Anomalous Hall Effect in Anisotropic Weyl Semimetals

Abstract

Various types of Anomalous Hall effects have been widely investigated in recent years [1, 2]. In most of the cases, the predicted or measured Hall signal has been attributed to spontaneous magnetic moment arisen from the dynamics of the current flow [1]. In this work, we aim to investigate the role of anisotropic band dispersion of chiral Weyl fermions in generating a transverse current resulting in a Hall signal. For this purpose, the simplest Weyl semimetal phase consisting of two Weyl nodes with opposite chirality is adopted and the angular dependence of conductance is calculated [3, 4]. Multi-terminal devices with circular geometry as illustrated in Fig. 1 are commonly used to probe the dependence of the signal on transport direction in experiments [1, 3]. Therefore, in the proposed model, we adopt the circular channel geometry and calculate the conductance for arbitrary transport directions.The Hamiltonian of the simplest Weyl system is as follows,H = V0 + Σi hki τ (viσi+wi),where σ’s are Pauli matrixes, v’s are velocities and their sign (τ = ±) carry the chirality of Weyl nodes. We assume symmetric velocities equal to vF = 106 m/s. The dispersion of Weyl fermion can be tilted along all three directions, and the strength of the tilt is denoted by wi. In the current work, two Weyl nodes having opposite chirality are also tilted along opposite directions. The tilt direction is chosen to lie along the high anisotropy axis, which is close to one of the crystal axes in most of the Weyl semimetals reported up to date [e.g., Refs. 3, 5].The angular dependence of the conductance [3, 4, 5] can be calculated based on transmission probability along an arbitrary direction and integrated over all transverse wave vectors based on the following equation,Gβ = G0 ∫ -(∂f/∂E)dE ∬ dSFS Tφ,γ cosφ cosγ.In the above equation, γ is the angle between Fermi wave vector k and the x-y plane, φ is the azimuthal angle with respect to the x-axis, G0 is the quantum conductance, dSFS is the infinitesimal element of the elliptical Fermi surface per unit variation of the coordinates φ and γ. The anisotropy in crystal symmetry and the bandstructure make the conductance dependent on the angle between the transport direction and the crystal axis. Based on the conductance profile, the voltage difference between the transverse electrodes can be found as follows,Vt = Vt1-Vt2 ∝ (S/ρ)[(Gt1-Gt2)/(L/√2)],where Gt1 = Gβ+(π/4) and Gt2 = Gβ-(π/4). The longitudinal potential difference can be calculated similarly by replacing the respective conductance value and distance between the voltage probes.As shown in Fig. 2 and predicted by the above equation, transverse voltage emerges at some angles which are related to the imbalance of Gt1 and Gt2. The highest Vt occurs at the maximum imbalance, where Gt1 and Gt2 coincide with the lowest and highest conductance points in the conductance profile shown in Fig. 1 (right). On the other hand, the conductance between source and drain is minimum in this direction as seen in Fig. 2. Our predicted results are quite consistent with the experimental results presented recently in literature [1] thus revealing a possible origin of the measured Hall signals in Weyl semimetals in the absence of a magnetic field. **