**Spin-orbit torque and surface states in topological insulators**

Topological insulators are a new class of materials introduced about a decade ago (see Qi & Zhang, Rev. Mod. Physics 83, 1057 (2011)), and presenting outstanding properties such as insulating bulk and spin-momentum locked surface states. These unique features suggest these materials could overwhelm traditional heavy metal - transition metal heterostructures for spin-charge conversion processes. As a matter of fact, the recent demonstration of extremely large current-driven torques in these systems has opened new avenues for condensed matter physics research [1]. Moreover, their disruptive potential has been beautifully demonstrated this year with the simultaneous observation of very low critical current magnetization switching at room temperature (Wang et al., Nat. Com. 8:1364; Han et al., Phys. Rev. Lett. 119, 077702; Mahendra et al., arXiv:1703.03822). In these experiments, the magnetization is reversed using current densities two to three orders of magnitude smaller than in conventional transition metal heterostructures! These discoveries are expected to foster thrilling developments in research directions such as spintronics, topological condensed matter and even THz physics.

Besides these very exciting experimental results, understanding the nature of the physical mechanisms involved in spin-charge conversion remains a challenging issue: while it is clear that spin-momentum locked surface states contribute to the torque, the debate about the role of the bulk states rests unsettled. In many publications, the very large torque efficiency observed experimentally is attributed to a “giant spin Hall effect” emerging from the bulk of the topological insulator. This is a crucial point because it boils down to question whether one needs a conductive topological insulator to ensure large spin-charge conversion, or, on the opposite, do only surface states matter? This poses fundamental interrogations about the true nature of spin-charge conversion in these systems.

Our research is currently developed towards two directions: (i) developing a proper description of spin transport in heterostructures involving topological insulators [2,3,4,5] and (ii) understanding the nature of interfacial orbital hybridization between the topological insulator surface states and a magnetic overlayer using first principles calculations (unpublished).

Schematics of the main effects that give rise to spin-orbit torque at the interface between a topological insulator and a ferromagnet: the interfacial spin-momentum locking gives rise to inverse spin galvanic effect, while bulk states produce spin Hall effect.

**Weyl Semimetals**

Topological materials offer a thrilling new perspective for next generation data storage and manipulation. The gapless surface states of topological insulators, as discussed, can be described as massless fermions and have strong spin-momentum locking leading to potential spintronics applications. However, an additional symmetry is required for these surface states to be topologically protected. Weyl semimetals on the other hand contain band crossings, so-called Weyl points, in their bulk with the surface states forming Fermi arcs (see Wan et al. PRB 83, 205101 (2011)). The states close to the Weyl points can be described as massless chiral fermions and thus contribute to several transport phenomena in the bulk like the intrinsic spin-Hall effect and negative magneto-resistance with the most intriguing one from a fundamental point of view being the chiral anomaly (see e.g., Burkov Science 350, 378 (2015)).

Weyl semimetals come in two types. In Type I semimetals the Fermi level crosses the Weyl points but the Fermi surface remains point-like, whereas for type II, the Fermi surface opens up and a finite density of states emerges. The spin-momentum locked surface states of these behave very differently from each other and our research focuses on them with the goal being their exploitation for spintronics applications. We study the spin transport properties of surface states for both Type I and Type II using primarily analytical tools. A Type I/Type II heterojunction can also lead to mind-blowing physics (but before we say more about that, let us make sure it works the way we think it does). We look at the bulk and interface transport using both analytical and numerical tools.

**Topological crystalline insulators**

Time-reversal symmetric topological insulators such as Bi2Se3 are classified as Ζ_2 topological materials. Topological invariants can also be defined beyond the description of the Ζ_2 topological invariant. One can define a topological invariant based solely on crystal symmetry considerations; this in turn characterizes topological materials commonly known as topological crystalline insulators (TCIs). This direction allows for finding topological insulators that need not have time reversal symmetry. The first experimental realization of a TCI is the IV-VI semiconductor SnTe (Hsieh et al., Nat. Comm. 3, 982 (2012)). SnTe has a rock-salt structure with its band gap being located at four different L points. For each plane defined by two such L points and the origin, one can define a topological invariant called the Mirror Chern number using only the reflection symmetry of the plane (110) mirror plane in the rock-salt structure. Then, non-zero Mirror Chern numbers define a TCI that cannot be adiabatically connected to a trivial insulator without closing the band gap. A consequence of this is the existence of topologically protected gapless surface states on certain surface terminations that respect the symmetry; such states differ in their nature from those in Ζ_2 topological insulators and are still being investigated.

Now, why are we interested in this class of systems? What make these materials particularly interesting are the role of crystalline symmetries and the emergence of a complex Fermi surface at the edges, displaying multiple Dirac cones. More specifically, since the surface states are protected by crystalline symmetries, not by time-reversal symmetry, there is a good change that such states are preserved in the presence of magnetism…At this point, you might see where we are going: we could use these state for robust spin-charge conversion!

**Group members**