Origins of the structural phase transitions in ${\mathrm{MoTe}}_2$ and ${\mathrm{WTe}}_2$

Layered transition metal dichalcogenides ${\mathrm{MoTe}}_2$ and ${\mathrm{WTe}}_2$ share almost similar lattice constants as well as topological electronic properties except their structural phase transitions. While the former shows a first-order phase transition between monoclinic and orthorhombic structures, the latter does not. Using a recently proposed van der Waals density functional method, we investigate structural stability of the two materials and uncover that the disparate phase transitions originate from delicate differences between their interlayer bonding states near the Fermi energy. By exploiting the relation between the structural phase transitions and the low energy electronic properties, we show that a charge doping can control the transition substantially, thereby suggesting a way to stabilize or to eliminate their topological electronic energy bands.

Figure 1

Schematic atomic structures of (a) the $\beta$ and (b) the $\gamma$ phase of ${\mathrm{MoTe}}_2$ and ${\mathrm{WTe}}_2$ projected on the $bc$ plane. $\mathbf{b}$ and $\mathbf{c}$ denote unit vectors of the primitive unit cell ($\mathbf{a}$ is perpendicular to the $bc$ plane). The solid line indicates the unit cell. The dark (red) and bright (gray) circles represent Mo (W) and Te atoms, respectively. Te atom being close to (away from) the transition metal plane is denoted by ${\mathrm{Te}}^{i\left(o\right)}$, respectively. For the $\beta \left(\gamma \right)$ phase, $d_1<d_3(d_1>d_3)$. The angle between $\mathbf{b}$ and $\mathbf{c}$ is (a) $\theta \simeq {94}^{\circ }$ and (b) $\theta ={90}^{\circ }$.

Figure 2

Optimized lattice parameters $a, b$, and $c$ for (a) the $\beta$ and (b) the $\gamma$ structures, obtained using different exchange-correlation functionals. Experimental values for $\beta -{\mathrm{MoTe}}_2$ and $\gamma -{\mathrm{MoTe}}_2$ (Ref. [32]) and those for $\gamma -{\mathrm{WTe}}_2$(Ref. [17]) are shown by horizontal solid lines, and horizontal dotted lines, respectively.

Figure 3

(a) Energy profile calculated using rev-vdW-DF2 with and without SOC and $U$ along the transition path from $\beta$ to $\gamma$ phase of ${\mathrm{MoTe}}_2$ and ${\mathrm{WTe}}_2$ with respect to the total energy of the $\beta$ phase. (b) Calculated free energy difference $\mathrm{\Delta }F=F_{\gamma }-F_{\beta }$ using rev-vdW-DF2 with SOC and without $U$, where $F_{\gamma \left(\beta \right)}$ is a free energy of $\gamma \left(\beta \right)$ phase.

Figure 4

Band structures of (a) $\beta -{\mathrm{MoTe}}_2$, (b) $\gamma -{\mathrm{MoTe}}_2$, (c) $\beta -{\mathrm{WTe}}_2$, and (d) $\gamma -{\mathrm{WTe}}_2$ using rev-vdW-DF2 method with SOC. The Fermi energy ($E_F$) is set to zero. The bands are plotted along $Y(0,\frac{1}{2},0)\to \mathrm{\Gamma }(0,0,0)\to X(\frac{1}{2},0,0)$ and $\mathrm{\Gamma }(0,0,0)\to A$(0,0,$\frac{1}{2}$). The bands projected onto the $d_{xz}$ and $d_{z^2}$ orbitals of Mo and $p_x$ and $p_z$ orbitals of Te are displayed with circles whose radii are proportional to the weights of each orbital. To visualize the bonding nature of valence bands, the wave functions at the $\mathrm{\Gamma }$ point are drawn for (e) ${\psi }_1$ and ${\psi }_2$ of $\beta -{\mathrm{MoTe}}_2$ and (f) ${\phi }_1$ and ${\phi }_2$ of $\beta -{\mathrm{WTe}}_2$ where blue (green) color denotes plus (minus) sign.

Figure 5

Calculated energy profile (with SOC and without $U$) along the transition path from $\beta$ to $\gamma$ phase of (a) $\mathrm{MoTe}2$ and (b) $\mathrm{WTe}2$ as a function of doping ($n_{3D}$) ranging from $-3.3\times {10}^{20}{\mathrm{cm}}^{-3}$ to $+3.3\times {10}^{20}{\mathrm{cm}}^{-3}$. The energy profiles with electron (positive) doping, hole (negative) doping, and neutral case are drawn by red, blue, and black lines, respectively. Doping density difference between the consecutive lines is $6.6\times {10}^{19}{\mathrm{cm}}^{-3}$.