Modeling Spin-Orbit Torques
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Chapter 6 Numerical Methods and Implementations
In this chapter, the numerical methods used in this work and their implementations are presented. Furthermore, simple cases are considered to test and validate the implementation of the methods.
6.1 Fermi Surface Quadratures
The BCs presented in the previous chapter require the evaluation of integrals over the Fermi surface. Assuming the Fermi surface is spherical, the wave vector \(\bm {k}\) can be expressed in spherical coordinates as
\(\seteqnumber{0}{6.}{0}\)
\begin{equation}
\bm {k} = k_F \left (\sin \theta \cos \phi , \sin \theta \sin \phi , \cos \theta \right ),
\end{equation}
where \(\theta \) is the polar angle, and \(\phi \) is the azimuthal angle of the wave vector. The in-plane component can then be expressed as \(\bm {k_\|} = k_\|(\cos (\phi ),\sin (\phi ),0)\), where \(k_\| =
k_F\sin (\theta )\). Thus, the integrals over the incident part of the Fermi surface can be expressed in terms of the polar and azimuthal angles:
\(\seteqnumber{0}{6.}{1}\)
\begin{equation}
\int _\mathrm {FS\in in} d\bm {k_\|} f(\bm {k_\|}) = \int _0^{k_F} dk_\| k_\|\int _{-\pi }^{\pi } d\phi f(k_\|,\phi ) = k_F^2\int _0^{\pi /2} d\theta \cos (\theta )\sin (\theta )\int _{-\pi
}^{\pi } d\phi f(\theta ,\phi ),
\end{equation}
where the final integration is over the unit hemisphere. To evaluate the integrals numerically, the polar and azimuthal angles are discretized into \(N\) and \(M\) points, respectively. Integration over the unit hemisphere can then be
approximated as
\(\seteqnumber{0}{6.}{2}\)
\begin{equation}
\int _0^{\pi /2} d\theta \sin (\theta ) \int _{-\pi }^{\pi } d\phi g(\theta ,\phi ) \approx \sum _{i=1}^{N}\sum _{j=1}^{M} 2\pi w_{ij} g(\theta _i,\phi _j),
\end{equation}
where the weight \(w_{ij}\) and the angles \(\phi _i\) and \(\theta _j\) are determined by the chosen quadrature rule.
To determine the accuracy of the numerical integration of the Fermi surface integrals, the spin-mixing conductance, given by the Fermi surface integral
\(\seteqnumber{0}{6.}{3}\)
\begin{equation}
\label {eq:num_spin_mixing_conductance} G_{\uparrow \downarrow } = \frac {e^2}{4\pi ^2\hbar }\int _\mathrm {FS\in in} d\bm {k_\|} [1-r^\uparrow (k_\|)r^\downarrow (k_\|)],
\end{equation}
is calculated using the majority/minority reflection coefficients \(r^{\uparrow /\downarrow } (k_\|)\) from the interface magnetism Hamiltonian from the previous chapter (Eq. (5.29 ) with \(u_R =0\)) and compared to the analytical solution presented in Appendix A . Using the
parameters: \(k_F = 16\,\mathrm {nm}^{-1}\), \(u_0 = 0.42645\), and \(u_m = 0.20055\), Eq. (A.1 ) yields
\(G_{\uparrow \downarrow } \approx (5.95 + 0.86i)\times 10^{14}\) \(\mathrm {Sm}^{-2}\). The absolute relative error (ARE) between the analytical and numerical solution, \(u\) and \(u_h\), respectively, is given by
\(\seteqnumber{0}{6.}{4}\)
\begin{equation}
\eta = \frac {\vert u-u_h\vert }{\vert u\vert }.
\end{equation}
(a) Lebedev
(b) Gauss-Legendre
(c) Midpoint
Figure 6.2: The distribution of integration points for the Lebedev (a), Gauss-Legendre (b), and midpoint (c) spherical quadrature rules, with \(1800\), \(1730\), and \(1800\) points respectively.
Figure 6.1 shows the ARE between the analytical and numerical solution for the spin-mixing conductance using the Lebedev, the Gauss-Legendre quadrature rule,
and the common midpoint method. The Lebedev quadrature rule provides a set of points and weights for integrating over the surface of a sphere, which are determined by enforcing the exact integration of spherical harmonics up to
a given order [81]. The Gauss-Legendre quadrature rule provides weights \(w_i\) and points \(x_i\) such that the numerical integration is exact for polynomials of degree \(2n-1\) over the interval \([-1,1]\) [82]. It is converted
into a spherical quadrature scheme by using the midpoint method for the azimuthal angle and the Gauss-Legendre rule for the polar angle. The distribution of the integration points for the different quadrature schemes is shown in
Fig. 6.2 . The mixing conductance obtained with the Lebedev quadrature rule is the most accurate out of the three methods, followed by the Gauss-Legendre rule
and the midpoint method. Assuming this holds for all the conductance and conductivity tensors, also when \(u_R \neq 0\), from here on, all numerical evaluations of the Fermi surface integrals are evaluated using the Lebedev
quadrature rule with \(469\) integration points (\(\eta = 0.0006 \%\)).
