Quasi-hydrostatic equation of state of silicon up to 1 megabar at ambient temperature

The isothermal equation of state of silicon has been determined by synchrotron x-ray diffraction experiments up to 105.2 GPa at room temperature using diamond anvil cells. A He-pressure medium was used to minimize the effect of uniaxial stress on the sample volume and ruby, gold and tungsten pressure gauges were used. Seven different phases of silicon have been observed along the experimental conditions covered in the present study.


Results
The structural evolution of crystalline Si from ambient pressure to 105.2 GPa has been investigated at ambient temperature under quasi-hydrostatic conditions. Over this pressure range, Si was observed to undergo six phase transitions four of which are associated with a coexistence region. In the following sections, the stability field and compression curve of each phase are described and compared with previous studies. The crystallographic changes occurring in the structures and the corresponding mechanisms of the phase transitions are well known, therefore they are not discussed in this paper. A comprehensive description of the phase transitions of the group IV elements has been previously presented by Katzke et al. 30 .
Si-I: diamond structure. At ambient pressure, the observed signal on the image plate presents a single crystal like texture corresponding to the 111, 311, 331, 422 and 511 reflections of the Si-I phase (diamond, Fd m 3 ). Under the present experimental conditions, the Si-I phase can be observed up to 13.9 GPa although a coexistence with the phases Si-II and Si-XI appears between 13.1 GPa and 13.5 GPa and will be discussed in detail later in the text.
During the data analysis, each (unsaturated) single crystal XRD peak was integrated individually and used to determine the lattice parameters for each reflection. The final value of the Si-I lattice parameter was then calculated as the average of all the measured ones.
In Table 1 the obtained results for Si-I are reported together with the corresponding pressure measured from the ruby fluorescence method 28 . Pressure measured from the compression curves of Au 31 and W 28 are also reported in the table to compare the reliability of the measured pressure and, to further constrain the pressure distribution inside the high pressure chamber during the experiment.
In Fig. 1 the corresponding normalized volumes per unit cell of the Si-I phase are plotted as a function of pressure together with the results obtained from previous studies. Si-I bulk modulus K 0 , its pressure derivative ′ K 0 and the volume V 0 at ambient pressure have been determined by a least-squares fit of the present pressure-volume data to a Rydberg-Vinet 32 and a third-order Birch-Murnaghan (BM3) EOS 1 .
The resulting values are summarized in Table 2. The uncertainties in the obtained values correspond to the 95 % confidence interval of the fitted values. When K 0 ′ is fixed to 4.24 as obtained in the ultrasonic measurement of McSkimin et al. 33 , the obtained fit values are in good agreement with the ultrasonic ones (see Table 2).
In Table 2 the values obtained in this experiment are compared to the ones obtained from previous studies together with the observed pressure domain of pure Si-I phase. It is interesting to observe that the K 0 obtained in the present study by fixing K 0 ′ to the ultrasonic values of Mcskimin at al. 33 , results 2.7% smaller than the ones calculated by Hennig et al. 34 in a DFT study using a Heyd-Scuseria-Ernzerhof screening Coulomb hybrid functional (HSE06). Whereas, the K 0 values obtained with DFT calculations using a generalized gradient approximation (GGA) and a local density approximation (LDA) in Wang et al. 35 are 9.5% and 2% lower than the one obtained in the present study (fixing ′ K 0 ), respectively. The EOS derived from the compression curve of Si-I in the present work ( Fig. 1) is in good agreement with the reported curve of McMahon et al. 6 . We note that in that work the authors report a phase transition to Si-II at 11.0 GPa, whereas in the present study no phase transition was observed before 13.1 GPa. A good agreement is also observed with the data of Senoo et al. 24 obtained with XRD experiment performed in a hydraulic press-driven cubic anvil apparatus using a 1:1 mixture of ethyl-and methyl-alcohols and, with the data obtained from an energy dispersive XRD experiment in 4:1 methanol:ethanol performed by Olijnyk et al. 10 .
Hu et al. 5,36 also studied the compression of Si in a XRD-DAC experiment in 4:1 methanol:ethanol mixture. Comparing their result with the present study it is clear they are making an overestimation, despite their using the values derived from the ultrasonic studies of Mcskimin et al. 33 . Furthermore such an overestimation becomes more evident with increasing pressure with a percentage difference from the ultrasonic EOS passing from 0.27% at 3 GPa to 1.48% at 12.6 GPa. This trend is normally associated to a non-hydrostatic condition in the high pressure chamber of a DAC such as a direct compression of the sample between the diamond anvils 37 .
Si-I, II and XI: first coexistence region. The Si-I phase (cubic diamond structure, Fd m 3 ) is stable up to a pressure of 13.1 GPa when new peaks, associated to the tetragonal β-tin (I4 1 /amd, Si-II) and the orthorhombic Imma (Si-XI) phases, develop in addition to the five reflections of the cubic diamond phase (see Fig. 2). A similar behaviour has been previously observed by McMahon et al. 6 , although they only discuss the possibility of a coexistence of the three phases due to the overlapping of several peaks that can belong to both the Imma and the β-tin but couldn't be discriminated due to signal resolution problems. The coexistence of these three phases, has also been predicted by DFT simulations by Yao and Klug 38 . They found that up to a pressure of 10 GPa, the diamond structure is the most enthalpically favourable. From 10 GPa up to 13.2 GPa, the β-tin and the Imma structures are the most stable structures, both with similar enthalpies, whilst above this range the Imma structure becomes more favourable. Table 3 reports the measured pressure domain of the Si-II and Si-XI phases together with the ones obtained from previous studies.  Reference   Under the present experimental conditions, the onset of the phase transition from Si-I to Si-II (13.1 GPa) is higher than previously observed in other static compression experiments. The corresponding pressure domain of Si-II is narrower and, actually corresponds to the observed coexistence region between Si-II and XI reported by McMahon et al. 6,23 . The onset of the phase transition to Si-XI is also in good agreement with the results of McMahon et al. 6,23 while the observed pressure domain of Si-XI appears to be slightly narrower but in agreement within the experimental errors.
Although the β-tin and Imma phase reflections are strongly overlapping, it was possible to determine the lattice parameters for all three phases at the transition pressure via a Le Bail fit. The results obtained from the fit together with the measured pressure for both Si-II and Si-XI phases are reported in Table 4.
In Fig. 2, the reported integrated diffraction patterns and the corresponding Le Bail fits, show the structural evolution of Si from 13.0 GPa to 16.3 GPa. It is important to emphasise that at 13.5 GPa only by considering the three phases (I,II and XI) together can explain the observed reflections. Also interesting is the different behavior of the 020, 211 and 121 reflections of the Si-XI phase from 13.5 GPa to 15.5 GPa. At 13.5 GPa, when the three phases coexist, the 020 reflection of Si-XI is closer to the 200, while the 211 and 121 are overlapped, forming a single broad peak. At 15.5 GPa, however, when only the Si-XI phase is observed, the 020 reflection appears to be closer to the 101 and the 211 and 121 are well separated in two distinct peaks as previously reported 6 . Detailed XRD refinements of the coexistence region are reported from Fig. S8 to Fig. S17 of the Supplementary Materials.
In the compression curves obtained for both phases (Fig. 3) we observe a plateau corresponding to the coexistence region. In the region with only the Si-XI phase, a decreasing trend in the compression curve is observed.   P6/mmm, Si-V) were observed in addition to those of the Imma phase. The two phases coexist from 15.8 to 16.1 GPa and from 16.3 GPa only the simple hexagonal phase was observed. Figure 5 shows the evolution of Si diffraction patterns at pressure above 16.3 GPa. A two phase Le Bail fit at the transition pressure of the Imma and simple hexagonal phases gave lattice parameters a = 4.748 Å, b = 4.479 Å, c = 2.551 Å (b/a = 0.943; c/a = 0.537) for Imma; and a = 2.563 Å and c = 2.381 Å (c/a = 0.929) for simple hexagonal. There was only a small change in volume of ~0.18% between the two phases. The phase transition from Imma to simple hexagonal is brought about by a further shift of the Si atoms along the [100] direction to form 6-fold symmetry chains parallel to the Imma α-direction/simple hexagonal c-direction. In the simple hexagonal structure, there is only one atom per unit cell (Wyckoff position 1a) which has a distorted 8-fold coordination environment (CN = 8; at 16.3 GPa 2 × 2.386Å, 6 × 2.553Å).
The simple hexagonal phase (Si-V) has been indexed up to 40.9 GPa. At 42.1 GPa, Si-V is replaced by the coexistence of Si-VI and Si-VII, the first with an orthorombic (Cmca) symmetry and the latter with a hexagonal (P6 3 /mmc) symmetry. This field of coexistence serve as transition to the pure Si-VII stability field that begins at 47.0 GPa (see Table 5). Si-VII is the only phase observed up to a pressure of 90.8 GPa, at which point peaks belonging to the cubic Fm m 3 make their first appearance together with the hexagonal structure. From 94.0 GPa, only the cubic (Si-X) phase is observed. The resulting compression curves are plotted in Fig. 4 Figure 3. Measured volume of Si-II and Si-XI as a function of pressure (solid black and green marks) compared to previous experimental studies (red solid marks) 6 . The solid line represent a fit of the present experimental data to a Vinet EOS, whereas the dashed line separates the coexistence region from the one phase region.   www.nature.com/scientificreports www.nature.com/scientificreports/ transition pressure is observed in the present study (with the only exception of the study of Hanfland et al. 16 ). Such a behaviour is attributed to the different hydrostatic condition experienced by the sample in various experiments.
The experimental compression curves for Si-V, Si-VII and Si-X were all fitted with both a Vinet and a BM3 EOS. A fit with the data obtained for the Si-VI phase was not possible as only three data points are available. During the fitting procedure, the V 0 of the high pressure phases have been calculated by extrapolating the data to ambient pressure. The parameters obtained from the fit are reported in Table 8. Concerning the P6/mmm phase, the results obtained from the Vinet and the BM3 fit agrees within the error bars. The observed differences in the absolute values are explained by the trade-off between the bulk modulus and its first derivative in pressure. The lower K 0 in the Vinet is compensated by a higher K 0 ′ . A K 0 smaller than the one of Si-I is consistent with a typical cubic-to-hexagonal structural transition. A similar trend is also obtained for the compression curve of Si-VII, when both equations can be used to describe the experimental data within the error bars. As expected, the fitting parameters for the two hexagonal structures are close to each other with only a minimal reduction of the unit cell volume at higher pressure. In the higher pressure range there is an increment in the bulk modulus value caused by the stabilization of the cubic Fm m 3 phase.

conclusions
In the present study, the compression curve of Si was investigated from ambient pressure up to 105.2 GPa at ambient temperature using angular dispersive XRD in DAC. The experiment was performed in quasi-hydrostatic conditions using He as pressure transmitting medium. The hydrostaticity of the experiment was qualitatively determined from an analysis of the X-ray diffraction pattern at the highest pressure reached. This showed a maximum deviation from expected lattice parameters, caused by macroscopic stress, of ca. 0.08%, which is within the uncertainty of the experiment. The pressure experienced by the sample was measured via the ruby fluorescence method and from the measured atomic volume of W and Au using the calibration from Dorogokupets et al. 28 . Seven different polymorphs of Si were observed in the investigated pressure range, in agreement with literature data [5][6][7][8][9][10][11]16,17,24,25 . The use of He as pressure transmitting medium had not lead to the formation of new Si phases in the investigated pressure range, although a general trend of each phase stability field shifting to higher pressure was observed. In particular, it is important to emphasise the observed behaviour of the well-known Si-I to Si-II phase transition [5][6][7][8][9][10][11] . In the present study the Si-II structure was never observed singularly, instead it was found to coexist in the entire pressure domain with the Si-XI phase.  www.nature.com/scientificreports www.nature.com/scientificreports/ In Fig. 6 the measured compression curve of Si from ambient to 105.2 GPa pressure is reported. The stability fields of the different phases of Si are represented with different colors and filling pattern in the figure. The solid black lines are obtained from the fit of the measured data to Vinet EOS extrapolated down to ambient pressure. The solid line of Si-VII phase starts at 42.1 GPa, even though experimental data for this phase are not reported in the graph. The Vinet EOS curve is extrapolated into the region between 42.1 and 48.6 GPa because in this range Si-VII phase peaks were observed together with the Si-VI phase peaks, indicating their coexistence. However, as the Si-VII phase peaks were not well resolved, it was impossible to include them in a LeBail refinement.
It is interesting to notice how, due to the hydrostatic condition of the actual experiment, four different coexistence regions have been observed in the investigated pressure range. This are: the coexistence between the Si-II and the Si-XI phase in the region between 13.1 GPa and 14.3 GPa; the coexistence between the Si-XI and the Si-V phase between 15.8 GPa and 16.1 GPa; the coexistence between the Si-VI and the Si-VII phase in the region between 42.1 GPa and 46.0 GPa.
The observed coexistence between the Si-VI and the Si-VII phase is in good agreement with the data of Hanfland et al. 16 although in their study, they observe a wider stability field for the Si-VI going from a coexistence with the Si-V phase at 38 GPa, to a coexistence with the Si-VII phase starting at 42.5 GPa and ending at above 47.5 GPa when only the Si-VII phase is observed.
The newly observed coexistence between the Si-VII and the Si-X phases in the region between 90.8 GPa and 94.4 GPa was probably not observed by Duclos et al. due to the absence of hydrostatic condition during their experiment 17 .
Finally, it is important to notice that these coexistence regions do not necessarily describe the thermodinamically stable fields since usually, solid-solid phase transitions (like those observed here) can be strongly affected by kinetic barriers 39 . Therefore, the phase boundaries experimentally determined in the present study might correspond to kinetic phase boundaries 40 . A further characterization of these coexistence region under releasing pressure will provide a better insight on the thermodynamic stability field of these phases.

Methods
The structural evolution of Si from ambient pressure to 105.2 GPa was investigated by synchrotron XRD experiments in DAC at ambient temperature. Four DACs were used during the course of this study, each fitted with a Re gasket. Gaskets were prepared by pre-indenting Re foils (200 μm initial thickness), which were then drilled by spark erosion to form the high-pressure chambers of the cells. A grain of Si (approx 4 μm each; 99.999% purity, Sigma Aldrich) was loaded into each chamber and a pressure standard was also added a few μm from the Si sample. Ruby spheres were used in all four cells as pressure standards, whilst in three of them an additional W or Au X-ray standard was included since these metals have high X-ray scattering powers and well characterised EOS, attested by the consistency between static, dynamic and ultrasonic measurements 27,41 . Finally the He pressure transmitting medium was loaded into the cell. The experimental conditions adopted in each run are reported in Table 9.
Diffraction data were collected at beamline I15 (Diamond Light Source, Oxon., UK) 42 using a monochromatic X-ray beam (λ = 0.4246 Å and λ = 0.3099 Å) and measured using a MAR345 area detector. The detector geometry was calibrated with a La B 6 standard using the powder calibration routines of the DAWN software suite 43 . Measurements were made at three different sample-detector distances (500.39 mm, 424.36 mm and 324.44 mm; determined from DAWN calibration), with the distance selected depending on the angular range necessary to follow changes in the structure under investigation. Masks were applied to the raw diffraction images on a per-image basis before they were azimuthally integrated using the processing tools in the DAWN suite 44 . Diffraction data were analyzed by Le Bail fitting using the routines of the TOPAS software suite 45 , literature values for the lattice parameters of each phase were used as a starting point for these refinements.
During the experiment, pressures inside the high-pressure chambers of the DACs were measured using ruby luminescence and the unit cell volume of either W or Au for runs ST1, ST2 and ST3 and ruby luminescence only for run ST1-2. Calibration data for these three pressure standards were taken from Dorogokupets et al. 28 . The entire set of integrated XRD patterns is reported in Figs S1-S3 of the Supplementary Materials. Hydrostatic conditions. As stated in Takemura and Dewaele 31 and Anzellini et al. 29 , it is desirable to achieve purely hydrostatic conditions in the sample chamber of a diamond anvil cell, which would be provided by a liquid   www.nature.com/scientificreports www.nature.com/scientificreports/ pressure transmitting medium. However, even helium becomes solid above about 12 GPa at room temperature and above this pressure, tends to induce non-hydrostatic stress in the sample chamber. Even at lower pressure, the stress can become non-hydrostatic if the sample bridges the anvils due to excessive thinning of the gasket or due to a large initial thickness of the sample. For this reason, interferometry was used to ensure that sample size and indent depth were of the correct relative dimensions to prevent the sample bridging the diamond culets.
A qualitative analysis of the hydrostatic conditions of the sample has been performed by comparing the measured d-spacing of Si at the highest pressure reached (105.2 GPa) to the theoretical (hydrostatic) one calculated using the lattice parameter obtained by a refinement of the entire XRD pattern 26 . Table 10 shows the measured and the calculated d-spacing of Si-X at 105.2 GPa together with their percentage deviation. The deviation of less than 0.08%, is within the experimental error of the present experiment. Such an error was estimated from the deviation between the measured and calculated d-spacing of the LaB 6 standard at ambient pressure. We can thus conclude that the non-hydrostatic stress is below the detection limit of our measurement. This is in agreement with the quantification of the macroscopic non-hydrostatic stress on metals in a helium pressure transmitting medium -i.e. 0.5 GPa at 150 GPa 46 .
A similar conclusion is obtained from the analysis of the pressure evolution of the R 1 -R 2 splitting and the full width half maximum (FWHM) of R 1 ruby fluorescence peaks (Fig. S6 of the Supplementary Materials (SM)) and the pressure evolution of the FWHM and normalized d-spacing of the W pressure gauge (Fig. S7 of SM), as explained in Takemura 26 .