# Structural and mechanical properties of stable and metastable phases in SixGe1-x alloys

Max Gerin, Vittoria Pischedda, Sylvie Le Floch, Silvana Radescu, Frederico Alabarse, Yann Le Godec, Erwann Jeanneau, Nicolas Guignot, David Amans, Denis Machon

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Max Gerin, Vittoria Pischedda, Sylvie Le Floch, Silvana Radescu, Frederico Alabarse, et al.. Structural and mechanical properties of stable and metastable phases in SixGe1-x alloys. Journal of Alloys and Compounds, 2025, 1041, pp.183590. 10.1016/j.jallcom.2025.183590. hal-05291448

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# Structural and mechanical properties of stable and metastable phases in SiₓGe₁₋ₓ alloys

Max Gerin<sup>a</sup>, Vittoria Pischedda<sup>a, *,</sup> Sylvie Le Floch<sup>a</sup>, Silvana Radescu<sup>b</sup>, Frederico Alabarse<sup>c</sup>, Yann Le Godec<sup>d</sup>, Erwann Jeanneau<sup>e</sup>, Nicolas Guignot<sup>f</sup>, David Amans<sup>a</sup>, Denis Machon<sup>g,h,**</sup>

## ARTICLE INFO

Keywords:

Si-Ge alloys

High-pressure

X-ray diffraction

Density Functional Theory

## ABSTRACT

Stabilization of metastable phases of silicon-geranium alloys are of both fundamental and applied interest. By combining experimental and theoretical approaches, we investigated different SiₓGe₁₋ₓ compositions (x = 0.5 and 0.3) under high-pressure (HP) conditions. Homogeneous Ge-dominant SiₓGe₁₋ₓ alloys are difficult to synthesize, which explains the paucity of literature on the subject. Such Ge-rich alloys were obtained by combining arc melting and solid-state synthesis. Then, in situ HP X-ray diffraction experiments were performed. During HP cycling, the investigated SiₓGe₁₋ₓ alloys follow a similar phase transition pathway of Si and Si-rich binaries. On compression, the stable cubic semiconducting phase transforms to the tetragonal β-tin metallic phase at around 10 GPa. Under decompression, the metallic β-phase first transforms to the rhombohedral r8 phase. After further decompression, the rhombohedral phase partially transforms to the body-centered cubic bc8 phase. Both r8- and bc8-phases coexist at ambient conditions in the recovered sample. The experimental results are compared with density functional theory simulations and discussed in relation with the previous published results on Si-rich alloy (Si₀.₈Ge₀.₂). This allows giving an overview of the structural and mechanical properties of different phases as well as the phase stability as a function of the concentration in SiₓGe₁₋ₓ alloys. This opens the way to select composition and phases to tune some physical properties.

## 1. Introduction

Silicon and germanium are widely used semiconductors. They both exhibit a rich polymorphism under pressure and temperature associated with a change in physical properties. They also can exist in different metastable phases after a pressure cycle, depending on the pressure conditions applied. Thus, under cold compression, the stable semiconducting diamond structure of silicon and germanium transform to

the metallic β-tin phase in the range of 8–10 GPa [1–6]. Upon decompression, this metallic phase can transform in different metastable phases depending on the thermodynamic conditions [7–11]. For instance, the β-tin phase of silicon transforms into a distorted body-centered rhombohedral structure (r8) at around 10 GPa, and, subsequently, at ~3 GPa this phase transforms to a body-centered-cubic structure (bc8) [10,11]. The transformation pathway for germanium during decompression is different. Under non-hydrostatic conditions

<footnote><sup>a</sup> Universite Claude Bernard Lyon 1, CNRS, Institut Lumiere Matiere, UMR5306, F-69100, Villeurbanne, France</footnote>

<footnote><sup>b</sup> Departamento de Fisica and Instituto de Materiales y Nanotecnologia, MALTA Consolider Team, Universidad de La Laguna, San Cristóbal de La Laguna, Tenerife E38200, Spain</footnote>

<footnote><sup>c</sup> Elettra Sincrotrone Trieste S.C.p.A., Basovizza, Trieste 34149, Italy</footnote>

<footnote><sup>d</sup> Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie (IMPMC), Sorbonne Université, UMR CNRS 7590, Muséum National d'Histoire Naturelle, IRD UMR 206, Paris 75005, France</footnote>

<footnote><sup>e</sup> Centre de Diffractométrie Henri Longchambon, Université Claude Bernard Lyon 1, Villeurbanne 69622, France</footnote>

<footnote><sup>f</sup> Synchrotron SOLEIL, Gif-sur-Yvette 91190, France</footnote>

<footnote><sup>g</sup> Laboratoire Nanotechnologies Nanosystèmes (LN2)-IRL3463, CNRS, École Centrale de Lyon, Université de Sherbrooke, INSA Lyon, Université Grenoble Alpes, École Centrale de Lyon, Sherbrooke, Québec J1K 0A5, Canada</footnote>

<footnote><sup>h</sup> Institut Interdisciplinaire d'Innovation Technologique (3IT), Université de Sherbrooke, 3000 Boulevard Université, Sherbrooke, Québec J1K 0A5, Canada</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>** Correspondence to: Laboratoire Nanotechnologies Nanosystèmes (LN2)-IRL3463, CNRS, Université de Sherbrooke, INSA Lyon, Université Grenoble Alpes, École Centrale de Lyon, Sherbrooke, Québec J1K 0A5, Canada.</footnote>

<footnote>E-mail addresses: vittoria.pischedda@univ-lyon1.fr (V. Pischedda), denis.machon@insa-lyon.fr (D. Machon).</footnote>
and slow unloading, the β-tin phase of germanium transforms to a tetragonal structure (st12) at around 8 GPa [1,8,12,13]. Instead, in case of fast hydrostatic unloading, the β-tin phase tends to form the r8 phase at ~7 GPa followed by a transformation to the bc8 phase at quite low pressure (~0.5 GPa) [7,8]. Then, bc8 phase rapidly transforms at ambient pressure to a hexagonal phase [2,14,15].

SiₓGe₁₋ₓ alloys offer synergistic properties that can be tuned by varying the composition. The variability of band gap [16] or thermal properties [17] has made SiₓGe₁₋ₓ alloys interesting materials for applications in energy conversion in the space industry [18,19], as well as in photonics [20] or in the microelectronics industry [21]. They have potential as a next-generation material for solar cells [22], or as a new high-capacity anode material for Li-ion batteries [23,24].

Applying pressure to the SiₓGe₁₋ₓ alloys is another effective way of tuning properties and exploring the stability of different structures. For example, the hexagonal structure of SiₓGe₁₋ₓ (x < 0.35) features a tunable direct bandgap desirable for optoelectronic applications [25] and can be synthesized at high pressure and high temperature (HP-HT) [26]. Compared to the amount of investigations of the pressure effects in silicon and germanium, there are far fewer theoretical [27–29] and experimental studies [30–32] on the behavior of stable and metastable phases of SiₓGe₁₋ₓ alloys under high pressure conditions. In a recent study on Si₀.₈Ge₀.₂ [33], a sluggish pressure-induced phase transition towards a β-tin metallic phase around 12 GPa was observed. Upon decompression, the transition pathway β → r8 → r8 + bc8 was reported. For Ge-dominant alloys (up to x = 0.13) [34], the alloys seem to follow the HP behavior of the main component Ge. In particular, on decompression, the beta β-phase transforms to the metastable st12 phase under non-hydrostatic conditions.

To further investigate the possible metastable phases in the Si-Ge system and to comprehensively study the structural and mechanical properties across a broader compositional range, we investigated two different SiₓGe₁₋ₓ compositions (x = 0.5 and x = 0.3) under high pressure conditions. This allows proposing a systematic view of the phase stability and variations of structural and mechanical properties as a function of the composition both from an experimental but also theoretical point of view.

## 2. Methods

### 2.1. Materials

For both samples (x = 0.3 and 0.5), the first preparation stage was identical: first, silicon powder from Strem chemicals (purity: 99.999 %) and germanium powder from Sigma Aldrich (purity: 99.999 %) were mixed in their respective atomic ratio using an agate mortar under protective atmosphere. The resulting mixture was pressed to form pellets 2 mm in height and 8 mm in diameter.

The second stage of synthesis was different for each composition. Si₀.₅Ge₀.₅ was synthesized using the solid-state synthesis process described in [33], which involves a series of 1-hour annealing steps at temperatures ranging from 1000 to 1200 °C, with increments of 100 °C.

The synthesis of Ge-rich alloys is more challenging. Indeed, the marked difference in melting temperature between Si (1410 °C) and Ge (938.2 °C) is the most critical parameter for synthesis of Ge-rich alloys. For solid-state synthesis of Ge-rich powder mixture two main difficulties are encountered: 1) annealing at temperature higher than Ge melting point results in fast melting of the majority of the sample and sputtering of the sample before reaction with silicon; 2) annealing at temperatures below the melting point of Ge does not facilitate an efficient reaction due to extremely low diffusion rates, resulting in no observable reaction even over several weeks, and exposing the sample to slow oxidation. Consequently, direct solid-state synthesis is not suitable for such synthesis. Therefore, to overcome these pitfalls, a first pre-mixing step is required. Arc-melting is a relevant technique to solve the described issues. The fast melting/quenching cycling under controlled atmosphere

allows overcoming the melting point differences between the two materials. End product is a dense sphere of pre-reacted powder. Even if high segregation persists, the obtained material has a higher melting point due to the formation of Ge-rich pre-reacted alloys. Solid-state reaction can therefore be used to increase homogeneity in germanium rich arc-melted materials. For instance, in the case of Si₀.₃Ge₀.₇, a two-day post-annealing process comprising four temperature stages from 950 to 1020°C produced a homogeneous sample.

### 2.2. *In situ* high pressure characterization

In situ powder synchrotron X-ray diffraction (XRD) measurements under HP conditions were performed on Si₀.₃Ge₀.₇ and Si₀.₅Ge₀.₅ samples at Xpress beamline Elettra Synchrotron Radiation Facility (Trieste, Italy). First experiments were performed on Si₀.₅Ge₀.₅ which was studied using a Paris-Edinburgh (PE) large volume press. The XRD patterns were recorded using an 80 µm focused monochromatic X-ray beam with a wavelength of 0.4958 Å. High-quality 2D diffraction patterns were acquired in typically 80 s of exposure using a PILATUS3 S 6 M detector (Dectris). For this experiment, Si₀.₅Ge₀.₅ powder sample was grinded and loaded inside a boron-nitride capsule of 0.8 mm inner diameter and 2 mm height, surrounded by a graphite furnace. This assembly was then inserted into an X-ray transparent gasket (boron-epoxy composite) of 5 mm outer diameter, which serves as a thermal and electrical insulator as well as the pressure transmitting medium (PTM). To reach desired pressures of ~10 GPa, two sintered diamond anvils were used. To ensure a good pressure estimation, the hydraulic pump pressure versus internal pressure was re-calibrated by using the silicon Equation Of State (EOS) [6] in the same configuration, which also was used to obtain the instrument parameters for the XRD patterns integration, giving a sample-detector distance of 593.75 mm. For Si₀.₃Ge₀.₇ sample a pressure cycle up to 18 GPa and back to ambient was performed using a membrane diamond anvil cell (DAC). The sample was placed inside a 250 µm chamber drilled in an indented stainless-steel gasket (thickness = 80 µm), along with a ruby acting as a pressure gauge. The pressure was calculated using the known dependence of the ruby luminescence line with pressure [35]. Oil daphne 7474 (hydrostatic conditions up to 3.7 GPa [36] was used as PTM. For the DAC measurements, CeO₂ was used to obtain the instrument parameters.

Additional *in situ* XRD measurements on Si₀.₅Ge₀.₅ at HP conditions were performed on the beamline PISCHE at SOLEIL synchrotron (λ= 0.3738 Å and PILATUS CdTe 2 M detector) using a membrane-driven Le Toullec-type Diamond-Anvil Cell (DAC) with 350 µm diameter diamond culets. The sample was placed inside a 150 µm chamber drilled in an indented stainless-steel gasket (thickness = 80 µm). 4:1 methanol-ethanol mixture was used as PTM and gold was used as internal pressure standard [37]. Two compression cycles were performed with the same DAC. The first cycle up to 20.6 GPa and down to 4 GPa was used to synthesize metastable phases and the second cycle up to 39.9 GPa was meant to study the behavior of the metastable phases under pressure including reversibility of the phase transformations.

### 2.3. Ab initio simulations

Theoretical calculations based on Density Functional Theory (DFT) were performed using the Vienna ab initio simulation package (VASP) within the projector augmented wave (PAW) scheme and the PBEsol generalized gradient approximation (GGA) for the exchange and correlation functional. The parameters of the calculation are similar to those used in previous works [33], with a cutoff energy of 375 eV, and 10x10x10 Monkhorst-Pack integration grids. In order to deal with the alloys, the structure generator of the ATAT cluster expansion method was used with cluster sizes up to 8 atoms, which is enough to obtain accurate cluster convergence for the Si₀.₅Ge₀.₅ and Si₀.₂₅Ge₀.₇₅ alloys.
# 3. Results and discussion

## 3.1. Si<sub>0.5</sub>Ge<sub>0.5</sub>

Fig. 1 shows the XRD patterns of Si<sub>0.5</sub>Ge<sub>0.5</sub> at selected pressures obtained from the PE press. The diffraction pattern at ambient conditions can be indexed with the diamond structure. Cell parameter, measured using LeBail refinement (Figure S1), is equal to 5.53(1) Å which agrees with the composition-weighted linear combination of the cell parameters of each element.

Upon compression, a shift of the peaks to higher angles, associated with the volume reduction, is observed. In addition, the widths of the main diffraction peaks (111), (220) and (311) increase up to ~6–7 GPa before having a strong reduction (Fig. 1b). This behavior seems to be common to all Si-Ge solid solutions and it is attributed to stress

relaxation due to the pre-transition effect as observed in Ge, Si and Si<sub>0.8</sub>Ge<sub>0.2</sub> [12,33]. At ~12.3 GPa, the highest pressure attained in this run, the diamond phase persists without noticeable trace of the high-pressure β-phase. The unit cell volume versus pressure curve can be adjusted using a second-order Birch-Murhaghan (BM) equation of state (EOS) giving a value of 84 (2) GPa for the bulk modulus K<sub>0</sub> in full agreement with the results of DFT simulations yielding K<sub>0</sub> = 82 GPa. This value is also consistent with weighted average of the reported bulk modulus for Si (K<sub>0</sub> = 96 GPa with a first derivative of K<sub>0</sub> with respect to pressure K' = 3.9 [6]) and Ge (K<sub>0</sub> = 74.9 GPa with K' = 3.0 [1]).

To investigate the behavior of Si<sub>0.5</sub>Ge<sub>0.5</sub> at HP conditions, further in situ XRD measurements by using a DAC were carried out at the synchrotron SOLEIL. XRD patterns at selected pressures are shown in Fig. 2a. The first pressure range (compression of the diamond phase) was rapidly covered. The phase transition from diamond to the tetragonal

<figure><img src="image_4.png" /><figcaption>Fig. 1. (a) Diffraction patterns of Si<sub>0.5</sub>Ge<sub>0.5</sub> up to 12.3 GPa obtained using a PE press at Xpress beamline, Elettra. The additional peaks observed in the X-ray diffraction patterns are due to the sample environment (BN crucible, graphite furnace). (b) FWHM evolution as function of pressure for the three most intense diffraction peaks. (c) V/V<sub>0</sub> evolution as a function of pressure obtained from this experimental work (black points) along with a second-order BM EOS with K' parameter fixed at 4 (red line) fitted to the data leading to K<sub>0</sub> = 84(2) GPa. This is compared to the results obtained by DFT simulations (blue line).</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 2. (a) XRD patterns of Si<sub>0.5</sub>Ge<sub>0.5</sub> at selected pressures (synchrotron SOLEIL). Note the transition from the diamond to the β phase which starts at around 10.5 GPa and both phases coexist up to 14 GPa. (b) Diffraction patterns of Si<sub>0.5</sub>Ge<sub>0.5</sub> in a DAC upon pressure release from 20.6 GPa. (c) V/V<sub>t</sub> plot of the β-phase in Si<sub>0.5</sub>Ge<sub>0.5</sub> as a function of pressure (points) along with a second-order Birch-Murnaghan EOS (red line). Data are compared with results from DFT simulations (blue line).</figcaption></figure>

β-tin structure for Si<sub>0.5</sub>Ge<sub>0.5</sub> is observed at ~10.5 GPa and is completed at 14 GPa. Compression up to 20.6 GPa does not induce additional phase transition. The pressure range of phase coexistence is similar to that of Si<sub>0.8</sub>Ge<sub>0.2</sub>, silicon and germanium [1,3,33] indicating kinetics effect related to the reconstructive character of the transition associated with a large volume jump. At 11.7 GPa, the obtained unit cell parameters for the β-phase are: a= 4.80(2) Å; c= 2.65(1) Å and V= 61.1(2) Å<sup>3</sup>. The volume jump at the diamond-to-β transition (11.7 GPa) is ~ 20% in Si<sub>0.5</sub>Ge<sub>0.5</sub>, a value close to the one observed in pure silicon (-22% [38]).

Fig. 2b shows the XRD patterns during the decompression from 20.6 to 4.4 GPa. The β-phase is the only phase observed down to ~9.2 GPa. At 8.6 GPa, a new phase that can be indexed as the rhombohedral r8 structure is observed. On further decompression, the amount of the β-phase decreases with decreasing pressure until its disappearance below 7.3 GPa. At 4.4 GPa, the refined unit cell parameters of the r8 phase are a,b= 9.49(3)Å and c= 5.60(3) Å.

To characterize the mechanical properties of the β-tin phase, the bulk modulus at the transition (K<sub>t</sub>) was determined using a modified 2nd order Birch-Murnaghan EOS

$$P(V) - P_t = \frac{3K_t}{2} \left( \left( \frac{V_t}{V} \right)^{\frac{7}{3}} - \left( \frac{V_t}{V} \right)^{\frac{5}{3}} \right) \quad (1)$$

Where P<sub>t</sub> is the pressure of the transition from the diamond to the

β-structure and V<sub>t</sub> is the volume of the β-phase measured at P<sub>t</sub>.

As shown in Fig. 2c, the obtained lattice volume as a function of pressure can be fitted using this modified 2nd order Birch-Murnaghan EOS with P<sub>t</sub>= 11.7 GPa and V<sub>t</sub>= 61.1(2) Å<sup>3</sup>. This yields a value K<sub>t</sub>= 138(1) GPa for the β-phase of Si<sub>0.5</sub>Ge<sub>0.5</sub>.

The DFT-simulated bulk modulus B of the stable HP β-tin phase obtained at 11.8 GPa using the standard Voigt-Reuss and Hill schemes upon the calculated elastic coefficients is 135 GPa, which is in good agreement with this experimental value.

To investigate the pressure behavior of the metastable phase r8, we did not decompress to ambient pressure but a second cycle of compression/decompression was performed already starting from 4.4 GPa.

The XRD patterns collected during the second pressure cycle of Si<sub>0.5</sub>Ge<sub>0.5</sub> up to 40 GPa are shown in Fig. 3. Upon compression, the transition from r8- to β-phase is observed at 11.3 GPa with an endpoint of the transition at 17.5 GPa. Above 30.9 GPa, the doublet near a 2θ angle of 13° merges into a single peak. This observation suggests a transformation into a new phase, possibly corresponding to the orthorhombic Imma structure. In silicon, this phase, named Si-XI, was reported in a narrow pressure region between ~13 and 16 GPa [6,39]. The Imma structure corresponds to an orthorhombic distortion of the β-tin structure and can be considered as a transition state between the β and simple hexagonal structure (Si-V) observed at higher pressure, in the range 16-41 GPa in silicon [6]. In germanium, this sequence of phases is
<figure><img src="image_6.png" /><figcaption>**Fig. 3.** XRD patterns of Si<sub>0.5</sub>Ge<sub>0.5</sub> obtained during the second pressure cycle. (a) Upon compression, the r8-phase fully transforms to a β-phase at 13.4 GPa. Above 30.9 GPa, a new phase indexed with an Imma structure is observed. (b) Upon decompression, the reverse β-Si<sub>0.5</sub>Ge<sub>0.5</sub> to r8-Si<sub>0.5</sub>Ge<sub>0.5</sub> transition is measured at 8.2 GPa.</figcaption></figure>

similar but it is observed at higher pressures. It was found that β-Ge is stable up to 75 GPa before transforming to the Imma structure. The latter transforms to the simple hexagonal phase at pressures above 81 GPa [40].

In Si<sub>0.5</sub>Ge<sub>0.5</sub>, we noticed that this Imma phase is stable at least up to 39.9 GPa, the highest pressure achieved in this run. A LeBail refinement of the orthorhombic Imma phase at 39.9 GPa gives the following cell parameters *a*= 4.70(3) Å, *b*= 4.42(3) Å and *c*= 2.52(3) Å (Figure S2).

Upon decompression from the Imma phase, the XRD pattern of the β-phase is recovered around 30.8 GPa (Fig. 3). This observation confirms the reversibility of transition Imma → β. The absence of hysteresis indicates the slightly first-order character of the transition. Upon further decompression, we observed the phase transition from the β to the r8-phase at 8.2 GPa which remains stable down to the lowest pressure reached (5.2 GPa). A bulk modulus of $K_0$= 76(1) GPa was obtained for r8-phase. This is fully consistent with the theoretical value of 76 GPa obtained using the Hill formalism.

Fig. 4 shows the calculated differences in enthalpy with respect to the diamond phase for the different structures considered for the Si<sub>0.5</sub>Ge<sub>0.5</sub> alloy, with the inset showing the corresponding energy-volume curves. The enthalpy of the β- and Imma-phases are very close, but with increasing pressure the Imma phase tends to be more stable in good agreement with the observations. In addition, the closeness of the r8 and bc8 phases at ambient pressure is shown indicating a possible coexistence of both phases after decompression as observed in silicon [10,11, 41], Si<sub>0.8</sub>Ge<sub>0.2</sub> [33] and Si<sub>0.3</sub>Ge<sub>0.7</sub> (see next section).

The observed transformation paths are different between the compression and decompression. During the compression, the system follows the thermodynamic path with a transformation from the diamond to the β-phase starting around 8 GPa with a sluggish character associated to the reconstructive character of the transition. On decompression, the β-structure does not back-transform to the diamond phase but follow a kinetically-preferred pathway towards the metastable phases r8 and bc8.

<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Difference in enthalpy with respect to the diamond phase (cd). In inset energy-volume curves for all studied structures ($V_o$ per unit formula = 42.505 Å³).</figcaption></figure>

### 3.2. Si<sub>0.3</sub>Ge<sub>0.7</sub>

Si<sub>0.3</sub>Ge<sub>0.7</sub> was studied under HP conditions by using DAC at Elettra synchrotron up to 18 GPa and back to ambient conditions (see Fig. 5) using oil Daphne 7474 as the pressure-transmitting medium [36].

The ambient condition diffractograms of Si<sub>0.3</sub>Ge<sub>0.7</sub> can be indexed with the diamond structure. Cell parameter measured using LeBail refinement is *a*= 5.58(1) Å (Figure S3) which agrees with the composition-weighted linear combination of the cell parameters of each element.

As the compression is carried out with a PTM only hydrostatic up to
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 5. (a) XRD patterns of Si<sub>0.3</sub>Ge<sub>0.7</sub> up to 18 GPa obtained by using a DAC at Xpress beamline, Elettra. The transition to the β-phase occurs at 11.8 GPa and is completed at 18 GPa. (b) XRD patterns of Si<sub>0.3</sub>Ge<sub>0.7</sub> during decompression from 18 GPa to ambient conditions. The transition from the β to the r8 phase occurs around 10.4 GPa. At ambient conditions, both r8 and bc8 phases coexist.</figcaption></figure>

3.7 GPa, using a second-order Birch-Murnaghan EOS with K' parameter fixed at 4 is not relevant. The alternative fitting procedure used here involves transformation of the V(P) data to the normalized or reduced stress-strain variables F and f (Figure S4) [42]. This gives a bulk modulus of K₀ = 79(2) GPa, in good agreement with the value of 75 GPa obtained by DFT simulations on Si₀.₂₅Ge₀.₇₅. The transition towards the β structure starts at 11.8 GPa and finished between 16 and 18 GPa (Fig. 5a).

Using Eq. (1), we determine a bulk modulus of K<sub>t</sub> = 127 (2) GPa for the β- phase of Si<sub>0.3</sub>Ge<sub>0.7</sub> (Figure S5). The number of data points is limited but this result is supported by the agreement with the DFT simulations using the standard Voigt-Reuss and Hill schemes with a calculated value of K = 126 GPa for Si<sub>0.25</sub>Si<sub>0.75</sub>.

Fig. 5b shows the XRD patterns collected under decompression from 18 GPa to ambient conditions. The transition from the β- to the r8-phase occurs around 10.4 GPa. At ambient conditions, both r8 and bc8 phases coexist. The volume contraction of the r8-phase during decompression is reported in the Fig. 6. A value of the bulk modulus K₀ = 73(1) GPa for r8- are obtained using a second-order Birch-Murnaghan EOS to fit the experimental data compared to a value of 73 GPa obtained by DFT for the alloy Si<sub>0.25</sub>Ge<sub>0.75</sub>.

<figure><img src="image_10.png" /><figcaption>Fig. 6. V/V₀ plot for the r8-phase of Si<sub>0.3</sub>Ge<sub>0.7</sub> as a function of pressure (points) along with a second-order Birch-Murnaghan EOS with K = 73(1) GPa. The DFT simulations obtained on Si<sub>0.25</sub>Ge<sub>0.75</sub> give also a value of K = 73 GPa leading to the superposition of the experimental and simulated 2nd order EOS.</figcaption></figure>

### 3.3. Composition dependence of structural and mechanical properties of metastable and stable phases

#### 3.3.1. Structural parameters and Mechanical properties

Fig. 7 shows the bulk modulus and cell parameter of the cubic diamond phase of Si<sub>x</sub>Ge<sub>1-x</sub> for x = 0, 0.3, 0.5, 0.8 and 1. Based on experimental results, the bulk modulus increases by 0.22(2) GPa per %Si and the cell parameter follows a Vegard's law with a variation of -2.4 (1).10<sup>-3</sup> Å per % of Si. These tendencies are in very good agreement with the DFT simulations showing similar linear variations. Some slight differences between experiments and simulations can be noticed especially for the low-silicon content alloys. The largest discrepancy between simulation and experimental results is found to the case of Ge in its diamond structure and is only of 7 % which is acceptable for such comparison simulation/experiment.

Regarding the β-phase, the cell parameters at P ≈ 12 GPa are reported as a function of the composition in Fig. 8. The variations are -2.9 (1).10<sup>-3</sup> and -1.8(1).10<sup>-3</sup> Å per % of Si for a-axis and c-axis, respectively. Bulk modulus as function of silicon content in the alloys roughly follows a linear variation with composition with a slope of 0.31(5) GPa per % of Si.

The experimental and simulated cell parameters evolution as function of pressure and bulk modulus of the r8 phase for different compositions are shown in Fig. 9. The evolution of the cell parameters are -4.6 (2).10<sup>-3</sup> and -2.7(4).10<sup>-3</sup> Å per % of Si for a-axis and c-axis,

<figure><img src="image_11.png" /><figcaption>Fig. 7. Experimental and simulated bulk modulus, and cell parameters of the cubic diamond phase vs the silicon content of the Si<sub>x</sub>Ge<sub>1-x</sub> alloys.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Fig. 8. Cell parameters (left figure) and bulk modulus (right figure) of the β-phase of SiₓGe₁₋ₓ alloys at around 12 GPa as functions of the silicon content.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 9. Cell parameters (left figure) and bulk modulus (right figure) of the r8-phase as functions of silicon content. Experimental data for compositions x = 0.3 and x = 0.5 were obtained from this work (cell parameters for x = 0.5 were linearly extrapolated at ambient pressure), while data for x = 0.8 were taken from [33]. Theoretical data for pure Si and Ge are also included for comparison.</figcaption></figure>

respectively. The bulk modulus of the r8 phase shows a linear variation as function of composition with an increase of 0.16(1) GPa per %Si.

The cell parameters of the bc8 phase recovered after pressure cycle and from DFT calculations are shown as function of composition in Fig. 10. The agreement between the experimental and theoretical values

are remarkable. Here again, a linear relationship between the cell parameter and the composition is observed with a variation of -3.3 (2)·10⁻³ Å per % of Si.

In conclusion, variations of physical parameters such as the bulk modulus and the cell parameters for four phases (cubic diamond, β-tin, r8 and BC8) of the SiₓGe₁₋ₓ system can be obtained using high-pressure experiments (Table 1). The degree of hydrostaticity may slightly affect the results mainly by broadening the uncertainties on the measurement of the physical properties. However, data are mainly obtained in quasi-hydrostatic regime and during decompression which minimizes the influence of non-hydrostatic stresses, as the pressure gradients and deviatoric stresses tend to relax during decompression. The excellent adequation with DFT simulations performed in hydrostatic configurations attests to the robustness of our results.

<caption>Table 1</caption>

Variation of physical properties (bulk modulus and cell parameters) with respect to the composition of SiₓGe₁₋ₓ for the four different phases (Cubic diamond, β-tin, r8 and BC8).

<table><thead><tr><th></th><th>∂K<sub>0</sub>/∂x (GPa per %Si)</th><th>∂a/∂x (10<sup>-3</sup> Å per %Si)</th><th>∂c/∂x (10<sup>-3</sup> Å per %Si)</th></tr></thead><tbody><tr><td>Cubic diamond</td><td>0.22(2)</td><td>-2.4(1)</td><td>-</td></tr><tr><td>β-tin</td><td>0.31(5)</td><td>-2.9(1)</td><td>-1.8(1)</td></tr><tr><td>r8</td><td>0.16(1)</td><td>-4.6(2)</td><td>-2.7(4)</td></tr><tr><td>BC8</td><td>-</td><td>-3.3(2)</td><td>-</td></tr></tbody></table>

<figure><img src="image_14.png" /><figcaption>Fig. 10. Cell parameters of the bc8 phase as a function of silicon content. The cell parameter of Si₀.₅Ge₀.₅ is from DFT simulations.</figcaption></figure>
### 3.3.2. Phase stability

Throughout this HP study, we observed that Si$_{x}$Ge$_{1-x}$, in all the compositions we studied, exhibits similar behavior under compression. Initially, they undergo a gradual transition over a wide pressure range to the metallic β-phase. The effect of composition on the transition pressure appears negligible, at least under the uncertainties of the experimental determination. DFT simulations indicate that between Si$_{0.25}$Ge$_{0.75}$ and Si$_{0.75}$Ge$_{0.25}$, the diamond to the β-phase transition pressure shifts only by 2 GPa (from 7 to 9 GPa). This is below the experimental determination of the transition pressure, specially in the case of this sluggish transformation.

The β phase appears to be stable up to 20.6 and 18 GPa for Si$_{0.5}$Ge$_{0.5}$ and Si$_{0.3}$Ge$_{0.7}$, respectively. However, here again, this difference in pressure is not significant enough to be associated with a clear composition effect.

On the other hand, the transition to the Imma-phase appears to be highly composition-dependent. The transition pressure in Si$_{0.5}$Ge$_{0.5}$ (~31 GPa) is found to be intermediate between the transition pressures observed in Si (~14 GPa) and Ge (75 GPa), suggesting a linear variation of the pressure of transition with composition. This aspect will be addressed by using a Landau approach in the following.

Interestingly, during decompression, all samples, including the Ge-rich compositions, follow a transition pathway similar to that of silicon. They first transition to a rhombohedral r8-phase. For Si$_{0.5}$Ge$_{0.5}$, during decompression from the β-phase, the r8 phase starts to appear at 9.2 GPa. For Si$_{0.3}$Ge$_{0.7}$, we observe this transition at 10.4 GPa. Then, both rhombohedral and the body-centered cubic bc8-phase coexist once recovered at ambient conditions.

These transition paths are consistent with those of silicon [11,41,43] and those previously measured in Si$_{0.8}$Ge$_{0.2}$ [33]. The stability domain of the β-phase in Ge-dominant alloys upon decompression appears to be wider than that of pure silicon, for which the lower limit is reported at 6.2 GPa [11,41], and wider than that of β-Si$_{0.8}$Ge$_{0.2}$, measured down to 5.4 GPa. For germanium, a similar transition pathway was observed in very specific conditions: during very fast unloading (a duration of less than a few seconds) in quasi hydrostatic conditions [8]. In this case, the transition to the r8 phase occurs at ~7 GPa and to the bc8 phase at less than 1 GPa [11].

In the following section, the previous results are interpreted using Landau theory of phase transition.

#### 3.3.2.1. Landau approach.
Analysis based on the Landau theory of phase transitions indicates that the variational potential for describing first-order phase transitions such as the diamond to the β-phase can be written as

$$F = F_0 + \frac{\alpha}{2}\eta^2 + \frac{\beta}{4}\eta^4 + \frac{\gamma}{6}\eta^6$$

Where η represents the order parameter implies in the mechanism of the phase transitions. The nature of the order parameter in the case of pressure-induced structural transformations in group IV have been described and listed elsewhere [44,45].

Regarding isothermal compression path, the parameter α is expressed as

$$\alpha = \alpha_0(P - P_c)$$

Where P is the applied pressure, Pc is the transition pressure between the two phases and α$_{0}$ is a numerical parameter.

In our study, we noticed that varying the composition in the alloys does not change the nature of the phase transformations but can have an effect on the thermodynamics stability of the different phase. This situation is different from the case of morphotropic transition in which a variation of composition is associated with a marked change in the structure.

In the current case, this shift of the line of transitions can be

integrated as a secondary order parameter. This has been done for example to incorporate the effect of interface energy on variation of the stability limit in nanomaterials [46].

As the concentration is a totally symmetry quantity, the potential can be written as

$$F = F_0 + \frac{\alpha}{2}\eta^2 + \frac{\beta}{4}\eta^4 + \frac{\gamma}{6}\eta^6 + \delta x \eta^2$$

Where δ represents the coupling between the primary order parameter η and the composition x considered as a secondary order parameter.

This equation can be modified as

$$F = F_0 + \frac{\alpha^*}{2}\eta^2 + \frac{\beta^*}{4}\eta^4 + \frac{\gamma^*}{6}\eta^6$$

With

$$\alpha^* = \alpha_0 \left( P - \left( P_c^0 - \frac{2\delta}{\alpha_0}x \right) \right)$$

Giving the form

$$P_c(x) = P_c^0 + \lambda x$$

This shows a linear deviation of the transition pressure with the concentration, the extent of the deviation depending on the value of the coupling parameter λ.

Interestingly, this simple approach brings a more formal explanation to a previous work describing temperature -concentration phase diagrams using the Landau theory of phase transitions [47]. To describe the concentration-dependence of the temperature-induced transformation (the liquidus and solidus), a double dependence of α on temperature and concentration had been presupposed. Here, we propose a theoretical support to this phenomenological assumption.

## 4. Conclusion

In this study, high-quality silicon-geranium alloys Si$_{0.5}$Ge$_{0.5}$ and Si$_{0.3}$Ge$_{0.7}$, which are notoriously challenging to synthesize, were successfully produced and investigated *in situ* using powder synchrotron XRD under high pressure and room temperature conditions. Pressure cycling allowed us to: i) explore the high-pressure polymorphism of these alloys; ii) stabilize metastable phases; and iii) determine the structural and mechanical properties of various phases observed. Combined with our previous investigation on Si$_{0.8}$Ge$_{0.2}$ [33] and DFT simulations, these results now offer a systematic understanding of how structural parameters and mechanical properties evolve with alloy composition. As a first approximation, we observed that both cell parameters and bulk modulus of all identified phases vary linearly with composition. These observations are further supported by DFT calculations. Phase stability in the alloys closely resembles that of pure Si, with the structural and elastic properties of the cubic, β, and r8 phases exhibiting linear scaling with Ge content. The pressures of transition show a linear variation with composition of the involved phases. Overall, this work provides a comprehensive overview of the thermodynamics, structure, and mechanics of Si$_{x}$Ge$_{1-x}$ alloys, which will be valuable for the growing range of applications based on these materials.

### CRediT authorship contribution statement

**David Amans:** Writing – review & editing, Validation, Supervision, Resources, Methodology, Conceptualization. **Nicolas Guignot:** Supervision, Resources, Formal analysis. **Sylvie Le Floch:** Writing – review & editing, Validation, Supervision, Methodology, Investigation. **Vittoria Pischedda:** Writing – review & editing, Validation, Supervision, Project administration, Methodology, Funding acquisition. **Max Gerin:** Writing – review & editing, Writing – original draft, Visualization, Investigation, Formal analysis. **machon denis:** Writing – review & editing, Writing –
original draft, Validation, Supervision, Funding acquisition, Conceptualization. **Erwann Jeanneau:** Validation, Supervision, Methodology. **Yann Le Godec:** Writing – review & editing, Validation, Supervision, Investigation, Formal analysis. **Frederico Alabarse:** Writing – review & editing, Validation, Supervision, Investigation, Formal analysis. **Silvana Radescu:** Writing – review & editing, Validation, Supervision, Methodology, Investigation, Formal analysis.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgment

LN2 is a joint International Research Laboratory (IRL 3463) funded and co-operated in Canada by Université de Sherbrooke (UdeS) and in France by CNRS as well as ECL, INSA Lyon, and Université Grenoble Alpes (UGA). It is also supported by the Fonds de Recherche du Québec (FRQ).

SR acknowledges the support of Spanish Ministerio de Ciencia e Innovación (DOI: 10.13039/501100011033) under Project PID2022-138076NB-C44 and Programa de Movilidad de Recursos Humanos (Plan Nacional de I+D+i), Grant No. PRX22/00287 as well as the computer resources used: Spanish Supercomputing Network (Red Española de Supercomputación) LaPalma Supercomputer, located at the Instituto de Astrofísica de Canarias.

The synchrotron X-ray diffraction experiments were performed at the Xpress beamline of Elettra Sincrotrone Trieste (proposal number: 20210068) and at the PSICHE beamline, SOLEIL (BAG, proposal number: 20210410). The research leading to this result has been supported by the project CALIPSOplus under Grant Agreement 730872 from the EU Framework Programme for Research and Innovation HORIZON 2020. We thank Boby Joseph for his technical assistance on the Xpress beamline.

We are grateful to Thomas Gaudisson for help on the experiment at Elettra, to Alain Polian and Jean-Paul Itié for advices and assistance at the experiment at SOLEIL. For synthesis of the sample we thank P. Veber and colleagues of the Université de Tours.

## Appendix A. Supporting information

Supplementary data associated with this article can be found in the online version at doi:10.1016/j.jallcom.2025.183590.

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