Dimensionality-dependent type-II Weyl semimetal state in ${\mathrm{Mo}}_{0.25}{\mathrm{W}}_{0.75}{\mathrm{Te}}_2$

Weyl nodes and Fermi arcs in type-II Weyl semimetals (WSMs) have led to lots of exotic transport phenomena. Recently, ${\mathrm{Mo}}_{0.25}{\mathrm{W}}_{0.75}{\mathrm{Te}}_2$ has been established as a type-II WSM with Weyl points located near Fermi level, which offers an opportunity to study its intriguing band structure by electrical transport measurements. Here, by selecting a special sample with the thickness gradient across two- (2D) and three-dimensional (3D) regimes, we show strong evidence that ${\mathrm{Mo}}_{0.25}{\mathrm{W}}_{0.75}{\mathrm{Te}}_2$ is a type-II Weyl semimetal by observing the following two dimensionality-dependent transport features: (1) a chiral-anomaly-induced anisotropic magnetoconductivity enhancement, proportional to the square of in-plane magnetic field ($B_{\mathrm{in}}^2$); and (2) an additional quantum oscillation with thickness-dependent phase shift. Our theoretical calculations show that the observed quantum oscillation originates from a Weyl-orbit-like scenario due to the unique band structure of ${\mathrm{Mo}}_{0.25}{\mathrm{W}}_{0.75}{\mathrm{Te}}_2$. The in situ dimensionality-tuned transport experiment offers an alternative strategy to search for type-II WSMs.

Figure 1

Characterization of synthesized ${\mathrm{Mo}}_{0.25}{\mathrm{W}}_{0.75}{\mathrm{Te}}_2$ flakes. (a), (b) Atomic configuration and optical image of synthesized ${\mathrm{Mo}}_{0.25}{\mathrm{W}}_{0.75}{\mathrm{Te}}_2$ flakes. (c) The high-resolution x-ray photoelectron spectroscopy (XPS) spectrum of as-synthesized ${\mathrm{Mo}}_{0.25}{\mathrm{W}}_{0.75}{\mathrm{Te}}_2$ flakes. The peak calculation results reveal that the composition of the synthesized alloy is stoichiometric ${\mathrm{Mo}}_{0.25}{\mathrm{W}}_{0.75}{\mathrm{Te}}_2$. (d) The elemental energy-dispersive x-ray spectrometry (EDX) mapping of ${\mathrm{Mo}}_{0.25}{\mathrm{W}}_{0.75}{\mathrm{Te}}_2$ flakes collected from the region are shown in the inserted TEM image, revealing a uniform distribution of constituent atoms. The scale bars in (b) and (d) are 20 $\mu \mathrm{m}$ and 100 nm, respectively.

Figure 2

Dimensionality-dependent magnetotransport characterization of the as-synthesized ${\mathrm{Mo}}_{0.25}{\mathrm{W}}_{0.75}{\mathrm{Te}}_2$ sample. (a), (d) Longitudinal magnetoresistivity ${\rho }_{\mathrm{xx}}$ of Seg-28 and Seg-8 in perpendicular magnetic field, respectively. The upper inset of (a) and (d) plots the zero-field resistivity of Seg-28 and Seg-8 as a function of temperatures. The region marked with a solid red line in the lower right inset of (a) and (d) is the optical image of Seg-28 and Seg-8 with the sample thickness of 28 and 8 nm. (b), (e) Longitudinal magnetoresistivity ${\rho }_{\mathrm{xx}}$ of Seg-28 and Seg-8 in parallel magnetic field, respectively. (c), (f) Magnetic field dependence of longitudinal magnetoresistivity ${\rho }_{\mathrm{xx}}$ of Seg-28 and Seg-8 at $T$ = 0.3 K with different tilt angles $\theta$. The inset is a schematic drawing of the tilt experimental setup, where $a$ and $c$ represents the crystallographic axis, and $\theta$ is the tilt angle between the magnetic field $B$ and the positive direction of the $c$ axis.

Figure 3

Chiral-anomaly-induced magnetoconductivity enhancement in Seg-28. (a) The zero-field longitudinal resistivity versus $T^3$. The black dotted line represents the $T^3$-fitting curve, and the yellow shaded region indicates the EEI-induced resistivity upturn. (b) Signature of chiral-anomaly-induced magnetoconductivity enhancement. The solid blue and green lines are the temperature dependence of magnetoconductivity measured at 14 and 0 T in-plane magnetic fields, respectively. The dashed and dotted lines are theoretical curves calculated with Eqs. (1) and (2), respectively. (c) Magnetic field dependence of the in-plane ${\sigma }_{\mathrm{xx}}$ collected at selected temperatures. The dashed lines are the corresponding theoretical curves calculated with Eq. (1). The inset shows the magnetic field dependence of the in-plane ${\sigma }_{\mathrm{xx}}$ measured at 0.3 K in a semilogarithmic scale. The black and yellow dashed lines represent the theoretical curves calculated with Eq. (5) with and without consideration of chiral anomaly contribution, respectively. (d) Magnetoconductivities measured with the in-plane magnetic field parallel ($B_{\mathrm{in}}\parallel a$) and perpendicular ($B_{\mathrm{in}}\parallel b$) to the current direction. The dashed line represents the fitting results of Eq. (1). The inset shows the magnetoconductivity difference between these two magnetic field directions. The agreement between the experimental data and the theoretical curve calculated with Eq. (4) suggests that the magnetoconductivity enhancement is due to chiral anomaly.

Figure 4

Schematic figure of Weyl-orbit-like orbit and Fermi surface contours of 30-layer and 10-layer ${\mathrm{Mo}}_{0.25}{\mathrm{W}}_{0.75}{\mathrm{Te}}_2$. (a) The conventional Weyl orbit formed by the Fermi arcs and Landau levels of Weyl points. (b) The Weyl-orbit-like (WOL) orbit consists of unique surface states in a thick (left) and a thin (right) sample. Calculated Fermi surface contours and orbital weight of top and bottom atoms in (c) 30-layer and (d) 10-layer ${\mathrm{Mo}}_{0.25}{\mathrm{W}}_{0.75}{\mathrm{Te}}_2$. In (c) and (d), the constant energy contours and magnitude of orbital weight are indicated by the black lines and the background gradient color. Two major surface states on the top surface are indicated as ${\mathrm{S}}_{\mathrm{t}1}$ and ${\mathrm{S}}_{\mathrm{t}2}$, and the major surface states on the bottom surface are indicated as ${\mathrm{S}}_{\mathrm{b}1}$. The original Fermi arc wave vector $k_{\mathrm{w}}$ considered in the conventional Weyl orbit is substituted by $k_{\mathrm{t}}$ in the WOL scenario.

Figure 5

Signature of Weyl orbit observed in the ${\mathrm{Mo}}_{0.25}{\mathrm{W}}_{0.75}{\mathrm{Te}}_2$ sample. (a) $d{\rho }_{\mathrm{xx}}/dB$ versus 1/$B_{\perp }$ = 1/($B\cos \theta$) at different tilt angles. All curves are vertically shifted for clarity and curves with large $\theta$ ($>{45}^{\circ }$) are horizontally shifted to compensate the additional phase shift (see Sec. V of the Supplemental Material [31]). Vertical dashed lines emphasize the positions of minima of quantum oscillations. The inset shows the angle dependence of FFT $\mathrm{frequency}{F}_1$. The red and blue dashed lines are the fitting results of the 2D surface states (1/$\cos \theta$) and the 3D ellipsoidal Fermi surface (with $k_c/k_a\sim 3$), respectively. (b) The pronounced quantum oscillations extracted from ${\rho }_{\mathrm{xx}}$ by subtracting a smooth background for $B\parallel c$ ($\theta$ = $0^{\circ }$) at 0.3 K. The inset is the optical image of different segments with Seg-32 (red), Seg-29 (blue), and Seg-26 (green). Seg-28 (cyan) is composed of Seg-29 and Seg-26. (c) Fast Fourier transform (FFT) spectra obtained from quantum oscillations shown in (b) at different segments. Seg-28 data is calculated from oscillations shown in Fig. S8. $F_1, F_2$, and $F_3$ represent three main oscillation frequencies. The inset shows the FFT of quantum oscillations in Seg-8 at 0.3 K. Only two main oscillation frequencies of $F_2^{\prime }$ and $F_3^{\prime }$ are observed. (d) Landau-fan diagram for quantum oscillations $\mathrm{\Delta }{\rho }_{\mathrm{xx}}(B$) measured at 0.3 K and $\theta$ = $0^{\circ }$ . The least-square fitting gives the intercepts of different segments, which are listed in the inset table with the calculated thickness. The left upper inset shows the new Weyl orbit in ${\mathrm{Mo}}_{0.25}{\mathrm{W}}_{0.75}{\mathrm{Te}}_2$.