Orbital contribution to the regulation of the spin-valley coupling in antiferromagnetic monolayer ${\mathrm{MnPTe}}_3$

In addition to the ferromagnetic materials with inversion symmetry breaking, the symmetric antiferromagnetic materials also exhibit intrinsic valley splitting due to the spin-valley coupling. Using first-principles calculations, we investigated the manipulation of valley splitting of antiferromagnetic monolayer ${\mathrm{MnPTe}}_3$ via biaxial strain. It is shown that two ${\mathrm{MnPTe}}_3$ monolayers with different structures are both stable antiferromagnetic semiconductors, and exhibit valley splitting between $K$ and $K^{\prime }$. The spin-valley coupling strength can be greatly tuned by in-plane strain, which is due to the changes of orbital composition of electronic state and the different contribution of two sublattice atoms. The proportions of $d$ orbitals of Mn atoms determine the orbital angular momentum of the electronic state, and the different contribution of different sublattices results in the change of Berry curvature at $K$ and $K^{\prime }$ points. The combination of two factors leads to the same changes of valley splitting and the proportion of the $d_{xz}$, $d_{yz}$ orbitals. These results of are some significance for the design of materials with greater valley splitting and the understanding of its mechanism.

Figure 1

(a), (b) Top and side views of monolayer ${\mathrm{MnPTe}}_3$ with octahedral structure ($T\text{−}{\mathrm{MnPTe}}_3$) and trigonal prismatic structure ($H\text{−}{\mathrm{MnPTe}}_3$). (c) Phonon spectrum of monolayer $H\text{−}{\mathrm{MnPTe}}_3$. (d) Variation of energy per unit cell of monolayer $H\text{−}{\mathrm{MnPTe}}_3$ with time during ab initio molecular simulation at room temperature.

Figure 2

Band structures of monolayer $T\text{−}{\mathrm{MnPTe}}_3$ without (a) and with (b) spin-orbit coupling. Band structures of monolayer $H\text{−}{\mathrm{MnPTe}}_3$ without (c) and with (d) spin-orbit coupling.

Figure 3

Spin-dependent Berry curvatures near $K$ and $K^{\prime }$ points of the top valence band and the bottom conduction band of monolayer T-${\mathrm{MnPTe}}_3$ (a), (b) and $H\text{−}{\mathrm{MnPTe}}_3$ (c), (d).

Figure 4

Effect of strain on the band edge for extracting the deformation potentials ($E_{\mathrm{DP}}$) of monolayer $T\text{−}{\mathrm{MnPTe}}_3$ (a) and $H\text{−}{\mathrm{MnPTe}}_3$ (b). (c) Strain-regulated band gaps at $K$ and $K^{\prime }$ points. (d) Tuning the valley splittings of top valence band and bottom conduction band via in-plane biaxis strain. Changes of $d$-orbital compositions at $K$ point of the TVB (e) BCB (f) of monolayer ${\mathrm{MnPTe}}_3$ as functions of the biaxial strain. Solid line and dotted line represent $T\text{−}{\mathrm{MnPTe}}_3$ and $H\text{−}{\mathrm{MnPTe}}_3$, respectively.

Figure 5

(a) Strain-regulated Berry curvature at $K$ or $K^{\prime }$ point of the top valence band and bottom conduction band. (b) Change of hopping parameter $t$ as a function of the in-plane biaxial strain. Valley splitting at the TVB (c) and the BCB (d) with different spin-splitting energy and spin-orbit coupling strength. Comparisons between the valley splitting at the TVB (e) and the BCB (f) obtained from the effective Hamiltonian model with or without $t$ and the first-principle calculation, when the parameters are $m_o=0.05\mathrm{eV}$, $m_u=0.1\mathrm{eV}$, and $\lambda =0.05\mathrm{eV}$.