardi1, V. Umansky1, P. Plochocka2, D. K. Maude2, and I. Bar-Joseph1 1 Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot, Israel and 2 Laboratoire National des Champs Magnétiques Intenses, CNRS-UJF-UPS-INSA, 38042 Grenoble, France (Dated: January 18, 2012) Resistively detected nuclear magnetic resonance is used to measure the Knight shift of the 75As nuclei and determine the electron spin polarization of the fractional quantum Hall states of the second Landau level. We show that the 5/2 state is fully polarized within experimental error, thus confirming a fundamental assumption of the Moore-Read theory. We measure the electron heating underradiofrequencyexcitation,andshowthatweareabletodetectNMRatelectrontemperatures down to 30 mK. 2 PACSnumbers: 73.43.Fj 1 0 2 The physics of interacting electrons at half integer fill- may also originate from the destruction of the 5/2 state n ing factor has intrigued researchers over the past two induced by the orbital coupling to the parallel magnetic a decades. It was found that the behavior at the first Lan- field [10–12]. Optical measurements also provided indi- J dau level can be understood in terms of non-interacting cations, which supported an unpolarized ground state. 7 composite fermions (CF) at zero magnetic field [1]. Nu- Raman experiments have shown diminishing of the spin 1 merous experimental findings confirm this picture, and flip mode, and were interpreted as an evidence for a par- establish the formation of a non-gaped state at ν =1/2. tially polarized second Landau level [13]. Recently, we ] l ThebehaviorinthesecondLandaulevelis,however,fun- reportedtheresultsofphotoluminescence(PL)measure- e - damentally different. Early experiments, dating back to ments [14] that show a dip in the Zeeman splitting in r t thelateeighties,showtheformationofafractionalquan- the vicinity of ν = 5/2, and can be interpreted as due s . tumHall(FQH)stateatν =5/2[2,3]. Thisobservation to a spin unpolarized ground state. However, a possi- t a has challenged the CF theory and triggered an intense ble complication in the interpretation of the optical data m theoretical effort. Early on Moore and Read (MR) have could arise from the role of the positively charged va- - suggested that a weak residual attractive interaction in lence band hole that is introduced by laser illumination. d the second Landau level gives rise to pairing of the CFs, The effect of this valence hole is not fully understood n and to the formation of a gaped state [4]. One of the and could, in principle, affect the observed results. All o c exciting aspects of MR theory is the predicted excita- theseexperimentalfindingsareinclearcontrastwiththe [ tion spectrum of the ν =5/2 state, which should consist results of numerical simulations which clearly show that of quasiparticles that obey non-Abelian braiding statis- thegroundstateshouldbespinpolarized[10]. Thereare 1 v tics. It was shown that this property may turn the 5/2 alsotransportdatawhichsupportaspinpolarizedstate; 8 state into a platform for quantum computing by means e.g theobservationofν =5/2athighmagneticfields[15] 8 of topological manipulations [5]. and resistively detected NMR measurements performed 4 at a relatively high excitation power [16]. This contro- 3 The examination of MR theory has become the focus versycallsforfurtherexperimentalworkusingadifferent . 1 of intensive experimental effort in recent years. A major experimental technique [17]. 0 support for its validity has been provided by the mea- 2 surement of the quasiparticle charge, found to be e/4, in In this work we use resistively detected nuclear mag- 1 netic resonance (NMR) to measure the Knight shift of agreement with the prediction of the MR theory [6, 7]. : the75Asnucleianddeterminetheelectronspinpolariza- v However, this finding is also consistent with other com- Xi peting theories [1], and further experimental support is tion of the FQH states of the second Landau level. We monitortheelectronheatingunderradiofrequency(RF) needed. The determination of the electron spin polar- r excitation, and show that we are able to detect an NMR a ization can provide this needed support. A central as- signal at electron temperatures down to 30 mK. We find sumption of the MR theory is that the electrons in the that the FQH states in the second Landau level, and in second Landau level are fully-polarized, and can there- particular the 5/2 state, are preserved under RF excita- foreformpairswithp-typesymmetry. Hence,confirming tion. We show that the 5/2 state is fully polarized, thus this point would provide a strong experimental evidence confirming a fundamental assumption of the MR theory. for the validity of the MR theory. Unfortunately, the Note that throughout this paper polarization refers to experiments realized so far to probe the polarization at the polarization of the partially occupied Landau level ν =5/2haveledtoambiguousresults: Tiltedfieldexper- at the Fermi energy. iments have shown that the ν = 5/2 gap decreases with tilt angle [8, 9], which could be interpreted in favor of The NMR technique is a powerful tool to measure the a spin depolarized ground state. However, this behavior electron spin polarization. It is based on the coupling 2 of the electron and nuclei spins via the hyperfine interaction. In the presence of an external magnetic field, the nuclei acquire an average polarization (cid:104)I(cid:105) and create a local magnetic field B ∝ (cid:104)I(cid:105) (Overhauser effect). N B acts exclusively on the electronic spin and has no N influence on the orbital motion of the electrons, so that the filling factor remains unchanged. The polarized electrons also create a local magnetic field B acting on the e nuclear spins. This field reduces the Larmor resonance frequency of the nuclei by K ∝ n P, an effect known S e as the Knight shift, where n is the local electron den- e sity and P is the electron spin polarization. Obtaining the signal from a 2D electron gas is however an experimental challenge, since the number of nuclei in a single quantum well is much smaller than the total number of nucleiinthebulkofthesample. Severaltechniqueshave been implemented to overcome this difficulty; increasing the effective thickness by using multiple quantum wells [18], increasing the nuclei polarization in the quantum well using optical pumping [19], or using a selective de- tecting scheme, which is specific to the electrons in the well, known as resistively detected NMR [20, 21]. This technique relies on the dependence of the longitudinal Figure 1: (a) The NMR excitation and detection protocol as resistance R on the Zeeman energy gap, which can be xx described in the text. (b) R as a function of gate voltage writtenasE =gµ (B+B ). Byapplyingaradiofre- xx Z B N showing the detection point (c) R versus applied radio fre- xx quency at the Larmor resonance frequency, it is possible quency with a power of -5 dBm; under excitation at ν =5/2 todepolarizethenuclearspinsandreducetheamplitude and detection at ν = 2.16 (red line) and a scan at the de- of B , and consequently decrease E . This, in turn, re- tection point (blue line). All measurements in this paper are N Z sults in a slight change of R , typically of the order of performed at B =5 T. xx a few percent. Attemptstodirectlyimplementthistechniquetomea- suretheelectronspinpolarizationatν =5/2havefailed; Fig. 1(b)). To excite the system, the filling factor is set no detectable change of R was measured throughout to the excitation point (e.g. ν = 5/2). After a short xx the quantum Hall plateau. Indeed, tilted field measure- waiting time ((cid:39) 30ms), the radio frequency is abruptly ments show that the longitudinal resistance at this fill- changed during the excitation time τexc, without chang- ing factor has a very weak dependence on E [9]. One ing the RF power. Then, the filling factor and the radio Z can,however,measuretheKnightshiftat5/2byquickly frequency are returned to their original values and Rxx switching to another filling factor at which R exhibits is measured at the detection point for a time interval of xx a strong dependence on EZ. This idea relies on the τdet. This sequence is repeated many times while vary- long relaxation time of the nuclei which decay back to ingtheexcitationradiofrequencysoastosweepthrough their original polarization on a timescale from a few tens theresonance. Afewimportantconsiderationsshouldbe of seconds to many minutes. It was implemented in mentioned here: Ref.[16]byapplyingshortRFpulseswhenthesystemis (i) The quick change of the filling factor between the atν =5/2. Wenoticed,however,thatapplyingtheseRF excitation and detection point can only be achieved by pulsescausesanonresonanttransientresponseduetothe applyingagatevoltage. Thesampleweusedisthesame heatingoftheelectronicsystemandgreatlydegradesthe we used in our previous work, in which we measured the signal to noise ratio for the detection. For this reason, PL [14], and has a 4 nm PdAu gate. we have developed a technique where a low RF power is (ii) The detection point should be close enough to the continuously applied. The improved signal to noise ratio fillingfactorunderconsideration,toavoidirreversiblede- allows us to work at much lower RF power, and there- terioration of the quality of the FQH states that occurs fore lower electronic temperature than the pulsed NMR under strong gating. We have selected to work at the technique of Ref.[16]. As will be shown this reduction in highendoftheν =2plateau,atν (cid:39)2.16. Inthisregion the RF power is critical for the measurement. R exhibitsasignificantdependenceontheZeemanen- xx The exact sequence used is shown in Fig. 1(a). The ergy and the NMR signal can be easily observed in a detection of the signal is always performed under off res- standard continuous wave experiment. onance RF at the detection filling factor ν = 2.16 (see (iii)TheRFpowercouplesalsototheelectronicsystem 3 and heats it up. Thus, one should conduct in parallel a measurement of the electron temperature and confirm that the quantum Hall state is not destroyed. (iv)Thelongtimeforfullthermalizationofthenuclei, whichat5Tcanbedays,mayinhibitperformingsequen- tialfrequencyscans. Thus,oneshouldprovidemeansfor fastnuclearrepolarizationbetweenfrequencyscans. This is effectively done by ramping the magnetic field to ∼11 T, at the end of each scan. (v)Wefoundthattherampingofthefieldcaninducea changeofthevalueoftheLarmorfrequencyofa(cid:39)1kHz. Hence, it is essential to always have a point to which each scan can be compared. This is done by performing a measurement at the detection point at the end of each scan. This procedure sets a limit on the precision of measuring the Knight shift to be 300 Hz. Figure 1(c) demonstrates the implementation of this technique. The red curve shows the R signal under xx excitation at 5/2 and detection at 2.16. It is seen that a well resolved dip with an amplitude of 4 Ω appears Figure 2: (a) Resistivity ρ at B = 5 T versus gate volt- xx at 36.24 MHz. The blue curve is a scan obtained by age V for various RF powers: +1 dBm, -2 dBm, -8 dBm, - G performing a regular resistively detected NMR measure- 11dBmand-14dBm. Theblackcurveisρ atT =13.5mK xx ment at the detection point (ν = 2.16). The fact that without RF power. (b) ρ at filling factor ν = 5/2 ver- xx the5/2dipappearsatlowerfrequencythanthesignalat sus the electron temperature. The dotted red line is a fit to ν = 2.16 is significant and a qualitative conclusion can ρxx =ρ0+ρ1exp[−∆/2T] giving ∆(cid:39)300 mK. (c) Electron temperature versus RF power. Below -14dBm, the tempera- already be drawn. Since the electron polarization can ture is limited by the current I = 30 nA. The red triangles only reduce the Larmor frequency (K ≤ 0 ) it follows S correspond to I =5 nA. that the ν = 5/2 FQH state cannot be fully depolarized,incontrastwiththeresultsofthePLmeasurements. Note that all our measurements are shifted compared to is only slightly below that measured at -14 dBm. the accepted value of the gyromagnetic ratio for 75As in bulk GaAs (7.29) due to the remanent field (-27 mT) of To evaluate the effect of the residual heating of the the magnet which reduces the effective magnetic field to electrons one needs to estimate the 5/2 gap. The dotted 4.973 T. lineinFig.2(b)isafittoρxx =ρ0+ρ1exp[−∆/2T]. Here Before presenting a quantitative analysis of the shift ρ0 =22 Ω is the lowest value of ρxx at zero temperature and its implications, one should verify that the observed which is different than zero for this sample, ρ1 = 120 polarization is not due to heating of the electrons by the Ω, and ∆ (cid:39) 300 mK. We can therefore conclude that RFpowerandthattheFQHstatestillexists. Figure2(a) the region below −5 dBm corresponds to the activation showsthelongitudinalresistanceofthesampleastheRF region, and by reducing the RF power we can determine power is reduced, from 1 dBm down to -14 dBm. It is thebehaviorofthespinpolarizationdowntoT/∆(cid:39)0.1. seen that the resistance at 7/3 and 5/2 increases as the Figure3showsthechangeinresistance,∆R,asafunc- RF power is increased until the minima become barely tion of radio frequency f −f at four filling factors, 5/3, 0 visibleat1dBm. Toobtainacalibrationoftheelectron 7/3, 5/2, and 8/3, where f is the resonance frequency 0 temperature we measured the dependence of ρ on the at ν = 2. Since the ν = 2 state is clearly depolarized, xx bath temperature at several filling factors without RF it will have a resonance frequency f i.e the bare (un- 0 excitation. WeshowinFig.2(b)thebehavioratν =5/2. shifted) Larmor frequency. On the other hand, the frac- This allows to obtain a calibration curve of the electron tionalquantumHallstateatν =5/3istheelectron-hole temperature as a function of the RF power (Fig.2(c)). It symmetric state of ν = 1/3 and is known to be fully is seen that at 0 dBm the electrons are heated up to 100 spinpolarized[22]. Aswewereunabletomeasureacon- mK. At the lowest RF powers the curve deviates from a vincing signal at ν =2 with our technique at low power lineardependence(solidline). Atthisregimetheelectron a lengthy series of calibration measurements were per- temperature is limited by the applied current (30 nA), formedathigher RFpowerusingapulsedNMR method and reducing it to 5 nA allows us to bring the electron similar to Ref.[16]. During the calibration procedure we temperaturedownto33mK.Theresidualheatingbythe simultaneously measured the signal at ν =5/3 to obtain RFpowerisdemonstratedinFig.2(a);thelowestcurveis a calibration of the maximum possible Knight shift cor- takenwithoutRFpoweranditisseenthatitsresistance responding to a full degree of polarization. As ν = 5/3 4 Figure3: Changeofthelongitudinalresistance∆Rasafunc- tion of radio frequency (f −f ) where f is the resonance 0 0 frequency at ν = 2. The RF power is here -11 dBm. Inset: Knight shift as a function of filling factor. The red dotted Figure4: (a)NMRsignalatν =5/2fordifferentRFpowers. linecorrespondstofullelectronpolarizationaccordingtothe (b)Electronspinpolarizationatν =5/2asafunctionofT/∆. shift obtained at ν =5/3. To further reduce the heat load on the sample at the lowest RF powers, the current is reduced to I = 1 nA during the excitationphase. NMRsignal(c)andtheresistivitytogether with the electron spin polarization (d) in the vicinity of ν = is also detected in the low power measurements this pro- 5/2 at RF power -14 dBm. Note the correspondence of the vides a direct calibration for f0 in our measurements. symbols in (a-b) and in (c-d). Additionally, this calibration was confirmed by measurements at ν =2/3 which can serve as a reference for both fully polarized and unpolarized states due to the forma- However, they seem to be in stark contradiction with tion of domains [23]. the results of the photoluminescence measurements [14], FromourmeasurementsweobtainadifferentK equal whichwereconductedonthesamesample. Wenotethat S to 5650±75 Hz, 7750±175 Hz and 10175±150 Hz for thetwomeasurementsprobedifferentaspectsoftheFQH ν =7/3, 5/2 and 8/3, respectively. In the inset of Fig. 3 state. The NMR measurement probes the spin polar- we plot K as a function ν; remarkably, we find that ization through its effect on the longitudinal resistance. S they reside on a straight line, implying the same degree SincethechangeinR arerelatedtotunnelingthrough xx of spin polarization. The red line corresponds to the saddle points between edges at the two sides of the sam- extrapolatedvalueoftheKnightshiftforafullypolarized ple [24, 25] - the effect is global and is related to the state according to its value at ν = 5/3. We can, thus, average spin polarization of the whole sample. The PL conclude that the electrons at the second Landau level measurement, on the other hand, is local in nature, and are fully polarized at the three filling factors 7/3, 5/2 samplestheimmediateneighborhoodofthevalencehole. and 8/3. The results of the PL measurement imply that this local The full polarization at 5/2 is robust and persists environment is depolarized. Recently, it was suggested throughout the temperature range that we have stud- that at ν = 5/2 Skyrmions, which consist of two quasi- ied, and throughout the Hall plateau, as demonstrated particleswithoppositespin,tendtoforminlocalminima in Fig.4. Figure 4(a) shows ∆R as a function of fre- of the disordered potential (formed by well width disor- quency for three RF powers, 1 dBm, -8 dBm and -14 derorremoteimpurities)[26]. Onecanthereforeimagine dBm, which correspond to electron temperatures of 110, twopossiblescenariosthatmightexplaintheoriginofthe 65, and 35 mK, respectively. It is readily seen that the observeddepolarization. Thefirstisthattheholesareat- resonance frequency does not change and the spin polar- tractedtothesamelocalminimaatwhichSkyrmionsare ization remains constant at 0.1<T/∆<0.35 (Fig.4(b). formed. The second is that the potential of the valence Figure 4(c-d) show a measurement at -14 dBm in the band hole, which induces a potential minima, could help vicinity of ν =5/2; no depolarization is observed within forming a Skyrmion, essentially playing the same role as the 5% precision of our measurement. the disorder in Ref.[26]. 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