Wigner crystalization about $nu$=3

a r X i v :c o n d -m a t /0307182v 1 [c o n d -m a t .m e s -h a l l ] 8 J u l 2003Wigner crystallization about ν=3R.M.Lewis 1,2,∗,Yong Chen 1,2,L.W.Engel 1,D.C.Tsui 2,P.D.Ye 1,2,L.N.Pfeiffer 3,and K.W.West 31NHMFL,1800E.Paul Dirac Dr.,Florida State University,Tallahassee,FL 32310,USA2Dept.of Electrical Engineering,Princeton University,Princeton,NJ 085443Bell Laboratories,Lucent Technology,Murray Hill,NJ 07974We measure a resonance in the frequency dependence of the real diagonal conductivity,Re[σxx ],near integer filling factor,ν=3.This resonance depends strongly on ν,with peak frequency f pk ≈1.7GHz at ν=3.04or 2.92close to integer ν,but f pk ≈600MHz at ν=3.19or 2.82,the extremes of where the resonance is visible.The dependence of f pk upon n ∗,the density of electrons in the partially filled level,is discussed and compared with similar measurments by Chen et al.[14]about ν=1and 2.We interpret the resonance as due to a pinned Wigner crystal phase with density n ∗about the ν=3state.The integer quantum Hall effect (IQHE)[1,2]is un-derstood to arise in two–dimensional electron systems (2DES)from gaps in the single particle density of states.At low temperatures,electron localization when the Fermi level resides in these gaps,gives rise to wide,deep minima in the diagonal conductivity,σxx ,and quantized values of the Hall conductance,σxy .In very clean sys-tems,there exists the possibility that electrons may be-come localized in a collective manner forming crystalline domains[3,4,5,6]pinned by disorder.Collective local-ization of electrons is known to occur deep in the lowest Landau level at filling factors ν<1/5where the Wigner crystal (WC)[7]becomes energetically favorable [8,9].Key observations of the WC have been provided by mea-surements of the real diagonal microwave conductivity,Re[σxx ],which shows a resonance at frequency,f ∼1GHz [10,11].The resonance is caused by the pinned WC domains oscillating in the disorder potential,i.e.a pinning mode[12,13].Recently,Chen et al.[14]have shown that a similar resonance exists at νclose to inte-gers 1and 2and attributed it to a crystal formed from the electrons in the uppermost occupied spin split Lan-dau level.In this paper,we detail measurements of a resonance in Re[σxx ]about ν=3.We interpret this resonance as due to a pinned crystal phase about ν=3.This resonance is sharp and has strong B dependence in agreement with Ref.[14].The dependence of f pk upon the partial den-sity in the uppermost occupied spin split Landau level is shown to agree well with measurements of f pk about ν=1and 2.It also shows that Landau level index is im-portant in determining the properties of integer quantum Hall Wigner crystals (IQHWC).Our measurements were performed on an MBE grown GaAs/AlGaAs 300˚A quantum well of density n =3.03×1011cm −2and mobility µ=2.4×107cm 2V −1s −1.The2lZ 0ln(P/P 0)where P is the transmittedpower,P 0is the power transmitted in the absence of a 2DES,Z 0=50Ω,and w2FIG.1:The real part of the conductivity,Re[σxx]versus B between3.5and6.5T at T≈50mK.Five frequencies,.3,.6, 1.0,1.5,and2GHz are shown.The data are offset for clarity.ν=3.04and3.19on the high side.In Fig.2B,the peak frequency f pk of the resonance is plotted for those traces to which a lorentzian curve can reasonably befit-ted.The f pk runs from600MHz atν=2.83to1.7GHz atν=2.91.Aboveν=3,f pk fall between1.8GHz at ν=3.04and about560MHz atν=3.18.Fig.2C shows the full width at half maximum,FWHM,taken from the same lorentzianfit.The resonance reaches its strongest (narrowest FWHM)atν=2.86and3.12.In both cases, FWHM≈650MHz.The errors on the FWHM are about 10%.Those on f pk are about6%and the error inνis less than±0.02.A resonanceof this kind in Re[σxx],as has been dis-cussed elsewhere[14,20],is naturally interpreted as a pin-ning mode[12]of an electronic crystal.The resonance ob-served here varies from well above the temperature(40 mK)at1.7GHz to well below it at650MHz(1GHz ×h/k B≈50mK)and persists to temperature well in excess of100mK.Hence,it is unlikely that individu-ally trapped electrons are its cause.The resonance is sufficiently sharp with Q=f pk/FWHM>1that some averaging of pinning potentials is occurring,indicating a large domain size.Finally,the data can be compared to the resonance seen in the Wigner Crystal regime,ν<1/5 [11],to the similar resonance seen aboutν=1and2[14],Re[σxx](µS)(A)FIG.2:(A)The real part of the conductivity,Re[σxx]versus f atfilling factors,ν,between3.78and4.24.(B)The peak frequency,f pk versusν.(C)Full width at half maximum, FWHM,versusν.and to the resonance[20]of the bubble phase.The resonance behavior seen here and in ref.[14]is very different from what has been seen in the IQHE min-ima before.In moderately clean[19]2DES over a similar range of frequency,1≤f≤10GHz,microwave conduc-tivity experiments[17,18]found Re[σxx]∝f,within the QHE minima.Those data were consistent with single particle hopping theory[16].In contrast,the pinning res-onance seen here is due to the collective motion of pinned crystalline domains.The dependence of the resonance onνis mainly due to a change in the density of electrons/holes in the crys-tal.Assuming thatfilled levels are inert,the remain-ing electrons/holes in the partiallyfilled/empty level above/belowν=3have density,n∗=n×ν∗/νwhere ν∗=ν−[ν]is the partialfilling of the level and in this case,[ν]=3.Infigure3we plot f pk versus|n∗|for the resonance aboutν=3,2,and1[21].Data for the resonance above integerν(electrons)are shown asfilled symbols and those from holes(just below integerν)are open symbols.The quantum numbers for Landau level index,N,and spin,σ,are also given.While f pk monotonically decreases as n∗increases in all cases,the data naturally bunch into2 groups.Aroundν=3(triangles),the f pk for electrons (1,↑)and holes(1,↓)fall close to each other.Both data3FIG.3:Peak frequency,f pk versus partial density,|n∗|,about integerfilling,n∗=n×ν∗/ν.Filled symbols represent elec-trons and open symbols holes.Data from the resonance about ν=1and2are included.The quantum numbers N andσare also given.sets agree well with data from the electrons nearν=2 (filled squares)(1,↓).All three occur in the N=1Landau level.However,the other three data sets were taken at ranges ofνwhich fall in the lowest Landau level.These are,nearν=1,electrons(0,↑)and holes(0,↓)and holes belowν=2(0,↓)the data for all of which follow a more gradual slope.These two distinct bunchings are due to differences between in the single particle wave function in the N=0and N=1Landau levels.In conclusion,we have presented data for an integer Wigner crystal phase of electrons whose density comes from the partialfilling of the N=1Landau level about ν=3.Our evidence is a strong resonance in Re[σxx]at νclose to3,with a range from2.83to2.91and again from 3.04to3.19.The best Q are measured atν=2.86and 3.12.The dependence of f pk uponν∗closely resembles that of the previously measured resonance nearν=1and 2and suggests that Landau Level index is of principal importance to the properties of the resonance.We thank Kun Yang,Nick Bonsteel,and Ravin Bhatt for stimulating discussions.This work is supported by a grants from the AFOSR and the NHMFL in house re-search program.[1]K.von Klitzing et al,Phys.Rev.Lett.45494(1980).[2]See reviews on the QHE see,The Quantum Hall Ef-fect,edited by R.E.Prange and S.M.Girvin(Springer-Verlag,New York,1987)and Perspectives in Quantum Hall Physics edited by S.Das Sarma and A.Pinczuk (Wiley and Sons,New York,1997).[3]A.H.MacDonald and S.M.Girvin,Phys.Rev.B33,4009(1986).[4]M.M.Fogler, A.A.Koulakov,and M.I.Shklovskii,Phys.Rev B54,1853(1996).[5]F.D.Haldane,E.H.Rezayi,and Kun Yang,Phys.Rev.Lett,85,5396(2001).[6]N.Shibata and D.Yoshioka,Phys.Rev.Lett.86,5755(2001).[7]See chapters by H.Fertig(3)and M.Shayegan(9)inPerspectives in ref.2.[8]m and S.M.Girvin,Phys.Rev.B.30,473(1984).[9]D.Levesque et al.Phys.Rev.B30,1056(1984).[10]L.W.Engel et al.Solid State Comm.,104167(1997).[11]P.D.Ye et al.,Phys.Rev.Lett.89,176802(2002).[12]H.Fukuyama and P.A.Lee,Phys.Rev.B18,535(1978).[13]B.G.A.Normand et al.,Phys.Rev.B463920(1992).[14]Yong P.Chen,R.M.Lewis,L.W.Engel,...cond-mat/0301579,to appear in Phys.Rev.Lett.[15]L.W.Engel et al.Phys.Rev.Lett.71,2638(1993).[16]D.G.Polyakov and B.I.Shklovskii,Phys.Rev.B48,11167(1993)and references therein.[17]R.M.Lewis and J.P.Carini,Phys.Rev.B64073310(2001).[18]F.Hohls et al.Phys.Rev.Lett.865124(2001).[19]Both experiments used samples ofµ∼5×105cm2V−1s−1.[20]R.M.Lewis,P.D.Ye,L.W.Engel,D.C.Tsui,L.N.Pfeiffer,and K.W.West,Phys.Rev.Lett.89136804 (2002).[21]The f pk data for resonances aroundν=1and2weremeasured in a separate cooldown on a different sample cut from the same wafer.。