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Observation of soft magnetorotons in bilayer quantum Hall ferromagnets Stefano Luin,1,2 Vittorio Pellegrini,1 Aron Pinczuk,2,3 Brian S. Dennis,2 Loren N. Pfeiffer,2 and Ken W. West2 1NEST-INFM and Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa (Italy) 2Bell Labs, Lucent Technologies, Murray Hill, New Jersey 07974 3Dept. of Physics, Dept. of Appl. Phys. and Appl. Math, 3 Columbia University, New York, New York 10027 0 (Dated: February 2, 2008) 0 2 Inelasticlightscatteringmeasurementsoflow-lyingcollectiveexcitationsofelectrondoublelayers n in the quantum Hall state at total filling νT=1 reveal a deep magnetoroton in the dispersion of a charge-densityexcitationsacrossthetunnelinggap. Therotonsoftensandsharpensmarkedlywhen J the phase boundary for transitions to highly correlated compressible states is approached. The 6 findings are interpreted with Hartree-Fock evaluations that link soft magnetorotons to enhanced 1 excitonic Coulomb interactions and to quantumphase transitions in the ferromagnetic bilayers. ] PACSnumbers: 73.43.Lp,78.30.-j,73.21.-b l l a h The quantum Hall states of the two-dimensional elec- ing characteristics and anomalous quantized Hall drag - s tron gas (2DEG) occur in high perpendicular mag- [14, 15, 16]. These remarkable results are interpreted as e netic fields that quantize the kinetic energy into dis- evidenceofaGoldstonemodeintheincompressiblephase m crete, highly-degenerate Landau levels (LLs). The en- andofcondensationofthe bilayersintomany-bodyexci- t. ergyscaleforCoulombinteractionsisheree2/ǫlB,where tonphases. Experiments that probe dispersivecollective a m lB=p¯hc/eBisthemagneticlengthandBtheperpendic- excitations andtheir softening as a function of∆SAS and ular magnetic field. The neutral quasiparticle-quasihole dcouldprovidedirectevidenceoftheimpactofexcitonic - excitations carry the fingerprints of electron interactions termsofinteractionsinthequantumphasetransitionsof d n [1, 2]. Low-lying collective modes of energies ω(q) and the ferromagnetic bilayers. o in-plane wave vector q are linked to the condensation Resonant inelastic light scattering methods have been c into highly correlatedstates that emerge in the presence employed in studies of very low-energy q ∼ 0 tunneling [ of strong electron interactions. Theoretical dispersions excitationswithspinreversalinelectronbilayersateven 1 ω(q) display characteristic magnetoroton (MR) minima integervaluesofν [17]. Thesestudieshaverevealedthat 5v at finite wave-vectors (q∼lB−1) that are due to excitonic excitonic interactiTons can drive changes in the quantum binding terms of the Coulomb interactions in the neu- 9 ground state and finite temperature transitions [18, 19]. tral pairs [3, 4]. It has been predicted that MRs can 2 We report here light scattering experiments that offer 1 soften and create instabilities leading to quantum phase directevidenceofsoftmagnetorotonsinthe CDE modes 0 transitions that transform the ground-states into highly across the tunneling gap of coupled electron bi-layers at 3 correlated electron phases [3, 5, 6, 7, 8, 9]. ν =1. MR modes at wavevectors q ∼ l−1 and modes 0 T B / Coupled electron bilayers at total Landau level fill- with larger wave vectors can be accessed in the resonant t a ing factor νT=1 exhibit a rich quantum phase diagram light scattering experiments due to breakdown of wave m due to the interplay of transition energies ∆ across vector conservation. The light scattering spectra with SAS - thetunneling gapwithintra-andinter-layerinteractions breakdown of wave vector conservation show maxima at d [5, 8, 10, 11, 12, 13]. Interactions drive quantum phase the critical points in the mode dispersion [20]. n transitions from the incompressible ferromagnetic quan- o We find that the light scattering spectra of low-lying c tized Hall phase, stable at low inter-layer spacing d or modes typically display three bands of CDE. One is the v: large ∆SAS, to a compressible phase that results from q ∼0excitation. TheothertwoareassignedtotheCDE the collapse of the many-body tunneling gap. In current i modes at critical points in the dispersion. The lowest X theories the phase transitionsare linked to soft rotonin- of these two is the critical point at the magnetoroton r stabilitiesinthecharge-density-excitations(CDE)across minimum with q ∼l−1, and the higher energy band is a B thetunneling gap[5,8]. WithintheHartree-Fockframe- the large density of states of modes with q ≫ l−1. The B workthemagnetorotoninstabilityisrelatedtointra-layer MR mode softens markedly when ∆ is reduced and SAS interactionsthatleadtolargeexcitonicbindingsbetween the double-layer system approaches the incompressible- quasiparticles and quasiholes. compressiblephaseboundary. Closetothisboundarythe Recent experimental studies of coupled electron dou- MR occurs at an energy significantly lower than ∆ . SAS ble layers in ν =1 ferromagnetic states focus on the Magnetoroton spectral lineshapes and temperature de- T very low ∆ region of the phase diagram, where inter- pendences display striking differences with those of the SAS layer Coulomb correlations are important. These stud- long-wavelength modes. Close to the phase boundary ies have displayed enhanced zero-bias inter-layer tunnel- theMRbandshowsextremenarrowingtoawidthofless 2 than 70 µeV at temperatures T ≃60mK. Wehaveinterpretedtheseresultswithintheframework (a) (c) T~60mK oHfFaA)titmhea-tdienpcelnuddeenstbrHeaarktdroewe-nFoocfkwaapvpervoexcimtoartcioonns(eTrvDa-- SAS SW C0 B nits) MR u tion in light scattering. The calculations reproduce the d b. MRenergiesandindicatethatthe sharpeningoftheMR 4 ar (b) MR y ( bapnahndadsmerabetfloreuixcntedslaesrmiyg.ennifitTcnhaeenastretcrhheeasnuingltcesosmuinnpcrtoehvseseirbmlseoi-gdcneoimfidcipsarpneetsrsseiibvoline- d/l23B QNHOEA B A C0 Intensit dencethatsofteningoftherotonsplaymajorrolesinthe QHE phasetransitionsofbilayersatνT=1andsuggestalead- 1 ing rolefor excitonic Coulombinteractionsintransitions 2SAS (410-26 e2/ l8B) 0.0 En0e.5rgy Shi1ft. 0(meV)1.5 between highly correlated phases. FIG.1: (a)Schematicrepresentationofthedoublequantum Results obtained in two modulation doped double wellandofthetwolowestconfinedstates. (b)Phasediagram quantum wells (DQWs) grown by molecular beam epi- fortheincompressible(QHE)-compressible(NOQHE)states taxy are presented. Samples consist of two 18nm GaAs ofthebilayeratν =1(from Refs.12,13);thedotsmarkthe wells separated by an undoped Al0.1Ga0.9As barrier positions of samples A and B. (c) Inelastic light scattering (7.5nm for sample A and 6.23nm for sample B). Fig- spectrafrom sample A(blackcurvesandlabels) andB(gray ure 1(a) shows the schematic profile of the bottom of curveand labels). SW is the spin-wave, C0 is the q∼0 CDE the conduction band in the DQWs. Dotted lines repre- mode, and MR themagnetoroton minimum. sent the energy of lowest symmetric and antisymmetric states. By design the samples have the relatively high Energy Shift (meV) ∆SAS of 0.32meV in sample A and 0.58meV in sample 0.0 0.5 1.0 1.5 2.0 2.5 B. Magneto-transportconfirms that both samples are in s) the quantum Hall side of the phase diagram as shown nit 1528.92 (a) u iinn sFaimg.p1le(bA).anTdot1a.l1s×he1e0t11decnms−it2ieisnasraem1p.2le×B1w01it1hcmm−o2- (arb. 1529.01 T = 60mK bilities larger than 106cm2/Vs. Inelastic light scatter- sity 1529.20 ing spectra are obtained in a back-scattering geometry n e 1529.30 withlightpropagatingalongthemagneticfield. Samples nt I aremountedina3He/4Hedilutioncryo-magneticsystem 1530.03 MR C0 L with optical windows, at a small tilt angle (20degrees) with respect to the incoming laser light. Accessible tem- q lB2 (b) (c) 2 q lB peratures are in the range 50mK–1.4K. For these mea- 1 1 surements the optical emission of a dye laser is tuned to 0 0 a frequency ωI close to the fundamental interband tran- 0.0 0.5 1.0 0.52 1.0 Energy Shift (meV) |M(q)| (arb. units) sitions of the DQW. Incident power densities are kept below <∼10−4W/cm2, and spectra are recorded using FIG. 2: (a) Inelastic light scattering spectra at νT=1 obtained in sample B for different incident photon energies (in a double monochromator, CCD multichannel detection meV). The band labelled L is luminescence. (b) Calculated and spectral resolution of 15µeV. dispersionofCDEmodesacrossthetunnelinggap. (c)Calcu- TheC0bandsshowninFig. 1(c)havesimilarenergies lated matrix element M(q) for inelastic light scattering. The and widths in the two samples, and also occur in spec- vertical dash-dotted line shows the energy of the q=0 CDE tra obtained at B=0. On this basis they are assigned to mode;thegrayareaindicatescontributionsoflargewavevec- q →0 CDE modes [21]. The structures labelled MR are tor modes with energies down to the roton minimum. The remarkably different in the two samples. They appear horizontal thin lines in (b) and (c) are at the magnetoroton (MR) wave vector. as a weak shoulder with a cutoff at 0.65meV in sample B, and as a sharp low-energy peak at 0.22meV in sample A. The spin wave (SW) at the Zeeman energy that enters in the expression of the dynamic structure E = 0.11meV also occurs in the low-energy spectra Z factor S(q,ω) of sample A. Figure 2(a) shows a resonant enhancement profile measured in sample B that reveals a characteris- |M(q)|2ω (q) ωΓ C S(q,ω)∝ , (1) tic outgoing resonance with the higher optical interband [ω2−ω2(q)]2+ω2Γ2 transitionoftheluminescencepeaklabelledL.Thespec- C tra from sample A display similar outgoing resonances. where Γ is the homogeneous broadening [10, 20]. This Figures 2(b) and (c) show the calculated dispersion TDHFA model defines a phase boundary for the insta- 2 ω (q) obtained within TDHFA and the |M(q)| factor bility at values ofd/l lowerthan the experimentalones C B 3 tonsandtheincompressible-compressiblequantumphase Energy Shift (meV) transition at ν =1. Figure 3(a) shows resonant inelas- W) 0.0 0.5 1.0 1.5 2.0 2.5 T ticlightspectraofCDEmodesinsampleAwithconven- sec 10 152M8.3R5 1529.30 1531.85 I (meV) (a) tional substraction of the background due to the laser s/ C0 and to the main magneto-luminescence. The results dis- nt u playclearlythe threebands ofCDE collectivemodes. In o c g -2 y (10 5 T ~ 60mK a1.d0d8itmioenV)totwthoeloCwDerE-eanterqgy≈ex0c(itCa0ti,odnassahr-edoctleteadrlylinseeeant. sit The lowest energy mode at 0.22meV is assigned to the n nte MR critical point in the dispersion. Its energy is much I lower than the MR in sample B. The MR in Fig. 3(a) 0 is extremely narrow, with a full width at half maximum q lB2 (b) (c) 2 q lB (FWHM)of∼0.06meV,whichisafactorofthreesmaller 1 1 thanthe FWHMofthe C0band. The peakat0.75meV, labelled∆ ,isthelargewavevectorCDEexcitation. The 0 0 g 0.0 0.5 1.0 0.52 1.0 ∆g and MR modes display a marked sensitivity on de- Energy Shift (meV) |M(q)| (arb. units) viations of magnetic field values from ν =1 in a manner T FIG. 3: (a) Light scattering bands of CDE in sample A at that is similar to the QH states. The assignments of the νT=1. The estimated background due to luminescence and C0 and MRbands in Fig.3(a) arealso supportedby the laser has been substracted. The incident photon energies ωI calculated dispersions shown in Fig. 3(b). are indicated in meV. ∆g labels the q → ∞ mode. (b) Cal- Significant insights are gained from a study of the q- culated dispersion of CDE modes; extrapolation to the long 2 dependence of |M(q)| . Comparison of the calculation wave-vector limit is shown as dotted line. (c) Calculated in- for sample A in Fig. 3(c) with that for sample B in elasticlightscattering“matrixelement”. Verticallines in(a) showthepeakpositionofCDEmodes. Thehorizontallinein Fig. 2(c) indicates that on reduction of ∆SAS, approach- (b) and (c) is at theMR wavevector. ing the phase transition, |M(q)|2 tends to peak sharply at the MR wavevector. These results suggest that, by predicting the softening and sharpening of the magne- by a factor of two. To correct for this discrepancy the toroton, the TDHFA, a leading-order many body calcu- d/lB parameters used in the calculations have been con- lation,providesaframeworktoanalyzemanifestationsof sistently adjusted to match the parameters in our two interactionsthat eventually leadto the soft-mode driven 2 samples. The matrix element |M(q)| acts as an oscilla- quantum phase transition [22]. torstrengthforinelasticlightscatteringbythecollective The interactions that interpret the softening and nar- excitations. Atthelowestorder,thelight-scatteringcross rowing of the MR mode in sample A would eventually section is proportional to the product of S(q,ω) and a drive the incompressible-compressible transition upon factor that incorporates resonant enhancements and op- further reductionof∆ . These interactionsarerelated SAS tical matrix elements. S(q,ω) is used to evaluate the to the excitonic Coulomb term that creates the roton in intensities of inelastic light scattering by CDE modes of theCDEmodedispersion. Itisconceivablethatsuchex- different wave vectors. In this evaluation the extent of citonic binding increases at lower values of ∆ due to SAS breakdown of wavevector conservation is treated as in enhanced overlap between the single-particle wavefunc- Ref. [20]. tions of symmetric and antisymmetric states. The comparison of spectra of sample B in Fig. 2 with TheTDHFAinterpretstheenergies,linewidthsandin- the TDHFA calculation confirms the assignment of the tensities of MRs in the light-scattering spectra. The re- band labelled C0 at 1.13meV as the long wavelength sults supportthe picture thatthe groundstate ofthe bi- CDE shifted above ∆ by dynamical many-body con- layersatν =1 evolvestowardsabroken-symmetrystate SAS T tributions. This peak is observed at temperatures of up caused by the collapse of the energy of tunneling excita- to 1.4K and over a relatively broad range of ωI and ex- tions [5, 8, 10]. The marked narrowing of the MR band plored magnetic fields (0.7<νT <1.2). The low-energy and its interpretation within the TDHFA suggest that shoulder with a cutoff at MR is observed only in a very the transition might be characterized by a roton wave narrow range of ωI and for B values very close to νT=1. vector qR ∼lB−1. Our results also support a scenario Thisstructureisassignedtoresonantinelasticlightscat- in which the instability is associated with the conden- teringprocesseswithbreakdownofwavevectorconserva- sation of neutral excitons [14, 15, 16, 23]. The exciton tion. Thelowestmeasuredenergyinthisrelativelybroad fluid would be linked to large densities of extremely low structure represents the magnetoroton in the dispersion energymagnetorotonmodesoftheincompressiblephase. of CDE modes. The MR temperature dependence measured in sam- The marked softening of the MR mode in sample A ple A suggests that a highly correlated QH state may seen in Fig. 1(c) suggests links between soft magnetoro- occur near the phase boundary. Figure 4 shows two rep- 4 could play roles in such transition. SW MR V)0.20 In conclusion, we observed magnetorotons in the dis- e m persions of low-lying charge-density excitations in ferro- M (0.15 magneticelectronbilayersatν =1. Softandsharpmag- H T nits) FW0.10 ntoe-tcoormotpornessseixbilsetqinuacnlotsuemprpohxaimseittyratnostithieonin.cTomheprkeesysibfelae-- u b. 0.05 tures of the experiments have been interpreted within ar 0.0 0.2 0.4 0.6 0.8 time-dependent Hartree-Fock approximation. These re- y ( T (K) sit sults suggest direct links between transitions in the elec- n tron quantum ground-state and the low-lying dispersive e nt T=0.2K collective modes. I We are grateful to S. Das Sarma, E. Demler, S. M. Girvin, S.H. Simon and D.W. Wang for critical reading T=0.6K of the manuscript and significant suggestions. Partial (a) support from CNR (Consiglio Nazionale delle Ricerche), INFM/E (Istituto Nazionale per la Fisica della Materia, 0.2 0.4 0.6 Energy Shift (meV) section E) and MIUR is also acknowledged. FIG. 4: Open circles show the temperature dependence of light scattering spectra in sample A. The background substraction is as in Fig. 3(a). The solid lines are results of fits with two lorentzians. Inset: temperature dependence of the FWHM for the spin-wave (SW) peak at the Zeeman en- [1] S.DasSarmaandA.Pinczuk,eds.,PerspectivesinQuan- ergy(blackemptysquares)andtheMRpeak(grayemptycir- tum Hall Effect (Wiley, NewYork,1997). cles). Thesolidlinesareguidesfortheeyesandtheerrorbars [2] R.E. Prange and S.M. Girvin, eds., The Quantum Hall arestandarddeviationsforresultsondifferentmeasurements Effect (Springer-Verlag, New York,1987). and with different background subtractions. [3] S. M. Girvin et al., Phys.Rev.B 33, 2481 (1986). [4] C. Kallin and B. I. Halperin, Phys. Rev. B 30, 5655 (1984). resentativespectrainwhichbothSWandMRmodesare [5] A.H.MacDonaldetal.,Phys.Rev.Lett.65,775(1990). observed. The linewidth of the two excitations have the [6] Y. N. Joglekar and A. H. MacDonald, Phys. Rev. B 65, 235319 (2002). verydifferenttemperaturedependenceshownintheinset [7] H. A.Fertig, Phys.Rev. B 40, 1087 (1989). toFig.4. TheFWHM’sareobtainedfromLorentzianfits [8] L. Brey, Phys. Rev.Lett. 65, 903 (1990). (solid lines in Fig. 4). For temperatures above 0.8K the [9] R. K. Kamilla and J. K. Jain, Phys. Rev. B 55, R13417 MR mode can no longer be observed and minor changes (1997). occurinthe SWpeak. Similartemperaturedependences [10] L. Brey, Phys. Rev.B 47, 4585 (1993). characterize the q → ∞ (∆ ) mode and the magneto- [11] G.S.Boebingeretal., Phys.Rev.Lett.64,1793(1990). g transport data. The characteristic temperature is here [12] Kun Yanget al., Phys.Rev. B 54, 11644 (1996). [13] S. Q. Murphyet al., Phys.Rev.Lett. 72, 728 (1994). muchsmallerthan∆ andmorethanafactorofthreebe- g [14] I. B. Spielman et al., Phys.Rev. Lett. 84, 5808 (2000). lowthe MRenergyinsampleA.The q ∼0(C0)mode is [15] I.B.Spielmanetal.,Phys.Rev.Lett.87,036803(2001). temperature-independent up to 1.4K. A similar anoma- [16] M. Kellogg et al., Phys.Rev.Lett. 88, 126804 (2002). lous behavior observed in activated magneto-transport [17] V. Pellegrini et al., Phys. 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