Manipulation of type-I and type-II Dirac points in PdTe2 superconductor by external pressure

TL;DR

This study uses first-principles calculations to explore how hydrostatic pressure affects Dirac points in PdTe2, revealing the transition from type-II to type-I Dirac cones and their coexistence with superconductivity.

Contribution

It demonstrates the pressure-induced evolution of Dirac cones in PdTe2, including the emergence of type-I Dirac points and the coexistence of both types under specific pressures.

Findings

Type-II Dirac points disappear at 6.1 GPa.

Type-I Dirac points emerge at 4.7 GPa.

Superconductivity slightly decreases with pressure.

Abstract

A pair of type-II Dirac cones in PdTe$_2$ was recently predicted by theories and confirmed in experiments, making PdTe$_2$ the first material that processes both superconductivity and type-II Dirac fermions. In this work, we study the evolution of Dirac cones in PdTe$_2$ under hydrostatic pressure by the first-principles calculations. Our results show that the pair of type-II Dirac points disappears at 6.1 GPa. Interestingly, a new pair of type-I Dirac points from the same two bands emerges at 4.7 GPa. Due to the distinctive band structures compared with those of PtSe$_2$ and PtTe$_2$, the two types of Dirac points can coexist in PdTe$_2$ under proper pressure (4.7-6.1 GPa). The emergence of type-I Dirac cones and the disappearance of type-II Dirac ones are attributed to the increase/decrease of the energy of the states at $\Gamma$ and $A$ point, which have the anti-bonding/bonding characters of interlayer Te-Te atoms. On the other hand, we find that the superconductivity of PdTe$_2$ slightly decreases with pressure. The pressure-induced different types of Dirac cones combined with superconductivity may open a promising way to investigate the complex interactions between Dirac fermions and superconducting quasi-particles.

Full text

I Introduction

Topological (Dirac/Weyl) semimetalsRN765 ; RN812 are new topological states of three-dimensional (3D) quantum matters, different from the topological insulators. In Dirac/Weyl semimetals, the linear band crossings are the fourfold/twofold degenerated points, whose low-energy excitations are the massless Dirac/Weyl fermions corresponding to the counterparts in the high-energy physics. By breaking the inversion symmetry or time-reversal symmetry, one Dirac fermion will transform into two Weyl ones with opposite chiralities in the Brillouin zone (BZ). In recent years, the discoveries of Dirac semimetals (such as Na3BiRN1051 ; RN1121 and Cd3As2RN1123 ; RN1130 ; RN1125 ) and Weyl semimetals (such as Y2Ir2O7,RN257 HgCr2Se4RN1187 and TaAsRN1132 ; RN1131 ; RN1133 ) in theories and/or experiments made Dirac and Weyl fermions widely concerned.

The Dirac/Weyl semimetals that directly correspond to the counterparts in the high-energy physics are usually called type-I Dirac/Weyl semimetals. However unlike the high-energy physics, the restriction of Lorentz invariance is not necessary in the condensed matter physics. Recently a great deal of attention was paid to look for new topological quasi-particles beyond the direct counterparts in the high-energy physics.RN1327 Soluyanov et al.RN762 proposed a new type of Weyl fermion (type-II) that contacts the bulk electron and hole pockets in the condensed matter systems. Afterwards, many type-II Weyl semimetals were discovered, such as WTe2,RN762 ; RN1135 ; RN1137 ; RN1138 MoTe2,RN206 ; RN991 ; RN1134 MoxW1-xTe2,RN992 ; RN656 Ta3S2,RN1182 TaIrTe4RN1180 and LaAlGe.RN1181 The type-II Weyl semimetals show exotic properties different from the type-I ones, such as direction dependent chiral anomaly,RN762 ; RN1118 ; RN1183 enhanced superconductivity,RN1114 ; RN1058 anti-chiral effect of the chiral Landau level,RN1184 and novel quantum oscillations.RN1185

Researchers are also trying to find the type-II Dirac fermions in the condensed matters. Recently, Huang et al.RN980 predicted that the type-II Dirac fermions protected by $C_{3}$ rotational symmetry can exist in the PtSe2 family materials, and similar proposals were also put forward by Le et al.RN1179 and Chang et al.RN1178 in the KMgBi and VAl3 family materials, respectively. Following Huang’s predictions, the evidences of type-II Dirac cones in PtTe2,RN1011 ; RN1107 PdTe2RN1107 ; RN1028 ; RN1111 and PtSe2RN1107 ; RN1142 were soon characterized in angle-resolved photoemission spectroscopy (ARPES) experiments by different groups. Similar to the type-II Weyl cones, the type-II Dirac cones are strongly tilted in some specific directions. Novel physical properties different from those in the standard type-I Dirac semimetals are expected in the type-II Dirac semimetals.RN980 ; RN1178 Interestingly, PdTe2 is also a superconductor with transition temperature ($T_{C}$) about 1.7-2.0 K.RN1028 ; RN750 ; RN1047 The coexistence of superconductivity and type-II Dirac points in PdTe2 makes it significantly different from other members of PtSe2 family materials, which could provide a possible platform to explore the interplay between superconducting quasi-particles and Dirac fermions.RN1107 ; RN1028 ; RN1111

Pressure can drive the topological phase transitions in the topological materials and can also assist to comprehend the nature of topological states at ambient pressure. In this work, we focus on studying the evolution of the Dirac points and superconductivity in PdTe2 under hydrostatic pressure by the first-principles calculations. Our results show that the pair of type-II Dirac points disappears at 6.1 GPa, while a new pair of type-I Dirac points emerges at 4.7 GPa. Under the pressure of 4.7-6.1 GPa, the type-II and type-I Dirac cones coexist. The evolution of the two types of Dirac cones can be understood by the bonding and anti-bonding characters near the Dirac cones. We also find that the superconductivity slowly decreases with the increase of pressure, meaning that there are abundant topological transitions together with superconductivity in PdTe2 under the external pressure.

II Method

The first-principles calculations based on density functional theory (DFT) were performed using QUANTUM-ESPRESSO package.RN82 The ultrasoft pseudo-potentials and the general gradient approximation (GGA) according to the PBE functional were used. The energy cutoff of the plane wave (charge density) basis was set to 50 Ry (500 Ry). The BZ was sampled with a $12\times 12\times 8$ mesh of $k$-points. The Methfessel-Paxton Fermi smearing method with a smearing parameter of $\sigma=0.02$ Ry was used. The lattice constants and ions were optimized using Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-newton algorithm. All of the band structure calculations are cross-checked by the VASP codes,RN1434 ; RN1433 and the results are consistent with each other.

III Results and Discussion

PdTe2 belongs to transition metal dichalcogenides (TMDCs), and has a layered structure with space group P3̄m1 ($1T$-CdI2 type structure) with one Pd atom and two Te atoms located at (0, 0, 0) and (1/3, 2/3, $\pm z_{Te}$) sites, respectively (Fig. 1(a)). One Pd atom and the nearest six Te atoms compose an octahedron. The lattice constant ratio $c/a$ ($\approx$1.27) is the smallest among the iso-structural TMDCs,RN665 leading to the octahedra largely distorted. The special $k$ points and the paths of BZ are shown in Fig. 1(b), and the corresponding symmetries are summarized in Table S1 in the Supplemental Material.SM_URL The calculated lattice constants $a$ and $c$ at ambient pressure are slightly overestimated by about $1.76\%$ and $1.36\%$ respectively, due to the underestimation of bond strength in GGA. No structural phase transition was found up to 27 GPa in experiment.RN1018 The optimized lattice constants under pressure are shown in Fig. 2.

The band structures along $\Gamma$-$A$ direction at ambient pressure without/with spin orbit coupling (SOC) effects are shown in Fig. 3. Without SOC, the $E$ band (denoted by green color) and $A_{1}$ (denoted by red color) band, which belong to the two different irreducible representations (IRs), cross each other near the $A$ point. The $E$ band is mainly composed by the Te-$p_{x}$+$p_{y}$ orbitals, while the $A_{1}$ band is mainly composed by the Te-$p_{z}$ orbitals (see Fig. S1 in the Supplemental MaterialSM_URL ), leading to the $A_{1}$ band (red) more dispersive than the $E$ band (green). When the SOC is included, the band structures change dramatically as shown in Fig. 3(b). All the $A_{1}$ bands transform into the two dimensional (2D) IR $\Delta_{4}$ bands, and all the $E$ bands split into $\Delta_{4}$ (2D IR) and $\Delta_{5+6}$ bands (the $\Delta_{5}$ (1D IR) and $\Delta_{6}$ (1D IR) bands are degenerated along the $\Gamma$-$A$ direction). The band crossing near the $A$ point is also inevitable with SOC, since the $\Delta_{4}$ (red) and $\Delta_{5+6}$ (green) bands belong to different 2D IRs. The strongly dispersive band $\Delta_{4}$ crosses the less dispersive band $\Delta_{5+6}$, resulting in a titled Dirac cone along $k_{z}$, which is the type-II Dirac cone previously proposed by theories and verified in ARPES experiments.RN1107 ; RN1028 ; RN1111 The type-II Dirac cone protected by $C_{3}$ rotational symmetry is tilted along $\Gamma$-$A$ direction but untilted on the $k_{x}$-$k_{y}$ plane. According to the crystal symmetry, there is another type-II Dirac point located at the opposite position of the BZ, thus there is a pair of type-II Dirac cones. The location of the pair of type-II Dirac points in PdTe2 is closer to the Fermi energy ($E_{F}$) than those of PtSe2 and PdTe2.RN980

Because the $\Delta_{4}$ band (red) and the upper $\Delta_{4}$ band (black) belong to the same IR, as a result a band inversion at $\Gamma$ point instead of a band crossing at $\Gamma$-$A$ direction occurs (Fig. 3(b)). Topological surface state located at $\overline{\Gamma}$ deeply below $E_{F}$ at -1.75 eV was observed in experiments, which was considered to be due to the band inversion between the two lowest two $\Delta_{4}$ bands in Fig. 3(b).RN1107 ; RN1111 ; RN750

The $A_{4}^{+}$ state moves upward and while the $A_{5+6}^{-}$ state moves downward relative to $E_{F}$ with pressure (Figs. 4(a)-(e)), thus this pair of type-II Dirac cones gradually disappears near the $A$ point. The $\Gamma_{4}^{+}$ state moves downward, while the $\Gamma_{5+6}^{-}$ state moves upward with pressure, so a new pair untitled Dirac points i.e. type-I Dirac points, from the same two bands emerges near the $\Gamma$ point. The energy evolution of the $\Gamma_{4}^{+}$, $\Gamma_{5+6}^{-}$, $A_{4}^{+}$ and $A_{5+6}^{-}$ states under pressure is summarized in Fig. 5(a), from which we can see that the pair of type-I Dirac cones emerges at 4.7 GPa, and the pair of type-II Dirac cones disappears at 6.1 GPa. Under the pressure of 4.7-6.1 GPa, the two kinds of Dirac points coexist (for instance the 3D band structures at 5 GPa in Figs. 4(f)-(h)). When pressure further increases, it remains only the pair of type-I Dirac cones. The energy and positions of the two kinds of Dirac points are shown in Figs. 5(b) and (c). The type-II and type-I Dirac points locate below and above the $E_{F}$ respectively, and both move downward with pressure. Under higher pressure (about 30 GPa) the pair of type-I Dirac points will move to $E_{F}$. In the coexistence range, the type-II and type-I Dirac points are well separated in the BZ, which can be easily detected in experiments.

To reveal why the type-II Dirac points disappear and type-I Dirac points emerge, the charge density of the $\Gamma_{4}^{+}$, $\Gamma_{5+6}^{-}$, $A_{4}^{+}$ and $A_{5+6}^{-}$ states in real space was calculated and shown in Fig. 6. The interlayer Te-Te atoms show a bonding character in the $\Gamma_{4}^{+}$ (Fig. 6(a)) and $A_{5+6}^{-}$ (Fig. 6(c)) states, while the interlayer Te-Te atoms show an anti-bonding character in the $\Gamma_{5+6}^{-}$ (Fig. 6(b)) and $A_{4}^{+}$ (Fig. 6(d)) states. Since the interlayers bind to each other via a weak van der Waals interaction, and the Pd atom orbitals are localized and the Te atom orbitals are extended in the $\Gamma_{4}^{+}$ and $A_{4}^{+}$ states, the Te atom orbitals are easier to be affected by pressure. Pressure decreases the atom distance and enhances the atom interaction, leading to the energy of the bonding states moving downward, and the energy of the anti-bonding states moving upward relative to $E_{F}$. The opposite bonding character of $\Gamma_{4}^{+}$ and $\Gamma_{5+6}^{-}$ states makes their energy move in opposite directions with pressure, and so do the $A_{4}^{+}$ and $A_{5+6}^{-}$ states. The similar bonding and anti-bonding character of states make the topological phase transitions or metal-insulator transitions in Bi2Se3 RN1150 and phosphoreneRN1259 ; RN1153 under pressure. Just as the statement in Ref. (36), the formation of the type-II Dirac cones is very general: just due to the different dispersion of two IRs bands. Pressure can manipulate the band inversion and band dispersion, so more type-II Dirac and type-I Dirac materials are expected to be found in TMDCs under pressure.

Due to the same crystal structure, PtSe2 and PdTe2 show similar band structures to PdTe2 (see Figs. 7(a) and 7(b)). However the differences also exist. The $\Gamma_{4}^{+}$ and $\Gamma_{4}^{-}$ states inverse in PtTe2 (Fig. 7(b)) and PdTe2 (Fig. 3(b)), but they do not inverse in PtSe2 (Fig. 7(a)). The inversion energy of the $\Gamma_{4}^{+}$ and $\Gamma_{4}^{-}$ states (1.41 eV) is the largest, and the energy difference of $\Gamma_{4}^{+}$ and $\Gamma_{5+6}^{-}$ states (0.21 eV) is the smallest in PdTe2 (see Fig. 7(c)). Due to the distinctive band structures, two kinds of Dirac points are easier to coexist in PdTe2 compared with the cases of PtSe2 and PtTe2. Taking PtTe2 as an example, the evolution of band structures under external pressure in PtTe2 (see Fig. S5 in the Supplemental MaterialSM_URL ) is similar to those of PdTe2. The $A_{4}^{+}$/$A_{5+6}^{-}$ state of PtTe2 moves upward/downward with pressure, and the pair of type-II Dirac points gradually disappears at about 10 GPa (see Fig. 7(d)). Meanwhile the $\Gamma_{4}^{+}$/$\Gamma_{5+6}^{-}$ state moves downward/upward with pressure, and a new pair of type-I Dirac points will emerge at higher pressure. But the type-II and type-I Dirac points do not coexist in PtTe2 under pressure. Similar evaluation of Dirac points in PtSe2 is summarized in the Supplemental Material.SM_URL

We also evaluated the phonon and electron-phonon coupling of PdTe2 under pressure. The phonon branches become more dispersive and shift to higher frequency with pressure. The calcultaed $T_{C}$ decreases slowly with the nearly linear rate of 0.13 K/GPa from 1.97 K at ambient pressure to 0.69 K at 10 GPa (see Fig. 8). The decrease of electronic density of states at $E_{F}$ ($N(E_{F})$) and the blueshift of phonon density of states ($F(\omega)$) contribute to the decrease of $T_{C}$. Though $T_{C}$ decreases monotonously with pressure, it is still in an experimentally detectable range. Especially, the $T_{C}$ is above 1.0 K in the coexistence range of two types of Dirac cones. The detailed discussions are in the Supplemental Material.SM_URL

The intrinsic topology property can drive the conventional superconductivity into nontrivial $p$-wave-like unconventional topological superconductivity via conventional phonon mediated mechanism,RN1379 ; RN1422 like doped Bi2Se3RN1379 ; RN1415 ; RN1312 ; RN1318 ; RN1380 and pressed Cd3As2.RN1308 ; RN1314 Thus the issue whether PdTe2 is a topological superconductor needs to be further investigated. The electron and hole pockets near the type-II Dirac points provide plentiful density of states than those of the type-I Dirac points due to the titled cones. Like the type-II Weyl points,RN1114 ; RN1058 the type-II Dirac points are favorable for both superconducting and topological superconducting carrier ratios.RN1028 As for the coexistence of type-I and type-II Dirac points in PdTe2, more superconducting carrier ratios could be expected, and more interesting superconductivity properties should appear, which is deserved to be investigated further.

In addition, because the lattice constants under pressure optimized by GGA show about 2$\%$ systematic overestimation with the experimental data,RN1018 we calculated the band structures using the experimental data to estimate the overestimation (see Fig. S11 in the Supplemental MaterialSM_URL ). The pairs of type-II and type-I Dirac cones can still coexist under proper pressure when the experimental lattice constants are adopt, and the transition pressure of the Dirac points may be smaller than that using the optimized structure data.

IV Conclusion

The type-I and type-II Dirac points in PdTe2 can be tuned by applying the external pressure. The pair of type-II Dirac points disappears with pressure at 6.1 GPa, while a new pair of type-I Dirac points stemming from the same bands emerges at 4.7 GPa. The two types of Dirac points can coexist under proper pressure (4.7-6.1 GPa). The increase of the $A_{4}^{+}$ state energy and the decrease of the $A_{5+6}^{-}$ state energy make the pair of type-II Dirac points gradually disappear. The increase of the $\Gamma_{5+6}^{-}$ state energy and the decrease of the $\Gamma_{4}^{+}$ state energy make the type-I Dirac points emerge. The decrease/increase of the ($\Gamma_{4}^{+}$ and $A_{5+6}^{-}$)/($\Gamma_{5+6}^{-}$ and $A_{4}^{+}$) states energy is attributed to the bonding/anti-bonding character of interlayer Te-Te atoms. Due to the distinctive band structures of PdTe2, the type-II and type-I Dirac points from two same bands can coexist in PdTe2 under appropriate pressures. The superconductivity weakens monotonously with pressure with the average rate of 0.13 K/GPa due to the decrease of $N(E_{F})$ and the blueshift of $F(\omega)$, but still in an experimentally detectable range. Given the abundant topological transitions and superconductivity under pressure, PdTe2 under pressure may be an interesting topological Dirac material. Further experimental verification and theory studies need to be addressed.

Acknowledgements.

This work was supported by the National Key Research and Development Program of China under Contract No. 2016YFA0300404, the National Nature Science Foundation of China under Contract Nos. 11674326, 11404340, 11404024 and U1232139, Key Research Program of Frontier Sciences of CAS (QYZDB-SSW-SLH015) and Hefei Science Center of CAS (2016HSC-IU011). The calculations were partially performed at the Center for Computational Science, CASHIPS. Quansheng Wu was supported by Microsoft Research, and the Swiss National Science Foundation through the National Competence Centers in Research MARVEL and QSIT. We also acknowledge Dr. Chi-Ho Cheung in National Taiwan University for the useful discussions.