Tunable topological phases in monolayer ${\mathrm{Pt}}_2{\mathrm{HgSe}}_3$ with exchange fields

We investigate topological phases of monolayer jacutingaite (${\mathrm{Pt}}_2{\mathrm{HgSe}}_3$) that arise when considering the competing effects of spin-orbit coupling (SOC), magnetic exchange interactions, and staggered sublattice potential $V$. The interplay between the staggered potential and exchange field offers the possibility of attaining different topological phases. By analyzing the Berry curvatures and computing the Chern numbers and Hall conductivities, we demonstrate that the system is time-reversal-symmetry-broken quantum spin Hall insulator when $m_b<{\lambda }_{\mathrm{so}}$, where $m_b$ is the exchange field operating on the bottom Hg sublattice and ${\lambda }_{\mathrm{so}}$ is the intrinsic SOC. For $m_b>{\lambda }_{\mathrm{so}}$ and in the presence of Rashba SOC, we find that the band gap at valley $K\left(K^{\prime }\right)$ is topologically trivial (nontrivial) with Chern number $\mathcal{C}=1$ and valley Chern number ${\mathcal{C}}_v=-1$, indicating that the system is valley-polarized quantum anomalous Hall insulator. We show that the topology of each valley is swapped (the Chern number becomes $\mathcal{C}=-1$) by reversing the sign of the exchange field. The system transitions to a valley-polarized metal and quantum valley Hall phase as $V$ increases. Along the phase boundaries, we observe a single Dirac-cone semimetal states. These findings shed more light on the possibility of realizing and controlling topological phases in spintronics and valleytronics devices.

Figure 1

(a) The band structure of monolayer ${\mathrm{Pt}}_2{\mathrm{HgSe}}_3$ near the $K$ and $K^{\prime }$ valleys in the absence of the Zeeman field $m_{+}$ and $V=0$. (b) The same as in (a) but with $m_{+}=0.3{\lambda }_{\mathrm{so}}$. The blue (red) lines are for spin-up (spin-down) states. The splitting of the spin-degenerate bands is $2m_{+}$. (c) The same as in (a) but with $V=0.6{\lambda }_{\mathrm{so}}$. The splitting of the spin-degenerate bands is $2V$. (d) Side view of the structure of ${\mathrm{Pt}}_2{\mathrm{HgSe}}_3$ on an antiferromagnetic (AFM) substrate. The exchange fields $m_b$ and $m_t$ are shown by the arrows.

Figure 2

(a) The band structure of monolayer ${\mathrm{Pt}}_2{\mathrm{HgSe}}_3$ near the $K$ and $K^{\prime }$ valleys for $m_b=1.35{\lambda }_{\mathrm{so}}$, $m_t=0.05{\lambda }_{\mathrm{so}}$, $V=0.3V_1$, and ${\lambda }_R=0$. The band gap ${\delta }_{K^{\prime }}^{\downarrow \uparrow }<0$. (b) The same as in (a) but with ${\lambda }_R=0.2{\lambda }_{\mathrm{so}}$. (c) Valley-Hall conductivities ${\sigma }_{xy}^K$, ${\sigma }_{xy}^{K^{\prime }}$; see Sec. 2. (d) Anomalous Hall conductivity ${\sigma }_{xy}$ for ${\lambda }_R=0$.

Figure 3

Evolution of the band structure of monolayer ${\mathrm{Pt}}_2{\mathrm{HgSe}}_3$ (a)–(e) with staggered exchange and Zeeman fields and increasing sublattice potential $V$. The parameters are $m_b=0.5{\lambda }_{\mathrm{so}}$, $m_t=0.05{\lambda }_{\mathrm{so}}$, ${\lambda }_{\mathrm{so}}=81.2\mathrm{meV}$, and ${\lambda }_R=0$. In (a) the system is a QSH insulator for $0<V<V_1$ where $V_1={\lambda }_{\mathrm{so}}-m_{-}$. After the gap closing and reopening of the spin-up channel at the $K^{\prime }$ valley for $V=V_1$ [shown in (b)], the system enters a VPM phase, shown in (c). After the second gap closing and reopening of the spin-down channel [shown in (d) at the $K$ valley for $V=V_2$ where $V_2={\lambda }_{\mathrm{so}}+m_{-}$, the system becomes QVH insulator shown in (e). (f)–(h) Hall conductivities for the spin-up and spin-down channels, ${\sigma }_{xy}^{\uparrow }$ and ${\sigma }_{xy}^{\downarrow }$, respectively, as functions of ${\epsilon }_F=E_F/{\lambda }_{\mathrm{so}}$. (i)–(k) Hall conductivities for each valley, ${\sigma }_{xy}^K$ and ${\sigma }_{xy}^{K^{\prime }}$ as functions of ${\epsilon }_F=E_F/{\lambda }_{\mathrm{so}}$.

Figure 4

Berry curvature distribution of the spin-up and spin-down valence bands, ${\mathrm{\Omega }}_{-}^{\uparrow }$ and ${\mathrm{\Omega }}_{-}^{\downarrow }$, respectively, around the $K$ and $K^{\prime }$ points for (a) the QSH phase and (b) the VPM phase in units of $a^2$, where $a=7.6$ Å is the lattice constant. Other parameters are the same as in Fig. 3. (c) The phase diagram and the band gaps vs $V/{\lambda }_{\mathrm{so}}$.

Figure 5

Spin-Hall conductivity for the same parameters as in Fig. 3 at different temperatures.

Figure 6

(a) Band structure near the $K$ and $K^{\prime }$ valleys for $m_b=1.5{\lambda }_{\mathrm{so}}$, $m_t=0.05{\lambda }_{\mathrm{so}}$, ${\lambda }_R=0.15{\lambda }_{\mathrm{so}}$, and $V=0$. (b), (c) Calculated $Q_{Ks}$ and $Q_{K^{\prime }s}$ as functions of the wave vector $k$. (d) Berry curvatures of the $s=+$ and $s=-$ valence bands, ${\mathrm{\Omega }}_{-+}$ and ${\mathrm{\Omega }}_{--}$, near the $K$ and $K^{\prime }$ valleys. (e)–(h) The same as in (a)–(d) but with opposite exchange fields $m_b=-1.5{\lambda }_{\mathrm{so}}$ and $m_t=-0.05{\lambda }_{\mathrm{so}}$.

Figure 7

(a) Chern numbers for each valence band ($s=\pm 1$) in the topological gap at the $K^{\prime }$ valley as a function of ${\lambda }_R$ for fixed $m_b=1.5{\lambda }_{\mathrm{so}}$. Their sum is always ${\mathcal{C}}_{K^{\prime }}={\mathcal{C}}_{K^{\prime }+}+{\mathcal{C}}_{K^{\prime }-}=1$ (solid, brown line). (b) Chern numbers for each valence band ($s=\pm 1$) at the $K^{\prime }$ valley as a function of the exchange field in the bottom sublattice $m_b$ for fixed Rashba SOC ${\lambda }_R=0.54{\lambda }_{\mathrm{so}}$. Their sum is again ${\mathcal{C}}_{K^{\prime }}=1$.

Figure 8

(a–e) Evolution of the band structure with increasing $V$ for ${\lambda }_R=0.15{\lambda }_{\mathrm{so}}$. The other parameters are the same as in Fig. 6. For $0<V<m_{+}$ the system is VP-QAH insulator whereas for $m_{+}<V<V_2$ it is VPM. At $V=V_2$ the gap ${\delta }_K^{\downarrow }$ of the spin-down channel at the $K$ valley closes and reopens and the system becomes QVH insulator.

Figure 9

(a) The phase diagram and the band gaps as functions of $V/{\lambda }_{\mathrm{so}}$. In the VP-QAH phase, which exist for $0<V<m_{+}$, the Chern number and valley Chern number are $\mathcal{C}=1$ and ${\mathcal{C}}_v=-1$. In the QVH phase, which exists for $V>V_2$, $\mathcal{C}=0$ and ${\mathcal{C}}_v=2$. Violet and red lines correspond to band gaps at the $K^{\prime }$ and $K$ valleys.