Interaction-induced shift of the cyclotron resonance of graphene using infrared spectroscopy
E. A. Henriksen, P. Cadden-Zimansky, Z. Jiang, Z. Q. Li, L.-C. Tung, M. E. Schwartz, M. Takita, Y.-J. Wang, P. Kim, H. L. Stormer
IInteraction-induced shift of the cyclotron resonance in graphene via infrared spectroscopy
E. A. Henriksen, ∗ P. Cadden-Zimansky,
1, 2
Z. Jiang, Z.Q. Li, L.-C. Tung, M. E. Schwartz, M. Takita, Y.-J. Wang, P. Kim, and H. L. Stormer
1, 4, 5 Department of Physics, Columbia University, New York, New York 10027, USA National High Magnetic Field Laboratory, Tallahassee, Florida 32310, USA School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA Department of Applied Physics and Applied Mathematics,Columbia University, New York, New York 10027, USA Bell Labs, Alcatel-Lucent, Murray Hill, New Jersey 07974, USA (Dated: October 26, 2018)We report a study of the cyclotron resonance (CR) transitions to and from the unusual n = 0 Landau level(LL) in monolayer graphene. Unexpectedly, we find the CR transition energy exhibits large (up to 10%) andnon-monotonic shifts as a function of the LL filling factor, with the energy being largest at half-filling of the n = 0 level. The magnitude of these shifts, and their magnetic field dependence, suggests that an interaction-enhanced energy gap opens in the n = 0 level at high magnetic fields. Such interaction effects normally havelimited impact on the CR due to Kohn’s theorem [W. Kohn, Phys. Rev. , 1242 (1961)], which does not applyin graphene as a consequence of the underlying linear band structure. PACS numbers: 78.66.Tr; 71.70.Di; 76.40.+b
The intriguing electronic properties of graphene in a strongmagnetic field were first demonstrated by the observation ofan extraordinary ‘half-integer’ quantum Hall effect [1–5], re-sulting from the existence of an unusual n = 0 Landau level(LL). Later evidence from transport measurements indicatesthat the four-fold degeneracy of this n = 0 level is completelylifted in high magnetic fields [6, 7], and a gapped state forms[8–11]. This state and the origin of degeneracy-breaking inthe n = 0 level is a matter of intense theoretical study, withelectron-electron interactions expected to play a critical role[12]. However, since the charge transport in earlier experi-mental works is either local, or dominated by the sample edge,direct access to the intrinsic properties of graphene in the bulkremains limited. Additionally, the diverging resistance nearhalf-filling of the n = 0 level prohibits investigation of gappedstates by thermal activation experiments. In this Letter, wereport a study of the n = 0 LL utilizing infrared (IR) mag-netospectroscopy, which is sensitive to the cyclotron orbitsof charge carriers throughout the entire graphene sheet. Asthe LL filling factor is changed, we observe unexpected andsizable shifts in the cyclotron resonance (CR) transition ener-gies. We interpret this as due to electron-electron interactionswhich create a gap in the n = 0 LL, thereby affecting theenergies of CR transitions to and from this level.In a perpendicular magnetic field, B , the LL structure ofgraphene has an unusual energy dependence on both B andthe LL index, n , described by [13–15]E n = sgn ( n ) (cid:112) | n | (cid:126) ˜ c/l B . (1)Here (cid:126) is Planck’s constant, ˜ c ∼ × m/s the band ve-locity, and l B = (cid:112) (cid:126) / ( eB ) the magnetic length. The index n runs over both positive and negative integers representingelectron and hole-like states, and includes a field-independent n = 0 LL which is located at zero energy. This level, in com-bination with a four-fold level degeneracy due to the two-fold s ( e / h ) -40 0 40V g s ( e / h ) R ( k ‰ ) -40 0 40 V g ( V )25(cid:181)m cd ba R xx R xy s xx s xy B = 18 T
FIG. 1: (a) Optical microscope image of sample A. (b) MeasuredHall (blue) and longitudinal (red) 4-wire resistances of sample A.(c) Calculated Hall (black) and longitudinal (green) conductivities,showing a Hall plateau forming at ν = 0 . (d) Schematic of ex-perimental setup showing IR light focused through graphene. Thevarying bolometer resistance tracks changes in the transmitted light. electronic spin and pseudo-spin (valley) degeneracies, leadsto a unique sequence of LL filling factors, ν = ± , ± .... .Here ν = 2 πn s l B , where n s is the carrier density. In con-trast to conventional 2D systems, in which even the lowest LLexperiences an energy shift with B , in graphene the n = 0 LL remains put at zero energy, is shared by the conductionand valence bands and, at charge neutrality for vanishing ν , isfilled half with electrons and half with holes.Motivated by recent experiments [6–11], we have per-formed a careful study of LL transitions to and from the n = 0 a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b level via CR [16–21]. The five samples, A-E, used in this workconsist of monolayer graphene exfoliated from Kish graphiteonto Si/SiO wafer substrates [22], with Cr/Au contacts de-fined by standard thin film processing for in situ measurementof sample transport properties. Sample sizes and mobilities(at a hole density of 1.7 × cm − ) range from 330-1,100 µ m and 3,000-17,000 cm /(Vs), respectively. The lightly-doped Si substrates transmit mid-IR light, but remain suffi-ciently conductive for use as a back gate to vary the carrierdensity. Samples are mounted at the focus of a parabolic mir-ror and cooled via He exchange gas in a probe inserted intoa liquid He cryostat. Broadband IR light from a Bruker IFS66 Fourier transform spectrometer reaches the graphene viadirect optics, and the transmitted light is detected by a com-posite Si bolometer placed immediately below the sample; seeFig. 1 (d). The samples are wedged to suppress Fabry-Perotinterference. We determine the CR spectrum at constant B field as in Ref. [17], recording IR transmission through thesample at two gate voltages, V g and V b . The change in trans-mission, ∆ T/T = 1 − T(V) / T(V b ), is measured over a rangeof V g ( ν ). The background scans, T(V b ), are at ν = ± .In Fig. 1 (a) we show an optical microscope image of ourhighest mobility monolayer graphene device, sample A, withmobility µ = 17 , cm /(Vs). The Hall and longitudinalresistances for this sample, and the calculated conductivities,are shown as a function of V g in Fig. 1 (b) and (c) respectively.A plateau in the Hall conductivity is clearly evident near ν =0 , as in Ref. [6, 8, 9].In Fig. 2 (a) we plot the change in IR transmission, ∆ T / T,through sample A for a range of ν at a field of B = 18 T. Frombottom to top, the filling factor decreases from ν = +4 / to ν = − / in steps of ∆ ν = − / . Physically, this amountsto shifting the Fermi level, E F , through a fixed LL spectrumstarting just above half-filling of the n = 1 LL and endingbelow half-filling of the n = − LL. The broad peaks corre-spond to the resonant excitation of carriers between the lowestLLs, via the n = − → and/or the → transition. Thesharper spikes at 157 meV and 196 meV are due to harmon-ics of 60 Hz electrical noise. Additional small features resem-bling peaks or shoulders on the lineshape are visible in severaltraces. These are suggestive of splittings of the CR line. Suchan observation would be intriguing and underscores the needfor higher quality samples, but here we focus on the main CRpeak energies traced by the dashed line in Fig. 2 (a).Closer inspection of the CR traces reveals a distinct asym-metry in the lineshape with the high-energy side falling offmore slowly than the low-energy side. This distortion ofthe Lorentzian resonance originates in multiple reflections be-tween the graphene and the wedged Si/SiO substrate [23].To extract the physical parameters for each resonance, wehave performed detailed curve fits by modeling the transmis-sion through a multilayer system including substrate phononmodes and multiple reflections/refractions at the graphene-SiO and SiO -Si interfaces. The dielectric properties of thesubstrate used in the fits are determined experimentally by acombination of IR spectroscopy and ellipsometry [24]. Cal- ν νννν ν FIG. 2: (a) The change in transmission, ∆ T / T, for CR traces inmonolayer graphene at fixed field B = 18 T over a range of ν .Traces are offset for clarity. The bottom trace is ν = 4 / ; ν ofsuccessive traces decreases by / . The dashed line is a guide to theeye, marking the peak energies determined by curve fits. (b) Exam-ple of a fit (solid red line) to the CR line shape at ν = 1 / using amultilayer IR transmission model. (c) CR peak energies plotted vs. ν . The area of the markers represents the relative intensity. culated fits are compared to the data in successive iterations,as the model resonance energy, intensity and width are varieduntil best fits are achieved as determined by inspection. Anexample is shown in Fig. 2 (b) for the ν = +1 / trace. Wenote the peak energies from this fitting fall at energies slightlybelow the intensity maxima [25].Far from remaining constant as the occupancy of the n =0 LL is changed, the CR exhibits a marked non-monotonicdependence on ν , resulting in a nearly symmetric “W”-shapeas shown in Fig. 2 (c). Two sharp minima in the CR energyoccur at ν = ± , with a more rounded maxima at ν = 0 thatis nearly 10% higher than the values at | ν | = 2 . We note theincrease in energy occurring for | ν | > is accompanied by aloss in intensity, with the peaks fading away beyond | ν | = 4 .This is due to the onset of the next LL transition, n = 1 → or − → − , as E F is shifted farther from charge neutralityand either the n = +1 or − LL becomes partially occupied.The considerable shifts seen in Fig. 2 (c) are not reflected inthe interband transitions that are excited simultaneously, albeitat higher energies. In Fig. 3 we show the CR data for ν = ± and in sample B over an extended energy range whichalso includes the first degenerate pair of interband transitions, n = − → and − → . The transitions responsible forthe two resonances in each trace are illustrated schematicallyin the upper left inset to Fig. 3. The interband peak (at 440meV) shifts by less than 1% as ν is changed. In contrast, andconsistent with the behavior seen in Fig. 2, the lower energypeaks (at 165-180 meV) show the same behavior as in sampleA, with a strong upshift ( ≈ %) in energy at ν = 0 .The increase in CR energy of the lowest LL transitions at ν = 0 depends on both B and the sample quality. Figure4 shows the B field dependence in all five samples at fieldsup to 31 T. Here we characterize the overall magnitude of the ν ννν ν FIG. 3: Two CR peaks visible in ∆ T / T traces from sample B, at B = 18 T. The underlying transitions are n = 0 → and/or n = − → near 170 meV (blue arrows), and n = − → +2 and n = − → +1 at 440 meV (red arrows). Traces are offset forclarity. Left inset: LL schematic showing transitions at ν = +2 ,when E F lies between the zeroth and first LLs. Right inset: proposedLL schematic at charge neutrality ( ν = 0 ), showing a gap in the n = 0 LL and the corresponding lowest energy CR transitions. shift, ∆ E , as the difference in CR energies at ν = +2 and ν = 0 . Open symbols show data from the highest mobilitysample, A, and filled symbols are data from samples B-E. Aclear trend of increasing ∆ E (up to 23 meV) with increasing B emerges. We find ∆ E is correlated with device quality:sample A shows the largest shifts, while lower mobility sam-ples have consistently smaller ∆ E . Although the broader res-onances in the lower mobility samples lead to wide error bars,in sample A the uncertainty is sufficiently small to explore afunctional dependence between ∆ E and B . In the inset toFig. 4, we plot the CR energies of the lowest LL transitionsvs. √ B at two fields for sample A. As suggested by Eq. 1,these data are well described by straight lines constrained topass through zero and differing only in slope, implying thata √ B dependence is followed in both cases. Further experi-ments in higher quality samples are needed to determine theprecise field dependence of ∆ E .We now discuss possible origins of the observed CR shiftat ν = 0 . In the basic model of non-interacting carriers, theCR energy is the difference of the initial and final LL energiesfrom Eq. 1. For | ν | < , two degenerate transitions occur, n = − → and → . For < ν < the n = 1 → alsooccurs, but with an energy reduced by a factor of √ thatlies outside our experimental window. Thus we expect to seeonly a single CR peak with energy independent of ν , whichclearly is not observed.Disorder certainly plays a role, as seen in Fig. 4. We note asimple model where LLs are broadened only by short-rangedscattering predicts an upshift in the CR for half-filled LLs[26], similar to our observation. However, one would expectsuch a CR shift to increase in more disordered samples, incontrast to the observed increase in ∆ E with increasing mo- C R E n e r g y ( m e V ) ˆB ( ˆT ) D E ( m e V ) B ( T ) ˆ8 ˆ18 n =0 n =+2 FIG. 4: B field dependence of the CR shift, ∆ E, between ν = +2 and , as a function of B for five samples. Open symbols are from thehigh mobility ( µ = 17 , cm /(Vs)) sample A resonances shownin Fig. 2; filled symbols show lower mobility samples ( µ ≤ , cm /(Vs)). The dotted line is the bare Zeeman splitting, µ B B . Er-ror bars are estimated by varying the best fit parameters. Inset: ν = 0 and +2 CR energies of sample A at two fields, plotted against √ B .Dashed lines are least-squares linear fits through zero. bility. Also, the disorder potential in our samples is likelydominated by charged impurities [27]. Screening may affectthe CR transitions as well. However, we expect increasedscreening to soften the CR [28], making for an M-shaped ν -dependence rather than the W-shape of Fig. 2 (c); or else tonarrow the resonance linewidth in half-filled levels [29] whichruns counter to the bulk of our data. In this context, we notethe broadening may be due to the linear dispersion alone [30].Single particle spin-related effects are also an unlikely ori-gin for our observations. Even at 31 T the bare Zeeman energyamounts to only 3.6 meV, several times smaller than the ob-served shifts (see Fig. 4). Also, as spin is conserved in theCR transition, simple Zeeman effects should not be observed.There remains the possibility of a ν -dependent g-factor en-hancement. Large enhancements have been seen in other 2Dsystems [31]. However, these appear only in transport datawith the interpretation invoking many-body effects.The linear dispersion of graphene implies the CR can be in-fluenced by many body effects [32]. In our experiments, theclear dependence of the CR energies on ν – and hence E F –and the magnitude of the shifts suggest that interactions atthe Fermi surface are an important ingredient. The contribu-tion of electron-electron interactions to the dispersion of low-lying inter-LL excitations has been calculated for graphene inRef. [33], and also in Ref. [32] where the focus is restrictedto those modes that contribute to CR. In these works, inter-actions are generally found to increase the excitation ener-gies. In particular, the calculations show that when ν > ( ν < − ), the n = 0 → ( n = − → ) transition de-velops a splitting where the higher energy peak continues toincrease in energy with increasing (decreasing) ν . Qualita-tively, this can describe the CR shifts we observe at | ν | > ,if we assume the splitting is masked in our data by the broad-ened lineshape. On the other hand, neither work predicts achange in the energy of optically active modes for | ν | < ,and so fail to capture the large upward shift observed in ourCR data near ν = 0 . Yet in spite of this, interactions seem tobe the origin for our observations: although at 18 T the bareCoulomb energy e / ( (cid:15)l B ) ≈ meV is much larger than theshifts we observe in Fig. 2, no other energy scale is available.Zhang et al have observed quantum Hall plateaus at ν = 0 and ± which they interpret as interaction-induced splittingsof the n = 0 level [6]. Naively, one could view our results asthe consequence of such a splitting, because when the Fermilevel passes through the n = 0 LL, the presence of a gap leadsto shifts in the transition energies to neighboring LLs; see theright inset to Fig. 3. This picture, while certainly an oversim-plification, is nonetheless appealing: it easily accounts for the ν - and B -dependence of the CR energies, and is consistentwith the fact that only those transitions involving the n = 0 LL are strongly affected.As to similar experiments, the CR data on epitaxialgraphene [21] have not yielded similar behavior. Previousexperiments on GaAs and Si/SiO
2D systems have shownunexpected shifts and splittings of the CR bearing some re-semblance to our data [34–36]. However in detail the com-parisons break down. An increased effective mass of the car-riers (decreased CR energy) seen for full LLs in 2D GaAswas attributed to a particular impurity doping, to band non-parabolicity, or to ν -dependent screening, all of which are un-likely to be the source of our observations in graphene. Theorigin of the anomalous CR peaks and splittings in the Si data[36] was never resolved, but in general these features had no B dependence, with higher quality samples showing weakeranomalies, both in contrast to our data.In conclusion, we have observed sizable shifts of the CRenergy as a function of the LL filling factor and applied B field for transitions involving the n = 0 level in monolayergraphene. We suggest our results indicate the opening of aninteraction-induced gap in the bulk of the graphene at ν = 0 which increases with increasing B field strength, and is de-tectable via CR as Kohn’s theorem does not apply in graphene.We wish to thank M. Koshino, Kun Yang, Y. Barlas, A.MacDonald, S. Das Sarma, M. Fogler, M.-Y. Chou and J.Checkelsky for helpful discussions. This work is supportedby the DOE (DE-AIO2-04ER46133, DE-FG02-05ER46215and DE-FG02-07ER46451), the NSF under DMR-03-52738and CHE-0117752, ONR (N000150610138), NYSTAR, theKeck Foundation, the Microsoft Project-Q, and the SRC-NRI-MIND. The IR work was performed at the National High Magnetic Field Laboratory, which is supported by NSF Co-operative Agreement No. DMR-0654118, by the State ofFlorida, and by the DOE. We thank J. Jaroszynski, S. Han-nahs, J.-H. Park, and E.C. Palm for experimental assistance. ∗ Electronic address: [email protected][1] K. S. Novoselov et al , Nature , 197 (2005).[2] Y. Zhang et al , Nature , 201 (2005).[3] Y. Zheng and T. Ando, Phys. Rev. B , 245420 (2002).[4] V. P. Gusynin and S. G. Sharapov, Phys. Rev. Lett. , 146801(2005).[5] N. M. R. Peres, F. Guinea, A. H. Castro Neto, Phys. Rev. B ,125411 (2006).[6] Y. Zhang et al , Phys. Rev. Lett. , 136806 (2006).[7] Z. Jiang et al , Phys. Rev. Lett. , 106802 (2007).[8] J. G. Checkelsky, L. Li, and N. P. Ong, Phys. Rev. Lett. ,206801 (2008).[9] A. J. M. Giesbers et al , cond-mat/0904.0948.[10] G. Li, A. Luican, and E. Andrei, Phys. Rev. Lett. , 176804(2009).[11] D. L. Miller et al , Science , 924 (2009).[12] For a review, see K. Yang, Sol. St. Comm. , 27 (2007).[13] J. W. McClure, Phys. Rev. , 666 (1956).[14] G. W. Semenoff, Phys. Rev. Lett. , 2449 (1984).[15] F. D. M. Haldane, Phys. Rev. Lett. , 2015 (1988).[16] V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, Phys. Rev.Lett. , 157402 (2007).[17] Z. Jiang et al , Phys. Rev. Lett. , 197403 (2007).[18] M. L. Sadowski et al , Phys. Rev. Lett. , 266405 (2006).[19] R. S. Deacon et al , Phys. Rev. B , 081406(R) (2007).[20] E. A. Henriksen et al , Phys. Rev. Lett , 087403 (2008).[21] M. Orlita et al , Phys. Rev. Lett. , 267601 (2008).[22] K. S. Novoselov et al , PNAS , 10451 (2005).[23] G. Abstreiter et al , Phys. Rev. B , 2480 (1976).[24] Z. Q. Li et al , Nat. Phys.
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