Theory of resonant photon drag in monolayer graphene
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Theory of resonant photon drag in monolayer graphene
M.V.Entin and L.I.Magarill
Institute of Semiconductor Physics, Siberian Branch of Russian Academy of Sciences, Novosibirsk, 630090, Russia
D.L.Shepelyansky
Laboratoire de Physique Th´eorique (IRSAMC), Universit´e de Toulouse, UPS, F-31062 Toulouse, France andLPT (IRSAMC), CNRS, F-31062 Toulouse, France (Dated: January 31, 2010)Photon drag current in monolayer graphene with degenerate electron gas is studied under in-terband excitation near the threshold of fundamental transitions. Two main mechanisms generatean emergence of electron current. Non-resonant drag effect (NDE) results from direct transfer ofin-plane photon momentum q to electron and dependence of matrix elements of transitions on q .Resonant drag effect (RDE) originates from q -dependent selection of transitions due to a sharpform of the Fermi distribution in energy. The drag current essentially depends on the polarizationof radiation and, in general, is not parallel to q . The perpendicular current component appears ifthe in-plain electric field is tilted towards q . The RDE has no smallness connected with q and existsin a narrow region of photon frequency ω : | ~ ω − ǫ F | < ~ sq , where s is the electron velocity. PACS numbers: 73.50.Pz, 73.50.-h, 81.05.ue
I. INTRODUCTION
Though the theoretical study of two-dimensional car-bon has a long history [1],[2],[3],[4] only after experimen-tal evidence of existence of graphene as a stable two-dimensional crystal [5], [6–8] this material became verypopular. The presence of zero gap and zero electronmass, combined with a rather high mobility at room tem-perature, makes graphene an unique material for variousfundamental and applied problems. At present grapheneis intensively studied both theoretically and experimen-tally (see e.g. reviews [9, 10]).The study of graphene optics (see [11],[12]) is stimu-lated by the prediction that the absorption in monolayergraphene should be determined by the fundamental con-stant α = e / ~ c [13], [14] and its experimental evidence[15]. The investigation of coupling between photons andelectrons in graphene attracts now an active interest ofthe community (see e.g. [16, 17]). An observation ofamplified stimulated terahertz emission from opticallypumped epitaxial graphene heterostructures has been re-ported recently [18]. However, the photoinduced cur-rents, namely, photon drag and photogalvanic effects ingraphene were beyond of interest of the researchers. Inthis paper we present the theoretical analysis of theseeffects.The study of light pressure on solids has rather longhistory. The simplest variant is an instantaneous trans-mission of photon momentum to electrons. This processis permitted for interband transitions or in presence ofthe ”third body”, for example, phonons, other electrons,impurities. For a free particle this process is forbidden byconservation laws. Small value of the photon momentummakes Nonresonant photon Drag Effect (NDE) extremelyweak.At the same time there exists a less known variant ofthis effect, namely Resonant photon Drag Effect (RDE) which has no weakness of usual NDE [19],[20],[21]. Res-onance drag occurs when some partial kinetic propertyof electron gas sharply depends on electron energy. Asmall photon momentum gives an increase of the electronenergy, that can drastically change the relaxation time.This leads to a significantly different contributions to theelectron current for electrons exited along or oppositelyto the photon direction. In [21] the situation was stud-ied for interband transitions in weakly doped GaAs whenthe electron energy approaches the energy of longitudi-nal optical phonon. In this case electrons exited alongthe direction of photon have larger energy than electronsin opposite direction. Hence, their energy can exceed thethreshold for emission of optical phonon: they quicklyemit phonons and stop, while the opposite electrons willmove freely till they collide with impurity. This gives riseto the appearance of charge flow in the direction oppositeto the light ray.Here we develop another idea for RDE based on asharp Fermi distribution which forbids the transitionsbelow the Fermi energy ǫ F . This idea is illustrated inFig.1. Electrons are excited from the hole cone to theelectron cone by photons with frequency ω and wavevector Q . The conditions for resonant transitions are sk + s | k − q | = ω , ~ s | k | > ǫ F , where k is the electronmomentum counted from the cone point, s ≈ cm/s isthe electron velocity, and q is a projection of the wavevector Q of radiation to the plane of graphene. The firstcondition determines ellipse in k plane, the second lim-its a part of this ellipse accessible for transitions. Thewave vector tilts the transitions towards its direction.Fig. 1 shows the case when the frequency is close to2 ǫ F . The electrons in the figure are excited from theright segment of the Fermi surface contour. This resultsin electron flow rightwards. Since q ≪ k F the RDEappears when the frequency is close to 2 ǫ F , namely if | ω − ǫ F | < sq . Inside this window the current of RDE FIG. 1: (Color online) Interband phototransitions in n-type graphene. Left panel: diagram of transitions in themomentum-energy space. The hole cone is shifted in k spaceby the photon wave vector q . The transitions are permittedonly above the Fermi level. Right panel: projection to themomentum plane. Filled circle represents the Fermi sea, theelliptic curve corresponds to the energy conservation equation s | k − q | + sk = ω ; only momenta outside the Fermi circle arepermitted corresponding to the right segment of the ellipticcurve. has no smallness connected with q and can be estimatedas j ∼ esτ πα P/ ( ~ ω ), where e and s are the electroncharge and the velocity, πα is the opacity of graphene, τ is the transport relaxation time and P is the light in-tensity. Physical meaning of this estimation is evident: τ παP/ ( ~ ω ) is the instantaneous density of exited elec-trons which conserve their momentum. Being multipliedby the current of individual electrons es , this quantitygives the current density.Below we determine both NDE and RDE for interbandtransitions in monolayer graphene with degenerate elec-tron gas. Due to graphene electron-hole symmetry resultsare applicable to n- and p-type graphene. In general therelaxation process for electrons and holes are differentthat breaks electron-hole symmetry. For concreteness,we consider the n-type graphene. In this case the meanfree time of excited electrons is much longer than that ofholes since due to different distance from the Fermi levelholes can easier emit phonons. Thus, the contribution ofholes will be neglected.Fig. 2 illustrates a possible experiment on excitationof the drag current in a suspended graphene sheet placedin ( x, y ) plane. Light with frequency ω , wave vector Q ( Q = ω/c ) and amplitude of electric field E illuminatesgraphene plane. We consider transitions near the conesingularity. In this case the current is determined by theprojections of the electric field and the wave vector ontothe graphene layer.[22] These quantities are E ≡ e E =( E p cos β, E s ) and q = (1 , Q sin β , where β is the angleof incidence, E s and E p are amplitude components of the j E x E y E qQ y x FIG. 2: (Color online) Sketch of proposed experiment (seetext for details). electric field E perpendicular and parallel to the incidentplane. We ignore small modification of field caused bythe layer. II. BASIC EQUATIONS
The current of photon drag effect can be expressed viathe probability of transition g ( k ) from the hole state witha momentum k − q to the electron state with momentum k and the electron velocity v ( k ) = s k /k as j = 4 e Z d k π v ( k ) τ g ( k ) , (1)where the coefficient 4 accounts for the valley and spindegeneracies. The dependence on the photon momentumresults from the momentum and energy conservation lawsand the matrix elements for transition. For simplicity weput below ~ = 1.The two-band Hamiltonian near the Dirac point isˆ H ( k ) = s (cid:18) k x − ik y k x + ik y (cid:19) = s k σ . (2)Here σ is the vector of the Pauli matrices. The eigenval-ues and eigenvectors of the Hamiltonian (2) are ǫ ± ( k ) = ± sk and Ψ ± ( k ) = (1 , ± e iφ k ) / √
2, where φ k is the po-lar angle of the vector k . The different signs corre-spond to electrons and holes. The interaction withthe wave is determined by the matrix elements of thevelocity ∇ k ˆ H ( k ) = s σ between the hole and elec-tron states with the momenta k − q and k : v − + =(Ψ − ( k − q ) ∗ s σ Ψ + ( k )), correspondingly.The transition probability g ( k ) is g ( k ) = πe ω | Ev + − | δ ( sk + s | k − q | − ω ) θ ( ǫ k − ǫ F ) , (3)where θ ( t ) is the Heaviside function. The expression forcurrent Eq.(1) can be rewritten as j = e E s πω Z d k τ k | k | a jk e j e k δ ( sk + s | k − q |− ω ) θ ( sk − ǫ F )(4)where a jk = 1 s v − + ∗ j v − + k . (5)We utilized the symmetry of the tensor a ij resulting toinclusion of the field polarization in the combinations e ∗ i e j + e ∗ j e i only and independence on the degree of cir-cular polarization. Hence, without loss of generality onecan consider the field as linear-polarized and e as real.Due to the smallness of the wave vector q , as comparedto the electron momentum, one can expand all quanti-ties in powers of q . Expanding by q we can write theargument of the delta-function as sk + s | k − q | − ω ≈ sk − ω − sq cos φ k (we choose the direction of axis xalong q ). At the same time, q is comparable with 2 sk − ω and we keep ourselves from subsequent expansion of thedelta-function.Expanding the tensor a ij , we have a xx = sin φ k (1 + qk cos φ k ) ,a yy = cos φ k − qk sin φ k cos φ k , (6)2Re( a xy ) = − φ k cos φ k − qk sin φ k cos(2 φ k )From Eq.(4) we obtain for components of the current j x = − J Z min (1 ,a ) − dx √ − x ττ × n e x [ − x (1 − x )(1 + bx ) + 2 b (1 − x )(1 − x )] + e y [ − x (1 + bx ) + 4 bx (1 − x )] o ; (7) j y = 2 J e x e y Z min (1 ,a ) − dx ττ p − x × n − x (1 − x )(1 + bx ) + 2 b (3 x − o . (8)Here we have introduced the following notations: J = e ~ c cE π ~ ω | e | τ s, τ = τ | k = k F , a = ω − ǫ F sq , b = sqω . If τ is independent on the energy of electrons then theintegration in Eq.(7) can be done directly. The currenthas different values inside and outside the region | ω − ǫ F | < sq . If | ω − ǫ F | < sq then we have j x = − J p − a ((1 − a ) e x + (2 + a ) e y ) , (9) j y = − J (1 − a ) / e x e y . (10)These values represent resonant photon drag RDE. Itremains constant if q →
0. The value of resonant currentis determined by J . For the photon flow cE / π ~ ω =10 cm − s − , τ = 10 − s, J = 1 . · − A/cm. Thisapproximately corresponds to a power of 0 . W/cm forphotons with energy 0 . eV . -1.0 -0.5 0.0 0.5 1.0-1.5-1.0-0.50.0 j RDE /J ( F )/sq FIG. 3: Resonant photon drag current in units of J versusnormalized frequency ( ω − ǫ F ) /sq . The solid curve shows thelongitudinal component of current j x , the field is polarizedalong the projection of the wave vector on the plane ( θ = 0)and j y at θ = π/
4. The dashed curve shows j x at θ = π/ If | ω − ǫ F | > sq , then there is only NDE current. Itis proportional to q : j x = J πsq ω (3 e x − e y ) , (11) j y = 32 J πsqω e x e y . (12)The value of NDE is significantly smaller then the RDEvalue.In agreement with the simple estimates the RDE hasalways the direction opposite to the direction of lightwave vector. Its polarization dependence is explained bythe dependence of the directional diagram of excitation:most of carriers are excited perpendicular to the polar-ization. At the same time the Fermi sea limits the tran-sitions by the direction of the photon wave vector. Thiscircumstances together determine lower x-component ofcurrent if e || q in comparison with the case e ⊥ q andalso the appearance of y-component of the RDE current.In agreement with the system symmetry, j y exists onlyif the polarization has both e x and e y components. TheRDE current exists in a narrow window | ω − ǫ F | < sq which shrinks if q →
0. But inside this window RDE ismuch stronger than NDE so the later can be neglectedin this window.The sign of x-component of NDE depends on polariza-tion. This contradicts to a simple assumption accordingto which the current is mainly determined by kicks whichphotons give to electrons. The origin of this difference isthe dependence of the directional diagram on the smallwave vector q via the parameter a ij : at some polariza-tions electrons prefer to be excited in opposite directionto q . This explains the change of sign.Fig. 3 demonstrates the dependence of RDE currentcomponents on the frequency in the window | ω − ǫ F | We have studied the electron contribution to the pho-ton drag current. In fact, in the considered systemthe hole contribution also presents. The symmetry be-tween holes and electrons in a neutral system means thatthese contributions double. However, the result will bechanged if to take into account the difference betweenelectrons and holes caused by their different excitationenergy: while electrons are generated near the Fermi en-ergy the holes appear well below the Fermi energy. Thisleads to a strong difference between the relaxation times. In high-mobility samples at low temperature the mo-mentum relaxation time near the Fermi energy is muchgreater than far from the Fermi energy. At the sametime, quick relaxation of excited electrons (holes) to theFermi energy due to electron-electron interaction (de-scribed by e-e relaxation time τ ee ) conserves their mo-menta up to the moment when excitations reaches thetemperature layer. This results in equality of holes andelectrons contributions to the current. And vice versa,electron-phonon relaxation can cancel the hole contribu-tion if τ e − ph ≪ τ ee , where τ e − ph is the time of energyrelaxation due to electron-phonon collisions. Thus, theobtained current should be multiplied by a factor 2 inthe case of quick e-e relaxation and be kept unchangedin the opposite case. We note, that when the Fermi en-ergy tends to zero the system becomes symmetric.The RDE exists in a narrow energy range ∆ ǫ ≈ ~ sq ≈ ~ ωs/c near the Fermi energy. This means that the RDEis visible for temperature T < ∆ ǫ . For photons with ~ ω = 0 . eV this gives T < K .The observation of the resonant photon drag in mono-layer graphene is accessible to the modern experimentaltechnique that allows to investigate interesting aspects ofcoupling between photons and electrons in this material. IV. ACKNOWLEDGMENTS We thank A.D.Chepelianskii for useful discussions.The work was supported by grant of RFBR No 08-02-00506 and No 08-02-00152 and ANR France PNANOgrant NANOTERRA; MVE and LIM thank Laboratoirede Physique Th´eorique, CNRS for hospitality during theperiod of this work. [1] P.R. Wallace, Phys. Rev. , (1947) 622;[2] J.C. Slonczewski and P.R. Weiss, Phys. Rev. , (1958)272.[3] D.P. DiVincenzo and E.J. Mele, Phys. Rev. B , (1984)1685.[4] T. Ando, T. Nakanishi, and R. Saito, J. Phys. Soc. Japan , (1998) 2857.[5] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang,Y. Zhang, S.V. Dubonos, I.V. Grigorieva, andA.A. Firsov, Science, , 666 (2004).[6] C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud,D. Mayou, T. Li, J. Hass, A.N. Marchenkov, E.H. Con-rad, P.N. First, and W.A. de Heer, Science , 1191(2006).[7] K. S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang,M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos,A.A. Firsov , Nature , 197 (2005).[8] Y. Zhang, J.W. Tan, H.L. Stormer, and P. Kim, Nature , 201 (2005).[9] A.K. Geim and K.S. Novoselov, Nature Materials , 183 (2007).[10] A.H. Castro Neto, F. Guinea, N.M.R. Peres,K.S. Novoselov, and A.K. Geim, Rev. Mod. Phys., , 109 (2009).[11] L.A. Falkovsky and A.A. Varlamov, Eur. Phys. J. B, ,281 (2007).[12] L.A. Falkovsky, Phys. Usp. , 887 (2008) [Usp. Fiz.Nauk, , 923 (2008)][13] T. Ando, Y. Zheng and H. Suzuura, J. Phys. Soc. Japan, , 1318 (2002).[14] V.P. Gusynin, S.G. Sharapov and J.P. Carbotte, Phys.Rev. Lett. , 256802 (2006).[15] R.R.Nair, P.Blake, A.N.Grigorenko, K.S.Novoselov,T.J.Booth, T.Stauber, N.M.R.Peres, A. K. Geim, Sci-ence, , 1308 (2008).[16] J.Z. Bern´ad, U.Z¨ulicke, and K. Ziegler,arXiv:1001.3239[cond-mat] (2010).[17] K. Ziegler and A. Sinner, arXiv:1001.3366[cond-mat](2010).[18] T.Otsuji, H.Karasawa, T.Komori, T.Watanabe, H.Fukidome, M.Suemitsu, A.Satou, and Victor Ryzhii,arXiv:1001.5075[cond-mat] (2010).[19] A. A. Grinberg, Zh. Eksp. Teor. Fiz. , 989 (1970) [Sov.Phys.-JETP , 531 (1970)].[20] A.M.Danishevskii, A.A.Kastal’skii, S.M.Ryvkin, andI.D.Yaroshetskii, Zh.Exp.Teor.Fiz. , 544 (1970) [Sov.Phys. JETP, , 292 (1970)].[21] V.L. Al’perovich, V.I. Belinicher, V.N. Novikov, andA.S. Terekhov, , 557 (1981) [Sov. Phys.-JETP Lett., , 573 (1981)]. [22] The vertical component of the electric field also inter-acts with electrons, however, its action is weaker by theparameter k F d , where d is the vertical distance betweendangling bonds of neighboring atoms. In fact, this com-ponent results in the dynamical splitting of these statesand can be included in the Hamiltonian as σ z eE z d/d/