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The most incompressible metal osmium at static pressures above 750 gigapascals

Abstract

Metallic osmium (Os) is one of the most exceptional elemental materials, having, at ambient pressure, the highest known density and one of the highest cohesive energies and melting temperatures1. It is also very incompressible2,3,4, but its high-pressure behaviour is not well understood because it has been studied2,3,4,5,6 so far only at pressures below 75 gigapascals. Here we report powder X-ray diffraction measurements on Os at multi-megabar pressures using both conventional and double-stage diamond anvil cells7, with accurate pressure determination ensured by first obtaining self-consistent equations of state of gold, platinum, and tungsten in static experiments up to 500 gigapascals. These measurements allow us to show that Os retains its hexagonal close-packed structure upon compression to over 770 gigapascals. But although its molar volume monotonically decreases with pressure, the unit cell parameter ratio of Os exhibits anomalies at approximately 150 gigapascals and 440 gigapascals. Dynamical mean-field theory calculations suggest that the former anomaly is a signature of the topological change of the Fermi surface for valence electrons. However, the anomaly at 440 gigapascals might be related to an electronic transition associated with pressure-induced interactions between core electrons. The ability to affect the core electrons under static high-pressure experimental conditions, even for incompressible metals such as Os, opens up opportunities to search for new states of matter under extreme compression.

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Figure 1: The dependence of the unit cell volume of Pt on pressure P.
Figure 2: Diffraction patterns of the samples compressed in dsDACs.
Figure 3: Experimental dependence of the unit cell volume and the ratio of the lattice parameters c/a of Os on pressure P.
Figure 4: Calculated electronic density of states (DOS) of Os as a function of energy E (relative to the Fermi energy EF).

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Acknowledgements

L.D. and N.D. acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG) and the Federal Ministry of Education and Research (BMBF), Germany. N.D. thanks the DFG for funding through the Heisenberg Program and the DFG project number DU 954-8/1, and the BMBF for grant number 5K13WC3 (Verbundprojekt O5K2013, Teilprojekt 2, PT-DESY). M.E., Q.F., and I.A.A. acknowledge support from the Swedish Foundation for Strategic Research programme SRL grant numbers 10-0026, the Swedish Research Council (VR) grant numbers 621-2011- 4426, the Swedish Government Strategic Research Area Grant Swedish e-Science Research Centre (SeRC), and in Materials Science “Advanced Functional Materials” (AFM). The work was supported by the Ministry of Education and Science of the Russian Federation (grant number 14.Y26.31.0005). The simulations were carried out using supercomputer resources provided by the Swedish national infrastructure for computing (SNIC). M.I.K. acknowledges financial support from the ERC Advanced grant number 338957 FEMTO/NANO and from NWO via a Spinoza Prize. Portions of this work were performed at GeoSoilEnviroCARS (Sector 13), Advanced Photon Source (APS), Argonne National Laboratory. GeoSoilEnviroCARS is supported by the National Science Foundation - Earth Sciences (EAR-1128799) and Department of Energy - GeoSciences (DE-FG02-94ER14466). This research used resources of the Advanced Photon Source, a US Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under contract number DE-AC02-06CH11357.

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Contributions

L.D., N.D., E.B., M.B., V.P., C.P., K.G., H.-P.L., and M. H. conducted the experiments. L.D. and N.D. processed the experimental data. M.E., Q.F., L.V.P., M.I.K., J.M.W., and I.A.A. performed the theoretical analysis. The manuscript was written by L.D. and I.A.A. with contributions from all other authors. All the authors commented on drafts and have approved the final version of the manuscript.

Corresponding authors

Correspondence to L. Dubrovinsky or I. A. Abrikosov.

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Extended data figures and tables

Extended Data Figure 1 Equations of state of W.

a, Comparison of different EOSs of W reported in Dewaele et al.54, Sokolova et al.23, and Litasov et al.55, and this work. Although the curves agree for pressures up to about 50 GPa, there is a substantial discrepancy for pressures around 0.5 TPa. b, Pressure dependence of the unit cell volume of W. Experimental data points (red solid dots represent data collected using a Au–W mixture, green diamonds using a Pt–W mixture) were fitted using the third-order Birch–Murnaghan EOS (blue solid line, V0 = 31.674(3) Å3 per unit cell, K300 = 307(2) GPa, K′ = 4.53(4)); data are equally well fitted with the Vinet EOS (V0 = 31.686(7) Å3 per unit cell, K300 = 302(1) GPa, K′ = 4.82(3)).

Extended Data Figure 2 Examples of the full-profile (GSAS) treated diffraction patterns.

a, Au–Pt mixture collected in a dsDAC at 482(5) GPa (Au pressure scale from ref. 20). Even at pressures in the proximity of 5 Mbar, powder X-ray diffraction data are sufficient to clearly resolve peaks of Au and Pt, and accurately determine lattice parameters of both metals. b, W–Pt mixture, unsuccessful experiment in a dsDAC. According to the W EOS, the pressure is 461(7) GPa, whereas according to the Pt EOS, it is 559(10) GPa. This inconsistency in pressures is a result of an inhomogeneous distribution of the two metals between secondary anvils in the dsDAC; such data cannot be used for constraining EOSs. This observation also shows that very large pressure gradients are possible in dsDACs. c, Mixture of Os (a = 2.5404(6) Å, c = 4.0386(10) Å), W (a = 2.9133(5) Å), and Au (a = 3.6657(6) Å) collected in a conventional DAC (Ne used as the pressure transmitting medium) at a pressure of 134(2) GPa.

Extended Data Figure 3 Experimental dependence of the unit cell volume of Os on pressure in comparison with EOSs reported in the literature.

The magenta line is from Occelli et al.3 (K300 = 421 GPa, K′ = 4.0) and the green line is from Cynn et al.4 (K300 = 463 GPa, K′ = 2.8). These data have been re-fitted using the ruby pressure scale as suggested in refs 20, 22 and references therein. Experimental data points (solid red dots) were fitted using the third-order Birch–Murnaghan EOS (blue line, V0 = 28.02(4) Å3 per unit cell, K300 = 399(6) GPa, K′ = 4.04(4)).

Extended Data Figure 4 Electronic band structure of Os at moderate compressions along the high-symmetry lines in the hcp Brillouin zone.

Energies are given relative to the Fermi energy, which is taken to be zero. Calculations are carried out at pressures of 0 GPa (a, b), 97.5 GPa (c, d), and 134 GPa (e, f). a, c, e, The k -resolved spectral functions A( k , ω) obtained with LDA+DMFT. b, d, f, The band structure obtained with LDA. In both cases, we used the experimental lattice parameters (a = 2.734 Å, c/a = 1.580 at 0 GPa; a = 2.578 Å, c/a = 1.589 at 97.5 GPa; a = 2.540 Å, c/a = 1.590 at 134 GPa). Our LDA+DMFT calculations predict that two ETTs occur in hcp Os upon compression. In a, a band at the Γ point is well below the Fermi energy at ambient pressure; however, it nearly touches the Fermi energy at 97.5 GPa, and the corresponding hole pocket has already appeared at 134 GPa, giving rise to a change of the Fermi surface topology, that is, to an electronic topological transition at about 101.5 GPa (Extended Data Fig. 5). Our LDA+DMFT calculations also indicate that another ETT at the L point should occur above 134 GPa, at about 183 GPa (Extended Data Fig. 5).

Extended Data Figure 5 The position of the relevant bands at the Γ (top) and L (bottom) points obtained from LDA+DMFT and LDA calculations as a function of pressure.

The positive values indicate appearance of the corresponding hole pockets. Energies are given relative to the Fermi energy. A closer examination of the band shape in the vicinity of the L point reveals that this hole pocket first appears along the L–H line and then extends to include the L point. To estimate more precisely the critical pressures for the ETTs, we plot the positions of the relevant bands at the Γ and L points with respect to the Fermi level and obtain the values of the critical pressures by interpolation. The LDA+DMFT calculations predict that the hole pockets at the Γ and L points appear at about 101.5 GPa and 183 GPa, respectively.

Extended Data Figure 6 k -resolved spectral functions A( k , ω) of Os at ultra-high compressions along the high-symmetry lines in the hcp Brillouin zone.

a, b, Calculations are carried out at pressures of 247 GPa (a) and 477 GPa (b) using LDA+DMFT. The experimental lattice parameters were used in the calculations (a = 2.449 Å, c/a = 1.596 at 247 GPa; a = 2.335 Å, c/a = 1.597 at 477 GPa). Energies are given relative to the Fermi energy, which is taken to be zero. Hole pockets are present at the Γ and L points at P = 247 GPa. Increasing the pressure to 477 GPa does not induce any new ETTs. Moreover, no features of the band structure suggest that new ETTs might be induced by further increase of the pressure within reasonable limits.

Extended Data Figure 7 Electronic band structure of Os at moderate compressions along the high-symmetry lines in the hcp Brillouin zone.

ad, Band structure of Os at 0 GPa (a, b) and at 134 GPa (c, d). Energies are given relative to the Fermi energy. In a and c, spin–orbit coupling has been included, whereas in b and d, the energies were obtained with the scalar-relativistic approximation. Experimental lattice parameters were used in the calculations (see Extended Data Fig. 4). Bands are shown in different colours for clarity.

Extended Data Figure 8 Electronic band structure of Os at extreme compressions along the high-symmetry lines in the hcp Brillouin zone.

ad, Band structure of Os at 247 GPa (a, b) and at 477 GPa (c, d). Energies are given relative to the Fermi energy. In a and c, spin-orbit coupling has been included, whereas in b and d, the energies were obtained with the scalar-relativistic approximation. Experimental lattice parameters were used in the calculations (see Extended Data Fig. 6).

Extended Data Figure 9 Comparison of the EOS for hcp Os calculated using different approximations within DFT with the experimental EOS measured in this work.

Shown are GGA calculations carried out by us using the RSPt method (black solid line) and by Sahu et al.56 (blue dashed line), as well as LDA calculations carried out by us using the Wien2k method (red solid line), by Cynn et al.4 (pink dashed line), and by Sahu et al.56 (red dashed line). The experimental EOS obtained in this work is shown with black circles. The different curves agree reasonably well for pressures up to about 50 GPa. In contrast, there is noticeable discrepancy at ultra-high pressure, and there is no theoretical EOS that accurately describes the experiment for pressures around 0.5 TPa. Local and semi-local approximations within DFT are insufficient to describe the PV relationship of Os under extreme conditions. Part of the reason for the disagreement between theory and experiment might be related to the improper account of the many-electron effects within the theory. In principle, LDA and GGA work better at high pressure; however, errors that might be important at low pressure47 could propagate through the EOS to the whole pressure range, owing to the use of all the calculated points in the fitting of the energy versus volume data by, for example, the third-order Birch–Murnaghan EOS used in this work. To justify this statement, consider the calculated EOS parameters, summarized in Extended Data Table 1. The equilibrium volumes calculated by us differ from the experiment (Extended Data Table 1) by less than 1.5%. On the other hand, the overestimation of the calculated bulk moduli (B) and their pressure derivatives (B′) is greater, about 10%. Our EOS parameters are within the range of theoretical parameters available in the literature, which are fitted for P < 100 GPa (Extended Data Table 1). We deal with a highly incompressible metal, for which typical DFT errors in B, and especially in B′, translate into large differences in PV relationships at ultra-high pressure. Even in this regime, the error in volume at a fixed pressure remains within typical DFT limits of about 2%–3%. However, the pressure calculated at fixed volume can differ by several tens of gigapascals. This difference is due to errors in B and B′ calculated at ambient pressure, coupled to a very high value of B. The use of more advanced theoretical methods could improve the calculated EOS. In ref. 47, a substantial reduction of B in isoelectronic hcp Fe is demonstrated using a LDA+DMFT approach. Here the effect is expected to be smaller, but may still be sufficient to improve the agreement with experiment. Our results demonstrate a need to further develop the electronic structure theory, with the experiment reported here providing a bench-mark for the theory. On the other hand, we consistently used experimental lattice parameters in the discussion of the electronic structure of Os in this study.

Extended Data Figure 10 Comparison of the ratio of the lattice parameters c/a for hcp Os.

The ratio is calculated using the RSPt method at T = 0 K with a PBE-GGA (red dashed line), and using Wien2k with a LDA (green line) and a PBE-GGA (blue line). The room-temperature experimental results obtained in this work are shown with filled black dots. Agreement between the calculated and experimental c/a ratio is typical for DFT calculations. The theoretical results do not show any peculiarity of the lattice-parameter ratio; however, the calculations are carried out at T = 0 K, whereas the experimental data are taken at room temperature. Therefore, a direct comparison between theory and experiment is nontrivial (Methods). In hcp metals, the effect of the electronic transitions on the lattice-parameter ratio should become visible at finite temperatures, owing to the peculiarities of the thermal expansion coefficients, which are anisotropic along different directions of the crystal lattice.

Extended Data Table 1 EOS parameters of Au, Pt, W, and Os

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Dubrovinsky, L., Dubrovinskaia, N., Bykova, E. et al. The most incompressible metal osmium at static pressures above 750 gigapascals. Nature 525, 226–229 (2015). https://doi.org/10.1038/nature14681

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