Radiation effects on the electronic structure of bilayer graphene Eric Su´arez Morell1 and Luis E. F. Foa Torres2 1Departamento de F´ısica, Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, Chile 2Instituto de F´ısica Enrique Gaviola (IFEG-CONICET) and FaMAF, Universidad Nacional de Co´rdoba, Ciudad Universitaria, 5000 Co´rdoba, Argentina. (Dated: January 16, 2013) Wereportontheeffectsoflaserilluminationontheelectronicpropertiesofbilayergraphene. By using Floquet theory combined with Green’s functions we unveil the appeareance of laser-induced gaps not only at integer multiples of (cid:126)Ω/2 but also at the Dirac point with features which are showntodependstronglyonthelaserpolarization. Trigonalwarpingcorrectionsareshowntolead 3 to important corrections for radiation in the THz range, reducing the size of the dynamical gaps. 1 Furthermore,ouranalysisofthetopologicalpropertiesatlowenergiesrevealsthatwhenirradiated 0 with linearly polarized light, ideal bilayer graphene behaves as a trivial insulator, whereas circular 2 polarization leads to a non-trivial insulator per valley. n a PACSnumbers: 72.80.Vp,72.10.-d,03.65.Vf J 5 1 I. INTRODUCTION ] l Among the many promises sparked by graphene re- al search during the last few years1,2, graphene optoelec- h tronics is perhaps one of the brightest3–6. From im- - proved power conversion of energy harvesting devices7 s e to novel plasmonics properties8,9, graphene and re- m lated materials offer an outstanding playground for te . study of light-matter interaction with many potential at applications3,4,10,11. m Recent studies pointed out the intriguing possibility of inducing bandgaps in monolayer graphene by illumi- - d nation with a laser field12–14. The peculiar electronic n structureofgrapheneanditslowdimensionalityarecru- o cial for the occurence of this effect. Further studies have c [ predicted observable changes in the conductance15,16 and optical properties17 , with a strong dependence on 1 laser polarization15,18, setting off many other interest- v ing studies19–23. Moreover, the possibility of controlling 2 topologicalinsulatorswithphotocurrents11,aswellasthe FIG. 1: (Color online) (a) Scheme of the considered setup 8 3 emergenceofnontriviallaser-inducedtopologicalproper- where a laser field is applied perpendicular to a graphene bi- 3 tiesandedgestates16,19,24,25,theso-calledFloquettopo- layer. Panels(b) and(c)showthequasienergyFloquetband structure for bulk bilayer without and with trigonal warping . logical insulators24,25, add more relevance to this area. 1 respectively. These plots are along k direction (k = 0), 0 Graphene’s thicker cousin, bilayer graphene (BLG), solid lines are for circularly polarized lxight with a freyquency 3 has also shown an enormous potential1,2, allowing for a of5THzandanintensityof0.5mW/µm2. Theunirradiated 1 tunablebandgap29 asrequiredfortheoperationofactive spectrum is shown with dashed lines. : devices. Notwithstanding, the studies mentioned in the v i last paragraph were all centered in monolayer graphene. X Only in Ref.30, the authors proposed irradiated bilayer r asavehicleforinducingavalleypolarizedcurrent. Here, a we focus on the electronic and topological properties of bilayer graphene illuminated by a laser with frequency ing from a trivial insulator to one with properties akin either in the THz or in the mid-infrared range. In the those of a topological insulator. Specifically, we show THzrange,trigonalwarping(TW)correctionsareshown that the low energy properties of BLG illuminated by to induce strong modifications in the theoretical predic- circularly polarized light can be described by a simple tions leading, besides qualitative changes in the spectra, effective Hamiltonian similar to the one of BLG with a to quantitative differences in the laser-induced gaps up bias. Our theoretical analysis shows that although the to a factor of two. system behaves as a trivial insulator in the presence of Moreover, we show that a laser field may also lead to linearlypolarizedlight, switchingthepolarizationtocir- polarization-tunabletopologicalpropertiesinBLGrang- culartransformsitintoanon-trivialinsulatorpervalley. 2 II. RESULTS AND DISCUSSION form and can be calculated as in13,15. To such end we compute the Floquet-Green function, defined as G = F A. Floquet theory applied to irradiated bilayer (ε1−HF)−1, from which the time-averaged DOS is ob- (cid:110) (cid:111) graphene. tained as DOS(ε) = −1Im Tr(G (ε)) , where π F 0,0 (G ) stands for the sub-block of the Floquet-Green F 0,0 The unit cell of bilayer graphene with Bernal stacking function corresponding to vanishing Fourier index. has two inequivalent sites labeled as A1, B1 on the top layer and A2,B2 on the bottom layer, they are arranged insuchawaythatatomB1liesontopofatomA2. Using the wave-functions Ψ=(ψ ,ψ ,ψ ,ψ )T for the K B. Laser-induced modifications of the Floquet A1 B2 A2 B1 valley and Ψ = (ψ ,ψ ,ψ ,ψ )T for the K(cid:48) valley, spectra B2 A1 B1 A2 an effective Hamiltonian for the low energy properties is given by31: In the following we will analyze the behavior of the quasi-energy spectra and the DOS for various laser in-   tensities, frequencies and polarization37. 0 v π 0 vπ† 3 While in monolayer graphene trigonal warping (TW) H0((cid:126)k)=ξv30π† vπ0† v0π ξ0γ1 , (1) ionntlryodautcheisghsmenaellrgcieosrr(e∼ct5i0o0nsmweVh)ic,hinbtehceomcaesenooftibcielaaybeler vπ 0 ξγ 0 1 graphene these corrections are stronger at low energies where ξ = 1(−1) for valley K(K(cid:48)), π = p + i p , where they lead to a splitting of the Dirac point into a √ √ x y v = ( 3/2)aγ /(cid:126), v = ( 3/2)aγ /(cid:126), a = 0.246 nm, structure with four pockets31 as shown in the inset of 0 3 3 Fig.2a. Here we show that these effects, that were ne- graphene lattice constant, γ = 3.16 eV, γ = 0.39 eV 0 1 glected in previous studies of irradiated bilayer, are in- and γ =0.315 eV. The hopping parameter γ is respon- 3 3 deed very important for radiation in the THz range. sible for the trigonal warping effects. Weapplylinearly/circularlypolarizedlightperpendic- Figure1showsthequasi-energydispersionalongapar- ular to the graphene bilayer as shown schematically in ticular k direction without (b) and with (c) the TW cor- Fig. 1a. The time-dependent field is introduced through rection in the presence of the electromagnetic field. The thesubstitution(cid:126)k →(cid:126)k+eA(cid:126)/(cid:126),wherethevectorpotential dashed lines in each figure shows the unirradiated case. is A(cid:126)(t)=A(cos(Ωt),cos(Ωt+φ)), where φ=0(π/2) for The field is expected to have the stronger effect at the crossing points which, due to the electron-hole symme- linear(circular) polarization. The Floquet theorem32–34 try, are located at integer multiples of (cid:126)Ω/2 above and provides an elegant route to handle this time-periodic below the Dirac point, as can be seen in Fig. 1b and Hamiltonian (H(t + T) = H(t) = H ((cid:126)k + eA(cid:126)(t)/(cid:126)), 0 1c. The time-dependent perturbation introduces a non- where T = 2π/Ω), it states that the solutions to the vanishing matrix element between the states at those time-dependent Schr¨odinger equation can be written as crossings, thereby lifting the degeneracies and opening Ψα((cid:126)r,t) = e−iεαt/(cid:126)φα((cid:126)r,t), where φα(t) = φα(t + T) the so called dynamical gaps12,13. The gap at the charge is time-periodic, the Floquet states can be further ex- neutrality point is a higher-order effect and will be ana- panded into a Fourier series φα(t) = (cid:80)einΩtφ(αn) and a lyzed in more details later. substitution in the Schr¨odinger equation gives: Figure 2 shows the DOS for bilayer graphene in the presence of either linearly (b) or circularly (c) polarized (cid:88)(H(n,m)−n(cid:126)Ωδ )|φ(n)(cid:105)=(cid:15) |φ(n)(cid:105), (2) light (5THz) with (solid-line) and without (dashed-line) n,m α α TW.TheDOSintheabsenceofradiationisshownin(a) m for reference. Although from the discussion before one where H(n,m) = 1 (cid:82)T dtH(t)ei(n−m)Ωt and (cid:15) is the so- mayexpectthemaincorrectionstoariseonlyclosetothe T 0 α called quasi-energy. Simple inspection shows that this Diracpoint,Figs. 2band2cshowthattheyemergeeven is an eigenvalue equation analog to the one for time- at the dynamical gaps for radiation in the THz range. independent systems. There are however two main dif- For linearly polarized light, the DOS in the vicinity of ferences: the role of the Hamiltonian is played by the (cid:126)Ω/2 exhibits a depletion area with a linear dispersion so-called Floquet Hamiltonian HF = H −i(cid:126)d/dt; and andasinglepointofvanishingDOS.Thisissimilartothe the states belong to an extended Hilbert space which is caseofmonolayergraphenefoundinRef.15 andisdueto the direct product between the usual Hilbert space and the fact that the gap depends on relative angle between the space of time-periodic functions with period T. It is k andthepolarizationvector,nogapemergeswhenthey straightforward to see that H(n,m) =H(n,m)−n(cid:126)Ωδ . are parallel. One can also notice that the roughly linear F n,m Thismethodhasbeenappliedtoavarietyofsystemsand dispersion around the dynamical gaps acquire a struc- inparticulartoacfieldssuchasalternatinggatevoltages ture with three narrow features on each side when TW in graphene35,36 beyond the adiabatic limit. correctionsareincluded. Thisisaconsequenceofthede- The time-averaged density of states (DOS) gives valu- formation of the iso-energy lines in the k -k plane due x y able information on the Floquet spectra in a compact to the TW corrections (see inset of Fig. 2a). 3 FIG. 2: (Color online) a) DOS for bilayer graphene with the laser turned off. The continous line corresponds to calcula- FIG.3: DOSasafunctionofenergyfor(a)5THz,(b)10THz tionsperformedincludingtrigonalwarping(TW)corrections and(c)mid-infrared30THzradiation. (a)and(b)correspond and the dashed line without them. Panel a)-inset depicts to a laser intensity of 0.5 mW/µ m2, (c) is computed for the iso-energy lines for the dispersion of bilayer graphene in 10 mW/µ m2. Solid (dashed) lines are for circular (linear) theabsenceofradiation,thestrongTWdistortionisevident. polarization. Noticethechangeinthehorizontalscalein(c). Panels b) and c) show the DOS as defined in the text for bi- The structure induced by the TW becomes smoothened as layer graphene in the presence of linearly (b) and circularly the frequency increases. The inset in (b) is a zoom around (c)polarizedlight(5THz)withanintensityof0.5mW/µm2. zero energy. Forcircularpolarization,twostrikingobservationsnot tally. Figure 3 highlights this for three different frequen- reported before should be emphasized: a) There is a gap cies a) 5 THz, b) 10 THz and c) 30 THz (which cor- opening at zero energy (which also occurs in the absence responds to the mid-infrared range) for linearly (dashed of TW but is much smaller and cannot be distinguished line) and circularly (solid line) polarized light. in the figure, see inset of Fig. 1-b); and b) the dynam- ical gap (which turns out to be linear in the field in- tensity as for monolayer graphene) is overestimated by almost a factor two when the TW corrections are not takenintoaccount. Akeyingredientbehindthesediffer- C. Effective low-energy Hamiltonian description and topological considerations encesisagainthebreakingoftherotationalsymmetryin thek -k planeevenforlowenergies. Thoughthegapat x y the charge neutrality point would require stringent con- Though more academic in nature, we now turn to ditions (being of about 0.3 meV for a laser intensity of an instructive analysis of the low-energy and topologi- 0.5mW/micronsquare), the physics described here may cal properties of irradiated BLG. Our main fundamental promptadditionalresearchandexperimentsthatmayal- question is: are there non-trivial laser-induced topolog- low directly or indirectly to unveil it. In contrast, the ef- ical states to be expected in bilayer graphene? To such fectsdescribedatthedynamicalgaps(±(cid:126)Ω/2)aremuch end, we are interested in obtaining an effective Hamil- strongerandshouldbeobservableinlowtemperatureex- tonian to describe low energies electronic properties for periments. Indeed the dynamical gaps are of the order low values of the light intensity in the spirit of Kitagawa of5Kfor10Thzradiationat0.5mW/µ2 andreachlarger andcoworkers25. Wewillconsideronlytheprocesswhen values (up to 30meV, or 350K) for 30THz radiation for one photon is absorbed (emitted) and then re-emitted a power of a few mW/µ2. (re-absorbed), in this case applying the continued frac- As one moves to higher frequencies, trigonal warping tionmethodandretainingonlythetermsoforderO(F2), effects become less noticeable, though the gaps may be- whereF =eA/(cid:126)andinthefollowing(cid:126)=1, theeffective come larger and therefore easier to observe experimen- time-independent Hamiltonian can be expressed as: 4 not the trigonal warping. For values of Ω (cid:28) γ the sec- 1 ond term in the expression of ∆ can be neglected and it Heff =H0+V−1Gˆ(−1,Ω)V+1+V+1Gˆ(+1,Ω)V−1 (3) gives a quite simple dependence of the gap with F(cid:48) and Ω(cid:48), Gap=2×ε F(cid:48)2. This expression shows an excellent where V = H(n,m) for n − m = ±1 and Gˆ(n,Ω) = 0 Ω(cid:48) ±1 agreement with numerical calculations in the frequency 1 represents the propagator of a particle with n (cid:15)+nΩ−H0 range considered to obtain Eq. 4. photons. Forcircularlypolarizedlight,thisresultsinthe FromthiseffectiveHamiltonianitisstraightforwardto following effective Hamiltonian: calculate the Berry curvature and the Chern number38. The curvature is given in polar coordinates by:  F2Ωv32 + F2γv22Ω v3π 0 vπ†  H =ξ v3π† 1 −F2Ωv32 − F2γv22Ω vπ 0 , Ω(k,θ)= ξη∆(4k2−s2) , (6)  0 vπ† 1 −F2Ωv2 ξγ1  2(∆2+k4+k2s2−2k3ξs cos3θ)3/2 vπ 0 ξγ1 F2Ωv2 andtheintegrationgivesanintegernon-zeroChernnum- (4) berpervalley,aquantumvalley-Hallstate39. TheChern where we have assumed γ1 (cid:29) Ω (cid:29) (cid:15). All the terms in number has opposites values for the two valleys for a the diagonal should be multiplied by a factor η =±1 to given handedness of polarization. A valley current will take into account left or right polarization of the light. be proportional to the Berry curvature40. Therefore a Strikingly, this effective Hamiltonian resembles the change in the handedness implies a change in the direc- Hamiltonian of bilayer graphene with a bias, but there tionofthevalleycurrents,asthesignoftheBerrycurva- aresomesubtledifferencesthatacarefulanalysisreveals. turechanges. Itprovidesaneffectivewaytocontrolthese One may argue that laser illumination introduces three valley currents. There have been some proposals about ingredients: First, it breaks the intra-layer symmetry thissubjectseeforinstanceRef.41,42. Ontheotherhand by introducing a term similar to Kane-Mele spin-orbit the structure of the Berry curvature reveals the impact term (F2v2/Ω)25,26 (if the layers were decoupled the of the trigonal warping: For low values of ∆ the shape system would have a gap solely due to this term); of the curvature shows a central dip with a topological second, it breaks the inversion symmetry between the charge Q=-1 and three peaks away from the center and two layers (similar to a potential difference between separated 120o with Q=1 each; in the K(cid:48) valley we have layers), an effect which also opens a gap. And third, the opposite behavior. This segregation might have an when a graphene-based system with a gap is exposed impactontheedgecurrentsofasystembasedonbilayer to circularly polarized light an asymmetry between the graphene and energies in the Teraherz range43. valleys is expected due to the breaking of inversion A completely different picture is obtained from irradi- symmetry, an effect similar to optical circular dichroism ating bilayer graphene with linearly polarized light, fol- for valleys instead of spins.27,28 The valley degree of lowing the same procedure as before, one obtains a gap freedom can be exploited generating valley dependent at k = 0, with a peculiar behavior, it does not depend currents as we argue below. explicitlyonΩneitheronγ , Gap=2×F2v2. TheChern 3 γ1 The gap at k = 0 is given by 2×(F2v32 + F2v2Ω), the number equals zero in every valley, thus the states are Ω γ2 topologically trivial. 1 relative importance of these two terms is set by the fre- quency Ω: for Ω in the THz range the trigonal warping term has a leading impact on the gap as previously no- III. CONCLUSIONS ticed in the discussion of Fig. 2. To evaluate the topological properties of this effective Insummary,theeffectsofalaserwithfrequenciesrang- Hamiltonianwereducetheprevious4×4toa2×2Hamil- ingfromTHztothemid-infraredontheelectronicstruc- tonian which describes the effective interaction between ture of bilayer graphene are analyzed, highlighting the thenon-dimersitesA1-B2. Consideringasbeforeγ (cid:29)(cid:15) 1 appeareance of laser-induced gaps and their dependence the new effective low energy Hamiltonian is given by: with the light polarization as well as the strong influ- enceoftrigonalwarpingcorrections. Forradiationinthe (cid:18) ∆ k2 −sξk (cid:19) THz range, trigonal warping in bilayer graphene tends H =ε0 k2−sξk −ξ−∆ ξ , (5) to decrease the size of the dynamical gaps at ±(cid:126)Ω/2, ξ −ξ thisisverydifferentfromthecaseofmonolayergraphene where ε0 = (γ3/γ0)2γ1 ≈ 4 meV, ∆ = ηF(cid:48)2(Ω1(cid:48) + γ3γ2Ω2(cid:48)), wthheerrmeotrrei,gowneaolbwtaarinpiangtimeffee-cintsdeapreenmdeuncthewffeeacktievre15H.aFmuirl-- 0 ξ = 1(−1) f√or valley K(K(cid:48)) , k± = (kx ± iky)/k0, tonianwhichservesasastartingpointforthedetermina- k = 2γ γ /( 3aγ2), a = 0.246 nm and F(cid:48), Ω(cid:48) are now tion of the topological properties of the associated low- 0 3 1 0 dimensionlessparametersgivenink andε unitsrespec- energy states. We find that while for both polarizations 0 0 tively. 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