Magnetism-mediated transition between crystalline and higher-order topological phases in NpSb

Merging the fields of topology and magnetism expands the scope of fundamental quantum phenomena with novel functionalities for topological spintronics enormously. Here, we theoretically demonstrate that ferromagnetism provides an efficient means to achieve a topological switching between crystalline and higher-order topological insulator phases in two dimensions. Using a tight-binding model and first-principles calculations, we identify layered NpSb as a long-awaited two-dimensional topological crystalline insulator with intrinsic ferromagnetic order with a band gap which is as large as 220 meV. We show that when ${\mathcal{M}}_z$ symmetry is preserved for the out of plane magnetization of this material, it exhibits a pair of gapless edge states along all boundaries and carries a nonzero mirror Chern number ${\mathcal{C}}_{\mathcal{M}}=1$. Remarkably, when rotating the magnetization into the plane a higher-order topological insulator phase with a parity-based invariant ${\nu }_{2\text{D}}=1$ is achieved, and in-gap topological corner states emerge. Our results pave the way to understanding and engineering topological insulating states in two-dimensional ferromagnets.

Figure 1

(a) Sketch of a two-dimensional (2D) square lattice, in which the mirror symmetries depend sensitively on the magnetization direction. The shaded area indicates the mirror symmetry ${\mathcal{M}}_z$, and the solid lines denote corresponding mirror planes of the ${\mathcal{M}}_x$, ${\mathcal{M}}_y$, ${\mathcal{M}}_{xy}$, and ${\mathcal{M}}_{x\overline{y}}$ symmetries. Edge spectra of the tight-binding model with an (b) out-of-plane ($\theta =0^{\circ }$) and (c) in-plane magnetization ($\theta ={90}^{\circ }$ and $\phi ={45}^{\circ }$) for the exchange field of $B=0.5t$. (d) Energy spectrum of the $40\times 40$ nanoflake of the model versus the magnitude of $B$. Corner states are highlighted with blue circles. Energy levels for a nanoflake with an (e) out-of-plane and (f) in-plane magnetization. Insets show the spatial weight of the edge states and corner states.

Figure 2

Crystal structure of (a) bulk and (c) monolayer NpSb. The unit cell of NpSb monolayer is indicated by black solid lines. (b) The Brillouin zone of the primitive unit cell (black solid lines) and $\sqrt{2}\times \sqrt{2}$ supercell (yellow dashed lines) of the monolayer. Projected one-dimensional Brillouin zone with marked high symmetry points is shown with a purple solid line. (d) Phonon spectra for NpSb monolayer, indicating that the NpSb monolayer is dynamically stable.

Figure 3

Band structure of (a) majority and (b) minority states of NpSb monolayer in the absence of SOC, and that of NpSb monolayer with (c) an out-of-plane and (d) an in-plane magnetization along x including SOC. The bands are orbitally weighted with the contribution of Sb-$p_{x/y}$ and Np-$d_{x^2-y^2}$ states. The Fermi level is indicated with a dashed line.

Figure 4

Evolution of Wannier charge centers (WCCs) for NpSb monolayer with an out-of-plane magnetization associated with (a) $+i$ and (b) $-i$ mirror eigenstates, indicating a nonzero mirror Chern number ${\mathcal{C}}_{\mathcal{M}}=1$. (c) Evolution of WCCs along $k_x$, confirming the topological nature of the gap with ${\mathbb{Z}}_2=1$. (d) Energy dependence of the spin Hall conductivity ${\sigma }_{xy}^S$, showing a quantized value within the energy window of the SOC gap. Inset shows the reciprocal-space distribution of spin Berry curvature within the SOC gap.

Figure 5

Energy bands of the semi-infinite NpSb nanoribbon with (a) out-of-plane and (b) in-plane ferromagnetism, revealing characteristic edge states for the TCI and HOTI phases, respectively. Colors range from purple to red representing the higher density of states at the edges. Energy levels for a NpSb nanoflake with the (c) out-of-plane and (d) in-plane ferromagnetism. Corner states in (d) are highlighted in blue circles. Insets show the probability of the edge states and corner states.