Induced spin textures at $3d$ transition metal–topological insulator interfaces

While some of the most elegant applications of topological insulators, such as the quantum anomalous Hall effect, require the preservation of Dirac surface states in the presence of time-reversal symmetry breaking, other phenomena such as spin-charge conversion rather rely on the ability for these surface states to imprint their spin texture on adjacent magnetic layers. In this Rapid Communication, we investigate the spin-momentum locking of the surface states of a wide range of monolayer transition metals ($3d-\mathrm{TM}$) deposited on top of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ topological insulators using first-principles calculations. We find an anticorrelation between the magnetic moment of the $3d-\mathrm{TM}$ and the magnitude of the spin-momentum locking induced by the Dirac surface states. While the magnetic moment is large in the first half of the $3d$ series, following Hund's rule, the spin-momentum locking is maximum in the second half of the series. We explain this trend as arising from a compromise between intra-atomic magnetic exchange and covalent bonding between the $3d-\mathrm{TM}$ overlayer and the Dirac surface states. As a result, while Cr and Mn overlayers can be used successfully for the observation of the quantum anomalous Hall effect or the realization of axion insulators, Co and Ni are substantially more efficient for spin-charge conversion effects, e.g., spin-orbit torque and charge pumping.

Figure 1

Band structure of a ferromagnet-TI bilayer as described by Eq. (1): (a) Uncoupled case ($t=0)$, and coupled case ($t\ne 0$) with (b) weak and (c) strong exchange $\mathrm{\Delta }$. The arrows represent the expectation value of the spin angular momentum operator at different $k$ points in the Brillouin zone, at the interfacial ferromagnet-TI bands. The red and blue colors correspond to the contribution from the ferromagnet and the TI, respectively.

Figure 2

The relativistic band structure of ${\mathrm{Mn}}_{\mathrm{fcc}}/{\mathrm{Bi}}_2{\mathrm{Se}}_3$ heterostructure together with the corresponding three components of the spin moment, $S_x$ (left), $S_y$ (center), and $S_z$ (right). (a) displays the total spin moment while (b) shows the spin moment projected on Mn-$3d$ orbitals.

Figure 3

(a) In-plane total and (b) projected spin texture of the ${\mathrm{Mn}}_{\mathrm{fcc}}-{\mathrm{Bi}}_2{\mathrm{Se}}_3$ heterostructure, represented in momentum space and at the Fermi level. The arrows are normalized for better visualization and their color scales with the magnitude of the in-plane projection, $\sim \sqrt{S_x^2+S_y^2}$. The calculation is performed at $E_{\mathrm{F}}-0.05\mathrm{eV}$.

Figure 4

The relativistic band structure of ${\mathrm{Co}}_{\mathrm{fcc}}-{\mathrm{Bi}}_2{\mathrm{Se}}_3$ heterostructure together with the corresponding three components of the spin moment, $S_x$ (left), $S_y$ (center), and $S_z$ (right). (a) displays the total spin profile while (b) shows the spin profile projected on Co-$3d$ orbitals.

Figure 5

(a) In-plane total and (b) projected spin texture of the ${\mathrm{Co}}_{\mathrm{fcc}}-{\mathrm{Bi}}_2{\mathrm{Se}}_3$ heterostructure, represented in momentum space and at the Fermi level. The representation is the same as in Fig. 3. The calculation is performed at $E_{\mathrm{F}}$.

Figure 6

Induced spin-momentum locking on the $3d-\mathrm{TM}$ element (blue histogram: unitless) and total magnetic moment (red histogram: in the units of ${\mu }_{\mathrm{B}}$) of a $3d-\mathrm{TM}–{\mathrm{Bi}}_2{\mathrm{Se}}_3$ slab upon varying the $3d-\mathrm{TM}$ overlayer. While the magnetic moment is maximum for the Mn overlayer, the induced spin-momentum locking is maximum for the Co overlayer.

Figure 7

Band structure of $3d-\mathrm{TM}–{\mathrm{Bi}}_2{\mathrm{Se}}_3$ for the full $3d-\mathrm{TM}$ series, projected on the $3d-\mathrm{TM}$ orbitals (top panel) and on the top Se orbitals (bottom panel). The orbital contribution is given by the green shading. The red dot represents the position of the Dirac point (DP) associated with the bottom surface and the dashed horizontal line indicates the position of the Fermi level $E_{\mathrm{F}}$.