Hybridization-induced interface states in a topological-insulator–ferromagnetic-metal heterostructure

Recent experiments demonstrating large spin-transfer torques in topological-insulator (TI)–ferromagnetic-metal (FM) bilayers have generated a great deal of excitement due to their potential applications in spintronics. The source of the observed spin-transfer torque, however, remains unclear. This is because the large charge transfer from the FM to the TI layer would prevent the Dirac cone at the interface from being anywhere near the Fermi level to contribute to the observed spin-transfer torque. Moreover, there is still not much understanding of the impact on the Dirac cone at the interface from the metallic bands overlapping in energy and momentum, where strong hybridization could take place. Here, we build a simple microscopic model and perform first-principles-based simulations for such a TI-FM heterostructure, considering the strong hybridization and charge-transfer effects. We find that the original Dirac cone is destroyed by hybridization, as expected. Instead, we find an interface state that we dub a “descendent state” that forms near the Fermi level due to the strong hybridization with the FM states at the same momentum. Such a descendent state carries a sizable weight of the original Dirac interface state, and thus it inherits the localization at the interface and the same Rashba-type spin-momentum locking. We propose that the descendent state may be an important source of the experimentally observed large spin-transfer torque in the TI-FM heterostructure.

Figure 1

(a) The dispersions of the FM slab (the left panel) and the TI slab (the right panel) before forming the heterostructure along in-plane momentum ${\mathbf{k}}_{\parallel }=(k_x,0)$. The upper and lower dashed lines represent the Fermi levels of the FM (${\mu }_{\text{FM}}$) and TI slab (${\mu }_{\text{TI}}$), respectively. (b) The band structure of the heterostructure in the presence of hybridization. The vertical and horizontal dashed lines represent the Fermi level $\mu$ and the Fermi momentum ${\tilde{\mathbf{k}}}_{\parallel }=(k_F,0)$, respectively. The hybridization strength $g_{\beta }\left({\mathbf{k}}_{\parallel }\right)$ is given in the text with ${\tilde{g}}_0=0.078$ eV and ${\tilde{g}}_1=0.098$ eV. The red and orange dots label the interface states that emerge outside of and survive within the FM band, respectively. Here, the interface states are defined as states consisting of a substantial weight $>48\%$ on the TI side that has more than $80\%$ of weight localized in the first $30\%$ of the TI slab away from the interface. (c) The wave function of the interface state in the heterostructure [red dots in (b)] at the Fermi level $\mu$ with momentum ($k_F,0$). For all subfigures, the slab thicknesses are $N_{\text{TI}}=80$, $N_m=40$, and the band-bending potential parameters are $V_0=1$ eV and $\eta =0.3$, respectively.

Figure 2

(a) The spin texture $(S_{x,{\tilde{\mathbf{k}}}_{\parallel }},S_{y,{\tilde{\mathbf{k}}}_{\parallel }})$ of the descendent state in the hybridized heterostructure at Fermi level $\mu$ with Fermi momenta ${\tilde{\mathbf{k}}}_{\parallel }=k_F(cos{\theta }_k,sin{\theta }_k)$. (b) The weight of the original Dirac interface state $|{\psi }_{{\tilde{\mathbf{k}}}_{\parallel }}^{\left(0\right)}\rangle$ carried by the eigenstates of the heterostructure along the momentum-cut at ${\tilde{\mathbf{k}}}_{\parallel }=(k_F,0)$ [the vertical dashed line in Fig. 1]. The red (upper) horizontal axis and the red curve are for the new interface state $|{\psi }_{{\tilde{\mathbf{k}}}_{\parallel }}\rangle$ at the Fermi level $\mu$, and the black (lower) horizontal axis and the black curve are for the rest of the eigenstates $|{\varphi }_{\gamma ,{\tilde{\mathbf{k}}}_{\parallel }}\rangle$.

Figure 3

Top views of two different interfaces for the TI-FM bilayer. Interface (a) has a lower energy than interface (b) by 0.07 eV upon the geometry relaxation. Only the topmost Se (small green), Bi (large purple), and Ni (medium blue) atoms at the interface are shown. The in-plane lattice vectors in real space are shown as red and green arrows.

Figure 4

(a) DFT-calculated band structure of the TI-FM bilayer along the $k_y$ axis and (b) zoom-in of (a) when the magnetization is along the $y$ axis. In (a) and (b), green (blue) symbols represent states localized into the Ni slab (the bottommost QL in contact with vacuum), while red symbols denote states localized into the topmost-1 QL, such as descendent states. Orange symbols represent the states localized at the topmost QL. (c) Relative electron density $\rho \left(z\right)$ profile (dimensionless) of the red and orange bands as a function of $z$ coordinates. The red is for the $\rho \left(z\right)$ value of the red band or descendent state computed at $k_y=-0.087$ ${\AA }^{-1}$ or $k_{\mathrm{F}}$ [red squares in (b)], while the maroon is for the $\rho \left(z\right)$ value of the orange band at $k_y=-0.1$ ${\AA }^{-1}$ [orange square near $-0.3$ eV in (b)].

Figure 5

DFT-calculated band structure of the TI-FM bilayer, same as Fig. 4, but showing only the states (a) with the weight of the topmost QL greater than 10%, and (b) with the weight of the topmost-1 QL greater than 10%. The color box scale is the percentage weight of the topmost QL for (a) and at the topmost-1 QL for (b).

Figure 6

(a) DFT-calculated spin texture of the descendent state projected onto the TI side at $E_F$ indicated as the red dashed line in Fig. 4 when the magnetization is along the $y$ axis. This spin texture can be compared to Fig. 2. (b) DFT-calculated spin texture of the descendent state projected onto the TI side at $E_F$ when the magnetization is along the $z$ axis.

Figure 7

DFT-calculated spin texture of the descendent state at $E_F$ when the magnetization is along (a) the $y$ axis and (b) the $z$ axis. Here the contributions of the Ni slab and the TI film are all included.