Coexistence of Superconductivity and Charge Density Wave in Tantalum Disulfide: Experiment and Theory
Y. Kvashnin, D. VanGennep, M. Mito, S. A. Medvedev, R. Thiyagarajan, O. Karis, A. N. Vasiliev, O. Eriksson, M. Abdel-Hafiez
CCoexistence of Superconductivity and Charge Density Wave in Tantalum Disulfide:Experiment and Theory
Y. Kvashnin, D. VanGennep, M. Mito, S. A. Medvedev, R. Thiyagarajan, O. Karis, A. N. Vasiliev,
6, 7
O. Eriksson,
1, 8 and M. Abdel-Hafiez
1, 2 Uppsala University, Department of Physics and Astronomy, Box 516, SE-751 20 Uppsala, Sweden Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Graduate School of Engineering, Kyushu Institute of Technology, Fukuoka 804-8550, Japan Max Planck Institute for Chemical Physics of Solids, D-01187 Dresden, Germany Institut f¨ur Festk¨orper- und Materialphysik, Technische Universit¨at Dresden, 01069 Dresden, Germany Ural Federal University, Yekaterinburg 620002, Russia Lomonosov Moscow State University, Moscow 119991, Russia School of Science and Technology, ¨Orebro University, SE-701 82 ¨Orebro, Sweden (Dated: July 30, 2020)The coexistence of charge density wave (CDW) and superconductivity in tantalum disulfide (2H-TaS ) at ambient pressure, is boosted by applying hydrostatic pressures up to 30 GPa, therebyinducing a typical dome-shaped superconducting phase. The ambient pressure CDW ground statewhich begins at T CDW ∼
76 K, with critically small Fermi surfaces, was found to be fully suppressedat P c ∼ P c , we observe a superconducting dome with a maximum superconductingtransition temperature T c = 9.1 K. First-principles calculations of the electronic structure predictthat, under ambient conditions, the undistorted structure is characterized by a phonon instabilityat finite momentum close to the experimental CDW wave vector. Upon compression, this instabilityis found to disappear, indicating the suppression of CDW order. The calculations reveal an elec-tronic topological transition (ETT), which occurs before the suppression of the phonon instability,suggesting that the ETT alone is not directly causing the structural change in the system. The tem-perature dependence of the first vortex penetration field has been experimentally obtained by twoindependent methods and the corresponding lower critical field H c was deduced. While a d waveand single-gap BCS prediction cannot describe our H c experiments, the temperature dependenceof the H c can be well described by a single-gap anisotropic s -wave order parameter. PACS numbers: 71.45.Lr, 11.30.Rd, 64.60.Ej
Coexistence of superconductivity with competingphysical phenomena such as magnetic or charge orderhas been of interest for the condensed matter commu-nity for a long time[1–3]. A commonly accepted argu-ment says that for the materials exhibiting competingground states, suppressing the magnetic or charge orderhelps to stabilize the superconducting (SC) phase. Thisis the case, for instance, in layered materials that arecomposed of two-dimensional (2D) building blocks, withperiodic modulations of the charge carrier density, so-called charge density waves (CDWs)[4–6]. Classic exam-ples are the members of the transition-metal dichalco-genide family (TMDs) MX , where M = Nb, Ti, Ta,Mo and X = S, Se. TMDs provide an ideal playgroundfor studying semiconductors, metals, and superconduc-tors in 2D using the same structural template[7–9]. Forall known quasi-2D superconductors[10], the origin andexact boundary of the electronic orderings and super-conductivity are still attractive problems. At ambientpressure and without intercalation or chemical substitu-tion, 2H-TaS , a prominent member of the vast familyof TMDs, exhibits both superconductivity and a canon-ical CDW phase transition whose mechanisms remaincontroversial, even after decades of research[11–13]. De-spite extensive studies, the current understanding of the microscopic origin of the SC mechanism and the CDWstate is not complete. The SC transition temperature( T c ) increases while the CDW lock-in temperature fallsdown with chemical doping[13], increasing thickness ofthe sample[14] and external pressure[15–17]. Several the-oretical mechanisms behind the formation of CDW havebeen proposed[18]. For TMDs, the following origins wereextensively discussed: Fermi surface nesting[19], sad-dle points near Fermi surface[20], exciton-phonon[21] orelectron-phonon coupling[22–26]. The most recent exper-imental evidence suggests that the latter plays a decisiverole for CDW stabilization in Ta systems[27, 28]. It isthus of profound importance to understand the interplaybetween electronic and crystal structure in 2H-TaS . Ad-ditionally, there is no general consensus on the origin ofSC pairing mechanisms in this material and further stud-ies are necessary to elucidate this issue.Whatever the proposed understanding of the relationbetween CDW and superconductivity is, it is importantto determine the exact dependence of T c and the CDWphase with pressure[11]. Within this scope, through acombined complementary experimental techniques sup-plemented with theoretical calculations on 2H-TaS , wederive a previously not discussed pressure-temperaturephase diagram. We explore external pressure as a tool a r X i v : . [ c ond - m a t . s up r- c on ] J u l Intensity (a.u.)
P h - P h E A ( a ) ( b )( c ) ( d ) R a m a n s h i f t ( c m - 1 ) P ( G P a ) 4 0 . 7 3 8 . 9 3 7 . 4 3 2 . 7 2 1 . 6 2 0 . 2 1 8 . 3 1 6 . 3 1 5 . 1 1 3 . 7 1 1 . 1 1 0 . 1 8 . 4 6 . 5 5 . 1 3 . 8 3 . 0 1 . 9 1 . 3 0 . 7 0 . 3
Intensity (a.u.)
R a m a n s h i f t ( c m - 1 ) T a S 2 M o S 2 M o T e 2 M o S e 2 R e S 2
T M D m a t e r i a l s
Raman shift (cm-1)
P ( G P a ) A E P h - P h dAg/dP (cm-1/GPa)
FIG. 1: (a) Raman scattering spectra of 2H-TaS under room temperature and ambient pressure. (b) Raman of spectra undervarious hydrostatic pressures up to 40 GPa. (c) Pressure dependence of various vibrational modes in 2H-TaS . Ph-Ph refers totwo-phonon mode. (d) Comparison of pressure coefficient (dA1g/dP) of the out-of-plane (A1g) Raman peaks of TaS , MoS ,MoSe , MoTe and ReS [29]. One can see that the out-of-plane mode of TaS presents the highest pressure coefficient amongall selected materials. to tune the phonon dispersions and thus the stability ofthe CDW phase. Pressure has long been recognized asa fundamental thermodynamic variable, and it is con-sidered a very clean way to tune basic electronic andstructural properties without changing the stoichiometryof a material[30, 31]. Our analysis shows that the tem-perature dependence of the lower critical fields, H c ( T ),is inconsistent with a simple isotropic s -wave type of theorder parameter but are rather in favor of the presence ofan anisotropic s -wave. These observations clearly showthat the SC energy gap in 2H-TaS is nodeless.Details about the high-pressure measurements, crys-tal structure, and first-principles calculations can befound in the Supplemental Material[32]. Raman re-sponse of 2H-TaS at ambient pressure and room tem-perature is presented in Fig. 1(a), where three regularphonons are observed: (i) a second-order peak due to two-phonon process at 180.3 cm − (ii) E g - an in-planevibrational mode at 288.1 cm − and A g -out-of-planemode at 405.4 cm − , and these values are well agreedwith the reported works[33, 34]. Figure 1(b) shows theRaman spectra of 2H-TaS under hydrostatic pressureup to 40 GPa. The diamond background of each datapoint was subtracted by baseline fittings. By applica-tion of pressure, all Raman modes loose their intensities,get wider and show blue shift. Moreover, we could ob-serve a splitting of the E g peak which may be due tothe pressure-induced structural phase transition in 2H-TaS . This mode is known to experience the discontinu-ity as a function of temperature, as one crosses T CDW .The positions of all the three peaks against pressure areshown in Fig. 1(c). Raman mode at 180.3 which rep-resents two-phonon evolves till 15.1 GPa, whereas otherpeaks exist till final pressure of 40 GPa. For the case E n e r g y ( c m ) =406 K =406 K =406 K M K 0200400 E n e r g y ( c m ) =52 K M K =52 K M K =52 K (a)(b) P=0 GPa P=5 GPa P=10 GPa K Γ M FIG. 2: Calculated Fermi surfaces (panel ”a”) and phonon dispersions (panel ”b”) for three different values of external pressure.The phonon dispersions were calculated for two different electronic temperatures, defined by the smearing parameter σ . TheFermi surface was plotted using XCrySDen software (see text for more details). of A g mode, it shows blue shift with deformation co-efficient of 2.83 cm − /GPa. Pressure coefficient of A g modes of similar TMDs are compared in a bar diagram[Fig. 1(d)], and it can be clearly seen that 2H-TaS is verysensitive to the hydrostatic pressure compared to otherTMDs. As ReS is vibrationally decoupled, its Ramanspectrum is less sensitive to pressure. On the other hand,as A g mode is vibrationally coupled with E g mode inthe case of TaS , pressure coefficient of A g mode of 2H-TaS is higher than in any other TMD materials. As thepressure increases, pressure coefficient of A g is reducedabove 20 GPa at which structural transition may be ex-pected. For instance, pressure would significantly reduceinterlayer distance, so adjacent layers will be coupled andoverlap of electron wavefunctions is stronger in TaS thanin MoS and band structure transformation may happenabove 20 GPa.In order to get a physical insight into the suppres-sion of the CDW phase under pressure, we have investi-gated the electronic structure by means of ab initio the-ory. Recently, first-principles calculations show that theelectron-phonon interactions depend on both the amountof applied strain and the direction in 2H-TaSe [36]. Inaddition, a sudden change in E g mode in 2H-TaSe isobserved[28]. The main results are shown in Fig. 2. Ac- cording to the results obtained for an undistorted 2H-TaS , the material undergoes a pressure-induced elec-tronic topological (so-called Lifshitz[35]) transition. Anadditional hole pocket around the Γ point emerges, asshown in blue on Fig. 2(a). This transition happens be-low 2.5 GPa, and upon further compression, at least upto 15 GPa, the Fermi surface topology is intact, while itsshape becomes slightly modified. The pressure evolutionof the calculated phonon spectra is shown in Fig. 2(b).At the equilibrium, and under small applied pressures,there is a phonon instability along the Γ − M directionat the wavevector close to experimental q CDW . Uponcompression, the instability is suppressed somewhere be-tween 5 and 10 GPa, indicating the suppression of theCDW order.Interestingly, the instability disappears after the ETT,which indicates that the Fermi surface nesting itself is notthe only driving force of CDW order, which is in line withother, more recent studies on 2H-TaS [27]. The resultsof our calculations for ambient pressure are in agreementwith Ref.[34]. These types of calculations are not able toproperly capture the CDW transition temperature, butshow a correct qualitative behaviour. As the electronictemperature increases, the phonon instability becomesless pronounced, but persists up the temperatures well
02 04 06 08 0
15 3.8 7.5 GPa23.5 r u n 1 ( a )( b ) ( c ) r u n 2 r u n 3 ( d )
M/H (arb. units)
M/H (arb. units) T ( K ) T ( K ) T (K) P ( G P a ) S CC D W
C D W + S C r u n 1r u n 2r u n 3r u n 4 P ( G P a ) T (K) FIG. 3: The temperature dependence of the DC-susceptibilitycomponents of 2 H -TaS measured in dc field with an ampli-tude of 30 Oe at elevated pressures. (a) M - T curves at pres-sures between 3.8 and 23.5 GPa in run 1. (b) M - H curvesat pressures between 6.2 and 29.5 GPa in run 2. (c) M - HT curves at pressures between 3.5 and 19.5 GPa in run 3.The experimental data of run 4 is presented in Fig. S6[32].The data were collected upon warming in different the dcmagnetic fields after cooling in a zero magnetic field. (d)The obtained pressure-temperature ( P - T ) phase diagram of2 H -TaS . Pressure dependence of the SC transition tem-peratures T c up to 30 GPa. The values of T c were deter-mined from the high-pressure resistivity and DC magneticsusceptibility[11, 32]. The temperature dependence on thedisappearance of the CDW as function of pressure, is shownas stars. above experimental CDW ordering temperature. Theauthors of Ref.[34] attribute this to the presence of ashort-range CDW state. We would like to note, how-ever, that the employed treatment of the temperatureeffects is potentially oversimplified and does not capturemany phenomena. The main reason is an incomplete de-scription of the electronic correlations within DFT. More-over, certain crystal structures are known to be stabilizeddue to anharmonic effects[37]. An explicit account of theelectron-phonon interaction which is expected to be quiteanisotropic[27] might induce strong modifications of bothelectronic and phononic spectra.The effects of 8.7 GPa illustrate a suppression of theCDW state and enhanced the T c with a very sharp dropof the resistivity up to 9.1 K[32]. Figure 3(a-c) showsthe temperature-dependent magnetic M (T) of the 2 H -TaS at pressures from 0 to 30 GPa in three runs. Thedome-like evolution of T c was constructed based on theobserved pressure-dependent magnetization data shownin Fig. 3(d), which explicitly shows the gradual suppres-sion of the CDW phase. As displayed in Fig. 3(a-c), itis clear that T c increased up to a pressure of 8.5 GPa,where it exhibits a maximum, then immediately begins - 1 0 1- 4- 2024 0 , 5 1 , 0 1 , 5 H c 1 T ( K ) pc ' T = 3 5 0 m K
0 O e4 0 08 0 01 5 0 0 ( a ) M (10-2 emu) H ( O e ) T ( K )( b ) ( c ) H c1 (Oe) k H z H ( O e ) M (103 emu/mol) H ( k O e ) T ( K ) pc† (arb. unit) ( d ) B C S
E x p . d a t a f r o m ( b )E x p . d a t a f r o m ( c ) d - w a v e Anisotropc s -wave (cid:214) Mt ( (cid:214) emu) H c 1 H ( O e ) ‰ R ‰ T = 6 0 m K H c 1 FIG. 4: (a) The temperature dependence of the complex AC-susceptibility components 4 πχ (cid:48) v measured in an AC field withan amplitude of 5 Oe and a frequency of 1 kHz. Data werecollected upon warming in different DC magnetic fields aftercooling in a zero magnetic field. The insets illustrate theisothermal magnetization M vs. H loops measured at 350 Kup to 1000 Oe applied along the c -axis and the imaginary partof AC at various frequency ( ν m ). (b) The SC initial part of themagnetization curves measured at various temperatures downto 60 mK. The inset depicts an example used to determine the H c value using the regression factor R , at T = 60 mK[32]. (c)The field dependence of √ M t at various temperatures. Thearrows indicate H c values that estimated by extrapolatingthe linear fit of √ M t to 0. (d) Phase diagram of H c for thefield applied parallel to the c axis. H c has been estimatedby two different methods from the extrapolation of √ M t to0 (open symbols) and from detecting the transition from aMeissner-like linear (closed symbols). The solid red line is thefitting curves using anisotropic s -wave approach. The dottedand dashed lines represent the d -wave and a single-gap BCSapproach, respectively. to turn down. This kind of dome-shaped curve is one ofthe hallmarks of high temperature superconductors, butmany mysteries around these types of domes remain tobe explained[38]. Deriving a solid picture of the originof the SC dome constitutes a major challenge. Apply-ing external pressure to the system simply modifies theinteratomic spacing, wavefunction overlap and electronicstructure, as well as the balance between kinetic energyand Coulomb interaction among the electrons. The SCstate certainly depends on these parameters, which is de-termined by both the pressure and the existence of theCDW state. Since both pressure and the CDW stateheavily influence T c , the competition between the twomight be the cause of the SC dome.At ambient pressure, 2 H -TaS exhibits a prominentCDW anomaly at 76 K. While superconductivity iswell distinguished by the resistivity and specific heatmeasurements[32], we further confirmed the bulk super-conductivity by performing low-temperature AC suscep-tibility, χ (cid:48) , measurements as illustrated in Fig. 4(a). T c of 1.2 K has been extracted from the bifurcation pointbetween χ (cid:48) v and χ (cid:48)(cid:48) v . One can clearly see that the max-imum of the temperature dependence of the imaginarypart of AC susceptibility, see Fig. 4(a) (right inset), shiftsto higher temperatures upon increasing the frequencywhich we attribute to the motion of vortices. The lowercritical field, H c , i.e. the thermodynamic field at whichthe presence of vortices into the sample becomes energet-ically favorable, is a very useful parameter providing keyinformation regarding bulk thermodynamic properties.Therefore, a reliable determination of the lower criticalfield, H c , from magnetization measurements has beendetermined. The most popular approach of determiningthe H c , is the point of deviation from a linear M(H)response, compared with the values obtained from theonset of the trapped magnetic moment [( M t Fig. 4(c)],(see[32] for more details). We have confirmed the ab-sence of the surface barriers in our case from the verysymmetric DC magnetization hysteresis curves M(H) at350 mK [Fig. 4(a) (left inset)]. The experimental valuesof H c were corrected by accounting for the demagneti-zation effects. Indeed, the deflection of field lines aroundthe sample leads to a more pronounced Meissner slopegiven by M/H a = − / (1 − N ), where N is the demag-netization factor and is found to be ≈ .
97. Taking intoaccount these effects, the absolute value of H c can beestimated by using the relation proposed by Brandt[39].The most intriguing feature in Fig. 4(d) is the upwardtrend with negative curvature over the entire tempera-ture range, similar features are reported in[40, 41].With the above understanding of the nature of su-perconductivity in 2 H -Ta S , We now turn to study itsgap symmetry and structure of the SC order parameter,which can be used to reveal the pairing mechanism. Theobtained experimental temperature dependence of H c shown in Fig. 4(d) was analyzed using the phenomeno-logical α -model. This model generalizes the temperaturedependence of gap to allow α = 2∆(0) /T c > .
53 (i.e. α values higher than the BCS value). The temperaturedependence of each energy gap for this model can be ap-proximated as: [42, 43]∆ i ( T ) = ∆ i (0)tanh[1 . . T ci T − . ], where ∆(0) is the maximum gap value at T =0. We adjust the temperature dependence of H c ,which relates to the normalized superfluid density as˜ ρ s ( T )= H c ( T )/ H c (0), by using the following expression: H c ( T ) H c (0) = 1 + 1 π (cid:90) π (cid:90) ∞ ∆( T,φ ) ∂f∂E EdEdφ (cid:112) E − ∆ ( T, φ ) , (1)where f is the Fermi function [exp( βE + 1)] − , ϕ is theangle along the Fermi surface, β = ( k B T ) − . The energyof the quasiparticles is given by E = [ (cid:15) + ∆ ( t )] . , with (cid:15) being the energy of the normal electrons relative to theFermi level, and where ∆( T, φ ) is the order parameter asfunction of temperature and angle. We used for the s -wave, d -wave the following expressions ∆( T, φ ) = ∆( T )and ∆( T, φ ) = ∆( T ) cos(2 θ ), respectively. The mainfeatures from the corrected H c values in Fig. 4(d) canbe described in the following way: (i) As a first stepwe compare our data to the single band s -wave and wefind a systematic deviation at high temperature data,(ii) More obvious deviations exist in the case of d -waveapproach[43]. This clearly indicates that the gap struc-ture of our system is more likely to be nodeless s -wave,(iii) Then, anisotropic s -wave is further introduced to fitthe experimental data. For the anisotropic s -wave, thefitting with the magnitude of the gap ∆ = 1.21 meVwith an anisotropy parameter ≈ s -wave order parameter presents a good de-scription to the data. We hence conclude that in Ta S the exotic SC gap structure is related to the Ta tubularsheets and that, even if the charge density wave is per-turbing those sheets in TaS , this CDW does not affectthe SC gap structure.The temperature-pressure phase diagram of Ta S isdemonstrated here to have a dome-like SC phase with amaximum SC transition temperature T c = 9.1 K. By em-ploying ab initio electronic structure theory, we were ableto investigate the temperature and pressure dependenceof the phonon spectrum. It is shown that, at ambientconditions, there is a phonon instability at the propaga-tion vector close to the q CDW wavevector. Furthermore,the temperature dependence measurements of the criti-cal field are consistent with single gap anisotropic s -wavesuperconductivity.YOK and MAH acknowledge the financial supportfrom the Swedish Research Council (VR) under theproject No.2019-03569 and 2018-05393. We acknowledgeRajeev Ahuja and Goran Karapetrov for helpful discus-sions. [1] T. Yokoya, T. Kiss, A. Chainani, S. Shin, M. Nohara andH. Takagi, Science , 2518-2520 (2001).[2] M. B. Maple, Appl. Phys. , 179 (1976). [3] Paul C. Canfield, Peter L. Gammel, and David J. Bishop,Physics Today (10), 40 (1998).[4] M. D. Johannes and I. I. Mazin, Phys. Rev. B , 165135(2008).[5] J. J. Hamlin, D. A. Zocco, T. A. Sayles, M. B. Maple, J.H. Chu, and I. R. Fisher, Phys. Rev. Lett. , 177002(2009).[6] D. A. Zocco, J. J. Hamlin, K. Grube, J. H. Chu,H.H.Kuo, I. R. Fisher, and M. B. Maple, Phys. Rev. B , 205114 (2015).[7] B. Sipos, A. F. Kusmartseva, A. Akrap, H. Berger, L.Foro, and E. Tutis, Nat. Mater. , 960 (2008).[8] Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman,and M. S. Strano, Nat. Nanotechnol. , 699 (2012).[9] A. K. Geim and I. V. Grigorieva, Nature (London) ,419 (2013).[10] A. A. Kordyuk, Low Temp. Phys. / Fiz. Nizk. Temp. ,417 (2015).[11] M. Abdel-Hafiez, X.-M. Zhao, A. A. Kordyuk, Y.-W.Fang, B. Pan, Z. He, C.-G. Duan, J. Zhao, and X.-J.Chen, Sci. Rep. , 31824 (2016)[12] D. C. Freitas et al., Phys Rev. B , 184512 (2016).[13] K. E. Wagner, E. Morosan, Y. S. Hor, J. Tao, Y. Zhu,T. Sanders, T. M. McQueen, H. W. Zandbergen, A. J.Williams, D. V. West, and R. J. Cava, Phys. Rev. B ,104520 (2008).[14] Y. Yu, F. Yang, X. F. Lu, Y. J. Yan, Y.-H. Cho, L. Ma,X. Niu, S. Kim, Y.-W. Son, D. Feng, S. Li, S.-W. Cheong,X. H. Chen and Y. Zhang, Nat. Nano. , 270 (2015).[15] C. Berthier, P. Molinie, and D. Jerome, Solid State Com-mun. , 1393 (1976).[16] Z.-H. Chi, X.-M. Zhao, H. Zhang, A. F. Goncharov, S.S. Lobanov, T. Kagayama, M. Sakata, and X.-J. Chen,Phys. Rev. Lett. , 036802 (2014).[17] B. Sipos, A. F. Kusmartseva, A. Akrap, H. Berger, L.Forro, and E. Tutis, Nat. Mat. , 960 (2008).[18] X. Zhu, J. Guo, J. Zhang, and E. W. Plummer, Advancesin Physics: X , 622 (2017).[19] J. A. Wilson, F. J. Di Salvo, and S. Mahajan, Phys. Rev.Lett. , 882 (1974).[20] T. M. Rice and G. K. Scott, Phys. Rev. Lett. , 120(1975).[21] J. van Wezel, P. Nahai-Williamson, and S. S. Saxena,Phys. Rev. B , 165109 (2010).[22] A. H. Castro Neto, Phys. Rev. Lett. , 4382 (2001).[23] F. Weber, S. Rosenkranz, J.-P. Castellan, R. Osborn, R.Hott, R. Heid, K.-P. Bohnen, T. Egami, A. H. Said, and D. Reznik, Phys. Rev. Lett. , 107403 (2011).[24] M. D. Johannes, I. I. Mazin, and C. A. Howells, Phys.Rev. B , 205102 (2006).[25] M. D. Johannes and I. I. Mazin, Phys. Rev. B , 165135(2008).[26] L. P. Gor’kov, Phys. Rev. B , 165142 (2012).[27] K. Wijayaratne, J. Zhao, C. Malliakas, D. Young Chung,M. G. Kanatzidis, and U. Chatterjee, J. Mater. Chem. C , 11310 (2017).[28] H. M. Hill, S. Chowdhury, J. R. Simpson, A. F. Rigosi,D. B. Newell, H. Berger, F. Tavazza, and A. R. HightWalker, Phys. Rev. B , 174110 (2019).[29] S. Tongay, H. Sahin, C. Ko, et al. Nat Commun , 3252(2014).[30] M. Abdel-Hafiez, M Mito, K Shibayama, S Takagi, MIshizuka, AN Vasiliev, C Krellner, H. Mao, Phys. Rev. B , 094504 (2018).[31] M. Abdel-Hafiez, Y. Zhao, Z. Huang, C. Cho, C. Wong,A. Hassen, M. Ohkuma, Y. Fang, B. Pan, Z. Ren, A.Sadakov, A. Usoltsev, V. Pudalov, M. Mito, R. Lortz, CKrellner, W Yang, Phys. Rev. B , 134508 (2018).[32] Supplementary Material is available for the experimentaland calculation details and the supporting results.[33] R. Grasset, Y. Gallais, A. Sacuto, M. Cazayous, S.Manas-Valero, E. Coronado, and M.-A. Measson, Phys.Rev. Lett. , 127001 (2019).[34] J. Joshi, H. M. Hill, S. Chowdhury, C. D. Malliakas, F.Tavazza, U. Chatterjee, A. R. HightWalker, and P.M.Vora, Phys. Rev. B , 245144 (2019).[35] I. M. Lifshitz, Sov. Phys. JETP , 1130 (1960).[36] S. Chowdhury, J. R. Simpson, T. L. Einstein, and A. R.H. Walker, Phys. Rev. Mat. , 084004 (2019).[37] P. Souvatzis, O. Eriksson, M. I. Katsnelson, and S. P.Rudin, Phys. Rev. Lett. , 095901 (2008).[38] J. Lu, O. Zheliuk, Q. Chen, I. Leermakers, N. E.Hussey, U. Zeitler, J. Ye. PNAS 201716781 (2018) DOI:10.1073/pnas.1716781115[39] E. H. Brandt, Phys. Rev. B , 11939 (1999).[40] C. Ren, Z.-S. Wang, H.-Q. Luo, H. Yang, L. Shan, H.-H.Wen, Phys. Rev. Lett. , 103046 (2009).[42] A. Carrington and F. Manzano, Physica C , 205(2003).[43] A. Carrington, and F. Manzano, Physica C385