materials Article
Perfect Topological Metal CrB2: A One-Dimensional (1D) Nodal Line, a Zero-Dimensional (0D) Triply Degenerate Point, and a Large Linear Energy Range Yang Li 1, *, Jihong Xia 1, *, Rabah Khenata 2, * and Minquan Kuang 3, * 1 2 3
*
Department of Physics, Chongqing University of Arts and Sciences, Chongqing 402160, China Laboratoire de Physique Quantique de la Matiere et de Modelisation Mathematique (LPQ3M), Universite de Mascara, Mascara 29000, Algeria School of Physical Science and Technology, Southwest University, Chongqing 400715, China Correspondence: [email protected] (Y.L.); [email protected]; (J.X.); [email protected] (R.K.); [email protected] (M.K.)
Received: 17 August 2020; Accepted: 23 September 2020; Published: 28 September 2020
Abstract: Topological materials with band-crossing points exhibit interesting electronic characteristics and have special applications in electronic devices. However, to further facilitate the experimental detection of the signatures of these band crossings, topological materials with a large linear energy range around the band-crossing points need to be found, which is challenging. Here, via first-principle approaches, we report that the previously prepared P6/mmm-type CrB2 material is a topological metal with one pair of 1D band-crossing points, that is, nodal lines, in the kz = 0 plane, and one pair of 0D band-crossing points, that is, triple points, along the A–Γ–A’ paths. Remarkably, around these band-crossing points, a large linear energy range (larger than 1 eV) was found and the value was much larger than that found in previously studied materials with a similar linear crossing. The pair of nodal lines showed obvious surface states, which show promise for experimental detection. The effect of the spin–orbit coupling on the band-crossing points was examined and the gaps induced by spin–orbit coupling were found to be up to 69 meV. This material was shown to be phase stable in theory and was synthesized in experiments, and is therefore a potential material for use in investigating nodal lines and triple points. Keywords: DFT; CrB2 material; linear band crossings; topological metal; spin–orbit coupling; P6/mmm; electronic structures
1. Introduction Topological semimetals/metals [1–15], including nodal point semimetals/metals, nodal line semimetals/metals, and nodal surface semimetals/metals, with nontrivial band topologies, have attracted significant attention in the area of condensed matter physics. Topological semimetals/ metals can be roughly classified based on the dimensionality of the band crossings. In nodal point topological materials, the bands cross at isolated points. For example, in Weyl semimetals/metals [12], the valence band and conduction band touch at an isolated nodal point in the momentum space of solids. Around the isolated Weyl points, the low-energy quasiparticles are described using Weyl fermions with a definite chirality. If the system is protected from proper crystal symmetry, the opposite chirality of the two Weyl points can coincide at a certain point, resulting in a Dirac point. To date, the investigation of nodal point semimetals/metals has not been limited to the cases of Weyl and Dirac nodal point materials [16–20] with two-fold and four-fold degenerated band crossings. Some novel nodal point materials with three-fold, six-fold, and eight-fold degenerated band crossings have also
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been predicted. Significant efforts have been devoted to the prediction of nodal-point materials that can feature three-fold [21–25], six-fold [26,27], and eight-fold degenerated [28] band crossings. In nodal line semimetals/metals [29–38] and nodal surface semimetals/metals [38–41], the bandcrossing points form one-dimensional (1D) nodal lines and two-dimensional (2D) nodal surfaces, respectively, in the momentum space of solids. The nodal line materials were first predicted in carbon-based networks [42]. Nodal line materials have recently emerged as an interesting research topic due to their many interesting electronic and optical properties and their association with nontrivial drum-head-like surface states. Research on topological nodal surface materials is still in the preliminary stage, and many more in-depth physical problems still need to be solved [39]. Although some materials have been predicted to be nodal line or nodal point materials via first-principle calculations, the number of predicted triply degenerate nodal point semimetals/metals and the number of nodal line semimetals/metals is still limited. Importantly, for almost all of the previously studied topological materials, the linear energy range of the band dispersion around the band-crossing points is usually smaller than 1 eV [43]. It should be emphasized that this small linear energy range for the band dispersion adds a certain degree of difficulty to the subsequent experimental research. In this work, by using first-principle approaches, we report a P6/mmm-type CrB2 topological metal with one pair of 1D band-crossing points, that is, 1D nodal lines, in the kz = 0 plane, and one pair of 0D band-crossing points, that is, triply degenerate points, along the A–Γ–A’ paths. CrB2 , possessing a P6/mmm crystal structure, is an existing material [44], where the structural stability has been examined using phonon dispersion. Remarkably, a large linear band dispersion can be found around these band crossings, which should facilitate further experimental investigations. If spin–orbit coupling (SOC) is turned on, these band-crossing points become gapped; however, the gapped points have energies up to 69 meV. It should be noted that the gaps induced by spin–orbit coupling are smaller than other predicted topological materials with nodal line states. The CrB2 topological metal is a good candidate for investigating the physical properties of nodal point and nodal line fermions, as well as the relationship between them. 2. Methods and Materials To calculate the electronic structure and the topological signature of the CrB2 material, first-principle approaches were realized by using the Vienna ab initio simulation package [45]. The generalized gradient approximation (GGA) of the Perdew–Burke–Ernzerhof (PBE) [46] function was adopted as the exchange-correlation potential. Moreover, the cutoff energy was set as 600 eV, and the Brillouin zone (BZ) was sampled using a Monkhorst–Pack k-mesh with a size of 12 × 12 × 11. To determine the nontrivial surface states in CrB2 , the WANNIERTOOLS [47] package was used in this study. It should be highlighted that the CrB2 was prepared previously via direct synthesis from the elements using a method reported by Post et al. in 1954 [44]. The experimental lattice constants of CrB2 are reported to be a = b = 2.97 Å and c = 3.07 Å. The crystal structure of CrB2 was completely relaxed and the optimized lattice parameters are a = b = 2.97 Å and c = 2.93 Å. These values match well with those found in the experiment. The atomic positions are Cr at (0, 0, 0) and B at (0.66666, 0.333333, 0.5) and (0.333333, 0.66666, 0.5), and the corresponding relaxed crystal models under different views are shown in Figure 1a,b. Some information, including the elasticity, ICSD IDs, and the calculated X-ray diffraction can be found in the Materials Project database [48]. Based on the crystal structure obtained for CrB2 , the dynamical stability of CrB2 was examined based on the calculated phonon dispersion along the Γ–M–K–Γ–A–L–H–A paths (see Figure 1c). Usually, if the system does not contain imaginary frequencies in the phonon dispersion, it can be regarded as having dynamic stability [49–51]. As shown in Figure 2, the absence of imaginary frequencies in the first BZ reflects its dynamic stability.
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Figure 1. (a,b) Structural models of P6/mmm CrB2 from different viewpoints and (c) the bulk Brillouin zone of CrB2.
Based on the crystal structure obtained for CrB2, the dynamical stability of CrB2 was examined based on the calculated phonon dispersion along the Γ–M–K–Γ–A–L–H–A paths (see Figure 1c). Usually, if the system does not contain imaginary frequencies in the phonon dispersion, it can be regarded having dynamic stability [49–51]. shown in viewpoints Figure 2, and the (c) absence imaginary Figure models of CrB from different the Brillouin Figureas1. 1. (a,b) (a,b)Structural Structural models ofP6/mmm P6/mmm CrB22As from different viewpoints and (c) the bulk bulkof Brillouin frequencies in the zone of of CrB CrB zone 22.. first BZ reflects its dynamic stability. Based on the crystal structure obtained for CrB2, the dynamical stability of CrB2 was examined based on the calculated phonon dispersion along the Γ–M–K–Γ–A–L–H–A paths (see Figure 1c). Usually, if the system does not contain imaginary frequencies in the phonon dispersion, it can be regarded as having dynamic stability [49–51]. As shown in Figure 2, the absence of imaginary frequencies in the first BZ reflects its dynamic stability.
Figure 2. Phonon Phonon dispersion dispersion curve curve for for CrB CrB22..
3. Electronic 2.1. Electronic Structures Structures Figure 3a,b 3a,b shows shows the the calculated calculated density density of of states states (DOS) (DOS) and and the the calculated calculated band band structure structure of of the the Figure CrB22 material, material, respectively. 3a,3a, one cancan seesee an energy peakpeak around the Fermi level, CrB respectively.As Asshown shownininFigure Figure one an energy around the Fermi indicating that the CrB exhibited metallic properties. Moreover, based on the projected DOS, the total 2 CrB2 exhibited metallic properties. Moreover, based on the projected DOS, level, indicating that the Figure 2. Phonon dispersion curve for dominated CrB2. density of states in the energy range from −2 to 0 eV was mainly by the Cr–d orbitals. the total density of states in the energy range from −2 to 0 eV was mainly dominated by the Cr–d Figure 3b shows the calculated band structure of CrB2 along the Γ–M–K–Γ–A–L–H–A paths. orbitals. 2.1. Electronic Structures From Figure3b 3b, we can seecalculated three obvious points, named P2, and P3, locatedpaths. along Figure shows the bandband-crossing structure of CrB 2 along the P1, Γ–M–K–Γ–A–L–H–A the M–K path, and Γ–Aobvious path, respectively. P1 and P2, were two-fold degenerated Figure 3a,b shows calculated density of statesFor (DOS) andnamed the they calculated band structure of the From Figure 3b,K–Γ we path, canthe see three band-crossing points, P1, P2, and P3, located along nodal points; however, for P3, they were triply degenerate nodal points. Surprisingly, around these CrB 2 material, respectively. As shown in Figure 3a, one can see an energy peak around the Fermi the M–K path, K–Γ path, and Γ–A path, respectively. For P1 and P2, they were two-fold degenerated three nodal points, the energy range of the linear band dispersion was more than 1 eV (see the areas level, indicating that thefor CrB 2 exhibited properties.nodal Moreover, based on the projected nodal points; however, P3, they weremetallic triply degenerate points. Surprisingly, around DOS, these withtotal anodal yellow background Figure 3b). This large linear range wasdominated much larger most the density of the states ininthe energy range from −2dispersion to 0energy eV was mainly bythan the areas Cr–d three points, energy range of the linear band was more than 1 eV (see the of the other proposed materials with a linear-type band dispersion [43]. Moreover, this large linear orbitals. with a yellow background in Figure 3b). This large linear energy range was much larger than most energy range promisingband candidate for experimental on the interesting physics Figure 3bmakes showsCrB the2 acalculated structure of CrB2 alongstudies the Γ–M–K–Γ–A–L–H–A paths. related to band From Figure 3b,crossings. we can see three obvious band-crossing points, named P1, P2, and P3, located along the M–K path, K–Γ path, and Γ–A path, respectively. For P1 and P2, they were two-fold degenerated nodal points; however, for P3, they were triply degenerate nodal points. Surprisingly, around these three nodal points, the energy range of the linear band dispersion was more than 1 eV (see the areas with a yellow background in Figure 3b). This large linear energy range was much larger than most
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of the other proposed materials with a linear-type band dispersion [43]. Moreover, this large linear energy2020, range CrB2 a promising candidate for experimental studies on the interesting physics Materials 13, makes 4321 4 of 10 related to band crossings.
(a)
(b)
Figure 3. (a) The total and projected density of states (PDOS) of CrB2 without considering the spin– orbit coupling (SOC) effect and (b) the calculated band structure of CrB2 without SOC. The energy at the Fermi level was set to zero.
Based on the hybrid examined the band structure (a) functional (HSE-06) method [52], we further(b) of CrB2 along the M–K–Γ–A paths. The HSE-06 method is usually used to obtain an accurate large Figure3.3.(a) (a)The Thetotal totaland and projected density of states (PDOS) of CrB 2 without considering the spin– Figure projected density of states (PDOS) of CrB without considering the spin–orbit 2 for range of the linear band dispersion and band-crossing points topological materials. The results 2 without SOC. The energy at orbit coupling (SOC) effect and (b) the calculated band structure of CrB coupling (SOC) effect and (b) the calculated band structure of CrB2 without SOC. The energy at the are given in Figure 4a, where one can see that all of the nodal points, that is, the two two-fold the Fermi to zero. Fermi levellevel was was set toset zero. degenerated points and a triply degenerate point, were retained. In addition, the large energy range of the band found close to these band-crossing points.examined Based on the functional (HSE-06) method [52], Based ondispersion thehybrid hybridwas functional (HSE-06) method [52],we wefurther further examinedthe theband bandstructure structure The band structure at the experimental lattice constants [44] was to also calculated based large on the of an ofCrB CrB22 along along the the M–K–Γ–A M–K–Γ–Apaths. paths. The The HSE-06 HSE-06 method method is is usually usually used used toobtain obtain anaccurate accurate large GGAofof method, and the results are given inband-crossing Figure 4b. By comparing Figure 3b and Figure we found range the band dispersion and band-crossing points for for topological materials. The4b, results are range thelinear linear band dispersion and points topological materials. The results that there was only a small difference between the two cases. In the linear band dispersion range, the given in Figure 4a, where canone see that the all nodal thatpoints, is, the two degenerated are given in Figure 4a, one where can all seeofthat of points, the nodal thattwo-fold is, the two two-fold energy bands of the two cases were basically consistent (see the areas with a yellow background points and a triply point, were retained. In addition, large energy rangeenergy of the band degenerated pointsdegenerate and a triply degenerate point, were retained.the In addition, the large rangein 3bwas and the areas with a green background in Figure 4b). dispersion found close to these band-crossing points. ofFigure the band dispersion was found close to these band-crossing points. The band structure at the experimental lattice constants [44] was also calculated based on the GGA method, and the results are given in Figure 4b. By comparing Figure 3b and Figure 4b, we found that there was only a small difference between the two cases. In the linear band dispersion range, the energy bands of the two cases were basically consistent (see the areas with a yellow background in Figure 3b and the areas with a green background in Figure 4b).
(a)
(b)
Figure 4. (a) The band structure of CrB2 calculated using the hybrid functional (HSE06) method and Figure 4. (a) The band structure of CrB2 calculated using the hybrid functional (HSE06) method and 2 using the generalized gradient approximation (GGA) method under (b) band structure CrBusing (b) thethe band structure of of CrB the generalized gradient approximation (GGA) method under 2 experimental lattice constants. experimental lattice constants.
2.2.The Topological Signatures (a)at the experimental lattice constants [44] was also (b) band structure calculated based on the GGA method, and the results are given in Figure 4b. By comparing Figures 3b and 4b, we found that there In this4.section, we discuss theoftopological signatures ofhybrid the three band-crossing points. The Figure (a) The band structure CrB2 calculated using the functional (HSE06) method and three was only a small difference between the two cases. In the linear band dispersion range, the energy 2 using the generalized gradient approximation (GGA)nodal method under (b) the bandpoints structure CrB band-crossing canofbe classified into two types: (i) two-fold degenerated points P1 and bandsexperimental of the two cases basically consistent (see the areas with a yellow background in Figure 3b latticewere constants. and the areas with a green background in Figure 4b). 2.2. Topological Signatures 4. Topological Signatures In this section, we discuss the topological signatures of the three band-crossing points. The three In this section, we discuss the topological signatures of the three band-crossing points. The three band-crossing points can be classified into two types: (i) two-fold degenerated nodal points P1 and band-crossing points can be classified into two types: (i) two-fold degenerated nodal points P1 and P2 and (ii) the triply degenerate nodal point P3. Based on the arguments made by Weng et al. [53], these doubly-degenerated crossings (i.e., P1 and P2) should belong in a line and these band-crossing
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Materials 2020,the 13, triply 4321 degenerate nodal point P3. Based on the arguments made by Weng et al. 5[53], of 10 P2 and (ii) these doubly-degenerated crossings (i.e., P1 and P2) should belong in a line and these band-crossing points cannot be seen as isolated points. However, for the triply degenerate nodal point, this can points cannot be seen as isolated points. However, for the triply degenerate nodal point, this can occur occur both in isolation and at nodal line connections. both in isolation and at nodal line connections. Let us first consider the two-fold band-crossing points P1 and P2. Based on the bulk BZ, as Let us first consider the two-fold band-crossing points P1 and P2. Based on the bulk BZ, as shown shown in Figure 1c, one can see that the P1 and P2 nodal points were all located in the kz = 0 plane. in Figure 1c, one can see that the P1 and P2 nodal points were all located in the kz = 0 plane. To prove To prove that P1 and P2 reside on a nodal line, the K-centered three-dimensional (3D) plotting of the that P1 and P2 reside on a nodal line, the K-centered three-dimensional (3D) plotting of the two bands two bands in the kz = 0 plane is given in Figure 5. The white line in Figure 5 shows the intersections in the kz = 0 plane is given in Figure 5. The white line in Figure 5 shows the intersections between the between the two bands, namely, an obviously closed line. As shown in Figure 5, we can see that the two bands, namely, an obviously closed line. As shown in Figure 5, we can see that the band-crossing band-crossing points belonging to the nodal line are in the kz = 0 plane, and this nodal line had a slight points belonging to the nodal line are in the kz = 0 plane, and this nodal line had a slight energy energy variation. More importantly, from this figure, we can conclude that there should be a series variation. More importantly, from this figure, we can conclude that there should be a series of two-fold of two-fold degenerated nodal points, similar to P1 and P2, and around these points, there appeared degenerated nodal points, similar to P1 and P2, and around these points, there appeared to be a large to be a large energy range for the linear band crossings. It should be noted that this property should energy range for the linear band crossings. It should be noted that this property should facilitate the facilitate the experimental detection of the signatures of these band crossings. To capture the nature experimental detection of the signatures of these band crossings. To capture the nature of the nodal of the nodal lines, we calculated their Berry phase, as shown in Supplementary Materials Figure S1. lines, we calculated their Berry phase, as shown in Supplementary Materials Figure S1. From these From these calculations, we found that the Berry phase hosted a jump around the nodal line in the kz calculations, we found that the Berry phase hosted a jump around the nodal line in the kz = 0 plane. = 0 plane.
Figure 5. 5. 3D 3D band band dispersion dispersion around around the the K K point point in in the the kkzz ==00plane. plane.The Thewhite whiteline lineshows showsthe the profile profile Figure of the nodal line. of the nodal line.
The 2D 2D plane plane figure figure of of the the K-centered K-centered nodal nodal line line (located plane) is is The (located in in the the mirror-invariant mirror-invariant kkzz == 00 plane) shown in Figure 6a. For the CrB system, there were two independent mechanisms that protected the shown in Figure 6a. For the CrB22system, there were two independent mechanisms that protected the nodal line first was thethe mirror symmetry, as theasCrB a horizontal mirror nodal line in inthe thekzkz==0 0plane. plane.The The first was mirror symmetry, the2 hosted CrB2 hosted a horizontal plane; the other was the inversion symmetry and time-reversal symmetry. As the CrB hosted the mirror plane; the other was the inversion symmetry and time-reversal symmetry. As the 2CrB2 hosted time-reversal symmetry, one more nodalnodal line could found the K’ point, aspoint, shownas inshown Figure 6b. the time-reversal symmetry, one more line be could be around found around the K’ in We shall now discuss the triply degenerate nodal point, P3, along the Γ–A direction. These triply Figure 6b. degenerate nodal points were formed by the crossing of a two-fold degenerated band and a non-degenerated band, which corresponded to the E1 and A1 irreducible representations, respectively, of the C6v symmetry. The 3D band dispersion along the A’–Γ–A paths in the ky = 0 plane is shown in Figure 7a, reflecting the occurrence of the triply degenerate nodal points. The arrows show the profile of the triply degenerate points. As the CrB2 system featured time-reversal symmetry, one pair of triply nodal points could be found, where one was located along the Γ–A path and the other was located along the Γ–A’ path. The position of the triply degenerate points in the 3D BZ is shown in Figure 7b.
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(a)
(b)
Figure 6. (a) 2D shape of the K-centered nodal line in the kz = 0 plane (highlighted by the white line) and (b) an illustration of one pair of nodal lines (highlighted by red lines) in the kz = 0 plane.
We shall now discuss the triply degenerate nodal point, P3, along the Γ–A direction. These triply degenerate nodal points were formed by the crossing of a two-fold degenerated band and a nondegenerated band, which corresponded to the E1 and A1 irreducible representations, respectively, of the C6v symmetry. The 3D band dispersion along the A’–Γ–A paths in the ky = 0 plane is shown in Figure 7a, reflecting the occurrence of the triply degenerate nodal points. The arrows show the profile (a) (b) of the triply degenerate points. As the CrB2 system featured time-reversal symmetry, one pair of triply Figure 6. of K-centered line kkzz ==0the by white line) nodal points could be found, one nodal was along Γ–A(highlighted path and the other was located Figure 6. (a) (a)2D 2Dshape shape of the thewhere K-centered nodallocated line in in the the 0plane plane (highlighted bythe the white line) ==0shown and (b) an one of nodal lines (highlighted by lines) the kkzzis along the Γ–A’ path. Theof position triply thein 3D BZ and (b) an illustration illustration of one pair pairof ofthe nodal linesdegenerate (highlightedpoints by red redin lines) in the 0plane. plane.in Figure 7b. We shall now discuss the triply degenerate nodal point, P3, along the Γ–A direction. These triply degenerate nodal points were formed by the crossing of a two-fold degenerated band and a nondegenerated band, which corresponded to the E1 and A1 irreducible representations, respectively, of the C6v symmetry. The 3D band dispersion along the A’–Γ–A paths in the ky = 0 plane is shown in Figure 7a, reflecting the occurrence of the triply degenerate nodal points. The arrows show the profile of the triply degenerate points. As the CrB2 system featured time-reversal symmetry, one pair of triply nodal points could be found, where one was located along the Γ–A path and the other was located along the Γ–A’ path. The position of the triply degenerate points in the 3D BZ is shown in Figure 7b.
(a)
(b)
Figure 7. 7. (a) (a) 3D 3D band band dispersion dispersion along along the the A’–Γ–A A’–Γ–A paths paths in inthe thekky == 00 plane. plane. The The arrows arrows show show the the Figure y profile of the triply degenerate points and (b) the position of the triply degenerate points in the 3D profile of the triply degenerate points and (b) the position of the triply degenerate points in the 3D Brillouin zone (BZ). The triply degenerate nodal points are highlighted using red balls. Brillouin zone (BZ). The triply degenerate nodal points are highlighted using red balls.
Finally, itit should should be be highlighted highlighted that that the the CrB CrB22 system system co-exhibited co-exhibited the band crossing Finally, the 0D 0D band crossing point, point, for example, the triply degenerate nodal points, and the 1D band-crossing points, that is, the nodal for example, the triply degenerate nodal points, and the 1D band-crossing points, that is, the nodal line states. states. Therefore, Therefore, this this material material can can be be seen seen as as having having good good potential potential prospects prospects for for studying studying the the line entanglement between between the the nodal nodal point point fermions fermions and and the the nodal nodal line line fermions fermions in inthe thefuture. future. entanglement (a) (b) 5. and thedispersion Effectofof Spin–Orbit Coupling 2.3.Surface Surface and the Effect Spin–Orbit Figure States 7.States (a) 3D band along the Coupling A’–Γ–A paths in the ky = 0 plane. The arrows show the profile of the triply degenerate points and (b) the position of the triply degenerate points in the 3D Usually, enjoy nontrivial surface states. TheThe K-centered nodalnodal line inline thein kz the = 0 plane Usually,1D 1Dnodal nodallines lines enjoy nontrivial surface states. K-centered kz = 0 Brillouin zone (BZ). The triply degenerate nodal points are highlighted using red balls.
was projected onto surface (001) (see Figure where the resultthe is given From Figure 8b, plane was projected onto surface (001) (see8a), Figure 8a), where resultinisFigure given 8b. in Figure 8b. From we can 8b, observe thatobserve the nontrivial states (highlighted by the arrow) around the Figure we can that thesurface nontrivial surface states (highlighted by appeared the arrow) appeared Finally, it should be highlighted that the CrB2 system co-exhibited the 0D band crossing point, band-crossing points (highlighted using yellow balls). These clear nontrivial surface states show for example, the triply degenerate nodal points, and the 1D band-crossing points, that is, the nodal promise for being detected in future experimental studies. line states. Therefore, this material can be seen as having good potential prospects for studying the entanglement between the nodal point fermions and the nodal line fermions in the future.
2.3. Surface States and the Effect of Spin–Orbit Coupling Usually, 1D nodal lines enjoy nontrivial surface states. The K-centered nodal line in the kz = 0 plane was projected onto surface (001) (see Figure 8a), where the result is given in Figure 8b. From Figure 8b, we can observe that the nontrivial surface states (highlighted by the arrow) appeared
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aroundthe theband-crossing band-crossing points (highlighted (highlighted using Materials 2020, 13, 4321 7 of 10 around points using yellow yellow balls). balls).These Theseclear clearnontrivial nontrivialsurface surfacestates states show promise for being detected in future experimental studies. show promise for being detected in future experimental studies.
Figure 8. (a) 3D bulk BZ and 2D surface BZ and (b) projected spectrum on the (001) surface of CrB2. Figure 8. (a) 3D bulk BZ and 2D surface BZ and (b) projected spectrum on the (001) surface of CrB2 . Figure 8. (a) 3D bulkpoints BZ and surface BZ and (b) projected on the (001) surface of CrB The band-crossing and2Dthe nontrivial surface states arespectrum highlighted by the yellow balls and2. The band-crossing points and the nontrivial surface states are highlighted by the yellow balls and The band-crossing points and the nontrivial surface states are highlighted by the yellow balls and arrow, respectively. arrow, respectively. arrow, respectively.
Finally, we we examined examined the the effect effect of of spin–orbit spin–orbit coupling 2 Finally, coupling on on the the electronic electronicstructure structureofofthe theCrB CrB 2 Finally, we examined the effect of spin–orbit coupling on the electronic structure of the CrB system. The of CrB 2 investigated using the GGA method and the SOC effect are shown 2 system. Theband bandstructure structure of CrB 2 investigated using the GGA method and the SOC effect are system. The structure of CrB 2 investigated using the GGApredicted method and the line SOCmaterials, effect are shown in Figure 9.band It should be noted almost all the previously nodal triply shown in Figure 9. It should bethat noted that almost all the previously predicted nodal line materials, indegenerate Figure 9. It should be materials, noted thatand almost all the previously predicted nodal line materials, triply nodal point SOC-induced gaps usually appear at these band-crossing triply degenerate nodal point materials, and SOC-induced gaps usually appear at these band-crossing degenerate nodal point materials, and SOC-induced gaps usually appear at these band-crossing points. For For the the CrB CrB2 system, the SOC-induced gaps for the P1, P2, and P3 points were approximately points. 2 system, the SOC-induced gaps for the P1, P2, and P3 points were approximately points. 2 system, the SOC-induced gaps for the P1, P2, and P3 points were 2approximately 61, 50, For andthe 69 CrB meV, respectively. 61, 50, and 69 meV, respectively. However, However, the the spin–orbit spin–orbit coupling couplinggaps gapsininthe theCrB CrB system systemwere were 61, 50, and 69some meV,ofrespectively. However, the spin–orbit coupling themeV) CrB22[54], system were smaller than the other predicted topological materials, such as gaps BaSn2in (>50 CaAgBi smaller than some of the other predicted topological materials, such as BaSn2 (>50 meV) [54], CaAgBi smaller than[55], some ofTiOs the other (>80 meV) and (>100predicted meV) [56].topological materials, such as BaSn2 (>50 meV) [54], CaAgBi (>80 meV) [55], and TiOs (>100 meV) [56]. (>80 meV) [55], and TiOs (>100 meV) [56].
Figure 9. Band structure of CrB2 along the M–K–Γ paths calculated using a GGA. The spin–orbit coupling effect considered. Figure 9. Band Bandwas structure of CrB CrB22 along alongthe the M–K–Γ M–K–Γ paths paths calculated calculated using using aa GGA. GGA. The The spin–orbit spin–orbit Figure 9. structure of coupling effect effect was was considered. coupling
Finally, it should be further emphasized that topological materials with a 1D nodal line, 0D triply degenerate and afurther large emphasized energy rangethat (>1topological eV) for thematerials linear band Finally, points, it should should be further emphasized that topological materials withdispersion 1D nodal nodalaround line,0D 0Dband triply Finally, it be with aa 1D line, triply crossings have very rarely been reported in realistic materials [43]. Therefore, experimental degenerate points, energy range (>1 eV) linear around band crossings degenerate points,and anda large a large energy range (>1 for eV)the for the band lineardispersion band dispersion around band confirmation of the novel properties proposed here is urgent. have very rarely been reported in realistic materials [43]. Therefore, experimental confirmation of the crossings have very rarely been reported in realistic materials [43]. Therefore, experimental novel properties proposed here is urgent. confirmation of the novel properties proposed here is urgent.
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6. Conclusions Using first-principle approaches, we report that P6/mmm CrB2 was found to be a perfect topological metal with many interesting properties, where the conclusions of this study are summarized as follows: 1. 2. 3. 4. 5. 6.
CrB2 is an existing material and was confirmed to be dynamically stable based on the calculated phonon dispersion. CrB2 featured two types of topological elements: (i) one pair of 1D nodal lines in the kz = 0 plane and (ii) one pair of 0D triply degenerate nodal points along the A’–Γ–A paths. These band-crossing points were very robust against the effects of spin–orbit coupling. The energy range of the linear band dispersion was very large around the band-crossing points. The nontrivial surface states were around the band-crossing points and were very clear. The large linear energy range, 0D and 1D band crossings, and obvious nontrivial surface states observed in CrB2 will facilitate the experimental detection of potential topological elements.
Supplementary Materials: The following are available online at http://www.mdpi.com/1996-1944/13/19/4321/s1, Figure S1: The Berry phase along the M–K–Γ k-path for closed nodal line. Author Contributions: Conceptualization, Y.L., M.K. and R.K.; Formal analysis, Y.L. and J.X.; Investigation, Y.L., M.K., J.X., and R.K.; Methodology, Y.L., M.K. and R.K.; Supervision, R.K., J.X., and K.M.; Writing—original draft, Y.L., M.K., and R.K. All authors have read and agreed to the published version of the manuscript. Funding: This work is supported by the Science and Technology Research Program of the Chongqing Municipal Education Commission (grant no. KJQN201801346) and the Chongqing University of Arts and Sciences Foundation (grant no. Z2011Rcyj05). R. Khenata acknowledges the financial support of the General Direction of Scientific Research and Technological Development (DGRSDT). Conflicts of Interest: The authors declare no conflict of interest.
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Perfect Topological Metal CrB2: A One-Dimensional (1D) Nodal Line, a Zero-Dimensional (0D) Triply Degenerate Point, and a Large Linear Energy Range Yang Li 1, *, Jihong Xia 1, *, Rabah Khenata 2, * and Minquan Kuang 3, * 1 2 3
*
Department of Physics, Chongqing University of Arts and Sciences, Chongqing 402160, China Laboratoire de Physique Quantique de la Matiere et de Modelisation Mathematique (LPQ3M), Universite de Mascara, Mascara 29000, Algeria School of Physical Science and Technology, Southwest University, Chongqing 400715, China Correspondence: [email protected] (Y.L.); [email protected]; (J.X.); [email protected] (R.K.); [email protected] (M.K.)
Received: 17 August 2020; Accepted: 23 September 2020; Published: 28 September 2020
Abstract: Topological materials with band-crossing points exhibit interesting electronic characteristics and have special applications in electronic devices. However, to further facilitate the experimental detection of the signatures of these band crossings, topological materials with a large linear energy range around the band-crossing points need to be found, which is challenging. Here, via first-principle approaches, we report that the previously prepared P6/mmm-type CrB2 material is a topological metal with one pair of 1D band-crossing points, that is, nodal lines, in the kz = 0 plane, and one pair of 0D band-crossing points, that is, triple points, along the A–Γ–A’ paths. Remarkably, around these band-crossing points, a large linear energy range (larger than 1 eV) was found and the value was much larger than that found in previously studied materials with a similar linear crossing. The pair of nodal lines showed obvious surface states, which show promise for experimental detection. The effect of the spin–orbit coupling on the band-crossing points was examined and the gaps induced by spin–orbit coupling were found to be up to 69 meV. This material was shown to be phase stable in theory and was synthesized in experiments, and is therefore a potential material for use in investigating nodal lines and triple points. Keywords: DFT; CrB2 material; linear band crossings; topological metal; spin–orbit coupling; P6/mmm; electronic structures
1. Introduction Topological semimetals/metals [1–15], including nodal point semimetals/metals, nodal line semimetals/metals, and nodal surface semimetals/metals, with nontrivial band topologies, have attracted significant attention in the area of condensed matter physics. Topological semimetals/ metals can be roughly classified based on the dimensionality of the band crossings. In nodal point topological materials, the bands cross at isolated points. For example, in Weyl semimetals/metals [12], the valence band and conduction band touch at an isolated nodal point in the momentum space of solids. Around the isolated Weyl points, the low-energy quasiparticles are described using Weyl fermions with a definite chirality. If the system is protected from proper crystal symmetry, the opposite chirality of the two Weyl points can coincide at a certain point, resulting in a Dirac point. To date, the investigation of nodal point semimetals/metals has not been limited to the cases of Weyl and Dirac nodal point materials [16–20] with two-fold and four-fold degenerated band crossings. Some novel nodal point materials with three-fold, six-fold, and eight-fold degenerated band crossings have also
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been predicted. Significant efforts have been devoted to the prediction of nodal-point materials that can feature three-fold [21–25], six-fold [26,27], and eight-fold degenerated [28] band crossings. In nodal line semimetals/metals [29–38] and nodal surface semimetals/metals [38–41], the bandcrossing points form one-dimensional (1D) nodal lines and two-dimensional (2D) nodal surfaces, respectively, in the momentum space of solids. The nodal line materials were first predicted in carbon-based networks [42]. Nodal line materials have recently emerged as an interesting research topic due to their many interesting electronic and optical properties and their association with nontrivial drum-head-like surface states. Research on topological nodal surface materials is still in the preliminary stage, and many more in-depth physical problems still need to be solved [39]. Although some materials have been predicted to be nodal line or nodal point materials via first-principle calculations, the number of predicted triply degenerate nodal point semimetals/metals and the number of nodal line semimetals/metals is still limited. Importantly, for almost all of the previously studied topological materials, the linear energy range of the band dispersion around the band-crossing points is usually smaller than 1 eV [43]. It should be emphasized that this small linear energy range for the band dispersion adds a certain degree of difficulty to the subsequent experimental research. In this work, by using first-principle approaches, we report a P6/mmm-type CrB2 topological metal with one pair of 1D band-crossing points, that is, 1D nodal lines, in the kz = 0 plane, and one pair of 0D band-crossing points, that is, triply degenerate points, along the A–Γ–A’ paths. CrB2 , possessing a P6/mmm crystal structure, is an existing material [44], where the structural stability has been examined using phonon dispersion. Remarkably, a large linear band dispersion can be found around these band crossings, which should facilitate further experimental investigations. If spin–orbit coupling (SOC) is turned on, these band-crossing points become gapped; however, the gapped points have energies up to 69 meV. It should be noted that the gaps induced by spin–orbit coupling are smaller than other predicted topological materials with nodal line states. The CrB2 topological metal is a good candidate for investigating the physical properties of nodal point and nodal line fermions, as well as the relationship between them. 2. Methods and Materials To calculate the electronic structure and the topological signature of the CrB2 material, first-principle approaches were realized by using the Vienna ab initio simulation package [45]. The generalized gradient approximation (GGA) of the Perdew–Burke–Ernzerhof (PBE) [46] function was adopted as the exchange-correlation potential. Moreover, the cutoff energy was set as 600 eV, and the Brillouin zone (BZ) was sampled using a Monkhorst–Pack k-mesh with a size of 12 × 12 × 11. To determine the nontrivial surface states in CrB2 , the WANNIERTOOLS [47] package was used in this study. It should be highlighted that the CrB2 was prepared previously via direct synthesis from the elements using a method reported by Post et al. in 1954 [44]. The experimental lattice constants of CrB2 are reported to be a = b = 2.97 Å and c = 3.07 Å. The crystal structure of CrB2 was completely relaxed and the optimized lattice parameters are a = b = 2.97 Å and c = 2.93 Å. These values match well with those found in the experiment. The atomic positions are Cr at (0, 0, 0) and B at (0.66666, 0.333333, 0.5) and (0.333333, 0.66666, 0.5), and the corresponding relaxed crystal models under different views are shown in Figure 1a,b. Some information, including the elasticity, ICSD IDs, and the calculated X-ray diffraction can be found in the Materials Project database [48]. Based on the crystal structure obtained for CrB2 , the dynamical stability of CrB2 was examined based on the calculated phonon dispersion along the Γ–M–K–Γ–A–L–H–A paths (see Figure 1c). Usually, if the system does not contain imaginary frequencies in the phonon dispersion, it can be regarded as having dynamic stability [49–51]. As shown in Figure 2, the absence of imaginary frequencies in the first BZ reflects its dynamic stability.
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Figure 1. (a,b) Structural models of P6/mmm CrB2 from different viewpoints and (c) the bulk Brillouin zone of CrB2.
Based on the crystal structure obtained for CrB2, the dynamical stability of CrB2 was examined based on the calculated phonon dispersion along the Γ–M–K–Γ–A–L–H–A paths (see Figure 1c). Usually, if the system does not contain imaginary frequencies in the phonon dispersion, it can be regarded having dynamic stability [49–51]. shown in viewpoints Figure 2, and the (c) absence imaginary Figure models of CrB from different the Brillouin Figureas1. 1. (a,b) (a,b)Structural Structural models ofP6/mmm P6/mmm CrB22As from different viewpoints and (c) the bulk bulkof Brillouin frequencies in the zone of of CrB CrB zone 22.. first BZ reflects its dynamic stability. Based on the crystal structure obtained for CrB2, the dynamical stability of CrB2 was examined based on the calculated phonon dispersion along the Γ–M–K–Γ–A–L–H–A paths (see Figure 1c). Usually, if the system does not contain imaginary frequencies in the phonon dispersion, it can be regarded as having dynamic stability [49–51]. As shown in Figure 2, the absence of imaginary frequencies in the first BZ reflects its dynamic stability.
Figure 2. Phonon Phonon dispersion dispersion curve curve for for CrB CrB22..
3. Electronic 2.1. Electronic Structures Structures Figure 3a,b 3a,b shows shows the the calculated calculated density density of of states states (DOS) (DOS) and and the the calculated calculated band band structure structure of of the the Figure CrB22 material, material, respectively. 3a,3a, one cancan seesee an energy peakpeak around the Fermi level, CrB respectively.As Asshown shownininFigure Figure one an energy around the Fermi indicating that the CrB exhibited metallic properties. Moreover, based on the projected DOS, the total 2 CrB2 exhibited metallic properties. Moreover, based on the projected DOS, level, indicating that the Figure 2. Phonon dispersion curve for dominated CrB2. density of states in the energy range from −2 to 0 eV was mainly by the Cr–d orbitals. the total density of states in the energy range from −2 to 0 eV was mainly dominated by the Cr–d Figure 3b shows the calculated band structure of CrB2 along the Γ–M–K–Γ–A–L–H–A paths. orbitals. 2.1. Electronic Structures From Figure3b 3b, we can seecalculated three obvious points, named P2, and P3, locatedpaths. along Figure shows the bandband-crossing structure of CrB 2 along the P1, Γ–M–K–Γ–A–L–H–A the M–K path, and Γ–Aobvious path, respectively. P1 and P2, were two-fold degenerated Figure 3a,b shows calculated density of statesFor (DOS) andnamed the they calculated band structure of the From Figure 3b,K–Γ we path, canthe see three band-crossing points, P1, P2, and P3, located along nodal points; however, for P3, they were triply degenerate nodal points. Surprisingly, around these CrB 2 material, respectively. As shown in Figure 3a, one can see an energy peak around the Fermi the M–K path, K–Γ path, and Γ–A path, respectively. For P1 and P2, they were two-fold degenerated three nodal points, the energy range of the linear band dispersion was more than 1 eV (see the areas level, indicating that thefor CrB 2 exhibited properties.nodal Moreover, based on the projected nodal points; however, P3, they weremetallic triply degenerate points. Surprisingly, around DOS, these withtotal anodal yellow background Figure 3b). This large linear range wasdominated much larger most the density of the states ininthe energy range from −2dispersion to 0energy eV was mainly bythan the areas Cr–d three points, energy range of the linear band was more than 1 eV (see the of the other proposed materials with a linear-type band dispersion [43]. Moreover, this large linear orbitals. with a yellow background in Figure 3b). This large linear energy range was much larger than most energy range promisingband candidate for experimental on the interesting physics Figure 3bmakes showsCrB the2 acalculated structure of CrB2 alongstudies the Γ–M–K–Γ–A–L–H–A paths. related to band From Figure 3b,crossings. we can see three obvious band-crossing points, named P1, P2, and P3, located along the M–K path, K–Γ path, and Γ–A path, respectively. For P1 and P2, they were two-fold degenerated nodal points; however, for P3, they were triply degenerate nodal points. Surprisingly, around these three nodal points, the energy range of the linear band dispersion was more than 1 eV (see the areas with a yellow background in Figure 3b). This large linear energy range was much larger than most
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of the other proposed materials with a linear-type band dispersion [43]. Moreover, this large linear energy2020, range CrB2 a promising candidate for experimental studies on the interesting physics Materials 13, makes 4321 4 of 10 related to band crossings.
(a)
(b)
Figure 3. (a) The total and projected density of states (PDOS) of CrB2 without considering the spin– orbit coupling (SOC) effect and (b) the calculated band structure of CrB2 without SOC. The energy at the Fermi level was set to zero.
Based on the hybrid examined the band structure (a) functional (HSE-06) method [52], we further(b) of CrB2 along the M–K–Γ–A paths. The HSE-06 method is usually used to obtain an accurate large Figure3.3.(a) (a)The Thetotal totaland and projected density of states (PDOS) of CrB 2 without considering the spin– Figure projected density of states (PDOS) of CrB without considering the spin–orbit 2 for range of the linear band dispersion and band-crossing points topological materials. The results 2 without SOC. The energy at orbit coupling (SOC) effect and (b) the calculated band structure of CrB coupling (SOC) effect and (b) the calculated band structure of CrB2 without SOC. The energy at the are given in Figure 4a, where one can see that all of the nodal points, that is, the two two-fold the Fermi to zero. Fermi levellevel was was set toset zero. degenerated points and a triply degenerate point, were retained. In addition, the large energy range of the band found close to these band-crossing points.examined Based on the functional (HSE-06) method [52], Based ondispersion thehybrid hybridwas functional (HSE-06) method [52],we wefurther further examinedthe theband bandstructure structure The band structure at the experimental lattice constants [44] was to also calculated based large on the of an ofCrB CrB22 along along the the M–K–Γ–A M–K–Γ–Apaths. paths. The The HSE-06 HSE-06 method method is is usually usually used used toobtain obtain anaccurate accurate large GGAofof method, and the results are given inband-crossing Figure 4b. By comparing Figure 3b and Figure we found range the band dispersion and band-crossing points for for topological materials. The4b, results are range thelinear linear band dispersion and points topological materials. The results that there was only a small difference between the two cases. In the linear band dispersion range, the given in Figure 4a, where canone see that the all nodal thatpoints, is, the two degenerated are given in Figure 4a, one where can all seeofthat of points, the nodal thattwo-fold is, the two two-fold energy bands of the two cases were basically consistent (see the areas with a yellow background points and a triply point, were retained. In addition, large energy rangeenergy of the band degenerated pointsdegenerate and a triply degenerate point, were retained.the In addition, the large rangein 3bwas and the areas with a green background in Figure 4b). dispersion found close to these band-crossing points. ofFigure the band dispersion was found close to these band-crossing points. The band structure at the experimental lattice constants [44] was also calculated based on the GGA method, and the results are given in Figure 4b. By comparing Figure 3b and Figure 4b, we found that there was only a small difference between the two cases. In the linear band dispersion range, the energy bands of the two cases were basically consistent (see the areas with a yellow background in Figure 3b and the areas with a green background in Figure 4b).
(a)
(b)
Figure 4. (a) The band structure of CrB2 calculated using the hybrid functional (HSE06) method and Figure 4. (a) The band structure of CrB2 calculated using the hybrid functional (HSE06) method and 2 using the generalized gradient approximation (GGA) method under (b) band structure CrBusing (b) thethe band structure of of CrB the generalized gradient approximation (GGA) method under 2 experimental lattice constants. experimental lattice constants.
2.2.The Topological Signatures (a)at the experimental lattice constants [44] was also (b) band structure calculated based on the GGA method, and the results are given in Figure 4b. By comparing Figures 3b and 4b, we found that there In this4.section, we discuss theoftopological signatures ofhybrid the three band-crossing points. The Figure (a) The band structure CrB2 calculated using the functional (HSE06) method and three was only a small difference between the two cases. In the linear band dispersion range, the energy 2 using the generalized gradient approximation (GGA)nodal method under (b) the bandpoints structure CrB band-crossing canofbe classified into two types: (i) two-fold degenerated points P1 and bandsexperimental of the two cases basically consistent (see the areas with a yellow background in Figure 3b latticewere constants. and the areas with a green background in Figure 4b). 2.2. Topological Signatures 4. Topological Signatures In this section, we discuss the topological signatures of the three band-crossing points. The three In this section, we discuss the topological signatures of the three band-crossing points. The three band-crossing points can be classified into two types: (i) two-fold degenerated nodal points P1 and band-crossing points can be classified into two types: (i) two-fold degenerated nodal points P1 and P2 and (ii) the triply degenerate nodal point P3. Based on the arguments made by Weng et al. [53], these doubly-degenerated crossings (i.e., P1 and P2) should belong in a line and these band-crossing
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Materials 2020,the 13, triply 4321 degenerate nodal point P3. Based on the arguments made by Weng et al. 5[53], of 10 P2 and (ii) these doubly-degenerated crossings (i.e., P1 and P2) should belong in a line and these band-crossing points cannot be seen as isolated points. However, for the triply degenerate nodal point, this can points cannot be seen as isolated points. However, for the triply degenerate nodal point, this can occur occur both in isolation and at nodal line connections. both in isolation and at nodal line connections. Let us first consider the two-fold band-crossing points P1 and P2. Based on the bulk BZ, as Let us first consider the two-fold band-crossing points P1 and P2. Based on the bulk BZ, as shown shown in Figure 1c, one can see that the P1 and P2 nodal points were all located in the kz = 0 plane. in Figure 1c, one can see that the P1 and P2 nodal points were all located in the kz = 0 plane. To prove To prove that P1 and P2 reside on a nodal line, the K-centered three-dimensional (3D) plotting of the that P1 and P2 reside on a nodal line, the K-centered three-dimensional (3D) plotting of the two bands two bands in the kz = 0 plane is given in Figure 5. The white line in Figure 5 shows the intersections in the kz = 0 plane is given in Figure 5. The white line in Figure 5 shows the intersections between the between the two bands, namely, an obviously closed line. As shown in Figure 5, we can see that the two bands, namely, an obviously closed line. As shown in Figure 5, we can see that the band-crossing band-crossing points belonging to the nodal line are in the kz = 0 plane, and this nodal line had a slight points belonging to the nodal line are in the kz = 0 plane, and this nodal line had a slight energy energy variation. More importantly, from this figure, we can conclude that there should be a series variation. More importantly, from this figure, we can conclude that there should be a series of two-fold of two-fold degenerated nodal points, similar to P1 and P2, and around these points, there appeared degenerated nodal points, similar to P1 and P2, and around these points, there appeared to be a large to be a large energy range for the linear band crossings. It should be noted that this property should energy range for the linear band crossings. It should be noted that this property should facilitate the facilitate the experimental detection of the signatures of these band crossings. To capture the nature experimental detection of the signatures of these band crossings. To capture the nature of the nodal of the nodal lines, we calculated their Berry phase, as shown in Supplementary Materials Figure S1. lines, we calculated their Berry phase, as shown in Supplementary Materials Figure S1. From these From these calculations, we found that the Berry phase hosted a jump around the nodal line in the kz calculations, we found that the Berry phase hosted a jump around the nodal line in the kz = 0 plane. = 0 plane.
Figure 5. 5. 3D 3D band band dispersion dispersion around around the the K K point point in in the the kkzz ==00plane. plane.The Thewhite whiteline lineshows showsthe the profile profile Figure of the nodal line. of the nodal line.
The 2D 2D plane plane figure figure of of the the K-centered K-centered nodal nodal line line (located plane) is is The (located in in the the mirror-invariant mirror-invariant kkzz == 00 plane) shown in Figure 6a. For the CrB system, there were two independent mechanisms that protected the shown in Figure 6a. For the CrB22system, there were two independent mechanisms that protected the nodal line first was thethe mirror symmetry, as theasCrB a horizontal mirror nodal line in inthe thekzkz==0 0plane. plane.The The first was mirror symmetry, the2 hosted CrB2 hosted a horizontal plane; the other was the inversion symmetry and time-reversal symmetry. As the CrB hosted the mirror plane; the other was the inversion symmetry and time-reversal symmetry. As the 2CrB2 hosted time-reversal symmetry, one more nodalnodal line could found the K’ point, aspoint, shownas inshown Figure 6b. the time-reversal symmetry, one more line be could be around found around the K’ in We shall now discuss the triply degenerate nodal point, P3, along the Γ–A direction. These triply Figure 6b. degenerate nodal points were formed by the crossing of a two-fold degenerated band and a non-degenerated band, which corresponded to the E1 and A1 irreducible representations, respectively, of the C6v symmetry. The 3D band dispersion along the A’–Γ–A paths in the ky = 0 plane is shown in Figure 7a, reflecting the occurrence of the triply degenerate nodal points. The arrows show the profile of the triply degenerate points. As the CrB2 system featured time-reversal symmetry, one pair of triply nodal points could be found, where one was located along the Γ–A path and the other was located along the Γ–A’ path. The position of the triply degenerate points in the 3D BZ is shown in Figure 7b.
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(a)
(b)
Figure 6. (a) 2D shape of the K-centered nodal line in the kz = 0 plane (highlighted by the white line) and (b) an illustration of one pair of nodal lines (highlighted by red lines) in the kz = 0 plane.
We shall now discuss the triply degenerate nodal point, P3, along the Γ–A direction. These triply degenerate nodal points were formed by the crossing of a two-fold degenerated band and a nondegenerated band, which corresponded to the E1 and A1 irreducible representations, respectively, of the C6v symmetry. The 3D band dispersion along the A’–Γ–A paths in the ky = 0 plane is shown in Figure 7a, reflecting the occurrence of the triply degenerate nodal points. The arrows show the profile (a) (b) of the triply degenerate points. As the CrB2 system featured time-reversal symmetry, one pair of triply Figure 6. of K-centered line kkzz ==0the by white line) nodal points could be found, one nodal was along Γ–A(highlighted path and the other was located Figure 6. (a) (a)2D 2Dshape shape of the thewhere K-centered nodallocated line in in the the 0plane plane (highlighted bythe the white line) ==0shown and (b) an one of nodal lines (highlighted by lines) the kkzzis along the Γ–A’ path. Theof position triply thein 3D BZ and (b) an illustration illustration of one pair pairof ofthe nodal linesdegenerate (highlightedpoints by red redin lines) in the 0plane. plane.in Figure 7b. We shall now discuss the triply degenerate nodal point, P3, along the Γ–A direction. These triply degenerate nodal points were formed by the crossing of a two-fold degenerated band and a nondegenerated band, which corresponded to the E1 and A1 irreducible representations, respectively, of the C6v symmetry. The 3D band dispersion along the A’–Γ–A paths in the ky = 0 plane is shown in Figure 7a, reflecting the occurrence of the triply degenerate nodal points. The arrows show the profile of the triply degenerate points. As the CrB2 system featured time-reversal symmetry, one pair of triply nodal points could be found, where one was located along the Γ–A path and the other was located along the Γ–A’ path. The position of the triply degenerate points in the 3D BZ is shown in Figure 7b.
(a)
(b)
Figure 7. 7. (a) (a) 3D 3D band band dispersion dispersion along along the the A’–Γ–A A’–Γ–A paths paths in inthe thekky == 00 plane. plane. The The arrows arrows show show the the Figure y profile of the triply degenerate points and (b) the position of the triply degenerate points in the 3D profile of the triply degenerate points and (b) the position of the triply degenerate points in the 3D Brillouin zone (BZ). The triply degenerate nodal points are highlighted using red balls. Brillouin zone (BZ). The triply degenerate nodal points are highlighted using red balls.
Finally, itit should should be be highlighted highlighted that that the the CrB CrB22 system system co-exhibited co-exhibited the band crossing Finally, the 0D 0D band crossing point, point, for example, the triply degenerate nodal points, and the 1D band-crossing points, that is, the nodal for example, the triply degenerate nodal points, and the 1D band-crossing points, that is, the nodal line states. states. Therefore, Therefore, this this material material can can be be seen seen as as having having good good potential potential prospects prospects for for studying studying the the line entanglement between between the the nodal nodal point point fermions fermions and and the the nodal nodal line line fermions fermions in inthe thefuture. future. entanglement (a) (b) 5. and thedispersion Effectofof Spin–Orbit Coupling 2.3.Surface Surface and the Effect Spin–Orbit Figure States 7.States (a) 3D band along the Coupling A’–Γ–A paths in the ky = 0 plane. The arrows show the profile of the triply degenerate points and (b) the position of the triply degenerate points in the 3D Usually, enjoy nontrivial surface states. TheThe K-centered nodalnodal line inline thein kz the = 0 plane Usually,1D 1Dnodal nodallines lines enjoy nontrivial surface states. K-centered kz = 0 Brillouin zone (BZ). The triply degenerate nodal points are highlighted using red balls.
was projected onto surface (001) (see Figure where the resultthe is given From Figure 8b, plane was projected onto surface (001) (see8a), Figure 8a), where resultinisFigure given 8b. in Figure 8b. From we can 8b, observe thatobserve the nontrivial states (highlighted by the arrow) around the Figure we can that thesurface nontrivial surface states (highlighted by appeared the arrow) appeared Finally, it should be highlighted that the CrB2 system co-exhibited the 0D band crossing point, band-crossing points (highlighted using yellow balls). These clear nontrivial surface states show for example, the triply degenerate nodal points, and the 1D band-crossing points, that is, the nodal promise for being detected in future experimental studies. line states. Therefore, this material can be seen as having good potential prospects for studying the entanglement between the nodal point fermions and the nodal line fermions in the future.
2.3. Surface States and the Effect of Spin–Orbit Coupling Usually, 1D nodal lines enjoy nontrivial surface states. The K-centered nodal line in the kz = 0 plane was projected onto surface (001) (see Figure 8a), where the result is given in Figure 8b. From Figure 8b, we can observe that the nontrivial surface states (highlighted by the arrow) appeared
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aroundthe theband-crossing band-crossing points (highlighted (highlighted using Materials 2020, 13, 4321 7 of 10 around points using yellow yellow balls). balls).These Theseclear clearnontrivial nontrivialsurface surfacestates states show promise for being detected in future experimental studies. show promise for being detected in future experimental studies.
Figure 8. (a) 3D bulk BZ and 2D surface BZ and (b) projected spectrum on the (001) surface of CrB2. Figure 8. (a) 3D bulk BZ and 2D surface BZ and (b) projected spectrum on the (001) surface of CrB2 . Figure 8. (a) 3D bulkpoints BZ and surface BZ and (b) projected on the (001) surface of CrB The band-crossing and2Dthe nontrivial surface states arespectrum highlighted by the yellow balls and2. The band-crossing points and the nontrivial surface states are highlighted by the yellow balls and The band-crossing points and the nontrivial surface states are highlighted by the yellow balls and arrow, respectively. arrow, respectively. arrow, respectively.
Finally, we we examined examined the the effect effect of of spin–orbit spin–orbit coupling 2 Finally, coupling on on the the electronic electronicstructure structureofofthe theCrB CrB 2 Finally, we examined the effect of spin–orbit coupling on the electronic structure of the CrB system. The of CrB 2 investigated using the GGA method and the SOC effect are shown 2 system. Theband bandstructure structure of CrB 2 investigated using the GGA method and the SOC effect are system. The structure of CrB 2 investigated using the GGApredicted method and the line SOCmaterials, effect are shown in Figure 9.band It should be noted almost all the previously nodal triply shown in Figure 9. It should bethat noted that almost all the previously predicted nodal line materials, indegenerate Figure 9. It should be materials, noted thatand almost all the previously predicted nodal line materials, triply nodal point SOC-induced gaps usually appear at these band-crossing triply degenerate nodal point materials, and SOC-induced gaps usually appear at these band-crossing degenerate nodal point materials, and SOC-induced gaps usually appear at these band-crossing points. For For the the CrB CrB2 system, the SOC-induced gaps for the P1, P2, and P3 points were approximately points. 2 system, the SOC-induced gaps for the P1, P2, and P3 points were approximately points. 2 system, the SOC-induced gaps for the P1, P2, and P3 points were 2approximately 61, 50, For andthe 69 CrB meV, respectively. 61, 50, and 69 meV, respectively. However, However, the the spin–orbit spin–orbit coupling couplinggaps gapsininthe theCrB CrB system systemwere were 61, 50, and 69some meV,ofrespectively. However, the spin–orbit coupling themeV) CrB22[54], system were smaller than the other predicted topological materials, such as gaps BaSn2in (>50 CaAgBi smaller than some of the other predicted topological materials, such as BaSn2 (>50 meV) [54], CaAgBi smaller than[55], some ofTiOs the other (>80 meV) and (>100predicted meV) [56].topological materials, such as BaSn2 (>50 meV) [54], CaAgBi (>80 meV) [55], and TiOs (>100 meV) [56]. (>80 meV) [55], and TiOs (>100 meV) [56].
Figure 9. Band structure of CrB2 along the M–K–Γ paths calculated using a GGA. The spin–orbit coupling effect considered. Figure 9. Band Bandwas structure of CrB CrB22 along alongthe the M–K–Γ M–K–Γ paths paths calculated calculated using using aa GGA. GGA. The The spin–orbit spin–orbit Figure 9. structure of coupling effect effect was was considered. coupling
Finally, it should be further emphasized that topological materials with a 1D nodal line, 0D triply degenerate and afurther large emphasized energy rangethat (>1topological eV) for thematerials linear band Finally, points, it should should be further emphasized that topological materials withdispersion 1D nodal nodalaround line,0D 0Dband triply Finally, it be with aa 1D line, triply crossings have very rarely been reported in realistic materials [43]. Therefore, experimental degenerate points, energy range (>1 eV) linear around band crossings degenerate points,and anda large a large energy range (>1 for eV)the for the band lineardispersion band dispersion around band confirmation of the novel properties proposed here is urgent. have very rarely been reported in realistic materials [43]. Therefore, experimental confirmation of the crossings have very rarely been reported in realistic materials [43]. Therefore, experimental novel properties proposed here is urgent. confirmation of the novel properties proposed here is urgent.
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6. Conclusions Using first-principle approaches, we report that P6/mmm CrB2 was found to be a perfect topological metal with many interesting properties, where the conclusions of this study are summarized as follows: 1. 2. 3. 4. 5. 6.
CrB2 is an existing material and was confirmed to be dynamically stable based on the calculated phonon dispersion. CrB2 featured two types of topological elements: (i) one pair of 1D nodal lines in the kz = 0 plane and (ii) one pair of 0D triply degenerate nodal points along the A’–Γ–A paths. These band-crossing points were very robust against the effects of spin–orbit coupling. The energy range of the linear band dispersion was very large around the band-crossing points. The nontrivial surface states were around the band-crossing points and were very clear. The large linear energy range, 0D and 1D band crossings, and obvious nontrivial surface states observed in CrB2 will facilitate the experimental detection of potential topological elements.
Supplementary Materials: The following are available online at http://www.mdpi.com/1996-1944/13/19/4321/s1, Figure S1: The Berry phase along the M–K–Γ k-path for closed nodal line. Author Contributions: Conceptualization, Y.L., M.K. and R.K.; Formal analysis, Y.L. and J.X.; Investigation, Y.L., M.K., J.X., and R.K.; Methodology, Y.L., M.K. and R.K.; Supervision, R.K., J.X., and K.M.; Writing—original draft, Y.L., M.K., and R.K. All authors have read and agreed to the published version of the manuscript. Funding: This work is supported by the Science and Technology Research Program of the Chongqing Municipal Education Commission (grant no. KJQN201801346) and the Chongqing University of Arts and Sciences Foundation (grant no. Z2011Rcyj05). R. Khenata acknowledges the financial support of the General Direction of Scientific Research and Technological Development (DGRSDT). Conflicts of Interest: The authors declare no conflict of interest.
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