Polarization of graphene in a strong magnetic field beyond the Dirac cone approximation Shengjun Yuan1, Rafael Rolda´n2, and Mikhail I. Katsnelson1 1Institute for Molecules and Materials, Radboud University of Nijmegen, NL-6525AJ Nijmegen, The Netherlands 2Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco E28049 Madrid, Spain (Dated: January 24, 2012) Inthispaperwestudytheexcitationspectrumofgrapheneinastrongmagneticfield,beyondthe Diracconeapproximation. Thedynamicalpolarizabilityisobtainedusingafullπ-bandtight-binding model where the effect of the magnetic field is accounted for by means of the Peierls substitution. The effect of electron-electron interaction is considered within the random phase approximation, from which we obtain the dressed polarization function and the dielectric function. The range of 2 validityoftheLandaulevelquantization withinthecontinuumapproximation isstudied,aswellas 1 the non-trivial quantization of the spectrum around the Van Hove singularity. We further discuss 0 the effect of disorder, which leads to a smearing of the absorption peaks, and temperature, which 2 activates additional inter-Landaulevel transitions induced by theFermi distribution function. n a J I. INTRODUCTION larities (VHS). In this paper, we present a complete theoretical study 1 2 One of the most remarkable features of graphene is of the density of states (DOS), the polarizability and di- its anomalousquantum Halleffect (QHE), whichreveals electric function of graphene in a strong magnetic field, ] the relativistic character of the low energy carriers in calculatedfromaπ-bandtight-bindingmodel. Themag- l al this material.1,2 In fact, the linear electronic dispersion netic field has been introduced by means of a Peierls h of graphene near the neutrality point leads to a rela- phase,3,26 and the effect of long range Coulomb interac- - tivistic quantization of the electrons’ kinetic energy into tion is accounted for within the random phase approxi- s e non-equidistantLandaulevels(LL),withthe presenceof mation(RPA). Our method allowsus to study the effect m a zero-energy LL, which is the characteristics spectrum of temperature, which leads to the activation of addi- for systems of massless Dirac fermions.3,4 As a conse- tional inter-LL transitions. We also study the effect of . at quence, the excitation spectrum and the screening prop- disorder in the spectrum, which leads to a smearing of m erties in graphene are different from those of a standard the resonance peaks. two-dimensional electron gas (2DEG) with a quadratic - d band dispersion, as it may be seen from the polarization n and dielectric function in the two cases.5–9 II. DESCRIPTION OF THE METHOD o The Coulomb interaction between electrons in com- c pletely filled LLs leads to collective excitations and to In this section we summarize the method used in the [ the renormalization of the electronic properties such as numericalcalculationofthe polarizabilityofgraphenein 1 the band dispersion and the Fermi velocity. These is- the QHE regime.48 A monolayer of graphene consists of v sues have been studied both theoretically10–16 and ex- twotriangularsublatticesofcarbonatomswithaninter- 4 perimentally, in the framework of cyclotron resonance atomic distance of a 1.42 ˚A. By considering only first 5 ≈ 4 experiments.17–21 However, most of the theoretical work neighbor hopping between the pz orbitals, the π-band 4 hasbeenbasedonthe continuumDiracconeapproxima- tight-binding Hamiltonian of a graphene layer is given . tion, which does not apply when high energy inter-LL by 1 transitions are probed.22 In recent experimental realiza- 0 2 tion of “artificial graphene”,23 a two-dimensonal nanos- H =− (tija†ibj +h.c.)+ vic†ici, (1) 1 tructure that consists of identical potential wells (quan- hXi,ji Xi : tum dots) arranged in a honeycomb lattice, the lattice v constant (a 130 nm) is much larger than the one in where a† (b ) creates(annihilates) anelectronon sublat- Xi graphene(a ∼ 0.142nm). This providesa wayto study tice A (iB) iof the graphene layer, and t is the nearest 0 ij ∼ r graphene in the ultra-high magnetic field limit, since a neighbor hopping parameter, which oscillates around its a perpendicularmagneticfieldin“artificialgraphene”cor- mean value t 3 eV.27 The second term of H accounts respondes to aneffective field which is (a/a )2 8 105 fortheeffecto≈fanon-sitepotentialv ,wheren =c†c is 0 ∼ × i i i i times larger than in graphene. Furthermore, the re- theoccupationnumberoperator. Forsimplicity,weomit cently developed techniques of chemical doping24 and the spin degree of freedom in Eq. (1), which contributes electrolytic gating25 have enabled doping graphene with onlythroughadegeneracyfactor. Inournumericalcalcu- ultrahigh carrier densities, where the band structure is lations, we use periodic boundary conditions. The effect no longer Dirac-like and one should take into account of a perpendicular magnetic field B = Bzˆ is accounted the full π-band structure including the Van Hove singu- by means of the Peierls substitution, which transforms 2 the hopping parameters according to26 R 2π j t t exp i A dl , (2) ij ij → Φ0 R · ! Z i M whereΦ =hc/eisthefluxquantumandAisthevector 0 q potential, e. g. in the Landau gauge A = ( By,0,0). We will calculate the DOS and the polarizatio−nfunction ky K’ Γ θ K of the system by using an algorithm based on the evo- lution of the time-dependent Schr¨odinger equation. For thiswewillusearandomsuperpositionofallbasisstates as an initial state ϕ (see e. g. Refs.28,29) | i ϕ = a c†c 0 , (3) | i i i i| i i X where ai are random complex numbers normalized as kx a 2 = 1. The DOS, which describes the number i| i| of states at a given energy level, is then calculated as a P Fouriertransformofthetime-dependentcorrelationfunc- FIG. 1: Constant energy contours obtained from the band tions dispersion Eq. (13). The thick black lines correspond to dis- persion at the VHS |ǫ| = t. Notice that the CEC are cen- d(ǫ)= 1 ∞ eiǫτ ϕe−iHτ ϕ dτ, (4) tered around the Dirac points for |ǫ| < t and around Γ for 2π h | | i |ǫ| > t. For illustrative reasons, the hexagonal BZ is shown Z−∞ in white. Forundopedgraphene, thevalenceand conduction with the same initial state defined in Eq. (3). The dy- bandstoucheach otherat theverticesof thehexagon,theso namical polarization function can be obtained from the calledDiracpoints(KandK’).TheVanHovesingularitylies Kubo formula30 at the M point, and we have defined θ as the angle between thewave-vectorq and the kx-axis. i ∞ Π(q,ω)= dτeiωτ [ρ(q,τ),ρ( q,0)] , (5) V h − i Z0 Long range Coulomb interaction is considered in the where V denotes the volume (or area in 2D) of the unit RPA, leading to a dressed particle-hole polarization cell, ρ(q) is the density operator given by Π(q,ω) χ(q,ω)= , (9) ρ(q)= c†c exp(iq r ), (6) 1 V (q)Π(q,ω) i i · i − Xi where andtheaverageistakenoverthecanonicalensemble. For 2πe2 the case of the single-particle Hamiltonian, Eq. (5) can V (q)= (10) κq be written as29 2 ∞ is the Fourier component of the Coulomb interaction in Π(q,ω)= dτeiωτ two dimensions, in terms of the background dielectric −V Z0 constant κ. Furthermore, the dielectric function of the Im ϕ nF (H)eiHτρ(q)e−iHτ [1 nF (H)]ρ( q) ϕ , system is calculated as × h | − − | i (7) ε(q,ω)=1 V (q)Π(q,ω). (11) − where The collective modes lead to zeroes of ε(q,ω), and their 1 dispersion relation is defined from n (H)= (8) F eβ(H−µ)+1 Re ε(q,ω )=1 V(q)Π(q,ω )=0, (12) pl pl is the Fermi-Dirac distribution operator, β = 1/k T − B whereT isthetemperatureandk istheBoltzmanncon- which leads to poles in the response function (9). The B stant, and µ is the chemical potential. In the numerical technicalitiesabouttheaccuracyofthenumericalresults simulations, we use units such that ~ =1, and the aver- have been discussed elsewhere.28,29,31 Here we just men- ageinEq.(7)isperformedovertherandomsuperposition tion that the efficiency of the method is mainly deter- Eq. (3). The Fermi-Dirac distribution operator n (H) mined by three factors: the time interval of the propa- F and the time evolution operator e−iHτ can be obtained gation, the total number of time steps, and the size of bythestandardChebyshevpolynomialdecomposition.29 the sample. The methodismoreefficientinthe presence 3 1.0 BB==2200TT 0.25 B=20T 0.8 0.20 S (1/t)0.6 S (1/t)0.15 O O D0.4 D0.10 0.2 0.05 0.0 0.00 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 E (t) E (t) 1.0 B=50T 0.25 B=50T 0.8 0.20 S (1/t)0.6 S (1/t)0.15 O O D0.4 D0.10 0.2 0.05 0.0 0.00 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 E (t) E (t) FIG.2: Left: DOSofamonolayerofgrapheneinamagneticfield,fortwodifferentvaluesofB. Right: zoomofthelowenergy region ofthespectrum. TheredverticallinesindicatethepositionoftheLLsinthecontinuumDiracconeapproximation,Eq. (19). Wehaveused a sample made of 4096×4096 atoms. of strong magnetic fields. Because for weak fields (e.g., structure is electron-hole symmetric, and the constant B <1T)theenergydifferencebetweenLLsbecomevery energycontours(CEC)obtainedfromEq. (13)areshown small,thismakesthatthetotalnumberoftimestepsand in Fig. 1. For undoped graphene (µ = 0), which is the the sizeofthesamplehavetobe largeenoughto provide bandfillingthatwewillconsiderallalongthispaper,the the necessary energy resolution in the numerical simula- Fermi surface consists of just six points at the vertices tion. of the BZ. In this case, the low energy excitations can be described by means of an effective theory obtained from an expansion of the dispersion Eq. (13) around III. DENSITY OF STATES AND EXCITATION the K points. This leads to an approximate dispersion SPECTRUM ǫ(k) λv k, where v =3ta/2 is the Fermi velocity. F F ≈ If we now consider the effect of a perpendicular mag- In this section we study the DOS and the excitation neticfield,theLandauquantizationofthekineticenergy spectrum of a graphene layer in a magnetic field, ne- leads to a set of LLs, which can be described from the glectingtheeffectofdisorderandelectron-electroninter- semiclassical condition3,32 action. The B =0dispersionrelationofthe π bandsob- 2π 1 Γ(C) tainedfroma tight-binding model with nearest-neighbor S(C)= n+ (15) l2 2 − 2π hopping between the pz orbitals is B (cid:18) (cid:19) where ǫ(k)=λtφk (13) | | where λ= 1 is the band index and S(C)= dkxdky (16) ± Z Z √3 ǫ(kx,ky)≤ǫn φk =1+2ei3kxa/2cos 2 kya . (14) is the area enclosed by the cyclotron orbit C in mo- ! mentum space [for circular orbits S(C) is just πk2], The band dispersion Eq. (13) consists of two bands that l = ~c/eB is the magnetic length, n is the LL index B toucheachotherinthe verticesofthe hexagonalBrilloin andΓ(C)istheBerryphase. Ingraphene,aswewilldis- p zone (BZ) (Fig. 1), which are the so called Dirac points. cuss in the next section, Γ(C)=π for orbits around the In the absence of longer range hopping terms, the band K and K’ points, and 0 for orbits around the Γ point.33 4 the number of atoms used in the calculation, as well as 3.0 the totalnumberoftime steps,whichdetermines the ac- B=500T 3 3 curacy of the energy eigenvalues. In order to check the 2.5 rangeof validity of the continuum approximation,in the DOS (1/t)2 DOS (1/t)2 right-hand side of Fig. 2 we show a zoom of the pos- 1 1 2.0 itive low energy part of the DOS for the two values of 1/t) 00.90 0.95 E1(.t0)0 1.05 1.10 20.40 2.45 E2(.t5)0 2.55 2.60 B, comparing the DOS obtained with the full π-band S (1.5 tight-binding model Eq. (4) [black lines] to the Dirac O cone approximationof Eq. (19) [verticalred lines]. Con- D trary to multi-layer graphenes, for which trigonal warp- 1.0 ing effects are important atrather low energies,34 we see that the deviations of the LL positions in the contin- 0.5 uum approximation Eq. (19) with respect to the full π-band model are weak even at energies of the order 0.0 of ǫ 0.3t 1eV, in agreement with magneto-optical -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 trans∼mission∼spectroscopy experiments.22 E(t) Amuchlessinvestigatedissue isthe effectofthe mag- netic fieldonthe DOSaroundthe VHS ǫ t. Forillus- FIG.3: DOSobtained fromEq. (4)atB =500T. Theinsets | |≈ trativereasonsweshowinFig. 3thenumericalresultsfor show a zoom of the DOS around the VHS |ǫ| ≈ t and at the DOS of a graphene layer at an extremely high mag- |ǫ|≈2.5t. netic field of B = 500T.49 At this energy the LL quan- tization is highly nontrivial because of the saddle point From the energy dependence of S(C) one can calculate in the band structure at which there is a transition from the energy of the Landau levels as CECsencirclingtheDiracpoints,toCECsencirclingthe Γ point,asit canbe seeninFig. 1. Becauseinthe semi- ǫ =S−1 2π n+ 1 Γ(C) , (17) classical limit, the cyclotron orbits in reciprocal space n l2 2 − 2π follow the CECs, we have that at the saddle point there (cid:18) B (cid:20) (cid:21)(cid:19) is a change in the topological Berry phase Γ(C) from where S−1(x) is the inverse function to S(x). Using Eq. Γ(C)= π fororbitsencirclingtheDiracpoints(ǫ <t) (17), it is easy to check that the LL quantization corre- to Γ(C)±= 0 for orbits encircling the Γ point (ǫ |>|t).33 sponding to a low energy parabolic band ǫ(k)=k2/2m | | b The different character of the cyclotron orbits at both (where m is the effective mass) with a Γ(C) = 0 Berry b sides of the saddle point leads to two series of LLs, with phase, is ǫn = ωc(n+1/2), where ωc = eB/mb is the different cyclotron frequencies ω = eB/m , that merge c b cyclotron frequency. On the other hand, a linearly dis- at the VHS, as it may be seen in the left-hand side inset persingbandastheoneforgrapheneleadstoaLLquan- of Fig. 3. Because of the effective mass m below the b tization around the Dirac points as VHS is largerthan the one aboveit (the band below the v saddle point is flatter than above it), the cyclotron fre- ǫ =λǫ =λ F√2n √Bn. (18) λ,n n lB ∝ quencies are also different ωc(|ǫ| < t) < ωc(|ǫ| > t), and consequentlythe LLs aremoreseparatedfor ǫ >t than | | for ǫ <t. The possibilityofplacingthe chemicalpoten- | | tialattheVHSwouldbringthechanceofstudyinghighly A. Density of states anomalous inter-LL transitions, due to the different sep- arationof the LLs above and below the VHS. Finally, at TheDOSclosetotheDiracpointcanbeapproximated an even higher energy, the LL quantization is quite sim- by27 ilar to that of a 2DEG with a parabolic dispersion, with asetofroughlyequidistantLLs,asitmaybeseeninthe 2A ǫ d (ǫ) c | | (19) right-hand side inset of Fig. 3 for ǫ 2.5t. However, Dirac ≈ π v2 | | ≈ F we emphasize that for realistic values of magnetic field, the LL quantization in graphene is inappreciable in this where A = 3√3a2/2 is the unit cell area. In Fig. 2 c range of energies, and the DOS for energies ǫ & 0.7t is we show the DOS for two different values of the mag- similar to the DOS at B =0,29 as it may be|se|en in Fig. netic field. The black line corresponds to the numerical 2. tight-binding result obtained from Eq. (4). Near ǫ = 0 we notice the presence of a zero energy LL surrounded by a set of LLs whose separation decreases as the en- B. Particle-hole excitation spectrum ergy increases, leading to a stacking of the LLs as we move away from the Dirac points. The presence of a finite broadening in these LLs is due to the energy reso- Theparticle-holeexcitationspectrum(PHES)fornon- lution of the numerical simulations, which is limited by interacting electrons, which is the part of the ω q − 5 0.14 0.14 -1 0.12 -1 0.12 q=0.1a -1 B=0 q=0.1a q=0.15a 0.10 B=0 q=0.15-a1-1 0.10 q=0.2a-1 2 ) B=0 q=0.2a -1 2 ) 1- a0.08 B=20T q=0.1a -1 1- a0.08 B=20T - (t0.06 BB==2200TT qq==00..125a-a1 - (t0.06 =30o m o m -I0.04 =30 -I0.04 0.02 0.02 (a) (b) 0.00 0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.1 0.2 0.3 0.4 0.5 (t) (t) 0.14 0.14 -1 0.12 -1 0.12 q=0.1a B=0 q=0.1a -1 -1 q=0.2a B=0 q=0.15a 0.10 -1 0.10 ) B=0 q=0.2a ) -1-2 (ta00..0068 BBB===555000TTT qqq===000...112a5a--a11-1 -1-2 (ta00..0068 B==050T m m =0 -I0.04 -I0.04 0.02 0.02 (c) (d) 0.00 0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 (t) (t) FIG. 4: −ImΠ(q,ω) for different values of wave-vector q and strength of the magnetic field B. The angle θ defines the orientation of the wave-vector in the Brilloin zone (see Fig. 1). Plots (a) and (c) show the polarization in the whole energy range. For comparison we show the polarization at B =0. In plots (b) and (c) we show the low energy part of the spectrum. The vertical black lines signal theenergy of the particle-hole processes expected from Eq. (20). plane where ImΠ(q,ω) is non-zero, defines the region of incides with the the one at B =0 at high energies. This the energy-momentum space where particle-hole excita- isduetothealmostnegligibleeffectofthemagneticfield tions are allowed. For undoped graphene (µ = 0), the on the DOS at energies ǫ & 0.7t for B . 50T, as we | | particle-hole excitations correspond to inter-band tran- saw in the previous section. This part of the spectrum sitions across the Dirac points. In Fig. 4 we show is dominated, as in the B = 0 case,31,36 by a peak of ImΠ(q,ω) for different values of wave-vectorand mag- ImΠ(q,ω) around ω 2t, which is due to particle-hole − ≈ netic field. Two different orientations of q are shown, transitions between states of the Van Hove singularities namely along the Γ-M and Γ-K directions. First, due to of the valence and the conduction bands at ǫ t and ≈ − the low energy linear dispersion relation and to the ef- ǫ t respectively. ≈ fectofthechiralityfactor(orwave-functionoverlap)that However, the low energy part of the spectrum is com- suppresses backscattering,we observe that the strongest pletely different to its zero magnetic field counterpart, contribution to the polarization is concentrated around and it is dominated by a series of resonance peaks at ω v q. In fact, at B = 0 it was shown35 that some given energy values. For the undoped case stud- F ImΠ≈(q,ω) q2(ω2 v2q2)−1/2, which implies an infi- ied here, the possible excitations correspond to inter-LL ∼ − F nite response of relativistic non-interacting electrons in transitionsof energyωn,n′ =ǫn′+ǫn, where n′ is the LL graphene at the threshold ω =v q. The main difference index of the particle in the conduction band (λ = +1) F of the PHES at finite magnetic field with respect to its and n is the LL index of the hole in the valence band B =0counterpartinthislowenergyrange,isthatinthe (λ = 1). In the continuum approximation, they have − B = 0 case Π(q,ω) presents a series of peaks of strong an energy 6 spectralweight,duetotheLLquantizationofthekinetic energy,thatwewilldiscussinmoredetailbelow. Weno- ωn,n′ =√2(vF/lB)(√n′+√n). (20) ticethat, forthe realisticvaluesofmagneticfieldusedin Fig. 4, the spectrum at finite magnetic field roughly co- The energy corresponding to each of these transitions is indicated by a black vertical line in Fig. 4(b) and 6 (d), where we show a zoom of ImΠ(q,ω) that corre- − 3.0 sponds to the low energyLL transitions about the Dirac -1 q=B=0 0.1a (a) points. Noticethat,incontrasttoastandard2DEGwith -1 2.5 q=B=0 0.15a a parabolicdispersionandequidistantLLs,the relativis- -1 q=B=0 0.2a tic quantization of the energy band in graphene makes 2.0 B=50T q=0.1a-1 -1 that in a fixed energy window athigh energies,there are B=50T q=0.15a / -1 morepossibleinter-LLexcitationsfromthelevelninthe m11.5 B=50T q=0.2a valencebandtotheleveln′ intheconductionband,than -I =30o at low energies. As a consequence, there is a stacking of 1.0 neighboring LL transitions as we increase the energy of the excitations, which manifests itself in a continuum of 0.5 possible inter-LL transitions from a given energy, which for the case of Fig. 4(b) at B =20T is ω/t&0.25. 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Contraryto what one cannaively expect, only at very (t) strong magnetic fields and for small values of ω and q, 2.5 the peaks of ImΠ(q,ω) occur at the energies given by q=0.1a-1 (b) Eq. (20). I−n fact, we can see in Fig. 4(d) that for 2.0 q=0.15a-1 -1 B =50T and for the smaller value of q shown (red line), q=0.2a thepeaksof ImΠ(q,ω)matchverywelltheenergiesfor − 1.5 B=50T the inter-LL transitions given by Eq. (20). However, at / o 1 =30 weaker magnetic fields and/or larger wave-vectors, the m peaks of the polarization function do not coincide any -I1.0 more with every of the inter-LL transitions given by Eq. (20). Infact,thereisaseriesofpeaksofImΠ(q,ω),corre- 0.5 spondingtoregionsofthePHESofhighspectralweight, whichcanbe understoodfromthe formofthe wavefunc- 0.0 tions of the electron and the hole that overlap to form 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 an electron-hole pair. A detailed discussion about the (t) structure ofthe PHESin graphenein comparisonwith a 2DEG can be found in Ref. 8. Here we just remember FIG. 5: (a) Loss function −Im 1/ε(q,ω) in the RPA, for that the modulus of the LL wavefunction, due to the ze- different valuesof wave-vector. Theresults for graphenein a ros of Laguerrepolynomials,presents a number of nodes magnetic field of B = 50T are compared to the B = 0 case. (b) Zoom of thelow energy part of thespectrum. that depend on the LL index n. On the other hand, the existence of an electron-hole pair will be possible if thereisafiniteoverlapoftheelectronandholewavefunc- tions, which will define the form factor for graphene in broad peak at ω 2t, associated to the π-plasmon.31 the QHE regime, n,n′(q). Because the node structure When the graphen∼e layer is subjected to a strong per- F of the single-particle wavefunctions will be transfered to pendicular magnetic field, Im[1/ε(q,ω)]presents a series n,n′(q)2, all together will lead to a highly modulated of prominent peaks at low frequencies, associated to col- |F | spectral weight in the PHES, as it is seen in Fig. 4(b) lective modes in the QHE regime, as it may be seen in and (d). Fig. 5(b). Similarly to the single-particle case discussed in Sec. IIIB, the strength of the peak is determined by theCoulombmatrixelementsV(q) n,n′(q)2,whichde- |F | IV. COLLECTIVE MODES pends stronglyonthe wave-vectorq. We emphasize that these modes can not be understood as a simple many- bodyrenormalizationofthedispersionlessinter-LLtran- In the previous section we have discussed the excita- sitions given in Eq. (20), because only the low energy tion spectrum in the absence of electron-electron inter- and long wavelength modes have their non-interacting action. In this section we include in the problem the effect of long range Coulomb interaction. The polariza- counterpart ωn,n′ associated to a specific single-particle electron-hole transition with well defined indices n and tion and dielectric functions are calculated within the n′. As we go to higher energies and/or weaker mag- RPA, Eqs. (9) and (11). Within this framework,the ex- netic fields, the relativistic LL quantization of graphene istence of collective excitations will be identified by the leads to a so strong LL mixing that the collective modes zeros of the dielectric function or equivalently by the di- vergences of the loss function Im[1/ǫ(q,ω)], which is cannot be labeled any more in terms of single-particle − excitations,38,39asinthecaseofa2DEGwithaquadratic proportional to the spectrum measured by Electron En- ergyLossSpectroscopy(EELS).37 In Fig. 5 we showthe dispersion and a set of equidistant LLs.40 lossfunctionofgrapheneinamagneticfield,ascompared In Fig. 6 we compare the real part of the dielectric to the one at B =0. At B =0 the main structure is the function Re ε(q,ω) for zero and finite magnetic field. 7 1.0 15 15 1.2 B=50T B=50T Clean B=0 B=0 10 10 0.8 vr=0.2t 1.0 tr=0.1t 0.8 Re 05 Re 05 1/t)0.6 B=50T 0.6 0.96 0.98 1.00 1.02 1.04 ( q=0.1a-1 q=0.15a-1 S -50.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -50.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 DO0.4 (t) (t) 15 0.2 B=50T B=0 10 (a) 0.0 Re 5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 E (t) 0 0.25 -1 q=0.2a -5 Clean 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 vr=0.2t (t) 0.20 tr=0.1t ) FIG. 6: Reε(q,ω) for B=0 (red lines) and B =50T (black 01/t.15 B=50T ( lines), for thevalues of thewave-vectorsused in Fig. 5. S O D 0.10 The zeros of Re ε(q,ω) correspond to the frequencies of 0.05 the undamped collective modes. Contrary to the B = 0 case, for which there is no collective modes for undoped (b) graphene at the RPA level, at B =0 we observe a num- 0.00 6 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 ber of well defined zeros for Re ε, which correspond to E (t) coherent and long-lived linear magneto-plasmons.14 The divergence of ImΠ(q,ω) at ω = vFq and the absence of FIG. 7: (a) DOS of clean graphene (black lines) and of dis- backscatteringingraphenemakethat,asweincreasethe ordered graphene with a random on-site potential (red lines) wave-vector q, the more coherent collective modes are and with a random renormalization of the hopping integrals defined also for higher frequencies, namely around the (blue lines). The inset shows the smearing of the VHS peak threshold ω v q, which is the frequency associated to duetodisorder. (b)Zoomofthelowenergypartofthespec- F the highest p≈eaks in Fig. 5(b). For even higher energies, trum. the main contribution to the modes of large frequencies areinter-LLtransitionsbetweenwellseparatedLLs,with shown to lead to a splitting of the n = 0 LL,41,42 origi- the subsequent reduction in the overlap in the electron- nated from the breaking of the sublattice and valley de- hole wavefunctions. Therefore, the collective modes will generacy. Other kinds of disorder as vacancies create suffer a stronger Landau damping as higher frequencies midgapstates thatmake the n=0LL to remainrobust, are probed. whereastherestofLLsaresmearedoutduetothe effect of disorder.29 In Fig. 7 we show the DOS of graphene in a perpendicular magnetic field of B = 50T for different A. Effect of disorder kind of disorder, as compared to the clean case. We let the on-site potential v to be randomly distributed (in- i Nowwefocusourattentionontheeffectofdisorderon dependently on each site i) between vr and +vr. On − the DOS and on the excitation spectrum of graphene in the other hand, the nearest-neighbor hopping tij is ran- the QHE regime. In general, disorder leads to a broad- dom and uniformly distributed (independently on sites eningofthe LLs,withextended(delocalized)statesnear i,j) between t tr and t+tr. At high energies, as we − the center of the original LL, and localized states in the have seen in Sec. IIIA, the DOS for this strength of tails. We consider here two different kinds of disorder, the magnetic field is quite similar to the DOS at B =0. namely random local change of the on-site potentials v , Therefore, as in the zero field case,29,31 the main effect i which acts as a chemical potential shift for the Dirac awaythe Diracpointisasmearingofthe VHSat ǫ =t, | | fermions, and random renormalization of the hopping as it is observed in the inset of Fig. 7(a). amplitudes t , due e. g. to changes of distances or an- WelldefinedLLsoccuraroundtheDiracpoint,andthe ij gles between the carbon p orbitals. They enter in the effectofdisorderonthepeaksisobservedinFig. 7,where z single-particle Hamiltonian as given in Eq. (1). The ef- weshowazoomofthelowenergypartoftheDOSaround fect of correlated long-range hopping disorder has been ǫ = 0. Both kinds of disorder (random on-site potential 8 0.10 0.14 0.14 00..0089 Cvrl=e0a.n2t 0.12 Cvrl=e0a.n2t 0.12 Cvrl=e0a.n2t 0.07 tr=0.1t 0.10 tr=0.1t 0.10 tr=0.1t -1-2 (ta)00..0056 Bq==05.01Ta-1 -1-2 (ta)00..0068 Bq==05.01T5a-1 -1-2 (ta)00..0068 Bq==05.02Ta-1 m0.04 =0 m =0 m =0 -I0.03 -I0.04 -I0.04 0.02 0.02 0.02 0.01 (a) (b) (c) 0.00 0.00 0.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 (t) (t) (t) 2.5 1.0 1.0 Clean Clean Clean 2.0 vr=0.2t 0.8 vr=0.2t 0.8 vr=0.2t tr=0.1t tr=0.1t tr=0.1t 1.5 B=50T 0.6 B=50T 0.6 B=50T 1/ -1 1/ -1 1/ -1 m q=0.1a m q=0.15a m q=0.2a -I1.0 -I0.4 -I0.4 0.5 0.2 0.2 (d) (e) (f) 0.0 0.0 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 (t) (t) (t) 15 Clean Clean vr=0.2t 10 vr=0.2t 10 10 tr=0.1t tr=0.1t Clean vr=0.2t 5 5 5 tr=0.1t Re 0 Re 0 Re 0 -5 B=50T-1 -5 B=50T -1 B=50T q=0.1a q=0.15a -1 q=0.2a -10 (g) (h) -5 (i) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 (t) (t) (t) FIG. 8: (a)-(c) Non-interacting single-particle excitation spectrum of graphene in a magnetic field of B =50T, as defined by −ImΠ(q,ω), for different kinds of disorder and for different values of the wave-vector q. The black vertical lines signal the energyoftheelectron-holeprocessesdefinedbyEq. (20). (d)-(f)Lossfunction−Im[1/ε(q,ω)]intheRPA.(g)-(i)Realpartof thedielectric function Reε(q,ω) in theRPA. andrandomhopping)leadstoasimilareffect,producing gies the position of the resonance peaks of disordered a broadeningof the LLs. We canalso observe,especially graphene coincides with the position for the clean case, for the highest LLs shown in Fig. 7(b), a tiny but ap- we observe a redshift of the resonance peaks as we con- preciableredshift ofthe positionofthe center ofthe LLs sider inter-LL transitions of higher energies. This is due with respect to its original position, in agreement with to the change in the position of the high energy LLs of previousworks.42,43 Thefullself-consistentBornapprox- disordered graphene with respect to the original LLs, as imationcalculationsforgraphenewithunitaryscatterers we have discussed previously [see Fig. 7(b)]. ofRef. 44leadedtoarathersignificantchangeinthepo- The presence of disorderwill also affect the dispersion sition of the LLs towards higher energies. However, the relation and coherence of the collective modes when the exacttransfer matrix anddiagonalizationcalculationsof effectofelectron-electroninteractionis included. Inpar- Ref. 45 found only a small shift of the LL position for ticular,theeffectofashortrangedisorderongraphenein very strong disorder. a magnetic field can lead, due to the possibility of inter- We now study the effect of disorder on the PHES. In valleyprocessesassociatedtothebreakdownofsublattice Fig. 8(a)-(c) we show ImΠ(q,ω) for different values symmetry,tothelocalizationofsomecollectivemodeson of q, and for different k−inds of disorder. First we no- the impurity.46 In Fig. 8(d)-(f) we show the effect of a tice a smearing of the resonance peaks associated to the randomon-sitepotentialorarandomhoppingrenormal- LL broadening due to disorder. Whereas for low ener- ization on the loss function. First, we can observe an 9 with electrons (holes) in the conduction (valence) band. 00..1102 TB===032000TK 00..2205 TB===035000TK NfuontcitcieonthtahtatattrTav=ers0esatnhdefnor=µ0=LL0,.OnFf(cǫo)urissej,utshteansutmep- S (1/t)00..0068 OS (1/t)0.15 bofernof(ǫa)c,tgivraowtesdaLsLwse,iwnhcricehasaertehtehtoesmepcreorassteudrebayntdh/eortaaisl DO D0.10 F 0.04 wedecreasethemagneticfield. Thepopulationeffectdue 0.02 0.05 to the thermal excitations of carriers has been observed 0.00 0.00 by far infrared transmission experiments.47 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 E (t) E (t) In Fig. 10 we show the single-particle polarization 0.12 T=1000K 0.25 T=1000K and the loss function for two values of temperature and 0.10 B==020T 0.20 B==050T magnetic field. At room temperature and for the rather DOS (1/t)00..0068 DOS (1/t)00..1105 satllroownegdmealegcntertoinc-fiheolldestcraonnssiitdieornesdairneotuhrecsaalmcuelaatsioinn, tthhee 0.04 zero temperature limit (see the top panels of Fig. 9). 0.02 0.05 Therefore, the peaks of ImΠ for T = 300 K are cen- 0.00 0.00 tered at the frequencies of inter-LL transitions marked -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 E (t) E (t) by the black vertical lines, which accounts only for the usualinter-bandtransitionsacrosstheDiracpoint. Fora FIG.9: DOSofcleangraphenefordifferentvaluesoftemper- considerable higher temperature of T = 103 K there are atureT andmagneticfieldB. Theshadedareaisasketchof additional electron-hole transitions (some of them intra- theFermi-Dirac distribution function for each case. band processes, especially important at low frequencies) whicharenowallowedduetotheeffectoftemperature,as markedby the redverticallines in Fig. 10(a)-(b). These important attenuation of the intensity peaks due to dis- thermally activated inter-LL transitions at high temper- order. Second, as we have discussed above, the position aturescontributetotheadditionalspectralweightofthe of the peaks of disordered and clean graphene coincides PHES of Fig. 10(a)-(b). Finally, in Fig. 10(c)-(d) we at low energies, but not at high energies, where the res- show the loss functions corresponding to the magnetic onance peaks of the loss function of disorderedgraphene fieldsandtemperaturesdiscussedabove. Aswehavedis- are shifted with respect to clean graphene. Although we cussedabove,thepeaksofIm1/εcorrespondtotheposi- obtain a similar renormalization of the spectrum for the tion of collective excitations. Here we find, in agreement two kinds of disorder considered here, we reiterate that withprevioustight-bindingandband-likematrixnumer- this effect is highly dependent on the type disorder con- ical methods,16 a weak but appreciable renormalization sidered, as well as the theoreticalmethod used to obtain ofthecollectivemodepeakpositionasafunctionoftem- the spectrum. perature. This temperature dependence of the collective Finally, we mention that the disorder LL broaden- mode is easily noticed by comparing the red and blue ing leads to an amplification of the LL mixing discussed peaks at ω 0.12t of Fig. 10(d). above. As a consequence, some collective modes which ≈ are undamped for clean graphene, start to be Landau damped due to the effect of disorder. Indeed, in Fig. V. CONCLUSIONS 8(g)-(i) we can see how Eq. (12), which is the condi- tion for the existence of coherent collective modes, is fulfilled more times for the clean case than for the dis- Inconclusion,wehavestudiedtheexcitationspectrum ordered membranes, for which the collective modes are of a graphene layer in the presence of a strong magnetic more highly damped. field, using a full π-band tight-binding model. The mag- neticfieldhasbeenintroducedbymeansofaPeierlssub- stitution, and the effect of long range Coulomb interac- B. Effect of temperature: thermally activated tion has been considered within the RPA. For realistic electron-hole transitions values of the magnetic field, the LL quantization leads to well defined LLs around the Dirac point, whereas the In this subsection we discuss the effect of temperature DOSathigherenergiesisratersimilartothe oneatzero on the excitation spectrum, which enters in our calcu- field. However, we have shown that in the ultra-high lations trough the Fermi-Dirac distribution function Eq. magnetic field limit,23 for which the magnetic length is (8). Temperaturecanactivateadditionalinter-LLtransi- comparable to the lattice spacing, the LL quantization tionsduetofreethermallyinducedelectronsandholesin aroundtheVanHovesingularityishighlynontrivial,with thesample. InFig. 9wesketchtheLLsofundopedclean two different sets of LLs that merge at the saddle point. graphene that are thermally activated, for two different Ourresultsforthepolarizationfunctionshowsthat,at strengthsofthe magneticfield. Forthis, thecorrespond- high energies, the PHES is dominated, as in the B = 0 ingFermi-Diracdistributionfunctionissketchedoneach case,31 by the π-plasmon, which is associated to the en- plot, indicating the LLs that can be partially populated hancedDOSatthe VHSofthe π bands. Thelowenergy 10 0.030 0.050 T=300K 0.045 T=300K 0.025 T=1000K T=1000K 0.040 -1-2 (ta)00..001250 Bq===3020.01oTa-1 -1-2 m(ta)0000....000022330505 Bq===005.01Ta-1 m0.010 -I0.015 I - 0.010 0.005 (a) 0.005 (b) 0.000 0.000 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 (t) (t) 0.10 0.4 0.08 T=300K T=300K T=1000K 0.3 T=1000K 0.06 m1/ B=20T-1 m1/0.2 B=50T-1 -I0.04 q=0.1oa -I q=0.1a =30 =0 0.1 0.02 (c) (d) 0.00 0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 (t) (t) FIG. 10: (a)-(b) −ImΠ(q,ω) for two different values of T and B. The black vertical lines denote the position of the inter-LL transitionsgivenbyEq. (20)whicharepossibleatT =0. Theredverticallinescorrespondtotheenergyofthermallyactivated inter-LL transitions. (c)-(d)Loss function −Im[1/ε(q,ω)] for thesame values of T and B. part of the spectrum is however completely different to transitions,effectwhichisespeciallyrelevantatlowmag- its zero field counterpart. The relativistic LL quantiza- netic fields. tion of the spectrum into non-equidistant LLs lead to a peculiar excitation spectrum with a strong modulation ofthe spectralweight,whichcanbe understoodinterms VI. ACKNOWLEDGEMENT of the node structure of the electron-hole wavefunction overlap.8 Furthermore, we have shown that the presence of disorder in the sample lead to a smearing of the reso- ThesupportbytheStichtingFundamenteelOnderzoek nance peaks of the loss function, andto an enhancement der Materie (FOM) and the Netherlands National Com- of the Landau damping of the collective modes. Finally, puting Facilities foundation (NCF) are acknowledged. we have studied the effect of temperature on the spec- We thank the EU-India FP-7collaborationunder MON- trum, andshownthatit canactivate additionalinter-LL AMI and the grant CONSOLIDER CSD2007-00010. 1 K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, 7 E. H. Hwang and S. D. Sarma, Phys. Rev. B 75, 205418 M. I. Katsnelson, I. V. 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