Quantum oscillations of the magnetic torque in the nodal-line Dirac semimetal ZrSiS
F. Orbani?, M. Novak, Z. Glumac, A. McCollam, L. Tang, I. Kokanovi?
QQuantum oscillations of the magnetic torque in the nodal-line Dirac semimetal ZrSiS
F. Orbani´c, M. Novak, Z.Glumac, A. McCollam, L.Tang, and I. Kokanovi´c ∗ Department of Physics, Faculty of Science, University of Zagreb, Croatia Josip Juraj Strossmayer University of Osijek, Croatia High Field Magnet Laboratory, Radbound University, Nijmegen, the Netherlands (Dated: February 4, 2021)We report a study of quantum oscillations (QO) in the magnetic torque of the nodal-line Diracsemimetal ZrSiS in the magnetic fields up to 35 T and the temperature range from 40 K down to 2 K,enabling high resolution mapping of the Fermi surface (FS) topology in the k z = π (Z-R-A) planeof the first Brillouin zone (FBZ). It is found that the oscillatory part of the measured magnetictorque signal consists of low frequency (LF) contributions (frequencies up to 1000 T) and highfrequency (HF) contributions (several clusters of frequencies from 7-22 kT). Increased resolution andangle-resolved measurements allow us to show that the high oscillation frequencies originate frommagnetic breakdown (MB) orbits involving clusters of individual α hole and β electron pockets fromthe diamond shaped FS in the Z-R-A plane. Analyzing the HF oscillations we unequivocally shownthat the QO frequency from the dog-bone shaped Fermi pocket ( β pocket) amounts β = 591(15)T. Our findings suggest that most of the frequencies in the LF part of QO can also be explainedby MB orbits when intraband tunneling in the dog-bone shaped β electron pocket is taken intoaccount. Our results give a new understanding of the novel properties of the FS of the nodal-lineDirac semimetal ZrSiS and sister compounds. I. INTRODUCTION
In recent years the discovery of Dirac and Weyl typeexcitations in the low-energy band dispersion of the Diracand Weyl semimetals (DSM and WSM) represents a ma-jor breakthrough in condensed matter physics [1–8]. Dueto their unique band topology, they show different exoticelectronic properties of technological and fundamentalinterest. Dirac semimetals can become Weyl semimet-als or can be driven to other exotic topological phasessuch as topological insulators and topological supercon-ductors by breaking certain symmetries which determinethe band topology of the material [9–17]. Unlike Diracor Weyl semimetals, where there are discrete touchingpoints between the valence and conduction band in thefirst, in a nodal-line semimetal there are symmetry pro-tected band degeneracies which form lines [closed loops oropen lines in the first Brillouin zone (FBZ)]. If the mate-rial posses time reversal and inversion symmetries, thesecrossing lines will be fourfold degenerate (analogous toDirac points in a DSM) and we are talking about a nodal-line Dirac semimetal (NLDSM). Due to their unique bandtopology, effects like charge order, magnetism, and su-perconductivity are theoretically predicted to occur inNLDSM materials [18–26].ZrSiS is a member of the MX (cid:48) X (cid:48)(cid:48) group of compounds[27], where M is a metal (Zr, Hf, Ta, Nb), X (cid:48) is a +2valence state of Si, Ge, As and X (cid:48)(cid:48) belongs to the chalco-gen group. ZrSiS and sister compounds have recentlygained a lot of attention due to the symmetry protectedcrossing of the conduction and valence bands which re-sults in the NLDSM phase. In the system with no spin- ∗ Corresponding author: [email protected] rotation symmetry [namely a system with spin-orbit cou-pling (SOC)] additional non-symmorphic crystal symme-tries (glide planes and a screw axis) are required to pro-tect the NLDSM phase [28, 29], which is the case in ZrSiSand sister compounds [30–33]. In ZrSiS there are sym-metry protected nodal lines running parallel to the k z direction in the FBZ (X-R and M-A directions) locateddeeper in the valence band. There is another set of nodallines forming a cage-like structure in the FBZ which arecloser to the Fermi energy. The later nodal lines arenot symmetry protected and thus susceptible to a smallgap opening due to SOC in ZrSiS. While most of thetransition-semimetal materials studied so far have lin-ear band dispersion up to a few hundred meV from theDirac node, in ZrSiS this energy range is as high as 2 eVin some regions of the FBZ. Thus, the primary criterionto observe exotic properties related to nodal-line Diracfermions, that the Fermi energy of the semimetal shouldremain within the linear dispersion region, is fulfilled inZrSiS. The main goal of this work is a detailed studyof the ZrSiS Fermi surface (FS) using cantilever torquemagnetometry for high quality home-grown crystals.Recent studies of the FS morphology in ZrSiS byangle-resolved photoemission spectroscopy (ARPES) andquantum oscillations (QO) measurements confirmed thenature and position of three-dimensional (3D) electronand quasi two-dimensional (2D) hole pockets, and an-other two small pockets with quantum limits at around10 and 32 T [30, 31, 34–42]. A calculated 3D represen-tation of ZrSiS FS can be seen in [41, 42]. Since in thiswork we analyze QO in the magnetic torque signal (dHvAoscillations) for magnetic fields in directions near to thecrystalline c axis (or near to the k z direction in momen-tum space), we show that the most relevant part of theFS for an explanation of the measurement data is a crosssection of the FS at k z = π (Z-R-A plane) in the FBZ, a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b schematically shown later in Fig. 5. It consists of four β electron and four α hole pockets in a diamond shapeconfiguration separated by a small gap (10-20 meV) dueto the small SOC.Very recently, a high magnetic field study of ZrSiSsemimetal has revealed interesting MB orbits and sig-nificant mass enhancement of the quasiparticles residingnear the nodal loop, whereas, in the sister compoundHfSiS, an effect of Klein tunneling in momentum spacebetween adjacent electron and hole pockets in the topZ-R-A plane of the FBZ has been reported [41–43].The FS of ZrSiS and sister compounds is still not fullyunderstood. In this paper, in an effort to fully under-stand the FS of ZrSiS in the top Z-R-A plane of the FBZand how it’s shape affects electron dynamics in high mag-netic field, we performed magnetic QO measurements us-ing the highly sensitive piezo-resistive cantilever torquetechnique at low temperature in magnetic fields up to35 T and mapped the experimental frequencies to thecalculated FS [41, 42]. From the FFT analysis of thehigh-resolution torque magnetometry data we obtained anumber of high oscillation frequencies in the range from7 kT up to 12 kT as well as their harmonics. We con-firmed that these arise from the MB orbit clusters ofthe individual α hole [oscillation frequency of 241(4) T]and β electron [oscillation frequency of 591(15) T] pock-ets. In [41], Pezzini et al . observed almost identical highoscillation frequencies from 7 up to 12 kT, but no har-monics around 22 kT. Most of the LF FFT spectra ofQO in ZrSiS (frequencies from ca
100 to 1000 T) is alsoexplained by MB orbits in the Z-R-A plane of the FBZ.The clearly observed LF set of QOs, which were not allidentified in the previous study [41] can be explained asan MB effect with a linear combination of α and β pock-ets and an additional β/ β elec-tron pocket. The γ = 415(8)T pocket could be explainedwithin the experimental error bar as an effect of MB with β/ α and β orbits of the FSin the top Z-R-A plane of the BZ. It is found that thedog-bone shaped Fermi electron pocket has an oscillationfrequency of 591(15) T, which is different from the resultin [42]. II. EXPERIMENTAL RESULTS
Single crystals of ZrSiS were grown by standard chem-ical vapour transport [44]. Their excellent quality isshown by their low- T in-plane resistivity of only 0.1 µ Ωcm. The ZrSiS crystals were characterized by x-raydiffraction which confirms a tetragonal PbFCl-like com-pound structure with
P4/nmm space group. The tem-perature dependence of the resistivity ρ ab of the crystalsat zero field shows metallic behavior with a residual re- a a + b (cid:215) B f i t q = 0 o q = - 1 o q = 1 o torque signal [mV] B [ T ] T = 2 K D torque [mV] B [ 1 / T ] m m a ) b ) a b sam ple q c ) B c b FIG. 1. a) The magnetic torque signals from a ZrSiS crystalat 2 K, for θ = − ◦ , θ = 0 ◦ , θ = 1 ◦ . The torque signalconsists of a B dependent background with QO contributionssuperimposed. Dashed line shows a + bB fit to the measuredmagnetic torque signal. The inset to FIG. 1 a) shows anenlarged view of isolated HF QO as a function of 1 /B for θ = − ◦ , 0 ◦ and 1 ◦ . b) Image of a typical piezoresistive chip withZrSiS crystal glued on the cantilever. c) A schematic pictureof sample and magnetic field configuration. By rotation ofthe sample around the crystalline a axis the angle θ betweenthe direction of magnetic field and the crystalline c axis wasvaried. sistivity ratio ( RRR = ρ /ρ . ) of around 80.The magnetic torque signal of the ZrSiS single crys-tal was measured in a He cryostat measurement systemwith a single-axis rotator option, using commercially-available piezoresistive cantilevers (SEIKO-PRC120), insteady fields up to 35 T and in the temperature rangefrom 2 to 40 K [45]. The direction of the applied mag-netic field is determined by simultaneously measuring theHall voltage of a Hall probe. The offset between the max-imum of the Hall probe signal and the c axis of the crystalis expected to be smaller than 1 ◦ and is neglected in theanalysis of the data. The magnetic torque signal wasmeasured for various angles θ between the crystalline c axis and the direction of the magnetic field, Figs. 1 (b)and (c). The measurement data were taken during upand down magnetic field sweeps from 0 to 35 T.The magnetic torque signals of a ZrSiS crystal in mag-netic fields up to 35 T at the temperature 2 K and angles θ = − ◦ , 0 ◦ and 1 ◦ , shown in Fig. 1 (a), highlights themain experimental observations: (i) the magnetic torquesignal is a superposition of a B dependent contribu-tion from crystal magnetism and multi-frequency QO, (ii)above the MB threshold field of around 13 T, the torquesignals reveal a series of HF QO, which are strongly sup-pressed by small angle misalignment between the direc-tion of magnetic field and the c crystalline axis [the insetto FIG. 1 a)]. The inset to Fig. 1 (a) shows an enlargedview of isolated HF QO contribution as a function of1 /B for angles θ = − ◦ , 0 ◦ and 1 ◦ . An image of theZrSiS single crystal glued on the end of the piezoresistivelever is shown in Fig. 1 (b), whereas Fig. 1 (c) displays a FF T a m p lit ud e [ a . u . ] F [kT]
ZrSiS, θ = 0 o l o w F A s u r f a ce F B s u r f a ce F . h a r m o n i c o f A s u r f a ce . h a r m o n i c o f B s u r f a ce FIG. 2. The FFT spectrum of QO in the measured magnetictorque of ZrSiS crystal for θ = 0 ◦ at 2 K in the magnetic fieldrange from 0 up to 35 T. The FFT frequency spectrum con-sists of LF contribution (frequencies up to 1 kT) and severalclusters of HF contributions (frequencies from 7-22 kT). Thehigh oscillation frequencies are attributed to different electronMB orbits as shown in Fig. 5. The first harmonic of the MBA + n α (yellow bar) and B + n α (blue bar) orbits are clearlyvisible and highlighted in brown and light blue, respectively. schematic view of the axis of rotation and the angle θ be-tween the direction of magnetic field and the c crystallineaxis. The magnetic torque signal of the crystal was alsomeasured at temperatures of 4.2, 10, 20 and 40 K. Differ-ent frequency contributions in the oscillatory part of themeasured magnetic torque data have been distinguishedby fast Fourier transform (FFT) analysis.The FFT spectrum of the measured magnetic torquesignal of the ZrSiS crystal at 2K for θ = 0 ◦ and the mag-netic field range from 0 to 35 T is shown in Fig. 2.From the peaks in the FFT spectrum we identify theQO frequencies F k , which are related to the extremalcross-sectional areas A k of the FS and the plane normalto the magnetic field direction via the Onsager relation F k = φ π A k with φ being flux quantum. As can be seenin Fig. 2, the FFT spectrum consists of the LF contri-bution (frequencies up to 1 kT) highlighted in green andtwo clusters (A and B) of high frequencies and their har-monics highlighted in different colors (frequencies from7-22 kT).The LF contributions to QO in the magnetic torqueof the same ZrSiS crystal at 2 K for angles θ = 0 ◦ and θ = 1 ◦ obtained from the measured magnetic torque datain different magnetic field ranges are shown in Figs 3 (a)and (b), respectively. As can be seen from the LF FFTspectrum for the magnetic field range from 0 to 7 T, onlyone peak at 241(4) T is observed, which corresponds tothe α hole pocket located at the vertex of the diamond-shaped FS in the top Z-R-A plane in the FBZ of ZrSiS(Fig. 5). For wider magnetic field ranges, the appearanceof the new peaks in the LF FFT spectrum is strongly dependent on magnetic field angle tilted from the c axisof the crystal, Fig. 3. Analyzing the FFT peaks for themagnetic field in the range from 0-35 T one can noticethat the FFT spectra consist of peaks at 241(4) T ( α ),591(15) T ( β ), 415(8) T ( γ ), 286(8) T ( η , for θ = 0 ◦ ) andpeaks separated from these peaks by a multiple of α , seeFigs. 3 (a) and (b).In the FFT spectra below 100 T we can clearly seefrequencies at 8 and 22 T, which are already observedin several studies [30, 31, 34–42, 46] and attributed tothe different parts of ZrSiS FS. These frequencies are notrelated to the MB effects discussed in this work and sofor clarity the origin of the x axis is set at 50 T in theFig 3. The amplitude of the most pronounced peak at8 T is independent of the magnetic field range at whichthe FFT is performed and of angle θ (for θ = − , , ◦ ).As such it is taken as a reference for normalization of theFFT spectra.The HF part of the FTT spectra of QO data in mag-netic torque signal of the ZrSiS crystal, for various tem-peratures listed in the main panel, is shown in Fig. 4 a).Figure 4 (a) is obtained by performing the FFT of anisolated HF contribution to QO shown in Fig. 4 (b) (QOwere isolated from the raw signal by subtracting the meanvalue of envelope curves of maxima and minima of HFQO). The HF FFT spectra consists of two main clustersof equidistant peaks separated by the α pocket frequencyof 241(4) T. These are labeled as A and B MB orbits inFIG. 4 a), their harmonics are shown in the inset to Fig.4 (a). Figure 4 (c) shows the HF FFT spectrum for angles θ = − ◦ , 0 ◦ , 1 ◦ and reveals much stronger suppression ofthe B + n α cluster by tilting the magnetic field directionwith respect to the c crystal axis than for the A + n α cluster. Our results confirm the results obtained in [41]but with an enhanced resolution so that the FS structurecan be studied in more detail. III. DISCUSSION
The FS of ZrSiS calculated using density functionaltheory (DFT) is presented in many papers [41, 42, 46].It is an open FS in the k z -direction (due to the quasi two-dimensional crystal structure of ZrSiS) built from severalelectron and hole pockets. States from the extremal crosssections of the FS and plane normal to the direction ofmagnetic field are responsible for QO. It turns out thatthe most relevant cross section for explanation of exper-imental results at θ close to 0 ◦ is the diamond-shapedFS in the k z = π (Z-R-A) plane of the FBZ, see Fig. 5.The Z-R-A plane of the FBZ consists of four electron ( β )pockets and four hole ( α ) pockets separated by a smallgap (10-20 meV) due to the small SOC in ZrSiS [41].Theoretically predicted value of α and β pocket frequen-cies in ZrSiS are α = 235 T and β = 596 T [41]. Thereare many experimental confirmations of the α frequencyin ZrSiS [36–42] while the β frequency is, so far, seen onlyin few cases [36, 41] at magnetic fields higher than 10 T. d - ad - 2 a d FFT amplitude [a.u.]
F [ T ] 0 - 7 T 0 - 1 3 T 0 - 2 0 T 0 - 3 5 T a bg q = 1 o aa a a d - 2 a d - a FFT amplitude [a.u.]
F [ T ] 0 - 7 T 0 - 1 3 T 0 - 2 0 T 0 - 3 5 T a bgh q = 0 o aaa aa d a )b ) FIG. 3. The LF FFT spectrum of QO in the magnetic torquesignal of ZrSiS crystal at 2 K, obtained for the different mag-netic field ranges and angles (a) θ = − ◦ and (b) θ = 0 ◦ . Fre-quencies α = 241(4) T and β = 591(15) T are contributionsfrom α and β pockets from the FS in the Z-R-A plane of theFBZ. Prominent high field peaks with frequencies γ = 415(8)T, η = 286(8) T and 180(5) T and other smaller peaks couldbe ascribed to various MB orbits, as explained in Sec. III B.It is interesting to notice that at high fields, a small change inangle θ leads to a strong change in FFT spectrum and severalof the peaks are α apart, giving a strong indication of the MBeffect. A. Magnetic breakdown
Most of the experimental results presented here canbe explained by the effect of MB. MB is an importantquantum complement to the semiclassical Lifshitz andOnsager theory of metals in which tunneling of chargecarriers between separated Fermi pockets in k -space istaken into account [47]. In the MB regime, charge car-riers tunneling through a momentum gap in k -space be-tween adjacent pockets leads to the observation of ex-tremal MB orbits consisting of combinations of closedextremal orbits whose effective area can be much largeror smaller than the area of the individual Fermi pock-ets in the given plane of the FBZ. In the case of MBtunneling, an electron moves classically along the FS ex- cept in the close vicinity of the MB gap which acts likea two-channel scattering center. Thus, an initial wave ofunitary amplitude entering a MB gap is separated into atransmitted wave with amplitude A t = iP / and a re-flected wave with amplitude A r = (1 − P ) / , where thetunneling probability, P , depends on the MB field, B MB ,according to P = e − B MB /B . The probability of an elec-tron tunneling through the MB k -space gap ( k g ) dependson the magnetic field B , k g and also on the local band-dispersion E ( k ). An estimate of the B MB can be madeusing the formula B MB = π (cid:126) e (cid:16) k a + b (cid:17) / , where a and b are the k -space radii of curvatures of the orbits on eachside of the gap [48]. Calculations show that variations inlocal FS curvature can easily change B MB by a factor of10 [49]. For multiple closed orbits all possible closed or-bits are additive and the contribution from each orbit ismultiplied by the MB reduction factor R MB = CA l t t A l r r ,where l t represents the number of MB points the orbittraverses by transmission, l r represents the number ofMB points the orbit traverses by reflection, and C is theweighting factor which depends on the symmetry of theorbit and represents the number of possible realizations ofthe given effective MB orbit. According to the directionof circulation of the electron momentum around individ-ual orbits in momentum space, which build an effectiveMB orbit, the resulting QO frequency associated withthe MB orbit will be the difference or sum of frequenciesassociated with individual orbits (different direction ofcirculation leads to the difference of frequencies and viceversa), Fig. 5 (b). TABLE I. Table of experimentally measured QO frequenciesshown in the FIG. 3Orbit F exp [T] Orbit F exp [T] α β + α β β − α γ ≈ β − α β − α η ≈ β γ − α δ ≈ η η + α δ − α δ − α B. LF spectra
In the LF FFT spectra in Figs 3 (a) and (b), for mag-netic fields larger than 20 T, a series of clearly resolvedpeaks are identified. Those are labeled as α , β , γ and η , and peaks that can be clearly matched with values β − α , β − α , β + α , γ − α and η + α . The α and β peaks correspond to magnetic oscillations of individualFermi pockets ( α hole petal pocket and β electron dog-bone pocket in Fig. 5) with frequencies of 241(4) T and591(15) T, respectively. The claim that α and β peakscorrespond to the individual FS orbits in the top Z-R-A B s u r f a c e q = 1 o q = 0 o FFT amplitude [a.u.]
F [ k T ] q = - 1 o A s u r f a c e
B + 4 a B + 3 a B + a A + 4 a A + 3 a A A + a FFT amplitude [a.u.]
F [ k T ]
2 K 4 . 2 K 1 0 K
A + 2 a B + 2 a b q = 0 o aaa FFT amplitude [a.u.]
F [ k T ] a A h a r m o n i c B h a r m o n i c b D torque [mV] B [ 1 / T ] 2 K 4 . 2 K 1 0 K q = 0 o c ) b )a ) FIG. 4. a) The FFT of HF contribution to QO in the magnetic torque signal of ZrSiS crystal measured at the differenttemperatures indicated in the main panel. The HF FFT spectrum consists of two main clusters of equidistantly separatedpeaks by the α hole pocket (A and B clusters, see Fig. 5) and their harmonics [see the inset to Fig. 4 (a)]. The calculated FFTspectra is explained by the effect of MB. According to the picture of MB orbits on the FS, A and B clusters of peaks and theirharmonics should be separated by 4 β and 8 β respectively. Panel (b) highlights an enlarged view of the HF QO in the magnetictorque signal of ZrSiS for the temperatures of 2, 4.2 and 10 K as a function of 1/B. c) FFT of HF contribution to QO in themagnetic torque signal of ZrSiS for angles θ = − ◦ , 0 ◦ and 1 ◦ . Figure illustrates strong angle dependence of the MB orbits. plane is discussed in more detail later when discussingthe HF FFT spectra. One can notice that γ and η peaksat 415(8) T and 286(8) T are in an excellent numericalagreement with putative relations γ ≈ ( α + β ) / η ≈ β/
2. The peak at 180(5) T could be seen as γ − α within the error bar. The peak δ with frequency near 600T, which makes the β peak broaden, can be attributedto the orbit 2 η . Associated peaks δ − α and δ − α arealso visible, Fig. 3. This suggests that most of the peaks,except α and β , can be associated with MB orbits. Onecan notice that, for the magnetic field range from 0 T to35 T, there is a strong angle dependence of the α , η and γ pockets as well as γ − α , β − α and β + α pockets forthe angle close to B parallel to the crystalline c axis, Figs3 (a) and (b). Note that at θ = 0 ◦ the signal from γ − α is large while the signals from α and γ are suppressed,whereas the opposite is true for θ = 1 ◦ . This is to beexpected for orbits which are strongly affected by MBbecause there will be a higher proportion of MB orbitswhen the FFT is performed over a wider range of fields.At low magnetic fields, i.e., for fields less than the MBfield, B < B MB , only α hole and β electron pocket con-tribute to QO. For B > B MB ( B MB may vary for differ-ent orbits) the MB-driven closed orbits appear at β − α , β − α , β + α , γ , γ − α , η + α . These are connectedby α orbits so that Landau bands may develop. In thecase of ZrSiS both α and β orbits contribute to the QOand the FFT spectrum of these oscillations becomes morecomplicated because the frequencies of α and β are in- commensurate, in general. Therefore, the frequency withwhich the β Landau band cross the Fermi level (as a func-tion of 1 /B ) may be less or more than the β frequencyplus and minus integer multiples of the α frequency. Theabove picture of QO differs from that in the standardLK approach [50, 51]. The principal difference is thatin LK theory an external magnetic field does not changethe energy spectrum of electrons and only determinesthe cross section of the FS. On the other hand, in theMB case the energy spectrum itself becomes a complexquasiperiodic function of the inverse magnetic field andstrongly influences the QO spectrum [51]. The QO fre-quency spectrum therefore becomes much more complexas MB gives rise to combined extremal orbits consistingof individual extremal orbits of the FS.Our further assumption is that there is an additionalintraband tunneling in the β pocket at the neck of thedog-bone shaped β pocket, Fig. 5 (c). In this case anelectron can orbit around nearly half of the β pocket,leading to a broader η ≈ β/ η should also be present. Thepeak associated with MB orbit δ ≈ η makes the β peakbroaden, which can be seen in Figs 3 (a) and (b). Onepossible combined MB orbit, in which an electron tunnelsonce through the neck of the β pocket, shown in Fig. 5(c), leads to the oscillation frequency β − α . Because β ≈ . α in ZrSiS this frequency amounts ≈ ( α + β ) / γ peak in the LF FFT spectra ofZrSiS. The peak observed at 348(7) T matches the dif-ference in frequencies of β − α demonstrating a ”figure ofeight” orbit, shown in Fig. 5 (b), that results from tun-neling between adjacent β electron and α hole pocketsin the Z-R-A plane in the FBZ of ZrSiS. Such a ”fig-ure of eight” orbit in the member of the same family ofmaterials HfSiS was reported for the first time in [43] asa manifestation of Klein tunneling in momentum space[49].All QO frequencies in the LF part of the FFT spectrashown in Fig. 3 are summarized in Table I.O’Brien et al . [49] showed that in the case of a type-IIWeyl semimetal, the amplitude of the breakdown tun-neling between the electron and hole Fermi pockets has avery strong dependence on the angle of the magnetic fieldwith respect to to the axis of the Weyl cones. Thus, astrong variation of the peak amplitudes in LF FFT within1 ◦ around θ = 0 ◦ , seen in Fig. 3, additionally confirmsour conclusion that most of the peaks in LF FFT comefrom MB orbits.Further, it is well known that the intensity of eachfrequency peak in the FFT depends on the strength ofdisorder in the crystals, which suppresses the amplitudeof oscillations. Thus, it depends also on the perimeterof the closed electron orbit. The peak broadening fromdisorder makes the β = 591(15) T pocket disappear inmagnetic fields lower than 13 T as shown in the Figs. 3(a) and (b). This is because larger broadening makesfaster oscillations (larger frequency) less visible. Thedisorder strength also affects the intensity of the otherfrequencies.In principle the mixing frequencies β ± α , γ ± α andtheir harmonics can also be produced by the effect oftorque interaction [52], where the response of electronsto the induced oscillating magnetic moment is also takeninto account. In large magnetic field the torque inter-action can become so strong causing a distortion of themagnetization signal which leads to an enhancement ofmain frequency harmonics amplitude in FFT spectra.The effect of magnetic interaction increases with angle θ . In our measured data no distortion of oscillating sig-nal is observed so this scenario has been ruled out in ourexperiment. C. HF spectra
Next we will focus on the HF part of the FTT spectraof QO data in the magnetic torque signal of ZrSiS crys-tal which is shown in Fig. 4 (a) for various temperatureslisted in the main panel. There are no individual orbits inthe DFT-derived FS with areas that match the observedhigh frequencies in FFT spectra. Thus, the peaks of theA + n α and B + n α clusters in the HF FFT spectra cor-respond to MB orbits that encircle the entire diamond-shaped FS in the Z-R-A plane in the FBZ of ZrSiS, seeFig. 5. The A + n α and B + n α groups of peaks in the HF spectrum correspond to orbits in which an electrontraverses β pocket over the inner (A surface) or outer (Bsurface) edge. HF MB orbits can additionally include 0to n α pockets on the path of an electron. This explainsthe equidistantly separated peaks by the α hole frequencyof 241(4) T inside A + n α and B + n α clusters. Accordingto this picture of MB electron orbits in momentum space,A + n α and B + n α clusters of peaks should be separatedby 4 β pockets, which is true in our case [for β = 591(15)T], see Fig. 4 (a). As can be seen in the inset to Fig. 4(a), the first harmonics of the A + n α and B + n α clustersare clearly visible in the HF FFT spectrum. Further, ac-cording to this MB picture, harmonics of the two clustersshould be separated by 8 β pockets, which is again truein our case [for β = 591(15) T], see inset to Fig. 4 (a).As well as in the A + n α cluster, individual peaks in thefirst harmonic of A + n α cluster are also separated by α ,not 2 α as one might naively expect. The first harmonicof the main frequency in QO comes from electrons whichtraverse the closed orbit in momentum space twice be-fore they scatter. By performing a double orbit aroundthe entire diamond shaped FS in the k z = π plane of theFBZ, an electron can again pick up from 0 to n additional α pockets. FIG. 5. Schematic representation of the FS of ZrSiS in theZ-R-A plane of the FBZ, with MB orbits highlighted with redlines [41]. FS consist of four β electron (light blue dog-boneshaped pocket) and four α hole pockets (dark blue pocket)in diamond shape configuration separated by a small gap dueto the small SOC in ZrSiS. (a) A red line indicates HF MBorbits where an electron encircles the entire FS in the Z-R-Aplane. An electron can traverse the β pocket over inner (Asurface) or outer edge (B surface) and pick up between 0 to n α pockets. This explains the equidistantly separated HF FFTpeaks by the α frequency inside A + n α and B + n α clusters.(b) The ”figure of eight” MB orbit connecting one electronand one hole pocket resulting in β − α frequency of QO. Itcan be seen as an example of Klein tunneling in momentumspace. (c) Example of MB orbit with tunneling through theneck of the dog-bone shaped β pocket. The MB orbit shownexplains the observed QO frequency γ = 415(8) T which canrepresented as a trajectory of the form β − α . HF QO begin to appear at nearly 13 T. Therefore,above 13 T electrons gain enough energy to tunnelthrough at least eight gaps between α and β pockets.One can assume that at this field the amplitudes of HFQO (amplitudes of HF FFT peaks) are dominantly de-termined by the weighting factor C (introduced in Sec.III A on MB). MB orbit A + 2 α (B + n α ) has the highestweighting factor, C = 6 (2 of the 4 α pockets can beselected in (cid:0) (cid:1) = 6 ways), so the peak with the highestamplitude in the A + n α (B + n α ) cluster of HF FFT isassigned to the A + 2 α (B + 2 α ) MB orbit.HF MB orbits have a strong angle dependence, whichcan be seen in Fig. 4 (c). Amplitude of B + n α clus-ter of orbits is strongly affected by the change of angle θ suggesting that for some angles θ electrons have a pre-ferred way of tunneling between α and β pockets (goingover inner or outer edge of the β pocket). This effect canprobably be associated with strong variation of peaksamplitude in LF FFT spectra within θ = 0 − ◦ seenin Figs. 3 (a) and (b). For some more quantitative con-clusion about the angle dependence (within θ = 0 − ◦ )of energy gap between α and β pocket a more detailedexperiment is needed (planned for the near future). IV. CONCLUSION
It has been shown that the measurement of QO in themagnetic torque, by the piezoresistive cantilever method,provide a very precise probe for the FS of the top Z-R-A ( k z = π ) plane of the FBZ of the NLDS ZrSiS. TheFS topology deduced from the magnetic torque measure-ments is in very good agreement with DFT calculations,consisting of four α hole and four β electron individualorbits which give rise to the diamond-shaped FS in thetop Z-R-A plane of the FBZ.We showed that A and B clusters of peaks in the HFFFT part correspond to MB orbits where electrons, tun-neling through at least eight gaps, encircle the entirediamond-shaped FS in the Z-R-A plane in the FBZ ofZrSiS. The more sensitive torque data presented here al-low us to see additional structures in the HF part of FFT around 17 and 21 kT arising from the first harmonic ofA and B clusters, which to the best of our knowledge isanother new result. Comparing the picture of MB orbits(Fig. 5) with measured A and B clusters of peaks andtheir harmonics, it is confirmed that the QO frequencyfrom the β electron pocket in FS of ZrSiS is β = 591(15)T, not 420 T as suggested in [42].FFT peaks corresponding to the individual α hole[ α = 241(4) T] and β electron [ β = 591(15) T] pocketsare both visible also in the LF part of FFT, Fig. 3. Dueto the dog-bone shape of the β electron pocket a reason-able assumption is that there will be intraband tunnelingthrough the neck of the β pocket. We showed that thisadditional tunneling could well account for the observedprominent LF peaks γ and η with a tunneling path givenin Fig. 5. Also all other peaks in the LF part of FFT(which arise at fields above 20 T) can be interpreted ascombined MB orbits that include one or more α pocketsbecause every peak has an associated peak separated by α = 241(4) T pocket, Fig. 3.The impact of changes in the Fermi energy and inter-layer interaction on the FS of ZrSiS (and other mem-bers of the same materials family) is still being inten-sively studied and the FFT spectra of QO in ZrSiS isstill not completely resolved [33, 35, 41–43]. In this workwe present one possible scenario of electron dynamics inthe MB regime in ZrSiS which is supported by all ourmeasurement results. Acknowledgments
This work was supported by Croatian Science Foun-dation under the project IP 2018 01 8912 and CeNIKSproject cofinanced by the Croatian Government and theEU through the European Regional Development Fund- Competitiveness and Cohesion Operational Program(Grant No. KK.01.1.1.02.0013). Measurements were per-formed at High Field Magnet Laboratory (HFML) in Ni-jmegen. We acknowledge T. Klaser for XRD measure-ments. We thank J. R. Cooper for useful discussions. [1] Z. K. Liu, B. Zhou, Y. Zhang, Z. J. Wang, H. M. Weng,D. Prabhakaran, S. K. Mo, Z. X. Shen, Z. Fang, X. Dai,Z. Hussain, and Y. L. Chen, Science (80- ) 343, 864(2014).[2] T. Liang, Q. Gibson, M. N. Ali, M. Liu, R. J. Cava, andN. P. Ong, Nat Mater 14, 280 (2015).[3] S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G.Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C.Lee, S.-M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B.Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia,and M. Z. Hasan, Science (80- ) 349, 613 (2015).[4] S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee, G.Chang, B. Wang, N. Alidoust, G. Bian, M. Neupane,C. Zhang, S. Jia, A. Bansil, H. Lin, and M. Z. Hasan, Nat Commun 6, 7373 (2015).[5] S.-Y. Xu, N. Alidoust, I. Belopolski, Z. Yuan, G. Bian,T.-R. Chang, H. Zheng, V. N. Strocov, D. S. Sanchez,G. Chang, C. Zhang, D. Mou, Y. Wu, L. Huang, C.-C. Lee, S.-M. Huang, B. Wang, A. Bansil, H.-T. Jeng,T. Neupert, A. Kaminski, H. Lin, S. Jia, and M. ZahidHasan, Nat Phys 11, 748 (2015).[6] L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos,and M. Soljacic, Science (80- ) 349, 622 (2015).[7] A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer,X. Dai, and B. A. Bernevig, Nature 527, 495 (2015).[8] S.-Y. Xu, I. Belopolski, D. S. Sanchez, C. Zhang, G.Chang, C. Guo, G. Bian, Z. Yuan, H. Lu, T.-R. Chang, P.P. Shibayev, M. L. Prokopovych, N. Alidoust, H. Zheng,