Introduction

In recent years, 5d transitional metal iridium ditelluride (IrTe2) has attracted much interest due to its peculiar structural/electronic phase transitions and emergence of superconductivity upon chemical doping or intercalation1,2,3,4,5,6,7,8. It was reported9 that IrTe2 undergoes a structural phase transition at about 250 K from a high-temperature trigonal () to a low-temperaure monoclinic (C2/m) structure, accompanied by temperature-dependent anomalies of electron resistivity and magnetic susceptibility. IrTe2 exhibits superconductivity when the structural phase transition is suppressed by chemical doping or intercalation1,2,3. Since the structural phase transition is characterized by the formation and breaking of Ir-Ir bonds along the b-axis, the superconductivity can be interpreted by bond-breaking-induced superconductivity1,10. On the other hand, it has been proposed to understand this phase transition down to the electron-structure level. Yang et al.2 suggested that the phase transition is a charge-orbital density wave type and superconductivity up to 3 K sets in as soon as the density wave transition is suppressed by Pd intercalation and substitution. Ootsuki et al.3 argued that the orbitally induced Peierls effect plays an important role in the charge-orbital instability and superconductivity of Ir1−xPtxTe2. Recent optical spectroscopic measurements and density functional theory calculations indicated that the transition is not driven by the density wave type instability but caused by the crystal field effect which further splits/separates the energy levels of Te (px and py) and Te (pz) bands6. A local bonding instability associated with the Te 5p states was suggested as a main factor dominating the structure and the superconductivity7,8.

It is well known that pressure can efficiently modify the atomic and electronic structures of materials, leading to the formation of novel materials with unusual physical properties and crystal structure can play a crucial role in the appearance of superconductivity. Recently, a pressure-induced superconductivity was reported in AuTe2 and its bulk superconductivity appears only in the high-pressure phase10. Therefore, it is highly desirable to search for new structural phases of IrTe2 under high pressures and investigate the origin of possible superconductivity.

In this work, we use crystal structural prediction techniques to find energetically stable/metastable structures of IrTe2 under high pressures up to 150 GPa. Several novel crystal structures for IrTe2 are predicted in different pressure regions, i.e., the one with 4 formula units (f.u.)/cell, the C2/m-2 one with 2 f.u./cell and the I4/mmm one with 1 f.u./cell. The electronic structure, phonon spectrum and electron-phonon (EP) coupling are investigated for each phase. The calculated results show that the band structures and Fermi surfaces (FSs) for the high-pressure phases have dramatically reconstructed. It is found that the and I4/mmm phases have relatively strong electron-phonon coupling, which will be of benefit to possible superconductivity. We also have carried out the variable-composition search. Newly found compounds such as IrTe3, IrTe and Ir3Te are thermodynamically and dynamically stable and have unusual electronic and phonon properties.

Results and Discussion

Structural and Bonding analysis

First, we use crystal structural prediction techniques to find energetically stable structures of IrTe2 under high pressures. Five types of crystallographic structures shown in Fig. 1(a) are found to be the stable/metastable phases with lower energies. The calculated enthalpy-pressure (ΔH-P) and volume-pressure (V-P) curves are plotted in Figs. 1(b) and (c) for the structures interested. From Fig. 1(b), we find that with increasing pressure, IrTe2 undergoes a series of structural phase transitions, from the C2/m to structure at about 20 GPa, then to the C2/m-2 structure at about 70 GPa and finally to the I4/mmm structure at about 100 GPa. The C2/m structure consists of two formula units, containing edge-sharing Ir-Te6 octahedra. Although we did not get any clue in advance and “predict” the phase from scratch, this structure actually has been synthesized in experiment11 and our theoretical transition pressure to it agree very well with the experimental value. It has a cubic lattice embodying vertex-sharing Ir-Te6 octahedra. An Ir ion locates at the center of the lattice and other Ir ions lie in the middle of the lattice edges. The C2/m-2 structure has the same symmetry as the C2/m one, but it exhibits a two-dimensional (2D) characteristic in the xz plane. The Ir and Te ions form an unique edge-sharing Ir-Te7 octahedron in the lattice. For the I4/mmm structure, each Ir ion is surrounded by eight Te ions, forming an Ir-Te8 cuboid and it consists of a stack of Ir-Te8 cuboid arranging along the z direction. It's interesting that upon pressure, the coordination number of Ir ions increases continuously, forming a series of atomic polyhedrons of Ir-Te6 → Ir-Te7 → Ir-Te8. From Fig. 1(c), one sees that there is a sudden drop in volume at each critical pressure, indicating a first-order structural phase transition there. The optimized structural parameters for these five IrTe2 phases discussed above are listed in Table I.

Table 1 Optimized structural parameters of five structures for IrTe2 calculated with PBE and DFT-D2 methods. The parameters of C2/m and C2/m-2 shown here correspond to the primitive cell. Note that the volumes are unified in one formula unit
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Figure 1
figure 1

Crystal structure and equation of states of the IrTe2 phases.

(a) Five types of crystallographic structures of IrTe2 and (b) enthalpy relative to that of the phase vs pressure and (c) volume vs pressure curves for various structures. Insets in (c) show different type of polyhedron formed under pressure.

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As found in many other system, such as CO12 and CO213, some of the IrTe2 phases present a laminar crystal structure, we did additional calculations including van der Waals (vdW) correction proposed by Grimme (DFT-D2)14. The obtained structural parameters are listed in Table I. The message is that the volume of unit cell for several structures shrinks due to the vdW corrections. The C2/m and structures under ambient condition have the largest percentage of contraction (about 5.3%) and we find this is mainly from the decreasing of the distance between the adjacent IrTe2 layers. However, the magnitude of contraction decreases with increasing pressure, reflecting that the vdW contribution under high pressure becomes small. Based on the electron localization function (ELF), as showned in the Supporting Informtion, we find there are small overlap between the electrons on the Te atoms in two adjacent layers, both in and C2/m structure, this somehow gives the reason of why the binding energy between the layers are quite large. Without vdW corrections, we obtain the binding energies are 219 meV for and 214 meV for C2/m structure. While with vdW corrections, we obtain 847 meV and 843 meV, respectively.

The calculated total and partial density of states (DOS), band structure with Ir-d and Te-p characteristics and Fermi surfaces are shown in Fig. 2. For the DOS of the C2/m structure, the Te px orbital exhibits a distinct peak located at the Fermi level (EF), indicating a small van Hove singularity (vHs) behavior. A recent comprehensive study of the electronic states has revealed the vHs origin of the phase transition between high temperature phase and low temperature C2/m phase in IrTe215. According to the orbital characteristics of the fat-band, two hybridized Ir 5d and Te 5p bands cross the Fermi level, forming two Fermi surfaces. Compared with the structure, the interlayer interaction of the C2/m structure and hence its band dispersion along the c axis (Γ-A direction) are suppressed, so that the 2D character of the FSs becomes enhanced. The reconfiguration in the topology of FSs along the kz direction appears to be consistent with what has been observed in angle-resolved photoemission spectroscopy measurements15,16,17.

Figure 2
figure 2

Electronic structures of the novel IrTe2 phases.

Total and partial density of states, fat-band structures with Ir-d and Te-p characteristics and Fermi surfaces for C2/m at ambient pressure (a) – (c), at 40 GPa (d) – (f), C2/m-2 at 80 GPa (g) – (i) and I4/mmm at 120 GPa (j) – (l).

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The structure exhibits quite unique electronic structure and Fermi surfaces. There are four bands crossing the Fermi level, which construct four complicated 3D FSs. A van-Hove-like peak lies below the Fermi level 0.1 eV, yielding a large DOS at the Fermi level (N(EF)) ~ 7 states/eV per unit cell, which is contributed mainly by the Ir 5d planar orbitals and the Te pz orbital. A pseudo gap appears around E = −1.9 eV, where a significant Fermi cone originates mainly from the Te p orbitals around the Γ point in the band structure. The Fermi level of the C2/m-2 structure lies at the valley bottom of the DOS curves and N(EF) ~ 1 states/eV per unit cell, exhibiting a bad metallic behavior. Two bands crossing the Fermi level form the FSs shown in Fig. 2(h). For the I4/mmm structure, the FSs arise from the four bands crossing EF. Around the Γ point there is a claw-like Fermi pocket with four-fold symmetry and the Fermi pockets at zone corners are shaped as irregular cylinders. At the same time, there is a reduction of DOS at EF for I4/mmm, N(EF) = 1.33 states/eV, compared with those in the low-pressure phases. The reconstructed band structures and FSs may greatly affect phonon properties and electron-phonon coupling.

Vibrational Analysis

The calculated phonon dispersions, projected phonon densities of states (PDOS) and Eliashberg spectral function α2F(ω) are shown from left to right in Fig. 3. In the left panel of Fig. 3, the distribution of the EP coupling for each phonon mode, λqν, is described by red circles, the radius of the circles being proportional to the magnitude of each λqν. For the C2/m structure, as shown in Fig. 3(a), the two gaps near 120 cm−1 and 150 cm−1 in the phonon spectrum divide the spectrum into three regions: the acoustic branches, middle-frequency and high-frequency optical branches. The acoustic modes extend to about 100 cm−1. The modes at the high frequency region are associated with the vibrations of Ir atoms beating against Te atoms. Whereas the modes at the middle frequency region are associate with the vibrations of Te-Te bond stretching. The obtained α2F(ω) curve exhibits two main humps below 120 cm−1, which originate from the acoustic Au modes.

Figure 3
figure 3

Calculated phonon dispersions, projected phonon densities of states (PDOS) and Eliashberg spectral function.

(from left to right) (a) C2/m at ambient pressure, (b) at 40 GPa, (c) C2/m-2 at 80 GPa and (d) I4/mmm phases at 120 GPa.

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For the structure at 40 GPa [see Fig. 3(b)], phonon frequencies extend up to ~ 280 cm−1 and the point group at the Γ point is Th, which can be decomposed as with Eg and Eu being double degenerate and Tg and Tu triple degenerate. There are two tiny gaps in the PDOS, respectively, near 125 cm−1 and 210 cm−1. The acoustic phonon modes exhibit a significant phonon softening along all the symmetric directions. Furthermore, we notice that the optical branches at the lower frequency side also become soften and merge with the acoustic branches. In the DFPT calculations, the Eliashberg spectral function depends directly on the EP matrix element, , which is defined as the variation of the self-consistent crystal potential. The low-frequency spectral function curve α2F(ω) has two peaks near 50 cm−1 and 100 cm−1, which are contributed by the acoustic and optical Tu modes. The latter originate from the out-plane breathing and rocking vibrations of the Ir and Te atoms. The integration of λ(ω) up to 100 cm−1 is near 0.7, almost 80% of the total EP coupling. In the phase, the softening of low-frequency modes and enhancement of the Eliashberg spectral function can significantly increase the EP coupling.

For the C2/m-2 structure at 80 GPa [see Fig. 3(c)], the point group at the Γ point is Ci, same as the C2/m phase, which can be decomposed as . Both acoustic and optical branches are hardened and the obtained EP coupling strength is very weak. For the I4/mmm at 120 GPa [see Fig. 3(d)], the phonon frequencies expand up to more than 300 cm−1 and the point group at the Γ point is D4h. The Γ modes can be decomposed as , where Eu and Eg modes are double degenerate. From the PDOS curves, one sees that the low-frequency branches below 125 cm−1 are mainly associated with the vibrations of the Ir atoms, whereas the high-frequency branches above 200 cm−1 originate mainly from the vibrations of the Te atoms. The α2F(ω) curve exhibits a distinct peak around ω = 100 cm−1 and the low-frequency phonons account for more than 60% of the total EP coupling. The calculated phonon vibration frequencies for each IrTe2 compound at the Γ point are listed in Table 2.

Table 2 Phonon mode frequencies in units of cm−1 at the Γ point for various crystal structure of IrTe2 with I: infrared active and R: Raman active
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Using the Allen-Dynes formula18,19

where ωln is the logarithmically averaged frequencies and taking the Coulomb parameter as µ* = 0.15, we evaluate the pressure-dependent Tc. The obtained λ and Tc as a function of pressure are plotted in Figs. 4 (a) and (b), respectively. It is found that the calculated value is equal to λ ~ 0.4 for the C2/m phase and λ ~ 0.25 for the C2/m-2 phase. Such weak EP couplings, together with bad metallic behavior in the C2/m-2 phase, result in almost vanishing Tc. Very interestingly, the predicted value of λ for the phase is enhanced up to ~ 1.5, yielding high-pressure induced superconductivity with Tc ~ 6 K. In the I4/mmm phase, the maximal λ is equal to 0.7 and it decreases with further increasing pressures.

Figure 4
figure 4

Calculated superconducting properties of IrTe2 phases as functions of pressure.

(a) Calculated electron-phonon coupling and (b) superconducting transition temperature.

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Variable-composition Search

We also investigated the phase stabilities of Ir-Te systems calculating the formation enthalpy of various IrxTe1−x compounds, as shown in Fig. 5. The formation enthalpy Ef of IrxTe1−x was calculated as

In the entire region of explored pressures, IrTe2 stays as a stable compound. The Ir-rich phase emerges at high pressure. Convex hull diagram in Fig. 5 features newly predicted thermodynamically stable compounds: IrTe3 at 20 GPa [Trigonal (space group R3c)]; IrTe at 70 GPa [Monoclinic (space group C2/c)] and 120 GPa [Orthorhombic (space group Cmcm)]; and Ir3Te at 120 GPa [Tetragonal (space group I4/mcm)]. Electronic densities of states show that all of these compounds are good metals. The lattice parameters including the vdW (DFT-D2) correction are shown in Table 3. In IrTe3 and two phases of IrTe, six Te atoms bonding with one Ir atom construct a Ir-Te6 octahedra. In Ir3Te, the fundamental unit is consisted by Te-topped cube with Ir atoms locating at the face centers (Fig.S1 in supplementary information). For all the structures, we calculated phonons and EP couplings and found all of them to be dynamically stable (Fig.S2). The EP couplings given by phonon calculations are from 0.3 to 0.5.

Table 3 Optimized structural parameters of IrTe3, IrTe and Ir3Te, calculated with PBE and DFT-D2 methods
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Figure 5
figure 5

Formation enthalpies of IrxTe1−x per atom with respect to Ir and Te for different Ir-Te phases.

The abscissa x is the fraction of Ir in the structures. The blue circles indicate the most stable structures at each stoichiometry.

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In summary, with crystal structural prediction techniques, it is found that, with increasing pressure, IrTe2 may undergo a series of structural phase transitions: and the latter three phases under high pressures are newly discovered. For various phases of IrTe2, the electronic band structures, phonon spectra and electron-phonon couplings have been studied. It is shown that there are relatively higher EP coupling constant in the high-pressure and I4/mmm. We hope such a pressure induced superconductivity in IrTe2 will stimulate new experiments on this material. The variable-composition search gives unexpected chemical compounds at high pressure with good metal feature. These results suggest that pressure is an effective way to overcome the kinetic barrier to formation in the synthesis of Ir-Te binary system.

Methods

Two crystal structure prediction methods, crystal structure analysis by particle swarm optimization (CALYPSO)20,21,22 and ab initio random structure searching (AIRSS)23,24 are employed. Both methods give the same best candidates independently for given chemical compositions and external conditions. The electronic structure calculations with high accuracy for the stable structures are performed using the full-potential linearized augmented plane wave (FP–LAPW) method implemented in the WIEN2K code25. The generalized gradient approximation (GGA)26 is applied to the exchange-correlation potential calculation. The muffin tin radii are chosen to be 2.5 a.u. for Ir and 2.33 a.u. for Te. The plane-wave cutoff is defined by RKmax = 7.0, where R is the minimum LAPW sphere radius and Kmax is the plane-wave vector cutoff. The phonon calculations are carried out by using a density functional perturbation theory (DFPT) approach through the Quantum-ESPRESSO program27. The cutoffs are chosen as 30 Ry for the wave functions and 300 Ry for the charge density. The electronic integration is performed over a 16 × 16 × 16 k-point mesh. Dynamical matrices and the EP couplings are calculated on a 4 × 4 × 4 q-point grid. A dense 24 × 24 × 24 grid is used for evaluating an accurate EP interaction matrix. Due to the large atomic numbers of Ir and Te, the spin-orbit coupling is included in our calculations.