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[Ryo Matsumoto](https://orcid.org/0000-0001-6294-5403), [Kazuki Yamane](https://orcid.org/0000-0002-0162-5411), Yoshikazu Mizuguchi, Rikuya Ishikawa, Kyohei Takae, Rei Kurita, Hidetomo Usui, Masahiro Ohkuma, [Kensei Terashima](https://orcid.org/0000-0003-0375-3043), [Yoshihiko Takano](https://orcid.org/0000-0002-1541-6928)

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[Pressure-induced superconductivity and robust Tc against external pressure in (Ge,Sn,Pb)Te](https://mdr.nims.go.jp/datasets/9cec05ae-30c1-42cc-b442-69c403adcd58)

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1 Pressure-induced superconductivity and 1 robust Tc against external pressure in (Ge,Sn,Pb)Te 2 *Ryo Matsumoto1, Kazuki Yamane1,2, Yoshikazu Mizuguchi3, Rikuya Ishikawa3, Kyohei Takae4,  3 Rei Kurita3, Hidetomo Usui5, Masahiro Ohkuma1, Kensei Terashima1, Yoshihiko Takano1,2 4  5 1Research Center for Materials Nanoarchitectonics (MANA), 6 National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan 7 2Graduate School of Pure and Applied Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, 8 Ibaraki 305-8577, Japan 9 3Department of Physics, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji, Tokyo 192-10 0397, Japan 11 4Department of Fundamental Engineering, Institute of Industrial Science, University of Tokyo, 4-6-1 12 Meguro-ku, Tokyo, 153-8505, Japan 13 5 Department of Applied Physics, Shimane University, 1060 Nishikawatsu-cho, Matsue, Shimane, 14 690-8504, Japan 15  16 *Corresponding author; Email: MATSUMOTO.Ryo@nims.go.jp 17  18  19  20 Abstract 21 The robustness of superconducting transition temperature (Tc) against external pressure in medium 22 entropy alloy (MEA) and high entropy alloy (HEA) -type compounds has attracted significant interest 23 with regards to the realization of stable superconducting applications. In this study, we have 24 synthesized Ge1/3Sn1/3Pb1/3Te belonging to MEA-type MTe, where the M site comprises only isovalent 25 group-14 elements, to depict a critical factor for the robustness of Tc. High-pressure electrical transport 26 measurements and structural analysis reveal that the high-pressure phase of CsCl-type cubic structure 27 exhibits the robustness of Tc against external pressure. Molecular dynamics simulation and density 28 functional theory calculation suggest that the glassy atomic vibration characteristic mainly contributes 29 to the appearance of the robustness. This insight accelerates the further development of unique 30 properties within HEA superconductors and their applications. 31  32   33  2 1. INTRODUCTION 34 High-entropy alloys (HEAs), composed of at least five elements with 5–35 at% in the composition [1-35 3], have attracted significant research attention because of their distinct mechanical properties, 36 including ultrahigh fracture toughness [4-5], remarkable resistance to degradation [6], and so on [7-37 9]. These attributes make HEAs promising candidates for innovative applications. The tunability of 38 charge, spin, and composition degrees of freedom within HEAs enables the design of functionality not 39 only in mechanical performance but also in physical properties, such as achieving low thermal 40 conductivity in thermoelectric materials [10], improved ionic conductivity [11], and enhancement of 41 superconducting properties [12]. The advancement of HEAs paves the path for the creation of high-42 functionality materials. 43 In 2014, Ta-Nb-Zr-Hf-Ti HEA was initially discovered to exhibit superconductivity with a 44 transition temperature (Tc) of 7.3 K [13]. One of the noteworthy characteristics of HEA 45 superconductors is the robustness of Tc against an applying external pressure. While the Tc of simple 46 substance Nb demonstrates a decreasing trend under external pressure, the HEA (TaNb)0.67(HfZrTi)0.33 47 maintains an almost constant Tc up to 190 GPa [14]. This robustness in superconductivity is believed 48 to be associated with a reduction in Gibbs’s free energy due to the elevated configuration-entropy of 49 mixing (ΔSmix), defined as − R ∑i cilnci, where R is the gas constant and ci is the atomic ratio of the 50 element (i). Nevertheless, the mechanism of the robustness of Tc in HEA superconductors remains an 51 unresolved matter. The investigation of the mechanism is significant importance for the design of 52 stable superconducting applications under compressive stress.  53  Recently, the concept of HEA superconductor has been successfully expanded to include 54 “HEA-type compounds” [15]. This extension has led to the development of various HEA-type 55 compound superconductors, such as NaCl-type metal chalcogenides MCh (M = Ag, In, Ge, Sn, Sb, Bi, 56 Pb; Ch = S, Se, Te) [16-19], layered systems of REO0.5F0.5BiS2 [20,21], high-Tc cuprates REBa2Cu3O7-57 d (RE: rare-earth elements) [22-24], and van der Waals material (Co, Au)0.2(Rh, Ir, Pd, Pt)0.8Te2 [25]. 58 Among them, medium entropy alloy (MEA) -type compound of Ag1/3Pb1/3Bi1/3Te (ΔSmix = 1.1R) and 59 HEA-type compound of Ag0.2In0.2Sn0.2Pb0.2Bi0.2Te (ΔSmix = 1.6R) exhibit the robustness in 60 superconductivity under external pressure in their high-pressure phase with a CsCl-type cubic 61 structure [26]. Here, the MEA is defined as the alloy with ΔSmix between 1.0R and 1.5R [3]. In contrast, 62 the mother material PbTe (ΔSmix = 0) and low entropy Sn0.5Pb0.5Te (ΔSmix = 0.69R) show a decrease 63 in Tc under compression due to a reduction in the electronic density of state (DOS) at Fermi energy 64 (EF). High-pressure X-ray absorption spectroscopy conducted on PbTe and Ag0.2In0.2Sn0.2Pb0.2Bi0.2Te 65 reveals that the DOS at EF for MTe are insensitive as ΔSmix increases [26]. Consequently, the 66 mechanism behind the robust superconductivity under pressure in MEA- and HEA-type MTe is not 67 solely comprehensible by considering alterations in the electronic states. Recent molecular dynamics 68 (MD) simulations and band calculations for HEA-type MTe illuminate a potential mechanism for the 69  3 robustness, involving unique electron-phonon coupling in conjunction with glassy atomic vibrations 70 and blurry electronic band dispersions [27]. 71 However, it remains uncertain whether the glassy phonon state or the blurry electronic 72 structure is the critical factor for this robustness. This is because of the fact that in the previous studies, 73 the constituent elements for M in MTe have been from various groups in the periodic table, for instance, 74 Ag, Pb, and Bi in Ag1/3Pb1/3Bi1/3Te (ΔSmix = 1.1R). Considering their valence states, MTe are basically 75 composed of M2+ and Te2− ions such as PbTe, while typical valence states of M site in Ag1/3Pb1/3Bi1/3Te 76 are Ag+, Pb2+ and Bi3+. In such a case, the substitution of Pb2+ with Ag+ and Bi3+ would substantially 77 influences both electronic and phonon states, and it is difficult to distinguish the possible main origin 78 of the observed robustness. 79 In this study, we synthesize Ge1/3Sn1/3Pb1/3Te (ΔSmix = 1.1R), where all the elements in the M 80 site belong to the same group with isovalent 2+. Here, we observe the occurrence of pressure-induced 81 superconductivity at high pressures. Remarkably, Ge1/3Sn1/3Pb1/3Te exhibits a similar robustness in Tc 82 against pressure as the case of Ag1/3Pb1/3Bi1/3Te and Ag0.2In0.2Sn0.2Pb0.2Bi0.2Te. Through MD 83 simulation and first-principles calculations, Ge1/3Sn1/3Pb1/3Te exhibits broadened vibration spectrum 84 without the presence of blurry electronic band dispersion. Our results imply that the origin of 85 robustness in MEA- and HEA-type MTe is attributed to the presence of glassy phonon states in a 86 crystal structure with enhanced ΔSmix.  87  88 2. Materials and methods 89 Polycrystalline samples of Ge1/3Sn1/3Pb1/3Te (ΔSmix = 1.1R) were prepared employing a high-pressure 90 annealing method. Starting materials of Ge powder, Sn grains, Pb grains, and Te chips were put into 91 an evacuated quartz tube in the stoichiometric compositions of Ge1/3Sn1/3Pb1/3Te. The sealed ampoule 92 was heated at 800°C for 15 hours, followed by furnace cooling. The obtained precursors were ground 93 into powders and pelletized into 5 mm diameter, and loaded into a hexagonal BN cell for the high-94 pressure annealing in a cubic-anvil-type 180-ton press to obtain NaCl-type structure. The annealing 95 conditions of pressure, temperature, and period are 3 GPa, 500°C, and 30 min, respectively. Powder 96 X-ray diffraction (XRD) at ambient pressure was performed on a Miniflex600 (RIGAKU) 97 diffractometer equipped with a CuKα radiation (λ = 1.5418 Å) by the θ-2θ method. Rietveld 98 refinement was conducted for obtained XRD pattern using RIETAN-FP software [28] to estimate the 99 lattice constant. The crystal structure image was drawn using VESTA software [29].  100 The crystal structure in the obtained sample under high pressure was investigated through XRD 101 measurements in a diamond anvil cell (DAC). These measurements were carried out utilizing 102 synchrotron radiation at the AR-NE1A beamline of the Photon Factory (PF) situated at the High 103 Energy Accelerator Research Organization (KEK). The X-ray beam was monochromatized to an 104 energy of 30 keV (λ = 0.4175 Å) and introduced to the sample in the DAC through a collimator with 105  4 50 μm diameter. The XRD patterns obtained were subsequently integrated into a one-dimensional 106 profile using IPAnalyzer [30], and the lattice constants were determined using PDIndexer [30]. High-107 pressure electrical transport measurements were executed within the DAC, employing a diamond 108 electrode [31-33], and were carried out in the physical property measurement system (PPMS, Quantum 109 Design). The sample itself and cubic BN powders were used as pressure-transmitting medium for 110 XRD analysis and electrical measurement, respectively. The pressure in DAC was estimated by ruby 111 fluorescence method [34] using an inVia Raman Microscope (RENISHAW).  112  The vibrations of atoms in MTe were investigated for MTe with NaCl-type structure using 113 MD simulations by using the Ewald summation method. Particles i and j interact via the Lennard-114 Jones (LJ) and Coulomb potentials.  115 𝑚𝒓̈𝑖 = ∑ 𝑭(𝒓𝒊𝒋)𝑗≠𝑖 , 116 𝑭(𝒓𝒊𝒋) = −𝜕𝑈LJ 𝜕𝒓𝑖𝑗⁄ + 𝑭𝑞, 117 𝑈LJ = 4𝜀 [(𝜎𝑖𝑗𝑟𝑖𝑗)12− (𝜎𝑖𝑗𝑟𝑖𝑗)6], 118 𝐹𝑞 = 𝑘𝑞𝑖𝑞𝑗𝒓𝑖𝑗/𝑟𝑖𝑗3, 119 where rij is the center-to-center displacement vector from particle j to i, rij is its absolute value, m is 120 the mass common to all the particles, ε is the coefficient for LJ potential, and σij = (σi + σj)/2, where σi 121 is the size of particle i, k is the Coulomb constant, and qi is the charge of particle i. The length of the 122 particles was normalized by the ionic diameter of Te2- (σTe = 4.42 Å). The particle sizes of Ag, Bi, Ge, 123 Sn, and Pb are 0.520, 0.466, 0.330, 0.421, and 0.538, respectively. Here, the atomic mass m = 127.6 124 g/mol and interatomic interaction ε = 295 kB J are assumed to be constant to elucidate the role of size 125 and charge dispersity. We randomly arranged 6912 cation particles and 6912 Te2- particles in a cubic 126 structure. In one direction, the number of unit cells was 12. The same values as in Ref. [27] were used 127 for the cutoff length and Ewald parameters. We calculate the vibrational density of state (VDOS) using 128 the Fourier transformation of the velocity autocorrelation function. Details of the calculations were 129 described in Ref. [27]. Here we note that NaCl-type structure is stable since this simulation contains 130 no defects. However, we consider that VDOS in NaCl-type structure is similar to that in CsCl-type 131 structure since both structures are cubic [27].  132 To evaluate the electronic band structure of CsCl-type MTe, density functional theory (DFT) 133 calculations using the Korringa-Kohn-Rostoker Green’s function method was performed in the 134 AkaiKKR package [35]. A coherent potential approximation was used to deal the disordered structure 135 [36]. For the calculation, the generalized gradient approximation parameterized by Perdew-Burke-136 Ernzerhof [37] was adapted, and spin-orbit coupling was included. The k-mesh of 13×13×13 was set, 137 and the width of the energy contour for complex integration was set to 1.5 Ry. The reliability of DFT 138 calculations was checked by the calculation of electronic structure in PbTe using WIEN2k [38]. 139  140  5 3. Results and discussion 141 Figure 1 (a) presents a powder XRD pattern for the obtained Ge1/3Sn1/3Pb1/3Te, along with the result 142 from Rietveld refinement. The green bars correspond to the peak positions of the determined crystal 143 structure, and the blue spectrum represents the differential curve in fitting. The analysis elucidates that 144 the sample crystallizes with a NaCl-type cubic structure (Fm-3m) devoid of any impurity phase. The 145 derived lattice parameters is a = 6.24887 Å with a reliability factor of Rwp = 5.753%. Notably, the 146 value of a is close to that of PbTe under high pressure of 5.9 GPa (6.2492 Å), where PbTe undergoes 147 an insulator-to-metal transition above 2 GPa [26]. This similarity can be attributed to the chemical 148 compression effect arising from the substitution of Pb with Ge and Sn, both of which have smaller 149 atomic radius. 150 Figure 1 (b) shows a temperature (T) dependence of resistivity (ρ) in Ge1/3Sn1/3Pb1/3Te at 151 ambient pressure. The ρ-T curve in the low-temperature region is well-fitted with the Bloch-Gruneisen 152 equation, as described by following equation [39], 153 ρ(T)=𝜌0+A (TθD)5∫x5(ex-1)(1-e-x)dxθD T⁄0 154 where ρ0 is the residual resistance, A is a characteristic constant, θD is the Debye temperature. The 155 parameters are determined as ρ0 = 1.493(4) mΩcm, A = 1.12(2), and θD = 125(1) K, as shown in the 156 inset. The well-fitted ρ-T curve indicates that Ge1/3Sn1/3Pb1/3Te exhibits metallic transport nature. The 157 θD is comparable with that of pristine PbTe [40]. In fig. 1 (c), the Hall resistivity of Ge1/3Sn1/3Pb1/3Te 158 is plotted as a function of the applied magnetic field at ambient pressure to confirm carrier 159 concentration. The carrier concentration has been estimated from the slope of Hall resistivity versus 160 magnetic field, using the formula ρH = (1/ned)H, where ρH is the Hall resistivity, n is the number of 161 carriers, e is the elementary charge, H is the magnetic field and d is the sample thickness. The ρH curve 162 displays a positive slope, indicating a p-type characteristic with a carrier concentration of 1.0×1021 163 cm-3. This carrier concentration significantly exceeds the typical value found in pristine PbTe, which 164 are on the order of 1016 cm-3 [41] with an insulating nature. The elevated carrier concentration in 165 Ge1/3Sn1/3Pb1/3Te is consistent with the metallic ρ-T behavior and is likely a result of the chemical 166 compression effect.  167  6  168 FIG. 1. (a) XRD patterns with Cu Kα radiation (λ = 1.5418 Å) of obtained Ge1/3Sn1/3Pb1/3Te with 169 the fitting result of Rietveld refinement. The green bars and blue spectrum indicate a peak 170 position of the determined structure and a differential curve for the fitting. (b) Temperature 171 dependence of resistivity in Ge1/3Sn1/3Pb1/3Te. The inset shows a result for the Bloch-Gruneisen 172 fitting. (c) Hall resistivity as a function of magnetic field at 1.8 K in Ge1/3Sn1/3Pb1/3Te. 173  174 Figure 2 (a) shows a temperature-dependent resistance (R) of Ge1/3Sn1/3Pb1/3Te under pressures 175 up to 21.1 GPa. Throughout all pressure regions, a metallic R-T curve is consistently observed. The R 176 value at 300 K continuously increases with pressure up to 8.2 GPa, after which it tends to a decreasing 177 trend with further compression, as indicated in the inset. Notably, the R-T curve exhibits a sharp drop 178 at 20.1 GPa and reaches zero at 1.8 K at 21.1 GPa. This drastic change in R-T curve indicates the 179 emergence of pressure-induced superconductivity in Ge1/3Sn1/3Pb1/3Te. As the pressure increases, the 180 R value at 300 K decreases until 30 GPa and then shows a saturating trend, as presented in fig. 2 (b). 181 The Tc demonstrates a monotonous increase as a function of pressure, reaching a plateau at 30.1 GPa, 182 where it remains nearly constant at 5.5 K. This behavior is quite similar to other HEA-type MTe 183 compounds [26,27]. 184 In fig. 2 (c), XRD patterns of Ge1/3Sn1/3Pb1/3Te at various pressures up to 37 GPa are displayed. 185 At 1.0 GPa, the observed pattern is indexed as a NaCl-type cubic structure, identical to the ambient 186 structure. The peaks with minor intensity gradually appear above 5 GPa, leading to a complete 187 transformation of the NaCl-type structure into the GeS-type structure at 9.0 GPa. Above 31.0 GPa, the 188 CsCl-type cubic structure gradually becomes predominant. The phase diagram, depicting the pressure-189 dependent volume for each structure, is illustrated in fig. 2 (d). This pathway of structural phase 190 transition is similar to that of other HEA-type MTe [26,27], including pristine PbTe [42]. Although 191 the stabilized region of the intermediate GeS-type structure is more extended in Ge1/3Sn1/3Pb1/3Te 192 compared to other MTe compounds, the underlying reason remains uncertain at present. 193  7  194 FIG. 2. Results of high-pressure experiments in Ge1/3Sn1/3Pb1/3Te. (a) Temperature dependence 195 of resistance under various pressures from 4.1 to 21.1 GPa and (b) 20.1 to 33.5 GPa. The inset 196 in (a) shows the pressure dependence of resistance at 300 K. (c) XRD patterns under various 197 pressures. (d) Pressure dependence of volume in each structure.  198  199 Figure 3 presents a comparison of the pressure-dependent Tc in PbTe (ΔSmix = 0) [26], 200 Ag1/3Pb1/3Bi1/3Te (ΔSmix = 1.1R) [26], and Ge1/3Sn1/3Pb1/3Te (ΔSmix = 1.1R). Pristine PbTe shows a 201 drastic decline in Tc above 15 GPa, with a rate of dTc/dP = −0.24 K/GPa, due to a reduction of 202 electronic DOS at EF under pressure. In contrast, Ag1/3Pb1/3Bi1/3Te exhibits a notably suppressed 203 decreasing rate of Tc, with dTc/dP = −0.052 K/GPa. This suppression, namely, robust Tc against 204 pressure is considered to be related to the glassy atomic vibrations or blurry electronic structures [27]. 205 Interestingly, Ge1/3Sn1/3Pb1/3Te, where all constituent elements in the M site belong to the same group 206 with a valence state of 2+, also displays a robust Tc with a decreasing rate of dTc/dP = −0.016 K/GPa. 207 This decreasing rate is comparable to those of Ag1/3Pb1/3Bi1/3Te, and Ag0.2In0.2Sn0.2Pb0.2Bi0.2Te. It 208 indicates that the robustness of Tc against pressure is primarily a result of the increased ΔSmix and 209 independent of the valence state of the elements within the M site. 210  8  211 FIG. 3. Comparison of the pressure-dependent Tc in PbTe (ΔSmix = 0), Ag1/3Pb1/3Bi1/3Te (ΔSmix = 212 1.1R), and Ge1/3Sn1/3Pb1/3Te (ΔSmix = 1.1R). The data of PbTe and Ag1/3Pb1/3Bi1/3Te are referred 213 from ref [26]. The dashed lines are guides for the eye. 214  215 According to the previous reports, one possible scenario for the robustness of Tc against 216 pressure involves the presence of glassy vibrational characteristics and/or blurry electronic structures 217 [27]. MD simulations for atomic vibrations and DFT calculations for electronic states are conducted 218 to delve into the mechanism behind the robust Tc. Figure 4 (a) shows the VDOS for PbTe, 219 Ag1/3Pb1/3Bi1/3Te, and Ge1/3Sn1/3Pb1/3Te calculated for the NaCl-type crystal structure. The VDOS 220 peaks attributed to atomic vibrations exhibit a flattened profile in Ag1/3Pb1/3Bi1/3Te compared to PbTe. 221 This broadening in the spectrum implies the presence of various vibrational modes in the material 222 without a distinct characteristic vibration mode. The broadened VDOS is indicative of a glassy 223 vibrational characteristic, which contributes to the robustness of Tc. The VDOS spectrum for 224 Ge1/3Sn1/3Pb1/3Te shares a similar flat feature with Ag1/3Pb1/3Bi1/3Te. The emergence of the glassy 225 vibrational characteristic is attributed to an enhancement in ΔSmix, regardless of the valence states in 226 the M site. Figure 4 (b) presents an electronic band structure for Ge1/3Sn1/3Pb1/3Te in CsCl-type crystal 227 structure with a=3.5Å. The band structure exhibits slight blurriness, indicating the presence of 228 localized splitting of electronic states. In contrast, the degree of blurriness is more pronounced in 229 Ag1/3Pb1/3Bi1/3Te [27]. The shape of the band structure in Ge1/3Sn1/3Pb1/3Te is close to that of PbTe 230 rather than Ag1/3Pb1/3Bi1/3Te. The results of MD simulation and DFT calculation collectively suggest 231 that the primary contributor to the emergence of robust Tc against external pressure in MEA- and HEA-232 type MTe is the presence of broadened vibrational characteristic due to the enhancement of ΔSmix. 233  9  234 FIG. 4. (a) Simulated VDOS spectra for PbTe, Ag1/3Pb1/3Bi1/3Te, and Ge1/3Sn1/3Pb1/3Te. (b) 235 Electronic band structures for Ge1/3Sn1/3Pb1/3Te at a=3.5Å. 236  237 IV. CONCLUSIONS 238 In this study, we synthesize Ge1/3Sn1/3Pb1/3Te belonging to MEA-type MTe family with ΔSmix = 1.1R, 239 and subsequently observe the emergence of pressure-induced superconductivity. The high-pressure 240 phase of the CsCl-type cubic structure shows almost constant pressure-dependent Tc, namely, the 241 appearance of the robustness of Tc against external pressure. Based on the MD simulations and DFT 242 calculations, it was turned out that the robust Tc persists in Ge1/3Sn1/3Pb1/3Te where the broadened 243 VDOS exists without the blurry electronic band dispersions. Our current study implies that the unique 244 phonon state due to the enhancement of ΔSmix plays a crucial role to induce the robust Tc in MEA- and 245 HEA-type MTe. This insight holds significant importance for advancing our understanding of the 246 mechanisms behind the robustness of Tc and in developing stable superconducting applications under 247 compressive stress. 248  249 ACKNOWLEDGMENTS 250 This work was partly supported by JSPS KAKENHI Grant Number 23K13549. The fabrication 251 process of diamond electrodes was partially supported by the NIMS Nanofabrication Platform in the 252 Nanotechnology Platform Project sponsored by the Ministry of Education, Culture, Sports, Science 253 and Technology (MEXT), Japan. 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