Emergence of Ferroelectricity at a Metal-Semiconductor Transition in a $1T$ Monolayer of ${\mathrm{MoS}}_2$

Using a combination of Landau theoretical analysis and first-principles calculations, we establish a spontaneous symmetry breaking of the metallic state of the $1T$ monolayer of ${\mathrm{MoS}}_2$ that opens up a band gap and leads to an unexpected yet robust ferroelectricity with ordering of electric dipoles perpendicular to its plane. Central to the properties of this thinnest known ferroelectric is a strong coupling of conducting states with valley phonons that induce an effective electric field. The current in a semiconducting $1T\text{−}{\mathrm{MoS}}_2$ channel can, thus, be controlled independently by changing its ferroelectric dipolar structure with a gate field, opening up a possibility of a class of nanoscale dipolectronic devices. Our analysis applies equally well to ${\mathrm{MoSe}}_2$, ${\mathrm{WS}}_2$, and ${\mathrm{WSe}}_2$, giving tunability in design of such devices based on two-dimensional chalcogenides.

Figure 1

Structure, Fermi surface, and phonon dispersion of $1T\text{−}{\mathrm{MoS}}_2$ monolayer. (a) Top view of the structure of a monolayer of ${\mathrm{MoS}}_2$ in $c1T$ polytypical form with octahedral coordination of Mo atoms. To distinguish between the two S planes, the S atoms in the top plane are denoted by smaller radii than the ones in the bottom plane. (b) Its Fermi surface exhibiting a weak nesting between sides of the triangular pockets centered at $K$ and $K^{\prime }$. The nesting vector ($\mathbf{q}$) is denoted by an arrow. (c) Phonon dispersion of the $c1T$ structure, with inset showing a zoomed-in view of the unstable modes near the $K$ point, and the strongest instability is at $q=K+\delta K$.

Figure 2

Structure, band structure, and comparison of $d1T$ with $c1T$. (a) Trimerization of Mo atoms in the distorted low symmetry $1T$ form with a $\sqrt{3}\times \sqrt{3}$ unit cell and (b) electronic structure of $d1T$ ${\mathrm{MoS}}_2$. (c) Displacement vectors (green arrows) of the $d1T$ phase with respect to the $c1T$ phase. (d) An isosurface of the difference in charge densities of ferroelectric $d1T$ state with up polarization and the $c1T$ state. Green color (light grey) denotes negative charge and blue (dark grey) denotes positive charge. The broken inversion symmetry in the charge density difference confirms ferroelectricity in the cell-tripled ground state structure.

Figure 3

Ferroelectric transition behavior, Landau free energy landscape, and metallic states of $d1T$. (a) Polarization ($P$) and dielectric susceptibility ($\chi$) as a function of temperature derived from Landau theory. (b) Variation of Landau free energy ($F$) with $\eta$. $F$ is minimized with respect to $c$ where $\eta =\{{\eta }_1,{\eta }_2=\sqrt{3}{\eta }_1,{\eta }_3=0,{\eta }_4=0,{\eta }_5=c\}$. Note that the cubic dependence of free energy on $\eta$, i.e., the first order nature of the phase transition, is reflected in the two unequal minima. (c) Contour plot of Landau free energy as a function of $\{{\eta }_1,{\eta }_2,0,0,-0.039\}$ at $T=0K$, and structures corresponding to the local minima each of which involves trimerization of Mo. Semi-infinite blue lines in the $\{{\eta }_1,{\eta }_2\}$ plane correspond to metallic states.

Figure 4

Ferroelectric domain wall in $1T\text{−}{\mathrm{MoS}}_2$. (a) Structure of the domain wall (blue dashed line) between up and down polarized states of $d1T$. (b) Variation of Khon-Sham potential at the interface of up and down polarized domains. The energy barrier at the domain wall (marked with dashed lines) separating domains with opposite polarization is $\approx 0.13\mathrm{eV}$. This barrier is estimated by taking the macroscopic average of the Kohn-Sham potential in the $yz$ plane for every point on the line perpendicular to the domain wall (i.e., the $x$ direction), which gives the variation in the Kohn-Sham potential across the interface of up and down polarized domains. (c) Schematic of the XNOR logic gate.