Spin-orbit-proximitized ferromagnetic metal by monolayer transition metal dichalcogenide: Atlas of spectral functions, spin textures, and spin-orbit torques in $\mathrm{Co}/{\mathrm{MoSe}}_2, \mathrm{Co}/{\mathrm{WSe}}_2$, and $\mathrm{Co}/{\mathrm{TaSe}}_2$ heterostructures

The heterostructures composed of an ultrathin ferromagnetic metal (FM) and a material hosting strong spin-orbit (SO) coupling are the principal resource for SO torque and spin-to-charge conversion nonequilibrium effects in spintronics. The key quantity in theoretical description of these effects is nonequilibrium spin density, which can appear on any monolayer of the heterostructure through which the current is flowing as long as the monolayer bands are affected by the native or proximity induced SO coupling. Here we demonstrate how hybridization of wave functions of Co layer and a monolayer of transition metal dichalcogenides (TMDs)—such as semiconducting ${\mathrm{MoSe}}_2$ and ${\mathrm{WSe}}_2$ or metallic ${\mathrm{TaSe}}_2$—can lead to dramatic transmutation of electronic and spin structure of Co within some distance away from its interface with TMD, when compared to the bulk of Co or its surface in contact with a vacuum. This is due to proximity induced SO splitting of Co bands encoded in the spectral functions and spin textures on its monolayers, which we obtain using noncollinear density functional theory (ncDFT) combined with equilibrium Green function (GF) calculations. In fact, SO splitting is present due to structural inversion asymmetry of the bilayer—i.e., just the presence of the interface—even if SO coupling within TMD monolayer is artificially switched off in ncDFT calculations. However, switching it on makes the effects associated with proximity SO coupling within Co layer about five times larger. Injecting spin-unpolarized charge current through SO-proximitized monolayers of Co generates nonequilibrium spin density over them, so that its cross product with the magnetization of Co determines SO torque. The SO torque computed via first-principles quantum transport methodology, which combines ncDFT with nonequilibrium GF calculations, can be used as the screening parameter to identify optimal combination of materials and their interfaces for applications in spintronics. In particular, we identify heterostructure two-monolayer-Co/monolayer-${\mathrm{WSe}}_2$ as the most optimal one, at least in the clean limit.

Figure 1

(a) Schematic view of heterostructures semi-infinite-Co(0001)/monolayer-TMD—where $\mathrm{TMD}={\mathrm{MoSe}}_2, {\mathrm{WSe}}_2$, or ${\mathrm{TaSe}}_2$—which are employed in calculations of spectral functions [Eq. (10)] and the corresponding spin textures. These quantities are computed on planes passing through Se and Co atomic layer at the interface, denoted by Se1 and Co1, respectively, as well as on the 4th ML of Co away from the interface which is denoted by Co4. Panel (b) shows the in-plane hexagonal Brillouin zone of heterostructures in panel (a) with high-symmetry points marked—$\mathrm{\Gamma }=(0,0,0), \mathrm{M}=(1/2,0,0)$ and $\mathrm{K}=(1/3,1/3,0)$ in the units of reciprocal lattice vectors of the supercell—which define the high-symmetry $k$ path employed in Fig. 3. (c) Lateral heterostructure 2ML-Co/monolayer-TMD, with unpolarized current injected parallel to its interface using small bias voltage $V_b$ along the $x$ axis in the linear-response regime. The heterostructure in (a) is assumed to be infinite (i.e., periodically repeated) in the $yz$ plane, while the heterostructure in (c) is infinite in the $xy$ plane. (d) Schematic view of 1H- and 1T-polytype structure of TMDs, in which transition metal atom has trigonal prismatic and octahedral coordination with Se atoms, respectively. 1T structure preserves the spatial inversion symmetry, whereas 1H structure does not.

Figure 2

Spatial profile of magnetic moments, averaged over all atoms within each atomic plane, for semi-infinite-Co/monolayer-TMD heterostructures illustrated in Fig. 1: (a) Co/1H-${\mathrm{MoSe}}_2$ (black circles); (b) Co/1H-${\mathrm{WSe}}_2$ (red squares); (c) Co/1H-${\mathrm{TaSe}}_2$ (green diamonds); and (d) Co/1T-${\mathrm{TaSe}}_2$ (blue triangles). Magenta and green solid boxes on the top of the panel show spatial extent of Co and TMD layers, respectively, as a guide to the eye.

Figure 3

Spectral function $A(E;k_y,k_z,x\in \left\{\mathrm{Co}4,\mathrm{Co}1,\mathrm{Se}1\right\})$ plotted along the high-symmetry $k$ path, $\mathrm{M}\text{−}\mathrm{\Gamma }\text{−}\mathrm{K}$ [Fig. 1], at the atomic plane passing through (a) Co4, (b) Co1, and (c) Se1, as indicated in Fig. 1, of Co/1H-${\mathrm{MoSe}}_2$ heterostructure. The magnetization of semi-infinite Co layer is $\mathbf{m}{}_{\mathrm{Co}}\parallel \widehat{x}$ [Fig. 1] and perpendicular to the interface. Panels (d)–(f) plot constant energy contours of $A(E-E_F=0;k_y,k_z,x\in \left\{\mathrm{Co}4,\mathrm{Co}1,\mathrm{Se}1\right\})$ at the Fermi energy $E-E_F=0$ and the corresponding spin textures within the monolayers Co4, Co1, and Se1, respectively, where the out-of-plane $S_x$ component of spin is indicated in color (red for positive and blue for negative values). Panels (g)–(l), (m)–(r), and (s)–(x) show the same information as panels (a)–(f) but for Co/1H-${\mathrm{WSe}}_2$, Co/1H-${\mathrm{TaSe}}_2$, and Co/1T-${\mathrm{TaSe}}_2$ heterostructure, respectively. The units for $k_y$ and $k_z$ are $2\pi /a$ and $2\pi /b$ where $a$ and $b$ are the lattice constants of the unit cell of the heterostructure.

Figure 4

The spatial profile of nonequilibrium spin density vector ${\mathbf{S}}_{\mathrm{CD}}\left(z\right)={\mathbf{S}}_{\mathrm{CD}}^{\mathrm{even}}\left(z\right)+{\mathbf{S}}_{\mathrm{CD}}^{\mathrm{even}}\left(z\right)$ [Eq. (9)] at the Fermi level for various 2ML-Co/monolayer-TMD heterostructures in Fig. 1 where monolayer TMD is: (a) 1H-${\mathrm{MoSe}}_2$; (b) 1H-${\mathrm{WSe}}_2$; (c) 1H-${\mathrm{TaSe}}_2$; and (d) 1T-${\mathrm{TaSe}}_2$. The magnetization of Co layer is perpendicular to the interface, $\mathbf{m}{}_{\mathrm{Co}}\parallel \widehat{z}$. Black, red, and green lines indicate $z,y,$ and $x$ components of ${\mathbf{S}}_{\mathrm{CD}}\left(z\right)$, respectively. Panels (e)–(h) and (i)–(l) show the same information as panels (a)–(d) but for $\mathbf{m}{}_{\mathrm{Co}}\parallel \widehat{y}$ and $\mathbf{m}{}_{\mathrm{Co}}\parallel \widehat{x}$, respectively. Vertical dashed lines mark the position of each atom plane.

Figure 5

(a)–(c) Direction of current-driven nonequilibrium spin density vector ${\mathbf{S}}_{\mathrm{CD}}^{\mathrm{odd}}$ [Eq. (9)] for magnetization ${\mathbf{m}}_{\mathrm{Co}}$ (thick cyan arrow) in Fig. 1 oriented along the $x, y$, and $z$ axis, respectively. Red, green, blue, and magenta arrows indicate the direction of ${\mathbf{S}}_{\mathrm{CD}}^{\mathrm{odd}}$ in Co/1H-${\mathrm{MoSe}}_2$, Co/1H-${\mathrm{WSe}}_2$, Co/1H-${\mathrm{TaSe}}_2$, and Co/1T-${\mathrm{TaSe}}_2$ heterostructures, respectively. Panels (d) and (e) are bar plots of the magnitudes $S_{\mathrm{CD}}^{\mathrm{odd}}=\left|{\mathbf{S}}_{\mathrm{CD}}^{\mathrm{odd}}\right|$ and $T_{\mathrm{CD}}^{\mathrm{odd}}=\left|{\mathbf{T}}_{\mathrm{CD}}^{\mathrm{odd}}\right|$, respectively, within 2 MLs of Co layer embedded into Co/1H-${\mathrm{MoSe}}_2$ (red bars), Co/1H-${\mathrm{WSe}}_2$ (green bars), Co/1H-${\mathrm{TaSe}}_2$ (blue bars), and Co/1T-${\mathrm{TaSe}}_2$ (magenta bars) heterostructures for different orientations of ${\mathbf{m}}_{\mathrm{Co}}$. The area of the common rectangular supercell of these heterostructures is denoted by $\square =4a\times 2\sqrt{3}a$, where $a$ is the lattice constant of the respective TMD.

Figure 6

The azimuthal ($\theta$) or polar ($\varphi$) angle dependence of the magnitude of the odd component of SO torque, $T_{\mathrm{CD}}^{\mathrm{odd}}\equiv \left|{\mathbf{T}}_{\mathrm{CD}}^{\mathrm{odd}}\right|$ [Eq. (8)], in 2ML-Co/monolayer-1H-${\mathrm{WSe}}_2$ heterostructure for: (a),(b) $\varphi =0^{\circ }$; (c),(d) $\theta ={90}^{\circ }$; and (e),(f) $\varphi ={90}^{\circ }$. In the left (right) column of panels ${\mathbf{T}}_{\mathrm{CD}}^{\mathrm{odd}}$ is computed with SO coupling in ${\mathrm{WSe}}_2$ being turned on (off) in ncDFT calculations. Blue curves fit numerically computed red dots using expansion in Eq. (13), with nonzero fitting parameters listed in Table 1.

Figure 7

The layer-resolved magnitude of the odd component of SO torque $T_{\mathrm{CD}}^{\mathrm{odd}}\equiv \left|{\mathbf{T}}_{\mathrm{CD}}^{\mathrm{odd}}\right|$ in $n$-ML-Co/monolayer-1H-${\mathrm{WSe}}_2$ heterostructures where Co layer consists of 2 MLs—as in the illustration in Fig. 1 and as studied in Fig. 6—or 4, 6, and 8 MLs. The unpolarized charge current is injected parallel to the interface [i.e., along the $x$ axis in Fig. 1] and it flows through each of $n$ MLs of Co layer. The total ${\mathbf{T}}_{\mathrm{CD}}^{\mathrm{odd}}$ on Co layer is the sum of contributions from each of these $n$ MLs of Co.