Silicon (Si) is readily abundant in nature and is currently the dominant material in the modern semiconductor and electronics industries. In its most thermodynamically stable form at ambient conditions, Si adopts the diamond cubic structure (DC-Si, Fd3̅ m, a = 5.431 Å) and has an inherent indirect band gap of 1.1 eV. 1,2 However, this indirect band gap limits the ability of DC-Si to be the major component of nextgeneration optoelectronic and photonic technologies, [3][4][5][6] promoting the search for new materials compatible with current CMOS-technologies and manufacturing processes that have desirable optical and electrical properties. 7 In recent decades, substantial research effort has been contributed toward the synthesis of other novel stable and metastable forms of Si with potentially useful properties, including improved light absorption and emission. Pure Si allotropes are particularly desirable as they would be relatively easy to incorporate into pre-existing technologies and with well-developed manufacturing processes. Pure Si allotropes that are recoverable to ambient conditions include BC8, R8, and hexagonal diamond (HD)-Si. 8 Other unique allotropes have been produced locally through confined micro-explosions. 9 The BC8-Si structure is a narrow-gap semiconductor, 10,11 while calculations indicate that R8-Si has a small indirect band gap of 0.24 eV. 12 HD-Si has a similar electronic structure to DC-Si, 13 but solid solutions with Ge may offer the possibility for tunable direct band gaps in the near-to mid-infrared. 7,14 While several Si allotropes are already known, calculations indicate that there are numerous others with desirable optical or electrical properties, and additional isolatable crystalline forms are yet to be synthesized.
Si-rich compounds have also been used as viable precursors for synthesizing novel Si allotropes with potentially desirable physical properties. 15 For example, Na-Si clathrate structures such as Na 24 Si 136 (Type-II clathrate, cubic Fd3̅ m) have been synthesized via thermal decomposition 16 and by using high pressure and temperature. 17,18 Na can be removed from the Type-II structure to produce Si 136 . 19,20 While Si 136 has a wide direct (or nearly direct) gap near 2 eV, optically forbidden transitions and difficulties associated with the production of high-quality crystals and films have hindered recent developments. 21,22 Similar to Si 136 , the open-framework allotrope Si 24 can be produced by removing Na from the high-pressure precursor Na 4 Si 24 (EuGa 2 Ge 4 -type structure, 23 orthorhombic Cmcm, a = 4.081 Å, b = 10.579 Å, c = 12.275 Å). 17,24 Unlike the Si clathrates, which contain polyhedral cages that tile three-dimensional space, Si 24 is a "clathrate-like" open-framework structure with one-dimensional channels along the crystallographic a-axis [see Fig. 1(a-c)]. This altered geometry allows for increased Na mobility and guest removal at much lower temperatures compared with Type-II clathrate. After Na-removal, the volume of the resulting Si 24 framework is slightly contracted (orthorhombic Cmcm, a = 3.818 Å, b = 10.692 Å, c = 12.637 Å), 25 as shown in Fig. 1(d). Si 24 possesses a quasi-direct band gap near 1.4 eV, 26,27 close to the ideal band gap for light absorption in the Shockley-Queisser limit for single-junction devices, 28 which suggests potential for efficient conversion of the solar spectrum compared to DC-Si. 27 Recently, large pure single crystals of Si 24 were demonstrated. The Na-free Si 24 structure showing the empty channels along the a-axis. These images were generated using VESTA-v3. 29 While Si 24 is metastable at ambient conditions and was shown to persist above 700 K, 25 its high-pressure stability, phase transition sequence, and bulk mechanical properties remain unknown. Previous openframework structures were shown to exhibit anomalous properties such as negative thermal expansion, [30][31][32] and certain optical phonons of Si 24 exhibit softening with pressure and negative Grüneisen parameters suggesting potential for anisotropic compression. 33 Previous studies on cubic Si 136 clathrate demonstrated a surprisingly high bulk modulus compared with DC-Si, 34 but it is unclear whether this structural stability will extend to orthorhombic Si 24 . In addition, it remains unconfirmed whether small gaseous atoms penetrate into the open-framework channels under pressure -the diameter of the large 8-membered ring is comparable to a He atom or H 2 molecule -similar to observations of other zeolite-type structures. [35][36][37] In this paper, we address these open questions by studying compressed Na 4 Si 24 and Si 24 using in situ synchrotron X-ray diffraction (XRD) measurements. We then quantify the mechanical compressibility parameters using a variety of pressure media (PM) and show that hydrostaticity strongly influences the transformation pressure of Si 24 and its prevalence to coexist with other high-pressure Si phases.
The Na 4 Si 24 precursor and subsequent Si 24 samples were prepared as discussed previously. 25,26 In short, a 6:1 molar DC-Si (powder Alfa-Aesar, 99.999%) to Na metal (Alfa-Aesar, 99.95%) mixture was prepared in an Ar glovebox and sealed within a boron nitride capsule for high-pressure synthesis at 9 GPa and 1125 K using a 14/8 multi-anvil assembly. 24 To prepare Si 24 , recovered Na 4 Si 24 samples were wrapped in a Ta pouch and placed inside a quartz tube under a dynamic vacuum of 3x10 -5 torr. The sample was then annealed at 125°C under vacuum for four days to generate Na-free Si 24 . 25 The resulting material was sonicated and rinsed in water to remove any residual Na salts on the surface of the Si 24 .
All high-pressure experiments were performed in diamond anvil cells (DAC) equipped with culets ranging between 500-600 μm in diameter. Re metal gaskets were used for all experiments. The holes acting as sample chambers were drilled into the pre-indented Re gaskets using an electric discharge machine. The maximum pressure reached in each experiment was dependent on the culet diameter and specific diamond anvil seat type. In situ pressures were measured using the calibrated shift of the R1 ruby fluorescence line, 38 and cross-referenced with the Ar-EoS 39 when possible. Five different compression runs were performed in total: Na 4 Si 24 compressed in Ar, Si 24 compressed in Ar (twice), Si 24 compressed in H 2 , and Si 24 compressed with no pressure medium (PM). The Na 4 Si 24 and Si 24 samples were first crushed into fine powders and then pressed into pellets (~50 μm in diameter and 10-20 μm thick) before being loaded into the sample chambers. All gas loadings (Ar and H 2 ) were performed in-house, and were initially loaded to ~0.1 GPa. For the Si 24 sample that was compressed without a PM, the crushed powder was inserted directly into the sample chamber, filling it almost entirely.
XRD measurements were performed at beamline 16-ID-B of the High-Pressure Collaborative Access Team (HPCAT), Advanced Photon Source, Argonne National Laboratory. A monochromatic X-ray beam with energy of ~30 keV and FWHM of approximately 4×6 μm 2 was focused on the sample. 40 The measurements of Na 4 Si 24 (in Ar) and one Si 24 experiment (in Ar) were collected on a 1-M Pilatus detector. The diffraction patterns from all other high-pressure experiments of Si 24 (in Ar, H 2 , and without a PM) were collected on a MAR-CCD detector. Samples were rotated in the beam from ω = -10° to 10° at 1°/s to improve powder averaging statistics. The sample-to-detector distance and other geometrical parameters were calibrated using Dioptas 0.5.0 41 in conjunction with a CeO 2 diffraction standard. 2D diffraction images were processed using Dioptas 0.5.0, and Pawley refinements were performed using GSAS-II 42 to determine lattice parameters. In general, these refinements were performed on data ranging between 2θ = 3 -16°, which includes approximately 50 Bragg peaks for either the Na 4 Si 24 or Si 24 phases (see Supplementary Fig. 1). A polynomial background function was removed using GSAS-II, and Ar Bragg peaks were also fitted above the Ar solidification pressure. Typical estimated standard deviations on refined unit cell parameters were below 0.001 Å.
Zero-pressure bulk moduli (B 0 ) and their derivatives with respect to pressure (B 0 ') were determined using the EoSfit7c software 43 using both 3 rd -order Birch-Murnaghan and Vinet equations of state. The uncertainty in V is determined from the GSAS-II LST output files, and the uncertainties in ruby pressure measurement are estimated to be ±2% for the experiments conducted in Ar and H 2 , and ±5% for Si 24 compressed with no PM. V 0 was set for all equation of state (EoS) refinements using the known values determined by powder-XRD at ambient conditions.
Powder-XRD patterns were collected with increasing pressure for the five high-pressure experiments. Figure 2 shows select diffraction patterns of Si 24 compressed in the H 2 PM up to 20.1 GPa. The signal diffracted from Si 24 remains intense up to 18.0 GPa, before weakening substantially. During compression, all Si 24 Bragg peaks shift to higher angle. Small variations in the relative intensities of Bragg peaks are due to small differences in powder averaging statistics between different locations in the cell (e.g., peaks between 2θ = 3 -8°). Bragg peaks remain relatively sharp to the highest-pressure conditions, indicating that anisotropic strain broadening is not significant, and that the molecular H 2 PM provides quasihydrostatic compression conditions over the full range tested. While Si 24 remains the dominant phase up to 18.0 GPa, clear signs of transformation toward other high-pressure Si phases are present, starting above 13 GPa. Bragg peaks from the -Sn-Si phase were detected at 13.8 GPa, which transforms to the Imma-Si phase at 14.7 GPa. The Imma-Si phase persists alongside Si 24 up to 17.3 GPa, above which simplehexagonal (SH)-Si begins to form. The signal diffracted from SH-Si grows stronger as pressure is further increased, and by 20.1 GPa, the Si 24 is almost completely gone, and the dominant phase is SH-Si.
The high-pressure phase transition sequence observed for Si 24 perfectly mirrors that of compressed DC-Si with slight differences in the onset pressures and coexistence ranges. A recent precision study on compressed DC-Si in quasi-hydrostatic helium marks the -Sn, Imma, and SH-Si onset pressures at 13.1, 13.1 and 15.5 GPa, respectively, with coexistence of the phases across the transition boundaries. 44 The similarity in the phase transition sequence between compressed DC-Si and Si 24 reflects the strong thermodynamic driving force toward the denser phases at high pressure (volume drops ~20% across the -Sn transition for DC-Si, even more for Si 24 ). Notably, DC-Si only persists to 13.1 GPa before transforming into denser phases. However, low-density Si 24 can persist up to 20 GPa under similar quasi-hydrostatic conditions. The difference in this persistence under similar quasi-hydrostatic conditions might be related to differences in kinetic barriers between the different starting allotropes. Previous phonon dispersion calculations predict that Si 24 is dynamically stable to at least 10 GPa. The degree of hydrostaticity was found to influence Si 24 phase transition and coexistence pressures, but did not affect the overall phase transition sequence, as summarized in Fig. 3. For the compression experiments performed in Ar (two combined datasets) and the experiment with no PM, the -Sn-Si phase formed earlier relative to the experiment conducted in H 2 , at 11.1 GPa and 11.3 GPa, respectively. The lower transition pressure is attributed to decreased hydrostaticity relative to H 2 . Indeed, this behaviour is similar to observations for compressed DC-Si [45][46][47] and for Si 136 clathrate compressed in less hydrostatic pressure media. 34 When compressed in Ar, -Sn-Si (from Si 24 ) transforms to Imma-Si at 15.5 GPa, and then into SH-Si at 17.6 GPa. The Si 24 experiment without a PM was stopped at 15 GPa, to prevent diamond-anvil culet damage.
In addition to having lower -Sn-Si onset pressures, Si 24 compressed in Ar and in no medium also persists to much lower pressures. As indicated by the thick black bars in Fig. 3, the highest pressure for which a refinable signal from Si 24 was measured was only 13.2 GPa for Ar and 14.3 GPa for no PM. In contrast, Si 24 compressed in H 2 yields a refinable diffraction pattern up to 18.0 GPa. These observations indicate that the Si 24 phase transformation is facilitated by non-hydrostatic environments. It is also important to note that DC-Si was never observed in any of the compression experiments. That is, the higher-pressure phases of Si form directly from compressed Si 24 and not through an intermediate DC-Si phase. Na 4 Si 24 compressed in Ar remained stable up to 18 GPa. However, unlike Si 24 under the same conditions, no other higher-pressure Si phases were observed to co-exist. This indicates that the structure remains intact and Na remains in place within the Na 4 Si 24 channels up to at least 18 GPa, preventing the formation of localized regions of pure Si that are able to transform to other higher-pressure phases. Furthermore, even though the Na 4 Si 24 peaks remain visible up to 18 GPa, the decreased intensity of peaks in the diffraction patterns measured above 13.2 GPa makes it impossible to refine lattice parameters quantitatively. Other filled type-I 48 and type-III 49 Si clathrate structures have shown large homothetic volume collapses and irreversible amorphization at even higher pressures. For other chemical systems, this phenomenon typically occurs at lower pressures than their empty structural analogues. 50 However, our results show that this does not occur below 18 GPa for Na 4 Si 24 in an Ar PM. Experimental verification of these pressure thresholds is outside of the scope of this study. Figure 4 shows the normalized decrease in the total unit cell volume (determined from refinement), V/V 0 , with respect to pressure for Na 4 Si 24 in Ar, and for Si 24 in Ar, H 2 , and with no PM. A small variation in V 0 was observed for Na 4 Si 24 (~0.25%) between synchrotron data and our in-house measurement of the current sample, which is due to a small variation in a 0 . 17 Notably, older samples (e.g., months after synthesis) exhibit smaller (up to 0.5%) values for a, consistent with Na-removal from the channels. It is known that Na diffuses out of the channels very slowly over time at ambient conditions, 25,51 and could initially reflect a small decrease along the a-axis. While the compression curve reported here reflects one distinct sample composition near Na 4 Si 24 , we note that other slightly shifted curves may be possible due to small changes in stoichiometry, although further measurements are needed to confirm this hypothesis.
The unit cell volume compression curves for Si 24 in H 2 (red), Ar (green), and with no PM (blue), alongside that of Na 4 Si 24 in Ar (black). Uncertainties in volume measurements are taken from GSAS-II LST output files. Uncertainties in pressure are estimated to be higher for the sample measured with no PM. 3 rd -order Birch-Murnaghan equations of state were determined using EoSfit7c and overlay their respective points on the plot. The uncertainties in both pressure and volume were taken into account during EoS fitting.
The compression curves presented in Fig. 4 show that Na 4 Si 24 is more compressible than Si 24 . To determine the quantitative behavior, both 3 rd -order Birch-Murnaghan and Vinet equations of state (EoS) were fitted to the compression data. The results are displayed in Table 1. For all compression curves, there is only a small difference between the values determined using each type of EoS. The B 0 of Na 4 Si 24 in Ar is 87(2) GPa and 86(2) GPa for the Birch-Murnaghan and Vinet fits, respectively. In both cases, B 0 ' determined from each fit is close to 4. The increased compressibility of Na 4 Si 24 relative to Si 24 is attributed to its metallicity, which tends to decrease the directionality of covalent bonds and overall rigidity of the framework. For the case of Na 4 Si 24 , the Na valence electrons are donated to silicon framework rendering the system metallic, whereas Si 24 possesses a balanced electron count and band gap near 1.4 eV. Further to this, the results showing that the Na-filled structure has a lower B 0 than empty Si 24 is expected. A similar trend has previously been observed for silicon clathrates 52 and has been predicted for carbon clathrates. 53 When compressed in quasi-hydrostatic H 2 , B 0 for Si 24 is 91(2) GPa for both the Birch-Murnaghan and Vinet EoS. At ambient conditions, the Si-Si bond lengths within the open-framework structure of Si 24 range from 2.33-2.41 Å and the bond angles vary between 93.7-139.5°, in comparison with bond lengths and bond angles in the perfectly tetrahedral DC-Si structure that are 2.35 Å and 109.5°, respectively. Also, the density of Si 24 is substantially lower than that of DC-Si, at 2.17 g/cm 3 relative to 2.33 g/cm 3 . 25 These structural factors make it no surprise that Si 24 is more compressible than DC-Si (B 0 = 97.9 GPa and B 0 ' = 4.24). 44 The lower B 0 of 91(2) GPa for Si 24 compressed in H 2 scales well with the reduction in density compared with DC-Si (~7%), and is similar to that of other open-framework or polyhedral Si structures with distorted bond lengths and angles. For example, the Si 136 structure which has a density of 2.15 g/cm 3 , was shown to have a B 0 of 90(5) GPa, 34 similar to Si 24 . The B 0 ' value determined for Si 24 in H 2 is larger than that of DC-Si (4.2 GPa 44 ), and consistent with the larger initial volume reduction of the open-framework structure.
The fact that the compression curve of Si 24 compressed in H 2 is nearly identical to that of Si 24 in Ar and with no PM suggests that H 2 is not able to enter the structure in the pressure range investigated, as has been proposed previously as a possibility. 54 If H 2 was able to enter the structure, we would expect to see an abrupt stiffening or flattening of the compression curve, as observed when small molecules penetrate the pores of other open-framework systems. [55][56][57][58] We find no evidence for H 2 incorporation under room temperature compression, although cannot fully exclude some nominal amount of penetration into the structure. A previous Raman study conducted at high pressure and high temperature also indicates that He penetration is unlikely up to 8 GPa and 400 K. 33 Table 1. Bulk moduli, B 0 , and their derivatives, B 0 ', with respect to pressure for Na 4 Si 24 in Ar, and for Si 24 in H 2 , Ar, and with no PM. Estimated uncertainties in the last digit are shown in parentheses, as determined from EoSfit7c. For all EoS fitting results shown here, V 0 was set to 515.87(5) Å for Si 24 and at 529.95(5) Å for Na 4 Si 24 , as determined from powder-XRD measurements.
Relative to compression in H 2 , B 0 of Si 24 appears lower when compressed without a PM up to 14 GPa, and even more so when compressed in Ar up to 13.2 GPa. For both of these data sets, the best-fit B 0 ' values are significantly higher. These trends hold for both the Birch-Murnaghan and Vinet EoS and demonstrate the role of hydrostaticity on compression. For comparison, the F-f plots of Si 24 and Na 4 Si 24 in Ar, and Si 24 in H 2 generated in Eosfit7c using the Birch-Murnaghan EoS are shown in supplementary Figs. 234, respectively. Indeed, previous studies have documented how non-hydrostatic conditions can affect the EoS for many systems, including Si. 44 The compression curves for Si 24 in all cases appear similar at low pressure, but begin to deviate at higher pressure. In the absence of a PM, a clear offset begins above ~5 GPa when compared with quasi-hydrostatic H 2 . Excluding the highest-pressure Ar points in the refinement yields EoS parameters that are in much better agreement with the case of H 2 (Table 1). We attribute these differences to the loss of hydrostaticity with increasing pressure. The (quasi)hydrostatic limit of Ar was previously shown to persist to about 9 GPa. 59 As mentioned above, we attribute the differences in phase transition pressures shown in Fig. 3 directly to these differences in hydrostaticity. Previous studies on high-pressure transformations in pure Si have also shown strong dependencies on hydrostaticity. 44 While the volumetric compressibilities of Na 4 Si 24 and Si 24 exhibit similar trends with their bulk moduli differing by ~7%, it is interesting to examine the behavior of individual lattice parameters and structural compression mechanisms. Figure 5 highlights the behavior of a, b, and c for both Na 4 Si 24 and Si 24 with respect to pressure. While the general trends for Si 24 are consistent for all pressure media conditions (with small deviations attributed to the non-hydrostaticity discussed previously), it is also clear that Si 24 and Na 4 Si 24 exhibit fundamentally different behavior along different crystallographic axes. Na 4 Si 24 is much more compressible in both the b and c directions, while it is much less compressible along the a direction. This behavior can be understood by the fact that channels in the structure propagate along the a-axis, and, when filled with Na ions, provide strong resistance to compression in this direction. Despite showing decreased compressibility along a, the overall increase in volumetric compressibility of Na 4 Si 24 is due to more compressible b and c axes, as compared with Si 24 .
The compression curves of the a, b, and c unit cell parameters with respect to pressure of Si 24 in H 2 (red), Ar (green), and with no PM (blue), alongside that of Na 4 Si 24 in Ar (black). Uncertainties in in the refined unit cell parameters are taken from GSAS-II LST output files. Uncertainties in pressure are estimated to be higher for the sample measured with no PM.
In all cases, the individual lattice parameters decrease monotonically over the entire pressure range tested, with no indication of negative linear compressibility at room temperature. The determination of B 0 for Si 24 allows for accurate determinations of mode Grüneisen parameters from high-pressure Raman spectroscopy, which previously assumed B 0 = 90 GPa. 33 The observation of negative Grüneisen parameters for Si 24 may indicate the possibility for low-temperature negative thermal expansion, as was observed previously for Si 136 , 30 motivating future studies on Si 24 over a broader range of pressure and temperature conditions.
In this work we experimentally determined B 0 and B 0 ' for and Si 24 and Na 4 Si 24 under quasi-and nonhydrostatic high-pressure conditions. EoS fits of compression curves show that the B 0 of Si 24 compressed in H 2 is 91(2) GPa, which is ~7% lower than the denser DC-Si structure, and is comparable to the B 0 of other open-framework Si allotropes, such as Si 136 . The compression curves also show that Si 24 compressed in H 2 and without a PM vary only subtly, suggesting that H 2 does not enter the Si 24 structure under high pressure at room temperature. Both Si 24 and Na 4 Si 24 structures exhibit comparable volumetric compressibilities, however there are differing axial trends, such as Na 4 Si 24 having a significantly higher incompressibility along the a-axis due to Na-filled channels along this direction. High-pressure XRD patterns reveal that the Na 4 Si 24 structure persists up to at least 18 GPa. Whereas, Si 24 partially transforms into high-pressure phases including -Sn, Imma and SH-Si, but is able to co-exist alongside these phases.
The results also show that the Si 24 to -Sn-Si pressure transformation threshold depends strongly on the choice of PM, and that the transformation is suppressed in more hydrostatic environments.