2.1. Electrical resistance
Figure 1a shows the temperature dependence of the electrical resistance of AgInSnPbBiTe5 measured using the conventional four-probe method at ambient pressure. Tczero was 1.8 K, which is slightly lower than that reported in previous research . We measured the electrical resistance (at National Institute for Materials Science) after several days since the sample was synthesized by HP annealing (at Tokyo Metropolitan University), and a slight aging effect of Tc was noticed in MTe superconductors synthesized under high pressure [28, 29]. We consider that the slight decrease in Tc can be understood by the change in the internal strains because the XRD patterns do not show a remarkable change after aging in the HP-synthesized MTe samples. Therefore, we continued to perform resistance measurements under high pressure using a diamond-anvil cell (DAC).
Figure 1b shows the temperature dependences of the electrical resistance of AgInSnPbBiTe5 measured under various pressures with a DAC. Normal-state resistance decreases with pressure, for pressures up to 9.3 GPa and increases with pressure for pressures above 15.5 GPa. An onset of the superconducting transition was observed at 2 K under 6.5 GPa, and the zero-resistance state was observed at Tczero = 2 K under 10.2 GPa. As shown in Fig. 2a, the Tconset monotonously increased with pressure until it reached 13.0 GPa, and then it became insensitive to pressure for pressures ranging from 13.0 GPa to 35.1 GPa ( P = 35.1 GPa is the chosen maximum pressure for the experiment). The results indicate that the Tc in the CsCl-type structure of AgInSnPbBiTe5 is independent of applied pressure, while lattice constant decreases with pressure. The highest Tczero and Tconset observed in the experiment on AgInSnPbBiTe5 are 4.5 K and 5.3 K, respectively. See Figure S1 for the determination criterion for Tconset.
To compare the pressure evolution of Tc in AgInSnPbBiTe5 with that of PbTe and AgPbBiTe3, the pressure dependences of Tconset for PbTe and AgPbBiTe3 were examined (see Fig. 2c for structural difference and Figs. S2 and S3 for the resistance measurements), and the resulting Tc-P plots are shown in Fig. 2b. PbTe is a semiconductor at ambient and low pressures, but exhibits a pressure-induced superconducting transition above 17 GPa in the CsCl-type structure . At higher pressures, Tc of PbTe monotonously decreases with pressure. In the case of AgPbBiTe3, superconductivity was observed at P > 2.6 GPa, and at this instance Tconset reached 6.5 K. The Tc for AgPbBiTe3 slightly decreases at high pressures. In AgInSnPbBiTe5, superconductivity is observed at low pressures as well because the low-pressure phase, having a NaCl-type structure, itself is a metal and shows superconductivity under ambient pressure (Figs. 1a and 2a). As demonstrated in the next section, the crystal-structure type under high pressure is CsCl-type for all the compounds. However, the trend of the pressure dependences of Tc in the CsCl-type structure exhibit a clear difference among PbTe, AgPbBiTe3, and AgInSnPbBiTe5. The main findings of this study are that the robustness of superconductivity to pressure in HEA-type AgInSnPbBiTe5 is similar to that observed in (TaNb)0.67(HfZrTi)0.33 . To validate the conclusion, we investigated the pressure evolutions of the crystal structure and the electronic structure for those MTe samples under high pressure.
2.2. Crystal structure
Figures 3a–3c show the pressure-dependent synchrotron X-ray diffraction (SXRD) patterns for the PbTe, AgPbBiTe3, and AgInSnPbBiTe5 samples, respectively. For all the SXRD patterns, we performed the Rietveld refinement to confirm the structural type and to evaluate the lattice constant. See Tables S1–S3 and Figs. S4–S6 for details on refinements. On the basis of the refinement results, we established structural phase diagrams under high pressure (Figs. 3d–3f) by plotting the pressure dependence of volume per unit formula (Z). For PbTe, the structural transition from NaCl-type to Pnma occurs at around 6.80 GPa, and the second transition to CsCl-type takes place at 14.28 GPa. The transition gradually occurred, hence the phase diagram contains mixed phases. The results on PbTe are consistent with the previous work by Li et al. . For AgPbBiTe3 and AgInSnPbBiTe5, similar phase diagrams were obtained, where the NaCl-type structure is stabilized up to ~ 10 GPa, and the Pnma phase is suppressed. The pressure where the CsCl-type phase is induced is common to the case of PbTe. In all the structural types including the CsCl-type phase, the lattice volume continuously decreases with pressure. Although the difference in the stability of the NaCl-type and Pnma structures may be related to the difference in lattice volume at ambient pressure, we consider that the Pnma phase is suppressed, and the NaCl-type phase is stabilized by the effect of alloying at the M site. We note that configurational entropy of mixing does not affect the structure of the CsCl-type phase, and lattice volume of the CsCl-type phase commonly decreases with pressure in three MTe sample.
2.3. Electronic structure
To examine the effects of pressure and HEA states on the electronic structure, we performed X-ray absorption spectroscopy with partial fluorescence mode (PFY-XAS) for PbTe and AgInSnPbBiTe5. See Fig. S7 for pressure dependences of spectra, and analysis results on the Pb-L3 and Bi-L3 spectra. In general, the absorption spectra at the Pb-L3 absorption edge are similar to those at the Bi-L3 absorption edge. On the basis of analogically referring to other Pb- or Bi-containing compounds [30, 31], we analyzed the spectra by assuming several peaks. An example of the fit for the PFY-XAS spectra at 27 GPa is shown in Fig. 4a. The PFY-XAS spectra were fitted by assuming some Voigt functions with an arctan-like background . In this study, we focus on the peaks of P1, P2, and P3, where peak P1 could be assigned as a dipole transition of the 2p3/2 electron into the 6s state, and the peaks shown as P2 and P3 correspond to 6d states of t2g and eg, respectively [32, 33]; we measure 2p3/2 → nd (n > 6) transitions at the Pb L3 absorption edge. There is a p density of states (p DOS) above the Fermi level, however, we mainly observe the dipole-allowed transitions of Pb 2p–6s and Pb 2p–6d, and therefore, the observed spectra do not reflect the Pb 6p DOS spectroscopically. The absorption spectra reflect the empty DOS above the Fermi level generally, with a core hole in the final state.
For PbTe with a NaCl-type structure, there is a narrow band gap, and the p orbitals of Pb and Te near the Fermi energy are hybridized. See Figure S8 for the calculated DOS for PbTe . The valence band is mainly composed of Te 5p orbitals, and also contains the contribution from Pb 6p and 6s, while the conduction band is mainly composed of Pb 6p orbitals, but also contains Te 5p contributions. For PbTe with a CsCl-type structure, band gap is totally closed, and the DOS near the Fermi energy is explained by the contributions from Pb 6p and 6s orbitals, as well as the Te 5p orbitals. Therefore, the pressure evolution of the 6s states which could be resolved in our high-resolution spectroscopy, may play an important role on the closing of the band gap as well as the emergence of superconductivity.
The pressure dependences of the intensity and the energy of peak P1 in PbTe is shown in Fig. 4b. The intensity of peak P1 in Fig. 4b gradually increases with pressure, up to about 5 GPa. The increase in the intensity of P1 indicates an increase in the amount of holes in the Pb 6s states, which indicates a modification of the band structure. In the middle-pressure phase, (between 5–15 GPa) the intensity of P1 does not show a significant change, while it increases remarkably in the HP phase (P > 15 GPa). We note that PbTe with a NaCl-type or Pnma structure is a semiconductor with a band gap at the low-pressure regime, and we also note that the metallic phase is induced in a CsCl-type structure for pressures above 15 GPa [22, 34, 35]. The increase of the intensity of P1 corresponds with the increase of the unoccupied 6s states of the Pb. This may correlate with the emergence of the superconductivity after the closing of the band gap at P > 15 GPa.
On the other hand, the energy of peak P1 shifts to a lower incident energy until it reaches a pressure level of 5 GPa. The incident energy and the intensity do not change in the middle-pressure range of 5–17 GPa, and they start decreasing again for pressures above 18 GPa as shown in Fig. 4b. The shift of the energy of peak P1 to a lower incident energy is explained by the upward shift of the Fermi level or change in the DOS at the Fermi level. Theory suggests that the energy shift of peak P1 may be influenced by the reduction of the band gap  and the theoretical band gap is in the same order as the energy shift of P1 at 5 GPa.
The intensity of P3 (Pb 6d DOS, eg) shows a trend which is similar to the intensity of P1. The intensity of P2 (Pb 6d DOS, t2g) on the other hand decreases with increasing pressure at P > 17 GPa (Fig. 4c). It is interesting that there is a large change in the electronic structure for pressures above 20 GPa, but the crystal structure still retains its CsCl-type structure in this pressure range. In PbTe, superconductivity suddenly appears above 18 GPa, and Tc decreases with pressure monotonically . The present result possibly suggests that the change in the electronic structure is not favorable for the superconductivity of MTe, when it has transitioned to the CsCl-type structure.
We also measured the PFY-XAS spectra at the Pb-L3 absorption edges for AgInSnPbBiTe5 as shown in Fig. S7. We observed similar trends in the pressure dependence of the electronic structures as those observed for PbTe. An example of the fit at 28.8 GPa is shown in Fig. 4d. The analysis results on P1, P2, and P3 are plotted in Figs. 4e and 4f. Figure 4e shows a gradual increase of the P1 intensity with pressure, which is similar to the case of PbTe. The trend of P2 is also similar for the entire pressure range, and that of P3 is basically similar between PbTe and AgInSnPbBiTe5. The PFY-XAS spectra at the Bi-L3 absorption edge were also taken, and the analysis results are summarized in Fig. S7. In AgInSnPbBiTe5 the pressure-induced change in the electronic structure seems to be common for Bi and Pb sites; the detailed results are shown under the Supporting Information section. In conclusion, the electronic structures of PbTe and AgInSnPbBiTe5 show a similar pressure dependence, even though the structure of PbTe does not change much in the middle-pressure range (Pnma + CsCl phase), which disappears in AgInSnPbBiTe5. Therefore, the difference in the robustness of superconductivity to pressure in the CsCl-type phase between PbTe and AgInSnPbBiTe5 cannot be explained by the pressure evolutions of crystal and electronic structures.