Nernst and magneto-thermal conductivity in a lattice model of Weyl fermions
Weyl semimetals (WSM) are topologically protected three dimensional materials whose low energy excitations are linearly dispersing massless Dirac fermions, possessing a non-trivial Berry curvature. Using semi-classical Boltzmann dynamics in the relaxation time approximation for a lattice model of time reversal (TR) symmetry broken WSM, we compute both magnetic field dependent and anomalous contributions to the Nernst coefficient. In addition to the magnetic field dependent Nernst response, which is present in both Dirac and Weyl semimetals, we show that, contrary to previous reports, the TR-broken WSM also has an anomalous Nernst response due to a non-vanishing Berry curvature. We also compute the thermal conductivities of a WSM in the Nernst () and the longitudinal () set-up and confirm from our lattice model that in the parallel set-up, the Wiedemann-Franz law is violated between the longitudinal thermal and electrical conductivities due to chiral anomaly.
After the theoretical prediction of topological insulators, and their subsequent experimental realization, the field of topological condensed matter has grown manifold (1); (2); (3); (4); (5); (6); (7); (8); (9). The topological order manifested in these systems is not associated with spontaneous breaking of a symmetry, but rather can be described by a topological invariant which is insensitive to a smooth deformation of the Hamiltonian. Usually the robust topological protection is associated with a non-zero spectral gap in the bulk of the system, and the presence of protected zero energy surface states is regarded as the hallmark of a non-trivial topological phase of matter. However, recently it has been proposed that systems in three spatial dimensions, in the presence of broken time-reversal (TR) and/or space-inversion (SI) symmetry, can also be topologically protected even without a bulk energy gap (10); (11); (12); (13); (14); (15); (16); (17); (18); (19); (20); (21). These are Weyl semimetals (WSM) - the nomenclature based on the Dirac/Weyl equation which is used to describe their low energy excitations (22).
A number of recent experiments have claimed to be able to observe the Weyl semimetal phase in an inversion asymmetric compound TaAs (23); (24); (25), and also in a 3D double gyroid photonic crystal (26), without breaking TR. Another route which can result in the experimental verification of the novel WSM phase is to first realize a 3D Dirac semimetal and then break time reversal symmetry by applying a magnetic field, which will split a Dirac cone into a pair of Weyl nodes. NaBi and CdAs were recently proposed to be Dirac semimetals (27); (28), and also have been confirmed experimentally by a series of experiments (29); (30); (31); (32); (33); (34); (35); (36); (37). In BiSb for also the Dirac semimetal phase has been predicted (38); (39); (40), and experimental signatures of realizing a WSM phase by breaking TR have been reported (41).
A simple WSM with broken time reversal symmetry can be desribed by a pair of linearly dispersing massless Dirac fermions governed by the Hamiltonian: , where is the vector of Pauli spin matrices defined in the space of two non-degenerate energy bands, is the Fermi velocity, and are the two band touching points separated from each other in momentum space by . It is essential that is non-zero to ensure that the system breaks TR and is topologically non-trivial, in which case and describe two Weyl fermions of opposite chirality. The two band touching Weyl points act as a source and a sink (monopole and anti-monopole) of Berry curvature, which acts as a fictitious magnetic field on the electron wave-function in momentum space (75). For , the two Weyl points collapse onto each other giving rise to a topologically trivial (i.e with a zero Berry curvature flux) massless degenerate Dirac fermion. The topological nature of a WSM leads to a host of interesting physics, for example Berry curvature induced anomalous transport, namely charge and thermal Hall conductivities (43); (44); (45); (46); (47); (48) and open Fermi arcs on surfaces (11); (49); (50); (51); (52); (53); (54); (55). Anomalous transport phenomena, however, have been already known to exist in a variety of systems which possess a non-trivial distribution of the Berry curvature flux (56); (57). In a WSM, more interestingly, each Weyl node is chiral, with the chirality quantum number protected by a quantized flux of the Berry curvature, also known as Chern flux, which results in another peculiar phenomenon known as chiral anomaly (or Adler-Bell-Jackiw anomaly) (10); (16); (58); (59). The chiral anomaly concerns with the nonconservation of chiral charge i.e. an imbalance of charge between two distinct species of chiral fermions in the presence of non-orthogonal applied electric and magnetic fields. Several transport signatures have been proposed to test chiral anomaly such as negative longitudinal magenoresistance (21); (16); (60); (61) and chiral magnetic effect(16); (63); (62) of which the former has been recently claimed to be observed in experiments (24); (37); (42); (41).
A Dirac node can be split into two Weyl nodes by breaking either the TR symmetry or SI symmetry. Figure 1 shows a linearly dispersing Dirac node split into a pair of Weyl nodes when TR symmetry is broken, and also shows the energy-band spectrum of a lattice model of Weyl fermions obtained by diagonalizing the Hamiltonian in Eq. 56. The simple model of Weyl semimetal described in the previous paragraph by breaks TR, however it is also possible to realize a Weyl system when TR is intact but inversion symmetry is broken (14); (65); (64). This implies that the system must host more than one flavor of pairs of Weyl fermions for the vector sum of to vanish. In the SI broken Weyl semimetal, because of TR symmetry there is no Berry curvature induced anomalous charge or thermal Hall effect in the absence of an external magnetic field. However, in the TR broken WSM, because of a finite Berry curvature flux through any plane intermediate between the Weyl nodes in the momentum space, the anomalous charge and thermal Hall conductivities are non-zero (44); (45); (46); (13).
In this paper, we work with a TR broken phase of WSM and consider its Nernst response. Experimentally, the Nernst effect measures the transverse electrical response to a longitudinal thermal gradient in the presence of a perpendicular magnetic field. The Nernst effect has been used as an important experimental probe in a number of physical systems such as high temperature cuprate superconductors (66); (67); (68); (69), and charge density waves (70); (71). Since the TR-broken WSM has a non-zero anomalous Hall response (i.e., non-zero Hall efffect induced by the Berry curvature even in the absence of a magnetic field) it is expected that the anomalous Nernst response will also be non-zero. This is because, the anomalous Hall conductivity and the anomalous Peltier coefficient , which measures the transverse electrical currrent in response to a longitudinal temperature gradient, are related by the celebrated Mott relation, . In turn, a non-zero implies an anomalous (zero field) Nernst coefficient given by . In recent work (45), however, based on a linearized model of a TR-broken Weyl semimetal, the anomalous Nernst response has been argued to be zero, because a linearized Weyl Hamiltonian with unbounded (or very high) ultraviolet cut-off of the Dirac spectrum produces . Here we show, from a lattice model of a WSM (with the lattice regularization providing a physical ultra-violet cut-off to the low energy Dirac spectrum) that the anomalous Peltier coefficient, and in turn the anomalous Nernst coefficient is finite and measurable in a physical time reversal breaking Weyl semimetal such as BiSb. In the main part of the paper, we use the semi-classical Boltzmann equations in the relaxation time approximation in the presence of a non-zero magnetic field and a Berry curvature, and derive the thermoelectric and charge conductivity tensors (both longitudinal and Hall) which we use to calculate both the conventional (i.e., magnetic field dependent) and topological (i.e. zero field) Nernst coefficients.
Additionally, we also investigate the thermal conductivity of a WSM based on the Boltzmann equation approach. Unlike earlier works which were based on a linearized WSM model, (46); (45) we employ a lattice Bloch Hamiltonian. With the Nernst experimental setup i.e. the temperature gradient applied perpendicular to the magnetic field , we find that the transverse magneto-thermal conductivity obeys the Wiedemann-Franz law (88) (i.e., the ratio of the thermal and electrical conductivity is the Lorenz number , both with and without the external magnetic field). In the parallel setup (), however, there is an additional dependence of the Lorenz number, thus violating the standard Wiedemann-Franz law for quasiparticles in a Landau Fermi liquid, arising from the chiral anomaly. Our results confirm that the violation of the Wiedemann-Franz law between the longitudinal magneto thermal and electrical conductivities (47) persist in the physically more transparent lattice model and is not an artifact of the linearized low energy model.
This paper is organized as follows: in Section II, we discuss the Boltzmann semi-classical approach to calculate the Nernst response in a Weyl semimetal. We derive expressions for both longitudinal and transverse charge () and Peltier () conductivity tensors, taking into account perturbative electric and magnetic fields, and a finite temperature gradient, for a Hamiltonian with a non-vanishing Berry curvature. Though the approach is general and can be applied for various configurations, we will compute our expressions relevant for the Nernst experimental setup. Section III concerns with the Nernst response in a linearly dispersing model of Dirac and Weyl fermions. We compute the magnetic field dependent Nernst response for a single Dirac node, and also for a pair of Weyl nodes which have a non-vanishing flux of the Berry curvature. Additionally Weyl fermions also exhibit an anomalous Nernst response even at zero magnetic field when one imposes a physical ultraviolet cut-off on the energy spectrum, and we show that this imposition gives a non-zero Peltier coefficient . Section IV concerns with the lattice WSM model and its Nernst response. In Section V, the magneto-thermal conductivity is analyzed, and the Wiedemann-Franz law is studied for orthogonal () and parallel () setups. We conclude in Section VI.
Ii Boltzmann formalism for Nernst response in a lattice Weyl semimetal
Nernst effect measures the transverse electrical response to a longitudinal thermal gradient in the presence of a finite magnetic field and absence of a charge current i.e. , where is defined to be the Nernst coefficient and is the temperature gradient applied along the axis. The use of three conductivity tensors, , , and suffices to relate the charge current and thermal current to an applied electric field and temperature gradient. We reserve the symbol for thermal conductivity tensor which will be the focus of Section V. We can write the following linear response equation
The tensors and are related to each other by Onsager’s relation: . In the absence of charge current (), we have . The Nernst coefficient can be derived to be
which is a function of thermoelectric tensor and charge conductivity tensor . We will evaluate and using semi-classical Boltzmann treatment in the relaxation time approximation, accounting for an external magnetic field and a finite Berry curvature.
A non-zero Berry curvature in a Bloch Hamiltonian acts like a fictitious magnetic field in the momentum space (57), which substantially modifies transport properties of the system, giving rise to anomalous behavior. Anomalous transport due to the Berry curvature has been crucial in understanding intrinsic Hall and Nernst conductivity in ferromagnetic materials(72); (73); (74); (75). The Berry curvature for a Bloch Hamiltonian is defined to be: , where is the Berry connection given by , for a Bloch eigenstate . In the presence of Berry curvature , the semi-classical equation of motion for an electron takes the following form(75); (76)
where is the crystal momentum, is the energy dispersion, and . The first term in Eq. 3 is the familiar relation between semi-classical velocity and the band energy dispersion . The second term is the anomalous transverse velocity term originating from . In the presence of electric and magnetic fields we have the standard relation: . These two coupled equations for and can be solved together to obtain (77); (78)
where . is also the prefactor which modifies the invariant phase space volume , giving rise to a non-commutative mechanical model (77), because the Poisson brackets of coordinates is non-zero. For brevity of notation, we will sometimes omit showing the explicit dependence of on and and instead write just . In Eq. 4 and Eq. 5, we have also defined to be the band-velocity. The second term in Eq. 4 gives rise to anomalous transport perpendicular to the applied electric field, while the third term gives rise to chiral magnetic effect. The third term in Eq. 5 (proportional to ) is the source of chiral anomaly, triggering negative magnetoresistance. It has been shown recently that negative magnetoresistance can be derived using the semi-classical equations of motion employing Boltzmann transport(60). Other recent works have also developed a modified Boltzmann equation, taking into account Berry curvature and chiral anomaly effects(78); (79); (80); (45); (46). In these works a linearized model of the WSM has been examined, i.e. a pair of Dirac nodes topologically protected by chirality quantum numbers.
In this paper we solve the Boltzmann equation in the presence of the Berry curvature and chiral anomaly terms for a lattice model of a WSM. The steady state Boltzmann equation in the relaxation time approximation is given by
where is the scattering time, is the equilibrium Fermi-Dirac distribution function, and is the distribution function of the system in the presence of perturbations. The scattering time can in general be a function of the crystal momentum i.e. , but we shall treat it as independent of momentum for simplicity.
We will first consider the case when and derive the longitudinal and anomalous Hall conductivities. The linear response relations between the charge current and the applied fields dictate:
The charge current in the presence of an electric field and a temperature gradient is given by(75)
In the above expression, . The quantity is entropy density for the electron gas. The first term in Eq. 8 is the current in response to an applied electric field , also accounting for the transverse anomalous velocity acquired by an electron wave-packet due to . The second term is the anomalous response to the temperature gradient , which can be obtained using the semiclassical wavepacket methods taking into account the orbital magnetization of the carriers arising from the finite spread of the wavefunction (75). It can also be derived by first calculating the transverse heat current in response to an electric field and then using Onsager’s relation (81). The heat current takes the following form after accounting for both normal and anomalous contributions(75); (82); (84); (83)
where is the polylogarithmic function of order 2, which is generally defined as for an arbitrary complex order , for a complex argument . The first term in Eq. 9 is the standard contribution to the heat current in the absence of Berry curvature. The second term is the Berry curvature mediated transverse response to electric field which can be understood by the following simple argument: in the presence of the Berry curvature and the electric field, the electron velocity acquires the additional anomalous term . Multiplying this velocity by the entropy density of the electron gas, we obtain this contribution to the transverse heat current (81). From Eq. 1 we can write the transverse response of and on the applied temperature gradient and electric field respectively as: , and . Comparing the coefficients and from Eq. 8 and Eq. 9, it is easy to note that they obey Onsager’s relation: , as expected. From Eq. 1, the anomalous response on an applied temperature gradient can be written as: . The quantity in Eq. 1 can be calculated as: , where (83)
We will further assume that the electric field and temperature gradient have non-zero components only along the direction. Substituting for from Eq. 12 in Eq. 8 for the current , and comparing the resulting expression with Eq. 7, the longitudinal components of the conductivity tensors and can be easily read to be
where . The transverse components are:
which are purely anomalous because we have assumed and are applied along the direction, and there is no magnetic field.
We now discuss the case of a finite magnetic field. We consider a particular configuration relevant for the experiments measuring Nernst coefficient i.e. , , and , although the approach will work for other configurations also, like the parallel setup discussed in Section V. The Boltzmann equation (Eq. 6 takes the following form after making substitutions for and from Eq. 4 and 5).
The following Ansatz is chosen for the distribution which also accounts for correction factor () due to a finite magnetic field.
The Boltzmann equation (Eq. 17) thus becomes
Imposing the condition that this equation must be valid for all values of , we find that , and the equation can be simplified to:
In order to solve the above equation, we introduce complex variables , and , and rewrite the equation in the following manner
where is the inverse band-mass tensor and Re() stands for the real part of . Eq. 21 can be solved for :
For convenience of notation, we rewrite and as: , incorporating into the remaining factors apart from of Eq. 22 and Eq. 23. Using Eq. 18 and the results for and , we can now explicitly write the distribution function as:
The expression for the charge current , in the presence of and , is also modified by the factor (45); (75), as we pointed out earlier that is the multiplicative factor which alters the phase space volume locally.
The temperature dependence of the zero-field anomalous contribution in Eq. 27 is hidden in the entropy density of the electron gas. Similarly, the electrical conductivity components (transverse and longitudinal) are obtained to be:
As a good check of our calculation we also recover the results for and found in Ref. (45) where the tensorial nature of is ignored and . The transverse components, i.e. Eq. 27 and 29, are a sum of two terms: the first term captures the effect of a finite which is further modified by the Berry curvature , due to the factors , and which are non-trivial functions of the Berry curvature. The first terms in Eq. 27 and Eq. 29 depend on the scattering time , and we call these as ‘modified’ -dependent Hall conductivities (because they are modified due to the Berry curvature). The Berry curvature also alters the expressions for longitudinal conductivities given in Eq. 26 and Eq. 28 because of the factor . In the limits when , the factor for a quadratic band dispersion, upto linear order in , (where is the cyclotron frequency). In the same limit, the factor upto zeroth order in , thus yielding the standard expression for and given in Eq. 13 and 14. In contrast, the second term in Eq. 27 and 29, which is Berry curvature dependent persists in the absence of a magnetic field, and is a purely anomalous contribution. We shall roughly examine the limit in which the factor significantly deviates from 1. Defining , where is the lattice constant, and is dimensionless, , where is the magnetic length and is dimensionless. For a magnetic field of 1T, , . The Berry curvature for a single linearly dispersing Weyl node centered at the origin is , thus for low magnetic fields and away from band touching point () in the momentum space (i.e. approximately in this case). This is expected, as qualitatively one understands that the effect of Berry curvature peaks when the energy band gap . When the effects of Berry curvature can be neglected, the following standard expressions are derived from Eq. 27 and 29 for Hall conductivities, keeping terms only upto linear order in :
In Section III, we will use the formula obtained for and to calculate the Nernst coefficient in Eq. 2, first analytically for a simple Dirac and Weyl linearized Hamiltonian, and then numerically in Section IV, for a lattice model of Weyl fermions.
Iii Nernst response in linearized model Dirac and Weyl systems
In this section, we will concern ourselves with the Nernst response of a linearized spectrum of Dirac and Weyl systems. We examine the magnetic field dependent transverse conductivities ( and ) for a linearly dispersing Dirac node, which are analytically tractable using the Boltzmann approach. We repeat the procedure for a pair of Weyl nodes taking into consideration the Berry flux modification of the normal -dependent conductivities.
iii.1 Nernst effect in a linearized Dirac Hamiltonian
As a warm up, we discuss the Nernst response of a single Dirac cone, with linear dispersion , where . The density of states for a single Dirac/Weyl node with unbounded linear dispersion (taking into account spin degeneracy) is given by
The density of states vanishes at the Dirac node, which gives rise to many unusual properties. Eq. 28 for the longitudinal conductivity, without the Berry curvature term, reduces to Eq. 13, which can be employed to analytically deduce the zero temperature conductivity for a Dirac Hamiltonian to be
where we have assumed the scattering time to be a phenomenological parameter independent of energy or momentum. For transverse magneto-conductivity, for a weak magnetic field , we derive the Hall conductivity for a linearized Dirac Hamiltonian using Eq. 30 to be
We note that , where is the band mass near the Fermi surface given by . At low temperatures, the thermoelectric tensor is related to the derivative of via the Mott relation (88)
Combining the Mott relation with Eq. 2, the Nenrst coefficient becomes (when )
where is the Hall angle. However Eq. 36 is valid only when , and does not diverge for as we shall see shortly.
The scattering time in Boltzmann conductivity is sensitive to the type of impurities in the system. For neutral short range or point-like impurities, the scattering time is given by:(85)
where is the density of the impurities, is the strength of the impurity potential, and is the Fermi wave-vector. Considering Thomas-Fermi (TF) screening, the scattering time for long-range ionic impurities at zero temperature is given by(85)
In the above expression, is the density of charged impurities, is the fine-structure constant, , where is the Thomas-Fermi wave-vector, and . The total scattering time is given by Matthiesen’s rule
For a linear Dirac Hamiltonian, , therefore the scattering time expression take the following form:
From Matthiesen’s rule, it is evident that the shorter time scattering process ( or ) will dominate the carrier transport, therefore near , the scattering from ionic impurities will primarily determine the conductivity, and for , it is scattering from the neutral point-like impurities that govern charge transport. For an arbitrary value of , the following expression for scattering time can be written, using expressions in Eq. 39, 40, and 41
where and are constants depending on the coefficients of and in Eq. 40 and 41, whose exact form is lengthy and not illuminating for our discussion. Using this expression for the total scattering time in Eq. 33, we obtain:
Similarly, from Eq. 34, we have
The Hall angle near reduces to:
and thus from Eq. 36, the Nernst coefficient at is given by:
Alternatively, the same conclusion can be reached by using the equation for the Nernst coefficient i.e. Eq. 2, and Eq. 35, 43, and 44, by substituting the exact expressions instead of using the Mott relation. Far away from the Dirac point at , the charge conductivities are
The Hall angle in this case no longer varies linearly with the Fermi energy (as in the case near ), but is instead given by . The Nernst coefficient for becomes
which approaches zero as is increased asymptotically, as expected from Fermi liquid theory, where the Nernst coefficient vanishes because of Sondheimers cancellation.(86)
The sign of the Nernst coefficient does not change with the sign of and is thus an even function of . This is because both and , which appear in the numerator of the expression for in Eq. 2 do not depend on sgn(). The plot in Figure 2 displays Nernst coefficient for a linear Dirac node obtained using Eq. 2, 13, 30, and 35. We have provided a physical ultra-violet cut-off to the low energy spectrum at with , , and . The chosen scattering time used for this calculation, and its scaling with is also shown in Figure 2. A regularized lattice model will however smoothly bound the dispersion in the Brillouin zone. Section IV will be devoted to evaluating the Nernst response of a WSM Hamiltonian defined on a lattice.
iii.2 Nernst effect for a pair of linearized Weyl fermions
A single Dirac node can be visualized as two Weyl nodes, which are topologically protected by chirality quantum number of opposite sign, coinciding with each other in energy-momentum space. Therefore the net flux of the Berry curvature, and henceforth the net chirality vanishes for a single Dirac node, resulting in a zero anomalous response for both the charge and thermoelectric conductivity. As a result, no anomalous Nernst response is also expected in a linear Dirac Hamiltonian. Using an external perturbation, a single band-touching point in a Dirac cone can be shifted into a pair of isolated Weyl points possessing opposite chirality quantum numbers. The external perturbation must break either time-reversal symmetry or inversion symmetry, and also lifts-up the degeneracy of the Dirac spinor. Our discussion will be centered upon the assumption that time-reversal symmetry is violated, which can be achieved using magnetic field as a perturbation. Also the Weyl points are assumed to occur at the same energy, thus there is no chiral chemical potential. One can also construct inversion asymmetric and TR invariant models of a WSM, but they are not of interest to us here because the anomalous Hall and Nernst response vanishes as the vector sum of the node separation becomes zero. Figure 3 shows Berry curvature plot of a Weyl semimetal in the plane, where we assumed the node separation is . At the origin is a Weyl node with chirality quantum number +1 which acts as a source of Berry flux. At we have another node with chirality quantum number -1 which acts a sink of Berry flux.
Let us first concern ourselves with the normal contribution to the Nernst effect i.e. due to an external magnetic field. As we pointed out earlier, this contribution is further modified due to effects of the Berry curvature, which are encoded in the factors , and , expressed in Eq. 26- 29. Adding up the contribution from both the Weyl nodes and keeping terms only upto linear order in the magnetic field (which is justified in the limit ), the zero temperature longitudinal electrical conductivity for the Weyl system becomes
For 1T, , (which is the typical value for in a WSM(11)), the factor . Thus away from , the expression reduces to Eq. 33 for a Dirac node, except upto an overall multiplicative factor of 2 (for two nodes). At , where the Fermi surface just reduces to a pair of Weyl points, . From Eq. 29, we can calculate the charge Hall conductivity for a WSM (again upto linear order in B):
Further away from , the above expression also reduces to Eq. 34 for Dirac node, after making the substitution for the cyclotron frequency . At , again . We can therefore conclude that away from the band touching point (which line-up with the fine tuning of chemical potential ), and for typical values of the Fermi velocity and weak magnetic field (such that the semiclassical Boltzmann approach is still valid) the deviation of the normal Nernst response due to the Berry curvature is negligible. In this limit the normal contribution to the Nernst response roughly reduces to the sum of individual contributions from the two Dirac nodes. We have also verified this conclusion through explicit numerical integration, even at finite temperatures, using Eq. 26- 29.
A Weyl system also exhibits anomalous Nernst response even at zero-field and thus the total Nernst signal must arise from both contributions. The anomalous Hall conductivity for a time-reversal broken Weyl semimetal is non-zero and varies linearly with the node separation ,(13) which can be obtained by integrating Eq. 29 with the correct regularization that is consistent with the broken symmetries in the presence of . (44)
The result in Eq. 52 suggests that the remains unaltered with temperature or for a finite . This result is strictly valid only for an unbounded linear dispersion of Dirac fermions. Mott formula (Eq. 35) then suggests that . If an upper physical cutoff on the energy of a Dirac node is imposed then is non-zero, because the contributions from the partially filled states generically will remain finite. Figure 4 shows the plot for as function of node separation for a linear WSM with an upper energy cut-off, and also the plot for as a function of upper energy cutoff, obtained using Eq. 27 and 29. Lowering the cutoff results in a finite non-zero , thus the anomalous Nernst response is also expected to be non-zero.
It is not entirely evident from Eq. 2, that the anomalous Nernst response will vanish for a linearized Weyl Hamiltonian with an unbounded dispersion when , because of the non-zero factor in the numerator of Eq. 2. However, using the results from the previous subsection we have from Eq. 43 that near ,