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Article Cite This: J. Phys. Chem. C 2018, 122, 10929−10938

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Metastable States in Pressurized Bulk and Mesoporous Germanium Abderraouf Boucherif,† Silvana Radescu,‡ Richard Arès,† Andrés Mujica,‡ Patrice Mélinon,§ and Denis Machon*,†,§ †

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Laboratoire Nanotechnologies et Nanosystèmes (LN2), CNRS UMI-3463, Université de Sherbrooke, Institut Interdisciplinaire d’Innovation Technologique (3IT), Sherbrooke QC J1K 0A5, Québec, Canada ‡ Departamento de Física and Instituto Universitario de Materiales y Nanotecnología, MALTA Consolider Team, Universidad de La Laguna, La Laguna 38200, Tenerife, Spain § Institut Lumière Matière, Université de Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5306, 69622 Villeurbanne, France S Supporting Information *

ABSTRACT: Combination of porosity and hydrostaticity during compression is used with a view to explore the energy landscape of germanium. In this work, pressure-induced phase transformations in mesoporous crystalline Ge has been investigated by in situ Raman spectroscopy. A pressure-induced amorphization to a low-density amorphous (LDA) state was observed prior to a reversible polyamorphic transformation between LDA and high-density amorphous states. These pressure-induced transformations show some similarities with the behavior previously reported in nanoparticles. Thermodynamics models developed in the case of nanoparticles are successfully used indicating that, in both cases, the large surface-to-volume ratio leads to an increase of the system energy and that mesoporous materials may be considered as the negative image of a collection of nanoparticles. However, an inhomogeneous stress distribution is expected in porous materials because of it being a network with hyperbolic geometry. A control experiment is presented using a reference bulk germanium sample. The diamond-to-β-tin transformation is observed starting at around 8.0 GPa. On decompression, the metastable ST12 (Ge-III) phase is observed. Ab initio simulations are used to assign and interpret Raman spectra of this phase.

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INTRODUCTION Mesoporous materials may be an alternative to the nanoparticle-based materials because of manufacturing (obtaining ceramics from powders of nanoparticles keeping the interesting related properties) and health implications (nanotoxicology). The large surface-to-volume ratio in both cases enhances properties related to surface effects such as catalysis, Li insertion, reactivity, batteries, and supercapacitors.1,2 However, interfaces and point defects at the surface induce additional contributions in the total energy of the system, leading to new energy landscapes that modifies phase stabilities and favors the emergence of new phases with potentially interesting properties. Creating mesoporous materials by porosification from the bulk is a way to create interfaces and defects, and mesoporous materials may be considered as the negative image of a collection of nanoparticles. A question that arises is whether the surface energy component of mesoporous materials is equivalent to that of a collection of nanoparticles and leads to similar metastable states? If the surface area may be considered as equivalent, the curvature of the interface at the nanoscale is different and may lead to other effects (Figure 1). Following the same interest of using high pressure, thermodynamics of mesoporous materials may be explored using pressure-induced transformations. Understanding of © 2018 American Chemical Society

Figure 1. Geometry of two systems with similar surface-to-volume ratios: a collection of nanoparticles and a mesoporous solid. In the case of nanoparticles, the stress may be continuously and smoothly distributed over the spherical shape. On the contrary, discontinuities in the mesoporous network with a hyperbolic geometry will lead to inhomogeneities in the stress distribution (here, the stress in B will be higher than that in A).

phase stability in materials with high surface-to-volume ratio is a key to control the properties associated with the different structures. Received: March 19, 2018 Revised: April 25, 2018 Published: May 1, 2018 10929

DOI: 10.1021/acs.jpcc.8b02658 J. Phys. Chem. C 2018, 122, 10929−10938

The Journal of Physical Chemistry C

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In the case of porous silicon, combination of mesoporosity and pressure allows obtaining metastable states such as lowdensity (LDA) and high-density (HDA) amorphous states.3 Thus, pressure-induced amorphization (PIA) was observed to occur for a porous nanocrystalline variety of p-doped Si compressed to P > 10 GPa at room temperature.3 Interestingly, the Raman spectrum did not correspond to that of a typical a-Si material based on tetrahedrally bonded units but instead reproduced the vibrational density of states of a highly coordinated metallic crystalline form such as Si-II (β-Sn structure). It was concluded that PIA had resulted in a direct transformation into the HDA polyamorph, which then backtransformed into the LDA form of a-Si during decompression. More recently, another study on mesoporous Si showed a different sequence of transformation:4 X-ray diffraction results indicate that π-Si, compressed in a metastable region, transforms to the primitive hexagonal crystalline phase at ∼20 GPa, and the formation of the HDA form of a-Si was confirmed during decompression of the high-pressure phase, and the HDA−LDA transformation occurred at around 4.5 GPa. Recompression of the LDA state led to the transformation to the primitive hexagonal structure around 18 GPa. These observations are an interesting indication that the nanostructuration due to porosity leads to the observation of metastable states and that, depending on the sample, the transformation pathway may vary. Silicon shows strong similarities with germanium but has been more extensively studied because of its technological interest. Similar to bulk silicon, germanium under pressure undergoes a phase transition from the diamond-type structure (Ge-I) to a metallic phase (β-tin structure also named Ge-II) around 10−11 GPa. This transformation is upshifted to pressures above 17 GPa for nanoparticles.5 In addition, highpressure investigations of 5 nm Ge nanoparticles, mainly using extended X-ray absorption fine structure, have concluded that nanoparticles progressively become amorphous under pressure from the surface to the core. A subsequent LDA-to-HDA polyamorphic transformation may be observed at 17.5 GPa.6 Such polyamorphism is also observed in bulk amorphous Ge7 at 11−12 and 7−5.5 GPa, respectively, during increasing/ decreasing pressure. On amorphous thin films, similar transformations are found but are dependent on the sample morphologies.8 It is worth noting that, despite all these works, no experimental Raman spectrum of the HDA state has been presented up to now except by simulations.7 In this work, we intend to study the high-pressure transformations of mesoporous Ge to compare with the literature results reporting the behavior in nanoparticles, bulk, and amorphous Ge. In addition, to infer the effect of hydrostaticity provided by the pressure-transmitting medium (PTM), a control experiment is also presented on the initial pdoped Ge wafer, before porosification. Ab initio simulations are used to characterize the Raman spectra of a metastable phase (Ge-III) appearing in this experiment. The observed pressure-induced transformations are described using theoretical models of increasing complexity. First, a model based on the Gibbs approach taking into account the interface energy gives an overview of the transformation in mesoporous and bulk germanium. Then, a Ginzburg−Landau approach introduces the kinetics of the transformation to discuss the competition between amorphization and polymorphic transformations.

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EXPERIMENTAL AND SIMULATION TECHNIQUES

The Raman experiment was carried out using a HORIBA LabRam HR Evolution Raman spectrometer operated with a 532 nm wavelength compatible with our high-pressure setup (diamond anvil cellDAC), which can detect an inelastic signal down to about 6 cm−1. Laser power was set at 5 mW at the entrance of the DAC to avoid heating. The beam was focused on the sample using a 50× objective, with a beam diameter of ∼2 μm at the sample. The scattered light was collected in backscattering geometry using the same objective. High pressure was generated using a membrane DAC with low-fluorescence diamonds. Mesoporous Ge samples were placed into a 125 μm chamber drilled in an indented stainless steel gasket. Paraffin oil was used as the PTM. We chose a viscous liquid to minimize the invasion of the nanopores. The pressure was probed by the shift of the R1 fluorescence line of a small ruby chip. All calculations were performed within the ab initio framework of the density functional theory, using the pseudopotential and plane waves method of calculation, as implemented in the VASP code.9−11 The parameters of the calculation are similar to those used in ref 12 for Ge. The 3d semicore electrons as well as the 4s and 4p valence electrons of Ge were dealt with explicitly in the calculations, for which a projector augmented wave scheme was used,13,14 with a kineticenergy cutoff in the plane wave expansions of 375 eV. The socalled PBEsol generalized gradient approximation (GGA) to the exchange−correlation (XC) functional was employed,15 although calculations with the GGA functional by Perdew et al. (PBE),16 as well as the local spin-density approximation (ref 17 as parametrized in ref 18) to the XC functional, were also performed for testing purposes. For the integration over the Brillouin zone of the ST12 (Ge-III) phase, we used 8 × 8 × 8 Monkhorst−Pack grids. The relaxation of the structural degrees of freedom of ST12, both internal parameters and cell parameters, was driven by the calculated values of the corresponding forces and components of the stress tensor, with the converged configurations having residual forces less than 5 meV/A and maximum anisotropy in the diagonal components of the stress tensor less than 0.1 GPa. All these calculations correspond to hydrostatic pressures spanning the region of interest and zero temperature, with the small effect of the zero point energy neglected. For the phonon calculations, we used an implementation of the density functional perturbation theory, with subsequent assignment of modes upon the symmetry analysis of the eigenvectors. Mesoporous germanium (see Figure 2) samples were prepared by electrochemical etching of the p-type (resistivity 12 × 10−6 Ω·cm) Ge substrate in hydrofluoric acid solution (HF(49%): ethanol (5:1, v/v)) under bipolar current during 120 min. A current density J+ = 1.5 mA/cm2 during 1 s was for etching, and a current density J− = −1.5 mA/cm2 during 1 s was for passivation. As a result, a ∼800 nm mesoporous layer was obtained with an average pore diameter ranging from 5 to 8 nm. More details about the process could be found in refs 19 and 20. After this treatment, the sample remains crystalline.21 Raman spectra of mesoporous Ge and initial Ge wafer are shown in Figure 3a. It is worth noting that despite the doping by Ga atoms, the initial Ge spectrum does not show any modification compared with an undoped bulk Ge sample (i.e., no Fano profile, no disordering effect). This is in agreement with previous measurements on p-doped Ge.22 10930


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dynamics simulations show that in the LDA state of Ge, the low-frequency band is as intense as the high-frequency one.7,25 This indicates that the nanostructure is getting more and more disordered with increasing pressure. This effect starts right from the first pressure steps, as a mesoporous structure is very sensitive to stress application. Compression leads ultimately to the disappearance of the initial strongest band at ∼300 cm−1 above 13.2 GPa. At higher pressure, spectra are dominated by a broad band at ∼65 cm−1 along with a shoulder at slightly higher wavenumbers. Spectra obtained at high pressure may be interpreted as a signature of an HDA state as the Raman spectra are in very good agreement with the molecular dynamics simulations.7 During, this transformation, the local fourfold coordination is transformed in an average HDA sixfold coordination associated with different local geometries.26 On decompression (Figure 4b), spectra associated with the HDA state remain down to ∼5 GPa where a high-frequency component appears. The spectra observed down to ambient pressure are typical from an LDA state.27 This evidenced a reversible polyamorphic transition in mesoporous germanium. The maxima of the low- and high-frequency peaks as a function of pressure are shown in Figure 5. At the first glance, at high frequency, the ν(P) data show a significant deviation from a linear dependence of the Raman frequency with pressure similar to the observations in compressed nanoparticles.5 Therefore, the data have been fitted with a secondorder polynomial

Figure 2. Typical TEM picture of mesoporous germanium showing a sponge-like morphology.

The main peak at 300 cm−1 is the T2g mode, the unique firstorder Raman-active mode expected for the diamond structure. After porosification, the Raman spectrum has changed drastically with (i) a change of the profile of the main peak and (ii) appearance of a broad band at ∼80 cm−1. The simulated phonon density of state (pDOS) of the diamond structure is in very good agreement with the recorded spectrum of the mesoporous Ge (Figure 3b). This is an indication that the q = 0 Raman selection rules are broken because of the appearance of defects (surface and point defects) during the porosification. In addition, the profile change of the highfrequency peak may be consistent with a phonon confinement model19 related to the reduced propagation path between nanopores.

ν = ν0 + A ·P + B ·P 2

where ν0 is the Raman frequency at zero pressure. This yields A = 6.3(3) cm−1·GPa−1 and B = −0.25(3) cm−1·GPa−2. Such parameters differ from the case of nanoparticles compressed in similar conditions5 where A = 3.9(3) cm−1·GPa−1 and B = −0.14(2) cm−1·GPa−2, and from the bulk Ge pressurized under hydrostatic conditions for which A = 3.85(5) cm−1·GPa−1 and B = −0.039(6) cm−1·GPa−2.28 It is worth noting that the pressure-induced shift of the peak is different from the one calculated for the pDOS of the cubic diamond phase Ge-I23 where the parameters are A = 4.8(5) cm−1·GPa−1 and B = −0.06 cm−1·GPa−2. The progressive softening of this vibrational mode in the mesoporous sample reflected by the value of the parameter B may be due to the progressive amorphization (revealed by the increase in intensity of the low-frequency peak) that may be triggered by the stress inhomogeneities generated by the geometry of the mesoporous network (Figure 1). Deviation

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RESULTS Mesoporous Germanium under Pressure. Raman spectra of mesoporous Ge during compression and decompression are shown in Figure 4. During the first range of compression (P < 12.0 GPa), the spectral change is dominated by a change in the relative intensity of the band centered around 80 cm−1. On the one hand, according to the ab initio simulations, such an effect is not expected under pressure in the pDOS of crystalline Ge.23 On the other hand, in amorphous Ge, a general property of this low-frequency peak is an increase in its amplitude as the structural order of the samples decreases.24 In addition, standard and ab initio molecular

Figure 3. (a) Raman spectra of Ge wafer before porosification and mesoporous Ge and (b) comparison of the Raman spectrum of mesoporous Ge along with the calculated pDOS of the diamond structure (Ge I). 10931


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Figure 4. Raman spectra of mesoporous Ge as a function of (a) increasing pressure, (b) decreasing pressure.

In amorphous Ge, the low-frequency vibrations have an acoustic-like nature.25 The pressure-induced shift is shown in Figure 5. This change is linear with pressure with apparently no marked effect due to the nonhydrostaticity, probably because of the broadness of the peak. The slope of ν(P) for this lowfrequency peak is negative, but this downshift is much lower than the one predicted by the ab initio simulations of the pDOS of the pressurized diamond structure.23 However, the simulated Grüneisen parameter, defined as γ =

B0 ∂ω · ∂P where ω0 T

( )

B0 is the

isothermal bulk modulus (B0 = 73 GPa for the amorphous Ge) and ω0 is the peak position at ambient pressure, lies in the range [−0.8; −0.9] in the frequency range of the measured lowfrequency peak.25 This value is in very good agreement with the experimental data that gives γ ≈ −0.8. This measurement fills an experimental deficiency noted by theoreticians as “no information reported in the literature about the response of the low-frequency modes of a-Ge to pressure and Grüneisen parameters of a-Ge”.25

Figure 5. Raman Peak positions as a function of pressure for mesoporous germanium.

from hydrostatic compression seems to be reflected by an increased value of the B parameter. This latter effect will be discussed when analyzing the control experiment in the next section.

Figure 6. Raman spectra of a Ge wafer as a function of pressure during (a) compression and (b) decompression. Asterisks indicate the signature of the Ge-I (diamond) phase. 10932


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Figure 7. (a) Peak position of the Ge wafer main peak as a function of pressure. The red line is a fit using a second-order polynomial on the whole pressure range. The black line is a second-order polynomial fit in the pressure range [0.1−4.0 GPa]. (b) Half-width at half-maximum of the peak in the Ge-I phase.

transition begins because of nonhydrostatic stresses. In addition, it is also worth noting that the peak position as a function of pressure shows a softening above 9 GPa (Figure 7a) probably because of mechanical relaxation during the phase transformation. The ν(P) data have been fitted with a quadratic relationship ν = ν0 + A·P + B·P2 in the pressure range [0−9.3 GPa.]. This yields A = 4.2(3) cm−1·GPa−1 and B = −0.10(3) cm−1·GPa−2. Surprisingly, these values are very close to the ones obtained by Sapelkin et al. when compressing nanoparticles in silicon oil.5 To take into account a possible effect of nonhydrostaticity, we performed a fit in the pressure range [0.1−4.0 GPa] below the limit of nonhydrostaticity.34 Using a second-order polynomial, the fitted values are A = 4.2(3) cm−1·GPa−1 and B = −0.04(7) cm−1·GPa−2, in good agreement with the data obtained by Olego and Cardona in hydrostatic conditions.28 Therefore, the anomalous behavior reported in the literature on nanoparticles and attributed to the surface effect may also be explained by the effect of shear stresses during compression as the same behavior is observed for a bulk sample compression under nonhydrostatic conditions. On decompression, a complete new spectrum is observed below 7.3 GPa (Figure 6b). It is well-known that decompressed metallic Ge-II does not follow the reverse structural sequence observed during compression.31 These spectra may be assigned to the “exotic” metastable Ge allotrope ST12 (also called GeIII). The ST12-Ge phase has a rich Raman spectrum and has been reported in the low-frequency region at ambient pressure for the first time only recently.35 The appearance of this metastable phase is favored by nonhydrostaticity.36 In addition to this phase, the T2g peak of the Ge-I (cubic diamond) structure is observed in spectra at 5.3 and 3.7 GPa, indicating an inhomogeneous mixture of the Ge-I and Ge-III phases. Despite their electronic and crystallographic structure similarities, the metastable phase usually obtained on decompression are different for Si and Ge. In Si, the BC8 structure is usually observed, whereas in the case of Ge, the ST12 structure dominates. This selectivity has been discussed in terms of kinetics barrier and relative phase stability. It appears that in Ge, the ST12 phase is more favorable than that in Si. This may be due to the bond energy that is different in the two elements. The stronger Si−Si bonds are harder and thus tend to rotate more rigidly, whereas the weaker Ge−Ge bonds are more susceptible to deformation in the form of bond twisting.37 Only recently, in ultrafast laser-induced confined

This result is a confirmation that, under pressure, the sample becomes amorphous. Therefore, compressing mesoporous germanium is not a simple homothetic reduction leading to expected pressure-induced shifts following the pDOS of the Ge-I structure. Compression induces a progressive disordering of the structure resulting in a PIA prior to a transformation into a HDA state and a polyamorphic transformation upon decompression. Control Experiment: Ge Wafer under Pressure. A highpressure experiment has been performed on the initial (before porosification) Ge wafer. A small piece of the wafer was loaded in the DAC with paraffin oil as the PTM. The goal of this experiment is to discriminate the effects of porosity from the effects of nonhydrostaticity. This latter one is due to the PTM that has been chosen viscous to minimize the liquid penetration into the porous structure. Raman spectra with increasing/ decreasing pressure are shown in Figure 6 and the fitted position and width of the T2g peak are shown in Figure 7. From 0.1 to 13.1 GPa, the T2g peak upshifts and no contribution at low frequency may be noticed. At pressure above 14.5 GPa, the transition to the Ge-II phase is observed with a change in the Raman spectra. The new spectra shows two modes (LO and TO on the low- and high-frequency range, respectively29). At first glance, the transition pressure (above 13.1 GPa) is higher than the one usually reported in the literature where a transition pressure between 10.0 and 11.0 GPa is reported under hydrostatic conditions30,31 using X-ray diffraction techniques. It has been demonstrated that nonhydrostaticity tends to decrease the transition pressure probably because of the ferroelastic character of the transition.32 However, the Raman scattering cross section of the two phases is very different as one is semiconductor (Ge-I) and the other one is metallic (Ge-II). Therefore, the scattered signal of the metallic phase may be very low compared with the one of the semiconductor even present in small quantity. Consequently, the reported transition pressure of 13.1 GPa in our study is more related to the disappearance of the diamond-structure phase. Recently, a study underlined the importance of the plastic deformation during the Ge-I → Ge-II phase transition and noticed that this effect leads to a narrowing of the Raman peak width.33 Extraction of the half-width at half-maximum of the T2g peak as a function of pressure is shown in Figure 7b. As reported by Yan et al.,33 the peak width decreases between ∼8 and ∼12 GPa, indicating the pressure range where the transition takes place. This is in agreement with the data from Yan et al. and with the lowered pressure at which the 10933


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The Journal of Physical Chemistry C microexplosion experiments that create far from equilibrium conditions, ST12 structure has been observed in silicon.38 The ST12 structure belongs to the P43212 space group (no. 96) with Ge atoms in 4a and 8b positions.39 Group theoretical analysis of the ST12 structure reveals 4A1 + 5B1 + 4B2 + 8E Raman-active optical phonon modes (21 modes). Numerical ab initio simulations have been performed to assign the spectrum and to better understand its features. First, it can be noticed that the 21 modes are distributed on a rather narrow frequency region. Consequently, some overlapping is expected and all modes cannot be resolved. In the experimental spectra obtained in the present study, only 12 peaks can be clearly observed and followed under pressure. The frequencies of the 21 Raman-active modes are indicated in Table 1 and comparison between experimental and calculated spectra at ambient pressure is displayed in Figure 8.

Figure 8. Experimental Raman spectra of decompressed germanium sample with a ST12 (Ge-III) structure along with the calculated Raman peak positions.

These results show that the pressure-induced Raman shifts and the phase transition mechanisms in the Ge wafer are sensitive to the hydrostatic conditions on compression (lowering of transition pressure and large pressure range where both phases coexist) or decompression (observation of metastable phases). ii) Using the same experimental conditions, the mesoporous diamond-type structure shows a progressive pressureinduced disordering that results in a PIA and, ultimately, in a transformation to an HDA state between 14.0 and 15.0 GPa. On decompression, a polyamorphic transformation occurs between the HDA state and LDA state. This HDA−LDA transformation is fully characterized for the first time using Raman spectroscopy, thanks to the lowfrequency access allowed by our Raman spectrometer. The Raman signal is in agreement with the numerical simulations available in the literature.7,25 The PIA followed by a polymorphic transformation upon decompression is similar to the observations in Ge nanoparticles. In the following, we discuss first the pressure-induced phase transformations using a classical thermodynamics approach. Then, the competition between the polymorphic transformation and the PIA is described using a Ginzburg−Landau approach that has been developed in the case of pressure-induced transformations in nanoparticles.

Table 1. Symmetry of the Vibrational Modes, Calculated Frequencies at Ambient Pressure, Calculated Pressure Coefficients, and Measured Frequencies at Ambient Pressure calculated pressure coefficients

symmetry

calculated position (cm−1) at ambient P

α (cm−1·GPa−1)

β (cm−1·GPa−2)

B1 E E B1 A1 B2 A1 E E B2 A1 B1 B2 E B1 E A1 E E B1 B2

56.5 77.2 81.9 87.1 89.2 91.3 98.8 149.7 186.3 191.1 193.2 211.7 212.9 222.1 229.3 245.3 271.9 271.8 276.8 280.7 292.1

−0.27(2) −0.27(5) 0.03(2) −0.56(2) 1.1(1) 0.78(2) 0.13(9) 0.57(2) 2.77(5) 1.69(3) 2.46(6) 2.01(8) 2.10(4) 2.11(6) 2.91(5) 3.00(5) 3.25(6) 3.82(8) 3.66(6) 4.2(1) 4.7(1)

−0.008(1) −0.018(2) 0.004(1) ∼0 −0.04(1) −0.009(1) 0.030(4) −0.011(1) −0.035(3) −0.016(1) −0.033(3) −0.046(4) −0.033(2) −0.029(3) −0.031(2) −0.027(3) −0.052(3) −0.058(4) −0.045(3) −0.057(5) −0.058(5)

measured position (cm−1) at ambient P 55(1) 77(1) 84(1) 87(1) 100(1) 150(1) 186(1) 191(1) 212(1)

228(1) 245(1) 273(1)

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DISCUSSION Metastable states of matter are controlled by the kinetics of phase transitions. Accessing such states requires energizing processes such as irradiation, pressurization, ball-milling, and their combination.40 Applying pressure to materials that have already been brought to a high free-energy state is a good opportunity to reach a highly metastable state.41 In the case of nanoparticles, the interface increases the total energy of the system and pressurization of nanoparticles leads to the observation of several metastable states as in the case of TiO2.42 In our case, nanoporosification creates interfaces and, consequently, defects that increase the energy of the material, as it happens in nanoparticles. Consequently, our experimental observations may be described using a classical thermodynamics approach and considering an interface energy.43 Figure 10 shows a schematic Gibbs free energy G(P) diagram that can summarize the transformations between the different states of Ge.

The peak positions have been simulated for different pressures. Comparison between the calculated and experimental data is shown in Figure 9, and the agreement is quite good. On a large pressure range, the peak positions have a clear quadratic dependence with pressure. The calculated positions as a function of pressure have been fitted using the relationships ν = ν0 + α·P + β·P2. Fitted α and β parameters are reported in Table 1. The summary of the experimental facts are as follows: i) Bulk Ge compressed in paraffin oil shows a transformation from the diamond-type (Ge-I) structure to the β-tin (Ge-II) phase starting at P ≈ 8 GPa and ending at ∼14.5 GPa. On decompression, a transition toward the metastable ST12 structure is observed. A part of the sample also transforms to the diamond-type structure. 10934


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Figure 9. Raman frequencies of the ST12 phase in Ge (Ge-III) as a function of pressure (lines: ab initio simulations; points: experimental measurements).

specific volume, enthalpy, and entropy are readily measured for solid amorphous materials and can be used to describe a noncrystalline substance that is in a metastable thermodynamic equilibrium.44 If the classical thermodynamic approach is useful in giving a general scheme describing the different transition pathways, it does not describe the kinetic reasons for this. Such aspects may be better taken into account using the Ginzburg−Landau theory of phase transition. Hence, the effect of the interface and defects on pressure-induced phase transitions in nanoparticles has been described theoretically recently using this approach.45,46 In this model, an additional term is introduced in the Landau potential, a term related to the inhomogeneity of the structure that will affect the kinetics of the transition. The use of this theoretical framework is specially adapted for porous materials that are heterogeneous materials. The strength of this approach is that it is possible to describe the polymorphic transition or the amorphization with the same kind of equation. First, the seminal work reported by Tolédano and Dmitriev47 demonstrates the possibility of defining a transcendental order parameter. Thus, the Landau approach to phase transitions can be extended to study any kind of structural phase transition and an order parameter η can always be determined. Hence, the GeI → Ge-II transition mechanism and the associated Landau free energy have been described by Katzke and Tolédano.48 In the Ginzburg−Landau approach, the Landau theory of phase transitions has been extended to take into account spatial derivatives of the order-parameter that may be associated with kinetics of the transition. Such a treatment was applied to the case of type-II superconductivity.49 Then, it was used to describe the incommensurate phase50 which shows some loss of the translational symmetry, and ultimately to describe amorphization processes.51,52 For a first-order phase transition as in the case of the Ge-I → Ge-II transformation, the master equation is given by

Figure 10. Free energy as a function of pressure. In the case of the bulk, the phase transformation occurs at the crossing of the respective free energies of Ge-I and Ge-II, defining the transition pressure. On decompression, the thermodynamics path is not reversible with the transformation to the Ge-III phase. In the case of mesoporous sample, defects (point defects and interface) induce an initial energy increase ΔGdef. Further energization process through pressure application leads to enter the amorphous states energy landscape.

Because G = U + PV − TS, the slope is approximately equal to the molar volume (V) at constant T. Considering the crystalline Ge, the free energy of Ge-I increases by V·ΔP during compression. At some point, its free energy crosses the one of Ge-II that is denser (and has a lower molar volume). This defines the transition pressure. On decompression, the same path may not be followed because of kinetics reasons. In our case, the decompression of Ge-II leads to the appearance of the metastable phase Ge-III. The role of kinetics and microscopic mechanisms on decompression has been detailed by Wang et al.37 If we consider the mesoporous Ge-I material, its initial free energy has been increased because of defects (such as interfaces). Subsequent compression allows reaching the amorphous energy landscapes. This series of transformation can be represented on the G(P) diagram by the intersection of the different G(P) curves associated with each amorphous state. Strictly speaking, the amorphous solid does not constitute a true phase and equilibrium thermodynamic arguments should not apply. However, thermodynamic quantities such as the

F = F0 + αη2 + βη 4 + γη6 + K(∇η)2

(1)

with K being the Ginzburg term related to the kinetics. By solving eq 1, it appears that the width of the transition ε (pressure range where low-pressure and high-pressure phases coexist) is proportional to K1/2.45 10935


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The Journal of Physical Chemistry C Two cases can be considered. In the first case, spatial inhomogeneity is limited and the polymorphic transition occurs. Then, the kinetics term (K) induced a broadening of the transition region. For instance, this parameter may take into account the effect of the pressure gradient generated during high-pressure experiment and nonhomogeneous compression due to the loss of hydrostaticity. Therefore, it may explain the broadening of the transition width observed in bulk Ge compressed under nonhydrostatic conditions (the width of the transition is more than 5 GPa in our case). In the second case, for the amorphization process, contrary to polymorphic transitions for which the order-parameter is associated with a single critical wave vector, the orderparameter associated with the crystal−amorphous transition varies continuously from one point to another.51 In the description of the crystalline to amorphous transformation under irradiation,51 the Ginzburg−Landau approach was used to derive that the radius of the amorphous region is given by the following relation ⎛ K ⎞1/2 rN = ⎜ ⎟ (CC − C N)−1/2 + r0 ⎝ α0 ⎠

term) is crucial. As in the case of nanoparticles, the interface and the associated defects favor a high K term leading to a slowing down of the polymorphic transition but favoring the amorphization. This study reveals that porosification is an interesting path using surface engineering of materials for accessing metastable states with potentially interesting properties. A control experiment on the germanium wafer before porosification was carried out. The nonhydrostaticity provided by the PTM induces a lowering of the transition pressure and an increased coexistence of the low- and high-pressure phases. This later effect may be also explained by the Ginzburg− Landau model as nonhydrostaticity introduces an inhomogeneity in the system. On decompression, a metastable crystalline state (Ge-III also known as ST12) is observed below ∼7 GPa in coexistence with Ge-I. Ab initio simulations of the pressureinduced shift of the Raman peaks have been performed, and the results are in good agreement with the experimental measurements.

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ASSOCIATED CONTENT

* Supporting Information

(2)

S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.8b02658. Simulated pressure-induced behavior of the p-DOS for the cubic-diamond structure (Ge-I) and peak positions of the low-frequency broad band as a function of pressure for the experimental spectra and the simulated p-DOS (PDF)

where α0 is a parameter of the Landau potential, K defines the Ginzburg term in eq 1, and r0 is the initial radius of the amorphous embryo. CN is the defect concentration at which the amorphous embryo nucleates, and CC is a critical concentration at which merging of amorphous embryos occurs (i.e., the percolation threshold is reached). On the one hand, the Ginzburg parameter K is related to the kinetics of a polymorphic transition, as it widens the transition region ε, that is, the coexistence between the low- and highpressure phases (ε ∝ K1/2), as demonstrated in ZnO under pressure.45 On the other hand, increasing K leads to a larger amorphous fraction as rN ∝ K1/2. Therefore, the kinetics factor tends to slow down the polymorphic transformation whereas it favors the amorphous state. This statement is in agreement with the general trend that amorphous states are kinetically favored states. Therefore, in mesoporous Ge, the creation of defects during the nanoporosification associated with inhomogenous stress distribution because of the hyperbolic geometry of pore network generates a high K factor that favors the apparition of the amorphous region to the detriment of the polymorphic transformation that is slowed down.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Denis Machon: 0000-0003-4627-6136 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS LN2 is a joint International Research Laboratory (Unité Mixte Internationale UMI 3463) funded and co-operated in Canada by Université de Sherbrooke (UdeS) and in France by CNRS as well as Université de Lyon (UdL, especially including ECL, INSA Lyon, CPE) and Université Grenoble Alpes (UGA). It is also associated with the French national nanofabrication network RENATECH and is supported by the Fonds de Recherche du Québec Nature et Technologie (FRQNT). Authors would like to thank Stéphanie Sauze and Arthur Dupuy for sample preparation, Guillaume Bertrand and Hubert Pelletier for technical help, Vincent Aimez, and Simon Fafard for fruitful discussions and would like to thank the Canadian Centre for Electron Microscopy for TEM image and the CECOMO platform at Université Lyon 1 for the use of the spectrometer. S.R. and A.M. acknowledged the support from MINECO Project no. MAT2016-75586-C4-3-P. This work was supported by the LABEX iMUST (ANR-10-LABX-0064) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). D.M. would like to thank the French “Ministère de lʼEnseignement supérieur, de la

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CONCLUSIONS Pressure-induced phase transitions in mesoporous germanium are reported and compared with initial germanium wafer before porosification used as a bulk germanium sample. The initially crystalline sample exhibits a Raman signature indicating the presence of defects (surface for instance) which is expected because of the interface. During compression, the degree of disordering increases and leads to an amorphous state (LDA) before undergoing a polyamorphic transformation to a HDA state above ∼14 GPa. The full Raman spectrum covering this HDA state is presented and agrees with the simulations. On decompression, a polyamorphic back-transformation from HDA to LDA is observed below ∼5.0 GPa. A formalism based on the Ginzburg−Landau theory explained the competition between polymorphic transformation and amorphization under pressure. These two mechanisms are described using a master equation where a parameter K (the Ginzburg 10936


Article

The Journal of Physical Chemistry C Recherche et de lʼInnovation” and “l’université de Sherbrooke” for their support.

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