Magnetoresistance of semimetals: The case of antimony

Large unsaturated magnetoresistance has been recently reported in numerous semimetals. Many of them have a topologically nontrivial band dispersion, such as Weyl nodes or lines. Here, we show that elemental antimony displays the largest high-field magnetoresistance among all known semimetals. We present a detailed study of the angle-dependent magnetoresistance and use a semiclassical framework invoking an anisotropic mobility tensor to fit the data. A slight deviation from perfect compensation and a modest variation with magnetic field of the components of the mobility tensor are required to attain perfect fits at arbitrary strength and orientation of magnetic field in the entire temperature window of study. Our results demonstrate that large orbital magnetoresistance is an unavoidable consequence of low carrier concentration and the subquadratic magnetoresistance seen in many semimetals can be attributed to field-dependent mobility, expected whenever the disorder length scale exceeds the Fermi wavelength.

Figure 1

(a) Magnetoresistance of various semimetals at $B=9$ T and $T=2$ K as a function of the mobility ${\mu }_0=\frac{1}{{\rho }_0(n+p)e}$ where $e$ is the charge of the electron, $n$ and $p$ are the electron and hole density, and ${\rho }_0$ the zero field resistivity at 2 K. ${\mu }_0$ is expressed in ${\text{Tesla}}^{-1}={10}^4{\mathrm{cm}}^2$ ${\mathrm{V}}^{-1}$ ${\mathrm{s}}^{-1}$. The points are extracted from the references [7, 8, 11, 12, 18, 19, 21, 22, 27, 28, 29, 30, 31, 32, 33] (see Ref. [34] for details). (b) Magnetoresistance of elemental semimetals antimony (j//trigonal and B//bisectrix), bismuth (j//bisectrix and B//trigonal) and ${\mathrm{WTe}}_2$ (j//α axis and B//$c$ axis) at $T=2$ K. RRR (residual resistivity ratio) is equal to $\frac{\rho (T=300\text{K})}{\rho \left(\text{2}\text{K}\right)}$.

Figure 2

Angular dependence of the magnetoresistance of antimony. (a),(b) Electron and hole Fermi surface of antimony according to the Liu and Allen tight-binding model. The unit of reciprocal length is $g=1.461$ Å${}^{-1}$. (c),(e),(g) Sketch of the Brillouin zone and the Fermi surface of antimony projected in the three planes of rotation ${\mathrm{P}}_1$ (trigonal, bisectrix), ${\mathrm{P}}_2$ (trigonal, binary), and ${\mathrm{P}}_3$ (binary, bisectrix). (d),(f),(h) ${\rho }_{ii}$ for $i=1$, 2, and 3 in the three planes of rotation ${\mathrm{P}}_1$, ${\mathrm{P}}_2$, and ${\mathrm{P}}_3$ in blue at $T=2$ K and $B=12$ T (the electric current is applied along the crystal axis perpendicular to the rotating plane) measured on the same sample. The red line is a fit using a theoretical model described in the text.

Figure 3

Transverse magnetoresistance of Sb at $T=2$ K along the axes of high symmetry: (a) ${\rho }_{11}$(B) for B//trigonal (in blue) and B//bisectrix (in red), (b) ${\rho }_{22}$(B) for B//trigonal (in blue) and B//binary (in red), (c) ${\rho }_{33}$(B) for B//binary (in blue) and B//bisectrix (in red), (d) ${\rho }_{12}$ for B//trigonal (in purple). The green lines correspond to the Mackey and Sybert's model assuming no change in mobility tensors with the magnetic field and a perfect compensation ($\frac{\mathrm{\Delta }n}{n}=0$). The dashed black lines correspond to the same model with a deviation from perfect compensation equal to $\frac{\mathrm{\Delta }n}{n}=7\times {10}^{-4}$ which allow us to capture the Hall response. The black lines correspond to the same model with a field dependence mobility tensor [reported in Fig. 4] and a deviation from perfect compensation $\frac{\mathrm{\Delta }n}{n}=7\times {10}^{-4}$.

Figure 4

Angular dependence of the magnetoresistance of Sb at $T=2$ K for $B=0.5$ T to 12 T in the three plans of rotations (a) ${\mathrm{P}}_1$, (b) ${\mathrm{P}}_2$, and (c) ${\mathrm{P}}_3$. The dot lines are fits using the semiclassical model described in the text.

Figure 5

Temperature and field dependence of mobility tensors of Sb: The hole and electron tensor component are, respectively, labeled (${\nu }_1,{\nu }_2,{\nu }_3$), (${\mu }_1,{\mu }_2,{\mu }_3$). (a) ${\nu }_i$ and ${\mu }_i$ for $i=1$, 2, 3 at $B=4$ T as a function of temperature. The black line is the mobility deduced from the temperature dependence of ${\rho }_{33}$ in the Drude picture. (b) Field dependence of ${\mu }_i$ and ${\nu }_i$ for $i=1$, 2, 3 at $T=2$ K. The raw data and the comparison with the fit can be found in Ref. [34].