Nernst and magnetothermal conductivity in a lattice model of Weyl fermions

Gargee Sharma, Pallab Goswami, and Sumanta Tewari

Phys. Rev. B 93, 035116 – Published 13 January 2016

Abstract

Weyl semimetals (WSMs) are topologically protected three-dimensional materials whose low-energy excitations are linearly dispersing massless Dirac fermions, possessing a nontrivial Berry curvature. Using semiclassical Boltzmann dynamics in the relaxation time approximation for a lattice model of time-reversal (TR) symmetry broken WSMs, 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 nonvanishing Berry curvature. We also compute the thermal conductivities of a WSM in the Nernst ( $\nabla T ⊥ B$ ) and the longitudinal ( $\nabla T ∥ B$ ) setup and confirm from our lattice model that in the parallel setup, the Wiedemann-Franz law is violated between the longitudinal thermal and electrical conductivities due to the chiral anomaly.

Received 18 October 2015

DOI:https://doi.org/10.1103/PhysRevB.93.035116

Authors & Affiliations

Gargee Sharma¹, Pallab Goswami², and Sumanta Tewari¹

¹Department of Physics and Astronomy, Clemson University, Clemson, South Carolina 29634, USA
²Condensed Matter Theory Center, University of Maryland, College Park, Maryland 20742, USA

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Vol. 93, Iss. 3 — 15 January 2016

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Images

Figure 1
Top: Linearized band dispersion for Dirac and Weyl semimetals ( $k_{z}$ is suppressed). (a) A doubly degenerate Dirac semimetal. (b) Transition from a Dirac semimetal to a Weyl semimetal (represented by a pair of Dirac cones separated by a finite momentum) by breaking of time-reversal symmetry. Bottom: Energy-band spectrum for the lattice model of Weyl fermions ( $k_{x}$ suppressed) described in Eq. (56). The two band touching points occur at $K_{\pm} = (0, 0, \pm π / 2)$ .
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Figure 2
Left: Plot of the Nernst coefficient $ϑ$ as a function of $μ / t$ for a single linearized Dirac node whose spectrum is bounded at $\pm 3.5 t$ . The energy parameter $t$ is chosen to be $t = 0.1 eV$ . Here $ϑ_{0}$ indicates the value of the Nernst coefficient at $μ = 0$ . Right: Chosen scattering time for the calculation in $p s$ as given by Eq. (42).
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Figure 3
Plot of Berry curvature of a Weyl semimetal in the $k_{z} = 0$ plane, where the node separation is $k_{0} = (0.5, 0.5, 0)$ . At the origin is a Weyl node of positive chirality, which acts as a source of Berry flux indicated by outgoing arrows, which flow towards $k_{0}$ , which acts as a sink of Berry flux indicated by incoming arrows.
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Figure 4
Left: Plot of anomalous $α_{x y}$ (in the units of $μ k_{B} e / ℏ$ ) for a bounded linear WSM with two bounded linear Dirac nodes vs the upper energy cutoff $E_{C}$ (in units of 10 meV), for $μ = - 0.7 t$ and $k_{0} = 0.25$ . As the upper energy cutoff for each bounded Dirac node increases, $α_{x y}$ decreases eventually becoming close to zero. Right: Anomalous Hall conductivity $σ_{x y}$ as a function of the node separation $k_{0}$ for a bounded linear WSM, where $σ_{x y}^{0}$ is the Hall conductivity when $k_{0} = 0.15$ .
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Figure 5
Results for the lattice WSM described by Eq. (56). Top panel (left): In red is the anomalous contribution to $α_{x y} / α_{x y}^{m}$ obtained using Eq. (16), and in blue is the anomalous contribution to $α_{x y}$ using Mott relation [Eq. (35)] displaying a reasonable agreement; $α_{x}^{m}$ is the value at $μ = t$ . Top panel (right): Anomalous Hall conductivity as a function of $μ / t$ [again $σ_{x y}^{0}$ is the value at $μ = 0)$ ] obtained using Eq. (15). Middle panel (left): Normal Nernst response $ϑ^{N} / ϑ_{0}$ as a function of $μ$ , where $ϑ_{0}$ is the Nernst coefficient at $μ = 0$ . Middle panel (right): Anomalous Nernst response $ϑ^{A} / ϑ_{m}$ as a function of $μ$ , where $ϑ_{m}$ is the anomalous Nernst coefficient at $μ = 0.3 t$ . Bottom panel (left): Total Nernst coefficient $ϑ^{T} / ϑ_{m}$ including both normal and anomalous response as a function of $μ / t$ , where $ϑ_{m}$ is the Nernst coefficient at $μ = 0.1 t$ . A slight asymmetry for the total Nernst signal about $μ = 0$ can be noted. Bottom panel (right): Chosen scattering time in $ps$ as a function of $μ$ , as given by Eq. (42). The chosen parameter values for this calculation are $T = 20 K, t = 0.1 eV, m = 0.6 t$ .
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Figure 6
Top panel (left): Longitudinal magnetothermal conductivity $κ_{x x}$ for the WSM lattice model as a function of $μ / t$ , where $κ_{x x}^{m}$ is the value at $μ = t$ . Top panel (right): Normal $B$ -dependent contribution to $κ_{x y}^{N}$ , where $κ_{x y}^{m}$ is the value at $μ = t$ . Bottom panel (left): Anomalous zero magnetic field contribution to $κ_{x y}^{A}$ , where $κ_{x y}^{0}$ is the value at $μ = 0$ . In all the three figures, we have plotted in red the thermal conductivity obtained using Eqs. (63) and (64), and in blue using Wiedemann-Franz law, Eq. (61) (all three plots are for transverse setup, i.e., $\nabla T ⊥ B$ ). Bottom panel (right): Plot of $Δ L (B) / L_{0} = [L (B) / L_{0} - 1]$ as a function of applied magnetic field $B$ (here $B_{0} = v_{F} k_{0} / 2 g$ ) when $\nabla T ∥ B$ showing an additional $B^{2}$ dependence of the Lorenz number arising from the chiral anomaly term $(Ω \cdot v)$ .
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