Engineering Weyl Phases and Nonlinear Hall Effects in ${\mathrm{T}}_d$-${\mathrm{MoTe}}_2$

${\mathrm{MoTe}}_2$ has recently attracted much attention due to the observation of pressure-induced superconductivity, exotic topological phase transitions, and nonlinear quantum effects. However, there has been debate on the intriguing structural phase transitions among various observed phases of ${\mathrm{MoTe}}_2$ and their connection to the underlying topological electronic properties. In this work, by means of density-functional theory calculations, we investigate the structural phase transition between the polar ${\mathrm{T}}_d$ and nonpolar $1{\mathrm{T}}^{\prime }$ phases of ${\mathrm{MoTe}}_2$ in reference to a hypothetical high-symmetry ${\mathrm{T}}_0$ phase that exhibits higher-order topological features. In the ${\mathrm{T}}_d$ phase we obtain a total of 12 Weyl points, which can be created/annihilated, dynamically manipulated, and switched by tuning a polar phonon mode. We also report the existence of a tunable nonlinear Hall effect in ${\mathrm{T}}_d$-${\mathrm{MoTe}}_2$ and propose the use of this effect as a probe for the detection of polarity orientation in polar (semi)metals. By studying the role of dimensionality, we identify a configuration in which a nonlinear surface response current emerges. The potential technological applications of the tunable Weyl phase and the nonlinear Hall effect are discussed.

Figure 1

Crystal structure of ${\mathrm{MoTe}}_2$ in (a) $1{\mathrm{T}}^{\prime }$, (b) designed ${\mathrm{T}}_0$, and (c) ${\mathrm{T}}_d$ phases. Green arrows denote the interlayer displacement direction parallel ($+$) or antiparallel ($-$) to $\overrightarrow{a}$, and $\lambda$ represents the interlayer displacement parameter (see text). Hollow circles ‘∘’ mark the inversion centers in the $1{\mathrm{T}}^{\prime }$ and ${\mathrm{T}}_0$ phases. (d) The phonon band structure of ${\mathrm{T}}_0$ phase shown within $[-20,80]\mathrm{c}{\mathrm{m}}^{-1}$. (e) The double-well potential energy profile of ${\mathrm{T}}_0$ phase as a function of the inversion symmetry breaking parameter $\lambda$. (f) A schematic representation showing the link between all ${\mathrm{T}}_d$ and $1{\mathrm{T}}^{\prime }$ phases via the reference structure ${\mathrm{T}}_0$ [11].

Figure 2

(a) Electronic band structure of ${\mathrm{T}}_d$-${\mathrm{MoTe}}_2$ calculated with (blue) and without (orange) SOC. (b) Enlarged view of bands calculated near the Fermi level $E_F$ along the $\mathrm{\Gamma }$-Y path highlighted by the magenta rectangle in (a). Gray ovals in the right panel of (b) show the regions near which Weyl crossings occur. (c) Band dispersions (with SOC) along a momentum cut parallel to $k_y$ passing through the WP (black dots) in question [50]; both W1 and W2 are of type II. The dotted black line represents $E_F$, and the green dotted line marks the energy of WPs situated at $k^{*}$. (d) Distribution of all WPs in the BZ for the ${\mathrm{T}}_d$-A phase; red/blue dots depict $+/-$ chiralities of the WPs.

Figure 3

Calculated Berry curvature (a) ${\mathrm{\Omega }}_x$ and (b) ${\mathrm{\Omega }}_y$ on the Fermi surface of ${\mathrm{MoTe}}_2$ in ${\mathrm{T}}_d$-A phase. Yellow (Blue) color represents positive (negative) Berry curvature. (c) Calculated BCDM of ${\mathrm{MoTe}}_2$ in ${\mathrm{T}}_d$-A phase. The nonvanishing $D_{xy}$ and $D_{yz}$ terms are plotted with respect to the chemical potential.