Chiral channel network from magnetization textures in two-dimensional ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$

When an atomically thin van der Waals magnet forms a long-period moiré pattern with a magnetic substrate, the sensitive dependence of interlayer magnetic coupling on the atomic registries can lead to moiré-defined magnetization textures in two-dimensional (2D) magnets. The recent discovery of 2D magnetic topological insulators such as ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ leads to the interesting possibility of exploring the interplay of such magnetization textures with topological surface states, which we explore here with a minimal model established for 2D ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$. The sign flip of the exchange gap across a magnetization domain wall gives rise to a single in-gap chiral channel on each surface. In the periodic magnetization textures, such chiral spin channels at the domain walls couple to form a network and superlattice minibands emerge. We find that in magnetization textures with closed domain-wall geometries, the formed superlattice miniband is a gapped Dirac cone featuring orbital magnetization from the current circulation in the closed loops of chiral channels, while in magnetization textures with open domain wall geometries, a gapless mini-Dirac cone is found instead. The miniband Bloch states feature a spatial texture of spin and local current densities, which are clear manifestations of the spin-momentum locked chiral channels at the domain walls. The results suggest a platform to engineer spin and current flows through the manipulation of magnetization domains for spintronic devices.

Figure 1

(a) Schematic of ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ bilayer and its surface state spectrum in AFM-$z$ (left) and AFM-$x$ (right) order. The black arrows and circles with cross and dot representing inward and outward describe the surface magnetization. The spectrum is gapped for AFM-$z$ and gapless for AFM-$x$ order. The spins (red and blue arrows) in the surface state are locked with their momentum. (b) Schematic of a magnetic domain wall in ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ bilayer and the corresponding topological domain wall states on top and bottom surfaces, with the spin (red arrows) locked to the velocity (blue circles with cross and dot representing inward and outward).

Figure 2

(a) Schematic of a moiré pattern formed by an ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ bilayer stacked on an FM medium layer ${\mathrm{CrI}}_3$ twisted on an AFM substrate ${\mathrm{MnPTe}}_3$. The green diamond denotes a moiré unit cell with periodicity $b$. Only one layer of Mn atoms from ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ are shown as the grey spheres, and the magnetic atoms on the FM medium layer and AFM substrate are denoted by the purple and green spheres. At the two local regions marked by the dashed circles, the Mn atoms in ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ and Cr atoms in ${\mathrm{CrI}}_3$ sit on top of the spin up and spin down sublattices of the AFM substrate, respectively. Consequently, the local ferromagnetic order in the ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ layer tends to be pinned in the up and down directions, respectively, at these two local regions by the proximity magnetic coupling from the substrate. (b), (c) The competition of this proximity field with the in-plane exchange and anisotropy magnetic interaction can turn the uniform magnetization order in the ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ bilayer into either an MB lattice (b), or a SK lattice (c) (see Appendix pp2 for detailed calculations of the magnetization textures). The $z$ component of the magnetization is color coded, and the in-plane component is denoted by the arrows near the domain walls, and the two examples of textures are reproduced using parameters from Ref. [38].

Figure 3

(a) The moiré minibands of the Dirac surface states subjected to the exchange field from an MB magnetization texture of moiré periodicity $b=10\mathrm{nm}$. The right panel shows a full dispersion around the Dirac point. (b), (d) LDOS $\rho \left(\mathit{r}\right)$ (background), electric current density $\mathit{j}\left(\mathit{r}\right)$ (black arrows) and in-plane components of local spin expectation $\mathit{s}\left(\mathit{r}\right)$ (red arrows) with wave vector $k=1/b$ in $y$ direction. The corresponding wave vector and band index are indicated by the red dots in (a). The black lines are the domain walls. The parameters used are $\hslash v=2\text{eV}\text{Å}$ and $g=25\mathrm{meV}$.

Figure 4

(a) The moiré minibands for an MB magnetization texture of moiré periodicity $b=50\mathrm{nm}$. The right panel shows a full dispersion around the Dirac point. (b) The velocity in $x$ and $y$ directions near the Dirac point as a function of moiré periodicity. (c)–(f) LDOS $\rho \left(\mathit{r}\right)$ (background), electric current density $\mathit{j}\left(\mathit{r}\right)$ (black arrows), and in-plane components of local spin expectation $\mathit{s}\left(\mathit{r}\right)$ (red arrows) with wave vector $k=1/b$ in $x$ and $y$ directions. The corresponding wave vector and band index are indicated by the red dots in (a). The black lines are the domain walls. The parameters used are $\hslash v=2\text{eV}\text{Å}$ and $g=25\mathrm{meV}$.

Figure 5

(a) The moiré minibands for a SK magnetization texture of moiré periodicity $b=10\mathrm{nm}$. The right panel shows a full dispersion around the Dirac point. (b)–(d) LDOS $\rho \left(\mathit{r}\right)$ (background), electric current density $\mathit{j}\left(\mathit{r}\right)$ (black arrows) and in-plane components of local spin expectation $\mathit{s}\left(\mathit{r}\right)$ (red arrows) with wave vector $k=0$ and $1/b$ in $y$ direction. The corresponding wave vector and band index are indicated by the red dots in (a). The wave function of conduction band can be derived from particle-hole symmetry, which is discussed in Sec. 4. The black lines are the domain walls. The parameters used are $\hslash v=2\text{eV}\text{Å}$ and $g=25\mathrm{meV}$.

Figure 6

(a) The moiré minibands for a SK magnetization texture of moiré periodicity $b=50\mathrm{nm}$. The right panel shows a full dispersion around the Dirac point. (b) The orbital angular momentum per morié supercell for the four topmost valence bands as a function of moiré periodicity. (c), (d) LDOS $\rho \left(\mathit{r}\right)$ (background), electric current density $\mathit{j}\left(\mathit{r}\right)$ (black arrows) and in-plane components of local spin expectation $\mathit{s}\left(\mathit{r}\right)$ (red arrows) for SK phase with wave vector $k=0$. The corresponding wave vector and band index are indicated by the red dots in (a). The black lines are the domain walls. (e) The distribution of orbital magnetic momentum of the first valence band in the mini-Brillouin zone with moiré periodicity $b=50\mathrm{nm}$. The parameters used are $\hslash v=2\text{eV}\text{Å}$ and $g=25\mathrm{meV}$.

Figure 7

Minibands in the MB magnetization texture (a) and in the SK magnetization texture (b) when the in-plane magnetization is considered. (c), (d) LDOS $\rho \left(\mathit{r}\right)$ (background), electric current density $\mathit{j}\left(\mathit{r}\right)$ (black arrows), and in-plane components of local spin expectation $\mathit{s}\left(\mathit{r}\right)$ (red arrows) for two representative Bloch states in (a) and (b), respectively (denoted by red dots). The black lines are the domain walls. The parameters used are $g^{\prime }=10\mathrm{meV}$ and $b=50\mathrm{nm}$.