Magnetotransport properties of ${\mathrm{MoP}}_2$

We report magnetotransport and de Haas–van Alphen (dHvA) effect studies on ${\mathrm{MoP}}_2$ single crystals, predicted to be a type-$\mathrm{II}$ Weyl semimetal with four pairs of robust Weyl points located below the Fermi level and long Fermi arcs. The temperature dependence of resistivity shows a peak before saturation, which does not move with magnetic field. Large nonsaturating magnetoresistance (MR) was observed, and the field dependence of MR exhibits a crossover from semiclassical weak-field $B^2$ dependence to the high-field linear-field dependence, indicating the presence of Dirac linear energy dispersion. In addition, a systematic violation of Kohler's rule was observed, consistent with multiband electronic transport. Strong spin-orbit coupling splitting has an effect on dHvA measurements whereas the angular-dependent dHvA orbit frequencies agree well with the calculated Fermi surface. The cyclotron effective mass $\sim 1.6m_e$ indicates the bands might be trivial, possibly since the Weyl points are located below the Fermi level.

Figure 1

(a) Crystal structure of ${\mathrm{MoP}}_2$. (b) The band structure for ${\mathrm{MoP}}_2$. (c) Fermi surface of ${\mathrm{MoP}}_2$; a pair of bowtielike Fermi pockets are located around $Y$ point in the Brillouin zone (BZ), and a pair of spaghettilike hole Fermi pockets are located at $X$ point of BZ.

Figure 2

(a) Temperature dependence of resistivity in different magnetic fields plotted on a log-log scale; the magnetic field is applied along $c$ axis, and electrical current is along $a$ axis. The grey vertical line is a guide to eye. (b) The temperature dependence of the $\text{MR}=\left[\rho \right(B)-\rho (0\text{T}\left)\right]/\rho \left(0\text{T}\right)\times 100\%$ at different fields (left), and the $\mathrm{\Delta }\rho =\rho \left(9\text{T}\right)-\rho \left(0\text{T}\right)$ (right). Panels (a) and (b) use the same legend. Inset shows $\rho \left(0\text{T}\right)$ fitted with Bloch-Grüneisen model.

Figure 3

(a) The MR vs magnetic field at different temperatures for ${\mathrm{MoP}}_2$ with $H\parallel c$. (b) A Kohler plot for ${\mathrm{MoP}}_2$. (c) Typical magnetic-field dependence of MR curves; the solid lines are fits using $\text{MR}=A\times H^n$. (d) Temperature dependence of $A$ (left) and $n$ (right) in $\text{MR}=A\times H^n$. (e) The field derivative of MR $\left[d\right(\mathrm{MR})/d\mathrm{B}]$ at different temperatures; solid lines show the criterion used to determine the critical field $B^{*}$. (f) Temperature dependence of the critical field $B^{*}$ (left); the black solid line is the fitting of $B^{*}$ by $B^{*}=\frac{1}{2e\hslash {\upsilon }_F^2}{(E_F+k_{B}T)}^2$. The red circle corresponds to high-field MR linear coefficient $A_1$.

Figure 4

(a),(b) Magnetic-field dependence of MR with field at different angles in $bc$ plane. The current is along $a$ axis and always perpendicular to the field. (c) Polar plot of the angular dependence of the MR with the field rotated in $bc$ plane at 2 K and 18 T. (d) Temperature dependence of $A$ (left) and $n$ (right) in $\text{MR}=A\times H^n$.

Figure 5

(a) dHvA oscillatory components at different temperatures obtained by subtracting smooth background. (b) FFT spectra for the dHvA oscillations in (a). (c) Temperature dependence of the oscillating amplitude at $F_{\alpha }=1261$ T, solid line is fitted using Lifshitz-Kosevich formula.

Figure 6

FFT spectra of dHvA oscillaiton with field rotated in the $bc$ plane (a) and in the $ab$ plane (b); the spectra are normalized and shifted vertically for clarity. (c),(d) Angular dependence of the oscillation frequency corresponding to the oscillation in (a) and (b), respectively. Red solid lines are fits with 2D model $F\left(0\right)/\text{cos}\left(\theta \right)$.