Magnetic domain walls in antiferromagnetic topological insulator heterostructures

We explore the emergence of spin-polarized flat bands at head-to-head domain walls (DWs) in topological insulator heterostructures with in-plane magnetization and interlayer antiferromagnetic coupling. We show in the framework of quantum well physics that, by tuning the width of a DW, one can control the functional form of the bound states appearing across it. Furthermore, we demonstrate the effect that the parity of the number of layers in a multilayer sample has on the electronic dispersion. The alignment of the magnetization vectors on the top and bottom surfaces of odd-layer samples affords particle-hole symmetry, leading to the presence of linearly dispersing topologically nontrivial states around $E=0$. By contrast, the lack of particle-hole symmetry in even-layer samples results in a gapped system, with spin-polarized flat bands appearing on either side of a band gap, with a characteristic energy well within terahertz energy scales. Such a system is a versatile platform for the development of spintronic devices and proposes one use in reconfigurable magnetic memory.

Figure 1

Schematic of a multilayer topological insulator (TI) heterostructure with in-plane magnetism and antiferromagnetic coupling between adjacent layers. In this paper, we will consider samples of infinite length along the $x$ axis. The magnetization in each layer on either side of the domain wall (DW; dashed line) is shown by black arrows. A sharp head-to-head (tail-to-tail, equivalently) DW is shown; however, it should be noted that competition between the magnetocrystalline and exchange energies will lead to a realistic DW having a finite width.

Figure 2

Electronic dispersion relations from the subblock Hamilonians $H^{+}$ (left) and $H^{-}$ (right). States are colored according to their spin-$z$ expectation value, $S_z=\langle \mathrm{\Psi }|{\sigma }_z|\mathrm{\Psi }\rangle$, with spin up and down given by yellow and blue, respectively.

Figure 3

(a) The electronic dispersion of a 50-nm-wide, two-layer system with $M_z=0$ and (b) the local density of states (LDOS) around the valence flat band showing states localized across the domain wall. The vertical position in the multilayer structure is shown along the $z$ axis, where for example, $t_1$ denotes the top surface of the first layer. The LDOS was calculated using Eq. (17) with $k_{B}T=1\mathrm{meV}$.

Figure 4

(a) The electronic dispersion of a 50-nm-wide three-layer sample with $M_z=5\mathrm{meV}$ and (b) the conduction flat band and (c) the valence flat band both showing states localized across the domain wall. Reflection symmetry around the midlayer of the sample demands that these states are located on opposite surfaces of the multilayer sample. The local density of states (LDOS) was calculated using Eq. (17) with $k_{B}T=1\mathrm{meV}$.

Figure 5

(a) The effect of $M_z$ on a two-layer system. From left to right, $M_z=10,15,25,\text{and}50\mathrm{meV}$. (b) The effect of $M_z$ on a three-layer system. From left to right, $M_z=15,25,50,\text{and}60\mathrm{meV}$.

Figure 6

Crystal structure of ${\text{VBi}}_2{\text{Te}}_4/{\text{EuBi}}_2{\text{Te}}_4$ along the (a) $a$ axis and (b) $c$ axis. The crystal structure is rhombohedral with the space group $R\overline{3}m$. The crystal is a layered structure formed of septuple layers (red box) separated by a van der Waals gap. The paramagnetic unit cell (black box) is shown in (a); however, it should be noted that the A-type antiferromagnetism of this material leads to a magnetic unit cell twice the length of that shown. Figure produced with vesta [36].