arXiv:0910.2286v1 [cond-mat.str-el] 13 Oct 2009 Theory of dissipationless Nernst effects. Doron L. Bergman1, * and Vadim Oganesyan2, ��� 1Department of Physics, California Institute of Technology, Pasadena, CA 91125 2Department of Engineering Science and Physics, College of Staten Island, CUNY, Staten Island, NY 10314 (Dated: October 13, 2009) We develop a theory of transverse thermoelectric (Peltier) conductivity, ��xy, in finite magnetic field ��� this particular conductivity is often the most important contribution to the Nernst thermopower. We demonstrate that ��xy of a free electron gas can be expressed purely and exactly as the entropy per carrier irrespective of temperature (which agrees with seminal Hall bar result of Girvin and Jonson). In two dimensions we prove the universality of this result in the presence of disorder which allows explicit demonstration of a number features of interest to experiments on graphene and other two-dimensional materials. We also exploit this relationship in the low field regime and to analyze the rich singularity structure in ��xy(B,T ) in three dimensions we discuss its possible experimental implications. PACS numbers: The magnetothermoelectric Nernst-Ettingshausen effect [1] has enjoyed renewed interest in recent years, first as a probe of superconducting fluctuations, and more generally, as a novel transport characterization of electronic correlations. Follow- ing the initial work on the cuprates[2] strong magnetother- moelectricity was found in a variety of interesting materi- als. While precise theoretical treatment is lacking for most of these cases, phenomenological descriptions in terms of con- ventional weak-field quasiparticle transport theory[3, 4, 5, 6] or effective classical hydrodynamic models [7, 8] have been used with varied degree of success[9]. In this letter we break from these earlier studies to treat finite field response directly, with no recourse to a low-field regime. Our chief accomplishment is the exact expression of the off-diagonal Peltier conductivity, ��xy, in terms of entropy of free fermionic carriers (see Eqs. 5, 6, 7 and 10). In two di- mensions we prove the universality of this expression (which also applies to Dirac fermions) in the presence of quenched disorder and compare it against available experimental data. In three dimensions we obtain, essentially in a closed form, the entire intricate singularity structure in ��xy (as a function of magnetic field and temperature) inherited from the Lan- dau level spectrum which bears strong resemblance to ther- moelectric phenomenology of graphite[10]. We also exam- ine the weak field limit, B ��� 0, where we predict a sim- ple, ��xy = ���s/B, dependence on magnetic field and entropy density s. Quite generally, ��xy is somewhat of a less stud- ied and, hence, poorly understood quantity, at least as com- pared to electrical conductivity or entropy. Thus, our basic re- sult directly linking ��xy and entropy (importantly, without in- voking the so called ���entropy currents��� used elsewhere in the literature[5, 11]) is useful both for simplifying computations but also for developing intuition. Even if only approximate in more realistic models (e.g. with inelastic processes ignored by us here), it gives some credence to empirical associations of strong Nernst signatures with singular rearrangements of electronic structure, e.g. phase transitions. Current flow in the presence of weak electric field and a k y kz g115g894g454g895 g17 z g115g894g454g895 g58 FIG. 1: (color online) Hall ���brick��� in a confining potential V (x) (above) and its Fermi surfaces showing three occupied Landau bands (depicted using different colors). The spectrum is discrete along kz axis and continuous along x. The chiral surface states occupy non- flat portions of the Fermi surface and flow along brick���s sides. small thermal gradient is determined by J = �� �� E ��� �� �� ���T , (1) JQ = T�� �� E ��� �� �� ���T , (2) where J, JQ, T, E, ��, ��, �� are respectively the charge and en- ergy currents (slightly modulated in space) temperature and electrical field strength electrical, Peltier and heat conduc- tivity tensors, respectively. Peltier conductivity is usually ex- tracted from electrical conductivity and thermopower, S = ��-1 �� ��, measured in a zero-current configuration. For convenience we consider a particular, Hall ���brick���, sample shape (see Fig. 1) of finite extent along x��� and z���axis, although our results will be independent of this as- sumption. Ignoring spin and possible valley quantum numbers for the time being the free electron Hamiltonian in a Landau

2 gauge is H = ��� planckover2pi12 2m bracketleftbig (���y ��� x/���B)2 2 + (���x)2 bracketrightbig + planckover2pi12kz 2 2m + V (x) , (3) where mz, m, V (x), B, ���e and ���B = radicalBig planckover2pi1 eB are the elec- tron masses in the �� z direction and x ��� y plane, confining potential along the x-axis, magnetic field along the z-axis, electron charge and magnetic length, respectively. The con- fining potential along the z-axis (not shown explicitly) is as- sumed to be featureless (hard walls), whereas one along the x-axis, V (x), has a more gradual rise as would be the case in graphite nano-ribbons[12, 13], for example. Consequently, the wave functions are standing waves along the z���axis, la- beled with a discrete set of kz���s. If V (x) = 0 the spec- trum of this problem consists of Landau levels dispersing with kz, ��n,kz = planckover2pi1��n + planckover2pi12kz 2 2mz , where ��n = eB m (n + ��0), with ��0 = 1/2 (but can be different generally[14]). Provided V (x) varies smoothly on scales set by the magnetic length, the spec- trum of the Hall brick can be reconstructed by an adiabatic shift of Landau bands at each kz and n, e.g. by absorbing V (x) into a spatially varying ��0. Thus, the Fermi sea of the Hall brick is a locus of points in the kz ��� ky plane where ��n,kz ��, the chemical potential (see Fig. 1). It is bounded by a set of closed Fermi surfaces each made up of two flat segments in the bulk (with kF,n = ��(�� z/ planckover2pi1) radicalBig ��-planckover2pi1��n 2mz ) and chiral surface states propagating in ���� y directions on opposite sides of the brick . While detailed structure of these surface states will depend on surface properties, we generally expect them to be robustly present (due to the necessary chirality) and posess interesting transport anisotropies between direc- tions along the equilibrium current direction (����) y and trans- verse to it, by analogy with chiral metals in layered quan- tum Hall systems[15, 16]. The analogy is not precise, since our chiral states are two dimensional, completely delocalized along the surface and smoothly connected to the gapless bulk (layered Hall states typically have mobility gaps in the bulk and strongly one dimensional surface states). Their wave- functions are peaked nearest to the surface for the smallest kz ��� ��/Lz component of the Fermi momentum (see fig. 1), which may allow for their characterization via coherent sur- face tunnelling. Detailed discussion of these chiral surface states will be presented elsewhere[17]. Bulk transport properties of the Hall brick can be com- puted easily by applying quantum Hall edge formalism for each transverse mode kz, e.g. following seminal works of Halperin[18] and Girvin and Jonson[19] on the Hall bar (which itself is a very thin Hall brick with only a single kz = ��/Lz state occupied). We apply both electric field and thermal gradient along the �� x which induces net currents along the �� y direction (see Eqs. 1 and 2). Recall[18, 19], that al- though microscopic current distributions depend sensitively on the details of the confining potential bulk transport coef- ficients computed by integrating over these distributions are independent of edge specifics, as they must be. The three non- zero Hall electric/Peltier/thermal conductivities can be written as ��xy = ��� e2 h C0, ��xy = kB e h C1 and ��xy = ��� kBT 2 h C2, re- spectively, with the help of Cq = ��� kz cq(kz)/Lz and cq(kz) = ��� summationdisplayintegraldisplay n ��� planckover2pi1��n+ planckover2pi12kz 2 2mz -�� d�� parenleftbigg �� kBT parenrightbiggq ���f(��) ����� . (4) Here and elsewhere, the Fermi function is denoted by f(��) = 1/(1 + e��/(kB T)), with kB the Boltzmann constant. These coefficients are for electrons, for holes ��xy and ��xy reverse sign. Explicit expressions for cq can be obtained by a further variable change d�� ��� df and a definition fn ��� f(planckover2pi1��n +���mz planckover2pi12kz 2 ��� ��), with a familiar result for c0(kz) = n fn and somewhat less familiar expressions for c1(kz) = ��� [fn log fn + (1 ��� fn) log(1 ��� fn)] and c2(kz) = ���n n bracketleftBig ��2 3 + fn log2(1/fn ��� 1) ��� log2(1 ��� fn) ��� 2Li2(1 ��� fn) bracketrightBig , where Li2(z) is the polylogarithm function. Thus, we find that, up to an overall prefactor, ��xy is the entropy per particle added over Landau bands and averaged over transverse modes: ��xy = kBe hLz summationdisplaysummationdisplay n kz [fn log fn + (1 ��� fn) log(1 ��� fn)] . (5) Eq. 5 is the basic observation from which the rest of this paper follows (thermal conductivity is left for future work[17]). We start by examining Eq. 5 in the semiclassical regime, with weak scattering and B ��� 0 ��� this regime may be realized in some semiconductors[20, 21]. Rewriting Eq. 5 in terms of entropy per volume (i.e. entropy density), s, we obtain ��xy = ���2�����B 2 e h s = ��� s B , (6) akin to ��xy = ��� ne B , where n is the density of particles. Ab- sent interactions, ��xy is even under charge conjugation, while ��xy is odd. At high temperature, for a non-degenerate gas with s = kBn, these results are completely consistent with the purely classical drift in crossed electric and magnetic fields, with entropic force[22], F = ���kB���T , and effective electric field is E = F/e. At low temperature, the original B = 0 Fermi surface of the problem is more or less intact with semi- classical wavepackets of quasiparticles precessing around it. Thus we expect an additional de Haas-van Alphen type mag- netooscillations on top of the 1/B envelope as well as con- comitant Shubnikov-de Hass traces in conductivity. Also, Boltzmann kinetics can be used[3] to interpolate ��xy(B) to very low fields where scattering dominates and ��xy ��� B. The two dimensional limit of Eq. 5 is obtained by omitting the sum over kz modes and setting fn = f(planckover2pi1��n ��� ��) ��xy = ekB h summationdisplay��� n=0 [fn log fn + (1 ��� fn) log(1 ��� fn)] (7) which shows a sequence of thermally broadened nearly sym- metric peaks of height ��� kBe h log 2 as a function of B or ��, centered at quantum critical points separating integer Hall