We report on ab initio studies of the spin Hall and spin Nernst effect in dilute ternary alloys. Our calculations are performed for a Cu host with different types of substitutional impurities. The obtained numerical results are well approximated by Matthiessen's rule relying on the constituent binary alloys. We show that the spin Nernst effect can be significantly more efficient in a ternary alloy with respect to the related binary alloys. Together with the application of Matthiessen's rule

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... opens an easy way to design materials for spintronics applications. The spin Hall effect (SHE), which was predicted by Dyakonov and Perel in 1971, 1 is one of the most promising research topics in the field of spintronics. It describes the separation of electrons with antiparallel spins lateral to an electrical current. 2 The direct experimental verification was provided 33 years later by Kato et al., 3 who observed the spin accumulation optically via Kerr rotation. However, an indirect measurement of the SHE was performed by Fert et al. 4 much earlier via studies of the anomalous Hall effect (AHE) in ternary alloys. Nowadays, the inverse SHE offers a simple method to detect a spin current via its conversion into a charge current. 5 The importance of the SHE for practical applications arises from the advantage to generate spin currents in nonmagnetic materials without spin injection from ferromagnets. Normally, three main contributions to the SHE, as well as for the AHE, 6 are discussed in literature. Namely, they are the intrinsic contribution due to the anomalous velocity 7,8 and the extrinsic skew-scattering 9,10 and side-jump 11 mechanisms. In dilute alloys the skew-scattering contribution is dominating. [12] [13] [14] In that limit the spin Hall conductivity depends strongly on the impurity type, which can even cause a sign change of the spin Hall current in one and the same host crystal. 15, 16 Recently, a related phenomenon, the spin Nernst effect (SNE), was studied theoretically. [17] [18] [19] [20] [21] [22] This phenomenon is connected to the rapidly emerging field of spin caloritronics. 23, 24 The SNE describes the creation of a transverse spin current by an applied temperature gradient, in contrast to an electric field used for the SHE. The mechanisms contributing to the SNE are the same as introduced for the SHE. Until now, the skew-scattering mechanism for both phenomena mentioned above was considered for binary alloys. 12, 15, 16, 20 In this Rapid Communication we present first-principles studies of the SHE and SNE in dilute Cubased ternary alloys. Due to the long spin diffusion length, together with the strong SHE and SNE reachable by impurity tailoring, 15, 16, 20, 25, 26 copper seems to be a good candidate for possible spintronic applications. Our work is motivated by the fact that in real systems more than one type of impurity can be present. Obviously, it is desirable to understand the influence of this to the considered phenomena. We will show that optimal combinations of different types of impurities in the same host material can enhance the generated spin current in comparison to the related binary alloys. Our investigated systems are Cu(A 1−w B w ) alloys, where a Cu host contains two different types of substitutional impurities labeled as A and B. In the considered dilute impurity limit, both charge and spin conductivity are inversely proportional to the impurity concentration. 9,10,12-15 For our studies we fix the total concentration of impurities at 1 at. % to obtain comparable results. Thus, the quantity w ∈ [0,1] describes the weighting between the impurities A and B. It implies for w = 0 and w = 1 the system reduces to the binary alloys Cu(A) and Cu(B), respectively. In our approach a fully relativistic Korringa-Kohn-Rostoker method 27 is used to obtain the electronic structure of the host and the impurity system. The transport properties are calculated within the semiclassical theory solving the linearized Boltzmann equation. 15,28 In the considered dilute limit, the impurities are assumed to be noninteracting and consequently the scattering cross sections can be added. Therefore, the microscopic transition probability of the ternary alloy Cu(A 1−w B w ) can be expressed by those of the related binary alloys Cu(A) and Cu(B) as 29 Here, P ss kk describes the scattering probability from an initial state {k,s} to a final state {k ,s }, where for each crystal momentum k there are two degenerate relativistic spin states labeled as s = + and s = −. 15,27 After solving the Boltzmann equation considering the corresponding spin-dependent microscopic transition probability, a conductivity tensor for each spin direction is calculated. 15 Within the two-current model, which is employed for our calculations, the charge conductivityσ and the spin conductivityσ s are represented bŷ This is a good approximation for a Cu host, where the electron spin polarization, expressed in units ofh/2, is higher than 0.99. 27 The results of our calculations presented below are obtained neglecting spin-flip transitions for the SNE but including them for the SHE, following to the corresponding approaches of Refs. 20 and 15. For a nonmagnetic cubic host with z as the global quantization axis, the spin-dependent conductivity tensors are