Spin-orbit torques in strained PtMnSb from first principles

We compute spin-orbit torques (SOTs) in strained PtMnSb from first principles. We consider both tetragonal strain and shear strain. We find a strong linear dependence of the field-like SOTs on these strains, while the antidamping SOT is only moderately sensitive to shear strain and even insensitive to tetragonal strain. We also study the dependence of the SOT on the magnetization direction. In order to obtain analytical expressions suitable for fitting our numerical ab initio results we derive a general expansion of the SOT in terms of all response tensors that are allowed by crystal symmetry. Our expansion includes also higher-order terms beyond the usually considered lowest order. We find that the dependence on the strain is much smaller for the higher-order terms than for the lowest order terms. In order to judge the sensitivity of the SOT to the exchange correlation potential we compute the SOT in both GGA and LDA. We find that the higher-order terms depend significantly on the exchange-correlation potential, while the lowest order terms are insensitive to it. Since the higher-order terms are small in comparison to the lowest order terms the total SOT is insensitive to the exchange correlation potential in strained PtMnSb.

Figure 1

Conventional unit cell of PtMnSb. The Pt, Mn, and Sb atoms each form an fcc lattice individually. The polar and azimuthal angles of the magnetization are denoted by $\theta$ and $\varphi$, respectively. By $\mathrm{\Phi }$ we denote the angle $\mathrm{\Phi }=\varphi -{45}^{\circ }$.

Figure 2

Angular dependence of the odd torkance in PtMnSb for several strains $\eta =(c-c_{\mathrm{cub}})/c_{\mathrm{cub}}$ obtained in GGA (filled circles) and LDA (filled triangles) when the electric current is applied along the [110] direction and when the magnetization is in-plane. $\mathrm{\Phi }$ is the angle between the magnetization and the [110] direction. The component of the odd torque pointing in the [001] direction is shown. Solid lines are fits to the GGA results according to Eq. (16).

Figure 3

Angular dependence of the odd torkance in PtMnSb for several strains $\eta =(c-c_{\mathrm{cub}})/c_{\mathrm{cub}}$ obtained in GGA when the electric current is applied in the [110] direction. The magnetization is rotated from the [001] direction ($\theta$=0) to the [110] direction ($\theta ={90}^{\circ }$). We show only the component of the torque that is parallel to the unit vector ${\mathit{e}}_{\theta }$ of the spherical coordinate system. Ab initio data are shown by symbols, while solid lines are fits according to Eq. (16).

Figure 4

PtMnSb: Expansion coefficients ${\beta }_i$ in Eq. (16) for several strains $\eta =(c-c_{\mathrm{cub}})/c_{\mathrm{cub}}$ when the odd torkance is obtained from GGA.

Figure 5

PtMnSb: Expansion coefficients ${\beta }_i^{\prime }$ in Eq. (23) for several strains $\eta =(c-c_{\mathrm{cub}})/c_{\mathrm{cub}}$ when the odd torkance is obtained from GGA.

Figure 6

PtMnSb: Expansion coefficients ${\beta }_i^{\prime }$ in Eq. (23) for several strains $\eta =(c-c_{\mathrm{cub}})/c_{\mathrm{cub}}$ when the odd torkance is obtained from LDA.

Figure 7

Odd torkance in shear-strained PtMnSb obtained in GGA. The current direction is along [100]. The magnetization is in-plane. $\varphi$ is the angle between the magnetization and the [100] direction. Ab initio data are shown by filled circles, while solid lines are fits according to Eq. (A9).

Figure 8

Dependence of the odd torkance on shear-strain in PtMnSb when the polar and azimuthal angles of magnetization are $\theta ={90}^{\circ }$ and $\varphi =0^{\circ }$, respectively.

Figure 9

Expansion coefficients ${\beta }_i$ in Eq. (A9) of the odd torkance in shear-strained PtMnSb obtained in GGA.

Figure 10

Angular dependence of the even torkance obtained within GGA in PtMnSb for several tetragonal strains $\eta =(c-c_{\mathrm{cub}})/c_{\mathrm{cub}}$ when the electric current is applied along [110] direction and when the magnetization is in-plane. $\mathrm{\Phi }$ is the angle between magnetization and the [110] direction. We show the component of the even torque that is parallel to the unit vector $e_{\varphi }$ of the spherical coordinate system. Ab-initio data are shown by filled circles, while solid lines are fits according to Eq. (19).

Figure 11

Angular dependence of the even torkance obtained within GGA in PtMnSb for several shear strains $\epsilon$ when the electric current is applied along the [100] direction and when the magnetization is in-plane. $\varphi$ is the angle between magnetization and the [100] direction. We show the component of the even torque that is parallel to the unit vector $e_{\varphi }$ of the spherical coordinate system. Ab initio data are shown by filled circles, while solid lines are fits according to Eq. (B2).