A scheme suggested in the literature to determine the symmetry-imposed shape of linear response tensors is revised and extended to allow for the treatment of more complex situations. The extended scheme is applied to discuss the shape of the spin conductivity tensor for all magnetic space groups. This allows in particular investigating the character of longitudinal as well as transverse spin transport for arbitrary crystal structure and magnetic order that give rise e.g. to the spin Hall, Nernst and the spin-dependent Seebeck effects. In addition we draw attention to a new longitudinal spin transport phenomenon occurring in certain nonmagnetic solids.

Symmetry and magnitude of spin-orbit torques (SOT), i.e., current-induced torques on the magnetization of systems lacking inversion symmetry, are investigated in a fully relativistic linear response framework based on the Kubo formalism. By applying all space-time symmetry operations contained in the magnetic point group of a solid to the relevant response coefficient, the torkance expressed as torque-current correlation function, restrictions to the shape of the direct and inverse response tensors are obtained. These are shown to apply to the corresponding thermal analogues as well, namely the direct and inverse thermal SOT in response to a temperature gradient or heat current. Using an implementation of the Kubo-Bastin formula for the torkance into a first-principles multiple-scattering Green's function framework and accounting for disorder effects via the so-called coherent potential approximation (CPA), all contributions to the SOT in pure systems, dilute as well as concentrated alloys can be treated on equal footing. This way, material specific values for all torkance tensor elements in the fcc (111) trilayer alloy system Pt | FexCo1−x | Cu are obtained over a wide concentration range and discussed in comparison to results for electrical and spin conductivity, as well as to previous work -in particular concerning symmetry w.r.t. magnetization reversal and the nature of the various contributions.

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For certain nonmagnetic solids with low symmetry the occurrence of spin-polarized longitudinal currents is predicted. These arise due to an interplay of spin-orbit interaction and the particular crystal symmetry. This result is derived using a group-theoretical scheme that allows investigating the symmetry properties of any linear response tensor relevant to the field of spintronics. For the spin conductivity tensor it is shown that only the magnetic Laue group has to be considered in this context. Within the introduced general scheme also the spin Hall and additional related transverse effects emerge without making reference to the two-current model. Numerical studies confirm these findings and demonstrate for (Au 1−x Pt x ) 4 Sc that the longitudinal spin conductivity may be on the same order of magnitude as the conventional transverse one. The presented formalism only relies on the magnetic space group and therefore is universally applicable to any type of magnetic order. The discovery of the spin Hall effect [1-3] (SHE) with its particular feature of converting a longitudinal charge current into a transverse spin current has sparked numerous studies that finally led to a deep understanding of many effects that are spin-orbit induced. Among them are the enigmatic anomalous Hall effect (AHE) that shares the same origin as the SHE and many new phenomena emerging from a coupling of spin, charge, and orbital degrees of freedom in electric fields as well as temperature gradients. Examples of these are the Edelstein effect (EE [4,5]) and the spincaloritronic pendants to the SHE and AHE, namely the spin and anomalous Nernst effects (SNE [6,7], ANE [8,9]), respectively. Many models have been formulated that aim to capture particular contributions to theses effects. For instance, the concept of the semiclassical Berry phase that can be determined on the basis of the band structure of perfect crystalline systems is connected to so called intrinsic contributions [10][11][12]. Extrinsic contributions arising from scattering at impurities in nonperfect systems can, for example, be obtained from diagrammatic methods [13] or Boltzmann transport theory [14].The aforementioned transport phenomena and their different contributions being linear in the driving fields should, in principle, be described using the fundamental Kubo formula for the corresponding response function [15],The effects then emerge from the characteristics of the underlying Hamiltonian, the pair of chosen operators for perturbation (Â j ) and observable (B i ), and the symmetry of the system. Due to the intractability of the problem to exactly solve the Kubo formula for a realistic system in practice one has to resort to approximations and/or models. However, irrespective of this problem one can still analyze the transformation properties of response tensors τ determined by the Kubo formula to make statements about which effects are in principle allowed, i.e., which nonvanishing tensor elements * sebastian.wimmer@cup.uni-muenchen.de † diemo.koedderitzsch@cu...

Quenching cubic high-temperature polymorphs of (GeTe)