Monolayer and bilayer ${\mathrm{PtCl}}_3$: Energetics, magnetism, and band topology

Two-dimensional (2D) magnetic materials hosting nontrivial topological states are interesting for fundamental research as well as practical applications. Recently, the topological state of 2D Weyl half-semimetal (WHS) was proposed, which hosts fully spin polarized Weyl points robust against spin-orbit coupling in a 2D ferromagnetic system, and single-layer ${\mathrm{PtCl}}_3$ was predicted as a platform for realizing this state. Here, we perform an extensive search of 2D ${\mathrm{PtCl}}_3$ structures, by using the particle swarm optimization technique and density-functional theory calculation. We show that the desired ${\mathrm{PtCl}}_3$ phase corresponds to the most stable one at its stoichiometry. The 2D structure also possesses good thermal stability up to 600 K. We suggest ${\mathrm{SnS}}_2$ as a substrate for the growth of 2D ${\mathrm{PtCl}}_3$, which has excellent lattice matching and preserves the WHS state in ${\mathrm{PtCl}}_3$. We find that uniaxial strains along the zigzag direction maintain the WHS state, whereas small strains along the armchair direction drives a topological phase transition from the WHS to a quantum anomalous Hall (QAH) insulator phase. Furthermore, we study bilayer ${\mathrm{PtCl}}_3$ and show that the stacking configuration has strong impact on the magnetism and the electronic band structure. Particularly, the $A{A}^{\prime }$ stacked bilayer ${\mathrm{PtCl}}_3$ realizes an interesting topological state—the 2D antiferromagnetic mirror Chern insulator, which has a pair of topological gapless edge bands. Our work provides guidance for the experimental realization of 2D ${\mathrm{PtCl}}_3$ and will facilitate the study of 2D magnetic topological states, including WHS, QAH insulator, and magnetic mirror Chern insulator states.

Figure 1

(a)–(d) Top and side views of 2D ${\mathrm{PtCl}}_3$ structures obtained from our structural search. The green shaded region in each figure indicates the primitive unit cell.

Figure 2

(a) Variation of total energy during the AIMD simulation. (b) Snapshot of single-layer ${\mathrm{PtCl}}_3$ lattice at the end of the simulation period.

Figure 3

(a) Top and (b) side views of 2D ${\mathrm{PtCl}}_3$ on the ${\mathrm{SnS}}_2$ substrate. (c) Band structure of the system. The red dot indicates the weight projected onto the ${\mathrm{PtCl}}_3$ layer. (d) Enlarged view of the band structure around the Fermi energy. One observes that the Weyl point of single-layer ${\mathrm{PtCl}}_3$ is preserved.

Figure 4

(a) Band structure and (b) Brillouin zone for the unstrained single-layer ${\mathrm{PtCl}}_3$. The red dots in panel (a) illustrate the location of the two fully spin-polarized Weyl points. (c) The magnetic easy axis is still along the armchair direction when uniaxial strain is applied in the zigzag direction. (d) The easy axis rotates to the zigzag direction for uniaxial strain ($>0.5\%$) applied along the armchair direction. In panels (c) and (d), the blue arrows indicate the strain direction. (e), (f) Band structures for the strained single-layer ${\mathrm{PtCl}}_3$. (e), (f) For uniaxial strain along the zigzag direction, the WHS state is preserved. (g), (h) Transition to QAH insulator state occurs for uniaxial strain applied along the armchair direction.

Figure 5

The edge spectrum corresponding to the case in Fig. 4 for the edge along the (a) zigzag and (b) armchair direction. The gapless chiral edge band for the QAH state can be observed.

Figure 6

Crystal structures for different stacking configurations of bilayer ${\mathrm{PtCl}}_3$. (a) Top and (b) side views of $AA$ stacking. (c) $A{A}^{\prime }$ stacking is obtained from $AA$ stacking by imposing the horizontal mirror reflection $M_z$ on the top layer. In the side view, some Cl atoms are made into transparent color style in order to indicate that they are at different $x$ coordinates as those Cl atoms without transparency. Comparing panels (b) and (c), one can see that $A{A}^{\prime }$ stacking preserves an $M_z$ symmetry, but $AA$ stacking does not. (d) Top and (e) side views of $AB$ stacking. (f) Like $A{A}^{\prime }$ stacking, $A{B}^{\prime }$ stacking is obtained from $AB$ stacking by flipping the top layer upside down. The arrows in the figures indicate the local spin direction. The green arrows are opposite the yellow arrows.

Figure 7

Band structure of bilayer ${\mathrm{PtCl}}_3$ with (a) $AA$, (b) $A{A}^{\prime }$, (c) $AB$, and (d) $A{B}^{\prime }$ stacking. Magnetism and spin-orbit coupling are fully included.

Figure 8

(a), (b) Berry curvature distribution for the two mirror subspaces. Edge spectra for (e) zigzag edge and (f) armchair edge, which exhibit a pair of gapless edge bands corresponding to the AFM mirror Chern insulator state.

Figure 9

(a)–(d) Band structures of $A{A}^{\prime }$ stacked bilayer ${\mathrm{PtCl}}_3$ with different interlayer distance $d$. For the gapped cases in panels (b)–(d), the system is in the AFM mirror Chern insulator state.