Spin-resolved electronic response to the phase transition in MoTe 2
Andrew P. Weber, Philipp Rüßmann, Nan Xu, Stefan Muff, Mauro Fanciulli, Arnaud Magrez, Philippe Bugnon, Helmuth Berger, Nicholas C. Plumb, Ming Shi, Stefan Blügel, Phivos Mavropoulos, J. Hugo Dil
SSpin-resolved electronic response to the phase transition in MoTe Andrew P. Weber , , , ∗ Philipp Rüßmann , Nan Xu , , Stefan Muff , , MauroFanciulli , , Arnaud Magrez , Philippe Bugnon , Helmuth Berger , Nicholas C.Plumb , Ming Shi , Stefan Blügel , Phivos Mavropoulos , , and J. Hugo Dil , Institute of Physics, École Polytechnique Fédérale de Lausanne, CH-1015, Lausanne, Switzerland Swiss Light Source, Paul Scherrer Institute, CH-5232 Villigen, Switzerland Donostia International Physics Center, 20018 Donostia, Gipuzkoa, Spain Peter Grünberg Institut and Institute for Advanced Simulation,Forschungszentrum Jülich and JARA, 52425 Jülich, Germany and Department of Physics, National and Kapodistrian University of Athens, 15784 Zografou, Greece (Dated: July 3, 2018)The semimetal MoTe is studied by spin- and angle- resolved photoemission spectroscopy toprobe the detailed electronic structure underlying its broad range of response behavior. A novelspin-texture is uncovered in the bulk Fermi surface of the non-centrosymmetric structural phasethat is consistent with first-principles calculations. The spin-texture is three-dimensional, bothin terms of momentum dependence and spin-orientation, and is not completely suppressed abovethe centrosymmetry-breaking transition temperature. Two types of surface Fermi arc are foundto persist well above the transition temperature. The appearance of a large Fermi arc dependsstrongly on thermal history, and the electron quasiparticle lifetimes are greatly enhanced in theinitial cooling. The results indicate that polar instability with strong electron-lattice interactionsexists near the surface when the bulk is largely in a centrosymmetric phase. PACS numbers: 64.70.K-, 81.30.-t, 68.35.Rh, 71.20.-b, 73.20.-r, 75.70.Tj, 79.60-i
MoTe exhibits a range of phenomena intersecting thephysics of polar lattice transitions, topological phases ofmatter, and novel magnetoelectric properties. The cen-trosymmetric 1T’ crystal undergoes a first order tran-sition into the noncentrosymmetric T d structural phaseupon cooling through T S ≈ K, with volume fractionsof both phases appearing within the - K range[1–5]. Such transitions are very rare in metals and al-low for control over the appearance of Weyl semimetalphases of matter (WSPs) and momentum dependentspin-polarization (spin-texture) that would be desirablefor spintronic applications [6]. Superconductivity pro-posed to be topologically non-trivial has been observed[7–10]. Like to WTe [11, 12], T d -MoTe is a type-II Weylsemimetal candidate [13, 14] and exhibits extreme trans-verse magnetoresistence (XMR) with turn-on behavior[15–17]. Simultaneous tuning of electronic properties andthe structural transition temperature and the breadth ofthe mixed-phase region is realized as a function of doping[3] and pressure/strain [5, 9, 18]. The sizes and shapesof the bulk electron Fermi pockets (EPs) and hole Fermipockets (HPs) are important to the electronic basis forthe properties of (Mo/W)Te [19], but there is growingrecognition that responses of electronic state vectors, de-scribed in terms of their spin and/or orbital projections,play a central role [16, 18, 20–22].The WSP is predicted to be sensitive to the lattice pa-rameters [13, 14, 23] and cannot exist in the centrosym-metric 1T’ crystal, wherein all of the bulk bands mustbe spin-degenerate. However, the electronic structure of1T’-MoTe ( T > T S ) observed in photoemission spec-troscopy appears much the same as that of T d -MoTe ( T << T S ) [24], although the decay of photoexcitedstates is clearly affected (likely due to loss of the WSP)[25]. Different reports on T d -MoTe favor the case of zero(trivial semimetal) [26], four [14, 27–29], or eight [23, 30–34] Weyl points (WPs) in the Brillouin zone (BZ) at lo-cations ranging from approximately 5 [33] to 55 meV [28]above the Fermi energy E F . The WPs impose subtle con-straints on surface Fermi arc dispersions in (Mo/W)Te systems [23, 35, 36], which have been taken as experi-mental signatures of the WSP [23, 24, 27, 29–39]. Twotypes of Fermi arc are present [23, 24, 28]. Small arcs areburied within the HPs and a large arc appears in the gapbetween the HPs and EPs. The large arc persists in 1T’-MoTe [24] and in the absence of WPs in WTe [28, 35],reinforcing the fact that Fermi arcs provide insufficient(although necessary) evidence of a WSP [40]. Quasipar-ticle scattering of the Fermi arcs is strongly affected bythe structural transition [29], however, this scattering oc-curs as a function of spin-texture and nesting conditionsrather than being directly related to the WSP [28].Spin-texture visualization provides a resource for un-derstanding scattering amplitudes, spin-transport [41],MR anisotropy [20], and the pairing order and criticalfield enhancement in superconductivity [42]. Spin- andangle-resolved photoemission spectroscopy (SARPES)was used to probe the spin-texture of T d -(Mo/W)Te in a few instances [24, 30, 34, 43], but only small ar-eas of momentum space were covered without measur-ing the full spin-polarization vector P . Here SARPESmeasurements and density functional theory (DFT) cal-culations reveal a spin-texture in the T d -MoTe Fermisurface that is three dimensional (3D) both in terms of a r X i v : . [ c ond - m a t . m t r l - s c i ] J u l -0.2 0.2-0.10.1 -0.10.1 k x ( Å - ) k z ( Å - ) k y ( Å - ) -0.2 0.2-0.10.1 -0.10.1 k x ( Å - ) k z ( Å - ) k y ( Å - ) -0.2 0.2-0.10.1 -0.10.1 k x ( Å - ) k z ( Å - ) k y ( Å - ) k x ( Å - ) k y ( Å - ) k x ( Å - ) k y ( Å - ) k x ( Å - ) k y ( Å - ) P i (a) (d)(b) (e)(c) (f)P x P y P z P x P y P z FIG. 1. (Color online) Results of first-principles calculationsfor T d -MoTe . False-color maps of (a) P x , (b) P y , and (c) P z on the full bulk Fermi surface and the corresponding averageof (d) P x , (e) P y , and (f) P z over the interval − π < k z < projected into the k x , k y plane. -0.2-0.100.10.2 0.40.30.20.1080757065605550 -0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4-0.4 -0.2 0 0.2 0.4656055 -0.4 -0.2 0 0.2 0.4 -0.2 0 0.2 P x -0.5 0 0.5 P y -0.2 0 0.2 P z EPsHPs b k x (Å –1 ) k y ( Å –1 ) (a) EPHP h n ( e V ) EP a g d (e) k x (Å –1 )k x (Å –1 ) k x (Å –1 ) k x (Å –1 ) EPEP (b) (c) (d) h n ( e V ) FIG. 2. (Color online) Photoemission data collected for T d -MoTe at T = 30 K. Photon energy dependence at E F of(a) ARPES intensity along Γ X (b-d) spin polarization mea-sured at k y ≈ . Å − for (b) P x , (c) P y , and (d) P z . (e)Symmetrized ARPES intensity at E F in the k x , k y plane. spin-orientation and momentum dependence. Small andlarge Fermi arc states persist at more than 90 K above T S . Their coherence improves significantly upon coolingthrough T S and the appearance of the large Fermi arcstate is affected by thermal history. An anomalous trendof Fermi arc self-energy through the transition and resid-ual spin-polarization in bulk electrons at T > T S suggestthat T d and 1T’ structural phases coexist near the surfaceat room temperature. A so-called hidden spin-texture[44] in 1T’-MoTe poses an interesting alternative expla-nation for the high temperature SARPES results.Details of the crystal synthesis and DFT calculationsare provided in the supplemental material (SM) [45]. Ex-periments were performed with the sample kept underultrahigh vacuum (UHV) (pressure < × − Pa) atvariable temperatures fully summarized in the SM. Tem-perature was measured using a Si diode near the sam-ple. Clean (001) surfaces were obtained by cleaving inUHV. High resolution spectra were obtained using a Sci-enta R4000 analyzer with instrumental angle and energyresolution better than 0.1° and 10 meV. SARPES mea-surements were done at the COPHEE endstation [46]with angle and energy resolution better than 1.5° and 75meV. No evidence of mixed (001) and ( ) terminations[23, 34] was found in our samples [47]. ARPES and quasi-particle interference results were consistent with only onetermination type [28].Fig. 1 captures the DFT-calculated T d -MoTe Fermisurface and its spin-texture, computed as an average overthe orbital degree of freedom. The HPs enclosing the Γ point of the BZ and EPs located further from the Γ point both exhibit high spin-polarization, reaching up to0.8 in total magnitude [45]. This indicates significant or-bital anisotropy when compared with, e.g., the Bi Se surface state ( | P | = 0 . [48]. The magnitudes of com-puted and measured spin-polarization have different sig-nificance, because polarized photons selectively excite orentangle orbital components of the electron wave function[45, 49, 50], but it will be shown that the spin-orientationstransform according to crystal symmetries in the sameway for both cases. In addition to time-reversal sym-metry, the space group contains one reflection M x andone glide reflection M y which take the spatial coordi-nates ( x, y, z ) to ( − x, y, z ) and ( x, − y, z + c/ , respec-tively, where c is the unit cell length along the directionperpendicular to the plane of the MoTe layers. The in-plane components of P are constrained by the M y and M x symmetries such that P x → − P x as k y → − k y and P y → − P y as k x → − k x , respectively. P z is constrainedto by both of these reflections and must also reverse upon ( k x , k y , k z ) → ( k x , k y , − k z ) . This is a novel property notfound in helical spin-textures that suppresses k z → − k z scattering [45]. The spin-polarization remains significantin all components when averaged over the lower half ofthe BZ as shown in Fig. 1(d-f). This is important forSARPES because the k z -resolution is limited to about -0.4-0.20.00.20.4 P a t T = K -0.4-0.20.00.20.4 P a t T = K k x ( Å –1 ) k x ( Å –1 ) I n t en s i t y ( a r b . u . ) Exp. I tot
Fit I tot g (–k y ) b (–k y ) b (+k y ) g (+k y ) -1-0.500.51 P P x P y P z S p i n - P o l a r i z a t i on P x T = 30 K P x fit result P x T = 300K P z T = 30 K P z fit result P z T = 300 K P y T = 30 K P y fit result P y T = 300 K T e m pe r a t u r e ( K ) I n t en s i t y ( a r b . u . ) -0.2 0.0 0.2 T = 300 K T = 280 K T = 260 K g ( - k y ) b ( - k y ) b ( + k y ) g ( + k y ) (a) (b) (c) (d) (e) (f) (g) (h) (i) (o)(j) (k) (l) (m) (n) (p)T = 300 KT = 30 K T = 300 K T = 300 K T = 300 K T = 300 KT = 30 K T = 30 K T = 30 K T = 30 K P x P y P z P x P y P z E F E F k y (Å –1 ) k y (Å –1 ) k y (Å –1 ) k y (Å –1 ) k x = 0.26 Å –1 k x = 0.26 Å –1 k x = 0.26 Å –1 k x = 0.26 Å –1 k x = 0.26 Å –1 k y (Å –1 ) k y (Å –1 ) k y (Å –1 ) k y (Å –1 ) k y (Å –1 ) k y (Å –1 ) E B ( e V ) E B ( e V ) C oo li ng E F E F FIG. 3. (Color online) (a-c) Spin-polarization momentum distribution curves at E F and k x = 0 . Å − . (d) Results ofvectorial spin analysis for the T = 30 K data, including peak intensities and spin components (inset). Temperature dependentmeasurements at (e-i) T = 300 K and (j-n) T = 30 K. (e,j) Fermi surfaces. (f,k) ARPES intensity in gray-scale and spin-polarizations in false-color (see inset) scale for (g,l) P x , (h,m) P y , and (i,n) P z mapped over E B ( k y ) at k x = 0 . Å − . Allspin-resolved data were collected using 20 eV photons from the same sample, which was cleaved and measured at 300 K andthen cooled. (o,p) Temperature dependence of high-resolution ARPES intensity at E F and k x = 0 . Å − collected using 67eV photons for a sample cleaved at 300 K. of the reciprocal lattice vector due to the finite prob-ing depth.Fig. 2(a) shows photon energy dependence of ARPESintensity at E F along Γ X . The states disperse with pho-ton energy, thus characterizing their bulk ( k z dispersive)origin [27]. The EP around k x = ± . Å − (yellowdashed-lines) produces the strongest spin-polarizationsignal seen in the photon energy dependent maps in Fig.2(b-d), which are taken along the same direction as in (a),but with a slight misalignment to k y ≈ . Å − . Thisallows P x and P z , which are reduced to zero by symme-try at k y = 0 , to be measured. The signs of ( P x , P y , P z ) for the electron states are (+ , + , − ) for positive k x and (+ , − , +) for negative k x , as enforced by the M x sym-metry. A dissection of the experimental Fermi surfacein the ( k x , k y )-plane is shown in Fig. 2(e) in which thecontours of bulk EPs (yellow dashed-lines), HPs (bluedashed-lines), large Fermi arc (green dashed-line), andsmall Fermi arc (red dashed-line) are indicated. Thestates making up the largest EP, the large Fermi arc, the largest HP, and the small Fermi arc are labelled α , β , γ , and δ , respectively.Fig. 3(a-d) shows SARPES measurements of the Fermisurface along the k y -direction for k x = 0 . Å − , whichcrosses through β and γ . The contributions of thesestates to the spin-polarization shown in Fig. 3(a-c) andintensity in Fig. 3(d) were disentangled quantitatively byvectorial analysis [51] for the case of T = 30 K. The mo-mentum distribution curves (MDCs) were fit using fourVoight peaks, two for β and two for γ , on a uniformunpolarized background and assuming | P | = 1 in eachpeak. The inset in Fig. 3(d) shows the P x , P y , and P z values obtained for each peak in green, orange, and pur-ple bars, respectively. The P x signal primarily originatesfrom hole-like states in this momentum cut, as seen in thebinding energy dependence of P x in Fig. 3(l). The fit re-sults show that β and γ have opposite signs of P y . In bothcases, the sign of P y is unchanged upon reversal of k y ,as required by the combination of M x and time-reversalsymmetry. The sign of P z in reverses upon k y → − k y in l ep = 0.134 Cooling | S ''(E F , T)| | S '' | ( m e V ) Temperature (K) E B ( m e V ) d T = 300 K (as cleaved) T = 340 K (annealed)k x (Å –1 )E B (meV) (a) (b)(c) (d) d b k x (Å –1 ) FIG. 4. (Color online) (a-b) ARPES intensity along k x at k y = 0 divided by the Fermi cutoff. (c) Energy distributioncurves of | Σ (cid:48)(cid:48) | in state δ at k y = 0 for different temperaturesas extracted from raw data. (d) Temperature-dependence of | Σ (cid:48)(cid:48) ( E F ) | . the case of β (which exhibits a negligible P x component inthis momentum cut). P x and P z both reverse sign across k y = 0 in γ . Both β and γ are constrained by bulk M y symmetry, which is broken on the T d -MoTe (001) sur-face [45]. The quality of the fit with | P | = 1 in each stateindicates fully coherent spin-orbital coupling at T = 30 K. Fig. 3(e-p) show measurements taken before and af-ter cooling from 300 K to 30 K. The spin-polarizationat E F for the two temperatures is also compared in Fig.3(a-c). Response to the temperature change is evidentin the lack of a coherent contribution from β and overallsuppression of P y and P z at 300 K. The P x signal of thehole-like states is retained through the full energy rangeseen in Fig. 3(g). At both temperatures, hole-like statescontribute an M-shape of + y -oriented spin in the energy-momentum maps of Fig. 3(h) and Fig. 3(m), as indicatedby dashed-lines, and z -polarization that switches across k y = 0 around E B = − . eV, as indicated by arrows inFig. 3(i) and Fig. 3(n). This serves as a faint signatureof T d order persisting at 300 K.Spin-integrated MDCs in Fig. 3(o-p) show the develop-ment of intensity in β upon cooling from 300 K, measuredat E F along the same momentum cut as in Fig. 3(a-c).The peak intensities rise upon cooling from 300 to 280K, but do not sharpen into clear, Lorentzian shapes until260 K is reached. One could say that β either lies above E F , is fully absent, or the signal is too broad and sup-pressed to be clearly observed at 300 K. ARPES spectraalong Γ X are shown divided by the Fermi cutoff in Fig4(a-b). For the case of a fresh surface prepared at 300K shown in Fig. 4(a), β is not visible. It is shown else-where that, as in Fig. 3(o-p), β does not clearly emerge in this momentum cut either until the sample is cooledto 260 K [45]. Fig. 4(b) shows that β , which presentsa line of intensity connecting the bulk electron and holestates (green arrow) [27, 45], persists after cooling to 120K and annealing to 340 K. It is shown elsewhere thatthe chemical potential irreversibly increases by about 30meV upon cooling through T S [45]. It is likely that thespectral function of β is broadened and suppressed byscattering in the initial condition, obscuring the signal.These effects may have been diminished by the bindingenergy shift and/or improved structural order after onethermal cycle. The signal is simply not clear enough inthe initial condition for further determination.The signal of δ , indicated by red arrows in Fig. 4(a-b),is clear at certain emission angles (negative k x ) for thiscase where p -polarized 67 eV photons are used. The steephole-like dispersion reaches above E B = 50 meV, which isaround the maximum energy expected for WPs [28]. Ad-ditional measurements show that δ corresponds to whatref. [23] referred to as a candidate topological surfacestate [45]. To investigate the response of electronic co-herence to cooling, the magnitude of the imaginary partof the photohole self-energy | Σ (cid:48)(cid:48) | was computed by multi-plying the group velocity with the peak half-width, usingraw ARPES data collected at different temperatures [45].There is a significant effect of noise on the results, butit can be appreciated from Fig. 4(c) that there is morearea under the distribution of | Σ (cid:48)(cid:48) ( E B ) | in the range − meV < E B < meV at 260 K (black bars) than at 240 K(gray bars). Of the possible scattering mechanisms, onlyelectron-phonon coupling (EPC) is expected to cause sig-nificant variation in | Σ (cid:48)(cid:48) ( E B ) | near E F [52]. In most met-als, the lifetime broadening at E F is proportional to theEPC constant λ ep as | Σ (cid:48)(cid:48) ( E F , T ) | = 2 πk B λ ep T , where k B is the Boltzmann constant [53]. The average of broaden-ing values extracted from the range E F ± k B T / is shownas | Σ (cid:48)(cid:48) ( E F , T ) | in Fig. 4(d), with the standard error ofthe mean shown as error bars. A linear fit in the 220-100K region obtains a weak dependence on temperature cor-responding to λ ep ≈ . plotted in as a red dashed-linein Fig. 4(d). Linear fitting in the 280-220 K region is un-physical ( | Σ (cid:48)(cid:48) ( E F , T = 0) | < ). There is a rapid changein EPC, or at least some form of scattering, upon coolingthrough T S . The strength of EPC has been reported tobe similar in 1T’ and T d -MoTe [5, 9], but new forms ofelectron-lattice interaction arise in the case of strong dis-order. For example, electron-phonon-impurity scattering[54, 55], wherein electron-impurity and electron-phononscattered paths interfere, can significantly contribute tothe self-energy, even at high temperatures [56].Noting that increased electron density stabilizes the T d structure [57, 58], it could be that the surface dipole sta-bilizes local T d order at temperatures well above T S , inanalogy to so-called negative dead layers in ferroelectricmaterials [59]. This would explain the observed resid-ual spin-polarization in bulk electrons. Alternatively,this observation could derive from a so-called R-2 hid-den spin-texture [44] that must exist in bulk 1T’-MoTe because centrosymmetry is absent in all of the latticesites [45]. However, a case of global 1T’ order at T > T S does not explain the anomalous lifetime broadening trendand one would expect a full lattice transition to producea qualitative change in the measured spin-orientationsthat is not apparent [45]. A mixed structural phasewould cause electrons to exist in mixed (non-coherent)states due to entanglement with variations in the lattice-polarity, thus decreasing the quasiparticle lifetimes andspin-polarization magnitudes as resolved by SARPES.The results best correspond to a case of local T d order,with at least one T d /1T’ phase boundary existing belowthe surface at T > T S . There is the added possibility ofT d domains of opposite or unequal lattice-polarity coex-isting in this region. Such cases are analogous to ferro-electric polar instability wherein the symmetry-breakingorder is short-ranged or fluctuates [3]. For MoTe , thisis synonymous with a mixed structural phase due to thefirst-order nature of the transition, but it has been sug-gested that the polar instability yields a dynamical orglass-like phase of matter with novel thermoelectric prop-erties [3]. The boundary motion is determined by the c -axis thermal gradient [4], which is well-defined in thecase of a cooled sample with an exposed surface. Bound-aries would move into the bulk upon cooling, leaving aglobally ordered sample with electronic coherence.In summary, the observed response to cooling the 1T’-MoTe crystal is a gain in electronic coherence that yieldsa clear view of Fermi arcs and the novel 3D spin tex-ture of T d -MoTe . The existence of finite P z must beconsidered in future discussions of the magnetoresponseproperties for T d -(Mo/W)Te materials. Both small andlarge Fermi arc states are observed at 340 K, where thevolume of the bulk is almost entirely in the 1T’ struc-tural phase [2]. Therefore, the existence of the Fermiarcs is independent of any global, bulk Weyl semimetalphase of matter. Precise determination of the crystalstructure near the surface (e.g. by scanning transmissionelectron microscopy) is vital for clarifying the relation-ship between the Fermi arcs and the Weyl and structuralphases, the anomalous changes in self-energy broadening,and the origin of the spin texture observed at 300 K. ACKNOWLEDGEMENTS
This work was supported by the Swiss National Sci-ence Foundation Project No. PP P _ (no/1),No. 200021-137783, No. PP P2_ , and NCCR-MARVEL. We thank Titus Neupert and Frank Schindlerat the University of Zürich for helpful discussions. P.R.,P.M., and S.B. gratefully acknowledge financial supportfrom the DFG (SPP-1666, Project No. MA 4637/3-1)and from the VITI project of the Helmholtz Association as well as computational support from the JARA-HPCSupercomputing Centre at the RWTH Aachen Univer-sity. ∗ [email protected][1] R. Clarke, Phil. Mag. B (1978).[2] S.-Y. Chen, T. Goldstein, D. Venkataraman, A. Rama-subramaniam, and J. Yan, Nano Letters , 5852 (2016).[3] H. Sakai, K. Ikeura, M. S. Bahramy, N. Ogawa,D. Hashizume, J. Fujioka, Y. Tokura, and S. Ishiwata,Science Advances (2016).[4] X.-J. Yan, Y.-Y. Lv, L. Li, X. Li, S.-H. Yao, Y.-B. Chen,X.-P. Liu, H. Lu, M.-H. Lu, and Y.-F. Chen, npj Quan-tum Materials , 31 (2017).[5] C. Heikes, I.-L. Liu, T. Metz, C. Eckberg, P. Neves,Y. Wu, L. Hung, P. Piccoli, H. Cao, J. Leao, J. Paglione,T. Yildirim, N. P. Butch, 1, and W. Ratcliff, (2018),arXiv:1804.09093.[6] J. He, D. Di Sante, R. Li, X.-Q. Chen, J. M. Rondinelli,and C. Franchini, Nature Communications , 492 (2018).[7] Y. Qi, P. G. Naumov, M. N. Ali, C. R. Rajamathi,W. Schnelle, O. Barkalov, M. Hanfland, S.-C. Wu,C. Shekhar, Y. Sun, V. Süß, M. Schmidt, U. Schwarz,E. Pippel, P. Werner, R. Hillebrand, T. Förster, E. Kam-pert, S. Parkin, R. J. Cava, C. Felser, B. Yan, and S. A.Medvedev, Nature Communications , 11038 (2016).[8] X. Luo, F. C. Chen, J. L. Zhang, Q. L. Pei, G. T. Lin,W. J. Lu, Y. Y. Han, C. Y. Xi, W. H. Song, and Y. P.Sun, Applied Physics Letters , 102601 (2016).[9] H. Takahashi, T. Akiba, K. Imura, T. Shiino, K. Deguchi,N. K. Sato, H. Sakai, M. S. Bahramy, and S. Ishiwata,Phys. Rev. B , 100501 (2017).[10] Z. Guguchia, F. von Rohr, Z. Shermadini, A. T.Lee, S. Banerjee, A. R. Wieteska, C. A. Marianetti,B. A. Frandsen, H. Luetkens, Z. Gong, S. C. Cheung,C. Baines, A. Shengelaya, G. Taniashvili, A. N. Pasu-pathy, E. Morenzoni, S. J. L. Billinge, A. Amato, R. J.Cava, R. Khasanov, and Y. J. Uemura, Nature Commu-nications , 1082 (2017).[11] A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer,X. Dai, and B. A. Bernevig, Nature , 495 (2015).[12] M. N. Ali, J. Xiong, S. Flynn, J. Tao, Q. D. Gib-son, L. M. Schoop, T. Liang, N. Haldolaarachchige,M. Hirschberger, N. P. Ong, and R. J. Cava, Nature , 205 (2014).[13] Y. Sun, S.-C. Wu, M. N. Ali, C. Felser, and B. Yan,Phys. Rev. B , 161107 (2015).[14] Z. Wang, D. Gresch, A. A. Soluyanov, W. Xie, S. Kush-waha, X. Dai, M. Troyer, R. J. Cava, and B. A. Bernevig,Phys. Rev. Lett. , 056805 (2016).[15] Q. L. Pei, W. J. Meng, X. Luo, H. Y. Lv, F. C. Chen,W. J. Lu, Y. Y. Han, P. Tong, W. H. Song, Y. B. Hou,Q. Y. Lu, and Y. P. Sun, Phys. Rev. B , 075132 (2017).[16] F. C. Chen, H. Y. Lv, X. Luo, W. J. Lu, Q. L. Pei, G. T.Lin, Y. Y. Han, X. B. Zhu, W. H. Song, and Y. P. Sun,Phys. Rev. B , 235154 (2016).[17] S. Thirupathaiah, R. Jha, B. Pal, J. S. Matias, P. K.Das, P. K. Sivakumar, I. Vobornik, N. C. Plumb, M. Shi,R. A. Ribeiro, and D. D. Sarma, Phys. Rev. B , 241105(2017). [18] J. Yang, J. Colen, J. Liu, M. C. Nguyen, G.-w. Chern,and D. Louca, Science Advances (2017).[19] I. Pletikosić, M. N. Ali, A. V. Fedorov, R. J. Cava, andT. Valla, Phys. Rev. Lett. , 216601 (2014).[20] J. Jiang, F. Tang, X. C. Pan, H. M. Liu, X. H. Niu, Y. X.Wang, D. F. Xu, H. F. Yang, B. P. Xie, F. Q. Song,P. Dudin, T. K. Kim, M. Hoesch, P. K. Das, I. Vobornik,X. G. Wan, and D. L. Feng, Physical Review letters ,166601 (2015).[21] Q. L. Pei, X. Luo, F. C. Chen, H. Y. Lv, Y. Sun, W. J.Lu, P. Tong, Z. G. Sheng, Y. Y. Han, W. H. Song, X. B.Zhu, and Y. P. Sun, Applied Physics Letters , 072401(2018).[22] L. Muechler, A. Alexandradinata, T. Neupert, andR. Car, Phys. Rev. X , 041069 (2016).[23] A. Tamai, Q. Wu, I. Cucchi, F. Bruno, S. Riccò, T. Kim,M. Hoesch, C. Barreteau, E. Giannini, C. Besnard,A. Soluyanov, and F. Baumberger, Physical Review X , 031021 (2016).[24] A. Crepaldi, G. Autès, A. Sterzi, G. Manzoni, M. Za-cchigna, F. Cilento, I. Vobornik, J. Fujii, P. Bugnon,A. Magrez, H. Berger, F. Parmigiani, O. V. Yazyev, andM. Grioni, Phys. Rev. B , 041408 (2017).[25] A. Crepaldi, G. Autès, G. Gatti, S. Roth, A. Sterzi,G. Manzoni, M. Zacchigna, C. Cacho, R. T. Chapman,E. Springate, E. A. Seddon, P. Bugnon, A. Magrez,H. Berger, I. Vobornik, M. Kalläne, A. Quer, K. Ross-nagel, F. Parmigiani, O. V. Yazyev, and M. Grioni, Phys.Rev. B , 241408 (2017).[26] D. Rhodes, R. Schönemann, N. Aryal, Q. Zhou, Q. R.Zhang, E. Kampert, Y.-C. Chiu, Y. Lai, Y. Shimura,G. T. McCandless, J. Y. Chan, D. W. Paley, J. Lee,A. D. Finke, J. P. C. Ruff, S. Das, E. Manousakis, andL. Balicas, Phys. Rev. B , 165134 (2017).[27] N. Xu, Z. J. Wang, A. P. Weber, A. Magrez, P. Bugnon,H. Berger, C. E. Matt, J. Z. Ma, B. B. Fu, B. Q. Lv,N. C. Plumb, M. Radovic, E. Pomjakushina, K. Conder,T. Qian, J. H. Dil, J. Mesot, H. Ding, and M. Shi,(2016), arXiv:1604.02116.[28] P. Rüßmann, A. P. Weber, F. Glott, N. Xu, M. Fanciulli,S. Muff, A. Magrez, P. Bugnon, H. Berger, M. Bode, J. H.Dil, S. Blügel, P. Mavropoulos, and P. Sessi, Phys. Rev.B , 075106 (2018).[29] A. N. Berger, E. Andrade, A. Kerelsky, D. Edelberg,J. Li, Z. Wang, L. Zhang, J. Kim, N. Zaki, J. Avila,C. Chen, M. C. Asensio, S.-W. Cheong, B. A. Bernevig,and A. N. Pasupathy, npj Quantum Materials , 2 (2018).[30] J. Jiang, Z. Liu, Y. Sun, H. Yang, C. Rajamathi, Y. Qi,L. Yang, C. Chen, H. Peng, C.-C. Hwang, S. Sun, S.-K.Mo, I. Vobornik, J. Fujii, S. Parkin, C. Felser, B. Yan,and Y. Chen, Nature Communications , 13973 (2017).[31] A. Liang, J. Huang, S. Nie, Y. Ding, Q. Gao, C. Hu,S. He, Y. Zhang, C. Wang, B. Shen, J. Liu, P. Ai, L. Yu,X. Sun, W. Zhao, S. Lv, D. Liu, C. Li, Y. Zhang, Y. Hu,Y. Xu, L. Zhao, G. Liu, Z. Mao, X. Jia, F. Zhang,S. Zhang, F. Yang, Z. Wang, Q. Peng, H. Weng, X. Dai,Z. Fang, Z. Xu, C. Chen, and X. J. Zhou, (2016),arXiv:1604.01706.[32] K. Deng, G. Wan, P. Deng, K. Zhang, S. Ding, E. Wang,M. Yan, H. Huang, H. Zhang, Z. Xu, J. Denlinger, A. Fe-dorov, H. Yang, W. Duan, H. Yao, Y. Wu, S. Fan,H. Zhang, X. Chen, and S. Zhou, Nature Physics ,1105 (2016). [33] L. Huang, T. M. McCormick, M. Ochi, Z. Zhao, M.-T. Suzuki, R. Arita, Y. Wu, D. Mou, H. Cao, J. Yan,N. Trivedi, and A. Kaminski, Nature Materials , 1155(2016).[34] M. Sakano, M. S. Bahramy, H. Tsuji, I. Araya, K. Ikeura,H. Sakai, S. Ishiwata, K. Yaji, K. Kuroda, A. Harasawa,S. Shin, and K. Ishizaka, Physical Review B , 121101(2017).[35] F. Y. Bruno, A. Tamai, Q. S. Wu, I. Cucchi, C. Bar-reteau, A. de la Torre, S. McKeown Walker, S. Riccò,Z. Wang, T. K. Kim, M. Hoesch, M. Shi, N. C. Plumb,E. Giannini, A. A. Soluyanov, and F. Baumberger, Phys-ical Review B , 121112 (2016).[36] I. Belopolski, D. S. Sanchez, Y. Ishida, X. Pan, P. Yu, S.-Y. Xu, G. Chang, T.-R. Chang, H. Zheng, N. Alidoust,G. Bian, M. Neupane, S.-M. Huang, C.-C. Lee, Y. Song,H. Bu, G. Wang, S. Li, G. Eda, H.-T. Jeng, T. Kondo,H. Lin, Z. Liu, F. Song, S. Shin, and M. Z. Hasan, NatureCommunications , 13643 (2016).[37] C. Wang, Y. Zhang, J. Huang, S. Nie, G. Liu, A. Liang,Y. Zhang, B. Shen, J. Liu, C. Hu, Y. Ding, D. Liu, Y. Hu,S. He, L. Zhao, L. Yu, J. Hu, J. Wei, Z. Mao, Y. Shi,X. Jia, F. Zhang, S. Zhang, F. Yang, Z. Wang, Q. Peng,H. Weng, X. Dai, Z. Fang, Z. Xu, C. Chen, and X. J.Zhou, Physical Review B , 241119 (2016).[38] Y. Wu, D. Mou, N. H. Jo, K. Sun, L. Huang, S. L. Bud’ko,P. C. Canfield, and A. Kaminski, Physical Review B ,121113 (2016).[39] J. Sánchez-Barriga, M. G. Vergniory, D. Evtushinsky,I. Aguilera, A. Varykhalov, S. Blügel, and O. Rader,Physical Review B , 161401 (2016).[40] N. Xu, G. Autès, C. E. Matt, B. Q. Lv, M. Y. Yao,F. Bisti, V. N. Strocov, D. Gawryluk, E. Pomjakushina,K. Conder, N. C. Plumb, M. Radovic, T. Qian, O. V.Yazyev, J. Mesot, H. Ding, and M. Shi, Phys. Rev. Lett. , 106406 (2017).[41] W. Qisheng, L. Jie, B. Jean, H. Chuang-Han, C. Kaim-ing, Y. Li, C. Shuai, W. Yang, Z. Wenfeng, W. Kaiyou,C. Tay-Rong, L. Hsin, C. Haixin, and Y. Hyunsoo, Ad-vanced Science (2018).[42] M. Smidman, M. B. Salamon, H. Q. Yuan, and D. F.Agterberg, Reports on Progress in Physics , 036501(2017).[43] B. Feng, Y.-H. Chan, Y. Feng, R.-Y. Liu, M.-Y. Chou,K. Kuroda, K. Yaji, A. Harasawa, P. Moras, A. Barinov,W. Malaeb, C. Bareille, T. Kondo, S. Shin, F. Komori,T.-C. Chiang, Y. Shi, and I. Matsuda, Phys. Rev. B ,195134 (2016).[44] X. Zhang, Q. Liu, J.-W. Luo, A. J. Freeman, andA. Zunger, Nature Physics , 387 (2014).[45] See Supplemental Material at [URL will be inserted bypublisher] for details.[46] M. Hoesch, T. Greber, V. Petrov, M. Muntwiler,M. Hengsberger, W. Auwärter, and J. Osterwalder,Journal of Electron Spectroscopy and Related Phenom-ena , 263 (2002).[47] A.P. Weber et al. (In Progress).[48] O. V. Yazyev, J. E. Moore, and S. G. Louie, Phys. Rev.Lett. , 266806 (2010).[49] Z. Xie, S. He, C. Chen, Y. Feng, H. Yi, A. Liang, L. Zhao,D. Mou, J. He, Y. Peng, X. Liu, Y. Liu, G. Liu, X. Dong,L. Yu, J. Zhang, S. Zhang, Z. Wang, F. Zhang, F. Yang,Q. Peng, X. Wang, C. Chen, Z. Xu, and X. J. Zhou,Nature communications , 3382 (2014). [50] K. Yaji, K. Kuroda, S. Toyohisa, A. Harasawa, Y. Ishida,S. Watanabe, C. Chen, K. Kobayashi, F. Komori, andS. Shin, Nature Communications , 14588 (2017).[51] F. Meier, H. Dil, J. Lobo-Checa, L. Patthey, and J. Os-terwalder, Physical Review B , 165431 (2008).[52] T. Valla, A. V. Fedorov, P. D. Johnson, and S. L. Hul-bert, Phys. Rev. Lett. , 2085 (1999).[53] P. Hofmann, I. Y. Sklyadneva, E. D. L. Rienks, and E. V.Chulkov, New Journal of Physics , 125005 (2009).[54] M. Reizer and A. Sergeev, Zh. Eksp. Teor. Fiz. , 2291(1987). [55] S. S. Yeh, J. J. Lin, J. Xiunian, and Z. Dianlin, Phys.Rev. B , 024204 (2005).[56] W.-C. Hsu, C.-C. Chen, Y.-H. Lin, H.-K. Lin, H.-T.Chiu, and J.-J. Lin, Nanoscale Research Letters , 500(2012).[57] H.-J. Kim, S.-H. Kang, I. Hamada, and Y.-W. Son, Phys.Rev. B , 180101 (2017).[58] R. He, S. Zhong, H. H. Kim, G. Ye, Z. Ye, L. Winford,D. McHaffie, I. Rilak, F. Chen, X. Luo, Y. Sun, andA. W. Tsen, Phys. Rev. B , 041410 (2018).[59] M. Stengel, D. Vanderbilt, and N. A. Spaldin, NatureMaterials8