Spin caloric transport refers to the coupling of heat with spin transport. Its applications primarily concern the generation of spin currents and control of magnetisation by temperature gradients for information technology, known by the synonym spin caloritronics. Within the framework of ab initio theory, new tools are being developed to provide an additional understanding of these phenomena in realistic materials, accounting for the complexity of the electronic structure without adjustable parameters. Here, we review this progress, summarising the principles of the density-functional-based approaches in the field and presenting a number of application highlights. Our discussion includes the three most frequently employed approaches to the problem, namely the Kubo, Boltzmann, and Landauer–Büttiker methods. These are showcased in specific examples that span, on the one hand, a wide range of materials, such as bulk metallic alloys, nano-structured metallic and tunnel junctions, or magnetic overlayers on heavy metals, and, on the other hand, a wide range of effects, such as the spin-Seebeck, magneto-Seebeck, and spin-Nernst effects, spin disorder, and the thermal spin-transfer and thermal spin–orbit torques.

The observation of the spin Nernst effect in platinum thin film is reported. This and the spin Hall effects are found to be of similar magnitude. The observation of the spin Hall effect1,2,3 triggered intense research on pure spin current transport4. With the spin Hall effect1,2,5,6, the spin Seebeck effect 7,8,9 and the spin Peltier effect 10,11 already observed, our picture of pure spin current transport is almost complete. The only missing piece is the spin Nernst (–Ettingshausen) effect, which so far has been discussed only on theoretical grounds12,13,14,15. Here, we report the observation of the spin Nernst effect. By applying a longitudinal temperature gradient, we generate a pure transverse spin current in a Pt thin film. For readout, we exploit the magnetization-orientation-dependent spin transfer to an adjacent yttrium iron garnet layer, converting the spin Nernst current in Pt into a controlled change of the longitudinal and transverse thermopower voltage. Our experiments show that the spin Nernst and the spin Hall effect in Pt are of comparable magnitude, but differ in sign, as corroborated by first-principles calculations.

We present a general first-principles approach to treat various linear response phenomena relevant for spintronics. It is based on a Kubo-Bastin formalism and implemented within the multiple-scattering Korringa-Kohn-Rostoker (KKR) Green's function method with the underlying electronic structure determined by density functional theory. The symmetric (e.g., longitudinal electronic transport) as well as the antisymmetric (e.g., transverse transport) parts of the response tensor are determined, including both the so-called Fermi-sea and the Fermi-surface contributions. To describe spin-orbit-induced phenomena, such as the anomalous and spin Hall effects, a fully relativistic description is employed. Exploiting the adopted Green's function method substitutional disorder in the full concentration range of alloys is treated within the coherent potential approximation, taking full account of occurring vertex corrections in the averaging procedure for the linear response quantities. Extrinsic (scattering related, e.g., side-jump and skew scattering) and intrinsic (band structure--related) contributions to the transport tensors are treated on equal footing. Other phenomena, such as Gilbert damping and spin-orbit torques, are particular cases of the general framework and their determination is briefly addressed. The versatility of the method is demonstrated by presenting results for the anomalous and spin Hall conductivities for elemental transition metals and their alloys.

Clusters built from individual iron atoms adsorbed on surfaces (adatoms) were investigated by atomic force microscopy (AFM) with subatomic resolution. Single copper and iron adatoms appeared as toroidal structures and multiatom clusters as connected structures, showing each individual atom as a torus. For single adatoms, the toroidal shape of the AFM image depends on the bonding symmetry of the adatom to the underlying structure [twofold for copper on copper(110) and threefold for iron on copper(111)]. Density functional theory calculations support the experimental data. The findings correct our previous work, in which multiple minima in the AFM signal were interpreted as a reflection of the orientation of a single front atom, and suggest that dual and triple minima in the force signal are caused by dimer and trimer tips, respectively.

We present a first-principles study of the extrinsic spin Hall effect in dilute metallic alloys and show how tailoring themagnitude of the spin Hall conductivity can be achieved bymaterials design concerning composition and varying the concentration of the alloy partners. An essential ingredient is the relative strength of the spin-orbit coupling of host and impurity partner. This is systematically studied by changing alloy composition and performing model calculations in which the spin-orbit coupling strength is scaled. The calculations reveal that changing the impurity concentration affects the extrinsic contributions, namely skew scattering and side jump, differently. This is put into the context of recent model calculations put forward by Fert and Levy [Phys. Rev. Lett. 106, 157208 (2011)]. A fully relativistic Kubo-St. reda formalism as implemented in the multiple-scattering KKR electronic structure method is used. The calculations were carried out for Pd, Pt, and Cu hosts doped with 4d including Ag and Cd and 5d impurities including Au and Hg.

We present a procedure for the separation of the intrinsic, side-jump, and skew-scattering contributions to the spin Hall conductivity within the ab initio Kubo-Středa approach. Furthermore, two distinct contributions to the side-jump mechanism, either independent of the vertex corrections or solely caused by them, are quantified as well. This allows for a detailed analysis of individual microscopic contributions to the spin Hall effect. The efficiency of the proposed method is demonstrated by a first-principles study of dilute metallic alloys based on Cu, Au, and Pt hosts.

The challenging problem of skew scattering for Hall effects in dilute ferromagnetic alloys, with intertwined effects of spin-orbit coupling, magnetism and impurity scattering, is studied here from first principles. Our main aim is to identify chemical trends and work out simple rules for large skew scattering in terms of the impurity and host states at the Fermi surface, with particular emphasis on the interplay of the spin and anomalous Hall effects in one and the same system. The predicted trends are benchmarked by referring to three different ab initio methods based on different approximations with respect to the electronic structure and transport properties.

Finite-temperature effects in the first-principles calculations of electronic transport up to now include almost exclusively only electronic temperatures by means of the Fermi-distribution function neglecting the influence of lattice vibrations. Here, employing the linear response Kubo formalism as implemented in a fully relativistic multiple-scattering Korringa–Kohn–Rostoker Green function method a systematic first-principles study of the anomalous Hall conductivity (AHC) of the 3d-transition metals Fe, Co and Ni is presented. It is shown that the inclusion of both correlations and thermal lattice vibrations is needed to give a material-specific description of the AHC in transition metals. The employed general framework will allow a first-principles description of other transverse transport phenomena treating correlations, finite temperatures and disorder on the same footing, giving valuable insights for experiments.

Two years after the prediction of a giant spin Hall effect for the dilute Cu(Bi) alloy [Gradhand et al., Phys. Rev. B 81, 245109 (2010)], a comparably strong effect was measured in thin films of Cu(Bi) alloys by Niimi et al. [Phys. Rev. Lett. 109, 156602 (2012)]. Both theory and experiment consider the skew-scattering mechanism to be responsible, however they obtain opposite sign for the spin Hall angle. Based on a detailed analysis of the obtained theoretical results, we propose that either the formation of extremely small clusters or the influence of interface roughness and grain boundaries decorated with Bi atoms are responsible for the observed phenomenon.

We present a detailed analysis of the skew-scattering contribution to the spin Hall conductivity using an extended version of the resonant scattering model of Fert and Levy [Phys. Rev. Lett. 106, 157208 (2011)]. For 5d impurities in a Cu host, the proposed phase shift model reproduces the corresponding first-principles calculations. Crucial for that agreement is the consideration of two scattering channels related to p and d impurity states, since the discussed mechanism is governed by a subtle interplay between the spin-orbit and potential scattering in both angular-momentum channels. It is shown that the potential scattering strength plays a decisive role for the magnitude of the spin Hall conductivity.

Two years after the prediction of a giant spin Hall effect for the dilute Cu(Bi) alloy [Gradhand et al., Phys. Rev. B 81, 245109 (2010)], a comparably strong effect was measured in thin films of Cu(Bi) alloys by Niimi et al. [Phys. Rev. Lett. 109, 156602 (2012)]. Both theory and experiment consider the skew-scattering mechanism to be responsible, however they obtain opposite sign for the spin Hall angle. Based on a detailed analysis of existing theoretical results, we explore differences between theory and experiment.

Employing the linear response Kubo formalism as implemented in a fully relativistic multiple-scattering Korringa-Kohn-Rostoker Green function method a systematic first-principles study based on density-functional theory (DFT) of the anomalous Hall conductivity (AHC) of the 3$d$-transition-metals Fe, Co and Ni is presented. To account for the temperature dependence of the AHC an alloy-analogy for a set of thermal lattice displacements acting as a scattering mechanism is used which is subsequently solved using the coherent potential approximation. Further, impurity scattering has been considered to elucidate the importance of an additional possible contribution to the AHC that might be present in experiment. The impact of correlations beyond the local spin-density approximation to the exchange-correlation functional in DFT is studied within the LSDA+$U$ approach. It is shown that both, the inclusion of correlations and thermal lattice vibrations, is needed to give a material-specific description of the AHC in transition-metals.

The transition from topologically nontrivial to a trivial state is studied by first-principles calculations on bulk zinc-blende type (Hg_1-xZn_x)(Te_1-xS_x) disordered alloy series. The random chemical disorder was treated by means of the Coherent Potential Approximation. We found that although the phase transition occurs at the strongest disorder regime (x≈ 0.5), it is still manifested by well-defined Bloch states forming a clear Dirac cone at the Fermi energy of the bulk disordered material. The computed residual resistivity tensor confirm the topologically-nontrivial state of the HgTe-rich (x<0.5), and the trivial state of the ZnS-rich alloy series (x>0.5) by exhibiting the quantized behavior of the off-diagonal spin-projected component, independently on the concentration x.

A fully relativistic description of the spin-orbit induced spin Hall effect is presented that is based on Kubo's linear response formalism. Using an appropriate operator for the spin-current density a Kubo-Středa-like equation for the spin Hall conductivity (SHC) is obtained. An implementation using the Korringa-Kohn-Rostoker band structure method in combination with the coherent potential approximation allow detailed investigations on various alloy systems. A decomposition of the SHC into intrinsic and extrinsic contributions is suggested. Accompanying calculations for the skew-scattering contribution of the SHC using the Boltzmann equation demonstrate the equivalence to the Kubo formalism in the dilute alloy regime and support the suggested decomposition scheme.

The substitutional series of Heusler compounds NiTi1−xMxSn where M =Sc,V and 0 x 0.2 were synthesized and investigated with respect to their electronic structure and transport properties. The results show the possibility to create n-type and p-type thermoelectrics within one Heusler compound. The electronic structure and transport properties were calculated by all-electron ab initio methods and compared to the measurements. Hard x-ray photoelectron spectroscopy was carried out and the results are compared to the calculated electronic structure. Pure NiTiSn exhibits massive “in gap” states containing about 0.1 electrons per cell. The comparison of calculations, x-ray diffraction, and photoemission reveals that Ti atoms swapped into the vacant site are responsible for these states. The carrier concentration and temperature dependence of electrical conductivity, Seebeck coefficient, and thermal conductivity were investigated in the range from 10 to 300 K. The experimentally determined electronic structure and transport measurements agree well with the calculations. The sign of the Seebeck coefficient changes from negative for V to positive for Sc substitution. The high n-type and low p-type power factors are explained by differences in the chemical-disorder scatteringinduced electric resistivity. Major differences appear because p-type doping Sc creates holes in the triply degenerate valence band at whereas n-type doping V fills electrons in the single conduction band above the indirect gap at X what is typical for all semiconducting transition-metal-based Heusler compounds with C1b structure.

The substitutional series of Heusler compounds NiTi1-xMxSn (where M=Sc, V and 0<x <= 0.2) were synthesized and investigated with respect to their electronic structure and transport properties. The results show the possibility to create n-type and p-type thermoelectrics within one Heusler compound. The electronic structure and transport properties were calculated by all-electron ab initio methods and compared to the measurements. Hard x-ray photoelectron spectroscopy was carried out and the results are compared to the calculated electronic structure. Pure NiTiSn exhibits massive "in gap" states containing about 0.1 electrons per cell. The comparison of calculations, x-ray diffraction, and photoemission reveals that Ti atoms swapped into the vacant site are responsible for these states. The carrier concentration and temperature dependence of electrical conductivity, Seebeck coefficient, and thermal conductivity were investigated in the range from 10 to 300 K. The experimentally determined electronic structure and transport measurements agree well with the calculations. The sign of the Seebeck coefficient changes from negative for V to positive for Sc substitution. The high n-type and low p-type power factors are explained by differences in the chemical-disorder scattering-induced electric resistivity. Major differences appear because p-type doping (Sc) creates holes in the triply degenerate valence band at T whereas n-type doping (V) fills electrons in the single conduction band above the indirect gap at X what is typical for all semiconducting transition-metal-based Heusler compounds with C1(b) structure.