Observation of surface states derived from topological Fermi arcs in the Weyl semimetal NbP
Ilya Belopolski, Su-Yang Xu, Daniel Sanchez, Guoqing Chang, Cheng Guo, Madhab Neupane, Hao Zheng, Chi-Cheng Lee, Shin-Ming Huang, Guang Bian, Nasser Alidoust, Tay-Rong Chang, BaoKai Wang, Xiao Zhang, Arun Bansil, Horng-Tay Jeng, Hsin Lin, Shuang Jia, M. Zahid Hasan
OObservation of surface states derived from topological Fermi arcsin the Weyl semimetal NbP
Ilya Belopolski ∗ , Su-Yang Xu ∗ , Daniel Sanchez ∗ , Guoqing Chang,
2, 3
Cheng Guo, Madhab Neupane, Hao Zheng, Chi-Cheng Lee,
2, 3
Shin-Ming Huang,
2, 3
Guang Bian, Nasser Alidoust, Tay-Rong Chang,
1, 6
BaoKai Wang,
2, 3, 7
Xiao Zhang, ArunBansil, Horng-Tay Jeng,
6, 8
Hsin Lin,
2, 3
Shuang Jia, and M. Zahid Hasan
1, 9 Laboratory for Topological Quantum Matter and Spectroscopy (B7),Department of Physics, Princeton University,Princeton, New Jersey 08544, USA Centre for Advanced 2D Materials and Graphene Research Centre,National University of Singapore, 6 Science Drive 2, Singapore 117546 Department of Physics, National University of Singapore,2 Science Drive 3, Singapore 117542 International Center for Quantum Materials,Peking University, Beijing 100871, China Condensed Matter and Magnet Science Group,Los Alamos National Laboratory, Los Alamos, NM 87545, USA Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan Department of Physics, Northeastern University,Boston, Massachusetts 02115, USA Institute of Physics, Academia Sinica, Taipei 11529, Taiwan Princeton Center for Complex Materials,Princeton Institute for Science and Technology of Materials,Princeton University, Princeton, New Jersey 08544, USA (Dated: October 4, 2018) ∗ These authors contributed equally to this work. a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Abstract
The recent experimental discovery of a Weyl semimetal in TaAs provides the first observation of aWeyl fermion in nature and demonstrates a novel type of anomalous surface state band structure,consisting of Fermi arcs. So far, work has focused on Weyl semimetals with strong spin-orbitcoupling (SOC). However, Weyl semimetals with weak SOC may allow tunable spin-splitting fordevice applications and may exhibit a crossover to a spinless topological phase, such as a Diracline semimetal in the case of spinless TaAs. NbP, isostructural to TaAs, may realize the first Weylsemimetal in the limit of weak SOC. Here we study the surface states of NbP by angle-resolvedphotoemission spectroscopy (ARPES) and we find that we cannot show Fermi arcs based on ourexperimental data alone. We present an ab initio calculation of the surface states of NbP andwe find that the Weyl points are too close and the Fermi level is too low to show Fermi arcseither by (1) directly measuring an arc or (2) counting chiralities of edge modes on a closed path.Nonetheless, the excellent agreement between our experimental data and numerical calculationssuggests that NbP is a Weyl semimetal, consistent with TaAs, and that we observe trivial surfacestates which evolve continuously from the topological Fermi arcs above the Fermi level. Basedon these results, we propose a slightly different criterion for a Fermi arc which, unlike (1) and (2)above, does not require us to resolve Weyl points or the spin splitting of surface states. We proposethat raising the Fermi level by >
20 meV would make it possible to observe a Fermi arc using thiscriterion in NbP. Our work offers insight into Weyl semimetals with weak spin-orbit coupling, aswell as the crossover from the spinful topological Weyl semimetal to the spinless topological Diracline semimetal.
A Weyl semimetal is a crystal which hosts Weyl fermions as emergent quasiparticles [1–8]. Although Weyl fermions are well-studied in quantum field theory, they have not beenobserved as a fundamental particle in nature. The recent experimental observation of Weylfermions in TaAs offers a beautiful example of emergence in science and grants access inexperiment to a wealth of phenomena associated with Weyl fermions in theory [9–13]. Weylsemimetals also give rise to a topological classification which is closely related to the Chernnumber of the integer quantum Hall effect [14]. In the bulk band structure of a three-dimensional sample, Weyl fermions correspond to points of accidental degeneracy, Weylpoints, between two bands. The Chern number on a two-dimensional slice of the Brillouinzone passing in between Weyl points can be non-zero. In addition, this Chern numberchanges when the slice is swept through a Weyl point. As in the quantum Hall effect, theChern number in a Weyl semimetal protects topological boundary modes. However, becausethe Chern number changes when the slice is swept through a Weyl point, the boundary modeshave the exotic property that they terminate in momentum space at the locations of Weylpoints. On the two-dimensional surface of a Weyl semimetal, the resulting surface stateband structure consists of topological Fermi arcs, with constant energy contours which donot form closed curves. In this way, Weyl semimetals provide the most dramatic exampleto date of an anomalous surface state band structure. While the first experiments on Weylsemimetals studied compounds with strong spin-orbit coupling (SOC), it is unclear howFermi arcs in a Weyl semimetal evolve in the limit of weak SOC [15–19]. Moreover, if SOCis ignored, ab initio numerical calculations predict that TaAs realizes a novel phase knownas a spinless topological Dirac line semimetal, with a touching of four bands along an entireclosed curve in the bulk Brillouin zone [20]. NbP is isostructural to TaAs and is predicted tobe a Weyl semimetal [9, 21]. However, NbP has much lower atomic number, placing it in thelimit of weak SOC. As a result, NbP provides the first known example in theory of a Diracline semimetal and further offers the opportunity to understand the crossover from a spinfulWeyl semimetal to a spinless Dirac line semimetal. The crossover to a spinless system is alsoassociated with a tunable spin-splitting in the bulk and surface band structure, which maybe useful in applications. The low spin-orbit coupling in NbP, the topologically non-trivialspinless phase of NbP and the availability of TaAs as a spin-orbit coupled cousin to NbPoffer the chance to realize a Dirac line semimetal, study a crossover to a Weyl semimetaland understand the evolution of Fermi arcs in the limit of weak spin-orbit coupling.Here we use vacuum ultraviolet angle-resolved photoemission spectroscopy (ARPES) tostudy the surface state band structure of the (001) surface of NbP. We do not directlyobserve topological Fermi arc surface states in the sense that all surface state constant-energy contours form closed curves. We are also unable to establish Fermi arcs by countingchiralities of edge modes on a loop in momentum space, a trick which has recently beenapplied to demonstrate a non-zero Chern number in TaP [18]. However, we present an abinitio calculation of the (001) surface states of NbP and find that we can reproduce withexcellent agreement all trivial surface states. Our calculation further shows that the spin-splitting in NbP is very small due to the low spin-orbit coupling, reducing the separationbetween Weyl points. In addition, we find that the most well-separated Weyl points in NbPare well above the Fermi level. Both of these facts make it difficult to demonstrate Fermiarcs by ARPES on the (001) surface of NbP. Nonetheless, the excellent agreement betweenour ab initio calculations and our ARPES spectra suggests that NbP is a Weyl semimetal,consistent with TaAs, NbAs and TaP. Our calculations further relate a trivial surface stateat the Fermi level to a Fermi arc ∼
26 meV above the Fermi level. In this sense, our ARPESspectra show trivial surface states which evolve continuously into topological Fermi arcsabove the Fermi level. We point out that if these states were accessible, then an ARPESexperiment could demonstrate that such a contour is a Fermi arc without resolving eitherthe two Weyl points to which it is attached or spin-splitting in the Fermi arc. Instead,it would be sufficient to consider the evolution of the surface states in binding energy.We emphasize that such an approach to demonstrating Fermi arcs is distinct from both adirect observation of an open constant-energy contour and the counting of chiralities of edgemodes. We summarize these three distinct ways of showing Fermi arcs in a Weyl semimetal.Lastly, we discuss spin-splitting in our ARPES data and we show that the trivial surfacestates have contributions from multiple atomic orbitals, giving rise to different surface statepockets which we observe do not hybridize. Our work provides insight into an inversion-breaking Weyl semimetal in the limit of a Dirac line semimetal, without spin-orbit coupling.Our work also offers a useful summary of the ways in which we can demonstrate in generalFermi arcs in Weyl semimetals.We first present the compound under study. Niobium phosphide (NbP) crystallizes ina body-centered tetragonal Bravais lattice, in point group C v (4 mm ), space group I md (109), isostructural to TaAs, TaP and NbAs [22–24]. The crystal structure can be understoodas a stack of alternating Nb and P square lattice layers, see Fig. 1a. Each layer is shiftedwith respect to the one below it by half an in-plane lattice constant, a/
2, in either the ˆ x orˆ y direction. The crystal structure can also be understood as arising from intertwined helicesof Nb and P atoms which are copied in-plane to form square lattices, with one Nb (or P)atom at every π/ C symmetry, where a C rotation followed by a translation by c/ d and P 3 p orbitals, respectively. However, an ab initio bulk bandstructure calculation along high-symmetry lines shows that NbP does not have a full gapbut is instead a semimetal, see Σ − Γ, Z − Σ (cid:48) , Σ (cid:48) − N in Fig. 1e, with the bulk Brillouinzone in Fig. 1f. In the absence of spin-orbit coupling, the band structure near the Fermilevel consists of four Dirac lines, shown in purple in Fig. 1g. These Dirac lines are protectedby two vertical mirror planes, shown in blue. After spin-orbit coupling is included, eachDirac line vaporizes into six Weyl points shifted slightly off the mirror plane, marked by theblack and white dots in Fig. 1g. Two Weyl points are on the k z = 0 plane, shown in red,and we call these Weyl points W . The other four, we call W . We note that on the (001)surface, two W of the same chirality project onto the same point of the surface Brillouinzone, giving rise to a projected Weyl point of chiral charge ±
2. The W give projections ofchiral charge ±
1, see Fig. 1h.On the basis of our ab initio results, we search for Weyl points and Fermi arcs in NbPby ARPES. First, we show that we observe surface states but not bulk states in vacuumultraviolet ARPES on the (001) surface of NbP. In our ARPES spectra, we observe a Fermisurface consisting of lollipop-shaped pockets along the ¯Γ − ¯ X and ¯Γ − ¯ Y lines and peanut-shaped pockets on the ¯ M − ¯ X and ¯ M − ¯ Y lines, see Fig. 2a-e. The spectra are consistent withthe other compounds in the same family, suggesting that these pockets are surface statesrather than bulk states. Because C symmetry is implemented as a screw axis in NbP, the(001) surface breaks C symmetry and the surface state dispersion is not C symmetric. Ourdata suggest that the peanut pockets at ¯ X and ¯ Y differ slightly, showing that we observesurface states. We note, however, that this effect is much weaker than in TaAs or TaP[11, 18]. This result could be explained by reduced coupling between the square latticelayers, restoring the C symmetry of each individual layer. In particular, we note that thelattice constants of NbP are comparable to those of TaAs, while the atomic orbitals aresmaller due to the lower atomic number. We expect this effect to be particularly importantfor surface states derived from the p x , p y , d xy and d x − y orbitals and indeed we observe no C breaking at all for the lollipop pockets, which arise from the in-plane orbitals [21]. Weconclude that we observe the surface states of NbP in our ARPES spectra.Because our ARPES spectra show the (001) surface states of NbP, we ask whether weobserve Fermi arcs. We see in Figs. 2a-e that both the lollipop and peanut pockets areclosed, so we observe no single disconnected arc. We also see no evidence of a kink in theconstant-energy contours, so we do not observe a pair of arcs connecting to the same W in adiscontinuous way. It is also possible that the two arcs approach the W with approximatelythe same slope. To exclude this possibility, we present a difference map of two ARPESspectra, at E B = 0 .
05 eV and E B = 0 . C , along ¯Γ − ¯ X − ¯ M − ¯Γ, which encloses netchiral charge +1, Fig. 2i, and (2) a small circular path, P , which encloses net chiral charge −
2, Fig. 2j. For each path, we label each spinless crossing with an up or down arrow toindicate the sign of the Fermi velocity. We find that going around either C and P we havenet zero chirality, showing zero Chern number on the associated bulk manifold. This resultis reasonable because our spectra simply consist of two overlapping hole pockets, illustratedin Fig. 2h.Next, we compare our experimental results to numerical calculations on NbP and showthat it is challenging to observe Fermi arcs in our spectra because of the low spin-orbitcoupling. We present a calculation of the (001) surface states in NbP for the P termination,at the binding energy of W , ε W = − .
026 eV, at the Fermi level and at the binding energyof W , ε W = 0 .
053 eV, see Figs. 3a-c. We also plot the Weyl point projections, obtainedfrom a bulk band structure calculation [21]. We observe surface states (1) near the mid-pointof the ¯Γ − ¯ X and ¯Γ − ¯ Y lines and (2) near ¯ Y and ¯ X . The surface states (1) form two Fermiarcs and two closed contours at ε W , see Fig. 3d,e. These states undergo a Lifshitz transitionnear ε F with the surface states (2), forming a large hole-like pocket below the Fermi level.The surface states (2) also form a large hole-like pocket. They contain within them, nearthe ¯ X and ¯ Y points, a short Fermi arc connecting each pair of W , see Fig. 3f,g. We notethe excellent agreement with our ARPES spectra, where we also see lollipops and peanutswhich evolve into trivial, closed, hole-like pockets below ε F . At the same time, we find in ourcalculation that the separation of Weyl points and the spin-splitting in the surface states issmall. This result is consistent with our ARPES spectra, which do not show spin-splitting inthe surface states near the Fermi level. We further find that the connectivity of Fermi arcs inthe calculation is consistent with the topological classification even as the spin degeneracy isrestored. In particular, two Fermi arcs connect to the W , but each Fermi arc is also pairedwith a nearly-degenerate closed contour, reflecting the low spin-splitting, see Figs. 3d,e.Similarly, each pair W are connected by one Fermi arc, whose spin-reversed partner is theFermi arc on the other side of the Brillouin zone connecting two other W , see Figs. 3f,g.In this way, the surface states obey both the topological protection of the non-zero Chernnumber as well as the spin degeneracy approximately enforced by low spin-orbit coupling.The small spin-splitting observed in our numerical calculations underlines the difficultyin observing topological Fermi arc surface states in NbP. The separation of the Weyl pointsis < . A − for the W and < . A − for the W , both well below the typical linewidthof our ARPES spectra, ∼ . A − . For this reason, we cannot resolve the momentum spaceregion between the W or the W to determine if there is an arc. We emphasize that wecannot surmount this difficulty by considering Fermi level crossings on P or C , as shownin Fig. 1f. It is obvious that if we cannot resolve the two Weyl points in a Fermi surfacemapping, then we also cannot resolve a Fermi arc connecting them in a cut passing throughthe Weyl points. In this way, on P we cannot verify the arc connecting the W and on P and C we cannot verify the empty region between the W . We point out that P fails forTaAs, TaP, NbAs and NbP due to the small separation of the W , despite recent claims thatthis path can be used to demonstrate a Weyl semimetal in TaAs and NbP [13, 15, 25].As an additional complication, it is difficult to use P or C because the Fermi level isbelow the Lifshitz transition for the W in NbP, see Fig. 4a. This invalidates any countingof chiralities of edge modes because for ε F < ε L , there is no accessible binding energy wherethe bulk band structure is gapped along an entire loop passing in between a pair of W ,illustrated in Fig. 4b by the broken dotted red line for ε = ε F . On the bright side, if wecould access ε > ε L , then it may be possible to demonstrate a Fermi arc in NbP withoutresolving the W and counting chiralities. In particular, while the Fermi arc may appear toform a closed contour due to the small separation of W , it could have a kink at the locationof the W . The Fermi arc would also tend to disperse in one direction with binding energy,in sharp contrast to a closed contour, which would tend to grow or shrink in all directions.Unlike the previous criteria which we considered to show Fermi arcs, this criterion does notdepend on resolving the W or the spin-splitting of the surface states. Lastly, we note thatwe can understand the lollipop pocket at the Fermi level as arising from the outer Fermi arcand trivial surface state at ε W , see again Fig. 3a. If we start from ε W and scan to deeperbinding energies, we see that both the arc and trivial surface state enlarge and eventuallyundergo a surface state Lifshitz transition with the ¯ X and ¯ Y pockets at ε ∼ ε F , see againFig. 3b. Below this Lifshitz transition, the lollipop pocket becomes a trivial hole-like pocketwith approximate spin degeneracy, as shown in Fig. 3c and illustrated in Fig. 2h. Inthis sense, although we do not observe a non-zero Chern number by counting chiralitiesof edges modes in NbP, the excellent correspondence between our ARPES spectra and abinitio calculation suggests that the trivial lollipop surface state evolves continuously intothe topological Fermi arcs above the Fermi level.We point out several other features of the (001) surface states of NbP. First, we note thatwe can observe a spin splitting in the lollipop pocket at E B ∼ . p and d orbitalsand the peanut pocket arises mostly from out-of-plane p and d orbitals. The suppressedhybridization may be related to the C symmetry of the (001) surface. In particular, wenote that the in-plane and out-of-plane orbitals transform under different representationsof C . The contributions from different, unhybridized orbitals to the surface states in NbPmay give rise to novel phenomena. For example, we propose that quasiparticle interferencebetween the lollipop and peanut pockets will be supressed in an STM experiment on NbP.Also, the rich surface state structure may explain why the bulk band structure of NbPis invisible to vacuum ultraviolet ARPES. Specifically, because the surface states take fulladvantages of all available orbitals, there are no orbitals left near the surface to participatein the bulk band structure. Other phenomena may also arise from the rich surface statestructure in NbP.Lastly, we compare the W and their Fermi arcs in NbP and TaAs. Unlike NbP, TaAshosts two Fermi arcs without any additional closed surface state contours. In addition, thespin splitting of the two Fermi arcs is much larger, see Fig. 4g. Similarly, the separationof Weyl points is ∼ W cross over into the adjacent Brillouin zone due to the fact that the Dirac line inTaAs is smaller than in NbP, as is seen in Fig. 4e,f. We see that the weak SOC of NbPprevents us from resolve the separation between Weyl points, so that we cannot directlyobserve Fermi arcs or demonstrate a non-zero Chern number by counting chiralities of edgemodes. However, we propose that even with weak SOC, if the Fermi level of NbP could beraised to the W , then it is possible to demonstrate a Fermi arc without resolving the W orthe spin splitting of the surface states. In particular, the Fermi arcs could show a kink at the W and would tend to disperse in one direction with binding energy. This criterion couldbe used to demonstrate Fermi arcs in Weyl semimetals which have yet to be discovered. Acknowledgements
We thank Makoto Hashimoto and Donghui Lu for technical assistance with ARPESmeasurements at SSRL Beamline 5-4, SLAC, Menlo Park, CA, USA. We also thank Nicholas0Plumb and Ming Shi for technical assistance with ARPES measurements at the HRPESendstation of the SIS beamline, Swiss Light Source, Villigen, Switzerland. [1] H. Weyl. Elektron und Gravitation.
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845 (1996).[25] D. F. Xu, et al . Observation of Fermi arcs in non-centrosymmetric Weyl semimetal candidateNbP. arXiv:1509.03847. f (cid:1850) (cid:1851)(cid:1852)(cid:1840) Γ(cid:3364)(cid:1850)(cid:3364) (cid:1839)(cid:3365)Σ (cid:4593) Σ(cid:1863) (cid:3051) (cid:1863) (cid:3052) (cid:1851) (cid:4593) (cid:1842) g (cid:1831) (cid:3003) e V h da cb NbP PNb (cid:3397)(cid:2779)(cid:3398)(cid:2779) (cid:3397)(cid:2778)(cid:3398)(cid:2778)(cid:3397)(cid:2779)(cid:3397)(cid:2779) (cid:3397)(cid:2779)(cid:3398)(cid:2779)(cid:3398)(cid:2779)(cid:3398)(cid:2779) (cid:3398)(cid:2778)(cid:3398)(cid:2778) (cid:3398)(cid:2778)(cid:3397)(cid:2778)(cid:3397)(cid:2778) (cid:3397)(cid:2778) (cid:1863) (cid:3051) e (cid:1851)(cid:3364) (cid:1863) (cid:3053) (cid:2024) / (cid:1855) (cid:1849) (cid:2869) (cid:1849) (cid:2870) FIG. 1:
Overview of the Weyl semimetal candidate NbP. (a) The crystal structure of NbP,which can be understood as a stack of square lattices of Nb and P, with a stacking pattern whichinvolves an in-plane shift of each layer relative to the one below it. (b) The crystal structure canalso be understood as a pair of intertwined helices of Nb and P atoms which are copied in-planeto form square lattice layers. The axis of the helix is the ˆ z direction of the conventional unitcell, the center of the helix is 1 / a/ √
2. (c) Photograph of the sample taken throughan optical microscope, showing a beautiful crystal. (d) An STM topography of the (001) surfaceof NbP, showing the high quality of the sample surface, with no defects within a 6.2 nm × − . . Ab initio bulkband structure calculation of NbP, using GGA exchange correlation functionals, without spin-orbitcoupling (SOC), showing that NbP is a semimetal with band inversions along Σ − Γ, Z − Σ (cid:48) andΣ (cid:48) − N . (f) The bulk Brillouin zone and (001) surface Brillouin zone of NbP, with high-symmetrypoints labeled. (g) Without SOC, NbP has four Dirac lines protected by two mirror planes (blue).With SOC, each Dirac line vaporizes into six Weyl points (black and white dots), two on the k z = 0plane (red), called W , and four away from k z = 0, called W . (h) Illustration of the Weyl pointprojections in the (001) surface Brillouin zone. Two W of the same chiral charge project onto thesame point in the surface Brillouin zone, giving Weyl point projections of chiral charge ±
2. Theseparation is not to scale, but the splitting between pairs of W is in fact larger than the splittingbetween pairs of W (see also Fig. 4h). Γ(cid:3364) (cid:1850)(cid:3364) (cid:1839)(cid:3365)(cid:1851)(cid:3364) (cid:1831) (cid:3003) (cid:3404) 0eV (cid:1831) (cid:3003) (cid:3404) 0.1eV (cid:1831) (cid:3003) (cid:3404) 0.2eV (cid:1831) (cid:3003) (cid:3404) 0.3eV (cid:1831) (cid:3003) (cid:3404) 0.4eV(cid:1831) (cid:3003) (cid:3404) 0.05eV (cid:2317)(cid:2330) (cid:2179) (cid:2779) (cid:2179) (cid:2778) (cid:1839)(cid:3365)Γ(cid:3364) (cid:1850)(cid:3364) (cid:1851)(cid:3364) a b c d ef (cid:2330)(cid:2317)Γ(cid:3364) X(cid:3365) M(cid:3365) Γ(cid:3364) i jh (cid:1863) || (cid:1344) (cid:2879)(cid:2869) (cid:1831) (cid:3003) e V (cid:1863) || (cid:1344) (cid:2879)(cid:2869) (cid:1863) || (cid:1344) (cid:2879)(cid:2869) (cid:1863) || (cid:1344) (cid:2879)(cid:2869) (cid:1863) || (cid:1344) (cid:2879) (cid:2869) (cid:1863) || (cid:1344) (cid:2879)(cid:2869) (cid:1863) || (cid:1344) (cid:2879)(cid:2869) (cid:1863) || (cid:1344) (cid:2879) (cid:2869) (cid:1863) || (cid:1344) (cid:2879)(cid:2869) (cid:1863) || (cid:1344) (cid:2879)(cid:2869) (cid:1863) || (cid:1344) (cid:2879)(cid:2869) g FIG. 2:
Surface states of NbP by ARPES. (a)-(e) Fermi surface by vacuum ultraviolet APRESon the (001) surface of NbP at binding energies E B = 0, 0 .
1, 0 .
2, 0 . . C symmetry breaking show that at vacuum ultravioletenergies the spectral weight is dominated by the surfaces states, not the bulk states of NbP. Weobserve lollipop and peanut-shaped pockets and find that both are hole-like. (f) Same as (a)-(e),but at E B = 0 .
05 eV and with additional decoration to illustrate the high symmetry points of thesurface Brillouin zone, the locations of the Weyl points and the two paths C and P on which wemeasure Chern numbers. (g) The difference of ARPES spectra at E B = 0 .
05 eV and E B = 0 . W in such a way as to create an apparently closed pocket.(h) Cartoon of the band structure, including the two paths C and P and the Weyl points whichthey enclose, and arrows indicating the chirality of each edge mode. (i) Band structure by ARPESalong the path C , with chiralities of edge modes marked by the arrows. Note that each arrowsecretly corresponds to two crossings, because we cannot observe spin splitting at the Fermi leveldue to the weak SOC of NbP. There are the same number of arrows going up as down, so theChern number is zero. This is easy to see because the path enters and exits two pockets. (j) Sameas (i) but along the path C . Again the Chern number is zero. (cid:3398)1/21/2(cid:3398)1/2 b cf a (cid:2013) (cid:3024)(cid:2870) (cid:3404) (cid:3398)0.026eV (cid:2013) (cid:3007) (cid:3404) 0eV (cid:2013) (cid:3024)(cid:2869) (cid:3404) 0.053eV e g (cid:3398)0.060.060.12 0.32 (cid:1863) (cid:3051) (cid:3398)0.008 0.0080.490.496 (cid:1863) (cid:3051) (cid:3051) (cid:3051) (cid:3051) d (cid:1849) (cid:2870) (cid:1849) (cid:2869) (cid:1849) (cid:2870) (cid:1849) (cid:2869) (cid:1849) (cid:2870) (cid:1849) (cid:2869) (cid:1849) (cid:2870) (cid:1849) (cid:2869) FIG. 3:
Numerical calculation of Fermi arcs in NbP.
First-principles band structure calcu-lation of the (001) surface states of NbP at (a) the energy of the W , above the Fermi level, (b)the Fermi level and (c) the energy of the W , below the Fermi level. The projected Weyl nodeson the (001) surface are denoted by the small red and white circles. In (a) we see (1) a small setof surface states near the W and (2) larger surface states near ¯ X and ¯ Y . Near the energy of (b)there is a Lifshitz transition between the surface states at (1) and (2), giving rise to lollipop andpeanut-shaped pockets. At (c) we see that the lollipop and peanut pockets enlarge, so they arehole-like. We also observe short Fermi arcs connecting the W . The numerical calculation showsexcellent overall agreement with our ARPES spectra. Specifically, we find trivial, closed, hole-likelollipop and peanut pockets below the Fermi level. (d) Zoom-in of the surface states around the W , indicated by the white box in (a). We find two Fermi arcs and two trivial closed contours,illustrated in (e). (f) Zoom-in of the surface states around W , indicated by the white box at thebottom of (c). We find one Fermi arc, illustrated in (g). (cid:1863) || (cid:3404) 0.9(cid:1344) (cid:2879)(cid:2869) (cid:1863) || (cid:3404) 0.85(cid:1344) (cid:2879)(cid:2869) (cid:1863) || (cid:3404) 0.8(cid:1344) (cid:2879)(cid:2869) (cid:1863) || (cid:3404) 0.75(cid:1344) (cid:2879)(cid:2869) a e (cid:1863) || (cid:3404) 0.7(cid:1344) (cid:2879)(cid:2869) f (cid:3397)(cid:1849) (cid:2870) (cid:3398)(cid:1849) (cid:2870) (cid:2013) (cid:3024) (cid:3404) (cid:3398)0.026 eV(cid:2013) (cid:3013) (cid:3404) (cid:3398)0.019 eV(cid:2013) (cid:3007) (cid:3404) 0 eV (cid:1863) (cid:3051) (cid:3398)0.060.060.12 0.32 gc Γ(cid:3364) (cid:1850)(cid:3364)(cid:1850)(cid:3364) d (cid:1863) || b (cid:2013) ∼ (cid:2013) (cid:3024) (cid:2013) (cid:3013) (cid:3407) (cid:2013) (cid:3407) (cid:2013) (cid:3024) (cid:2013) (cid:3404) (cid:2013) (cid:3007) (cid:3407) (cid:2013) (cid:3013) (cid:3397)(cid:1849) (cid:2870) (cid:3398)(cid:1849) (cid:2870) (cid:1863) || (cid:1863) || NbPTaAs (cid:1863) || (cid:1863) || (cid:2024) / (cid:1853) (cid:1863) || (cid:2024) / (cid:1853) hi (cid:1863) || (cid:1344) (cid:2879)(cid:2869) (cid:1863) || (cid:1344) (cid:2879)(cid:2869) (cid:1863) || (cid:1344) (cid:2879)(cid:2869) (cid:1863) || (cid:1344) (cid:2879)(cid:2869) (cid:1863) || (cid:1344) (cid:2879)(cid:2869) (cid:1863) || (cid:1344) (cid:2879)(cid:2869) (cid:1863) || (cid:1344) (cid:2879)(cid:2869) (cid:1863) || (cid:1344) (cid:2879) (cid:2869) (cid:1831) (cid:3003) e V (cid:1831) (cid:3003) e V (cid:1831) (cid:3003) e V TaAs (cid:1851)(cid:3364)Γ(cid:3364) (cid:1851)(cid:3364)Γ(cid:3364) FIG. 4:
Comparison between NbP and TaAs. (a) Relevant energies of the W compared tothe Fermi level in the numerical calculation. One consequence of the small spin-splitting is that theLifshitz transition between the W is only ∼ .
007 eV below the energy of the W and ∼ .
019 eVabove the Fermi level. (b) Because ε F < ε L , it makes no sense to calculate the Chern number on C and P , because the two-dimensional band structure corresponding to that cut is not an insulator.This is illustrated by the interrupted dotted red line in the last row of (b). (c) Surface states byARPES along ¯ X − ¯Γ − ¯ X , showing a spin splitting below the Fermi level. (d) Same as (c) butwith guides to the eye to mark the spin splitting. (e). Fermi surface by APRES at E B = 0 . C symmetry of the (001) surface of NbP.(g) Equivalent of Fig. 3d for TaAs. We see two co-propagating arcs with large spin-splitting dueto the large SOC. Plot of the positions of the Weyl point projections in (h) NbP and (i) TaAs. Wesee that the separation of the Weyl points is ∼∼