Tunable trimeric charge density wave phase with multiferroicity in the two-dimensional transition metal dichalcogenides

Charge density wave (CDW) is an important physical phenomenon to understand the intrinsic electron-phonon coupling and electron correlations in the low-dimensional systems. Theoretical and experimental studies have shown that there are abundant CDW phases in the two-dimensional transition metal chalcogenides. In this paper, a trimeric CDW phase is predicted by introducing distortion according to the imaginary frequency vibration mode of phonon spectrum in a $1T\ensuremath{-}M{X}_{2}$ ($M=\mathrm{Nb}$, Ta; $X=\mathrm{S}$, Se) lattice. In the trimeric $M{X}_{2}$ (t-$M{X}_{2}$) the three nearest neighbor $M$ atoms aggregate to form a trimer, accompanied by the asymmetric displacement of the $X$ atoms in the out-of-plane direction. Due to the breaking of inversion symmetry, the ferroelectric polarization occurs in the out-of-plane direction, resulting in the coexistence of magnetism, ferroelectricity, and CDW. More interesting, under the electron doping, t-$M{X}_{2}$ becomes more stable than the star-of-David phase with the ferromagnetic ground state, which have been predicted theoretically and verified experimentally. In addition, it is found that the ground state of t-$M{X}_{2}$ monolayer is an antiferromagnetic semiconductor with a direct band gap. Therefore, by the electron doping we can control CDW phase transition from an antiferromagnetic trimeric t-$M{X}_{2}$ to a ferromagnetic star-of-David phase, realizing the simultaneous electrical control of the CDW and magnetism. Furthermore, the CDW phase transition induced by the electron doping can be explained well by the band alignment theory. This study not only enriches the phase diagram of the transition metal dichalcogenides, but also predicts the possibility of multiferroicity with coexistence and coupling of multiple charge and magnetic orders.