Lorentz-violating type-II Dirac fermions in transition metal dichalcogenide PtTe2

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 Lorentz-violating type-II Weyl/Dirac fermions could be realized in topological semimetals. The recent realization of type-II Weyl fermions raises the question whether their spin-degenerate counterpart—type-II Dirac fermions—can be experimentally realized too. Here, we report the experimental evidence of type-II Dirac fermions in bulk stoichiometric PtTe2 single crystal. Angle-resolved photoemission spectroscopy measurements and first-principles calculations reveal a pair of strongly tilted Dirac cones along the Γ-A direction, confirming PtTe2 as a type-II Dirac semimetal. Our results provide opportunities for investigating novel quantum phenomena (e.g., anisotropic magneto-transport) and topological phase transition.

In three dimensional topological Dirac semimetals, the low energy quasiparticle excitations are fermions described by the massless Dirac equation [10], and such fermions are protected by certain crystalline symmetries in addition to time-reversal and inversion symmetries [11][12][13]. The generic Hamiltonian for a Dirac fermion is H( k) = i=x,y,z j=0,x,y,z k i A ij σ j ,where k is wave vector in momentum space, σ 0 is 2 × 2 unit matrix, and σ j (j=x,y,z ) are the three Pauli matrices. The dispersion ± ( k) = i=x,y,z are doubly degenerate as required by the presence of time-reversal and inversion symmetries [8]. The degeneracy will be lifted when time-reversal or inversion symmetry is broken, and each Dirac fermion splits into a pair of Weyl fermions with opposite chiralities. The linear term T ( k) tilts the Dirac cone and the relative maganitude of T ( k) and U ( k) can be used to classify the topological nature of the Dirac or Weyl semimetals [8,9]. For type-I Dirac [1,2] and Weyl semimetals [3,4], T ( k) is negligible compared to U ( k), and massless Dirac cones with linear dispersions (see schematic drawing in Fig.1(a)) are observed with isolated Dirac or Weyl points at the Fermi energy. When T ( k) > U ( k) along a certain direction, the Dirac cone is strongly tilted ( Fig. 1(b)), leading to an intrinsic Lorentz violation. The fermions in this case emerge at the topologically protected points between electron and hole pockets, and there are finite density of states at the Fermi energy. The different band topology also leads to distinguished magnetotransport properties. For example, while type-I Dirac and Weyl semimetals exhibit negative magnetoresistance along all directions [5][6][7], the magnetotransport in type-II semimetals is expected to be extremely anisotropic [8,14,15]. Although type-II Weyl fermions have been reported recently [16][17][18][19][20][21][22], their spin-degenerate counterparts -type-II Dirac fermions still remain to be realized. Here by combining angle-resolved photoemission spectroscopy (ARPES) and first-principles calculations, we report the first discovery of Lorentz-violating type-II Dirac fermions in a transition metal dichalcogenide PtTe 2 crystallizes in the CdI 2 -type trigonal (1T) structure with P3m1 space group (No. 162). The crystal structure is composed of edge-shared PtTe 6 octahedra with PtTe 2 layers tiling the ab plane (see Fig. 1(c,d)). The isostructural PtSe 2 monolayer film has been found to be a semiconductor with a gap of 1.2 eV [23] and exhibits helical spin texture with spin-layer locking induced by the local dipole field induced R-2 Rashba effect [24]. Here we focus on the topological property of the semimetallic PtTe 2 bulk crystal, and we note that similar topological property is also expected in bulk PtSe 2 . Figure 1(e) shows the hexagonal bulk Brillouin zone (BZ) and projected surface BZ onto the (001) surface. The Raman spectrum in Fig. 1(f) shows the E g and A 1g vibrational modes at ∼ 110 cm −1 and 157 cm −1 respectively, which are typical for 1T structure [25]. The sharp X-ray diffraction peaks ( Fig.1(g)) and low-energy electron diffraction (LEED) pattern ( Fig. 1(h)) confirm the high quality of the single crystals.
The overview of the band structure of PtTe 2 near the fermi energy E F is shown in Fig. 2. obvious one is centered at the Γ point, which is formed by a V-shaped dispersion touching a flatter Λ-shaped dispersion (pointed by red arrow). The calculated projected spectral weight along the two high symmetric directions is shown in Fig. 2(c,d) for comparison. The cone-like 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 ( Fig. 2(e,f)). It is also clearly revealed in the three dimensional electronic structure shown in Fig. 2(g). The evolution of this cone with the out-of-plane 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. 2(c)) show that this cone has bulk properties, and there are 5 sharp surface states connecting the gapped Dirac cone here. The third conical dispersion is at much deeper energy between -2.0 eV and -2.6 eV (pointed by gray arrow in Fig. 2(a,b)).
This cone corresponds to sharp surface states in the calculated dispersion (Fig. 2(c)). This cone connects the gapped bulk bands, similar to the Z 2 topological surface states observed in PdTe 2 [26]. 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 Γ-K and Γ-M directions at different photon energies.
The corresponding k z values are obtained using an inner potential of 13 eV, determined by comparing the experimental data with theoretical calculation. Figure 3(a-e) shows the measured dispersions. The calculated bulk band dispersions at each k z value are overplotted on the curvature image in Fig. 3(f-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 ≈ 0.35 c * , which is labeled as "D" in Fig. 1(d). Away from this special point along the Γ-A direction, the valence and conduction bands begin to separate, and the separation becomes larger as k z moves further away from 0.35 c * . The strong k z dependence confirms its three-dimensional nature. In addition to the bulk bands discussed above, there are surface states between -0.5 eV to -1 eV at the BZ center (highlighted by yellow dashed line in Fig. 3(f-j)) and at deeper energy (gray dashed line) which do not change with photon energy.  Fig. 3(l). The type-II characteristics are revealed by plotting the dispersion as a function of k z (Fig. 3(m)) where a strongly tilted Dirac cone at the D point is revealed. The type-II characteristics is also reflected in the constant energy contours (Fig. 3(n)). Three dimensional intensity map E-k x -k z shows an electron pocket (red dashed lines) and a hole pocket  In addition, there is another band inversion at the A point at ∼ −2.5 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 8 cone is formed by two valence bands with Te-p orbitals (highlighted by red color). These two bands show linear dispersions in the vicinity of D along both the in-plane (S-D-T) and out-of-plane (Γ-A) directions, confirming that it is a three-dimensional Dirac cone. This band crossing is unavoidable, because these two bands belong to different representations (∆ 4 and ∆ 5+6 ), respectively, as determined by the C 3 rotational symmetry about the c axis [11]. The different irreducible representations prohibit hybridization between them, resulting in the symmetry-protected band crossing at D = (0, 0, 0.346c * ). As each band is doubly degenerate, the band crossing forms the four-fold degenerate Dirac point. We also calculated the energy contours by tuning the chemical potential to the Dirac point, as shown in Fig. 4(b). 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. 4(d)). By tuning the chemical potential above or below E D (Fig. 4(c,e)), we find that the hole and electron pockets disconnect, confirming that they only touch at the Dirac point.
In conclusion, by combining both ARPES measurements and first-principles calculations, we provide first direct evidence for the realization of type-II 3D Dirac fermions in single crystal PtTe 2 . Such type-II Dirac fermions violate Lorentz invariance and do not have counterpart in high energy physics. The realization of type-II Dirac semimetal provides a new platform for investigating various intriguing properties different from their type-I analogues, e.g. anisotropic magneto-transport properties.

Methods
Sample growth High quality PtTe 2 single crystal was obtained by self-flux 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 flame-sealed in a vacuum of ∼ 1 Pa. The tube 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 to crystallize PtTe 2 in Te flux. The excess Te was centrifuged isothermally after 2 days. Theoretical calculation All first-principles calculations are carried out within the framework of density-functional theory using the Perdew-Burke-Ernzerhof-type [27] generalized gradient approximation for the exchange-correlation potential, which is implemented in the Vienna ab initio simulation package [28]. A 8 × 8 × 6 grid of k points and a default plane-wave energy cutoff are adopted for the self-consistent field calculations. Spin-orbit coupling is taken into account in our calculations. We calculate the surface spectral function and Fermi surface using the surface Green's function method [29] based on maximally localized Wannier functions [30] from first-principles calculations of bulk materials.