Fractional quantum Hall effect energy gaps have been measured as a function of Zeeman energy. The gap at ν = 1/3 decreases as the g factor is reduced by hydrostatic pressure. This behavior is similar to that at ν = 1 and shows that the excitations are spinlike. At small Zeeman energy, the excitation is consistent with the reversal of 3 spins and may be interpreted as a small composite Skyrmion. At 20 kbar, where g has changed sign, the 1/3 gap appears to increase again

Foxon, CT

Harris, JJ

Leadley, DR

Maude, DK

Nicholas, RJ

Portal, JC

Utjuzh, AN

UCL Discovery

VOLUME 79, NUMBER 21 P HY S I CA L REV I EW LE T T ER S 24 NOVEMBER 19974Fractional Quantum Hall Effect Measurements at Zero g FactorD. R. Leadley,1 R. J. Nicholas,2 D.K. Maude,3 A.N. Utjuzh,4 J. C. Portal,3 J. J. Harris,5 and C. T. Foxon61Department of Physics, University of Warwick, Coventry, CV4 7AL, United Kingdom2Department of Physics, Clarendon Laboratory, Parks Road, Oxford, OX1 3PU, United Kingdom3Grenoble High Magnetic Field Laboratory, MPI-CNRS, 25 Avenue des Martyrs BP 166, F-38042 Grenoble Cedex 9, France4Russian Academy of Sciences, Institute of High Pressure Physics, 142092 Troitsk, Moscow Region, Russia5Department of Electronic and Electrical Enginering, University College, London, WC1E 7JE, United Kingdom6Department of Physics, Nottingham University, University Park, Nottingham, NG7 2RD, United Kingdom(Received 16 June 1997)Fractional quantum Hall effect energy gaps have been measured as a function of Zeeman energy. Thegap at n ­ 1y3 decreases as the g factor is reduced by hydrostatic pressure. This behavior is similarto that at n ­ 1 and shows that the excitations are spinlike. At small Zeeman energy, the excitationis consistent with the reversal of 3 spins and may be interpreted as a small composite Skyrmion. At20 kbar, where g has changed sign, the 1y3 gap appears to increase again. [S0031-9007(97)04626-7]PACS numbers: 73.40.Hm, 72.20.Jv, 73.20.DxThe two-dimensional electron gas in a high magneticfield is an excellent test bed for studying electron-electroninteractions. In recent years our understanding has beengreatly simplified by the composite fermion (CF) model,which maps the fractional quantum Hall effect (FQHE) ofelectrons onto an integer quantum Hall effect (IQHE) ofCFs [1,2]. In this model, the physics of the state at fillingfactor n ­ 1y3, where there is one completely occupiedCF Landau level (LL), is explained by analogy with theIQHE state at n ­ 1. The other principal FQHE statesat n ­ pys2p 1 1d can similarly be explained by theinteger states at n ­ p. While the ground states are quitewell understood, the same is not true for the excited stateswhich are responsible for conduction when the Fermienergy lies in a mobility gap.The state at n ­ 1 is an itinerant ferromagnet witha spontaneous magnetization. Consequently, the activa-tion energy gap deduced from transport measurementsis found to be much larger than the single particle Zee-man energy sZE ­ gmBBd [3] and is instead dominatedby the exchange energy Ec ­ e2y4pelB (lB ­ph¯yeB isthe magnetic length). Furthermore, it has recently beenshown optically [4] and electrically [5,6] that the excita-tions at this point are probably charged spin texture exci-tations which, in the limit of vanishing ZE, are Skyrmions[7,8]. Here we examine the CF analog n ­ 1y3 (the com-posite fermion ferromagnet). Our measurements suggestthat in this limit the composite fermion excitation has aSkyrmion-like character.Although spin was initially ignored in the CF modelit is very important, especially when the Landé g factoris small. This is the case in GaAs where the ZE has asimilar magnitude to the energy gaps between CF levels.These gaps arise from electron-electron correlations andscale with Ec, but can be treated as cyclotron gaps due toorbital motion of the CFs in an effective magnetic fieldBp ­ B 2 B1y2. Bp is zero at n ­ 1y2, where the gaugefield exactly balances the external field of B1y2. However,246 0031-9007y97y79(21)y4246(4)$10.00the ZE is determined by the total external field, and atBp ­ 0 it still has a finite value of gmBB1y2. This is anessential difference from the IQHE, where the ZE andcyclotron energy are both zero at B ­ 0. Hence CF LLsof the two spin states may cross as the ZE and magneticfield are varied, leading to the observed disappearance andreemergence of fractions [9–11].At n ­ 1y3 the ground state will always be fully spinpolarized, but the states at n ­ 2y3 or 2y5 may either bepolarized or unpolarized depending on the relative sizesof the ZE and CF LL gaps. Similarly, the excitations mayeither involve spin flips or be spin preserving transitionsbetween CF LLs. At n ­ 1y3 and small ZE, i.e., verylow magnetic fields or small g factor, we expect a spin fliptransition to the lowest CF LL state with the opposite spin.The interesting question which we address is whether thisis a single spin flip of one CF or a collective phenomenon,i.e., a Skyrmionic excitation of the CFs, which we willrefer to as a composite Skyrmion.In our experiments, g is tuned through zero to favorSkyrmion formation by applying hydrostatic pressure ofup to 22 kbar [12]. In GaAs the magnitude of g isreduced from 0.44 and passes through zero at ,18 kbar.Previously, we used this to investigate the changingenergy gaps of the mixed spin states around n ­ 3y2[9]. Here we demonstrate that the gap at n ­ 1y3 isindeed a spin gap, with excitations consistent with flipping,3 spins at small ZE. This suggests that compositeSkyrmions can be formed at n ­ 1y3 when the electrong factor is sufficiently small. By contrast, the gap atn ­ 2y5 is consistent with a single particle excitation.The samples studied were high quality GaAsyGa0.7Al0.3As heterojunctions grown at Philips Re-search Laboratories, Redhill. Samples G586, G627,and G902 have undoped spacer layers of 40, 40, and20 nm. At ambient pressure and 4 K, their respectiveelectron densities after photoexcitation are 3.3, 3.5,and 5.7 3 1015 m22 with corresponding mobilities of© 1997 The American Physical SocietyVOLUME 79, NUMBER 21 P HY S I CA L REV I EW LE T T ER S 24 NOVEMBER 1997300, 370, and 200 m2yVs. Data from similar samplesmeasured without applied pressure are included fromRef. [13]. The samples were mounted inside a non-magnetic beryllium copper clamp cell [14], and thepressure was measured from the resistance change ofmanganin wire. The absolute values quoted are accurateto 61 kbar, but between data points the variation is lessthan 60.2 kbar. The pressure cell was attached to a toploading dilution refrigerator probe, allowing temperaturesas low as 30 mK to be obtained.Increasing the pressure reduces the electron density,and above 13 kbar no electrons were present in thedark at low temperature. They could be recovered byillumination with a red LED, but the illumination timerequired to get a constant density roughly doubled forevery 2 kbar increase in pressure. The highest pressurestudied was 22 kbar, but no conductivity could bemeasured despite prolonged illumination. The samplerequired several hours for the density to stabilize beforequantitative measurements could be made, during whichit varied by less than 1% over the full temperature range.Above 13 kbar the data from G586 was recorded with adensity of 0.44 6 0.06 3 1015 m22 which puts n ­ 1y3at 5.4 T. At lower pressures where the sample wasmeasured in the dark, the density was slightly higher.For G627 and G902 the data was recorded in the range0.77 1.23 3 1015 m22, i.e., n ­ 1y3 at 9–15 T.The magnetoresistance rxx of sample G586 at 40 mKis shown for pressures between 10 and 20 kbar in Fig. 1.The abscissa is 1yn which removes the remaining smalldensity variation. The feature at n ­ 1y3 weakens asthe pressure is increased and completely disappears at18.7 kbar. In the 20 kbar trace a dip in rxx is again evi-dent, suggesting the gap at 1y3 is recovered. Meanwhile,the feature at n ­ 2y3 remains approximately constant,which is an important indication that pressure does notdenigrate the sample quality and destroy the FQHE.FIG. 1. Magnetoresistance of sample G586 showing n ­ 1y3becoming weaker as the pressure increases, but recovering at20 kbar.Figure 2 shows the temperature and pressure varia-tion of the 1y3 minimum, defined as Dr ­ frxxs‘d 2rxxsT dgyrxxs‘d, where rxxs‘d is the resistivity at thesame field taken from a high temperature trace wherethere is no longer a minimum. It is clear that at higherpressures progressively lower temperatures are requiredto see a 1y3 minimum, showing that the energy gap Egdecreases strongly with pressure. We obtained values ofEg by fitting the temperature dependence to the Liftshitz-Kosevich (LK) formula, in which Dr ~ Xy sinhX andX ­ 2p2kTyEg. This procedure, described in Ref. [13],measures the gap between LL centers and so is less sen-sitive to changes in disorder. The LK formula is onlyvalid at relatively higher temperatures before the resis-tivity minima approach zero, and is typically used in thetemperature range Egy15 , kT , Egy5. For n ­ 1y3,we have also measured the activation energy D from anArrhenius plot of rxx ­ r0 exps2Dy2kT d. By contrast,this only uses data at the lowest temperatures. The pres-sure variation of both Eg and D is shown for n ­ 1y3 inFig. 3(a). This shows good agreement between the twomethods measured in different temperature ranges. Thereis a constant difference between the two values such thatEg ­ D 1 G, which we ascribe to a constant LL broad-ening of G ­ 1.3 K. A similar value of G was previ-ously found by extrapolating the activation energy gapsfor a series of fractions [15]. At pressures above 16 kbarthe value of D deduced approaches zero since the minimain rxx do not reach zero and cease to be activated at thelowest temperatures. At this point the energy gaps havebecome comparable to the broadening, and only the LKmethod is able to measure the gap values.FIG. 2. Temperature dependence of the n ­ 1y3 minimum.Eg is obtained from fits to the LK formula (dashed lines).4247VOLUME 79, NUMBER 21 P HY S I CA L REV I EW LE T T ER S 24 NOVEMBER 1997FIG. 3. (a) Eg deduced from fitting to the LK formulacompared with the activation energy D for sample G586 atn ­ 1y3. (b) Eg scaled by Ec , showing how the gap atn ­ 1y3 decreases, 2y5 increases, and 2y3 remains almostconstant as the pressure is increased.In Fig. 3(b) values of Eg at n ­ 1y3, 2y3, and 2y5 areshown in units of Ec to allow comparison with theory.This scaled data shows the same trends as the raw data andit is seen that the gaps at 1y3 (2y5) decrease (increase) withpressure over this range. Experimentally the feature at 1y3vanishes between 17 and 19 kbar, which is just where gis predicted to pass through zero. By vanishing, we meanthat 1y3 is weaker than 2y5 and a separate minimum cannotbe observed, although the 2y5 minimum has a pronouncedtail on the high field side from the residual 1y3 feature.While an upper limit can be set on the 1y3 gap, we cannottell if it has completely collapsed. At 20 kbar a 1y3 featurecould be seen in the lowest temperature data, but it was notpossible to obtain an accurate value for the energy gap asthe minimum could not be followed to higher temperatures.From the temperature dependence of rxx at the field of1y3, an energy gap of 0.017Ec results which is consistentwith the fraction being established again once g haschanged sign.Since g varies with both pressure and density, theZE must also be scaled to compare data from differentsamples. Figure 4 shows the gaps at 1y3 and 2y5 forall the samples studied as a function of ZE. Both axesare scaled by Ec making the abscissa h ­ gmBByEc, theratio which determines the Skyrmion size and energy [16].The data for 1y3 falls into two distinct groups. Withjhj . 0.01, mostly from ambient pressure data, the gapscales only with the Coulomb energy. This behavior issimilar to that observed at n ­ 1 [5] and shows that theFQHE state at n ­ 1y3 has a Coulomb gap, which may4248FIG. 4. (a) Energy gap at n ­ 1y3 for all the samples as afunction of the Zeeman energy. The line shows the energyrequired to flip 3 spins. (b) The energy gap at n ­ 2y5. Theslope of these lines corresponds to a single spin flip.correspond to either the spin wave or more probably theCF gap. For jhj , 0.01, using data taken above 9 kbar,there is a spin gap proportional to ZE. The line onFig. 4(a), with a gradient of 3, fits the data very well atsmall h. This corresponds to an energy gap of 3gmBBand indicates an excitation involving three spin reversals.This excitation could be a small composite Skyrmion, aspredicted by theory. In a rough estimate Sondhi et al. [8]suggested that a Skyrmion formed at n ­ 1y3 and occur-ring at 1 T should contain “a couple of reversed spins.”They also estimate the Skyrmion–anti-Skyrmion pair gapas 0.024Ec at g ­ 0. The minimum gap we obtain is0.01Ec, which compares well when account is taken ofthe typical 50% reduction in Coulomb energies found incalculations where finite thickness is included [17]. Ina more detailed calculation the energy required to cre-ate an anti-Skyrmion at n ­ 1y3, i.e., the energy to re-move one spin at fixed magnetic field, was found to beE1y3yEc ­ 0.069 1 0.024 exps20.38R0.72d 1 jhjR [18].The number of reversed spins R in the composite Skyrmioncan be found by minimizing this expression, and we seethat R ­ 1 for jhj . 0.004; R ­ 3 at jhj ­ 0.002 andR ­ 6 at jhj ­ 0.001. These numbers cannot be directlycompared with our experiments at a fixed particle number,where the excitation is a Skyrmion–anti-Skyrmion pair,because they do not include creation of the quasiparticleSkyrmion or finite thickness effects. Nonetheless, theyallow us to estimate relevant energy and size scales. Itis clear that composite Skyrmions will always be smallfor experimentally accessible parameters and that a sizeof 3 spins provides good agreement between experimentand theory in the region of jhj ­ 0.002. The experimentVOLUME 79, NUMBER 21 P HY S I CA L REV I EW LE T T ER S 24 NOVEMBER 1997suggests, however, that the minimum gap for SkyrmionicCF excitations is much less than half of the gap at large ZE.This is substantially different from the prediction of a 50%reduction in the gap due to infinite sized Skyrmions for theanalogous IQHE state at n ­ 1 [8], showing that analo-gies between these ferromagnetic states must be treatedcarefully.Turning to the data obtained at n ­ 2y5 [Fig. 4(b)],there are two distinct regions that cross over at h .20.006. For jhj . 0.006 the gap decreases as the sizeof the spin splitting decreases, and for jhj , 0.006 thegap increases again. This suggests a level crossing andfinds a straightforward explanation in the CF picture.The n ­ 2y5 FQHE gap occurs when two CF LLs arefull. When the ZE is small these will be the lowest LLsof the two opposite spin ladders, thus the excitation atn ­ 2y5 is a spin flip from an unpolarized ground state.As the ZE increases, the spin reversed ladder moves uprelative to the other spin and the 2y5 gap decreases.When the second and third levels cross over there is atransition to a fully polarized ferromagnetic ground state,and the gap might be expected to vanish. A furtherincrease of ZE opens the n ­ 2y5 gap again, until thespin flip is no longer the lowest excitation and the gapeventually saturates at h¯vpc . The slopes of unity observedon Fig. 4(b) show that this model describes the data well,although we have not accessed large enough jhj to reachthe saturation region. The exception is in the immediateneighborhood of h ­ 20.006 where a finite gap remains.The position of the crossover is somewhat puzzlingsince in a single particle picture it would be expectedat gmBB ­ h¯vpc . This is clearly not the case as ourprevious work [13] shows h¯vpc , 0.03Ec at n ­ 2y5,putting the cross over at h ­ 20.03. We observe thisat much smaller jhj which suggests that the exchangecontribution may stabilize the ferromagnetic ground statecompared to the unpolarized state even at very small ZE[19]. It may also lead to a finite gap at the crossoverbetween ferromagnetic and unpolarized states due todifferences in the nature and energy of the excitationsfrom the two different ground states.While the gap at n ­ 2y3 appears to be approximatelyconstant over the range of pressure in Fig. 3, the fieldat 2y3 is only half that at 1y3 which makes the rangeof ZE insufficient to draw definitive conclusions. Whenthe scaled data at large ZE from other samples withjhj . 0.01 is included, the gap decreases by gmBB ina manner similar to 2y5. In the CF model, 2y3 and2y5 are expected to behave in a very similar way asthey both have the same CF LLs structure. We do notsee an obvious minimum in the region 20.01 , h ,0, although the scatter is larger than for 2y5. Whilewe cannot see the levels cross over for 2y3, this haspreviously been observed when the Zeeman energy isincreased by tilting the magnetic field, but only for thelowest density samples [20]. Interestingly, the tilted fieldmeasurements did not see the crossover for 2y5, so it canbe seen that a combination of experimental techniques isrequired for the complete study of the FQHE.In summary, we have measured the FQHE gaps atn ­ 2y3, 2y5, and 1y3 under conditions where theZeeman energy can be tuned through zero. For theferromagnetic state at n ­ 1y3 the energy gap decreaseddramatically as the ZE was reduced. At small ZE, theexcitation appears to consist of 3 reversed spins which weinterpret as a small composite Skyrmion. The behavior issimilar to that of the most easily accessible quantum Hallferromagnet state at n ­ 1, and is in general agreementwith theoretical predictions. 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Nicholas et al., Semicond. Sci. Technol. 11, 1477(1996).[10] R. R. Du et al., Phys. Rev. Lett. 75, 3926 (1995).[11] R. G. Clark et al., Phys. Rev. Lett. 62, 1536 (1989); J. P.Eisenstein et al., ibid. 62, 1540 (1989); L.W. Engel et al.,Phys. Rev. B 45, 3418 (1992).[12] N. G. Morawicz et al., Semicond. Sci. Technol. 8, 333(1993).[13] D. R. Leadley et al., Phys. Rev. B 53, 2057 (1996).[14] M. I. Eremets and A.N. Utjuzh, High-Press. Sci. Technol.2, 1597 (1993).[15] R. R. Du et al., Phys. Rev. Lett. 70, 2944 (1993).[16] Some workers use the symbol g˜ for h. In Ref. [7],g˜ ­ hy2.[17] F. C. Zhang and S. Das Sarma, Phys. Rev. B 33, 2903(1986); T. Chakraborty et al., Phys. Rev. Lett. 57, 130(1986).[18] R. K. Kamilla, X. G. Wu, and J. K. Jain, Solid StateCommun. 99, 289 (1996).[19] G. F. Giuliani and J. J. Quinn, Solid State Commun. 54,1013 (1985).[20] R. G. Clark et al., Surf. Sci. 229, 25 (1990).4249

Fractional quantum Hall effect measures at zero g factor

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Fractional quantum Hall effect measures at zero g factor

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