We show that the pseudospin being an additional degree of freedom for
carriers in graphene can be efficiently controlled by means of the
electron-electron interactions which, in turn, can be manipulated by changing
the substrate. In particular, an out-of-plane pseudospin component can occur
leading to a zero-field Hall current as well as to polarization-sensitive
interband optical absorption.Comment: 4 pages, 2 figure

Schliemann, John

Trushin, Maxim

English

arXiv

We show that the pseudospin, being an additional degree of freedom for carriers in graphene, can be efficiently controlled by means of the electron-electron interactions which, in turn, can be manipulated by changing the substrate. In particular, an out-of-plane pseudospin component can occur leading to a zero-field Hall current as well as to polarization-sensitive interband optical absorption

University of Regensburg Publication Server

Pseudospin in Optical and Transport Properties of Graphene

We show that the pseudospin being an additional degree of freedom for
carriers in graphene can be efficiently controlled by means of the
electron-electron interactions which, in turn, can be manipulated by changing
the substrate. In particular, an out-of-plane pseudospin component can occur
leading to a zero-field Hall current as well as to polarization-sensitive
interband optical absorption

arXiv:1105.0580v1  [cond-mat.mes-hall]  3 May 2011Pseudospin in optical and transport properties of grapheneMaxim Trushin and John SchliemannInstitute for Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany(Dated: May 4, 2011)We show that the pseudospin being an additional degree of freedom for carriers in graphene canbe efficiently controlled by means of the electron-electron interactions which, in turn, can be manip-ulated by changing the substrate. In particular, an out-of-plane pseudospin component can occurleading to a zero-field Hall current as well as to polarization-sensitive interband optical absorption.Introduction. Electrons in solids demonstrate a hugevariety of behavior which can be described by an appro-priate effective Hamiltonian depending on the host crys-tal. For the charge carriers in graphene being an one-atom-thick planar sheet of carbon atoms densely packedin a honeycomb crystal lattice, the effective Hamilto-nian at low energies turns out to be formally equivalentto the massless two-dimensional Dirac Hamiltonian[1, 2]H0 = ~v0(σxkx + σyky), where v0 ≈ 106ms−1 is the ef-fective “speed of light”, k is the two-component particlemomentum, and σx,y are Pauli matrices. In the origi-nal Dirac Hamiltonian the Pauli vector ~σ represents thespin orientation of a spin-1/2 particle which can be de-tected in Stern–Gerlach-like experiments. The carriersin graphene do also have additional degree of freedomknown as pseudospin which is formally equivalent to thetrue spin of massless fermions but originates entirely fromthe peculiarity of the honeycomb lattice whose elemen-tary cell has a basis of two atoms. As consequence thepseudospin is not linked with the internal magnetic mo-ment of an electron and does not directly interact withthe external magnetic field prohibiting Stern–Gerlachtype experiments. In contrast to that, we predict in thisletter situations where the pseudospin manifests itself inobservable quantities and can be detected in transport aswell as optical measurements on graphene Hall bars.First of all we show that the exchange electron-electroninteraction can alter the pseudospin orientation in a verybroad range. In an eigenstate of H0 the pseudospin is al-ways parallel to the wave vector forming the well-knownradial texture in the xy-plane. As we shall see shortly,the exchange interactions can turn the pseudospin tex-ture to the out-of-plane phase with the out-of-plane angledepending on the absolute value of the particle momen-tum. This is due to the huge negative contribution to theHartree–Fock ground state energy from the valence band(i. e. “antiparticle” states) which cannot be neglectedin graphene because of the zero gap (i. e. zero effectivemass of carriers) and large effective fine structure con-stant α∗ = e2/(ε~v0) where ε is the dielectric constantdepending on the environment[3]. The exchange contri-bution to the ground state energy has previously beenstudied in both monolayer and bilayer graphene regard-ing properties such as the electronic compressibility[4–6]and ferromagnetism[7–9], but the importance of the in-terplay between pseudospin and electron-electron inter-actions has been recognized only recently in bilayers[10].Having established the possibility to create an out-of-plane pseudospin orientation by means of the exchangeinteraction, we apply the Boltzmann approach to derivethe electrical conductivity tensor which turns out to haveHall components even though the external magnetic fieldis absent. The mechanism of this phenomenon is in-timately linked to the pseudospin-momentum couplingwhich can be read out immediately from the HamiltonianH0. Similar to the skew scattering of electrons on impu-rities in spin-orbit coupled systems partly responsible forthe anomalous Hall effect,[11, 12] the carriers in graphenedo also skew to one side of the Hall bar as long as theirpseudospin has non-zero out-of-plane component. Thiseffect has been intensively studied[13–15] assuming thatthe out-of-plane component occurs due to the band gapopened by spin-orbit coupling[13] which, however, seemsto be weak in graphene[2]. We emphasize that neitherspin-orbit coupling nor an external magnetic field is nec-essary to obtain a Hall current in graphene being in thepseudospin out-of-plane phase.Experimental manifestations of the pseudospin are notlimited to the electron skew scattering phenomenon butcan also be seen in the interband optical absorption. Per-forming optical measurements on graphene[16] one canobtain direct information regarding conduction and va-lence band states without advanced sample processingnecessary for transport investigations. Moreover, the pe-culiar properties discovered so far make graphene a verypromising material for optoelectronic applications[17].Optical absorption via the direct interband optical tran-sitions in graphene has been investigated in the seminalpaper[18] as well as in [19–24], but the mechanism consid-ered there lies essentially in the two-dimensional natureand gapless electronic spectrum and does nor directly in-volve the pseudospin orientation. Here we show that, dueto the out-of-plane pseudospin orientation, the interbandabsorption is sensitive to the light polarization. In par-ticular, it can be substantially reduced or enhanced ascompared to its universal value πe2/~c just by switchingthe helicity.Exchange interactions. The Coulomb exchange Hamil-2tonian is given byHexch(k) = −∑κ′∫d2k′4π2U|k−k′||χκ′k′〉〈χκ′k′ | (1)with U|k−k′| = 2πe2/ε|k−k′| and κ′ = ± being the bandindex. The eigenstates of H0 +Hexch can be formulatedas Ψk±(r) = eikr|χ±k〉 with the spinors being |χ+k〉 =(cosϑk, sinϑkeiϕ)T , |χ−k〉 = (sinϑk,− cosϑkeiϕ)T , andtanϕ = ky/kx. Thus, a non-zero out-of-plane componentof the pseudospin corresponds to ϑk 6= π/2. To diago-nalize H0 +Hexch the following equation for ϑk must besatisfied[25]~v0k cosϑk +∑κ′∫d2k′8π2κ′U|k−k′| [cosϑk′ sinϑk−− sinϑk′ cosϑk cos(ϕ′ − ϕ)] = 0 (2)where the integration goes over the occupied states. Notethat the conduction and valence states are entangled,and the latter cannot be disregarded even at positiveFermi energies. Thus, in order to evaluate the integralsin Eq. (2) a momentum cut-off Λ is necessary. Its value≃ 0.1nm−1 is usually chosen to keep the number of statesin the Brillouin zone fixed[7], but our outcomes do not de-pend on any particular choice of Λ. Substituting x = k/Λwe arrive at4πx cosϑkα∗= (3)2pi∫0dϕ′1∫kF /Λdx′x′cosϑk′ sinϑk − sinϑk′ cosϑk cosϕ′√x2 + x′2 − 2xx′ cosϕ′.The momentum cuf-off is obviously much larger than theFermi momentum kF at any reasonable electron doping,and therefore we can to set the lower integral limit tozero. There is a trivial solution with ϑk = π/2 indepen-dent of k, and a non-trivial one ϑ(k) which is shown inFig. 1 for different α∗. These two solutions represent totwo phases with different total ground state energies Eintot(Eouttot ) for the in-plane (out-of-plane) pseudospin phase.The difference ∆Etot = Eintot − Eouttot per volume reads∆Etot~v0Λ3= − 2π1∫0dx′x′2(1− sinϑk′)− α∗8π32pi∫0dϕ2pi∫0dϕ′1∫0dx1∫0dx′xx′(1− sinϑk′ sinϑk) cos(ϕ′ − ϕ)− cosϑk′ cosϑk√x2 + x′2 − 2xx′ cos(ϕ− ϕ′)(4)where we take into account spin and valley degeneracyby a factor of 4. The energy difference for α∗ ∼ 1 issmall because the integrand in Eq. (4) is always mul-tiplied by x′ and therefore vanishes at x′ → 0, but atlarger x′ the ϑk′ gets close to π/2, and the integrandvanishes again. The Inset in Fig. 1 shows, however,that strong electron-electron interactions make the out-of-plane phase energetically preferable. The estimatesof α∗ for clean graphene vary from 2 (Ref. [3]) to 2.8(Ref. [7]) and are on the borderline of the out-of-planephase. Moreover, the presence of disorder can changethis qualitative picture essentially[7].The single-particle spectrum is given byE±(x)~v0Λ= ±x sinϑk − α∗4π2pi∫0dϕ′1∫0dx′x′×1∓ (cosϑk′ cosϑk + sinϑk′ sinϑk cosϕ′)√x2 + x′2 − 2xx′ cosϕ′, (5)and the group velocity can be written as v± =(v± cosϕ, v± sinϕ)T with v± beingv±v0= ± sinϑk + α∗4π2pi∫0dϕ′1∫0dx′x′x(1− cosϕ′)×1∓ (cosϑk′ cosϑk + sinϑk′ sinϑk cosϕ′)(x2 + x′2 − 2xx′ cosϕ′) 32 . (6)The dispersion law (5) is depicted in Fig. 2 for grapheneplaced on SiO2 substrate. The exchange interactionsshift the bands down to lower energies, but, most im-portantly, they open a gap between the valence and con-duction band as soon as the system changes to the pseu-dospin out-of-plane one phase. The gap at k = 0 equalse2Λε1∫0dx′ cosϑk′ . Note that the group velocity (6) van-ishes at small momentum k/Λ≪ 1 as long as the systemis in the out-of-plane phase corresponding to the almostflat bands close to k = 0 shown in the inset of Fig. 2.From now on we assume n-doping so that the Fermi en-ergy is always higher than the bottom of the conductionband.Zero-field Hall current. To describe the Hall conduc-tivity due to skew scattering we utilize the semiclassical32α*∆totΛE    / hv 03ΛSiO 0 0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1icepi/2pi/4Out−of−plane anglek /air 0  2  4  6  8  100.20.10FIG. 1: The pseudospin out-of-plane angle ϑ(k) for differ-ent environments numerically calculated from Eq. (3). Thecorresponding values of the substrate-dependent effective finestructure constant α∗ are taken from Ref. [3]. The inset showsthe total ground state energy difference (4) between the in-plane and out-of-plane phases for different effective fine struc-ture constant α∗ = e2/ε~v0. Increasing α∗ makes the out-of-plane phase more preferable.Λ   E / hv 0α  = 0.8*Λk / −1.5−1−0.5 0 0.5 1 0  0.2  0.4  0.8  1 0.6 0 0.1−0.4−0.3−0.5FIG. 2: The dispersion law E±(k) in the in-plane (dashedcurves) and out-of-plane (solid curves) phases for α∗ = 0.8corresponding to SiO2 substrate[3]. The curves for bothphases coincide for momenta larger than a certain criticalvalue where ϑk = pi/2 becomes independent of k, see Fig. 1.The inset shows the gap region in detail.Boltzmann approach which allows a physically transpar-ent interpretation of this mechanism[12, 14]. In generalthe anomalous Hall conductivity contributions can beclassified by their mechanism: (i) The intrinsic contribu-tion is due to the anomalous velocity (being non-diagonalwith respect to the band index, see Refs. [27, 28]) of carri-ers which is coupled to the equilibrium part of the distri-bution function. (ii) The side-jump contribution followsfrom coordinate shifts during scattering events. It occursin the non-equilibrium part of the distribution function aswell as in the anomalous velocity[12, 14]. (iii) The skewscattering contribution is independent of the coordinateshift and of the anomalous velocity. It occurs when thescattering rate is asymmetric with respect to the initialand final states and, therefore, must be considered be-yond the first Born approximationThe first two conductivity contributions do not dependon disorder but on the out-of-plane angle ϑk and canbe adopted from [14]. The skew scattering can be de-scribed using the interband incoherent Boltzmann equa-tion where the anomalous velocity is neglected but thescattering probability is calculated up to the third or-der in the short-range scattering potential which is char-acterized by the momentum-independent Fourier trans-form V . In linear order in the homogeneous electric fieldE this equation reads −eEvk[−∂f0(Ek)/∂Ek] = I[f1k ],where f0(Ek) is the Fermi-Dirac function, f1kis the non-equilibrium addition, and vk, Ek are taken from the up-per cases of Eqs. (6,5). The collision integral can bewritten as I[f1k] =∫d2k′(2pi)2wkk′(f1k′− f1k) with wkk′ be-ing the scattering probability. We divide wkk′ into twoparts. The first one is proportional to the cosine of thescattering angle and calculated up to the second orderin V . The second one is proportional to the sine of thescattering angle and calculated up to the third order inV . These two parts correspond to the conventional andskew scattering respectively which can be alternativelyexpressed in terms of the momentum relaxation times1τ‖= (1 + 3 cos2 ϑk)nikV24~2vk,1τ⊥= cosϑk sin2 ϑknik2V 38~3v2k(7)Here, ni is the concentration of such scatterers. Note,that in contrast to the non-interacting case neither ofτ ’s diverges at k = 0 because of the k-dependent groupvelocity (6).The solution of the Boltzmann equation can be foundin the form[26–28]f1k = eEvk(ax cosϕ+ bx sinϕay cosϕ+ by sinϕ)[−∂f0(Ek)∂Eκ](8)Substituting (8) into the kinetic equation one can findeasily ax = by = τ‖τ2⊥/(τ2‖ + τ2⊥) and bx = −ay =τ2‖ τ⊥/(τ2‖+τ2⊥). Including the valley and spin degeneracythe conductivity at zero temperature readsσ =2e2hkvkτ‖τ⊥τ2‖ + τ2⊥(τ⊥ −τ‖τ‖ τ⊥)∣∣∣∣k=kF(9)Since τ⊥ ∝ 1/V 3 whereas τ‖ ∝ 1/V 2 it is naturalto assume τ⊥ ≫ τ‖, and σyx ∼ σxxτ‖/τ⊥ which canvary in a quite broad range depending on the scat-tering parameter and Fermi momentum. In particu-lar, at lower carrier concentrations one can approximateτ‖/τ⊥ ∼ (kFϑ2kF V )/(8~vkF ). Note, that the Hall cur-rent changes to opposite direction if the sample becomesp-doped because the electrons and holes having antipar-allel pseudospins skew to the opposite sides of the Hallbar.4Interband optical absorption. From H0 one can de-duce the following interaction Hamiltonian between theelectromagnetic wave and carriers in graphene Hint =ev0c (σxAx + σyAy) which couples the vector potential Aand pseudospin ~σ. As consequence, the inter-band tran-sition matrix elements turn out to be sensitive to the lightpolarization and pseudospin orientations in the initialand final states. To be specific we assume monochromaticlight of frequency ω, normal incidence (i.e. zero momen-tum transfer from photons to electrons), and circular po-larization (fulfilling Ax = ±iA/√2, Ay = A/√2). Theprobability to excite an electron from the valence bandto an unoccupied state in the conduction band can becalculated using the golden-rule. Finally, the absorptionP can be calculated as a ratio between the total elec-tromagnetic power Wa absorbed by graphene per unitsquare and the incident energy flux Wi = ω2A2/4πc. Af-ter some algebra we obtain the following expressionP =πe2~c16Λv0ω∞∫0dxx{sin4 ϑk2cos4 ϑk2}δ(E+ − E− − ~ω~v0Λ).(10)In the non-interacting limit the absorption equals to theuniversal value pie2~c , as expected[18]. Note, that the inte-grand in Eq. (10) contains the multipliers sin4(ϑk/2) andcos4(ϑk/2) for two opposite helicities of light. At smallk/Λ the absorption of the right-hand circularly polarizedlight is substantially reduced whereas it is facilitated inthe opposite case. Moreover changing the excitation en-ergies ~ω one can investigate the dependence θ(k) shownin Fig. 1. In the in-plane phase with ϑ = π/2 the ab-sorption does not depend on light polarization but itsabsolute value is renormalized due to the exchange inter-actions.Conclusions. We have demonstrated that the pseu-dospin being until now rather uncontrollable and al-most unmeasurable quantity can be “unfrozen” by theexchange electron-electron interactions (1) and play anessential role in optical and transport properties ofgraphene. Thus, the pseudospin can be seen as an ad-ditional degree of freedom similar to the true spin butunaffected by the magnetic field directly. We hastento say that the Hartree-Fock approximation employedhere has generically a tendency to overestimate orderingsuch a the pseudospin out-of-plane polarization. We be-lieve, however, that the pseudospin eigenstates |χ±k〉 de-rived above are much more robust because their specialpseudospin-momentum entangled structure stems fromthe free Hamiltonian H0, and the electron-electron inter-actions do only modify it. 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Pseudospin in optical and transport properties of graphene

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