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We present the microscopic theory of the spin Nernst effect, which is a transverse spin current directly induced by a temperature gradient, employing the linear response theory with Luttinger's gravitational potential method. We consider a generic, non-interacting electron system with randomly distributed impurities and evaluate the spin current response to the gravitational potential. Our theory takes into account a contribution of the local equilibrium current modified by Luttinger's gravitational potential and is thus consistent with the thermodynamic principle that thermal responses should vanish. The Ward-Takahashi identities ensure that the spin Nernst current is well-behaved at low temperatures in any order of the random impurity potentials. Furthermore, we microscopically derive the spin-current version of Mott's formula, which associates the spin Nernst coefficient with the spin Hall conductivity. The spin-current version of the St\v{r}eda formula is also discussed. To demonstrate these findings, the spin Nernst current of three-dimensional Dirac electrons is computed. Our theory is general and can therefore be extended to interacting electron systems, where Mott's formula no longer holds., Comment: 16 pages, 4 figures

We study the intrinsic spin Hall effect of a Dirac Hamiltonian system with ferromagnetic exchange coupling, a minimal model combining relativistic spin-orbit interaction and ferromagnetism. The energy bands of the Dirac Hamiltonian are split after introducing a Stoner-type ferromagnetic ordering which breaks the spherical symmetry of pristine Dirac model. The totally antisymmetric spin Hall conductivity (SHC) tensor becomes axially anisotropic along the direction of external electric field. Interestingly, the anisotropy does not vanish in the asymptotic limit of zero magnetization. We show that the ferromagnetic ordering breaks the spin degeneracy of the eigenfunctions and modifies the selection rules of the interband transitions for the intrinsic spin Hall effect. The difference in the selection rule between the pristine and the ferromagnetic Dirac phases causes the anisotropy of the SHC, resulting in a discontinuity of the SHC as the magnetization, directed orthogonal to the electric field, is reduced to zero in the ferromagnetic Dirac phase and enters the pristine Dirac phase., Comment: 12 pages, 5 figure

Electron hydrodynamics is currently known to emerge only when electron-electron interaction dominates over the momentum-nonconserving scatterings of electrons, where the electron transport is described by a hydrodynamic equation. Here we show that electron transport in electron systems with the spin Hall effect is also given by the hydrodynamic equation, whose kinetic viscosity is determined by the spin diffusion length and the transport lifetime. The electric current vorticity is proportional to the spin accumulation due to the spin Hall effect in two-dimensional systems. We demonstrate by solving the hydrodynamic equation in a two-dimensional system with a cavity, combined with micromagnetic simulation for an attached chiral magnetic insulator, that the spin accumulated near the boundary of the cavity creates a magnetic skyrmion. Our findings and demonstration shed light on a novel aspect of electron hydrodynamics and spin transport., Comment: 6 pages, 4 figures

The quantum field theoretical approach with the Kubo formula has successfully captured spin torques, such as spin-transfer torques and spin-orbit torques, for continuum systems. We examine the field theoretical approach to current-induced spin-transfer torques in a magnetic junction system. We first give a brief overview of the field theoretical approach to spin torques. Then, we consider a five-layers system consisting of three nonmagnetic metal layers separated by two ferromagnetic metal layers and apply an electric field perpendicular to the layers. We demonstrate that the Slonczewski-type spin-transfer torque, or shortly the Slonczewski torque, on the magnetizations in ferromagnetic layers is obtained by evaluating nonequilibrium electron spin density, based on the linear response theory with the Green function method. The obtained coefficient of the Slonczewski torque has a quantum oscillation at absolute zero temperature, which has not been mentioned before. A field-like torque accompanied by the Slonczewski torque is also evaluated., Comment: 11pages, 5 figures, any comments are welcome

The spin Hall effect of a Dirac Hamiltonian system is studied using semiclassical analyses and the Kubo formula. In this system, the spin Hall conductivity is dependent on the definition of spin current. All components of the spin Hall conductivity vanish when spin current is defined as the flow of spin angular momentum. In contrast, the off-diagonal components of the spin Hall conductivity are non-zero and scale with the carrier velocity (and the effective $g$-factor) when spin current consists of the flow of spin magnetic moment. We derive analytical formula of the conductivity, carrier mobility and the spin Hall conductivity to compare with experiments. In experiments, we use Bi as a model system that can be characterized by the Dirac Hamiltonian. Te and Sn are doped into Bi to vary the electron and hole concentration, respectively. We find the spin Hall conductivity ($\sigma_\mathrm{SH}$) takes a maximum near the Dirac point and decreases with increasing carrier density ($n$). The sign of $\sigma_\mathrm{SH}$ is the same regardless of the majority carrier type. The spin Hall mobility, proportional to $\sigma_\mathrm{SH}/n$, increases with increasing carrier mobility with a scaling coefficient of $\sim$1.4. These features can be accounted for quantitatively using the derived analytical formula. The results demonstrate that the giant spin magnetic moment, with an effective $g$-factor that approaches 100, is responsible for the spin Hall effect in Bi.

Spin currents proportional to the first- and second-order of the electric field are calculated in a specific tight-binding model with time-reversal symmetry. Specifically, a tight-binding model with time-reversal symmetry is constructed with chiral hopping and spin-orbit coupling. The spin conductivity of the model is calculated using the Boltzmann equation. As a result, it is clarified that the first-order spin current of the electric field vanishes, while the second-order spin current can be finite. Furthermore, the spin current changes its sign by reversing the chirality of the model. The present results reveal the existence of spin currents in systems with time-reversal symmetry depending on the chirality of the system. They may provide useful information for understanding the chirality-dependent spin polarization phenomena in systems with time-reversal symmetry.

We study the Seebeck effect in the three-dimensional Dirac electron system based on the linear response theory with Luttinger's gravitational potential. The Seebeck coefficient $S$ is defined by $S = L_{12} / L_{11} T$, where $T$ is the temperature, and $L_{11}$ and $L_{12}$ are the longitudinal response coefficients of the charge current to the electric field and to the temperature gradient, respectively; $L_{11}$ is the electric conductivity and $L_{12}$ is the thermo-electric conductivity. We consider randomly-distributed impurity potentials as the source of the momentum relaxation of electrons and microscopically calculate the relaxation rate and the vertex corrections of $L_{11}$ and $L_{12}$ due to the impurities. It is confirmed that $L_{11}$ and $L_{12}$ are related through Mott's formula in low temperatures when the chemical potential lies above the gap ($|\mu| > \Delta$), irrespective of the linear dispersion of the Dirac electrons and unconventional energy dependence of the lifetime of electrons. On the other hand, when the chemical potential lies in the band gap ($|\mu| < \Delta$), Seebeck coefficient behaves just as in conventional semiconductors: Its dependences on the chemical potential $\mu$ and the temperature $T$ are partially captured by $S \propto (\Delta - \mu) / \kB T$ for $\mu > 0$. The Seebeck coefficient takes the relatively large value $|S| \simeq 1.7 \,\mathrm{m V/K}$ at $T \simeq 8.7\,\mathrm{K}$ for $\Delta = 15 \,\mathrm{m eV}$ by assuming doped bismuth., Comment: 11 pages, 5 figures

Using response theory, we calculate the charge-current vortex generated by spin pumping at a point-like contact in a system with Rashba spin-orbit coupling. We discuss the spatial profile of the current density for finite temperature and for the zero-temperature limit. The main observation is that the Rashba spin precession leads to a charge current that oscillates as a function of the distance from the spin-pumping source, which is confirmed by numerical simulations. In our calculations, we consider a Rashba model on a square lattice, for which we first review the basic properties related to charge and spin transport. In particular, we define the charge- and spin-current operators for the tight-binding Hamiltonian as the currents coupled linearly with the U(1) and SU(2) gauge potentials, respectively. By analogy to the continuum model, the spin-orbit-coupling Hamiltonian on the lattice is then introduced as the generator of the spin current., Comment: 10 pages, 6 figures

Employing unbiased large-scale time-dependent density-matrix renormalization-group simulations, we demonstrate the generation of a charge-current vortex via spin injection in the Rashba system. The spin current is polarized perpendicular to the system plane and injected from an attached antiferromagnetic spin chain. We discuss the conversion between spin and orbital angular momentum in the current vortex that occurs because of the conservation of the total angular momentum and the spin-orbit interaction. This is in contrast to the spin Hall effect, in which the angular-momentum conservation is violated. Finally, we predict the electromagnetic field that accompanies the vortex with regard to possible future experiments., Comment: 12 pages, 11 figures

Electron transport in magnetic orders and the magnetic orders dynamics have a mutual dependence, which provides the key mechanisms in spin-dependent phenomena. Recently, antiferromagnetic orders are focused on as the magnetic order, where current-induced spin-transfer torques, a typical effect of electron transport on the magnetic order, have been debatable mainly because of the lack of an analytic derivation based on quantum field theory. Here, we construct the microscopic theory of spin-transfer torques on the slowly-varying staggered magnetization in antiferromagnets with weak canting. In our theory, the electron is captured by bonding/antibonding states, each of which is the eigenstate of the system, doubly degenerates, and spatially spreads to sublattices because of electron hopping. The spin of the eigenstates depends on the momentum in general, and a nontrivial spin-momentum locking arises for the case with no site inversion symmetry, without considering any spin-orbit couplings. The spin current of the eigenstates includes an anomalous component proportional to a kind of gauge field defined by derivatives in momentum space and induces the adiabatic spin-transfer torques on the magnetization. Unexpectedly, we find that one of the nonadiabatic torques has the same form as the adiabatic spin-transfer torque, while the obtained forms for the adiabatic and nonadiabatic spin-transfer torques agree with the phenomenological derivation based on the symmetry consideration. This finding suggests that the conventional explanation for the spin-transfer torques in antiferromagnets should be changed. Our microscopic theory provides a fundamental understanding of spin-related physics in antiferromagnets.

The interaction between spin and nanomechanical degrees of freedom attracts interest from the viewpoint of basic science and device applications. We study the magnon current induced by the torsional oscillation of ferromagnetic nanomechanical cantilever. We find that a finite Dzyaloshinskii-Moriya (DM) interaction emerges by the torsional oscillation, which is described by the spin gauge field, and the DM interaction leads to the detectably-large magnon current with frequency same as that of the torsional oscillation. Our theory paves the way for studying torsional spin-nanomechanical phenomena by using the spin gauge field., Comment: 5 pages, 4 figures

We study valley-dependent spin transport theoretically in monolayer transition-metal dichalcogenides in which a variety of spin and valley physics are expected because of spin-valley coupling. The results show that the spins are valley-selectively excited with appropriate carrier doping and valley polarized spin current (VPSC) is generated. The VPSC leads to the spin-current Hall effect, transverse spin accumulation originating from the Berry curvature in momentum space. The results indicate that spin excitations with spin-valley coupling lead to both valley and spin transport, which is promising for future low-consumption nanodevice applications., Comment: 6 pages, 5 figures

We investigate an interfacial spin-transfer torque and $\beta$-term torque with alternating current (AC) parallel to a magnetic interface. We find that both torques are resonantly enhanced as the AC frequency approaches to the exchange splitting energy. We show that this resonance allows us to estimate directly the interfacial exchange interaction strength from the domain wall motion. We also find that the $\beta$-term includes an unconventional contribution which is proportional to the time derivative of the current and exists even in absence of any spin relaxation processes., Comment: 5 pages, 2 figures

We theoretically investigate a manipulation method of nonequilibrium spin accumulation in the paramagnetic normal metal of a spin pumping system, by using the spin precession motion combined with the spin diffusion transport. We demonstrate based on the Bloch-Torrey equation that the direction of the nonequilibrium spin accumulation is changed by applying an additional external magnetic field, and consequently, the inverse spin Hall voltage in an adjacent paramagnetic heavy metal changes its sign. We find that the spin relaxation time and the spin diffusion length are simultaneously determined by changing the magnitude of the external magnetic field and the thickness of the normal metal in a commonly-used spin pumping system.

We show theoretically that conversion between spin and charge by spin-orbit interaction in metals occurs even in a non-local setup where magnetization and spin-orbit interaction are spatially separated if electron diffusion is taken into account. Calculation is carried out for the Rashba spin-orbit interaction treating the coupling with a ferromagnet perturbatively. The results indicate the validity of the concept of effective spin gauge field (spin motive force) in the non-local configuration. The inverse Rashba-Edelstein effect observed for a trilayer of a ferromagnet, a normal metal and a heavy metal can be explained in terms of the non-local effective spin gauge field.

As a strongly spin-orbit coupled metallic model with ferromagnetism, we have considered an extended Stoner model to the relativistic regime, named Dirac ferromagnet in three dimensions. In the previous paper~[Phys. Rev. B 90, 214418 (2014)], we studied the transport properties giving rise to the anisotropic magnetoresistance~(AMR) and the anomalous Hall effect~(AHE) with the impurity potential being taken into account only as the self-energy. The effects of the vertex corrections~(VCs) to AMR and AHE are reported in this paper. AMR is found not to change quantitatively when the VCs is considered, although the transport lifetime is different from the one-electron lifetime and the charge current includes additional contributions from the correlation with spin currents. The side-jump and the skew-scattering contributions to AHE are also calculated. The skew-scattering contribution is dominant in the clean case as can be seen in the spin Hall effect in the non-magnetic Dirac electron system., Comment: 21 pages, 7 figures

We investigate the spin Hall effect (SHE) of electrons described by the Dirac equation, which is used as an effective model near the $L$-points in bismuth. By considering short-range nonmagnetic impurities, we calculate the extrinsic as well as intrinsic contributions on an equal footing. The vertex corrections are taken into account within the ladder type and the so-called skew-scattering type. The intrinsic SHE which we obtain is consistent with that of Fuseya et al. [J. Phys. Soc. Jpn. 81, 93704 (2012)]. It is found that the extrinsic contribution dominates the intrinsic one when the system is (semi)metallic. The extrinsic SHE due to the skew scattering is proportional to $\Delta / n_{\rm i} u$, where $2\Delta$ is the band gap, $n_{\rm i}$ is the impurity concentration, and $u$ is the strength of the impurity potential., Comment: 12 pages, 2 figures, submitted to J. Phys. Soc. Jpn

We propose a model ferromagnet based on the Dirac Hamiltonian in three spatial dimensions, and study its transport properties which include anisotropic magnetoresistance (AMR) and anomalous Hall (AH) effect. This relativistic extension allows two kinds of ferromagnetic order parameters, denoted by $\bm{M}$ and $\bm{S}$, which are distinguished by the relative sign between the positive- and negative-energy states (at zero momentum) and become degenerate in the non-relativistic limit. Because of the relativistic coupling between the spin and the orbital motion, both $\bm{M}$ and $\bm{S}$ induce anisotropic deformations of the energy dispersion (and the Fermi surfaces) but in mutually opposite ways. The AMR is determined primarily by the anisotropy of the Fermi surface (group velocity), and secondarily by the anisotropy of the damping; the latter becomes important for ${\bm M}=\pm{\bm S}$, where the Fermi surfaces are isotropic. Even when the chemical potential lies in the gap, the AH conductivity is found to take a finite non-quantized value, $\sigma_{ij} = -(\alpha /3\pi^2 \hbar) \epsilon_{ijk} S_k $, where $\alpha$ is the (effective) fine structure constant. This offers an example of Hall insulator in three spatial dimensions., Comment: 17 pages, 6 figures