Semirealistic tight-binding model for spin-orbit torques

We compute the spin-orbit torque in a transition-metal heterostructure using Slater-Koster parametrization in the two-center tight-binding approximation and accounting for $d$ orbitals only. In this method, the spin-orbit coupling is modeled within Russel-Saunders scheme, which enables us to treat interfacial and bulk spin-orbit transport on equal footing. The two components of the spin-orbit torque, dissipative (dampinglike) and reactive (fieldlike), are computed within Kubo linear response theory. By systematically studying their thickness and angular dependence, we were able to accurately characterize these components beyond the traditional inverse spin galvanic and spin Hall effects. We find that the conventional fieldlike torque is purely interfacial. In contrast, we unambiguously demonstrate that the conventional dampinglike torque possesses both an interfacial and a bulk contribution. In addition, both fieldlike and dampinglike torques display substantial angular dependence with strikingly different thickness behaviors. While the planar contribution of the fieldlike torque decreases smoothly with the nonmagnetic metal thickness, the planar contribution of the dampinglike torque increases dramatically with the nonmagnetic metal thickness. Finally, we investigate the self-torque exerted on the ferromagnet when the spin-orbit coupling of the nonmagnetic metal is turned off. Our results suggest that the spin accumulation that builds up inside the ferromagnet can be large enough to induce magnetic excitations.

Figure 1

Schematics of the diatomic chain model. The atoms of the bottom chain (gray) possess both $p_z$ (a) and $p_x$ (b) orbitals, while the atoms of the top chain have only $p_z$ orbitals. The phase acquired by Bloch electrons hopping from one orbital to the other is also given.

Figure 2

(a) Schematics of the bcc heterostructure composed of a ferromagnetic metal (blue) deposited on top of a nonmagnetic metal (gray). For simplicity, we consider that both metals possess the same lattice parameter. (b) Hopping parameters at the interface between the ferromagnet and the nonmagnetic metal. $t_1$ stands for the nearest-neighbor hopping and $t_2$ stands for the second-nearest-neighbor hopping.

Figure 3

Spin-resolved density of states of Fe(5)/W(7) projected on the $d$ orbitals and calculated by (a) density-functional theory and (b) our tight-binding model (with $\mathrm{\Gamma }=10$ meV). The blue shaded region corresponds to W (nonmagnetic layer) while the red shaded region corresponds to Fe (magnetic layer). The vertical dashed lines in (b) correspond to two cases of interest discussed in Sec. 3a.

Figure 4

Spin-resolved band structure for FM(5)/NM(7) bilayer: (a) $s_x$ along ${\overline{\mathrm{Y}}}^{\prime }-\overline{\mathrm{\Gamma }}-\overline{\mathrm{Y}}$, (b) $s_y$ along ${\overline{\mathrm{X}}}^{\prime }-\overline{\mathrm{\Gamma }}-\overline{\mathrm{X}}$, and (c) $s_z$ along $\overline{\mathrm{X}}-\overline{\mathrm{\Gamma }}-\overline{\mathrm{Y}}$. Blue and red colors refer to opposite sign of the spin momentum.

Figure 5

Nonequilibrium in-plane spin-density profile per unit electric field across the FM(5)/NM(7) bilayer calculated at Fermi energy $E_{\mathrm{F}}$. The shaded blue area refers to the FM region and the shaded yellow area refers to the NM region. The black solid line is the spin density when the spin-orbit coupling of both ferromagnetic and nonmagnetic layers is turned on, and the red solid line is the spin density when only the spin-orbit coupling of the ferromagnetic layer is on. Here the magnetization points perpendicular to the plane, along $\mathbf{z}$.

Figure 6

Dependence of the two torque components, (a) fieldlike torque and (b) dampinglike torque, as a function of the homogeneous broadening $\mathrm{\Gamma }$, for different values of the transport energy, $E-E_{\mathrm{F}}=1$ eV (black), $E-E_{\mathrm{F}}=0$ eV (blue), and $E-E_{\mathrm{F}}=-1$ eV (red). The inset displays the slab conductivity. Here the magnetization points perpendicular to the plane, along $\mathbf{z}$.

Figure 7

Transport properties of the heterostructure upon varying the nonmagnetic layer thickness while keeping the ferromagnetic layer thickness at 5 ML (left panels), and upon varying the ferromagnetic layer thickness while keeping the nonmagnetic layer thickness at 20 MLs (right panels). This figure shows the thickness dependence of (a), (c) the fieldlike torque and (b), (d) the dampinglike torque. The curves are calculated for a magnetization pointing along $\mathbf{z}$ and for various disorder strengths, $\mathrm{\Gamma }=10$ meV (black), $\mathrm{\Gamma }=20$ meV (blue), $\mathrm{\Gamma }=50$ meV (red), and $\mathrm{\Gamma }=100$ meV (green).

Figure 8

Efficiency of the (a) fieldlike torque and (b) dampinglike torque as a function of the nonmagnetic layer thickness. The curves are calculated for a magnetization pointing along $\mathbf{z}$ and for various disorder strengths, $\mathrm{\Gamma }=10$ meV (black), $\mathrm{\Gamma }=20$ meV (blue), $\mathrm{\Gamma }=50$ meV (red), and $\mathrm{\Gamma }=100$ meV (green).

Figure 9

Legendre expansion coefficients as a function of the thickness of the nonmagnetic layer. These coefficients correspond to (a) the conventional fieldlike torque, (b) the conventional dampinglike torque, (c) the planar fieldlike torque, and (d) the planar dampinglike torque.

Figure 10

Spin-orbit torque components upon varying the nonmagnetic layer thickness for a ferromagnetic layer thickness of 5 ML (left panels) and upon varying the ferromagnetic layer thickness for a nonmagnetic layer thickness of 20 MLs (right panels). The black curves are computed assuming spin-orbit coupling is present in both nonmagnetic and ferromagnetic layers, whereas the blue curve assumes it is present only in the ferromagnetic layer. The curves are calculated for a magnetization pointing along $\mathbf{z}$ and for $\mathrm{\Gamma }=50$ meV.

Figure 11

Dependence of the (a) anomalous Hall conductivity, (b) fieldlike, and (c) dampinglike components of the spin-orbit torque upon varying the spin-orbit coupling ${\xi }_{\mathrm{so}}$. The black lines represent the case where the spin-orbit coupling of the ferromagnet is set to zero, whereas the blue lines represent the case where the spin-orbit coupling of the nonmagnetic metal is set to zero. In this calculation, we set the nonmagnetic metal thickness to 40 MLs and the ferromagnet thickness to 7 ML. The curves are calculated for a magnetization pointing along $\mathbf{z}$ and for $\mathrm{\Gamma }=50$ meV.