Magnetoresistance in ZrSi$X$ ($X=$ S, Se, Te) nodal-line semimetals

TL;DR

The study addresses MR in the ZrSiX nodal-line semimetal family by combining first-principles calculations with semiclassical Boltzmann transport in the relaxation-time framework, treating the analysis through the lens of Fermi-surface geometry rather than topology. MR features, including the butterfly-shaped anisotropy in ZrSiS/Se and a peanut-shaped MR in ZrSiTe, are reproduced and attributed to open orbits and electron–hole compensation, with the dimensionless parameter $B\tau$ guiding regime analysis. Hall resistivity is explained by competition between carriers of different cyclotron masses, with reported $m^*_h$ and $m^*_e$ values aligning with observed sign changes. Overall, the work demonstrates magnetotransport as a geometry-sensitive probe of Fermi surfaces in nodal-line semimetals and emphasizes its role alongside ARPES and quantum oscillations in assessing topological contributions to transport.

Abstract

We present a comprehensive first-principles study of the magnetoresistance in ZrSi$X$ ($X=$ S, Se, Te) topological nodal-line semimetals. Our study demonstrates that all primary features of the experimentally measured magnetoresistance in these materials are captured by our calculations, including the unusual butterfly-shaped anisotropic magnetoresistance. This anisotropic magnetoresistance can be accurately reproduced using the semiclassical Boltzmann transport theory without introducing any information on the topological nature of bands or the concepts of topological phase transition. Considering the complex structure of the Fermi surface in these topological materials, we develop a theoretical description explaining the features observed in magnetoresistance measurements. Additionally, the atypical Hall resistance can be interpreted by the same semiclassical approach. Our findings establish magnetotransport as a powerful tool for analyzing the geometry of the Fermi surface, complementing angle-resolved photoemission spectroscopy and quantum oscillation measurements. This approach is demonstrated to be particularly useful for determining the role of non-trivial topology in transport properties.

Figures (7)

• Figure 1: (a) Fermi surface of ZrSiS. Blue sheets denote the hole pockets, while the pink ones are electron pockets. The four tube-shaped (blue) Fermi surface sheets extending along the $c$ axis may result in open orbits when magnetic field is applied in the $a$-$b$ plane. (b) A typical cross section of the Fermi surface for B$\parallel$[001] passing through the (0, 0, $\pi$/c) $k$-point. The the blue and pink curves on the right are the corresponding orbits in real space indicated by the same color arrows, respectively. (c),(d) Typical cross sections of the Fermi surface (hole pockets) passing through $k$-points (0.8$\pi$/a, 0, 0) and (0.84$\pi$/a, 0, 0)), respectively, for B$\parallel$[110]. The blue curves are the corresponding orbits in real space. (e),(f) Typical cross sections of the Fermi surface (hole pockets) passing through the (0.8$\pi$/a, 0, 0) and (0.84$\pi$/a, 0, 0) $k$-points, respectively, for B$\parallel$[100]. The blue curves show corresponding orbits in real space.
• Figure 2: Calculated anisotropic magnetoresistance of ZrSiS for currents along (a) the $a$, (b) $b$ and (c) $c$ axes, and magnetic field rotating in the $b$-$c$, $a$-$c$ and $a$-$b$ planes, respectively. Angles are expressed in degrees ($^{\circ}$). In panels (a) and (b), the maximum resistivity occurs at approximately $\theta = \pi /4 = 45^{\circ}$ at low magnetic field, and at nearly $\theta = \pi/5 = 36^{\circ}$ at large magnetic field. It has an apparent fourfold symmetry, especially at low magnetic field.
• Figure 3: (a) Difference between resistivities of electron and hole charge carriers $\rho_{yy}^e - \rho_{yy}^h$ when the magnetic field is rotated from $\theta = 0^{\circ}$ (along the $c$ axis) to $\theta = 42^{\circ}$ (close to the [110] direction). (b) Resistivity $\rho_{zz}$ as a function of magnetic field orientation. Field dependences of resistivity $\rho_{zz}$ for magnetic field oriented along (c) the $a$ axis and (d) the [110] direction. The curves in red and blue show individual resistivities $\rho_{zz}$ of electron and hole charge carriers. Individual resistivities of hole charge carriers (blue curve in panels (c) and (d)) show nearly quadratic dependence, indicating that the open-orbit mechanism is dominant.
• Figure 4: (a) Fermi surface of ZrSiTe. Hole and electron pockets are shown in blue and pink, respectively. There are three bands, i.e. one hole band and two electron bands. Although the Fermi surface viewed along the $z$ axis retains the diamond shape, all pockets are significantly different from those in ZrSiS. (b) Fermi surface cross sections (in black) and the corresponding orbits in real space (in pink). The magnetic field is rotated from $\theta = 0^{\circ}$ (top) to $\theta = 75^{\circ}$ (bottom) by interval of $15^{\circ}$, showing that the content of concave segments is reducing. The arrows mark orientations in which the orbits are followed in the real space, revealing that the content of anticlockwise segments is decreasing as magnetic field is rotating away from the $c$ axis. (c) and (d) Typical cross sections of the Fermi surface of ZrSiTe and the corresponding orbits in real space (blue curves). The cross section in panels (c) pass through points $(0.32\pi/a, 0, 0)$ with $B \parallel [100]$, and in panels (d) pass through points $(0.34\pi/a, 0.34\pi/a, 0)$ with $B \parallel [110]$.
• Figure 5: Calculated magnetoresistivity anisotropy of ZrSiTe for current oriented along (a) the $b$ and (b) $c$ axes and magnetic field rotated in the $a$-$c$ and $a$-$b$ planes, respectively. MR in panel (a) changes to peanut-shaped in contrast with that of ZrSiS and ZrSiSe while retaining its four-fold symmetry in (b).