Could a laser field lead to the much sought-after tunable bandgaps in
graphene? By using Floquet theory combined with Green's functions techniques,
we predict that a laser field in the mid-infrared range can produce observable
bandgaps in the electronic structure of graphene. Furthermore, we show how they
can be tuned by using the laser polarization. Our results could serve as a
guidance to design opto-electronic nano-devices.Comment: 4 pages, 3 figures, to appear in Applied Physics Letter

Hernán L. Calvo

Horacio M. Pastawski

Luis E. F. Foa Torres

Pastawski H. M.

Stephan Roche

Applied Physics Letters

English

arXiv

Calvo, Hernán L.

Pastawski, Horacio M.

Roche, Stephan

Foa Torres, Luis E. F.

Publikationsserver der RWTH Aachen University

Tuning laser-induced band gaps in graphene

3 páginas, 3 figuras.Could a laser field lead to the much sought-after tunable band gaps in graphene? By using Floquet theory combined with Green's functions techniques, we predict that a laser field in the mid-infrared range can produce observable band gaps in the electronic structure of graphene. Furthermore, we show how they can be tuned by using the laser polarization. Our results could serve as a guidance to design optoelectronic nanodevices.We acknowledge support by SeCyT-UNC and ANPCyT.
LEFFT acknowledges the support from the Alexander
von Humboldt Foundation and ICTP, Trieste.Peer reviewe

Digital.CSIC

Tuning laser-induced band gaps in grapheneHernán L. Calvo, Horacio M. Pastawski, Stephan Roche, and Luis E. F. Foa Torres  Citation: Appl. Phys. Lett. 98, 232103 (2011); doi: 10.1063/1.3597412 View online: http://dx.doi.org/10.1063/1.3597412 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v98/i23 Published by the American Institute of Physics.  Related ArticlesFirst-principles study of the transport behavior of zigzag graphene nanoribbons tailored by strain AIP Advances 2, 012103 (2012) Hybrid W-shaped graphene nanoribbons: Distinct electronic and transport properties J. Appl. Phys. 110, 124312 (2011) Annular wave packets at Dirac points in graphene and their probability-density oscillation J. Chem. Phys. 135, 224707 (2011) Excitation of discrete and continuous spectrum for a surface conductivity model of graphene J. Appl. Phys. 110, 114305 (2011) Abnormal electronic transport and negative differential resistance of graphene nanoribbons with defects Appl. Phys. Lett. 99, 192102 (2011)  Additional information on Appl. Phys. Lett.Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 24 Jan 2012 to 161.111.180.191. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsTuning laser-induced band gaps in grapheneHernán L. Calvo,1,2 Horacio M. Pastawski,1 Stephan Roche,3,4 and Luis E. F. Foa Torres1,a1Instituto de Física Enrique Gaviola (IFEG)–CONICET, FaMAF, Universidad Nacional de Córdoba,Ciudad Universitaria, 5000 Córdoba, Argentina2Institut für Theorie der Statistischen Physik, RWTH Aachen University, D-52056 Aachen, Germany3CIN2 (ICN–CSIC), Catalan Institute of Nanotechnology, Universidad Autónoma de Barcelona,Campus UAB, 08193 Bellaterra (Barcelona), Spain4Institució Catalana de Recerca i Estudis Avançats (ICREA), 08070 Barcelona, SpainReceived 6 March 2011; accepted 2 May 2011; published online 7 June 2011Could a laser field lead to the much sought-after tunable band gaps in graphene? By using Floquettheory combined with Green’s functions techniques, we predict that a laser field in the mid-infraredrange can produce observable band gaps in the electronic structure of graphene. Furthermore, weshow how they can be tuned by using the laser polarization. Our results could serve as a guidanceto design optoelectronic nanodevices. © 2011 American Institute of Physics.doi:10.1063/1.3597412More than a century ago, the use of alternating currentsac sparked a revolution that changed our modern world.Today, the use of ac fields has reached the nanoscale.1,2 Here,the interplay between the quantum coherence of the elec-trons, inelastic effects, and dynamical symmetry breakingoffers fascinating opportunities for both basic research andapplications. The coherent destruction of tunneling3 andquantum charge pumping,4–6 i.e., the generation of a dc cur-rent at zero bias voltage due to quantum interference,7 areprominent examples of the wealth of phenomena driving thisrapidly advancing area of research. By means of time-dependent gates,8,9 surface acoustic waves,10 or illuminationwith a laser field,11 experiments have probed different facetsof ac transport and light-matter interaction. On the otherhand, theoretical insights keep opening exciting directions,from interaction-induced quantum pumping12 to everlastingoscillations.13Among many other benefits, the advent of graphene14and carbon nanotubes15 provides an outstanding arena for thestudy of ac transport in highly coherent, low dimensionalsystems. Adiabatic quantum pumps,16,17 ac controlled Fabry–Pérot resonators,18 and photodetectors19 are among the newbreed of carbon-based devices. Furthermore, a few remark-able studies on Dirac fermions interacting with linearly20,21and circularly22,23 polarized monochromatic light pointed outstriking nonperturbative nonadiabatic effects; a laser fieldcould lead to the opening of dynamical gaps20 in graphene.For circularly polarized light, a further gap was shown todevelop at the Dirac point.22 Many open questions remain asfollows: which experimental setup would unveil these phe-nomena and how could we tune them?In this letter, we aim at elucidating these questions byanalyzing the interaction between electrons and a monochro-matic laser field of arbitrary polarization in graphene. Usinga Floquet approach one finds that a laser of frequency induces the lifting of degeneracies between Floquet statesof the combined electron–photon system. Here, a carefulanalysis allows for the tuning of a feasible parameter rangelaser frequency, power, and polarization. Our predictionsshow that these effects are within the reach of mid-infraredlaser technology in a transport setup, thereby opening prom-ising prospects for graphene-based optoelectronic devices.In graphene, the low energy electronic states contribut-ing to transport are close to the Dirac points K and K. Sincewe consider a clean sample and given that the ac field doesnot introduce any intervalley coupling, we can describe bothpoints separately. In the K-valley, those states can be accu-rately described by the k ·p approximation through the enve-lope wave function = A ,BT, where the two compo-nents refer to the interpenetrating sublattices A and B.24 Thetime periodic electromagnetic field, with period T=2 /, isincluded as a monochromatic plane wave traveling along thez-axis, perpendicular to the plane defined by the graphenesheet. The magnetic vector potential is thus written as At=ReA0e−it, where A0=A01,ei refers to the intensityand polarization  of the field. For this choice of parameters,=0 yields a linearly polarized field At=A0 cos txˆ+ yˆ while = /2 results in a circularly polarized fieldAt=A0cos txˆ+sin tyˆ. Therefore, the grapheneelectronic states in the presence of the ac field are encoded inthe Hamiltonian,Ht = vF · p − eAt . 1Here vF106 m /s denotes the Fermi velocity and = x ,y the Pauli matrices describing the pseudospin de-gree of freedom.As will be made clear later on, a correct description ofour problem crucially requires a solution valid beyond theadiabatic approximation. Hence, the Floquet theory2 repre-sents a suitable approach. The resulting picture is that ofan effective time-independent Hamiltonian in a higher-dimensional space, the so-called Floquet space, defined asthe direct product between the usual Hilbert space and thespace of 2 /-time periodic functions. This space isspanned by the states 	k ,n, where k is the electronicwave vector, n is the Fourier index, and the subindex refersto the alignment of the pseudospin with respect to the mo-mentum. In this basis, the problem to solve is identical to thetime-independent Schrödinger equation with Floquet Hamil-tonian HF=H−it. The power of recursive Green’s functiontechniques25 can be exploited to obtain both the dc compo-nent of the conductance and density of states DOS from theso-called Floquet Green’s functions.26aElectronic mail: lfoa@famaf.unc.edu.ar.APPLIED PHYSICS LETTERS 98, 232103 20110003-6951/2011/9823/232103/3/$30.00 © 2011 American Institute of Physics98, 232103-1Downloaded 24 Jan 2012 to 161.111.180.191. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsPrevious studies20,22 have considered lasers either in thefar infrared =29 meV Ref. 20 or in the visiblerange.22 In the first case, the predicted gaps were of about6 meV and the authors considered the photocurrents gener-ated in a p-n junction. On the other hand, for a laser in thevisible range, the photon energy of about 2 eV is much largerthan the typical optical phonon energies 170 meV, severecorrections to the transport properties due to dissipation ofthe excess energy via electron–opticalphonon interactionsare expected. Besides, appreciable effects in this last caserequired a power above 1 W /	m2, which could compromisethe material stability. To overcome both limitations we quan-titatively explore the interaction with a laser in the mid-infrared range =5–10 	m, where photon energies can bemade smaller than the typical optical phonon energy whilekeeping a much lower laser power. Furthermore, we showthat the polarization, whose role was overlooked, can be usedas a control variable.Figure 1 shows how the dc DOS defined as in Ref. 22evolves as the polarization changes. Already for linear polar-ization upper panel one can find some surprises not re-ported before; i close to the Dirac point, the DOS is en-hanced as compared to the case without laser red dottedline. The DOS increases linearly but with a different slope.ii Close to  /2, stronger effects lead to a depletion ofthe states but without reaching a full gap. The counterpart ofthis depletion is given by the peaks in the DOS around thedepletion area. More accurate calculations based on a tight-binding model in a micrometer-sized sample interacting witha linearly polarized field confirm these findings top panel.The panels in Fig. 1 show how these features changewith the laser polarization; i close to the Dirac point a gapopens and the structure of the side peaks is severely modi-fied. ii The depletion areas around  /2 transform intogaps whose widths are maximum for circular polarization.To rationalize the behavior observed in Fig. 1 we resortto the Floquet picture explained above. A scheme with therelevant states close to the Dirac point is shown in Fig. 2. Onthe left panel, one can see the dispersion relation for thestates 	k ,n for n=0 black solid lines and n=1 reddashed lines, the scheme on the right panel shows the statesrepresented by circles and their corresponding interactionswith full lines. The effects of the ac field are expected to bestronger at the crossing points between these lines, leading tothe opening of energy gaps at those points. From a geometri-cal argument one can see that the crossing of the states dif-fering in one photon lies exactly at  /2. These degenera-cies are lifted by the ac field leading to the gaps observed inFig. 1. This mechanism is reminiscent of an inelastic Braggreflection as found as a result of electron-phonon interactionin carbon nanotubes.27 An estimation of the resulting gapgives:k=/2vF  eA0vF1 − cossin2 , 2where =tan−1ky /kx. Equation 2 shows that the effect islinear in the field strength. Interestingly, after summing upover all directions in the two-dimensional 2D k-space, wesee that no net gap opens in the linearly polarized case sincein the orientation = /4 both states become degenerate.However, as can be seen in Fig. 1, there is a strong modifi-cation in the DOS around  /2. Changing the polarizationaway from the linear case, one sees that a dynamical gap20opens and reaches its maximum max23 meV 46 meVfor I=32 mW /	m2 130 mW /	m2 in the circularly polar-ized case.FIG. 1. Color online Scheme of the considered setup, where a laser fieldwith =140 meV is applied perpendicular to a graphene monolayer. DOSfor a linear, b =0.125, c =0.375, and d circular polarizations.The black solid line is for I=32 mW /	m2 while the red dashed line corre-sponds to I=130 mW /	m2. For linear polarization these results are com-pared with those of a tight-binding calculation for a system with 5104channels blue dashed and purple dash-dotted lines. For reference the zero-field DOS is shown in solid gray.-hΩ0v1 -v1v2-v2kε+,0ε+,-1-hΩ0hΩhΩ(a) (b)ε-,1εkε-,0+hvFk -hvFkFIG. 2. Color online a Scheme of the quasienergies as a function of k. bRepresentation of the Floquet Hamiltonian for the k ·p approach. Here, thedirect hopping v1=eA0vF /2cos +e−i sin  sets the transition amplitudebetween Floquet states conserving pseudospin. In contrast, the off-diagonalterm v2=ieA0vF /2e−i cos −sin  introduces an inelastic backscatter-ing process that simultaneously enables both m and pseudospin transitions.232103-2 Calvo et al. Appl. Phys. Lett. 98, 232103 2011Downloaded 24 Jan 2012 to 161.111.180.191. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsNow, let us account for the phenomenon occurring at theDirac point. Oka and Aoki Ref. 22 predicted that circularlypolarized light would induce a further gap around the Diracpoint, in agreement with our Fig. 1d. Here we see that thesestrong modifications extend all the way up to linear polariza-tion. The leading contribution comes from the states connect-ing the degenerate ones at the Dirac point there are four ofthese paths as can be seen by analyzing Fig. 2b. The cal-culated gap readsk=0 8Rev1v2 = 2eA0vF2sin  , 3which gives a quadratic dependence with the field strengthand is inversely proportional to the frequency. Note also thatthere is no dependence of the gap with the orientation of thek-vector. The polarization yields a maximum gap for thecircular polarization while no net gap opens for the linearlypolarized case. However in Fig. 1a noticeable correctionsaround the Dirac point are observed even for linear polariza-tion. A deeper analysis shows that this is due to the field-induced lifting of the degeneracies between the states	k ,1 around =0.Up to now, we have shown how the laser field modifiesthe electronic structure of 2D graphene that would be ob-served in any experiment carried out over a time much largerthan the period T. A relevant question is if these effectswould be observable in a transport experiment. To suchend we compute the transport response at zero temperatureusing Floquet theory26 applied to a  orbitals Hamiltonian.The electromagnetic field is accounted through the Peierls’substitution, which introduces an additional phase in thehopping ij connecting adjacent sites ri and r j; ij=0 expi2 /0rirjAt ·dr, where 02.7 eV is thehopping amplitude at zero field and 0 is the quantum ofmagnetic flux. For computational convenience we use anarmchair edge structure with the vector potential At=Ax cos txˆ+Ay sin tyˆ.Calculating the two-terminal dc component of the con-ductance as a function of Fermi energy is not, a priori, com-putationally easy. An efficient solution is achieved by de-composing the Hamiltonian into independent transversalchannels as explained in Ref. 18. Although this decomposi-tion is preserved by the interaction with the laser only forlinear polarization Ay =0, it is already enough to give aflavor on the transport effects. Figure 3 gives the computeddc conductance2,26 for a stripe 1 	m wide and 1 	m long.One can appreciate in Fig. 3 that the strong depletion areasobserved in the DOS of Fig. 1 are indeed mirrored in the dccomponent of the conductance. Accordingly, the predicteddynamical gaps can be unveiled through transport measure-ments.In summary, we have shown that it is possible to use alaser field in the mid-infrared range to tune the electronicstructure of graphene and its electrical response. The modi-fications are predicted to arise both around the Dirac pointand at  /2. Moreover, since the results are strongly de-pendent on the laser polarization, it may be used as a controlparameter. We encourage experimentalists to pursue this ex-citing line of research.We acknowledge correspondence with J. Kono, discus-sions with G. Usaj and support by SeCyT-UNC and ANP-CyT. LEFFT acknowledges the support from the Alexandervon Humboldt Foundation and ICTP, Trieste.1G. Platero and R. Aguado, Phys. Rep. 395, 1 2004.2S. Kohler, J. Lehmann, and P. Hänggi, Phys. Rep. 406, 379 2005.3F. Grossmann, T. Dittrich, P. Jung, and P. Hänggi, Phys. Rev. Lett. 67, 5161991.4D. J. Thouless, Phys. Rev. B 27, 6083 1983.5P. W. Brouwer, Phys. Rev. B 58, R10135 1998.6B. L. Altshuler and L. I. Glazman, Science 283, 1864 1999.7M. Büttiker and M. Moskalets, Lect. Notes Phys. 690, 33 2006, andreferences therein.8M. Switkes, C. M. Marcus, K. Campman, and A. C. Gossard, Science283, 1905 1999.9M. D. Blumenthal, B. Kaestner, L. Li, S. Giblin, T. J. B. M. Janssen, M.Pepper, D. Anderson, G. Jones, and D. A. Ritchie, Nat. Phys. 3, 3432007; B. Kaestner, V. Kashcheyevs, S. Amakawa, M. D. Blumenthal, L.Li, T. J. B. M. Janssen, G. Hein, K. Pierz, T. Weimann, U. Siegner, and H.W. Schumacher, Phys. Rev. B 77, 153301 2008.10P. J. Leek, M. R. Buitelaar, V. I. Talyanskii, C. G. Smith, D. Anderson, G.A. C. Jones, J. Wei, and D. H. Cobden, Phys. Rev. Lett. 95, 2568022005.11A. Srivastava, R. Srivastava, J. Wang, and J. Kono, Phys. Rev. Lett. 93,157401 2004.12F. Reckermann, J. Splettstoesser, and M. R. Wegewijs, Phys. Rev. Lett.104, 226803 2010.13S. Kurth, G. Stefanucci, E. Khosravi, C. Verdozzi, and E. K. U. Gross,Phys. Rev. Lett. 104, 236801 2010.14A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K.Geim, Rev. Mod. Phys. 81, 109 2009; N. M. R. Peres, ibid. 82, 26732010.15R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physical Properties ofCarbon Nanotubes Imperial College Press, London, 1998; J.-C. Charlier,X. Blase, and S. Roche, Rev. Mod. Phys. 79, 677 2007.16E. Prada, P. San-Jose, and H. Schomerus, Phys. Rev. B 80, 2454142009.17R. Zhu and H. Chen, Appl. Phys. Lett. 95, 122111 2009.18C. G. Rocha, L. E. F. Foa Torres, and G. Cuniberti, Phys. Rev. B 81,115435 2010; L. E. F. Foa Torres and G. Cuniberti, Appl. Phys. Lett. 94,222103 2009.19F. Xia, T. Mueller, Y.-m. Lin, A. Valdes-Garcia, and P. Avouris, Nat.Nanotechnol. 4, 839 2009.20S. V. Syzranov, M. V. Fistul, and K. B. Efetov, Phys. Rev. B 78, 0454072008.21F. J. Lopez-Rodriguez and G. G. Naumis, Phys. Rev. B 78, 201406R2008.22T. Oka and H. Aoki, Phys. Rev. B 79, 081406R 2009.23D. S. L. Abergel and T. Chakraborty, Appl. Phys. Lett. 95, 062107 2009.24T. Ando, Semicond. Sci. Technol. 15, R13 2000; D. P. DiVincenzo andE. J. Mele, Phys. Rev. B 29, 1685 1984.25H. M. Pastawski and E. Medina, Rev. Mex. Fis. 47, 1 2001.26L. E. F. Foa Torres, Phys. Rev. B 72, 245339 2005.27L. E. F. Foa Torres and S. Roche, Phys. Rev. Lett. 97, 076804 2006;Phys. Rev. B 78, 035412 2008.FIG. 3. Color online dc conductance through a graphene stripe of 1 	m1 	m in the presence of a linearly polarized laser as a function of theFermi energy.232103-3 Calvo et al. Appl. Phys. Lett. 98, 232103 2011Downloaded 24 Jan 2012 to 161.111.180.191. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions

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Tuning laser-induced bandgaps in graphene

Tuning laser-induced bandgaps in graphene

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