Exchange intervalley scattering and magnetic phase diagram of transition metal dichalcogenide monolayers

We analyze magnetic phases of monolayers of transition metal dichalcogenides that are two-valley materials with electron-electron interactions. The exchange intervalley scattering makes two-valley systems less stable to the spin fluctuations but more stable to the valley fluctuations. We predict a first-order ferromagnetic phase transition governed by the nonanalytic and negative cubic term in the free energy that results in a large spontaneous spin magnetization. Finite spin-orbit interaction leads to the out-of-plane Ising order of the ferromagnetic phase. Our theoretical prediction is consistent with the recent experiment on the electron-doped monolayers of ${\mathrm{MoS}}_2$ reported by Roch et al. [Nat. Nanotechnol. 14, 432 (2019)]. The proposed first-order phase transition can also be tested by measuring the linear magnetic field dependence of the spin susceptibility in the paramagnetic phase which is a direct consequence of the nonanalyticity of the free energy.

Figure 1

Electron spectrum of monolayers of TMDs described by the Hamiltonian defined in Eq. (1). Blue (red) bands correspond to spin up (down) along the $z$ axis, marked by up (down) arrows, $2\alpha$ is the SOI gap. Effect of the electron-electron interaction is schematically illustrated by the chemical potentials ${\mu }_{\pm }^{\uparrow }$ and ${\mu }_{\pm }^{\downarrow }$ which are different from the Fermi level $E_F$ of the noninteracting 2DEG (dashed-dotted).

Figure 2

Matrix elements of the electron-electron interaction: (a) the intravalley matrix element $v_1\left(q\right)$, (b) the direct intervalley matrix element $v_2\left(q\right)$, and (c) the EIV matrix element $v_3\left(q\right)$. Other nonzero matrix elements can be obtained by changing signs of all valley indexes. Here $q\lesssim k_F\ll q_0$, where $k_F$ is the Fermi momentum and $\tau {\mathit{q}}_0$ is the position of the center of the valley $\tau$.

Figure 3

Free energy $F\left(M\right)=a{M}^2+b{M}^4+c{\left|M\right|}^3$ per unit area of the interacting 2DEG as a function of order parameter $M$ with $b>0$. (Left) Standard GL free energy with $c=0$. At $a>0$ the minimum of $F$ is at $M=0$, at $a<0$ the minimum is at $M\ne 0$, the critical point $a_{\mathrm{cr}}=0$ corresponds to the second-order phase transition. The dashed arrow shows the evolution of minimum at $a<0$. (Right) The GL theory with nonanalyticity caused by $c<0$. At $a>a_{\mathrm{cr}}=c^2/4b$, the global minimum is at $M=0$. At $a=a_{\mathrm{cr}}$, the ferromagnetic $M\ne 0$ and paramagnetic $M=0$ minima are degenerate manifesting a first-order phase transition. At $a<a_{\mathrm{cr}}$, the ferromagnetic minimum is the global one, and its evolution is shown by the dashed arrow. Further evolution of the local paramagnetic minimum $M=0$ at $a<a_{\mathrm{cr}}$ is irrelevant.

Figure 4

Second-order diagrams for the grand canonical potential $\mathrm{\Omega }$ beyond the RPA approximation, see Appendix pp1. Blue wavy (dashed) line corresponds to the $v$ ($u$) component of the effective interaction, index ${\tau }_i$ (${\sigma }_i$) labels the valley (spin). The black solid lines correspond to the bare electron Green function.

Figure 5

First-order interaction correction to the grand canonical potential $\mathrm{\Omega }$. Blue wavy (dashed) lines correspond to the $v$ ($u$) components of the effective interaction. The black solid lines correspond to the electron Green function (A1).

Figure 6

The RPA series for the grand canonical potential $\mathrm{\Omega }$. Blue wavy lines represent here both components $v$ and $u$ of the effective interaction for brevity.