Highly anisotropic interlayer magnetoresitance in ZrSiS nodal-line Dirac semimetal

We instigate the angle-dependent magnetoresistance (AMR) of the layered nodal-line Dirac semimetal ZrSiS for the in-plane and out-of-plane current directions. This material has recently revealed an intriguing butterfly-shaped in-plane AMR whose origin is not well understood. Our aim was to understand the mechanism behind this peculiar shape of AMR and also to probe AMR in the out-of-plane current direction. In contrast to the in-plane AMR, the polar out-of-plane AMR shows a surprisingly different response with a pronounced cusplike feature. The maximum of the cusplike anisotropy is reached when the magnetic field is oriented in the $a\text{−}b$ plane. Moreover, the AMR for the azimuthal out-of-plane current direction exhibits a very strong fourfold $a\text{−}b$ plane anisotropy. Combining the Fermi surfaces calculated from first principles with the Boltzmann's semiclassical transport theory, we reproduce all the prominent features of the unusual behavior of the in-plane and out-of-plane AMR. We can conclude that the dominant contribution the cusplike AMR lies in open orbits of the hole pocket and, in general, AMR is strongly influenced by charge compensation effect and the off-diagonal conductivity tensor elements, which give rise to peculiar butterfly-shaped AMR. Finally, the semiclassical model was also able to clarify the origin of strong nonsaturating (subquadratic) transverse magnetoresistance observed in this material, as an effect of imperfect charge-carrier compensation and open orbits.

Figure 1

(a) Temperature-dependent resistivity for the in-plane (${\rho }_{ab}$) and out-of-plane (${\rho }_c$) current directions of the two single crystals (S1 and S2, respectively) of ZrSiS. In both directions, resistivity shows metallic behavior with moderately strong low-$T$ anisotropy ${\rho }_c/{\rho }_{ab}{|}_{1.8K}\approx 50$. The low-$T{\rho }_{ab}\approx 0.1\mu \mathrm{\Omega }\mathrm{cm}$ shows excellent material quality. (b) For both orientations, high-$B$ and low-$T$ transverse magnetoresistance (MR at 1.8 K) shows a comparable magnitude, reaching almost 20 $000\%$ at $B=10$ T. The inset shows the tetragonal crystal structure of ZrSiS (space group $P4/nmm$). Adjacent layers of S atoms are van der Waals bonded and thus form natural cleavage planes. (c) The intra-AMR at constant $B$ for the in-plane current direction (along $a$ axis) shows an unusual butterflylike pattern for $B$ rotated in the $c\text{−}a$ plane. (d) Detailed low-$T$ angular spectroscopic mapping of MR at 10 T for the in-plane current direction provides an overview of the shape of the AMR in 3D, with current along the $a$ axis. The mapping is preformed by scanning $\theta$ for a fixed value of $\varphi$.

Figure 2

(a) Polar inter-AMR for several constants $B$ at $T=1.8$ K and $\varphi =0^{\circ }$, with the current in the out-of-plane direction (along the $c$ axis) for sample S3, shows strikingly different behavior than the intra-AMR. $B$ is rotated in the $c\text{−}a$ plane. The AMR shows a pronounced increase in magnitude as $B$ is rotated toward the $a\text{−}b$ plane and forms a cusplike feature. The inset shows the temperature dependence of polar inter-AMR. The anisotropy becomes very weak around 100 K. (b) Polar intra-AMR for several different directions along the azimuthal ($\varphi$) angle reveals a strong $a\text{−}b$ plane anisotropy.

Figure 3

(a) Azimuthal inter-AMR ($a\text{−}b$ plane field rotation) for sample S2 at several different $B$. The azimuthal AMR shows a strong fourfold in-plane anisotropy with the maxima for $B$ oriented along the high-symmetry in-plane axes and the minima at the bisector axes (odd multiples of $\varphi =\pi /4$). At around 5 T, in the vicinity of the bisector axis, the AMR becomes truncated (flat) in contrast to the 2 T scan which has a quadratic-like shape. For $B$ close to the high-symmetry axes, quantum oscillations can be observed. (b) It is interesting to notice (sample S3 at 10 T) that the truncation of the inter-AMR has a substructure with a dip when $B$ is slightly misaligned with respect to the $a\text{−}b$ plane and a more complicated structure for $B$ in the $a\text{−}b$ plane. The substructure can not be due to quantum oscillations, since the peak positions do not change with the field. The substructure probably originates from a peculiarity in the FS shape. (c) Temperature dependence of the azimuthal inter-AMR for sample S2. Above 100 K the AMR becomes weakly anisotropic and at round 200 K it starts following the classical sinusoidal behavior. (d) Details of wide-angle spectroscopic mapping of the inter-AMR of sample S3 at 10 T. The sketch depicts the measurement geometry.

Figure 4

(a) Calculated angular resistivity ${\rho }_{xx}\left(\theta \right)$ (red solid line) at constant $B$ for the intralayer charge transport. The current and magnetic field orientations are indicted. The calculations reproduce the twofold symmetry butterfly-shaped magnetoresistance in good agreement with experimental data (black open circles). The number of experimental points has been reduced for clarity. Green and blue solid lines represent diagonal contributions of the hole (${\rho }_{xx}^h$) and electron (${\rho }_{xx}^e$) pocket individually. (b) The calculated MR for magnetic field along the $c$-axis ($I\left|\right|a$) follows the $B^{1.68}$ dependence, which agrees perfectly with the measured exponent of 1.67(1), pointing to the charge-carrier compensation mechanism of the observed large transverse MR. (c) Fermi surface used for calculating the conductivity tensor. The FS consists of four isolated electron pockets at $k_z=\pi /c$ and four hole pockets. The hole pocket hosts open orbits and has protruding armlike features in the $k_z=0$ plane. (d), (e) The model (red line) agrees well with experimental data taken (open circles) for polar and azimuthal experimental inter-AMR reproducing all distinctive features. Green (${\rho }_{zz}^h$) and blue (${\rho }_{zz}^e$) lines depict diagonal matrix elements of the resistivity for the hole and electron pockets, respectively. (f) MR values at the cusp maxima for the polar inter-AMR reproduced from Fig. 2 for different values of $B$. The slope corresponds to 1.92(5), which is close to 1.98 predicted by our model.