Crystal facet orientation and temperature dependence of charge and spin Hall effects in noncollinear antiferromagnets: A first-principles investigation (2024)

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  Figure 1

  (a) Crystal structure and spin configuration of cubic nc-AFM $\mathrm{M}{\mathrm{n}}_3\mathrm{Pt}$. (b) and (c) are the top views of $\mathrm{M}{\mathrm{n}}_3\mathrm{Pt}$ from (111) and (001) planes. The red, green, and blue arrows along different crystallographic axes indicate the $x$, y, and $z$ axes in the Cartesian coordinate. (b) and (c) are denoted as config1 and config2, which can be used to study the charge and spin Hall effect for (111) and (001)-oriented $\mathrm{M}{\mathrm{n}}_3\mathrm{Pt}$, respectively.

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  Figure 2

  The Bloch spectra function (BSF) of cubic $\mathrm{M}{\mathrm{n}}_3\mathrm{Pt}$ calculated at 0 K (a), 100 K (b), 200 K (c), and 350 K (d). The horizontal white dashed lines indicate the Fermi energy.

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  Figure 3

  (a) The calculated temperature dependence of longitudinal resistivity ${\rho }_{xx}$. The experimental longitudinal resistivity for single-crystal bulk $\mathrm{M}{\mathrm{n}}_3\mathrm{Pt}$ is shown in green open diamond for comparison [47]. The inset is the experimental $M\left(T\right)$ curve of $\mathrm{M}{\mathrm{n}}_3\mathrm{Pt}$ we use for considering spin-fluctuation scattering. (b) The calculated temperature-dependent $T$-odd (blue triangles, refer to left axis) and $T$-even (red diamonds, refer to right axis) charge Hall conductivity of $\mathrm{M}{\mathrm{n}}_3\mathrm{Pt}$ (001) in config2. $T$-odd experimental data are shown in blue open diamonds for comparison [47]. (c) The total $T$-odd and $T$-even charge Hall conductivity as a function of $\theta$ for $\mathrm{M}{\mathrm{n}}_3\mathrm{Pt}$ (001) at room temperature 300 K. The inset shows the definition of angle θ. (d) The calculated $T$-odd (blue triangles, refer to left axis) and $T$-even (red diamonds, refer to right axis) charge Hall conductivity as a function of longitudinal conductivity ${\sigma }_{xx}$. The dashed lines are the fitting curves based on scaling relation ${\tilde{\sigma }}_{xy}\sim a^{\prime }{\sigma }_{xx}^2+b^{\prime }{\sigma }_{xx}+c^{\prime }$ and ${\overline{\sigma }}_{xy}\sim a{\sigma }_{xx}+b$, respectively.

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  Figure 4

  Comparison of spin Hall effect for SOT applications: (a) Conventional nonmagnetic heavy metals like Pt and Ta may generate spin current along $z$ direction with only $S_y$ spin polarization by applying charge current $J_{\mathrm{c}}$ along $x$ direction. (b) Representative nc-AFM such as $\mathrm{M}{\mathrm{n}}_3\mathrm{Pt}$ (001) could produce spin current with all three spin polarizations $S_x,S_y$, and $S_z$, which is beneficial for field-free switching of perpendicular magnetization.

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  Figure 5

  The calculated spin Hall conductivity ${\sigma }_{zx}^x$ (a), ${\sigma }_{zx}^y$ (b), and ${\sigma }_{zx}^z$ (c) of $\mathrm{M}{\mathrm{n}}_3\mathrm{Pt}$ (001) in config2 as a function of temperature. (d)–(f) are the scaling relation between ${\sigma }_{zx}^x,{\sigma }_{zx}^y$, and ${\sigma }_{zx}^z$ and the longitudinal conductivity ${\sigma }_{xx}$, where the dashed lines are linear fittings by ${\sigma }_{zx}^{x,y,z}\sim a{\sigma }_{xx}+b$. (g)–(i) are the in-plane anisotropic spin Hall conductivity (refer to left axis) and the corresponding spin Hall angle (refer to right axis) for $\mathrm{M}{\mathrm{n}}_3\mathrm{Pt}$ (001) at room temperature (300 K).

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  Figure 6

  Comparison of the calculated spin Hall conductivity ${\sigma }_{zx}^z$ relevant for spin current with out-of-plane $S_z$ spin polarization for $\mathrm{M}{\mathrm{n}}_3\mathrm{Pt}$ (001) at 300 K and the reported experimental results for $\mathrm{PtT}{\mathrm{e}}_2/\mathrm{WT}{\mathrm{e}}_2/$ [51], $\mathrm{M}{\mathrm{n}}_3\mathrm{Ir}$ [50], $\mathrm{M}{\mathrm{n}}_3\mathrm{GaN}$ [52], $\mathrm{Ru}{\mathrm{O}}_2$ [53], and $\mathrm{WT}{\mathrm{e}}_2$ [54].

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