The Zintl compound TlInTe2is an intriguing material because of its outstanding thermoelectric properties at ambient pressure. Interestingly, it has recently been found that TlInTe2exhibits a V-shape dependence of the superconducting critical temperature (
Chain crystal structures from the TlSe family, such as TlInSe 2 , TlGaTe 2 , and TlInTe 2 , exhibit exciting properties at ambient pressure [1][2][3][4]. For example, TlInTe 2 has an exceptionally low value of lattice thermal conductivity (∼0.5 W/mK at room temperature) [5], and TlInSe 2 is considered to have outstanding thermoelectric properties [6]. However, relatively limited research has been conducted on this material class under pressure [7][8][9][10].
According to the first high-pressure Raman study on TlInTe 2 performed by S Ves in 1990, it was suggested that a structural transition from the tetragonal to hexagonal phase may occur in the 7-17 GPa range, but the type of symmetry in the new phase could not be determined [7]. The next work, conducted 30 years later, used Raman spectroscopy, x-ray diffraction, and transport measurements along with first-principles crystal structure prediction to investigate pressure-induced structural and electronic transitions [10]. It was found that the system undergoes a superconducting transition at 5.7 GPa with a reported critical temperature (T c ) of 3.8 K. Further compression resulted in a decrease in the critical temperature to a minimum of 2.9 K at 10 GPa, followed by an increase to a maximum observed value of 4.3 K at 25 GPa. This V-shape behavior of T c was correlated with the pressure dependence of the Raman active A g mode, where the T c decreases (increases) as the A g mode hardens (softens), respectively.
The potential role of the A g mode is compelling because it is well known that, at ambient conditions, TlInTe 2 is an anharmonic system, where the anharmonicity is considered to be responsible for the ultra low value of the lattice thermal conductivity [1,5,[11][12][13]. Additionally, a recent theoretical study employing a minimal model of superconductivity that takes into account the anharmonic decoherence of the optical phonons predicted a non-monotonic behavior of T c under pressure, in qualitative agreement with the V-shape curve observed experimentally [14].
The focus of this ab initio study is to analyze the structural, electronic, vibrational, and superconducting properties of TlInTe 2 as a function of pressure. First, we show that the tetragonal phase remains dynamically stable within the 0-30 GPa range, consistent with the latest experiments [10]. Second, in stark contrast to the experimental results that found a V-shaped pressure-induced dependence of T c , we find a dome-shaped pressure-induced dependence with a maximum T c value of 0.23 K at 18 GPa, approximately 20 times smaller than the largest experimental value. To further understand this discrepancy, we investigated the effect of anharmonicity on both the phonon dispersion and the phonon lifetimes at 0 and 14 GPa. Our theoretical results show that, at 0 K, the anharmonic renormalization of the harmonic phonon frequencies is negligible and the low-frequency phonons have extremely high lifetimes, which suggest that anharmonic effects should play a minor role in the superconducting properties of TlInTe 2 . We also analyzed the effect of doping and found that T c values in the experimental range could, in principle, be reached but only for large doping levels.
First-principles calculations were carried out with the Quantum ESPRESSO (QE) package [15,16]. We employed the optB86b-vdW functional [17][18][19][20][21] and optimized norm-conserving Vanderbilt pseudopotentials (ONCVPSP) [22] from the Pseudo Dojo library [23] generated with the relativistic Perdew-Burke-Ernzerhof parametrization [24]. The 5d 10 6s 2 6p 1 orbitals for Tl, 4d 10 5s 2 5p 1 for In, and 4d 10 5s 2 5p 4 for Te were included as valence electrons. A plane wave kinetic-energy cutoff of 80 Ry for the wavefunctions and 400 Ry for the charge density and potential were used. For the Brillouin-zone integration of the 8-atom unit cell, we used a Γ-centered 12 × 12 × 12 k-mesh [25] with a Methfessel-Paxton smearing [26] width of 0.01 Ry. The atomic positions and lattice parameters were optimized until the total energy was converged within 10 -6 Ry and the force on each atom was less than 10 -4 Ry/Å. The dynamical matrices and the linear variation of the self-consistent potential were computed using density-functional perturbation theory (DFPT) [27] on the irreducible set of a regular 4 × 4 × 4 q-mesh. The electron-phonon (e-ph) interactions were first evaluated on a Γ-centered 12 × 12 × 12 k-mesh and an irreducible set of a regular 4 × 4 × 4 q-mesh. For each q-point, the e-ph matrix elements were linearly interpolated to a denser 60 × 60 × 60 k-mesh [28].
The EPW [29][30][31] code was used to investigate the e-ph interactions and estimate the critical temperature for select pressures. The electronic wavefunctions required for the Wannier-Fourier interpolation [32,33] were calculated on a Γ-centered 8 × 8 × 8 k-mesh. Thirty maximum localized Wannier functions (sp 3 hybridized orbitals for Tl, In, and Te) were used to describe the electronic structure. Uniform 60 × 60 × 60 k-point and 30 × 30 × 30 q-point grids were employed in the e-ph coupling calculations.
Anharmonic effects can be computed through approaches based on stochastic sampling, ab initio molecular dynamics, explicit computation of higher order interatomic force constants (IFCs), or special displacements [34][35][36][37][38][39][40][41][42]. Here, the ALAMODE package [39,43,44] was used to compute anharmonic phonon frequencies and phonon linewidths through the real-space supercell apporach employing a 2 × 2 × 2 supercell of 64 atoms. This required the extraction of the second-, third-, and fourth-order IFCs. The second-order IFCs were extracted by performing displacements of 0.01 Å for atoms in the supercell and computing the Hellmann-Feynman forces for the displaced configuration. The anharmonic third-and fourth-order IFCs were extracted in a similar way as the second-order terms but with a displacement of 0.04 Å. There were no cutoff radii specified for the computation of the second-and third-order terms, including all interactions, however, the fourth-order term used a cutoff radius to only include nearest neighbor interactions for every atomic pair. The IFCs that contain a combination of atoms whose pair is larger than the cutoff radius is set to zero. The fitting error for the fourthorder IFCs was 0.20% at 14 GPa. Setting the nearest neighbor distances up to 8 Bohr required the calculation of over 800 displacements. Using the default distance of 10 Bohr would have significantly increased the computational cost and would have required calculating over 3500 displacements for a 64 atom unit cell. Considering the substantial computational cost for a larger cutoff radius, and the small fitting error achieved for nearest neighbors, the use of the smaller cutoff seems appropriate in this case. The anharmonic phonons were obtained by solving the self-consistent phonon (SCPH) equation. Both the q-mesh and the inner q′-mesh in the SCPH were set to 4 × 4 × 4. The phonon linewidths were calculated by taking the imaginary part of the anharmonic self-energy on a 10 × 10 × 10 q-mesh.
At ambient conditions, TlInTe 2 adopts the tetragonal crystal structure, with space group I4/mcm (No. 140) [45]. The unit cell consists of covalently bonded tetrahedral chains of [InTe 2 ] along the c-axis, as shown in figure 1 [46]. The tetrahedrons connect to each other by the corner shared Te atoms, with the In atom residing in the middle of the tetrahedron. The Tl atoms occupy the center of a distorted octagon formed by 8 Te atoms (2 different side lengths), with the Tl and Te atoms ionically bonded. The weak bonding of the Tl atoms is characteristic of rattler structures where loosely bound atoms in a regular periodic crystal lattice have large amplitude vibrations. The rattling model has been demonstrated in TlInTe 2 [5] and shown to be typical for materials with low lattice thermal conductivity [47,48].
A possible tetragonal to hexagonal structural phase transition in the 7-17 GPa range was pointed out in [7], but the Raman measurements did not allow for the determination of the symmetry in the new phase. On the other hand, a recent study did not find any evidence of a phase transition from the x-ray diffraction patterns taken up to 33.5 GPa, but predicted that the tetragonal structure undergoes a phase transition to a body-centered cubic Pm m 3 phase at 50 GPa through an intermediate distorted orthorhombic Pbcm phase that appears at 37.5 GPa [10]. We find that the tetragonal structure indeed remains dynamically stable up to 30 GPa, the maximum pressure investigated in our work, consistent with the latest study [10]. In addition, as shown in figure 2, the present theoretical results for the pressure dependence of in-plane and out-of-plane lattice parameters are in very good agreement with the theoretical and experimental data reported in [10], confirming that there is no structural phase transition in the 0-30 GPa pressure range.
According to experimental measurements, TlInTe 2 is a semiconductor with an indirect band gap of approximately 1.0 eV [49][50][51], and undergoes a semiconductor-to-semimetal transition at 4 GPa [10]. Our density functional theory (DFT) calculations return a lower band gap value of 0.29 eV and a semiconductor-tosemimetal transition at 2 GPa, as expected within the PBE scheme employed in this work. We also carried out electronic structure calculations with the modified Becke-Johnson (MBJ) exchange-correlation potential [52] that has been shown to provide a good description of band gaps in crystalline solids. With MBJ, we find that the band gap increases to 0.71 eV, in agreement with previous DFT calculations [10,53,54], and the semiconductor- to-semimetal transition shifts to 4 GPa. However, the role of the MBJ potential is secondary when describing the electronic structure in the vicinity of the Fermi level (E F ) in metallic systems.
To confirm this, we performed band structure calculations at 0 GPa (semiconducting) and 24 GPa (metallic), with and without the MBJ potential. As shown in figure S1 in the Supplemental Material [55], the MBJ potential at 0 GPa does a good job of reproducing the band features of the PBE calculation while shifting the conduction and valence bands away from the Fermi level. Meanwhile, at 24 GPa, there is no appreciable difference in the band structure near the Fermi level. As a result, the density of states (DOS) at the Fermi level for the two sets of calculations are almost identical for pressure points where the system is in the metallic regime. Lastly, band structure calculations at 0 and 24 GPa with the inclusion of spin-orbit coupling (SOC) indicate that the SOC has a negligible effect (see figure S1 in the Supplemental Material [55]). Since our primary interest is to investigate the superconducting properties of TlInTe 2 , which is predominately influenced by the electronic structure at the Fermi level, we performed all subsequent calculations at the PBE level without the inclusion of SOC.
The evolution of the electronic band structure and total DOS as pressure increases from 6 to 28 GPa is depicted in figure 3. At 6 GPa, the valence and conduction bands cross the Fermi level in the vicinity of the M and Z high-symmetry points and along the X-P direction, respectively. Around 14 GPa, the conduction band along the Γ-Z direction is also pushed down below the Fermi level, and the valence and conduction bands start to overlap. This results in an over two fold increase in the DOS at E F . Past this point, the DOS changes very little with increasing pressure. From the projected DOS (see figure S2 in the Supplemental Material [55]), we infer that in the 6-28 GPa pressure range, the main contribution at the Fermi level comes from Te 5p, In 5s, and Tl 6p states. Our band structure results are in good agreement with previous electronic calculations both at ambient and high pressure [10,54].
Next, we investigated the effect of pressure on the Fermi surface topology of TlInTe 2 [56]. In a number of systems [57][58][59][60][61], the emergence or disappearance of superconductivity has been associated with a Lifshitz transition [62]. This transition is manifested as a subtle change in the Fermi surface topology (e.g., the appearance or disappearance of a pocket or a neck on the Fermi surface) driven by external parameters, such as doping or pressure, and without breaking the crystal symmetry. Our theoretical analysis clearly shows that the crystal structure remains the same up to 30 GPa, but the topology of the Fermi surface evolves as a function of pressure, as shown in figure S3 in the Supplemental Material [55]. We observe that at 6 GPa the Fermi surface is comprised of individual electron and hole pockets. As pressure increases, the electron pockets connect and grow tubular necks forming an umbrella-shaped Fermi surface. It has been proposed that the appearance of these tubular necks could be responsible for the onset of superconductivity in TlInTe 2 [10].
Figure 4 displays the calculated phonon dispersion and atomic projected phonon density of states (PHDOS) at 6, 14, 21, and 28 GPa. The partial decomposed PHDOS shows that the spectrum can be separated into three frequency regions with different atomic contributions. The low-frequency region up to about 10 meV is linked to vibrations from all three types of atoms. The intermediate-frequency region from approximately 10 to 20 meV has little to zero contribution from the lighter In atoms and is characterized almost entirely by Te vibrations. The high-frequency region above 20 meV has evenly mixed contributions from the In and Te atoms with no contribution from the heavier Tl atoms. As pressure increases, there is a general hardening of all phonon branches across the Brillouin zone, except for the A g Raman active mode that has a dome-shaped dependence, peaking around 10 GPa, as shown in figure S4 in the Supplemental Material [55]. Our results capture well the pressure trends of the A g and E g modes observed experimentally, the former being slightly downshifted by about 1-2 meV. This level of underestimation is comparable to the one found for the A g mode in other systems [63,64].
We further analyze the Eliashberg spectral function, α 2 F(ω), and the e-ph coupling strength, », in conjunction with the PHDOS. The comparison between α 2 F(ω) and PHDOS in figure 4 shows that, for all calculated pressures, the modes below 15 meV are responsible for more than two thirds of the total e-ph coupling strength of the system. The remaining third mainly comes from the intermediate-frequency modes of Te, with only a small part from the high-frequency modes of In and Te. We note that in the intermediate region lies the aforementioned Raman active A g mode whose dome-shaped frequency dependence and its broadening linewdith as pressure increases has been suggested to lead to strong e-ph coupling and anharmonicity, and therefore was used to explain the V-shape T c behavior as a function of pressure extracted from transport measurements in [10]. Based on the calculated », we find that the A g mode does not play a major role in the e-ph coupling in TlInTe 2 . In addition, it has been shown that the strong anharmonicity in this material is linked to the low-frequency rattling mode of Tl atoms [5,12], and, from the projected PHDOS, we see that the A g mode is mainly associated with the vibration of the Te atoms. Our anharmonic calculations presented later also show that the A g mode does not appear to be influenced by anharmonic effects. Figure 5 summarizes the dependence of the DOS at the Fermi level and » as a function of pressure. Compression, from 6 to 15 GPa, results in an increase in » as the DOS at the Fermi level increases. Past this pressure point, the general hardening of all phonon modes triggers a decrease in ». The corresponding critical temperatures estimated with the Allen-Dynes modified McMillian formula [65,66] and a Coulomb pseudopotential of ¼ * = 0.1 using QE are shown in figure 5(c). Superconductivity calculations for select pressures performed with EPW give similar » and T c values. We find that TlInTe 2 does not exhibit superconductivity until 10 GPa, and the T c only reaches a maximum value of 0.23 K at 18 GPa. These results differ from the experimental findings in two significant ways: the V-shape curve of superconductivity is not recovered and the critical temperature is off by an order of magnitude.
The significant anharmonicity in the rattling motion of Tl atoms enclosed within the cage-like Thompson cubes formed by surrounding Te atoms has been proven to be a major contributor to the remarkably low lattice thermal conductivity observed in TlInTe 2 [1,5], as it results in ultra-low phonon lifetimes [12,13]. To this end, we calculated the phonon frequencies including quartic anharmonic effects within the SCPH theory. As shown in figure S5 in the Supplemental Material [55], the phonon spectra at 14 GPa computed within the harmonic approximation with DFPT in QE and frozen-phonon in ALAMODE are found to be in good agreement. Taking into account aharmonic effects, we observed that the phonons between 5 and 20 meV are slightly hardened with increasing temperature, while the high-frequency optical phonons remain relatively unchanged, in agreement with calculations at 0 GPa in [12]. These findings suggest that anharmonic effects are not expected to affect the superconducting properties of TlInTe 2 . To further support this point, we also calculated the phonon lifetimes at 0 and 300 K for 0 and 14 GPa. The comparison in figure 6 demonstrates that the lifetimes of low-frequency phonons at 0 K are much larger than those at 300 K for both pressure values. Since smaller lifetimes are linked to anharmonic effects, this plot implies that while TlInTe 2 might be strongly anharmonic at room temperature, the anharmonicity is dampened at low temperatures.
We further explored whether doping could enhance the superconducting transition temperature and bring the results in line with experimental observation. We simulated electron doping using a jellium model for two carrier densities of 0.25 and 0.50 electrons/u.c. Figures S6 and S7 in the Supplemental Material [55] show the band structures and phonon dispersion relations for the doped compound at 6, 10, 14, and 21 GPa. Across all pressures, the DOS at E F for the smaller doping remains comparable to the undoped compound. This behavior changes for the higher doping as a new state crosses the Fermi level along the N-X direction, leading to a more than 25% increase in the DOS at E F . In the case of phonons, there is a general softening of all modes which becomes more pronounced as doping increases. The cumulative effect of the increase in the DOS at the Fermi level and the softening of the phonon modes results in a significant enhancement of the predicted e-ph phonon coupling and critical temperature. As shown in figure 7, the calculated T c is of the same order of magnitude as the experimental value, but the T c dependence with pressure does not display the V-shape behavior found experimentally, instead decreasing almost linearly with increasing pressure independent of the doping concentration. Pressure effects on the critical temperature have been found to lead to vastly different behaviors depending on the system [67][68][69].
Lastly, we analyzed closely the superconductivity results presented by Yesudhas et al [10]. As illustrated in figure 6(b) of [10], the material only enters in a superconducting state with zero resistance at 19.5 GPa. Additionally, in the 5.7-19 GPa interval, there is a finite resistance (see figure 6(a) of [10]) which suggests that in this pressure range the system could be in a mixed state characteristic of a type-II superconductor [70]. This latter observation is also supported by the resistance versus temperature curves for varying magnetic fields at 10.1 GPa in figure 6(c) of [10], showing that as the system goes in this mixed state, the resistance exhibits a gradual decrease as the temperature is lowered instead of a discontinuous drop to zero [70]. We also note that the V-shape pressure dependence of T c was obtained using the onset T c [10] (i.e., the temperature where the resistance starts to drop), while the calculated critical temperature corresponds to the zero-resistance T c . Other factors, such as the appearance of competing electronic phases under pressure, non-hydrostatic pressure conditions in the experimental setup, and the presence of phase inhomogeneities in the sample, can also notably affect the critical temperature and may explain the discrepancy with theory [64,[71][72][73][74].
We performed an ab initio study to investigate the structural, electronic, vibrational, and superconducting properties of TlInTe 2 under pressure. Contrary to experiment, we did not find a V-shape T c behavior with pressure, and, in addition, the estimated T c is an order of magnitude smaller than the experimental values. Doping could, in principle, bring the T c within the experimental range, but the required doping levels are quite large. Furthermore, our calculations of the phonon spectra and phonon lifetimes with anharmonic effects demonstrate that anharmonicity is not expected to affect the superconducting properties of TlInTe 2 . We conclude by asserting that further experimental and theoretical studies are required to elucidate not only the pressure range in which TlInTe 2 is superconducting and the dependence of T c with pressure, but also to explore whether other mechanisms may contribute to superconductivity in this system.