Highly anisotropic interlayer magnetoresistance in ZrSiS nodal-line Dirac semimetal

TL;DR

This study investigates the highly anisotropic interlayer magnetoresistance in ZrSiS, revealing unique angle-dependent behaviors and explaining them through combined experimental measurements and theoretical modeling.

Contribution

It provides a comprehensive analysis of the anisotropic magnetoresistance in ZrSiS, combining experiments with first-principles calculations to explain unusual AMR features.

Findings

Distinct out-of-plane AMR with cusp-like features

Strong four-fold in-plane anisotropy of AMR

Estimated relaxation time of 2.6×10^{-14} s and mean free path of 15 nm

Abstract

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 that is not well understood. Our measurements of the polar out-of-plane AMR show a surprisingly different response with a pronounced cusp-like feature. The maximum of the cusp-like anisotropy is reached when the magnetic field is oriented in the $a$-$b$ plane. Moreover, the AMR for the azimuthal out-of-plane current direction exhibits a very strong four-fold $a$-$b$ plane anisotropy. Combining the Fermi surfaces calculated from first principles with the Boltzmann's semiclassical transport theory we reproduce and explain all the prominent features of the unusual behavior of the in-plane and out-of-plane AMR. We are also able to clarify the origin of the strong non-saturating transverse magnetoresistance as an effect of imperfect charge-carrier compensation and open orbits. Finally, by combining our theoretical model and experimental data we estimate the average relaxation time of $2.6\times10^{-14}$~s and the mean free path of $15$~nm at 1.8~K in our samples of ZrSiS.

Full text

Square net crystal structures have been of a considerate interest in the structural solid-state chemistry.Tremel_1987 Introducing non-trivial topology and Dirac fermions to the field of condensed-matter physics has started a surge in the discovery of new materials with linear energy dispersion.Wang_2013 ; Hasan_2010 ; Hsieh_2009 ; Borisenko_2014 ; Neupane_2014 ; Liu_2014 ; Huang_2015 ; Lv_2015_2 ; Lv_2015 Among many square-net structures, two phases have emerged as especially interesting from the topological point of view. The ATB2 phase, where A stands for alkali or rare-earth metal with +2 oxidation state, T is a 3$d$ transition metal and B a pnictogen group element. Typical representatives are Ca/Sr/Ba-MnBi2 that harbor quasi-2D Dirac fermions with a highly anisotropic band dispersion in the Bi-based atomic plane along with antiferromagnetic ordering in the Mn plane.Jo_2014 ; Feng_2014 ; Li_2016 ; Zhang_2016 ; Wang_2016 Replacing the alkali earth metal with Eu leads to an additional inter-layer decoupling and the formation of the half-integer quantum Hall effect.Masuda_2016

Another interesting phase is MX${}^{{}^{\prime}}$X${}^{{}^{\prime\prime}}$, which incorporates a large group of compounds,Tremel_1987 where M is a metal (Zr, Hf, Ta, Nb), X${}^{{}^{\prime}}$ is a +2 valence state of Si, Ge, As and X${}^{{}^{\prime\prime}}$ belongs to the chalcogen group. The prototypical representative of this group is ZrSiS which is the subject of this work. ZrSiS and isostructural compounds have recently gained a lot of attention due to the glide and screw symmetry protected crossing of the conduction and valence bands and thus resulting in a nodal-line Dirac semimetal (NLDSM) phase. NLDSMs are the topological phases related to the 3D Dirac and Weyl semimetals with the difference being that the conduction and valence bands do not cross only at isolated points in the $k$ space, but form loop or nodal-line degeneracies that give rise to new and interesting physical phenomena.Rui_2018 ; Ramamurthy_2017 ; Fang_2015 ; Rudenko_2018 ; Burkov_2018 ; Huh_2016

In the absence of spin-orbit interaction (SOI), ZrSiS has one set of nodal lines close to the Fermi energy (EF) and another set located deep in the valence band.Schoop_2016 ; Rudenko_2018 It has been argued that the nodal lines located in the vicinity of EF are protected by $C_{2v}$ symmetry and are thus susceptible to a degeneracy lifting due to the SOI that is effectively transforming the system into a weak topological insulator.Xu_2015 On the other hand, the deep-lying nodal lines are topologically protected by the non-symmorphic symmetry. ZrSiS has several theoretically predicted unique properties among 3D Dirac semimetals, including the large interval of linear dispersion (reaching almost 2 eV) without the presence of any trivial bands and the high degree of electron-hole symmetry.Schoop_2016 ; Xu_2015

Recent studies of the Fermi surface (FS) morphology in ZrSiS by means of angle-resolved photoemission spectroscopy (ARPES) and quantum oscillations measurements (QOM) have confirmed the nature and the position of the two pockets: a large electron and a smaller hole pocket.Schoop_2016 ; Neupane_2016 ; Fu_2017 ; Hosen_2017 ; Matusiak_2017 ; Hu_2017 The electron pocket has a 3D nature, whereas the hole pocket shows a quasi-2D signature.Zhang_2018 ; Ali_2016 ; Singha_2017-2 The QOM have also revealed the signature of another very small pocket with a puzzling Berry phase and the ultraquantum limit at around 10 T.Matusiak_2017 ; Hu_2017 Due to its small size and small charge-carrier concentration, as well as its weakly elongated ellipsoidal shape observed by QOM, this pocket is most likely irrelevant for the observed charge transport effects under the rotation of magnetic field.Abrikosov Furthermore, a high-magnetic field study of ZrSiS has revealed an interesting magnetic-breakdown effect and an unusual mass enhancement, whereas in the sister compound HfSiS an effect of Klein tunneling between electron and hole pockets was detected.Pezzini_2017 ; Delft_2018 The magnetoresistance is large and unsaturated with a sub-quadratic magnetic field dependence as frequently observed in the Dirac and Weyl semimetals.Singha_2017-2 ; Zhang_2018 ; Xiong-2015 ; Liang-2015 On the other hand, constant field angular-dependent magnetoresistance (AMR) measurements have observed an unexpected and intricate butterfly-shaped anisotropy for current applied along the in-plane axes.Ali_2016 ; Wang_2016-2 ; Zhang_2018 ; Hu_2016

In this paper, we present a detailed study of magnetic field ($B$) and temperature ($T$) dependence of the AMR in ZrSiS single crystals for current oriented along high-symmetry directions. The AMR has proved to be a very powerful tool for studying the FS shape of 3D and quasi-2D (q-2D) systems, with many materials exhibiting non-classical behavior.Klauder_1960 ; Collaudin_2015 ; Balicas_2005 We have performed out-of-plane AMR measurements (inter-AMR) with current along the $c$-direction and in-plane AMR measurements (intra-AMR) with current along the $a$($b$)-axis. To the best of our knowledge, ZrSiS and related compounds have not been previously characterized for the current oriented along the $c$ direction. For current along the $c$-direction polar inter-AMR reveals a large cusp-like anisotropy which becomes pronounced close to the $a$-$b$ plane. Additionally, the azimuthal inter-AMR shows a strong $a$-$b$ plane anisotropy with four-fold symmetry and a minimum at an angle corresponding to an odd multiples of $\pi/4$. In the case of the intra-AMR, the polar scan displays a previously observed butterfly-like shape.

To understand the striking difference of the AMR response for the in-plane and out-of-plane current directions and to elucidate the role of the putative q-2D hole pocket, we have employed a theoretical transport model based on the Fermi surface calculated from first principles combined with the semi-classical Boltzmann equation. Using this model we were able to reproduce all features observed in experimental data. Our model explains the intricate butterfly-like AMR and strongly anisotropic inter-AMR in terms of charge-carrier compensation due to electron and hole pockets and the effect of strong off-diagonal elements in the conductivity tensor. Furthermore, by using the model we clarify the origin of large sub-quadratic non-saturating magnetoresistance as an effect of the imperfect charge-carrier compensation. Finally, combining the experimental AMR results and the theoretical transport model we were able to refine the shape of the Fermi surface and estimate the average scattering time and the mean free path.

Single crystals of ZrSiS were grown by chemical vapor transport and show excellent quality with a low-$T$ in-plane resistivity of only 0.1 $\mu\Omega$cm.Ali_2016 Due to their layered crystal structure, ZrSiS commonly grows in as plate-like crystals, with a thickness of around 100 $\mu$m. By optimization of the synthesis procedure we managed to obtain samples of sufficient thickness (in the mm range) which also allowed us to measure the out-of-plane transport properties. All measured samples S1, S2, and S3 are cut from the same bigger single crystal whose homogeneity was verified before cutting. The zero $B$ temperature dependence of the resistivity for the in-plane current direction $\rho_{ab}$ of sample S1 shows metallic behavior [Fig.1a] with a residual resistivity ratio (RRR=$\rho_{300K}/\rho_{1.8K}$) of around 80. The out-of-plane resistivity $\rho_{c}$ of sample S2 shows also metallic behavior but with a considerably higher resistivity contributing to a moderate anisotropy $\rho_{c}/\rho_{ab}$ of around 50 at the lowest measured temperature. The almost identical $T$ profile for the in-plane and out-of-plane transport points towards the coherent intra-layer transport, which is expected in transport dominated by a 3D-FS.

The transverse magnetoresistance (MR)foot003 given in Fig. 1b shows a very strong response reaching almost 20 000 $\%$ at 10 T for both orientations (S1: $B||c$, $I||a$, and S2: $B||a$, $I||c$). A strong, non-saturating MR is a commonly observed property of 3D Dirac and Weyl semimetals arising from multiband transport of high mobility charge carriers. The MR of samples S1 and S2 has a sub-quadratic $B$ dependencefoot002 which has been recently associated with a $B$ dependent mobility.Fauque_2018 ; Zhao_2018 Sample S1 displays strong Shubnikov-de Haas (SdH) quantum oscillations with frequencies corresponding to 8.5 T and 241 T. ARPES measurements unambiguously related the higher frequency (241 T) to the q-2D tube-like hole pocket, whereas the position and exact shape of the pocket with the smaller frequency is not yet determined. Due to its small size and weakly elongated ellipsoidal shape its contribution to the total magnetotransport of the charge carriers should be negligible. SdH oscillations observed in sample S2 are less pronounced and composed of several frequencies (17 T, 23 T and 170 T).

Under the polar rotation (angle $\theta$) of sample S1 at $B=9$ T the transverse intra-AMR,foot003 shown in Fig. 1c) exhibits a peculiar four-fold butterfly-shaped angular dependence with the angle of maximum resistivity at odd multiples of $\theta=\pi/4$. The origin of this peculiar intra-AMR has been elusive since previously this kind of AMR was only observed in magnetically-ordered materials where its origin is not purely orbital.Jovan_2010 By using the framework of the semiclassical transport model we were able to understand the origin of the intra-AMR in terms of the charge-carrier compensation effects of the electron and hole pockets. Figure 1d displays a detailed spectroscopic mapping of the intra-AMR transport. When $B$ is tilted away from the $c$-axis the AMR increases showing the butterfly-shaped profile for all values of the azimuthal angle $\phi$. On the other hand, for the in-plane rotation ($a$-$b$ plane) the intra-AMR is small and the anisotropy is weak.

Figure 2a presents details of the polar scan of the inter-AMR at several discrete values of $B$ between 2 T and 10 T for current along the $c$-axis for sample S3. For the longitudinal configuration ($B$ and current are along the $c$-axis) the MR is small for all measured $B$, which is expected due to the vanishing Lorentz force. In the model of a single spherical FS it should be zero. By tilting $B$ away from the $c$-axis (in the $c$-$a$ plane) the AMR shows a strong increase in magnitude that becomes more pronounced with increasing $B$ and forms a cusp-like feature. The anisotropy ratio of the MR for the longitudinal and transverse $B$ orientations is around 50 at 10 T. The strong increase of inter-AMR when $B$ is close to the in-plane direction is a feature commonly observed in q-2D materials and it is associated with coherent intra-layer transport, i.e. a small warping of the 2D Fermi surface.Hanasaki_1998 ; McKenzie_1998 The coherence peak is usually accompanied by Yamaji oscillations in inter-AMR, but is not detected in our samples.Yagi_1990 The small oscillations in the AMR profile at higher $B$ originate from the quantum oscillations, which is supported by the field dependence of the oscillations peak positions. Recently, in several Dirac semimetals with square-net structure, the peak-like response has been observed for $B$ close to the $a$-$b$ plane, which was explained by q-2D FSJo_2014 ; Wang_2016 or by the inter-layer quantum transport governed by the Dirac point.Liu-2017 The inset in Fig.2a shows temperature dependence of the inter-AMR for sample S2 at 9 T. By increasing $T$ the cusp-like shape of the inter-AMR weakens and, at around 200 K, it acquires the classical sinusoidal shape. Performing the same polar scans but now for different angles $\phi$, we observe an indication of a strong in-plane anisotropy [see Fig. 2b]. The polar inter-AMR becomes significantly weaker as $\phi$ approaches $\pi/4$.

The azimuthal inter-AMR ($\phi$ angle rotation) given in Fig. 3a shows a strong fourfold $a$-$b$ plane anisotropy with the maxima positioned along the high-symmetry axes $a$ and $b$ and minima along the bisector axes (odd multiples of $\pi/4$). The strength of the anisotropy (ratio of the maximum and minimum values) is almost $B$ independent in the measured range between 5 T and 9 T and its value is around 4. This is in contrast to the anisotropy of the polar inter-AMR that is continuously growing with $B$. The observed $a$-$b$ plane anisotropy is fairly large. For comparison, it is roughly a factor of 2 larger than that in Sr2RuO4.Omichi_2000 ; Balicas_2005 For larger $B$ ($>5$ T), the AMR becomes truncated close to the bisector axis, whereas at 2 T the truncation is not observed. The effect of quantum oscillations is clearly seen at 9 T close to the high-symmetry axes. Closer inspection reveals that the truncation has a complicated structure, details for sample S3 at 10 T are shown in Fig. 3b. When $B$ is slightly misaligned with respect to the in-plane orientation, the dip in the AMR appears at the bisector axis, whereas when $B$ is in-plane, the truncated part shows oscillating-like behavior which is $B$ independent and thus cannot be related to quantum oscillations. This unusual behavior is probably related to the local morphology of the FS. Considering the temperature dependence of the azimuthal inter-AMR it can be seen that, above 100 K, the truncated part disappears [see Fig. 3c] and the anisotropy of the in-plane AMR weakens. In Fig. 3d we present a detailed spectroscopic mapping of the inter-AMR. It can be seen that the AMR pattern has a twofold symmetry for $\theta$-angle rotation and fourfold symmetry for $\phi$-angle rotation. The maximum in the AMR only appears when $B$ is oriented along the high-symmetry in-plane directions.

In order to achieve a deeper understanding of the unusual AMR response in ZrSiS we have employed numerical modeling using the semiclassical Boltzmann transport theoryLiu_2009 ; ShengNan_2018 as implemented in WannierTools open-source software package.WannierTools A detailed description of the methodology along with representative examples is given in Ref. ShengNan_2018, . We have used Fermi surfaces calculated using DFT to obtain the Boltzmann conductivity tensor $\sigma_{ij}$ for electron and hole pockets which was then transformed into the total resistivity tensor $[\rho_{ij}]=[\sigma^{e}_{ij}+\sigma^{h}_{ij}]^{-1}$. Analyzing individual contributions of electron and hole pockets will help us clarifying the physical mechanism underlying the discussed megnetotransport properties.

As a first step, in Fig. 4a we present the transverse angular resistivity $\rho_{xx}(\theta)$ calculated at a fixed magnitude of $B$ for the intra-layer current direction. The calculations reproduce very well the experimental data (black open circles), and in particular the two-fold symmetry butterfly-like shape. The green and blue lines in Fig. 4a represent individual contributions of the electron ($\rho^{e}_{xx}$) and hole ($\rho^{h}_{xx}$) pockets, respectively. Closer inspection reveals that the total angular resistivity is not a simple sum of electron and hole (parallel) channels. Only for the high-symmetry directions, i.e. $B||c$ and $B||b$, the total calculated resistivity is smaller that the resistivity of the contributing channels, giving rise to a noticeable drop in the intra-AMR. For other directions we cannot apply the parallel channel rule due to a significant contribution of the off-diagonal elements of the conductivity tensor, from which $\rho_{xx}(\theta)$ is obtained.

Next, we use our model to understand the mechanism of the observed large non-saturating transverse MR for the in-plane current direction. While non-saturating MR with $B^{2}$ dependence is usually assigned to systems with perfect electron-hole compensation, deviations from the ideal quadratic scaling often observed in the high mobility systems with electron and hole pockets has recently been attributed to the field-dependent mobility.Fauque_2018 The transverse MR for $B||c$ scales as $B^{1.68}$, which is in excellent agreement with the measured $B^{1.67(1)}$ dependence [Fig. 4b]. The strong orbital MR in our case comes from the imperfect compensation effect between electron and hole pockets.

We now proceed with the analysis of inter-AMR with current oriented along the $c$-direction for two distinct measurement configurations – polar ($\theta$-scan) and azimuthal ($\phi$-scan) ones. Fig. 4d shows the calculated polar angular resistivity $\rho_{zz}(\theta)$ (red solid line) for the current and field orientations defined in the figure. The calculated resistivity has a strong angular dependence with a cusp-like shape and a maximum being reached for the in-plane oriented field. Comparison of the calculated and measured (black solid circles) angular dependences again shows good agreement. To understand the origin of this unusual behavior we examine the individual contributions of the electron and hole pockets, $\rho^{e}_{zz}$ and $\rho^{h}_{zz}$, respectively. Comparison of the calculated magnetoresistivity $\rho_{zz}(\theta)$ with individual components again shows that the total resistivity cannot be described by combining two parallel channels and the off-diagonal elements play a significant contribution. Both matrix elements have a strong peak for $B$ close to the in-plane orientation, but of distinct origins. The peak in $\rho^{h}_{zz}$ originates from the open orbits, whereas the peak in $\rho^{e}_{zz}$ comes from the electron pocket deviates strongly from the ideal free-electron spherical shape, i.e. it results from its flatness.

As a next step, we aim at understanding the $a$-$b$ plane anisotropy of the intra-AMR [Fig. 4e]. Good agreement of the calculated and experimental angular dependences is achieved as in the previous cases. Comparing $\rho_{zz}(\phi)$ with the contributions of individual pockets $\rho^{h}_{zz}(\phi)$ (green line) and $\rho^{e}_{zz}(\phi)$ (blue line), allows us to conclude that the angular dependence of $\rho_{zz}(\phi)$ is mostly due to the hole pocket. The truncated part of the azimuthal AMR most likely also has its origin in the hole pocket since $\rho^{h}_{zz}$ shows anomalous behavior close to $\pi/4$. Furthermore, in Fig. 4f we have traced the cups maxima [from Fig. 2a] as a function of $B$ obtaining the $B^{1.92(5)}$ scaling. Our calculations predict a comparable value of 1.98 and both results are very close to the quadratic behavior expected for the open-orbit mechanism. Even though we have observed almost quadratic dependence, the azimuthal in-plane anisotropy indicates that besides the dominant effect of open orbits, the charge-carrier compensation effect also plays an important role since open orbits cannot account for the observed azimuthal angular dependence. Recently, strong increase in the inter-AMR close to the $a$-$b$ plane has been reported in several similar square-net Dirac materials.Jo_2014 ; Wang_2016 It has been argued that in these materials the peak-like structure originates form the small closed orbits on the side of the corrugated q-2D FS,Ghannadzadeh_2017 ; Hanasaki_1998 whereas in our case strong increase in the AMR is mostly due to the open orbits.

Combining the measured data with the theoretical model we are now able to comment on the average relaxation time $\tau$ and the FS shape. By using the constant relaxation time approximation and fitting the semiclassical Boltzmann model to the Hall measurementsSup we obtained $\tau\approx 2.6\times 10^{-14}$ s. From the ARPES measurements, the Fermi velocity is estimated to be around $5-6\times 10^{5}$ m/s resulting in the mean free path $l=15$ nm.Sup Assuming that at 2 K impurity scattering is the dominant charge-carrier relaxation mechanism we can estimate impurity concentration of ca. $10^{17}$ cm*-3*. Recent publication on ZrSiS proposed several different FS morphologies.Fu_2017 ; Pezzini_2017 ; Ali_2016 These discrepancies can be related to the sensitivity of the FS shape calculated using DFT to the exact position of $E_{\mathrm{F}}$ and various details of the methodology, such as pseudopotentials used in the calculations. The reported FS shapes have similar structure of the electron and hole pockets in the $k_{z}=\pi/c$ plane, but differ significantly in the $k_{z}=0$ plane. Fig. 4c shows our calculated FS characterized by the hole pocket consisting of an elongated tube-like structure that gives rise to open orbits. In the $k_{z}=0$ plane the hole pocket has protruding ”arm-like” features extending along the X-X lines. Good agreement between all the peculiar features of the calculated and experimental AMR provides a strong indication that the calculated FS reproduces the real one.

In conclusion, we have presented a detailed study of the angular magnetoresistance (AMR) in the nodal-line Dirac semimetal ZrSiS. We have determined the low-temperature, zero-field anisotropy between the in-plane and out-of-plane directions to be moderately strong with typical values around 50. The AMR was measured in two configurations, for current oriented in-plane along the $a$-axis (intra-AMR) and out-of-plane along the $c$-axis (inter-AMR). The intra-AMR shows an unusual butterfly-like shape previously reported by other authors. The inter-AMR shows a strong cusp-like shape anisotropy for polar angle rotation, with a maximum achieved for the magnetic field oriented in the $a$-$b$ plane. Additionally, the azimuthal angle rotation shows strong anisotropy with a four-fold symmetry, with the minimum at odd multiples of $\phi=\pi/4$. In order to understand this intricate AMR we have employed a theoretical model based on the Fermi surface calculated from first principles and the Boltzmann semiclassical theory. The model successfully reproduced all observed features of both the inter- and intra-AMR. Furthermore, our model explains the sub-quadratic dependence of transverse magnetoresistance as an effect of imperfect charge-carrier compensation for the in-plane case, and an open-orbit mechanism combined with charge-carrier compensation for the out-of-plane current direction. We were able to estimate the average relaxation time to be around $\tau\approx 2.6\times 10^{-14}$ s, the mean free path $l\approx 15$ nm and more accurately determine the Fermi surface shape.