Second-order topological insulators and tunable topological phase transitions in honeycomb ferromagnets

Intrinsic magnetic topological insulators categorized by bulk-boundary correspondence are of significant fundamental and technological importance in topotronics. Yet the topological phase transition with variation of the bulk-boundary correspondence remains elusive. Here, using a tight-binding model and first-principles calculations, we demonstrate that $2H\text{−}M{\mathrm{Br}}_2$ $(M=\mathrm{Ru} \text{and} \mathrm{Os})$ monolayers are intrinsic ferromagnetic (FM) second-order topological insulators (SOTIs) distinguished by the emergence of well-localized nontrivial corner states. Remarkably, with giant valley polarization, we point out the possibilities of the two-dimensional FM SOTIs for displaying a rich topological phase diagram; that is, topological phase transitions from FM SOTIs to quantum anomalous Hall insulators and then to normal insulators emerge by engineering the valleys. The obtained quantum anomalous Hall effect is characterized by a nonzero Chern number $\mathcal{C}=\pm 1$ and one chiral edge state. Our results not only uncover a general framework to tune the bulk-boundary correspondence but also motivate a technological avenue to bridge valleytronics and magnetic topology with potential applications in topotronics and valleytronics.

Figure 1

Band structures and corresponding Berry curvatures for the 2D honeycomb ferromagnets with multiple topological states of (a) and (b) a second-order topological insulator (SOTI) with ${\epsilon }_1=1$ and ${\epsilon }_2=2.4$, (c) and (d) a quantum anomalous Hall insulator (QAHI) with ${\epsilon }_1=1$ and ${\epsilon }_2=3.5$, and (e) and (f) a normal insulator (NI) with ${\epsilon }_1=1$ and ${\epsilon }_2=4.2$. Variation of the energy gaps at the $K$ and $K^{\prime }$ valleys versus (g) ${\epsilon }_1$ and (h) ${\epsilon }_2$. Topological phase transitions from a SOTI to a QAHI and then to a NI occur as the gap closure emerges at $K$ and $K^{\prime }$ sequentially. (i) Energy spectrum of a triangular nanoflake of the SOTI phase, where the occupied corner states are marked by red dots. The inset shows the total charge distribution of the three occupied corner states. (j) Chiral edge state of a semi-infinite ribbon, demonstrating the QAHI phase.

Figure 2

(a) Top and side views of $2H\text{−}M{\mathrm{Br}}_2$ ($M$=Ru and Os) monolayers. The violet and orange spheres represent Ru (Os) and Br atoms, respectively. The inset shows the first Brillouin zone with high-symmetry points indicated. (b) Band structures of the $2H\text{−}{\mathrm{RuBr}}_2$ monolayer without SOC. Red and black lines indicate the spin-up and spin-down states, respectively. (c) Orbitally resolved band structures of the $2H\text{−}{\mathrm{RuBr}}_2$ monolayer with SOC in the out-of-plane FM configuration. (d) The edge state of a semi-infinite ribbon with an armchair edge. Energy spectrum of a triangular armchair nanoflake for $2H\text{−}{\mathrm{RuBr}}_2$ (e) without strain and (f) with 2% compressive strain, where the occupied corner states are marked by red dots. Insets show the total charge distribution of the three occupied corner states.

Figure 3

(a) The calculated energy gaps in the $K$ and $K^{\prime }$ valleys and the global energy gap for $2H\text{−}{\mathrm{RuBr}}_2$ as a function of strain $\eta$. (b) Band structure evolution and (c) corresponding orbitally resolved band structures under a strain of $0\%–5\%$. Band inversions emerge in the $K$ and $K^{\prime }$ valleys sequentially. (d) Energy dependence of the anomalous Hall conductivity ${\sigma }_{xy}^A$ for the $2H\text{−}{\mathrm{RuBr}}_2$ monolayer under a $2.5\%$ strain, suggesting a quantized value within the energy window of the energy gap. The inset shows the $K$-space distribution of Berry curvatures. (e) The edge spectrum for the QAH and topological trivial states under strains of $2.5\%$ and $5\%$, respectively.

Figure 4

(a) The calculated energy gaps in the $K$ and $K^{\prime }$ valleys for $2H\text{−}{\mathrm{OsBr}}_2$ as a function of strain $\eta$. (b) The energy spectrum of a triangular armchair nanoflake for $2H\text{−}{\mathrm{OsBr}}_2$ with a SOTI phase, where the occupied corner states are marked by red dots. The inset shows the total charge distribution of the three occupied corner states. (c) Anomalous Hall conductivity ${\sigma }_{xy}^A$ as a function of the position of the Fermi level $E_F$ for the $2H\text{−}{\mathrm{OsBr}}_2$ monolayer under a strain of $\eta =2.7\%$. The inset shows the $K$-space distribution of Berry curvatures. A quantized value within the energy window of the nontrivial gap is, indeed, obtained. (d) The corresponding edge spectrum for the QAH state.