doi: /PhysRevB

Transcription

1 doi: /PhysRevB

2 PHYSICAL REVIEW B 78, Cylotron radiation and emission in graphene Takahiro Morimoto, 1 Yasuhiro Hatsugai, 2,3 and Hideo Aoki 1 1 Department of Physis, University of Tokyo, Hongo, Tokyo , Japan 2 Institute of Physis, University of Tsukuba, Tsukuba , Japan 3 Department of Applied Physis, University of Tokyo, Hongo, Tokyo , Japan Reeived 27 Deember 2007; published 27 August 2008 Peuliarity in the ylotron radiation and emission in graphene is theoretially examined in terms of the optial ondutivity and relaxation rates to propose that graphene in magneti fields an be a andidate for realizing the Landau-level laser, proposed deades ago H. Aoki, Appl. Phys. Lett. 48, DOI: /PhysRevB PACS numbers: Di, b I. INTRODUCTION There has been an inreasing fasination with the physis of graphene, a monolayer of graphite, as kiked off by the experimental disovery of an anomalous quantum Hall effet QHE. 1,2 The fasination omes from a ondensed-matter realization of the massless Dira-partile dispersion on lowenergy sales in the honeyomb lattie, 1,3 6 whih is behind all the peuliar properties of graphene. In magneti fields this appears as unusual Landau levels, where i the Landau levels = n, n: Landau index are unevenly spaed, ii the ylotron frequeny =2e/v F B is proportional to B rather than to B, and iii there is an extra Landau level right at the massless Dira point E=0, whih is outside the Onsager s semilassial quantization. 7 While various transport measurements as exemplified by the QHE have been extensively done, optial properties have also begun to be measured. For example, Sadowski et al. 8 have performed a Landau-level spetrosopy for a large graphene sample. Inter-Landau-level transitions are observed at multiple energies, whih are due to a peuliar optial seletion rule n n1 as opposed to the usual n n1, as well as to the uneven Landau levels. Now, if we look at the QHE physis, ylotron emission from the QHE system in nonequilibrium has been one important phenomenon. Experimentally, this typially appears as a strong ylotron emission from the hot spot: a singular point in a Hall-bar sample where the onvergene of eletri lines of fore puts the eletrons out of equilibrium. 9 For a bulk nonequilibrium effet, one of the present authors proposed theoretially a Landau-level laser for nonequilibrium QHE systems. 10 The basi idea is simple enough: we an exploit the unusual oalesene of the energy spetrum into a series of line spetrum Landau levels to realize a laser from a spontaneous emission if we an make a population inversion, where the photon energy =ylotron energy in this ase is tunable and falls on the terahertz region for B10 T. However, the most diffiult part is the population inversion, sine if we, e.g., optially pump the system, the exitation would go up the ladder of equidistant Landau levels indefinitely. This has motivated us to put a question Fig. 1: will the graphene Landau levels, with uneven spaing among their peuliarities, favor in realizing suh a population inversion? In this Brief Report, we show that this is indeed the ase by atually alulating the optial ondutivity as well as the relaxation proesses. The message then is that graphene is a andidate for the Landau-level laser. II. OPTICAL CONDUCTIVITY IN GRAPHENE Low-energy physis around the Fermi energy in graphene is desribed by the massless Dira Hamiltonian, 4 H 0 = v F , where v F is the veloity of the Dira eletron at E F, x i y with =p+ea, A is the vetor potential representing a uniform magneti field B=rotA, and the 44 matrix is spanned by the hirality and K,K Fermi points. In magneti fields the energy spetrum is quantized into Landau levels, n = sgnn n, = 2 v F = v F 2eB, for a lean system, where n=0,1,... is the Landau index and = /eb is the magneti length. Here we onsider realisti systems having a disorder with the self-onsistent Born approximation SCBA introdued by Ando 4,11 to alulate the optial ondutivity. FIG. 1. Color online Cylotron absorption green and emission red proesses shematially depited for the ordinary a and graphene b quantum Hall systems. Blak lines represent the band dispersion, while blue lines the Landau levels /2008/787/ The Amerian Physial Soiety

3 PHYSICAL REVIEW B 78, DOS (D hω /l 2 ) Energy (ε/ hω ) FIG. 2. A typial density of states for graphene for an intermediate disorder / =0.25 here. The optial ondutivity is given by = e2 d i f Trj Im Gj G + + G + Trj G G j Im G, 4 where,=x,y, f as the Fermi distribution, and G =Gi with a positive infinitesimal. For Green s funtion G, with the self-energy n = 2 4 n sgnn n n 1 in the SCBA, the Landau-level broadening is given by 2 4 =n 0V 2 0 /4 2 if we assume for simpliity a random potential arising from short-ranged satterers V = i V 0 r r i. The light absorption rate is then related to the imaginary part of the dieletri funtion =1 +i xx / 0 so we an look at Re xx. In order to disuss the optial ondutivity in graphene, we need the urrent matrix elements aross Landau levels. The eigenfuntions of the Hamiltonian 1 ditate an unusual seletion rule n n= 1 in plae of the ordinary n n = 1 with 11 j x n,n = vf C n C n sgnn n 1,n + sgnn n+1,n, j y n,n = ivf C n C n sgnn n 1,n sgnn n+1,n, where C n =1 n=0 or 1/ 2 otherwise. We have numerially obtained the Green s funtion and the optial ondutivity. While in usual ases the broadened Landau levels are simultaneously merged or separated as is varied, there is a striking differene for graphene, where the Landau levels n are unevenly spaed, so that the broadened Landau levels overlap to a lesser extent as we go to the entral one n 0 as typially depited in Fig. 2. Namely, for an intermeditate value of /, only the n=0 Landau level stands alone while the other levels form a ontinuous spetrum. We now look at the result for the optial ondutivity in Fig. 3 for the Fermi energy at F =0 energy for the Dira point, where eah resonane peak an be assigned to an allowed transition with the seletion rule Eq. 5. The largest peak around / =1 orresponds to the transition between n=0 1, while the peaks at higher frequenies 5 FIG. 3. Color online Optial ondutivity against is shown for temperatures k B T/ =0 2.0 for a fixed value of / =0.25 with F =0. ome from the transition aross the Fermi energy n n1. If we turn to the temperature dependene in the figure, we notie a peuliar phenomenon: there is a peak in the region / 1 that grows, rather than deays, for higher T. We an identify this as oming from the unusual Landau levels in graphene: as T is raised with the Fermi distribution funtion beoming longer tailed higher Landau levels begin to be oupied, whih enables the transitions among higher Landau levels n n1 to take plae. While this would not ause other lines to appear for equidistant Landau levels, this does so for the unequally spaed Landau levels n for / 1 transitions. So we an identify this property as one hallmark of the massless Dira dispersion. Previously, the optial ondutivity has been obtained by Gusynin et al., 12,13 who have derived the analytial expression for the optial ondutivity, but the self-energy from the disorder was set to a onstant while we have alulated the self-energy self-onsistently with SCBA. Sadowski et al. 8 also presented a similar expression for the ondutivity with a onstant self-energy as well. The present result qualitatively agrees with these results, but the findings here are, first, the full dependene on the k B T/, inluding the growing of low-frequeny peaks at low temperatures. Seond, we point out that the situation as depited in Fig. 2 should be interesting for the ylotron resonane and emission in nonequilibrium situations indued by, e.g., an optial pumping with laser beams. Namely, the eletrons exited to higher energies will relax down to the n=1 level aross the ontinuum spetrum, so that the population inversion aross n=0 and n1 should be easier to be realized. III. RELAXATION PROCESSES To quantify the above idea, we have to onsider the relaxation proesses whih should ontrol the population inversion. For the ordinary quantum Hall systems, the relaxation proesses have been extensively disussed. Speifially, Chaubet et al. 14,15 disussed dissipation mehanisms, where spontaneous photon radiation and oupling with phonons are examined on the basis of Fermi s golden rule. Other dissipation proesses suh as eletron-eletron or impurity satter

4 ings, whih onserve the total energy, do not ontribute to inter-landau-level proesses in the absene of external eletri fields while Chaubet et al. have foused on effets of finite eletri fields in the QHE breakdown where inter- Landau-level proesses are involved. So we extend the disussion by Chaubet et al. to relaxation proesses in graphene. We first estimate the effiieny of the photon emission with Fermi s golden rule, W i f = 2 ih intf 2 f i. Here i i is the wave funtion energy in the initial state while f stands for the final states and H int is the interation Hamiltonian between the eletromagneti field and eletrons, H int = e 2 0 V e ik r e va + e ik r e va, where 0 is the permittivity of the vauum, V is the volume of the spae where photons exist, is the frequeny of the light with mode and the wave number k, e is the polarization vetor, a is the annihilation operator of light, and v is the veloity of eletrons. When the wavelength of light is muh larger than the ylotron radius, as is usually the ase, we an drop the e ik r term and we have W i f = 2 V 2 e 2 d V ivf2 + f i =4 ivf 2, 7 where is the veloity of light, =e 2 /4 0 is the finestruture onstant, and we put = f i to be the ylotron energy. A peuliarity of graphene appears in the urrent matrix element Eq. 5, for whih the rate of the spontaneous emission, with nvn+1=c n C n+1 v F for graphene plugged in, reads W graphene n+1 n =2 v F v F 2 n =0, 6 2 n 0. 8 This expression, another key result here, shows that the spontaneous emission rate depends linearly on the ylotron energy and quadratially on the Fermi veloity. This is in sharp ontrast to the ordinary QHE systems suh as the twodimensional eletron gas 2DEG realized at, e.g., GaAs/ AlGaAs, interfaes. In this ase the veloity matrix element nvn+1 2 =n+1/2m should be plugged in Eq. 7, whih yields GaAs W n+1 n PHYSICAL REVIEW B 78, =2n +1 m 22. This reveals a dramati differene between graphene and usual 2DEG, where the emission rate in the latter is proportional to the square of the ylotron energy. We an quantitatively realize the differene: the ylotron energies are = eb/m 1.7 mev GaAs, v F 2eB 37 mev graphene, where on the right-hand sides we put B=1 T, adopted the value of graphene Fermi veloity v F = m/s, 8 and the GaAs effetive mass m 0.067m e. Hene the ylotron energy in graphene is an order of magnitude larger, whih reflets B for the Dira dispersion, while the energy is usually proportional to B. If we plug these in Eqs. 8 and 9, we end up with W i f B s 1 GaAs, B s 1 graphene, where the right-hand sides are for B=1 T. A onspiuous differene, B 2 in the former and B, should sharply affet the behavior and the spontaneous photon emission rate is orders of magnitude enhaned in graphene in moderate magneti fields as in the above numbers quoted for B=1 T. This indiates that the present system is indeed favorable for a realization of the envisaged Landau-level laser. Now, the dissipation proess whih ompetes with the photon emission is the phonon emission proess, whih has been disussed for the onventional QHE systems, in the ontext of the breakdown of the QHE. 14 The phonon emission rate is also obtained from Fermi s golden rule if we replae the eletron-light interation with the eletronphonon interation. If we first onsider aousti phonons, the dissipation rate is proportional to the extent of the overlap between initial and final wave funtions both in usual and graphene QHE systems, whih yields a fator e q2, with q the phonon wave number and the magneti length. In usual QHE systems the ylotron energy is 1 mev and the magneti length = /eb30 nm for B=1 T, while the aousti-phonon wave number is 1 Å 1, so that the overlap fator is exponentially small. The situation is similar in graphene sine the magneti length = /eb is the same. So the aousti-phonon emission should be negligible in graphene as well as in weak eletri fields. When the applied laser eletri field is so intense 1 kv/m that the Landau levels are distorted and the overlap fator grows, the phonon emission may begin to ompete with the photoemission. Are there any other fators that distinguish graphene from 2DEGs? In this ontext, we an note that Chaubet et al. have further pointed out the following. In an eletron system onfined to a 2D, a wave funtion has a finite tail in the diretion normal to the plane and the phonon emission is enhaned through the oupling of the tail of the wave funtion and perpendiular phonon modes, whih propagate normal to the 2D system in the substrate. 15 In this way the phonon emission an ompete with the spontaneous emission in usual

5 QHE systems. In ontrast, a graphene sheet is an atomi monolayer with only a loose oupling with the substrate. We an also onsider a ombined effet of sattering of eletrons by aousti phonons and impurity sattering, whih may ompensate for the momentum transfer q of phonons, and hene the overlap fator e q2. 16,21 To be preise, graphene itself should have phonon modes that inlude the out-ofplane modes and their effets are interesting future problems. As for optial phonons, their energies are known to be higher than 100 mev for q0 in graphene 17 so that optial phonons do not ontribute to the dissipation for B a few tesla with 40 mev. Overall, we onlude that the dissipation due to aousti phonons will be small in graphene in the weak eletri-field regime. When the pumping laser intensity is not too strong to invalidate the present treatment but strong enough for the population inversion, the present reasoning should apply and we an expet effiient ylotron emissions from graphene. Entirely different, but interesting, is the problem of Anderson loalization arising from the disorder. While this is out of sope of the present work, we an expet deloalized states, with diverging loalization length at the enter of eah Landau level, are present as inferred from the QHE observation and this poses an interesting future problem. The situation should also depend on whether the disorder is short range or long range; but, in ordinary QHE systems, a sum rule guarantees the total intensity of the ylotron resonane intat. 10 As for the ripples known to exist in atual graphene samples 18, the n=0 Landau level remains sharp PHYSICAL REVIEW B 78, whih is topologially proteted sine the slowly varying potential does not destroy the hiral symmetry 19 while other levels beome broadened 20 and this should favor the situation proposed in the presented Brief Report. 22 IV. SUMMARY To summarize, we have disussed the ylotron radiation from graphene QHE system. We onlude that unusual uneven Landau levels, unusual ylotron energy, and unusual transition seletion rules in graphene all work favorably for a population inversion envisaged for the Landau-level laser. An estimate of the photon emission rate shows that the emission rate is of the orders of magnitude more effiient than in the ordinary QHE system, while the ompeting phonon emission rate is not too large to mar the photon emission. Important future problems inlude the examination of the atual lasing proesses inluding the avity properties, oupling of eletrons to the out-of-plane phonon modes, et. ACKNOWLEDGMENTS We wish to thank Andre Geim for his illuminating disussions. This work has been supported in part by Grants-in-Aid for Sientifi Researh on Priority Areas from MEXT, Physis of new quantum phases in superlean materials Grant No for Y.H., and Anomalous quantum materials Grant No for H.A. 1 K. Novoselov, A. Geim, S. Morozov, D. Jiang, M. Katsnelson, I. Grigorieva, S. Dubonos, and A. Firsov, Nature London 438, Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, Nature London 438, J. MClure, Phys. Rev. 104, Y. Zheng and T. Ando, Phys. Rev. B 65, V. P. Gusynin and S. G. Sharapov, Phys. Rev. Lett. 95, N. M. R. Peres, F. Guinea, and A. H. Castro Neto, Phys. Rev. B 73, L. Onsager, Philos. Mag. 43, M. L. Sadowski, G. Martinez, M. Potemski, C. Berger, and W. A. de Heer, Phys. Rev. Lett. 97, K. Ikushima, H. Sakuma, S. Komiyama, and K. Hirakawa, Phys. Rev. Lett. 93, H. Aoki, Appl. Phys. Lett. 48, T. Ando, J. Phys. So. Jpn. 38, V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, Phys. Rev. Lett. 98, V. P. Gusynin and S. G. Sharapov, Phys. Rev. B 73, C. Chaubet, A. Raymond, and D. Dur, Phys. Rev. B 52, C. Chaubet and F. Geniet, Phys. Rev. B 58, Loalized states around point defets were deteted reently with STM in B Ref. 21 and their radius was found to be omparable to the magneti length 30 nm. 17 S. Pisane, M. Lazzeri, F. Mauri, A. C. Ferrari, and J. Robertson, Phys. Rev. Lett. 93, J. Meyer, A. Geim, M. Katsnelson, K. Novoselov, T. Booth, and S. Roth, Nature London 446, Y. Hatsugai, T. Fukui, and H. Aoki, Phys. Rev. B 74, A. Giesbers, U. Zeitler, M. Katsnelson, L. Ponomarenko, T. Mohiuddin, and J. Maan, Phys. Rev. Lett. 99, Y. Niimi, H. Kambara, T. Matsui, D. Yoshioka, and H. Fukuyama, Phys. Rev. Lett. 97, We an also note that the n=0 Landau level in graphene has an exatly E=0 edge mode, of a topologial origin, right at the enter Ref. 19, whih may also affet the photon absorption/ emission