Thermal equation of state of ruthenium characterized by resistively heated diamond anvil cell.

The high-pressure and high-temperature structural and chemical stability of ruthenium has been investigated via synchrotron X-ray diffraction using a resistively heated diamond anvil cell. In the present experiment, ruthenium remains stable in the hcp phase up to 150 GPa and 960 K. The thermal equation of state has been determined based upon the data collected following four different isotherms. A quasi-hydrostatic equation of state at ambient temperature has also been characterized up to 150 GPa. The measured equation of state and structural parameters have been compared to the results of ab initio simulations performed with several exchange-correlation functionals. The agreement between theory and experiments is generally quite good. Phonon calculations were also carried out to show that hcp ruthenium is not only structurally but also dynamically stable up to extreme pressures. These calculations also allow the pressure dependence of the Raman-active E2g mode and the silent B1g mode of Ru to be determined.

www.nature.com/scientificreports www.nature.com/scientificreports/ In contrast with other 4d transition metals, like Mo and rhenium (Re) [14][15][16][17][18] , to the best of our knowledge, only few studies have been focused their attention on the characterization of pure Ru under extreme conditions. In particular, a study was performed in 1965 by Bucher et al. 19 to characterize the effect of pressure on the superconductivity of Ru. The compression curve of Ru was experimentally measured up to 25 GPa by Clendenen et al. 20 in a pressure-cylinder apparatus 21 . More recently, Cynn et al. 22 studied Ru up to 56 GPa in diamond anvil cell (DAC), using argon (Ar) as pressure transmitting medium and energy-dispersive X-ray diffraction (ED-XRD) technique. In contrast with the results reported by Cynn et al., conclusions extracted from studies on iron-ruthenium alloys under high pressure suggest that a hcp-fcc transition should be expected in the range of the tenths of GPa 23 . On the other hand, a few theoretical works have been performed on hcp Ru. In one study, the stability field has been explored by means of elastic constant calculations 24 and in a second study by first-principles calculations 25 . Information on the high-pressure and high-temperature behaviour of Ru is still lacking 5 ; in particular, regarding its melting curve and thermal equation of state.
In this work, the structural stability of Ru has been investigated at room-temperature, up to 150 GPa by angular dispersive (AD) XRD, using helium as pressure medium. High-pressure (HP) high-temperature (HT) studies have also been performed to extend its structural and chemical characterization up to 975 K. The experiments have been combined with first-principle calculations of the structural and lattice dynamic properties. Results evolution at ambient temperature. Two different experimental runs were carried out at ambient temperature at the extreme conditions beamline (I15) of Diamond Light Source under quasi-hydrostatic conditions. Table 1 reports the obtained results. During Run1 the pressure was increased from ambient up to 97.5 GPa with a maximum step of 2 GPa between consecutive pressures.
During Run2 the pressure was increased rapidly up to 92.2 GPa and a finer pressure step was then used until reaching 151 GPa, the highest pressure covered in the present experiment. Under the present experimental conditions, Ru maintains its hcp phase. Figure 1 shows the raw and the integrated AD-XRD patterns of Ru collected at the lowest and the highest pressure achieved at ambient temperature. In both cases, it is possible to observe the 100, 002, 101, 102, 110, 103, 112, 201, 004, 202 and 104 reflections belonging to the hcp structure of Ru. Few parasitic reflections coming from the Re gasket are also visible (labelled with red asterisks), this is caused by the X-ray beam tails obtained on the first table of I15. However, the quality of the Ru XRD patterns is not affected by the presence of the signal from Re.
At ambient pressure, the raw XRD image shows a signal coming from a sample composed of multiple crystals of Ru, and from a Re foil. At the highest pressure achieved (151 GPa), the signal from the Re foil is similar to the one obtained at ambient pressure, whereas, the XRD of Ru shows the formation of some texturing, with small broadening in the diffraction peaks. This behavior is typically associated with the formation of micro-stress in the sample 26 . A Williamson-Hall analysis of our data indicates that the ξ parameter 27 , which is a measure of peak broadening due to strain, changes from 0.5(1) × 10 −3 at ambient pressure to 0.6(1) × 10 −3 at 151 GPa, indicating that strains due to deviatoric stresses are small in our experiments up to the highest pressure. No evidence of structural transitions or distortions have been observed at RT up to 151 GPa.
A qualitative analysis of the hydrostatic conditions of the sample has been performed by comparing the measured d-spacing of Ru at the highest pressure reached (151 GPa) to the theoretical (hydrostatic) one calculated using the lattice parameter obtained by a refinement of the entire XRD pattern 26 . Table 2 shows the interplanar spacing measured at the highest pressure reached in this experiment. They deviate from the spacing calculated using the lattice parameters obtained by the refinement of the whole XRD pattern by less than 0.008%. This is within the experimental error of the present experiment. Therefore, we can conclude that the non-hydrostatic effects are below the detection limit of the present measurements. That is in agreement with the quantification of the macroscopic non-hydrostatic stress on metals embedded in helium that reaches 0.5 GPa at 150 GPa 28 .
In Fig. 2, the obtained compression curve is reported together with the c/a evolution as a function of pressure. The corresponding bulk modulus K 0 , its pressure derivative ′ K 0 and the volume V 0 at ambient temperature have been determined from a least-square fit of the entire set of data to a Rydberg-Vinet 29 and a third-order Birch-Murnaghan (BM3) equation of state (EoS) using the EOSFit7c software 30 . The obtained values are reported in Table 3. The data obtained by Cynn et al. 22 with an ED-XRD DAC experiment performed using Ar as pressure transmitting medium are also reported for comparison in Fig. 2. Although their data results more scattered than the present one, both compression curves show a similar trend.
In the inset of Fig. 2, it can be seen how the c/a ratio of Ru increases with pressure, showing a tendency to approach the ideal c/a value for a hcp structure, thus increasing the packing efficiency of the crystal structure under compression. This phenomenon is consistent with the elasticity theory 31 , according to which, for a hcp ) ( ) 33 13 11 12 , where G is the shear modulus and C ii are the elastic constants. According to the present DFT calculations, the first term of the previous equation is positive in the entire range of the investigated pressure domain. Therefore ∂ ∂ < . c a V ln( / )/ ln( ) 0 This means that the c/a ratio of Ru increases as the volume decreases (i.e. pressure increases) which is exactly what is experimentally observed here. In addition, apparent slope changes can also be observed in the experimentally determined c/a of Fig. 2. However, as the observed variations are smaller than the accuracy of the measurements, they can probably be caused by experimental artifacts. It is important to point here how the c/a variation of Ru from ambient pressure to 150 GPa is only 0.6%. This indicates that the compression of Ru occurs with little changes in its an-isotropic properties.
The c/a evolution with pressure of Ru contrasts with that of other hcp metals like Os 32 and Re 17 . On one hand, Os presents a positive slope in the c/a evolution up to 150 GPa. After this critical pressure, the slope changes into a negative one. This phenomenon is due to an Electronic Topological Transition (ETT) predicted theoretically around that pressure 33 . On the other hand, Re presents a negative slope in the c/a evolution between www.nature.com/scientificreports www.nature.com/scientificreports/ a(Å) c(Å) c/a V (Å 3 ) P (GPa) a(Å) c(Å) c/a V (Å 3 ) P (GPa)  17 . This behaviour is probably caused by a recently discovered electronic transition called Core-Level Crossing (CLC) 1    www.nature.com/scientificreports www.nature.com/scientificreports/ 4f levels. Furthermore, the absence of anomalies in the c/a evolution in Ru rules out the presence of any other pressure-induced electronic transitions up to 150 GPa. This could be extrapolated to Technetium (Tc) as an additional hcp metal with a similar electronic configuration and a similar ionic radius. Regarding the bulk modulus, it is noticeable that this parameter is very similar in Re (352 GPa 17 ), Ir (339 GPa 2 ), and Ru (323 GPa (this work)) for experiments carried out using He as pressure medium. Being Os surrounded by these three elements in the periodic table and having a similar electronic configuration, it is then striking that it has been reported to be the material with the lowest experimentally determined compressibility (462 GPa 22   www.nature.com/scientificreports www.nature.com/scientificreports/ The compression curves and c/a evolution obtained in the present study by DFT using several exchange-correlation functionals, including LDA 35 , GGA-PBE 36 , GGA-PBEsol 37 , and meta-GGA SCAN 33 , are also reported in Fig. 2. In this case, the functional that displays an overall best agreement with the experiments is the GGA-PBEsol, hence most of the results that we present in this work are obtained with it. In particular, the difference between the bulk modulus experimentally measured and the one obtained via PBEsol functional is less than 10%. Furthermore, the pressure evolution of the c/a ratio is well described by the PBEsol for pressure higher than 30 GPa, confirming the anisotropic nature of the compressed hcp Ru. The obtained phonon dispersion at ambient and 160 GPa are shown in Fig. 3. The dispersion at ambient pressure qualitatively agrees with that reported by Heid et al. 38 In particular, at ambient pressure we observe anomalies in all branches in the vicinity of the M point of the Brillouin zone, with three branches being nearly degenerate and one of them being dispersionless. Our results show that hcp Ru is dynamically stable up to 160 GPa. In particular, all the minima from the phonon dispersion (with the exception of the acoustic modes at Gamma) moves towards higher frequencies as pressure increase, suggesting that phonon instabilities are not expected for hcp Ru even beyond 160 GPa. In addition, the anomalies around M are attenuated at 160 GPa and the degeneracy between branches partially broken.  Table 3. EOS parameters of Ru measured and calculated in different experiment. The volume V 0 , the bulk modulus K 0 and its pressure derivative ′ K 0 are listed. Experimental methods and EOS formulation are specified. *Fixed parameter. PTM: Pressure transmitting medium. BM3: third order Birch-Murnaghan. ED-XRD: Energy dispersive X-ray diffraction. www.nature.com/scientificreports www.nature.com/scientificreports/ The fact that the phonon dispersion at both pressures have all branches with positive frequencies, indicates that the hcp Ru is not only structurally, but also dynamically stable, i.e. it does not present phonon instabilities. Since the hcp structure has two atoms per primitive cell, it has six possible vibrations. However, at the Γ point, there are two pairs of degenerate modes (two optical and two acoustic branches). Based upon group-theory analysis, it is known that modes of the hcp structure at Γ are: A 2u + B 1g + E 2g + E 1u . The A 2u mode and the degenerate E 1u are the acoustic modes. The degenerate E 2g optical mode is Raman active, and the B 1g optical mode is a silent mode. We have calculated the pressure dependence of the optical modes which is shown in Fig. 4. The results for the Raman-active mode agree quite well with the measurement made by Olijnyk et al. up to 60 GPa 39 , thus they can be considered a good prediction for the pressure dependence of the Raman mode and the silent mode up to 160 GPa. The good agreement between the quasi-harmonic calculations and the experiments is an indication that anharmonic contributions are negligible in Ru at RT. Notice that the Raman active E 2g mode is a shear mode in which successive hexagonal planes of the hcp structure beat against each other 40 . In contrast, the silent B 1g correspond to bending vibrations within the hexagonal plane 40 . In general, in-plane vibrational frequencies are more energetic than the out-of-plane ones. This is consistent with the results of our calculations (B 1g has a higher frequency than E 2g ). The fact that the E 2g mode is associated to an inter-planar vibration along the c-axis, makes possible to link it with the C 44 elastic constant 39 ; basically C 44 is proportional to the square power of the frequency of the E 2g mode. According to this hypothesis, C 44 should increase by a 60% from ambient pressure to 160 GPa, which is consistent with calculations reported by Lugovskoy et al. 24 The fact that C 44 is expected to have a positive pressure derivative rules out a possible phase transition from hcp to cubic structure.
High pressure -high temperature evolution. XRD experiments performed up to 56 GPa and 960 K have shown that Ru remains in the hcp structure with no evidence of phase transitions or structural distortions. Furthermore, the obtained data do not show any formation of Ru carbides 41 and/or oxides 42 . The pressure and temperature dependence of the Ru unit-cell obtained from four isotherms together with the RT results are shown in Fig. 5. Also results from previous high-and room-temperature studies at ambient pressure are included 43,44 . The results from the HT experiments carried out using KCl as pressure transmitting medium are consistent with both the one obtained at room temperature using He as pressure transmitting medium and the one obtained in previous studies 43,44 . As described before, the RT P-V results are well represented by a third-order Birch-Murnaghan EoS 45 with K 0 = 323(1) GPa, ′ K 0 = 4.15 (2) and V 0 = 27.122(6) Å 3 . Regarding the HT results, the isotherms shown in Fig. 5 can be described using the Birch-Murnaghan isothermal formalism 46 . The obtained results are shown in the figure. In the fit of the P-V-T EoS the above given values for K 0 , ′ K 0 and V 0 have been considered as fixed. The bulk modulus has been assumed to present a linear dependence from the temperature 3 : A similar functional dependence has been assumed for the thermal expansion 3 : being α 0 , α 1 and β the only free parameters in the fit. As it can be seen in Fig. 5, the used approximations are sufficient for describing the pressure and temperature dependence of the volume in the P-T range explored in this study. The obtained values for the fitting parameters are α 0 = 2.2(1) 10 −5 K −1 , α 1 = 7.5(5) 10 −9 K −2 and β = −1.1(2) 10 −2 GPa/K. These values are comparable with those reported for the hcp phase of Ti and Zn in a similar temperature range 47,48 . The fact that a linear dependence in T for the thermal expansion and bulk modulus accounts well for the experimental results up to 960 K indicates that Ru behaves quasi-harmonically up to this temperature. www.nature.com/scientificreports www.nature.com/scientificreports/ conclusion In this work the stuctural and chemical evolution of Ru has been studied under HP-HT conditions combining AD-XRD and DAC techniques. The data have been collected along five different isotherms to a maximum of 151 GPa at ambient temperature and 21 GPa at 960 K, respectively. The collected data allowed a thermal EoS to be determined following a Birch-Murnaghan formalism. In the investigated P-T domain, only the hcp phase of Ru has been observed. The ambient temperature data have been obtained under quasi-hydrostatic conditions using He as pressure transmitting medium. First principle simulations based on DFT have been performed using different exchange correlation functionals to better constrain the structural and vibrational properties of Ru at ambient temperature. The PBEsol functional provided the best agreement with the experimental data when considering the ambient temperature EoS and the pressure dependence of the c/a ratio. The obtained vibrational modes confirm hcp Ru to be structurally and dynamically stable up to at least 160 GPa. Methods experimental. In this work, two series of AD-XRD experiments were performed: one at the Materials Science and Powder Diffraction beamline BL04 at the ALBA synchrotron 49 , and one at the Extreme Conditions beamline I15 at the Diamond Light Source (DLS) synchrotron 50 . For the experiment performed at DLS, two membrane DACs were equipped with diamonds with culet sizes ranging from 200 to 100 μm. Whereas for the experiments performed at ALBA, four membrane DACs were equipped with diamonds with culet sizes ranging from 300 to 200 μm. In both cases, gaskets where prepared from pre-indented and sparkle-erosion drilled Re foils. Sample loadings were performed under an inert atmosphere to prevent sample oxidation or other possible chemical reactions. For the experiment performed at DLS, few grains of Ru (approximately 4 μm each; 99.999% purity, Sigma Aldrich) were loaded at the centre of the DAC high pressure chamber. A ruby chip was placed few μm away from the sample and used as pressure gauge. Finally, the He pressure medium was loaded into the cell.
For the experiment performed at ALBA, the Ru powder was compressed into pellets using two diamond anvils. The obtained pellets were cut and loaded in the DAC high pressure chambers between two FIB-cut KCl disks. The KCl disks, acting as pressure transmitting medium as well as pressure gauges, were oven dried at 200 °C for two hours before being loaded in the DAC.
Diffraction data were collected at I15 with a monochromatic X-ray beam (λ = 0.4246 Å) and measured using a MAR345 area detector at a distance of 300.84 mm. During the experiment, pressure inside the high pressure chamber of the DAC was determined from the ruby luminescence method using the calibration of Dorogokupets et al. 51 .
At ALBA, XRD patterns were collected using an SX165 Rayonix charge-coupled device camera, with a monochromatic beam (0.4246 Å) at a sample-to-detector distance of 205.84 mm. The DACs were heated using 240-V Watlow coiled-cable heaters, which were wrapped around the outside of each DAC, and which can operate continuously at a power density greater than 4.65 W/cm 2 . These heaters are capable of inducing temperatures above 900 K in samples 52 . For temperature measurements, K-type thermocouples, with an accuracy <0.4% were www.nature.com/scientificreports www.nature.com/scientificreports/ attached to the piston of the DAC, close to one of the diamond anvils. The DAC was housed inside a dedicated water-cooled vacuum vessel which was evacuated using a rotary-backed turbo pump. An evacuated environment prevents oxidation of the diamonds at high temperatures and removes the effects of convective heating. Typically, vacuum pressures of ~10 −4 -10 −5 mbar were achieved within the vessel during the course of the experiments. Water cooling ensured the vessel remained cool (approximately at room temperature (RT)) relative to the heated DAC and any thermally induced movement of the sample with respect to the x-ray beam was thus minimized. This apparatus has been successfully used in previous HP-HT AD-XRD DAC experiments at different synchrotron facilities [52][53][54] .
The sample pressure was determined from the XRD patterns of KCl according to the thermal EOS of Dewaele et al. 55 .
Before each heating run, the sample was pressurized at RT to approximately 3 GPa and then heated up while collecting XRD data. Once the target temperature was reached, isothermal compressions were performed, and XRD patterns were collected every 1-2 GPa.
In both cases, the detector geometry was calibrated with a LaB 6 standard using the powder calibration routine of the DIOPTAS software suite 56 . Masks were applied to the raw diffraction images on a per image basis before they were azimuthally integrated in DIOPTAS. The obtained diffraction data were analysed by Le Bail fitting using the routines in TOPAS software suite 57 , literature values for the lattice parameters of Ru were used as starting point for these refinements. computational methods overview. First-principles calculations based on density functional theory (DFT) have been performed to analyze the equation of state and structural and vibrational properties of hcp Ru under pressure. The calculations were performed with the VASP code 58 by using projector augmented-wave (PAW) method potentials 59 . Four different approaches have been tested in the simulations: Local-density approximation (LDA), generalized-gradient approximation with Perdew-Burke-Ernzerhof (GGA-PBE) and Perdew-Burke-Ernzerhof for solids (GGA-PBEsol) functionals, and meta-GGA with strongly constrained and appropriately normed semilocal functionals (SCAN). The electronic states 4p 6 5s 1 4d 7 are considered as valence. Wave functions are represented in a plane-wave basis truncated at 650 eV. By using these parameters and dense k-point grids of 16 × 16 × 12 for integration within the first Brillouin zone (IBZ), energies are converged to within 1 meV per formula unit. In the geometry relaxations, a tolerance of 0.01 eV ⋅ Å −1 is imposed in the atomic forces. Ab initio phonon frequencies are calculated with the small-displacement method 25,60 in order to assess the vibrational stability of highly compressed hcp Ru and estimate the P-dependence of the corresponding Raman mode. In the small-displacement approach, the force-constant matrix is calculated in real-space by considering the proportionality between atomic displacements and forces. The quantities with respect to which the phonon calculations are converged include the size of the supercell, size of the atomic displacements, and numerical accuracy in the sampling of the IBZ. The settings providing a quasi-harmonic free energies converging to within 5 meV per formula unit are the following: 4 × 4 × 3 supercells (where the figures indicate the number of replicas of the unit cell along the corresponding lattice vectors), atomic displacements of 0.02 Å, and k-point grids of 4 × 4 × 4. The value of the phonon frequencies are obtained with the PHON code developed by Alfè 61 . In using this code, the translational invariance of the system is exploited to impose the three acoustic branches to be exactly zero at the center of the Brillouin zone, and apply central differences in the atomic forces.

Data Availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request. www.nature.com/scientificreports www.nature.com/scientificreports/