Search for promising structures and superconductivity in high-pressure Sr through first-principles calculations

Pressure-induced structural phase transitions of Sr have been investigated for more than 60 years, and five phases have been reported experimentally. At ambient pressure, Sr takes a face-centered cubic (fcc) structure (Sr-I). With increasing pressure, the fcc structure transforms to a body-centered cubic (bcc) structure at 3.5 GPa (Sr-II),^1,2 and the bcc structure transforms to the β-tin structure $(I 4 1/amd)$ at 24 GPa (Sr-III).³ Next, the β-tin structure transforms to a monoclinic structure with the space group Ia and 12 atoms per unit cell at 38 GPa (Sr-IV),⁴ and the Ia structure then transforms to an incommensurate phase with a host–guest (HG) structure at 46 GPa (Sr-V) and is stable up to 57 GPa. The HG structure then transforms to another accompanying phase at 57 GPa and persists up to at least 75 GPa.⁵ 

Several computational studies on compressed Sr have been conducted,^6–9 and there remains several discrepancies between the experimental and computational results. One computational work showed that the β-tin structure is not the lowest enthalpy structure compared to the bcc or Ia structures in the pressure range of 26–38 GPa suggested by experiments for the β-tin phase.⁶ Based on the enthalpy, the sequence of the phase transitions of Sr with increasing pressure was suggested to be fcc, bcc, and Sr-IV. Another study showed that a hexagonal close-packed (hcp) structure takes lower enthalpy than the β-tin, Ia, and HG structures.⁷ In this work, Srepusharawoot et al. used the Sr-V supercell containing 110 atoms with 14 guest and 10 host unit cells with $I4/mcm$ symmetry. They used an incommensurate ratio (γ) of 1.4 in the c axis compared with 1.4041 from an experiment.¹⁰ 

Kim et al. performed a structure search to find the stable structures of Sr up to 50 GPa with simulation cells containing 2, 3, 4, 8, 9, 10, and 12 atoms.⁸ They found an orthorhombic structure with space group Cmcm symmetry above 25 GPa bringing into question the stability of the HG structure suggested by McMahon et al.⁵ They also calculated the superconducting transition temperatures T_c up to around 55 GPa assuming the Cmcm and bcc structures. Since the difference between the Cmcm and hcp structures is small, it is acceptable to consider the hcp structure as suggested by Srepusharawoot et al.,⁷ however in either case, the suggested structures are not consistent with the experiments.

Tsuppayakorn-aek et al. reported that the discrepancy between the experimental and computational results for the β-tin structure can be explained using the screened exchange local density approximation (sX-LDA).⁹ However, this is the only case that could be explained by this functional. Hence, it is not sufficient for a final conclusion. Furthermore, the results of the enthalpy were shown for only the hcp, bcc, and β-tin structures, and the effects of considering sX-LDA were not discussed for the other structures of interest for compressed Sr.

Katzke and Tolédano investigated the phase transition mechanisms of alkaline-earth metals theoretically in the framework of an extension of the Landau approach to reconstructive phase transitions.¹⁰ They interpreted all the high-pressure phases as resulting from the full or partial realization of the Burgers and Bain deformation mechanisms. It was further suggested that the high-pressure HG structures of Sr and also Ba reflect the competition between these mechanisms.

Desgreniers et al. showed that in the high-pressure phases of Ba, the incommensurate and self-hosting Ba-IV phase is a high-temperature phase realized at room temperature, while there is a much simpler highly symmetric structure at comparable pressure values at low-temperatures, the Ba-VI phase.¹¹ Their calculations predicted Ba-VI to be dynamically stable at high pressures and to be superconductive. Moreover, Jackson et al. reported that T_c of the Ba-VI phase was higher than that of the Ba-IV phase.¹² These results go on to show the potential of cryogenic compression in revealing rich phase diagrams under high pressures.

In light of these findings around the incommensurate HG phase of Ba, and the discrepancies between experiments and theoretical calculations of high-pressure phases of Sr, especially around the experimentally observed highest-pressure HG Sr-V phase, there is a possibility of a highly symmetric structure on the low-temperature and high-pressure region of the incommensurate Sr-V phase. Thus far, to the best of our knowledge, no investigations have been carried out on Sr in the high-pressure region above 100 GPa using first-principles calculations. Very recently, Ito et al. reported two new high-pressure phases at low temperatures (Sr-VI and Sr-VII) of compressed Sr after the Sr-V phase and observed the highest T_c of 13 K at 165 GPa experimentally.¹³ Although they have not been able to identify the structure of the Sr-VI phase, they have identified the Sr-VII phase as having an hcp structure.

In trying to investigate the low temperature phases of compressed Sr after the Sr-V phase, we performed a structure search based on an original method developed by us¹⁴ in the high-pressure region from around 40–2400 GPa. In addition, we further investigated the phonon profiles and T_c associated with these structures. In particular, the pressure dependence of T_c of the Sr-VI phase is important in determining the actual structure. In this work, we determined a group of plausible structures for the Sr-VI phase and their pressure dependence of T_c, which will be useful for comparison with detailed experimental results when elucidating the actual structure. We also confirmed the Sr-VII phase as having an hcp structure. Herein, we report and discuss these results.

We performed the first-principles electronic structure calculations based on density functional theory (DFT) to determine the crystal and electronic structures of compressed Sr. We used Quantum-Espresso package¹⁵ to perform the total energy calculations, structural relaxations, and phonon calculations with the Perdew–Burke–Ernzerhof (PBE)-type generalized gradient approximation (GGA)¹⁶ as an exchange-correlation functional and ultrasoft pseudopotentials (USPP).¹⁷ The valence electron configuration of Sr is considered to be ten electrons consisting of 4s²4p⁶5s². The cutoff radius of the local potential was set to 1.9 a.u., the cutoff energy of the wave function E_cut was 80 Ry, and the cutoff energy for the charge density was set twelve times E_cut. The Brillouin zone for each structure was sampled using Monkhorst–Pack k-point meshes of 24 × 24 × 24 for four-atom systems, 12 × 12 × 12 for 12 atom systems, 8 × 8 × 8 for 16 atom systems, 4 × 4 × 4 for 32 atom systems, and 2 × 2 × 2 for 64 atom systems. In determining these values, test calculations were performed to ensure that the total-energy precision was sufficient to estimate the differences in the energy and enthalpy between the relaxed structures. The errors were found to be up to 3 meV/atom, and in many cases, smaller. Occasionally, E_cut = 120 Ry was used for validation purposes in some cases; however, there were no essential differences between the obtained results. Ensuring the computational precision with the aforementioned parameters, we performed an extensive structure search using an original method developed by us.^14,18

Initial structures from the space group index 75 (P4) to 230 $(Ia 3 ¯d)$ were generated for a unit cell consisting of four atoms. These cover the tetragonal, trigonal, hexagonal, and cubic structures as the initial structures. The subsequent structural relaxations were performed assuming only the Bravais lattice. In addition, we performed a structure search in a random manner at the given pressure values. We did not perform structural relaxation for all the randomly generated initial structures. Instead, we picked a set of low enthalpy structures obtained from a self-consistent field (SCF) calculation and performed a full optimization on those chosen ones only. We generated 800 initial structures and selected 20 low-enthalpy structures for further investigations.

For phonon calculations, E_cut was set to 80 Ry for the fcc, hcp, and double hcp (dhcp) structures, and k-point meshes of 24 × 24 × 24, 24 × 24 × 18, and 24 × 24 × 8 were used for the fcc, hcp, and dhcp structures, respectively. In addition, q-point meshes of 8 × 8 × 8, 6 × 6 × 4, and 6 × 6 × 2 were used for the fcc, hcp, and dhcp structures, respectively.

Figure 1 shows the calculated enthalpies of plausible structures hcp, Cc, I4/mcm, C2/m, Pnma, Cmc2₁, and P2₁/c for compressed Sr with respect to the enthalpy of the hcp structure as a function of pressure from 40 to 200 GPa. In obtaining these results, a structure search was performed (except for Cc) covering the space groups mentioned earlier with cells containing four atoms. It is noted that below 60 GPa and above 140 GPa, the structures labeled “Pnma,” “Cmc2₁,” and “P2₁/c” reduced to the hcp structure as a result of a structural relaxation. The enthalpies labelled “hcp,” “I4/mcm,” and “C2/m” remain the same even after a structural relaxation.

Below 60 GPa, the lowest enthalpy structure was the hcp one. To confirm the difference between the Cmcm and hcp structures around which some controversies exist, a structural relaxation was performed using the Cmcm structure as the starting structure, which resulted in the hcp structure. It is reasonable to suggest that the stable structure is obtained from first-principles calculations in the pressure region of Sr-IV is the Cmcm structure or its special case, the hcp structure. We conjecture that Sr-IV has the hcp structure. This is reasonable considering the results of previous works,^6–8 and the fact that the cutoff energy of 80 Ry used in our calculations is larger than that of 36.75 Ry (500 eV) used in previous works.

Sr-V is an incommensurate phase and should be explored using cells containing more atoms. Hence, the structure searches were performed using cells containing 12, 16, and 32 atoms, using a random search at 40 and 100 GPa. The C2/m structure was found as the lowest enthalpy structure through the random structure-search based method and the Pnma structure as the second lowest one using cells containing 32 atoms at 100 GPa. The I4/mcm structure was found as the lowest enthalpy structure and the P2₁/c, Pnma, and Cmc2₁ structures were found as the second, third, and fourth lowest enthalpy ones using cells containing 16 atoms at 100 GPa. The Pnma structure was found as the lowest enthalpy structure and the hcp one as the second lowest enthalpy structure using cells containing 12 atoms at 100 GPa. However, the C2/m and I4/mcm structures could not be found using smaller cells. Other than these structures, the Cc structure was also obtained by optimizing the Fdd2 structure containing 16 atoms per unit cell shown by Whaley-Baldwin et al. as the approximate structure of the HG structure.²⁰ These seven structures including hcp are the candidate structures for consideration for the Sr-V and Sr-VI phases in this pressure region.

From about 60 to 160 GPa, the C2/m, I4/mcm, and Cc structures are obtained as one group of lowest-enthalpy structures as plausible candidates for the Sr-V phase. The crystal structures of C2/m, I4/mcm, and Cc prepared using VESTA²¹ are shown in Figs. 2(a)–2(c). Even though the enthalpy is low, the Cc structure is dynamically unstable based on the phonon calculations within the harmonic approximation. The I4/mcm structure is also dynamically unstable, only with the exception at 100 GPa. The I4/mcm structure found here has the same space group as the structure shown in Ref. 5, which might correspond to the case where the value of the incommensurate ratio of the HG structure may be approximated by 16 atoms. The C2/m structure is dynamically stable in the entirety of the plotted pressure range and is the most promising candidate for the Sr-V phase as will be discussed later together with the accompanying T_c values and their conformity with the experimental results. The atomic positions of the C2/m structure at 100 and 120 GPa and that of the I4/mcm structure at 100 GPa are listed in Tables I–III in the supplementary material, respectively. The difference in atomic positions in the C2/m structure above and below 100 GPa is the 4e (1/4,1/4,0) or 4f (1/4,1/4,1/2) sites.

From 60 to 120 GPa, the Pnma, Cmc2₁, and P2₁/c structures are obtained as another group of the second-lowest enthalpy structures being potential candidates for the Sr-VI phase. The crystal structures of Pnma, Cmc2₁, and P2₁/c prepared using VESTA²¹ are shown in Figs. 2(d)–2(f). Within this group, the Pnma structure has the same space group as the Ba-VI phase.¹¹ In considering the experimental results of Ref. 13 and the similarity with Ba-VI, the Pnma structure may be the most promising candidate for the Sr-VI phase. The Pnma and Cmc2₁, and P2₁/c structures are similar to each other. In fact, only when the tolerance value of the atomic positions of FINDSYM^22,23 identifying a space group is increased from 0.01 to 0.1, Cmc2₁ is distinguished from Pnma. With respect to enthalpy, these three structures are considered to be the same. In the powder diffraction patterns calculated using VESTA,²¹ the patterns of these structures are very similar to each other as shown in Figs. S2(a)–S2(c) in the supplementary material.¹¹ The lattice constants, atomic positions, and atomic volumes of the Pnma, Cmc2₁, and P2₁/c structures at 100 GPa are listed in Tables I–III, respectively. The atomic positions are almost the same as the Ba-VI phase.¹¹ At 100 GPa, the pattern of the Cmc2₁ structure perfectly matches that of the Pnma structure. The deviation of the Cmc2₁ or P2₁/c structure from the Pnma structure is also very small. Therefore, it may be reasonable to consider these as representing the Pnma structure at 100 GPa. However, it is noted that at other pressure values, the structures show different T_c values, which will be discussed below.

Figure 3 shows the calculated T_c in this work for the hcp, I4/mcm, C2/m, Pnma, Cmc2₁, and P2₁/c structures from 80 to 500 GPa. The experimental data of T_c of compressed Sr up to 60 GPa were published by Mizobata et al.,²⁴ and the computational results were reported by Kim et al.⁸ There is no information on T_c of compressed Sr over 60 GPa. In the pressure region shown by Mizobata et al., T_c increased up to about 8 K. In our work, T_c increases up to around 9.3 K in the C2/m structure at 100 GPa, and then decreases with pressure. This is similar to the trend observed in the Sr-V phase.¹³ Hence, we conjecture C2/m to be the most plausible approximate structure for the Sr-V phase.

In the Pnma structure, T_c increases up to about 15.9 K as the pressure increases up to 140 GPa. In the Cmc2₁ structure, T_c increases up to about 15.3 K as the pressure increases up to 120 GPa, and then decreases with pressure. In the P2₁/c structures, T_c increases up to about 11.5 K as the pressure increases up to 120 GPa. These three plausible structures Pnma, Cmc2₁, and P2₁/c are aggregated into the hcp structure at approximately 160 GPa (or around 140 GPa). It is particularly important to note that the pressure dependence of T_c for Pnma, Cmc2₁, and P2₁/c is mostly consistent with that of the experimental results obtained for Sr-VI.¹³ This is a promising evidence that the structure of Sr-VI is either Pnma, Cmc2₁, or P2₁/c. If it is the same as for Ba,¹¹ the Pnma structure can be considered the most plausible structure.

The pressure dependence of T_c for the three structures in Fig. 3 shows that at 100 GPa, T_c is nearly identical. At this point, the Pnma structure is the most plausible candidate since the XRD patterns shown in the supplemental material (Fig. S2 in the supplementary material) almost overlap. However, at 120 and 140 GPa, T_c behaves quite differently in these three structures. Considering the experimental results where T_c increases with pressure within the Sr-VI phase and reaches a maximum of 13 K at 165 GPa in the hcp Sr-VII phase,¹³ the Pnma structure in our calculations shows a higher T_c of 15.9 K at a relatively lower pressure of 140 GPa, while the Cmc2₁ structure reaches a maximum at 120 GPa and shows a sharp drop at 140 GPa. On the other hand, the P2₁/c structure shows an increase in T_c up to 120 GPa (11.5 K) and then becomes the hcp structure at 140 GPa. A careful comparison of the pressure dependence of T_c based on our calculations with detailed experimental results is needed to ascertain the actual structure in the Sr-VI phase.

At approximately 160 GPa, the lowest enthalpy structure becomes the hcp structure and represents the Sr-VII phase. This is consistent with the experimental results of Ito et al.¹³ The enthalpies of the Cc, I4/mcm, and C2/m structures have higher enthalpy above around 160 GPa, and it is reasonable not to consider these as potential structures with further compression. Based on our structure search, there are three structures that should be compared, the hcp, dhcp, and fcc structures above 160 GPa pressurizing up to 2400 GPa as shown in Fig. 4. The plotted results were obtained through a structure search covering the tetragonal, trigonal, hexagonal, and cubic structures with cells containing four atoms. Moreover, to further validate the results, the structure search was performed at 600, 1200, 1600, 2000, and 2400 GPa using cells containing 16 atoms, which rendered results consistent with those obtained through smaller cells. The sequence of the phase transitions with increasing pressure is hcp (≈160 GPa) → dhcp (≈1000 GPa) → fcc (≈2200 GPa). The structural transitions from hcp to dhcp and then from dhcp to fcc indicate that as the pressure increases, the stacking of the close-packed structure from ABAB to ABAC and from ABAC to ABCABC results in lower enthalpy. The structural stability of these structures is confirmed through phonon calculations within a harmonic approximation, and the results are shown in Fig. 5. In the restricted calculations using small cells, structures with low enthalpy may not necessarily be dynamically stable; however, within the harmonic approximation, these structures are dynamically stable up to 2400 GPa.

After the transition to the hcp structure around 160 GPa, T_c decreases with pressure. Although not plotted in Fig. 3, as long as the hcp structure stays intact, the superconducting state survives beyond 1000 GPa with a decreasing T_c. At 1000 GPa, T_c is 0.65 K (λ = 0.371, ω_log= 664.4 K). In Fig. 4, the hcp structure transforms to the dhcp structure at approximately 1000 GPa. In the dhcp structure, the superconducting state is stable and T_c at 1000 GPa is 1.1 K (a = 2.1667 Å, c = 7.0549 Å, λ = 0.397, ω_log= 639.6 K) and T_c at 2000 GPa is 0.04 K. (a = 2.0041 Å, c = 6.5234 Å, λ = 0.292, ω_log= 811.7 K). The dhcp structure transforms to the fcc structure at approximately 2200 GPa. The superconducting state of the fcc structure diminishes at around 2400 GPa, and T_c at 2400 GPa is 0.001 K (a = 2.77 Å, λ = 0.246, ω_log= 882.3 K), and the value of λ is much smaller.

We performed a structure search for highly compressed Sr using the first-principles calculations and determined the structures from 40 to 2400 GPa. In particular, the plausible candidate structures for the Sr-V and Sr-VI (low-temperature) phases from about 60–160 GPa and for the Sr-VII phase above 160 GPa were determined. The lowest and second lowest enthalpy groups in the pressure range of 60–160 GPa are considered to be the plausible structures corresponding to the Sr-V and Sr-VI phases, respectively. For these structures, the pressure dependence of T_c was calculated, and the results are consistent with the recent experimental results.¹³ Based on our calculated structures and the conformity of T_c with the experimental results, C2/m is the most plausible approximate structure for the Sr-V phase and Pnma, Cmc2₁, and P2₁/c are the most plausible candidates for the low-temperature Sr-VI phase. The Pnma structure shares the same space group structure with the Ba-VI phase. More detailed experimental investigations are required together with our results to ascertain the actual structure in the Sr-VI phase. Approximately above 160 GPa, the hcp structure is the most plausible candidate for the Sr-VII phase according to our calculations, and this is consistent with the recent experimental results. The hcp structure is dynamically stable up to around 1000 GPa and transforms to the dhcp structure, and the dhcp structure then transforms to the fcc structure at around 2200 GPa. The superconducting state remained intact through these structural phase transitions, although with a decreasing T_c with increasing pressure.

See the supplementary material for information on the 100 GPa structures in the text and the XRD patterns calculated with VESTA using them.

This work was supported by JSPS KAKENHI, a Grants-in-Aid for Scientific Research (C) (Nos. 19K037170, 23K04518a, and 24K069590).

The authors have no conflicts to disclose.

Masaaki Geshi: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Project administration (lead); Resources (lead); Software (lead); Supervision (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Hiroki Funashima: Methodology (supporting); Software (equal); Writing – original draft (supporting); Writing – review & editing (supporting). Gayan Prasad Hettiarachchi: Formal analysis (supporting); Investigation (supporting); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.