Nonequilibrium spin density and spin-orbit torque in a three-dimensional topological insulator/antiferromagnet heterostructure

We study the behavior of nonequilibrium spin density and spin-orbit torque in a topological insulator/antiferromagnet heterostructure. Unlike ferromagnetic heterostructures where the Dirac cone is gapped due to time-reversal symmetry breaking, here the Dirac cone is preserved. We demonstrate the existence of a staggered spin density corresponding to a dampinglike torque which is quite robust against scalar impurities when the transport energy is such that the transport is confined to the topological insulator surface. We show the contribution to the nonequilibrium spin density due to both surface and bulk topological insulator bands. Finally, we show that the torques in topological insulator/antiferromagnet heterostructures exhibit an angular dependence that is consistent with the standard spin-orbit torque obtained in Rashba system with some additional structure arising from the interfacial coupling.

Figure 1

(a) Schematic of an AF/TI heterostructure. Red and blue spheres denote up and down magnetic moments of the AF layer and the green spheres denote the TI. (b) Top view of an AF lattice. The gray boundary denotes the unit cell. $a$ is the interatomic distance. (c) Corresponding first Brillouin zone (gray region) with lattice vectors ${\mathit{L}}_1$ and ${\mathit{L}}_2$ and high-symmetric points $M, \mathrm{\Gamma }$, and $X$ for the AF/TI heterostructure.

Figure 2

Band structure of a heterostructure with 10 layers of TI and 5 layers of AF with a coupling strength (a) $t_C=0$, (b) $t_C=0.5$, and (c) $t_C=0.75$. The red, green, blue, and black colors correspond to the contributions from AF layers, bottom TI layer, interfacial TI layer, and bulk TI layers, respectively. The gray region shows the gap due to exchange splitting for the decoupled AF.

Figure 3

(a) Band structure, nonequilibrium spin components (b) $S_x$, (c) $S_y$, and (d) conductivity for sites 1 to 12 with out-of-plane magnetization. The red, blue, green, and black colors in (a) show the contribution from AF, interfacial TI, bottom TI, and bulk TI bands. The horizontal dashed lines denote the peaks of spin densities. In (b), (c), and (d) the vertical dashed green lines show the AF layers and the thick green line indicates the position of the interface. The corresponding configuration is shown in the bottom panel where the blue and red boxes denote AF sites with up and down magnetic moments, and the green boxes correspond to TI sites.

Figure 4

(a) Band structure and spin texture at site (b) ${\mathrm{AF}}_2^{\mathrm{B}}$, (c) ${\mathrm{AF}}_1^{\mathrm{B}}$, and (d) ${\mathrm{TI}}_1^{\mathrm{B}}$ (see bottom panel in Fig. 3) at different energies [(1) $E/A=0.156$, (2) $E/A=0.582$, (3) $E/A=0.724$]. The arrow indicates the in-plane component and the color of the arrow represents the out-of-plane component. The dotted lines correspond to the degenerate bands.

Figure 5

Contribution from Fermi surface (orange) and Fermi sea (green) for the $B$ sublattice of the first three AF layers. The equilibrium $S_z$ component is shown in the boxed regions where red and blue colors correspond to negative and positive values. The maximum value in the red box is 0.35 and the maximum value in the blue box is 0.1.

Figure 6

Band structure of a Rashba ferromagnet, Eq. (14), along $k_x$ with projected (a) $S_y$, (b) $S_z$, (c) Berry curvature, and (d) integrated Berry curvature or anomalous Hall coefficient. The dashed lines show the exchange gap. Here we use $\hslash =1, m_0=0.5, \alpha =1$, and $\mathrm{\Delta }=0.05$.

Figure 7

(a) $S_x^{\text{stg}}$, (b) $S_y^{\text{tot}}$, and (c) ${\sigma }_{xx}^{\text{tot}}$ of the AF layer as a function of the coupling strength whose value is shown in the color bar. (d) The band structure of AF/TI with $t_C=0.5A$ (blue) and $t_C=0$ (red). The dashed lines show three different regions I, II, and III.

Figure 8

Energy dependence of (a) $S_x^{\text{stg}}$, (b) $S_y^{\text{tot}}$, and (c) ${\sigma }_{xx}^{\text{tot}}$ of the AF layer as a function of the on-site energy of the AF layer with $t_C=0.5A$. The inclined dashed line shows the region spanned by the decoupled AF bands.

Figure 9

(a) $S_x^{\text{stg}}$, (b) $S_y^{\text{tot}}$, and (c) ${\sigma }_{xx}^{\text{tot}}$ as a function of broadening. Different line colors correspond to different energies as shown in (d) along with the band structure.

Figure 10

Variation of (a)–(c) ${\tau }_{FL}$ and (d)–(f) ${\tau }_{DL}$ with respect to $\theta$ [(a) and (c) and (d) and (f)] and $\varphi$ [(b) and (e)]. The inset value corresponds to the fixed angle. (g), (h), and (i) The $S_x, S_z$, and $S_y$ components, where solid and dashed lines denote the contribution from AF sites with up and down spin. The colors represent different energies as described in Fig. 9.

Figure 11

Variation of ${\tau }_D$ in the $xy$ plane for (a) $t_C=0.25$, (b) $t_C=0.50$, and (c) $t_C=0.75$ and corresponding nonequilibrium $S_z$ (d)–(f). Different colors correspond to the energies at which $S_z$ has peaks. The band structures are shown in Fig. 2.