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Spin-Hall conductivity (SHC) of fully relativistic (4x4 matrix) Dirac electrons is studied based on the Kubo formula aiming at possible application to bismuth and bismuth-antimony alloys. It is found that there are two distinct contributions to SHC, one only from the states near the Fermi energy and the other from all the occupied states. The latter remains even in the insulating state, i.e., when the chemical potential lies in the band-gap, and turns to have the same dependences on the chemical potential as the orbital susceptibility (diamagnetism), a surprising fact. These results are applied to bismuth-antimony alloys and the doping dependence of the SHC is proposed.
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Spin-Hall conductivity $sigma_{{rm s}xy}$ and orbital susceptibility $chi$ are investigated for the anisotropic Wolff Hamiltonian, which is an effective Hamiltonian common to Dirac electrons in solids. It is found that, both for $sigma_{{rm s}xy}$ and $chi$, the effect of anisotropy appears only in the prefactors, which is given as the Gaussian curvature of the energy dispersion, and their functional forms are equivalent to those of the isotropic Wolff Hamiltonian. As a result, it is revealed that the relationship between the spin Hall conductivity and the orbital susceptibility in the insulating state, $sigma_{{rm s}xy}=(3mc^2/hbar e)chi$, which was firstly derived for the isotropic Wolff Hamiltonian, is also valid for the anisotropic Wolff Hamiltonian. Based on this theoretical finding, the magnitude of spin-Hall conductivity is estimated for bismuth and its alloys with antimony by that of orbital susceptibility, which has good correspondence between theory and experiments. The magnitude of spin-Hall conductivity turns out to be as large as $esigma_{{rm s}xy} sim 10^4 {Omega}^{-1}{rm cm}^{-1}$, which is about 100 times larger than that of Pt.
Bismuth crystal is known for its remarkable properties resulting from particular electronic states, e. g., the Shubnikov-de Haas effect and the de Haas-van Alphen effect. Above all, the large diamagnetism of bismuth had been a long-standing puzzle soon after the establishment of quantum mechanics, which had been resolved eventually in 1970 based on the effective Hamiltonian derived by Wolff as due to the interband effects of a magnetic field in the presence of a large spin-orbit interaction. This Hamiltonian is essentially the same as the Dirac Hamiltonian, but with spatial anisotropy and an effective velocity much smaller than the light velocity. This paper reviews recent progress in the theoretical understanding of transport and optical properties, such as the weak-field Hall effect together with the spin Hall effect, and ac conductivity, of a system described by the Wolff Hamiltonian and its isotropic version with a special interest of exploring possible relationship with orbital magnetism. It is shown that there exist a fundamental relationship between spin Hall conductivity and orbital susceptibility in the insulating state on one hand, and the possibility of fully spin-polarized electric current in magneto-optics. Experimental tests of these interesting features have been proposed.
We have studied the charge to spin conversion in Bi$_{1-x}$Sb$_x$/CoFeB heterostructures. The spin Hall conductivity (SHC) of the sputter deposited heterostructures exhibits a high plateau at Bi-rich compositions, corresponding to the topological insulator phase, followed by a decrease of SHC for Sb-richer alloys, in agreement with the calculated intrinsic spin Hall effect of Bi$_{1-x}$Sb$_x$ alloy. The SHC increases with increasing thickness of the Bi$_{1-x}$Sb$_x$ alloy before it saturates, indicating that it is the bulk of the alloy that predominantly contributes to the generation of spin current; the topological surface states, if present in the films, play little role. Surprisingly, the SHC is found to increase with increasing temperature, following the trend of carrier density. These results suggest that the large SHC at room temperature, with a spin Hall efficiency exceeding 1 and an extremely large spin current mobility, is due to increased number of Dirac-like, thermally-excited electrons in the $L$ valley of the narrow gap Bi$_{1-x}$Sb$_x$ alloy.
Existing investigations of the anomalous Hall effect i.e. a current flowing transverse to the electric field in the absence of an external magnetic field) are concerned with the transport current. However, for many applications one needs to know the total current, including its pure magnetization part. In this paper, we employ the two-dimensional massive Dirac equation to find the exact universal total current flowing along a potential step of arbitrary shape. For a spatially slowly varying potential we find the current density $mathbf{j}(vec r)$ and the energy distribution of the current density $mathbf{j}^varepsilon(vec r)$. The latter turns out to be unexpectedly nonuniform, behaving like a $delta$-function at the border of the classically accessible area at energy~$varepsilon$. To demonstrate explicitly the difference between the magnetization and transport currents we consider the transverse shift of an electron ray in an electric field.
We study the mechanisms of the spin Hall effect (SHE) and anomalous Hall effect (AHE) in 3$d$ ferromagnetic metals (Fe, Co, permalloy (Ni$_{81}$Fe$_{19}$; Py), and Ni) by varying their resistivities and temperature. At low temperatures where the phonon scattering is negligible, the skew scattering coefficients of the SHE and AHE in Py are related to its spin polarization. However, this simple relation breaks down for Py at higher temperatures as well as for the other ferromagnetic metals at any temperature. We find that, in general, the relation between the SHE and AHE is more complex, with the temperature dependence of the SHE being much stronger than that of AHE.
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