Strong Nernst-Ettingshausen effect in folded graphene Friedemann Queisser and Ralf Schu¨tzhold∗ Fakult¨at fu¨r Physik, Universita¨t Duisburg-Essen, Lotharstrasse 1, 47057 Duisburg, Germany (Dated: January 18, 2013) We study electronic transport in graphene under the influence of a transversal magnetic field B(r) = B(x)ez with the asymptotics B(x ) = B0, which could be realized via a folded → ±∞ ± graphene sheet in a constant magnetic field, for example. By solving the effective Dirac equation, we find robust modes with a finite energy gap which propagate along the fold – where particles 3 and holes move in opposite directions. Exciting these particle-hole pairs with incident photons 1 would then generate a nearly perfect charge separation and thus a strong magneto-thermoelectric 0 (Nernst-Ettingshausen) or magneto-photoelectric effect – even at room temperature. 2 n PACSnumbers: 72.80.Vp,78.67.Wj,85.80.Fi. a J 7 Introduction The Nernst-Ettingshausen effect [1] de- 1 scribes the generation of an electric current (or voltage) by a temperature gradient in the presence of a magnetic ] B l field. Suchthermoelectriceffectsfacilitatethedirectcon- l a version of thermal into electric energy and thus are of h general interest. Obviously, the C (charge), P (parity), - FIG. 1. Sketchof the considered set-up. s and T (time reversal) symmetries must be broken for e such an effect to occur. One way to achieve this is a m magnetic field in a suitable geometry: trajectories of op- Eigen-modes We consider length scales (e.g., curva- at. posite charge carriers are bent to antipodal directions. ture radius of fold) far above the lattice spacing of However, the mean free path in usual materials is too m graphene 0.25 nm and energies of 1 eV or below. In short to generate an efficient charge separation in that ≈ thislimit,wemaydescribethelow-energybehaviorbyan - d way – at least at room temperature. For example, the effective Dirac equation in 2+1 dimensions (~=q =1) e n classical cyclotron radius r = mev/(qeB) of a free elec- o tron at room temperature in a magnetic field B of one iγµ(∂ +iA )Ψ=0, (1) µ µ c Tesla r = (µm) is much larger than the typical mean [ O with xµ = [v t,x,y], where v 106m/s is the Fermi free path (in the nanometer range). Thus, the Nernst- F F 1 Ettingshausen effect is strongly suppressed by multiple velocity [14]. The Dirac matrice≈s γµ = [σz,iσy, iσx] − v acting on Ψ = [ψ ,ψ ] are related to the Pauli matrices scattering events and dissipation etc. 1 2 2 σx,y,z. In the Landau gauge, the vector potential A = 4 This motivates the study of graphene [2–6], since this µ [0,0,A(x)] generates the magnetic field B(x) = ∂ A(x) 1 system offers a comparably long mean free path and x with the asymptotics B(x )= B . 4 a large electron mobility, a linear (pseudo-relativistic) →±∞ ± 0 . Inviewofthetranslationsymmetryintandy,wecan 1 dispersion relation at low energies (i.e., near the Dirac make the separation ansatz for the modes 0 points), anda verylargeFermivelocityv 106m/s [3], F 3 see also [7, 8]. In this case, the pseudo-re≈lativistic cy- Ψ(t,x,y)=exp iEt+iky ΨE,k(x), (2) 1 {− } clotron radius at room temperature in a magnetic field : v of one Tesla is much smaller (some tens of nanometers). arriving at the two coupled equations i X In this regime, quantum effects should be taken into ac- iv [∂ +k+A(x)]ψE,k(x)=EψE,k(x) count – even at room temperature [4]. F x 2 1 ar In the following, we consider folded graphene in a ivF[∂x−k−A(x)]ψ1E,k(x)=Eψ2E,k(x). (3) transversal magnetic field, see Fig. 1. In principle, the Hence, we can choose ψE,k(x) to be real, for example, 1 foldingofgraphenehasalreadybeenrealizedexperimen- while ψE,k(x) is imaginary. We observe a particle-hole tally, see, e.g., [9, 10]. This set-up is advantageous since 2 symmetry since replacing E E and ψE,k ψE,k we avoid real edges in graphene which are typically not → − 2 → − 2 yields a new solution Ψ−E,k =σzΨE,k =(ΨE,k)∗. perfect and contain cracks or other defects which might Thetwofirst-orderequations(3)canbecombinedinto inducescattering,couplingtovibrationaldegreesoffree- one second-order equation dom, or further unwanted effects. Form a theoretical pointofview,these edgescanonlybe describedinideal- v2[k+A(x)+∂ ][k+A(x) ∂ ]ψE,k =E2ψE,k,(4) ized cases, e.g., via effective boundary conditions which F x − x 1 1 then depend on the concrete realization (e.g., zigzag or and analogously for ψE,k with ∂ ∂ . This equa- 2 x ↔ − x armchair structure [11–13]). tion can be cast into the form of a one-dimensional 2 Schr¨odingerequation ψE,k =E2ψE,k withtheHamil- where we have used the normalization Ψ Ψ = 1. Hk 1 1 h E,k| E,ki tonian = v2( ∂2 + ) containing the effective po- Together with Eq. (7) we find that particles with E > 0 Hk F − x Vk tential = [k +A(x)]2 +A′(x). Since this Hamilto- and holes with E < 0 have the opposite current (and k V nian is self-adjoint Hk = Hk† and the potential Vk has groupvelocity),i.e.,allparticles(withk >−Amin)move the asymptotics (x ) = , we get a complete to the right and all holes move to the left. In this way, k V → ±∞ ∞ set of discrete, orthonormal, and localized (in x) eigen- one obtains a (nearly) perfect charge separation. functions ψE,k(x) for every value of k. These modes are Asymptotics It is illustrative to study the two limit- 1 non-degenerateforeachk,i.e.,theenergybandsE(k)do ing cases k . For large and positive k, the po- → ±∞ not cross [15]. Due to the particle-hole symmetry, each tential k can be approximated by k k2 +2kA(x). V V ≈ of these eigen-functions ψE,k(x) corresponds to a pair of Thus, to lowest order in k, we obtain E vFk, i.e., 1 ≈ ± modes Ψ±E,k(x) of the original problem (3) with oppo- these modes propagate with a speed close to the Fermi velocity. Going to the next order in k, we may expand site energies. Furthermore, with = k +A(x) ∂ , k x we may write = v2 † whDich shows that − is A(x)arounditsminimumatx0 wherethemagneticfield Hk FDkDk Hk B(x ) = 0 vanishes A(x) A +A′′(x )(x x )2/2 non-negative (and thus E is real). In addition, k can- 0 ≈ min 0 − 0 H and obtain harmonic oscillator eigen-functions centered not have a zero eigen-value E = 0 since the correspond- ing ψ1E=0,k(x) must satisfy Dkψ1E=0,k = 0, which gives athtexp0o[taesnstuimalibneghtahvaetsAas′′k(xB0′)(x6=)0,]t.hSeimncoedtehsearsetiffstnreosnsgolyf ψE=0,k(x) exp kx+ dxA(x) and analogously for 0 ψ1E=0,k(x).∝Due to{ the aRsymptotic}s B(x )= B localized around x0 for large k and basically propagate 2 →±∞ ± 0 along the x0-line where the magnetic field vanishes. For and thus A(x ) B x, this solution is not nor- → ±∞ ∼ 0| | fixedandlargek,these modes haveequidistantvaluesof malizableandthus isstrictlypositiveforanyk. Ergo, Hk E where the distance scales with B′(x0). the modes do always have a finite energy gap E =0. 6 Forlargeandnegativek-values,pthe minima ofthe po- Current The current density of the modes reads tential are given by A(x ) + k = 0 and thus the k ± jµ =v Ψ γµΨ =v Ψ† γ0γµΨ . (5) modes aVre localized at large and nearly opposite values E,k F E,k E,k F E,k E,k of x k/B due to A(x ) B x. In this The zeroth component j0 = vFρ is simply given by the regim±e∼,A±(x|)isa0p|proximatelyl→ine±ar∞and∼thu0s|w|erecover density ρ = ψE,k 2 + ψE,k 2. As one would expect, | 1 | | 2 | the harmonic oscillator eigen-functions corresponding to jx vanishes identically since ψE,k(x) is real and ψE,k(x) 1 2 the usual (pseudo-relativistic) Landau levels in a con- imaginary, cf. Eq. (3). Using the same argument, the stant magnetic field [16]. Note, however, that the eigen- current density in y-direction simplifies to functions ψE,k(x) are linear superpositions of the Lan- ∗ 1 jy =ivF(cid:16)ψ2E,k(cid:17) ψ1E,k−h.c.=−2ivFψ1E,kψ2E,k. (6) dEa.uInlevtehlisscleimntiet,retdheateixg+en-aennderxg−ieswEithdothneostadmeepeennderogny Fromthe triangleinequality (2ab a2 + b2 ), we may k anymore (En = v √2B n with n N) and thus the infer jy vFρ, i.e., the speed| of|t≤he| as|soc|iat|ed charge current Jy alsLo van±ishFes. H0ence these∈modes are not so | | ≤ carriers is at most the Fermi velocity vF (as expected). interesting for our purpose. The total current in y-direction can be obtained by Matrix elements Now we arein the positionto study 2v2 the excitation of particle-hole pairs by incident photons Jy = dxjy = F dxψE,k[k+A(x)]ψE,k, (7) Z − E Z 1 1 (in the infra-red or optical regime). In second quantiza- tion, the interaction Hamiltonian reads where we have used v ψE,k = iEψE,k from Eq. (3). For the lowest E2 modFeDsk(fo1r a given k2), i.e., the upper- Hˆ = dxdy ΨˆγµAˆ Ψˆ . (9) int Z µ most negative mode and the lower-most positive mode, the wave-function ψ1E,k(x) corresponds to the ground where the photon field operator Aˆµ contains the cre- srteamte[1o5f])H.kSianncdeohnenecceanitriespneoant-tzheerosafmoreallilnexo(fnaordgeutmheenot- afrteiqonuenancyd ωa,nnwihaivleantiuomnboeprekra,toarnsdaˆp†ωo,kla,σrizaantdionaˆωσ,k.,σTfhoer for ψ2E,k(x), the integrand jy = −2ivFψ1E,kψ2E,k is non- Dirac field operator Ψˆ is a linear combination of the an- zeroforallxandhencethecurrentJy isfinite. Butother nihilation operators for particles cˆ Ψ and the E>0,k E>0,k lmarogdeesencoouugldhhkav>e JyA=0=at sommine kA-v(xal)ue,.thHeowinetveegrr,afnodr creation operators for holes cˆ†E′<0,k′ΨE′<0,k′. − min − { } If we now consider the transition matrix elements ainlltxheanadbotvheuseqtuhaetciounrrψen1Et,kh[kas+aAfi(nxi)t]eψv1Ea,lkueis. positive for hout|Uˆint|ini with an initial photon |ini=aˆ†ω,k,σ|0i and a final particle-hole pair out = cˆ† cˆ† 0 , we Furthermore, the current Jy is related to the slope | i E>0,k E′<0,k′| i get to first order in perturbation theory dE/dk ofthe dispersionrelation,i.e., the groupvelocity: Writing Eq. (3) as HˆE,k|ΨE,ki=E|ΨE,ki, we find AEω,,kk,;σE′,k′ = √12ω Z dtdxdy ΨE,kγµAσµΨE′,k′ × dHˆ dE Jy = ΨE,k E,k ΨE,k = , (8) e+iEt−ikye−iωt+ik·re−iE′t+ik′y, (10) −h | dk | i −dk × 3 where Aσ encodes the polarization of the photon. As Now, the integrand in the matrix elements (11) be- µ usual, the t-integral gives δ(ω E + E′), i.e., energy haves as ψE,k(x)ψE′,k(x) ψE,k(x)ψE′,k(x) for the two conservation. Since the wavelen−gth of the photons un- photon pol1arizatio2ns. Ins±erti2ng Eq.1(12) and integrat- der consideration (in the optical or infra-red regime) is ing over x, we see that the matrix elements (11) be- much larger than the typical length scales of the elec- tweenmodesofthesamepseudo-parityvanishforphoton tronic modes in graphene, we may neglect the photon polarizations in x-direction whereas the transition be- wavenumberk. Therefore,they-integralyieldsδ(k k′), tween modes of opposite pseudo-parity is forbidden for − i.e.,weexciteparticle-holepairswiththesamewavenum- the other polarization. ber k=k′. The remaining x-integral reads Yetanothersetofselectionrulescanbeobtainedinthe asymptotic regimes. For k we only get transitions AEω=,kE;E−′E,k′′,=kk≈0,σ ∝Z dxΨE,kγµAσµΨE′,k′. (11) between modes of opposite→ene∞rgies(due to the orthogo- nality of the harmonic oscillator eigen-functions). In the Let us first assume Aσ = const and consider the tran- opposite limit k , we recover the well-known [16] µ → −∞ ssuitciohnabsetthweeeunppmeor-dmesosotfnthegeastaivmeemEo2de(i(.efo.,rEa =giv−enEk′)), pnropNertwiehseroefwtheeonLlayngdeatutrleavneslistioEnLns =for±nvF√n2B01n.with ∈ → ± and the lower-most positive mode, cf. Fig. 2. In this Polarization dependence So far, we have discussed case, we may use the aforementioned particle-hole sym- the caseAσ =constinEq.(11). Thisis certainlyagood µ metry Ψ = σzΨ and simplify the integrand via approximation if the polarization of the incident photon −E,k E,k ΨE,kγµΨE′,k = ΨE,kγµσzΨE,k. Inserting γ1 = iσy and pointsinydirection,i.e.,isalignedwiththesymmetryof γ2 = iσx and using the properties of the Pauli ma- ourset-up. However,fortheother(x)polarization,Aσ in µ − trices, we see that the matrix element for the photon Eq.(11)shouldbereplacedbythelocalprojectionofthe polarization in x-direction Ax yields the same expres- photon wave function Aσ onto the graphene plane, i.e., µ sion as in the current Jy, cf. Eq. (5), and vice versa. become x-dependent Aσ(x). The profile of Aσ(x) then µ µ Consequently, the matrix elements (11) vanish for the depends on the incidence angle of the photon. If the photon polarization in y-direction, but yield a non-zero photon is incident from top, i.e., propagates parallel to contributionforthephotonpolarizationinx-direction,at theexternalmagneticfieldk B,thetwographenesheets k leastifk islargeenough[cf.thediscussionafterEq.(7)]. (topandbottom)haveoppositeprojections. ThusAσ(x) µ Moreover, the modes with large currents J and thus is anti-symmetric Aσ( x) = Aσ(x) and the above se- y µ − − µ large group velocities dE/dk do also have large matrix lection rules are reversed. If the photon propagates per- elements, which enhances the magneto-thermoelectricor pendicularlythroughthefold(k B),wegetasymmet- ⊥ magneto-photoelectric effect we are interested in. ric projection function Aσ( x) = Aσ(x) which vanishes µ − µ Pseudo-parity Further selection rules arise if we as- far away from the folding region (i.e., for large x). In | | sume reflection symmetry B( x) = B(x) and thus this case,the aboveselectionrules do stillapply, but the − − A( x)=A(x) which yields the additional symmetry matrix elements might be reduced a bit. − Example profile In order to visualize the behavior of ψE,k( x)= iψE,k(x)=i ψE,k(x), (12) themodesbymeansofaconcreteexample,letusconsider 1 − ± 2 PE,k 2 a magnetic field of the following form wherewecall = 1thepseudo-parityofthismode. E,k Recalling thePparticle±-hole symmetry Ψ = σzΨ , B(x)=B0tanh(αx), (13) −E,k E,k we find −E,k = E,k. The pseudo-parity of a given where 1/α measures the width of the fold. For α , mode caPn be dete−rmPined easily for large and positive we get a step function B(x)=B sign(x) with the v→ect∞or 0 kS,incwehetrhee wweavhea-fvuencitψio2En,kψ≈1E,kv(Fxk)ψo1Ef,kth/Ee lofrwoemstEpqo.sit(i3v)e. peqouteantitoianl(A3()xc)a=n bBe0|sxo|l,vcefd. [e1x7a].ctIlyn(tphiiescleiwmiiste,)thinetmeromdes mode (for k ) corresponds to the ground state of parabolic cylinder functions, cf. [18]. Incidentally, the → ∞ of a harmonic oscillator, it is Gaussian and symmet- spectrum for such a step function B(x)=B sign(x) can 0 ric ψ1E,k(−x) = ψ1E,k(x). Hence this mode has an also arise for some edge states [12, 13]. even pseudo-parity E,k = +1. The wave-function However, such a step function can only be a good ap- P ψE,k(x)ofthenextmodecorrespondstothefirstexcited proximation if k is not too large and if the curvature 1 state of aharmonic oscillatorand thus is anti-symmetric radiusofthegraphenefoldismuchsmallerthanthetyp- ψE,k( x)= ψE,k(x),whichgivesanoddpseudo-parity ical magnetic length scale ℓ = 1/√B. For one Tesla, 1 − − 1 B = 1 and so on. Together with the above result we get ℓ 26 nm while the radius of curvature cannot E,k B P − ≈ = we find that, for a fixed k, the pseudo- be too small since it should be much largerthan the lat- −E,k E,k P −P parity of the modes alternates if we go up and down in tice spacing 0.25 nm. Thus, let us consider a finite α ≈ energy. Assumingthatthe modesdeformcontinuouslyif and take α = 1/ℓ as an example. The spectrum can B k changes[i.e.,thatA(x)issufficientlywell-behaved],we then be obtainednumerically andis giveninFig.2. The may deduce an alternating pseudo-parity for all k. spectraforothervaluesofαarequalitativelysimilar. As 4 demonstratedabove,thetwolowestmodesaremonoton- incidenceangleofthephotons,whichshouldenableusto ically increasing/decreasing, whereas the higher modes distinguishthis effect from otherphenomena experimen- can have dE/dk = 0 at some small k-values. For large tally. Furthermore, we find that those modes with com- k , we recover the asymptotics discussed above. parably large group velocities (i.e., large currents) tend | | to have large matrix elements (at least for low-energy transitions) and thus are more strongly coupled to the incident photons (i.e., “nature favors our goal”). Outlook: electric field If we apply an additional elec- 2 tricfieldperpendiculartothefoldandthemagneticfield, we get an electrostatic potential Φ(x) = βv A(x) with F γ some constant β. If we have β < 1 (i.e., if the elec- ε 0 tric field is sub-critical), we m|ay| transform Φ away by an effective Lorentz boost in y-direction with a veloc- ity v = βv where v plays the role of the speed of boost F F 2 light[19]. IntheLorentzboostedframe,wegetthesame − modes as discussed above, but with a reduced magnetic fieldB′ =B 1 β2. Sincethisfieldentersthecharac- 4 2 0 2 4 0 0 − − − teristic energypscale via vF√2B0, the dispersion relation κ after transforming back to laboratory coordinates reads FIG.2. Dispersion relationofthelowest bandswithκ=kℓB E E′ =E(1 β2)3/4 kvFβ, (14) and E =εvF√2B0 and sketch of thephoto-absorption. → − − i.e., the spectrum in Fig. 2 is compressed and tilted. Conclusions Via the effective Dirac equation (1), we Acknowledgements FruitfuldiscussionswithA.Lorke studied the low-energy behavior of electronic excitations and M. Schleberger are gratefully acknowledged. 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