Magneto-transport signatures in periodically-driven Weyl and multi-Weyl semimetals
Introduction
Recently, there has been an upsurge in the explorations of condensed matter systems exhibiting multiple band-crossing points in the Brillouin zone (BZ), which host gapless excitations. These include the Weyl semimetals (WSMs) [1] and the multi-Weyl semimetals (mWSMs) [2], which are three-dimensional (3d) semimetals having non-trivial topological properties in the bandstructures, responsible for giving rise to various novel electrical properties (e.g., Fermi arcs). The nodal points behave as sinks and sources of the Berry flux, i.e., they are the monopoles of the Berry curvature. Since the total topological charge over the entire BZ must vanish, these nodes must come in pairs, each pair carrying positive and negative topological charges of equal magnitude. This also follows from the Nielsen–Ninomiya theorem [3]. The sign of the monopole charge is often referred to as the chirality of the corresponding node. While WSMs have a linear and isotropic dispersion, and their band-crossing points show Chern numbers (i.e., values of the monopole charges) , mWSMs exhibit anisotropic and non-linear dispersions, and harbour nodes with Chern numbers (double-Weyl) or (triple-Weyl). It can be proved mathematically that the magnitude of the Chern number in mWSMs is bounded by , by using symmetry arguments for crystalline structures [2], [4], [5]. Due to the non-trivial topological structure of these systems, novel optical and transport properties, such as circular photogalvanic effect [6], circular dichroism [7], negative magnetoresistance [8], [9], planar Hall effect (PHE) [10], magneto-optical conductivity [11], [12], [13], [14], and thermopower [15], can emerge.
There has been unprecedented advancement in the experimental front, where WSMs have been realized experimentally [9], [16], [17], [18], [19] in compounds like TaA, NbA, and TaP. These materials have been reported to have topological charges equal to . Compounds like and have been predicted to harbour double-Weyl nodes [2], [4], [20]. DFT calculations have found that nodal points, in compounds of the form (where A = Na, K, Rb, In, Tl, and X = S, Se, Te), have Chern numbers [21]. Dynamical/nonequilibrium topological semimetallic phases can also be designed by Floquet engineering [22], [23], [24], [25].
When a conductor is placed in a magnetic field , such that it has a nonzero component perpendicular to the electric field (which has been applied across the conductor), a current is generated perpendicular to the - plane. This current is usually referred to as the Hall current, and the phenomenon is the well-known Hall effect. A generalization of this phenomenon is the PHE, when there is the emergence of a voltage difference perpendicular to an applied external , which is in the plane along which and lie [cf. Fig. 1(a)]. The planar Hall conductivity, denoted by in this paper, is dependent on the angle between and . In contrast with the canonical Hall conductivity, PHE does not require a nonzero component of perpendicular to . In fact, a nonzero PHE is exhibited by ferromagnetic materials [26], [27], [28], [29], [30] or topological semimetals (which possess non-trivial Berry curvature and chiral anomalies), in a configuration in which the conventional Hall effect vanishes (because , , and the induced transverse Hall voltage, all lie in the same plane). Similar to the PHE, the planar thermal Hall effect [also referred to as the planar Nernst effect (PNE)] is the appearance of a voltage gradient perpendicular to an applied temperature () gradient (instead of an electric field), which is co-planar with an externally applied magnetic field [cf. Fig. 1(b)].
There have been extensive theoretical [15], [31], [32], [33] and experimental [9], [34] studies of the transport coefficients in these planar Hall set-ups for various semimetals. Examples include longitudinal magneto-conductivity (LMC), planar Hall conductivity (PHC), longitudinal thermo-electric coefficient (LTEC), and transverse thermo-electric coefficient (TTEC) (also known as the Peltier coefficient). In this paper, we will compute these magneto-electric and thermo-electric transport coefficients for WSMs and mWSMs, subjected to a time-periodic drive (for example, by shining circularly polarized light with frequency ). We will use a semi-classical Boltzmann equation approach for calculating these properties.
A widely used approach to analyse periodically driven systems, where the time-independent Hamiltonian is perturbed with a periodic potential, is the application of Floquet formalism [35], [36], [37], [38], [39], [40]. The approach relies on the fact that a particle can gain or lose energy in multiples of (quantum of a photon), where is the driving frequency. Since the time()-dependent Hamiltonian satisfies , where , we perform a Fourier transformation. When is much larger than the typical energy bandwidth of the system, we can combine the Floquet formalism with Van Vleck perturbation theory, to obtain an effective perturbative potential of the form: Here, denotes the Fourier mode of the Hamiltonian.
The paper is organized as follows: In Section 2, we show the low-energy effective Hamiltonians for the WSMs and mWSMs, and then write down the modifications needed to capture the properties of periodically driven systems. In Section 3, we use the semi-classical Boltzmann equations to derive the magneto-electric transport coefficients for PHE. We perform similar computations in Section 4 to determine the thermo-electric coefficients for PNE. In Section 5, we discuss our results and their implications. Finally, we conclude with a summary and outlook in Section 6.
Section snippets
Model and formalism
The low-energy effective Hamiltonian in the vicinity of a single multi-Weyl node, with topological charge , can be written as [2], [4], [5], [41], [42]: where , , and . Furthermore, and are the Fermi velocities in the direction and -plane, respectively, and is a system-dependent parameter with the dimension of momentum. As usual, is the vector of the Pauli
Longitudinal and transverse magneto-conductivities
In this section, we will consider the PHE set-up [cf. Fig. 1(a)], and evaluate the LMC and PHC. Using Eqs. (2.8), (2.9), the expressions for the LMC and PHC are given by [32], [50], [51], [52]: respectively. We have used an “approximately equal to” () sign, because we have ignored the contribution from the correction factors arising from external magnetic field. This is justified because
Longitudinal and transverse thermo-electric coefficients
In this section, we will consider the PNE set-up [cf. Fig. 1(b)], and evaluate the LTEC and TTEC. For the sake of completeness, we review the generic derivation for these transport coefficients in Appendix B.
Using the modifications of Eqs. (2.8), (2.9) for a temperature gradient along the -axis [viz., ], instead of an electric field, the LTEC and the TTEC are given by: respectively.
Discussions and physical interpretation of the results
In this section, we will discuss the results obtained for the various transport coefficients. With the choice of parameter values listed in Table 1, the condition is always satisfied, and hence the validity of the semi-classical Boltzmann transport theory is justified. While the LMC and PHC expressions have been evaluated at and in the small and large limit, the TTEC and LTEC have been computed in the large and large regime.
LMC and PHC – For large , both the LMC and the
Summary and outlook
In this paper, we have evaluated various transport coefficients for WSMs and mWSMs in the planar Hall and planad thermal Hall set-ups, when the system is perturbed by a periodic drive. We have used the low-energy effective Hamiltonian for a single node, and using the Floquet theorem, we have obtained the leading order corrections in the high frequency limit. This serves as a complementary signature for these semimetal systems, in addition to studies of other transport properties (see, for
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
We thank Tanay Nag, for suggesting the problem, and Surajit Basak, for participating in the initial stages of the project.
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