Abstract
Topological semimetals have recently attracted extensive research interests as host materials to condensed matter physics counterparts of Dirac and Weyl fermions originally proposed in high energy physics. Although Lorentz invariance is required in high energy physics, it is not necessarily obeyed in condensed matter physics, and thus Lorentzviolating typeII Weyl/Dirac fermions could be realized in topological semimetals. The recent realization of typeII Weyl fermions raises the question whether their spindegenerate counterpart—typeII Dirac fermions—can be experimentally realized too. Here, we report the experimental evidence of typeII Dirac fermions in bulk stoichiometric PtTe_{2} single crystal. Angleresolved photoemission spectroscopy measurements and firstprinciples calculations reveal a pair of strongly tilted Dirac cones along the ΓA direction, confirming PtTe_{2} as a typeII Dirac semimetal. Our results provide opportunities for investigating novel quantum phenomena (e.g., anisotropic magnetotransport) and topological phase transition.
Introduction
In three dimensional topological semimetals, the low energy excitations—Dirac or Weyl fermions—are described by the massless Dirac equation^{1}. Dirac fermions are protected by certain crystal symmetries^{2,3,4}. When timereversal or inversion symmetry is broken, a spindegenerate Dirac fermion splits into two Weyl fermions, and the topological surface states (TSS) evolve from a closed Fermi surface to open Fermi arcs^{5, 6}. The generic Hamiltonian for Dirac and Weyl fermions is H(k) = T(k) ± U(k), where U(k) is a potential component and T(k) is a linear kinetic term that tilts the Dirac cone. The relative magnitude of T(k) and U(k) can be used to classify the topological nature of the Dirac or Weyl semimetals^{7}. For typeI Dirac^{8, 9} and Weyl semimetals^{6, 10, 11}, which obey the Lorentz invariance, T(k) < U(k) and isolated Dirac or Weyl points with linear Dirac cones are expected at the Fermi energy (see schematic drawing in Fig. 1a). When T(k) > U(k) along certain momentum direction, the Lorentz invariance is violated and strongly titled Dirac cones emerge at the topologically protected touching points between electron and hole pockets (Fig. 1b). Such Lorentzviolating Dirac fermions are classified as typeII Dirac fermions. The classification and comparison of typeI and typeII Dirac/Weyl semimetals are summarized in Table 1.
The different band topology can also lead to distinct magnetotransport properties. While typeI Dirac and Weyl semimetals exhibit negative magnetoresistance along all directions^{16,17,18}, the magnetotransport properties of typeII semimetals are expected to be extremely anisotropic and negative magnetoresistance is expected only along momentum directions with T(k) < U(k)^{7, 19}. Although typeII Weyl semimetals have been reported recently^{14, 15, 20,21,22,23,24}, typeII Dirac semimetals still remain to be realized experimentally. This is particularly important since typeII Dirac semimetal stands at the critical point of topological phase transition and can be tuned to Weyl semimetal or topological crystalline insulator via crystal distortions or magnetic doping.
Here we fill in the last missing element in the classification of Dirac and Weyl semimetals. By combining angleresolved photoemission spectroscopy (ARPES) measurements and firstprinciples calculations, we report the experimental evidence of typeII Dirac fermions in bulk PtTe_{2} single crystal with a pair of strongly tilted Dirac cones.
Results
Structural characterization
PtTe_{2} crystallizes in the CdI_{2}type trigonal (1 T) structure with P \(\overline 3 \) m1 space group (No. 162). The crystal structure is composed of edgeshared PtTe_{6} octahedra with PtTe_{2} layers tiling the ab plane (Fig. 1c, d). The isostructural PtSe_{2} film with one monolayer thickness is a semiconductor with a 1.2 eV gap^{25} and exhibits helical spin texture with spinlayer locking^{26}. Here we focus on the topological property of the bulk semimetallic PtTe_{2} crystal, and we note that similar topological property is also expected in bulk PtSe_{2} ^{27}. Figure 1e shows the hexagonal bulk Brillouin zone (BZ) and projected surface BZ onto the (001) surface. The Raman spectrum in Fig. 1f shows the E_{ g } and A_{1g } vibrational modes at ~ 110 and 157 cm^{−1} respectively, which are typical for 1 T structure^{28}. The sharp Xray diffraction (XRD) peaks (Fig. 1g) and lowenergy electron diffraction (LEED) pattern (Fig. 1h) confirm the high quality of the single crystals.
Electronic structure
The overview of PtTe_{2} band structure measured by ARPES near the Fermi energy (E _{F}) is shown in Fig. 2. Figures 2a, b show ARPES data taken along two highsymmetry directions ΓM and ΓK at photon energy of 22 eV. There are a few conical dispersions in the ARPES data. The most obvious one is centered at the Γ point, which is formed by a Vshaped dispersion touching a flatter Λshaped dispersion (pointed by red arrow). The calculated projected spectral weight along the two highsymmetric directions is shown in Figs. 2c, d for comparison. The conelike dispersion discussed above corresponds to continuous states in the calculation, suggesting that this band is from the bulk states. This cone shows up as a pocket around the Γ point in the measured and calculated Fermi surface maps (Figs. 2e, f). It is also clearly revealed in the three dimensional electronic structure shown in Fig. 2g. The evolution of this cone with the outofplane momentum k _{ z } and its topological property are the main focus of this work. The second conical dispersion is located between the Γ and M points (labeled as M′), and it is gapped at the Dirac point slightly below E _{F} (pointed by white arrow). Calculated dispersions (Fig. 2c) show that this cone has bulk properties, and there are sharp surface states connecting the gapped Dirac cone here. The third conical dispersion is at much deeper energy between − 2.0 and − 2.6 eV (pointed by gray arrow in Figs. 2a, b). This cone corresponds to sharp surface states in the calculated dispersions (Fig. 2c), and it connects the gapped bulk bands, similar to the Z_{2} topological surface states observed in PdTe_{2} ^{29}. The comparison between the measured and calculated band structure shows a good agreement with multiple conical dispersions arising from both the bulk bands and surface states.
To reveal the bulk versus surface properties of these Dirac cones, we show in Fig. 3 ARPES data measured along the ΓK and ΓM directions using different photon energies. The corresponding k _{ z } values are calculated using an inner potential of 13 eV^{30}, which is determined by comparing the experimental data with theoretical calculations. Figure 3a–e shows the measured dispersions. The calculated bulk band dispersions at each k _{ z } value are overplotted on the curvature image in Figs. 3f–j. A good agreement is obtained for the bulk Dirac cone at the Γ point and its evolution with k _{ z }. The Dirac point discussed above is at k _{ z } = 0.35c ^{*} (c ^{*} = 2π/c), which is labeled as D in Fig. 1e. Away from this special point along the ΓA direction, the valence and conduction bands begin to separate, and the separation becomes larger when k _{ z } moves further away from 0.35c ^{*}. The strong k _{ z } dependence confirms its threedimensional nature. We note that at k _{ z } = 0.30c ^{*}, some signatures of the dispersion at k _{ z } = 0.35c ^{*} are also observed (pointed by gray arrow), suggesting that there is significant k _{ z } broadening due to a finite escape depth of photoelectrons^{31}. In addition to the bulk bands discussed above, there are surface states between −0.5 to −1 eV at the BZ center (highlighted by yellow dashed line in Figs. 3f–j) and at deeper energy (gray dashed line) that do not change with photon energy.
Figure 3k shows the zoomin dispersions at the Dirac point. The conical dispersion can be clearly observed by following the peaks in the momentum distribution curves (MDCs) in Fig. 3l. The typeII characteristics are revealed by plotting the dispersion as a function of k _{ z } (Fig. 3m) where a strongly tilted Dirac cone at the D point is revealed. The intensity of the lower branch in the Dirac cone is weaker than the upper one, which is attributed to dipole matrix element effect^{32}. We note that a gaplike feature appears at the band crossing point. We have taken a fine k _{ z } step of 0.012c ^{*} (corresponding to photon energy step of 0.3 eV, see Supplementary Fig. 1) to exclude misalignment. A more likely possibility is the k _{ z } broadening due to the finite escape depth of photoelectrons. Namely, the measured dispersion is the averaged dispersion over a finite k _{ z } window (from the penetration depth λ ~ 5 Å, k _{ z } broadening is estimated to be Δk _{ z } ≈ 1/λ ≈ 0.2 Å^{−1}, ~17% of the BZ), and the contribution of dispersion away from the Dirac node could contribute to a gaplike feature, considering the strong k _{ z } dispersion in this material. A more conclusive explanation requires further studies. The typeII characteristic is also reflected in the constant energy contours (Figs. 3n, o). The three dimensional intensity map E–k _{ x }–k _{ z } shows an electron pocket (red arrow) and a hole pocket (blue arrow) approaching each other near the Dirac point energy, which is another important signature of typeII Dirac cones. This anisotropic touching between the electron and hole pockets contributes finite density of states around the Dirac point and this is distinguished from the vanishing density of states at the Dirac point in typeI Dirac semimetal.
Theoretical calculation
In order to further reveal the topological nature of surface states and typeII Dirac cone in PtTe_{2}, we present the firstprinciples calculations of electronic structure and perform symmetry analysis. Figure 4a shows the calculated band structure along both the inplane SDT and outofplane ADΓ directions through the D point. Due to the band inversion between \(\Gamma _4^ + \) and \(\Gamma _4^  \) at the Γ point, there is a topologically nontrivial gap between them which gives rise to the surface states connecting the gapped cone structure at the M′ point. In addition, there is another band inversion at the A point at ~ − 2 eV, which leads to the existence of the deep surface state as mentioned above. We have calculated the Z _{2} invariants for bands below these two gaps, confirming the nontrivial topology of them. The bulk Dirac cone is formed by two valence bands with Tep orbitals (highlighted by red color). These two bands show linear dispersions in the vicinity of D along both the inplane (SDT) and outofplane (ΓA) directions, confirming that it is a threedimensional Dirac cone. This band crossing is unavoidable, because these two bands belong to different representations (Δ_{4} and Δ_{5 + 6}) respectively, as determined by the C _{3} rotational symmetry about the c axis^{2}. The different irreducible representations prohibit hybridization between them, resulting in the symmetryprotected band crossing at D = (0,0,0.346c ^{*}). As each band is doubly degenerate, the band crossing forms the fourfold degenerate Dirac point. We also calculated the energy contours by tuning the chemical potential to the Dirac point, as shown in Fig. 4b. It is clear that there is a hole pocket in the BZ center (red color), while the much more complicated electron pockets (green color) are composed of a large outer pocket and a small inner one. The hole pocket and the small electron pocket touch each other at two Dirac points as shown in the isoenergy counter in the k _{ x }–k _{ z } plane (Fig. 4d). By tuning the chemical potential above or below E _{D} (Figs. 4c, e), we find that the hole and electron pockets disconnect, confirming that they only touch at the Dirac point.
Discussion
By combining both ARPES measurements and firstprinciples calculations, we provide the first direct experimental evidence for the realization of typeII 3D Dirac fermions in single crystal PtTe_{2}. While there are trivial bands at the Dirac nodes of typeII semimetals, their magnetotransport response is different from Dirac fermions. As long as the topologically nontrivial bands cross the Fermi level, they can contribute to distinctive transport behavior even in the presence of trivial bands^{33, 34}. Recent transport measurements have revealed a nontrivial Berry phase in a sister compound PdTe_{2} with the Dirac node also below E_{ F } ^{35}, and similar transport properties can be expected in PtTe_{2} if the Fermi level is tuned so that both bands forming the Dirac cone cross the Fermi level. By further tuning the Dirac node to the Fermi energy, more intriguing transport properties (e.g., angledependent negative magnetoresistance) can be further revealed. Doping PtTe_{2} by Ir is one possible solution. IrTe_{2} has the same crystal structure as PtTe_{2} at room temperature, yet with the Dirac node above E_{ F } (Supplementary Fig. 2). By substituting Ir for Pt in Pt_{1−x }Ir_{ x }Te_{2}, it is possible to fine tune the Dirac point to the Fermi energy. Expanding the c axis lattice constant either by tensile strain^{27} or appropriate intercalation^{36} can also shift the Dirac node toward the Fermi level, and finding a suitable intercalant is critical along this line. Our study paves the way for designing and realizing a number of similar typeII Dirac materials in same chemical group (PCh _{2}, P = Pt,Pd; Ch = Se,Te)^{27}. Moreover, the realization of typeII Dirac semimetal provides a new platform for investigating various intriguing properties different from their typeI analogues such as anisotropic magnetotransport properties.
Methods
Sample growth
High quality PtTe_{2} single crystal was obtained by selfflux methods. High purity Pt granules (99.9%, Alfa Aesar) and Te lump (99.9999%, Alfa Aesar) at a molar ratio of 2:98, were loaded in a silica tube, which is flamesealed in a vacuum of ~ 1 Pa. The sample was heated at 700 °C for 48 hours to homogenize the starting materials. The reaction was then slowly cooled to 480 °C at 5 °C h^{−1} to crystallize PtTe_{2} in Te flux. The excess Te was centrifuged isothermally after 2 days.
ARPES measurement
ARPES measurements were taken at BL13U of Hefei National Synchrotron Radiation Laboratory, BL9A of Hiroshima Synchrotron Radiation Center under the proposal No.15A26 and our home laboratory. The crystals were cleaved in situ and measured at a temperature of T ≈ 20 K in vacuum with a base pressure better than 1 × 10^{−10} torr.
Theoretical calculation
All firstprinciples calculations are carried out within the framework of densityfunctional theory using the PerdewBurkeErnzerhoftype^{37} generalized gradient approximation for the exchangecorrelation potential, which is implemented in the Vienna ab initio simulation package^{38}. A 8 × 8 × 6 grid of k points and a default planewave energy cutoff (230 eV in this case) are adopted for the selfconsistent field calculations. We use the MethfesselPaxtontype smearing method with a width of 0.2 eV. Spinorbit coupling is taken into account in our calculations. We calculate the surface spectral function and Fermi surface using the surface Green’s function method^{39} based on maximally localized Wannier functions^{40} from firstprinciples calculations of bulk materials.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant No. 11334006, 11427903 and 11674188), Ministry of Science and Technology of China (Grant No. 2015CB921001, 2016YFA0301001 and 2016YFA0301004) and Beijing Advanced Innovation Center for Future Chip (ICFC).
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S.Z. designed the project. M.Y., K.Z., E.Y., W.Y., K.D., G.W., H.Z. and S.Z. performed the ARPES measurements and analyzed the ARPES data. H.H., W.D. performed theoretical calculation, K.Z., Y.W. prepared the single crystals. H.Y. discussed the data. M.Y. and S.Z. wrote the manuscript, and all authors commented on the manuscript.
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Yan, M., Huang, H., Zhang, K. et al. Lorentzviolating typeII Dirac fermions in transition metal dichalcogenide PtTe_{2} . Nat Commun 8, 257 (2017). https://doi.org/10.1038/s41467017002806
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DOI: https://doi.org/10.1038/s41467017002806
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