Hidden type-II Weyl points in the Weyl semimetal NbP
HHidden type-II Weyl points in the Weyl semimetal NbP
Shu-Chun Wu, Yan Sun, Claudia Felser, and Binghai Yan ∗ Max Planck Institute for Chemical Physics of Solids, D-01187 Dresden, Germany Department of Condensed Matter Physics, Weizmann Institute of Science, 7610001 Rehovot, Israel (Dated: August 24, 2017)As one of Weyl semimetals discovered recently, NbP exhibits two groups of Weyl points with onegroup lying inside the k z = 0 plane and the other group staying away from this plane. All Weylpoints have been assumed to be type-I, for which the Fermi surface shrinks into a point as the Fermienergy crosses the Weyl point. In this work, we have revealed that the second group of Weyl pointsare actually type-II, which are found to be touching points between the electron and hole pockets inthe Fermi surface. Corresponding Weyl cones are strongly tilted along a line approximately 17 ◦ offthe k z axis in the k x − k z (or k y − k z ) plane, violating the Lorentz symmetry but still giving rise toFermi arcs on the surface. Therefore, NbP exhibits both type-I ( k z = 0 plane) and type-II ( k z (cid:54) = 0plane) Weyl points. I. INTRODUCTION
Weyl semimetals (WSMs) were recently found infour transition-metal monopnictides, TaAs, TaP, NbAsand NbP, by theoretical predictions and angle-resolvedphotoemission spectroscopy (ARPES) . In the bulk,these materials exhibit Weyl points through which con-duction and valence bands cross each other linearly in thethree-dimensional momentum space. At the Weyl point,the Fermi surface usually shrinks into a point. The quasi-particle excitations near the Weyl point behave effec-tively as a kind of massless relativistic particles known asWeyl fermions in the standard model. These Weyl pointsare monopoles of the Berry curvature field and thus leadto topological boundary states on the surface. The topo-logical surface state connects a pair of Weyl points withopposite chiralities, resulting in an unclosed curve in theFermi surface, called the Fermi arc. Fermi arcs serve ahallmark for the detection of Weyl fermions by ARPESand scanning tunneling microscopy . Soon transportexperiments have been extensively studied on these ma-terials for large magnetoresistance (MR) , the chiralanomaly effect , the gravitational anomaly effect ,optical response , and even catalysis .Very recently a new type of WSMs were antici-pated and discovered in MoTe , called the type-II WSM. The Weyl cone is strongly tilted so that theFermi velocity reverses its sign along the tilting direction(see Fig. 1). The type-II WSM exhibit a very differ-ent Fermi surface from a normal WSM (referred to astype-I) at the Weyl point. It has finite Fermi pockets,where the touching point between the electron and holepockets is the type-II Weyl point. Although it also ex-hibits Fermi arcs, the type-II Weyl cone is expected todemonstrate direction-dependent chiral anomaly effect and large photocurrents .For long, TaAs, TaP, NbAs and NbP are believed toexhibit topologically equivalent band structures as thetype-I WSM, where the length of Fermi arcs as well asthe Weyl point separation varies as the spin-orbit cou-pling (SOC) is different . In this work, we report FIG. 1. Schematic diagrams of different types of Weyl points.(a) The type-I Weyl cone. When the Fermi energy is suffi-ciently close to the Weyl points, the Fermi surface shrinks toa point. (b) The type-II Weyl cone. When the Fermi energycrosses the Weyl point, the Weyl point become the touchingpoint between hole and electron pockets in the Fermi surface,due to the strong tilting of the Weyl cone. that a group of Weyl points in NbP, which has the weak-est SOC among four compounds, are actually type-II bybulk and surface band structure calculations. This factis overlooked in previous research, mainly because theseWeyl points lie above the Fermi energy and cannot be di-rectly accessed in ARPES and transport measurements,and also because the tilting direction of the Weyl cone isaway from ordinary crystallographic axes.
II. METHODS
The band structure calculations were performed withthe density-functional theory within the generalized gra-dient approximation, which is implemented in the Vi-enna ab-initio simulation package ( vasp ) . The SOCwas included. We project the ab initio wave functions a r X i v : . [ c ond - m a t . m t r l - s c i ] A ug to the localized Wannier functions and constructed thetight binding Hamiltonian for bulk NbP, to interpolatethe band structure and Fermi surface in the Brilloui zoneby dense k -grids. For the surface states, we used twomodels. One is a semi-infinite surface model constructedby the tight-binding parameters based on the bulk Wan-nier orbitals. The other is a slab model with a thicknessof 15 bulk unit-cells, calculated in a fully ab initio way.Two models give different Fermi arc states due to differ-ent boundary conditions, but exhibit the same topologyin the Fermi surface. III. RESULTS
The energy dispersion of NbP presents as a semimetal-lic state with tiny electron and hole pockets cuttingthe Fermi level. Due to inversion symmetry breaking,spin degeneracy splits for all the bands , as shownin Fig. 2(a). Based on the calculated band structurewe have achieved good agreement with experimentalmagnetic quantum-oscillation measurements, which con-firmed the accuracy of our calculations ? ? ? . Consistentwith previous reports, there are totally twelve pairs ofWeyl points in the whole Brillouin zone (BZ), with fourpairs in the k z = 0 plane (W1) and the others in theplane around k z ∼ ± π/c (W2), which are 57 meV bellowand 5 meV above the Fermi level in energy space, respec-tively ? ? ? ,8 . The Fermi surface topology revealed thatW1-type Weyl points are type-I. ? .The energy dispersion crossing the W2 Weyl pointalong high symmetry directions of k x , k y and k z presentvery good type-I features, as presented in Fig. 2(b-d).That should be a simple reason why W2 Weyl pointswas assumed to be a type-I. However, it cannot excludethe possibility that the W2 Wely cone is strongly tiltedalong some low-symmetric direction.For further identifying the feature of the Weyl point inNbP, we analyzed the Fermi surface accordingly. As itis already known from quantum oscillation and previouscalculations, when the Fermi level is 5 meV bellow Weylpoints, both of the electron and hole pockets present asbanana shape with strong anisotropy ? . The Fermi sur-faces can be separated into two branches in the reciprocalspace without any touching point. However, if we shiftup the chemical potential exactly to the energy of W2Weyl points , it is found that electron and hole pocketstouches each other in k z ∼ ± π/c planes. Here, W2 Weylpoints are exactly the touching points, presenting clearlya type-II feature. Taking the Fermi surfaces with M y symmetry as the example, Fig. 3(a), we can see that theelectron and hole pockets linear touching each other alongthe direction of 17 ◦ off the k z axis. So the Weyl coneis strongly titled in the k y = k Wy planes (the position ofWeyl point W2 is ( k Wx , k Wy , k Wz )). As shown in Fig. 3(b),The 2D cross-section of Fermi surfaces in k y = k Wy planeforms a linear crossing but not shrink to a point. We alsoanalysed the energy dispersion crossing the Weyl point along the titled direction, Fig. 3(c), which is completelydifferent from that along high symmetry lines as given inFig. 2(b-d). FIG. 2. (a) Bulk band structure of NbP. (b)-(d) The bandstructure along (b) k x , (c) k y and (d) k z directions. The redline is the highest valence band and the green line is the lowestconduction band. k x and k y are in unit 2 π/a , and k z is inunit 2 π/c . The type-II Weyl point is further confirmed by the sur-face state. Though both type-I and type-II Weyl pointscan induce the non-closed surface Fermi arc, the shapesof them are different. In type-I WSMs, the bulk Fermisurface shrinks to a point at Weyl point. Hence, whenthe energy is fixed at the Weyl point, the Fermi arc willterminate at two isolated points without any bulk den-sity of states. While in type-II WSMs, one can expect toobserve the linear touching of surface projected electronand hole pockets at the Weyl points, where are termina-tions of the Fermi arcs. In general, anion-terminated (P)surface in (001) direction was usually reported for theas-cleaved surface in ARPES for NbP . Since NbPis not a layered material, the chemical bonding is verystrong, and the charge redistribution plays an importantrole for the detailed shape of the surface Fermi arc states.Therefore, in this work we have first analysed the surfacestate by a tight-binding semi-infinite model, which pro-vides a clear understanding of the topological Fermi arcstate. Further, we also studied the surface state withsurface charge redistribution taken into account by fully ab-initio slab calculations. Tight-binding and ab-initio show different surface states, but the same topology.For convenience we just focus on one pair of Fermiarcs near the M y mirror plane. The surface band struc-ture with P termination along ¯k x with fixed k y = k Wy isgiven in Figure 4(a). Since two pairs of W2 points withthe same chirality are projected to the same point in the(001) 2D BZ, two Fermi arcs are expected from one pairof projected Weyl points. From surface energy disper- FIG. 3. (a) Eight pairs of Weyl points are denoted by greenand yellow dots to represent opposite chirality of type-II Weylpoints, and four pairs of Weyl points are denoted by blueand red dots to represent opposite chirality of type-I Weylpoints. Red and green pockets represent hole and electronpockets, respectively. The black dash line indicates the specialdirection for non-chiral anomaly. (b) The 3D band structureof the W2 Weyl cone in the k x -k z plane is heavily tilted, whichshows the type-II character of the W2 nodes in NbP. Theenergy contour at the bottom corresponds to the Fermi levelat W2 Weyl point. Two bands of upper and lower cones areindicated as green and red lines. The tilted line in (a) is alsoprojected into the contour. (c) The band structure along thetilted line in (a). The cross-section of the 3D band structureshows that the W2 node lies 5 meV above the intrinsic Fermilevel E f (neutral). k coordinates are in unit of reciprocalvectors. sion in Fig. 4(a) one can easily see that the conductionbands and valence bands linearly touching each other atthe projected Weyl point, with two surface bands cross-ing this points. Since no other surface band appears,we expect that these two states are just the Fermi arcsrelated states. For further understanding, we analyzedthe projected surface state with chemical potential ly-ing at the Weyl point, as shown in Fig. 4(b). One pairof projected Weyl points with opposite chirality are pre-sented as the linear touching point of electron and holepockets, and two clear non-closed Fermi arcs originatedfrom the linear touching points. Therefore, the type-IIWeyl point are directly confirmed by the co-existence oflinear touching of the projected bulk Fermi surfaces andcorresponding Fermi arcs.Though the half-infinite tight binding model providesthe correct understanding for the topological surface Fermi arc, the calculated surface state does not consis-tent with experiment ARPES measurement due to thelack of inclusion of surface charge redistribution. There-fore, in order to stimulate the realistic surface states,we also employed ab-initio calculations with a thick slabmodel. Though two kinds of calculations give the sametopology, where one close loop with one projected Weylpoint inside crosses surface FSs twice, the details of Fermiarc states are very different. In the former calculations,the two Fermi arcs extended in two opposite directionsalong k x , whereas two surface state just locate at thesame side of Weyl points in the fully ab-initio calcula-tions, see Fig. 4(c). In the slab model, the lengths ofsurface Fermi arcs increases and stay away from the bulkpockets. Therefore, if one can dope the sample with moreelectrons, it is expected to observe the linear touching be-tween projected electron and hole pockets at Weyl points,with surface Fermi arcs terminated at these two linearcrossing points. FIG. 4. (a) The band structure along ¯k x (¯k y = 0 . k x and k y are in unit 2 π/a . IV. SUMMARY
In conclusion, we have predicted the existence of type-II Weyl fermions in NbP based on electronic band struc-ture calculcations. We revealed that the Weyl cone istilted strongly along a specific direction, breaking theLorentz symmetry. Since the tilting direction is awayfrom ordinary a and c axes, one may still expect thechiral anomaly effect along a or c directions. Since thetype-II Weyl point is only 5 meV above the Fermi level,weakly electron doping may lead to the observation ofthe type-II Weyl points by ARPES. ACKNOWLEDGMENTS
This work was financially supported by the ERC (Ad-vanced Grant No. 291472 ”Idea Heusler”). B.Y. ac- knowledges the financial support of the Ruth and Her-man Albert Scholars Program for New Scientists in Weiz-mann Institute of Science and the German-Israeli Foun-dation (GIF Grant no. I-1364-303.7/2016). ∗ [email protected] X. G. Wan, A. M. Turner, A. Vishwanath, and S. Y.Savrasov, Phys. Rev. B , 205101 (2011). G. E. Volovik,
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