Theory of Spin Torques Emerging from Band Topology

Abstract

Band inversion of electrons due to strong spin-orbit coupling (SOC) is commonly seen in magnetic materials. It characteristically appears in topological insulators (TIs) with inverted bandgaps [1] and Weyl semimetals (WSMs) with pointlike band crossings [2], while metallic systems are also capable of having band inversion. In momentum space around the SOC-induced band inversion, the electron spin is often correlated to its momentum, known as spin-momentum locking (SML). Moreover, the electron system acquires nontrivial band topology characterized by the Berry curvature, which serves as an effective magnetic field in momentum space. It is well understood that the band topology gives a substantial contribution to anomalous transport phenomena, such as the anomalous Hall effect (AHE) and the spin Hall effect (SHE), due to the anomalous velocity of electrons triggered by the Berry curvature. Recent experimental attempts have been successful in synthesizing materials having both nontrivial band topology and magnetism (magnetic TIs, WSMs, etc.), and it is an important question if their magnetic functionalities are also governed by their band topology.In this talk, I show from theoretical aspect that the current-induced spin torques in such magnetic topological materials are influenced and occasionally enhanced due to their band topology. After reviewing the current status of theoretical and experimental studies on magnetic topological materials, I will introduce some of my recent works.From the general point of view, I first propose that the momentum-space topology gives rise to an "intrinsic" contribution to the spin torques, which is robust against disorder and thermal fluctuation similarly to the intrinsic AHE and SHE [3]. This intrinsic torque originates from electron spin polarization converted from the anomalous velocity by SML, and is distinct from the conventional spin-transfer torque (STT) and the spin-orbit torque (SOT) driven by electron spins in transport current. In particular, I point out the intrinsic torque exerting on magnetic textures, such as domain walls (DWs), which we call the "topological Hall torque (THT)". The THT is induced by the cooperation of the real-space magnetic texture and the momentum-space Berry curvature, and can emerge dominantly in bulk crystals without building any complex heterostructures.As a typical example, I show our microscopic calculation result of current-induced torques with the low-energy effective model of magnetic WSM [4]. The band crossing points, called "Weyl points", serve as source and sink of the Berry curvature in momentum space, accompanied with SML of electrons around them (see Fig. 1). The Weyl electrons are therefore capable of generating a strong intrinsic torque. From the microscopic calculation in response to an externally applied electric field, we have derived the SOT, the STT, and the THT exerted by the Weyl electrons. The THT gets enhanced at a sharp magnetic DW, due to the localized states of Weyl electrons formed at the DW [5,6].For more realistic situations, I also show the phenomenological calculation of the THT in a metallic ferromagnet with band inversion [3]. In response to an external electric field applied to a DW (see Fig. 2), we have obtained a large THT. The THT acts on a DW in the same form as the conventional nonadiabatic STT, but with the unusual size of the nonadiabaticity parameter β≈2, which will enable an efficient control of DW motion by electric field. Experimental measurability of the THT in the metallic ferromagnet SrRuO3, in connection with the temperature dependence in its AHE [7], will also be mentioned.  Schematic image of Weyl points emerging from band inversion. The Berry curvature (magenta curves) emerges in the vicinity of the Weyl points.  Schematic image of the setup of magnetic DW in our calculations [3]. We consider an external electric field Eext applied across the DW, and calculate the torque in response to Eext.