Formation of distinctive nanostructured metastable polymorphs mediated by kinetic transition pathways in germanium Open Access

Group IV elements are of significant importance in both fundamental science and technological applications, owing to the existence of promising metastable polymorphs with various nanostructures characterized by excellent electronic and optical properties.^1–5 Metastable polymorphs with specific crystal symmetry and nanostructures are often synthesized by a nucleation mechanism, depending on the heat treatment applied. On the other hand, diverse nanoscale structures may emerge as a result of different transition paths and nucleation kinetics under high pressures, leading to observable disparities in distinctive macroscopic properties.^6–8 The nanosized structures formed by pressure treatment have been confirmed to play a pivotal role in the application of metastable polymorphs.^9–11 Hence, exploration of nanostructured metastable materials under high pressures is of great scientific importance. For example, sp³ amorphous diamond, as distinct from crystalline diamond, has been synthesized at high pressures and temperatures, and analysis of its microstructure has revealed structural ordering on the atomic level.^12,13 In the case of germanium (Ge), although its nanostructured metastable polymorphs may have excellent properties with potential applications, they have received only limited research attention. Consequently, the underlying nanostructuring mechanisms and transition kinetics remain relatively obscure.¹⁴ 

The formation of metastable polymorphs with intriguing nanostructures is contingent upon several factors, including pressure, decompression rate, temperature,^2,15 and stress.^16–18 In the case of silicon (Si), the decompression of β-Sn Si has been observed to result in multiple transition paths governed by temperature, decompression rate, and stress, leading to the emergence of diverse distinctive product structures.^19–29 The formation of metastable allotropes in both Si and Ge follows similar phase transition kinetics.^{22–24,30–35} However, Ge exhibits an additional phase transition pathway, namely, the transformation of β-Sn Ge into a tetragonal st12 structure.^36,37 The conditions of the formation of st12 Ge remain ambiguous, primarily because of the intricate interplay between thermophysical factors (temperature, decompression rate, and pressure) that collectively impact the kinetic pathways and the atomistic mechanisms.³⁶ On the other hand, first-principles calculations and X-ray diffraction (XRD) experiments suggest an enhanced participation of the d orbital in the valence band when Ge is compressed above 7.7 GPa, while decompression of β-Sn Ge leads to an abrupt change in electron density at 10–11 GPa.^38,39 Nevertheless, experimental confirmation of whether the crystal transition pathway is associated with an electronic change remains to be obtained.²⁴ 

In this paper, we present detailed studies of the formation of metastable Ge phases with distinctive nanostructures through kinetic transition pathways of high-pressure metallic β-Sn Ge under rapid decompression at different decompression rates. High-resolution transmission electron microscopy (HRTEM) reveals diverse nanosized structures of metastable phases, which are affected by a combination of decompression rate, temperature, and stress. X-ray absorption fine structure (XAFS) indicates that electronic and crystal transitions appear simultaneously in the formation of bc8/r8, but an electronic transition occurs prior to the crystal transformation of β-Sn Ge into the st12 phase. The analyzed results demonstrate that nucleation kinetics plays a crucial role in the formation of nanosized metastable structures through the transition paths.

High-pressure β-Sn Ge was prepared by compressing diamond cubic (dc) Ge up to at least 14 GPa. Metastable polymorphs can be obtained by rapid decompression of high-pressure β-Sn Ge at different unloading rates. The decompression rates of st12 Ge, bc8 Ge, and a-Ge are consistent with the conditions of the fast in situ synchrotron XRD experiments described below. The experimental details can be found in the supplementary material. HRTEM images reveal that three product metastable phases exhibit distinctive microstructures (Fig. 1). For st12 Ge, the lattice fringes are regularly arranged, showing a long-range ordered crystal structure with a high degree of crystallinity [Fig. 1(a)]. The interfringe distances of 0.45, 0.30, and 0.27 nm can be indexed to the crucial (111), (201), and (112) lattice planes of st12 Ge, respectively. We selected regions of 50 × 50 nm² and applied fast Fourier transformation (FFT) to obtain scattering signals in momentum space. The FFT patterns indicate that all regions belong to a single crystal with a large grain size and a low number of defects. By contrast, the bc8/r8 phase shows a disordering atomic arrangement between adjacent grains with different crystal orientations and amorphous features in the grain boundaries [Fig. 1(b)]. Interfringe distances of 0.34 nm in HRTEM images and FFT patterns are observed, which can be indexed to the (200) lattice plane of bc8 Ge. However, the interfringe distances of 0.32 and 0.30 nm can be indexed to the (002) and (101) lattice planes of hexagonal diamond (hd) Ge. In agreement with previous reports,^23,31 the metastable product phase of bc8/r8 Ge is found to persist for a short time under ambient conditions and can transform to hd Ge spontaneously. a-Ge does not possess long-range periodicity as exhibited in st12 and bc8/r8, but a considerable number of randomly oriented clusters are visible [Fig. 1(c)]. The clusters are nanosized paracrystalline structures with grain size ranging from 0.8 to 2.5 nm, in which the average fringe distance is ∼0.32 nm, close to that of dc Ge. The diffuse halos of the FFT patterns, corresponding to the selected areas, confirm the overall amorphous feature. The grain size distribution for the product metastable phases at a given decompression rate is estimated by the two-dimensional projection area in the HRTEM images. Evidently, st12 exhibits larger grain sizes (30–90 nm) with a limited number of grains [Fig. 1(d)] in the observable region, whereas a-Ge exists in the form of clusters characterized by smaller (0.8–2.5 nm) yet abundant nuclei [Fig. 1(e)]. The bc8/r8 phase features grain sizes of approximately 3–17 nm, falling in between these two extremes [Fig. 1(f)].

Fast in situ synchrotron XRD measurements show that metastable Ge phases with distinctive nanosized structures form through rate-dependent transition pathways of high-pressure metallic β-Sn Ge under decompression [Figs. 2(a)–2(c)]. At room temperature, β-Sn Ge transforms gradually to crystalline st12 Ge at ∼9.5 GPa under very slow decompression (<0.001 GPa/s) [Fig. 2(a)]. Upon rapid decompression at a relatively high rate (∼40 GPa/s), β-Sn Ge transforms to the r8 phase at 8.9 GPa, followed by transformation to the bc8 phase [Fig. 2(b)]. The rate-dependent formation of the st12 and bc8/r8 phases at room temperature is consistent with the results of previous studies.^31,36 In comparison, under rapid decompression with a higher rate (∼4 TPa/s) at room temperature, β-Sn Ge transforms directly to an amorphous state with the appearance of two characteristic diffraction halo peaks at ∼3.58 and ∼1.96 Å⁻¹ [Fig. 2(c)]. The formation of a-Ge has only been reported previously by slow decompression of β-Sn Ge at low temperature or by shock compression of dc Ge. This is the first time that amorphous Ge has been observed to form under rapid decompression of β-Sn Ge at room temperature. Figure 2(d) compares the diffraction patterns of the product metastable phases. The st12 and bc8 phases show characteristic features of orientational texture with sharp Bragg peaks, while a-Ge shows a monotonous pattern with broad diffraction halos. The st12 and bc8 phases can be well refined with the tetragonal (space group P4₃2₁2) structure with lattice parameters of a = 5.923(1), c = 6.967(4), and V = 244.415(4), and a body-centered-cubic (Ia-3) structure with a = 6.924(1) and V = 331.948(1) (Fig. S1, supplementary material), respectively. The atomic volume of st12 Ge is ∼2% smaller than that of bc8, and ∼10% smaller than that of dc Ge at ambient pressure. The Ge–Ge distance is calculated to be 2.295(1) Å in st12 Ge, which is slightly shorter than the 2.455(1) Å in the bc8 phase, but significantly longer than the 1.996(1) Å in dc Ge.

As summarized in Fig. 2(e), the temperature coupled with the decompression rate (or strain rate) affects the transition paths of β-Sn Ge. Three distinct transition regions are unveiled in the rate–temperature diagram. Here, the strain rate is defined by (dV/dt)/V=(dP/dt) (dV/dP)/V=(dP/dt)/B, where B is the bulk modulus of β-Sn Ge at pressure P, and dP/dt is the decompression rate. The unit cell volumes of β-Sn Ge as a function of time were obtained directly from time-resolved XRD data (Figs. S2–S12, supplementary material). For the formation of st12 Ge, the phase transformation occurs in a narrow rate–temperature regime, namely, under very slow decompression near or above room temperature. Transformation from β-Sn to st12 Ge was not observed at experimental strain rates below 200 K (Fig. S3, supplementary material). Above room temperature (i.e., 373 K and above), XRD confirms the transformation of β-Sn Ge to dc Ge regardless of the presence of the st12 phase as an intermediate, as long as the thermal energy is sufficient to overcome the kinetic barrier (Figs. S4 and S5, supplementary material). On the other hand, the stress, combined with temperature and decompression (or strain) rate, is found to have a significant influence on the kinetic transitions of β-Sn Ge under decompression (Fig. S6, supplementary material). In contrast to hydrostatic/quasi-hydrostatic conditions, keeping the rate and temperature unchanged at room temperature under nonhydrostatic conditions (without a pressure medium) leads to the formation of st12 Ge. This may be due to the fact that the stress reduces the kinetic barrier for heterogeneous nucleation in the phase transformation from β-Sn to st12 Ge.^40–43 

There are relatively large regions in the temperature–rate phase diagram for the β-Sn Ge to bc8/r8 transition and amorphization. A boundary separating the crystal–crystal transition and amorphization [black dashed line in Fig. 2(e)] is found. Close to this boundary, a mixture of bc8 and a-Ge coexists (Fig. S7, supplementary material). As found in Si and ice, the boundary indicates the existence of a threshold strain rate for the crystal–crystal transformation and amorphization at a given temperature.^26,44 That is above the threshold, β-Sn Ge transforms to a-Ge, and below it, rapid decompression leads to formation of the bc8/r8 phase (Figs. S8–S12, supplementary material). It should be noted that the threshold rate can shift, depending on the stress (i.e., the hydrostatic condition). This is because the stress may change the onset transition pressure and modify the transition barrier.

Despite the combined effect of temperature, rate, and stress on the transition kinetics of β-Sn Ge, the three distinct transition pathways seem to occur in certain pressure ranges, related to the specific unit cell volumes of β-Sn Ge. Figure 2(f) summarizes the onset transition volume of β-Sn Ge at various temperatures and different decompression rates. Three distinct regions are revealed. It is found that β-Sn Ge transforms to st12 Ge in a narrow volume range with a small lattice volume (∼66.8–67.8 Å³), close to the phase boundary of the dc and β-Sn Ge, while the transition from β-Sn to bc8/r8 occurs within ∼67.8–70.2 Å³, corresponding to the intermediate pressure region, and β-Sn Ge collapses into a-Ge at a lattice volume of ∼71.2 Å³. It should be noted that the onset transition lattice volume of β-Sn Ge for amorphization is independent of the decompression rate and temperature. This is different from the rate- and temperature-dependent transformation of β-Sn Ge to the crystalline phases (st12 and r8/bc8). As indicated in the previous study of Si, the phase transformations of β-Sn Ge to the crystalline phases are thermally activated, showing a temperature- and rate-dependence. On the other hand, amorphization is derived from mechanical collapse due to lattice instability.

Compared with the kinetic paths of β-Sn Si, one more path from β-Sn Ge to the st12 phase was observed. This observation raises a question regarding the underlying mechanism for the formation of the st12 phase. Previous studies have shown there is an electronic transition involving the d orbitals occurring at ∼10.7 GPa.³⁸ This pressure is close to the phase boundary of β-Sn and dc Ge and is located in the pressure region where the β-Sn to st12 transition occurs. Does the involvement of the empty d orbitals lead to the crystal structural transition? To investigate the correlation between the electronic effect and crystal structural transitions, we conducted XAFS measurements under decompression.

The detailed near-edge absorption fine structure (XANES) and extended X-ray absorption fine structure (EXAFS) data measured during decompression are shown in Figs. S13–S17 in the supplementary material. The k-weighted χ(k) EXAFS signal obtained by subtracting the embedded-atom absorption background from the measured absorption coefficient and normalized by the edge step magnifies the variation of β-Sn Ge under decompression (Figs. S15 and S16, supplementary material). The oscillations in the EXAFS χ(k) give well-defined peaks in R space on Fourier transform [Fig. 3(b)]. The single scattering peak between the Ge atom and its nearest neighbors is about 2.0 Å [R₁, Fig. 3(b)]. The pressure-induced changes in the positions of R₁ and the absorption edge energy E₀ [Fig. 3(a)] correspond to the evolution of the crystal and electronic structures, respectively. Figures 3(c) and 3(d) show the variations in R₁ and E₀ as functions of pressure in the phase transformations from β-Sn Ge to the st12 and bc8/r8 phases, respectively, under decompression. These reveal two distinct paths for the electronic and crystal structural evolution from β-Sn Ge to the metastable phases under decompression. Upon decompression [Fig. 3(c)], E₀ starts to increase rapidly at ∼13 GPa, preceding the abrupt change of R₁ at ∼10.5 GPa. This indicates that the electronic transition may occur prior to the crystal transformation from β-Sn Ge to st12. β-Ge features a combination of sp³-hybridized covalent bonding and partial metallic bonding within a tetragonal crystal structure (space group I4₁/amd). In the ab plane, Ge atoms form four covalent bonds via sp³ hybridization, while along the c axis, the s and $dz2$ orbitals hybridize to form two metallic bonds.^38,39 With increasing pressure, the enhanced participation of d orbitals progressively weakens the sp³ covalent bonding, making it more susceptible to phase transitions during decompression. The st12 Ge adopts a simple tetragonal structure (space group P4₃2₁2) and can be regarded as a distorted derivative of β-Ge. When the contribution of d orbitals in β-Ge undergoes a sudden decrease, the electronic density along the c axis experiences a rapid drop, leading to structural instability and triggering the transformation to the st12 phase. In comparison, under hydrostatic conditions, E₀ and R₁ start to change almost simultaneously with decreasing pressure [Fig. 3(d)], suggesting concomitant changes in the electronic and crystal structures in the β-Sn Ge to bc8/r8 transition. bc8 Ge possesses a body-centered cubic (bcc) structure (space group Ia-3) and structurally resembles a distorted diamond cubic (dc) phase. During decompression, the reduced participation of d orbitals leads to the gradual disappearance of metallic bonding between Ge atoms, enabling covalent bonds to reestablish their dominance. This simultaneous evolution of electronic and crystal structures facilitates the transition from β-Ge to bc8 Ge.

The HRTEM, XRD, and XAFS measurements demonstrate the diverse nanostructured metastable phases that form via distinct transition pathways of β-Sn Ge under different thermophysical conditions. Here, we tentatively interpret the results based on classical nucleation and growth theory.^45–47 The onset transition strain ε = (V₀ − V_tr)/V_tr and over-depressurization ΔP = P_tr − P₀ of β-Sn Ge are determined by the decompression rate, where V_tr and V₀ are the volumes of β-Sn Ge at thermal equilibrium pressure P_tr and onset transition pressure P₀, respectively. Figure 4(a) summarizes the initial transition strain of β-Sn Ge as a function of ΔP obtained from Fig. 2(f). The transformation of β-Sn Ge into the st12 phase occurs within a relatively narrow range of low over-depressurization, indicating a small value of ε. As the over-depressurization increases, it leads to the transformation of β-Sn Ge to bc8/r8. With further over-depressurization, ε gradually increases until it reaches a critical value of ∼0.053. Beyond this critical threshold, ε remains constant, signifying the limit of the over-depressurization and structural metastability of β-Sn Ge for volume expansion beyond the equilibrium boundary. These results demonstrate that product metastable phases with unique microstructures and size distribution form in specific ranges of ε and ΔP.

The formation of each grain corresponds to a nucleation and growth process at the expense of the parent β-Sn Ge. According to the size distribution, the grain number for a certain amount of the sample is N_st12 < N_bc8/r8; that is a small grain size means a greater grain number. Both N_st12 and N_bc8/r8 vary with rate and temperature, and both are proportional to the nucleation rate I. According to classical nucleation theory,^45–47 the homogeneous nucleation rate is determined by the Arrhenius equation:^50,51 I ∼ exp (−Q/k_BT), where Q is the free energy barrier in the formation of nuclei. Q is proportional to γ³/ΔG,² where γ and ΔG are the interfacial energy and the difference in Gibbs free energies between the parent β-Sn Ge and the new product phases. ΔG is the thermodynamic driving force in the phase transition. To a first-order approximation, ΔG ≈ ΔVΔP, where ΔV is the activation volume of the phase transformation.⁴⁸ Therefore, I is proportional to exp(−C/ΔP²), where C is a constant related to temperature, ΔV, and γ. The schematic relationship between nucleation rate and over-depressurization at room temperature shown in Fig. 4(b) provides a qualitatively explanation of the grain number as a function of over-depressurization. It can be seen that under slow decompression, a low over-depressurization will result in a low probability of nucleation with a small number of nuclei, corresponding to the formation of st12 Ge with large grain size and small grain number. The low nucleation rate is favorable for heterogeneous nucleation. I increases rapidly with increasing ΔP. Moderate over-depressurization will lead to a high probability in the nucleation process; that is a greater number of grains with smaller grain sizes will be found in the formation of the bc8/r8 phase. However, the nucleation rate is expected to approach a maximum value with increasing over-depressurization. It can be speculated that at the critical over-depressurization (or strain), that massive homogeneous nucleation occurs in the formation of the bc8/r8 phase, similar to the nucleation catastrophe observed in the melting of superheated crystals and the crystallization of supercooled liquids at the kinetic stability limit.^52–55 In this case, the crystal nuclei of the bc8/r8 phase are unable to grow by amalgamating adjacent nuclei, owing to the large interfacial energy between adjacent grains and different orientations. As a result, an amorphous state consisting of nanosized clusters as observed in the HRTEM measurements is formed. Further studies are required to confirm whether the amorphization of β-Sn Ge corresponds to the homogeneous nucleation catastrophe at the critical over-depressurization (or the limit of structural stability) in the solid transformation.

We have observed three transition pathways for the formation of distinctive nanostructured metastable phases under decompression of β-Sn Ge at different decompression rates in the HRTEM images. Fast in situ XRD analysis has revealed the combined effect of temperature, decompression rate, and stress on the formation of metastable st12, bc8/r8, and a-Ge. The XAFS results indicate changes in the electronic structure related to the participation of d orbitals in bonding, facilitating the transformation of β-Sn Ge into the st12 phase. These results demonstrate that the formation of nanostructured metastable phases is strongly correlated with nucleation rate. On the basis of rate- and temperature-dependent transformation kinetics, we have identified three nucleation mechanisms governed by over-depressurization:

1. Heterogeneous nucleation: under slow decompression, low over-depressurization (or strain) results in a reduced nucleation rate, leading to the formation of st12 Ge with larger grain sizes and fewer grains.

2. Homogeneous nucleation: as overpressure increases, the nucleation rate rises, resulting in the formation of the bc8/r8 phase characterized by a higher number of grains with smaller sizes.

3. Nucleation catastrophe: at critical over-depressurization, a sudden and uniform nucleation event occurs, accompanied by nucleation mutations at the limit of dynamic stability.

Although these findings are specific to metastable Ge, the observed kinetic transition pathways and nucleation mechanisms, as well as the influence of external parameters such as pressure, temperature, and stress, can be generalized to other materials. By extending these insights, it should be possible to explore novel strategies for synthesizing and applying metastable materials with tailored properties, thereby opening new avenues for materials discovery and design. This work not only advances our fundamental understanding of phase transitions, but also provides a framework for engineering metastable materials with unique functional characteristics.

Details of the experiments and data processing are provided in the supplementary material.

The authors thank Curtis Kenney-Benson, Zhen Liu, Yong Jiang, and Shengqi Chu for technical support in the XRD and XAFS experiments and analysis of data. This work was partially supported by the National Nature Science Foundation of China (NSFC) (Grant No. 11974033). Xuqiang Liu acknowledges support from the National Postdoctoral Foundation Project of China under Grant No. GZC20230215 and the National Nature Science Foundation of China under Grants No. 12404001. The XRD measurements at room and high temperatures were performed at the 4W2 HP-Station of the Beijing Synchrotron Radiation Facility (BSRF) and beamline 15U1 of the Shanghai Synchrotron Radiation Facility (SSRF). In situ high-pressure, low-temperature XRD measurements were conducted at sector 16 ID-B, HPCAT of the Advanced Photon Source, and were supported by DOE-NNSA under Award No. DE-NA0001974.

The authors have no conflicts to disclose.

M.L. and X.L. contributed equally to this work.

M.L. and C.L. conceived the research and co-supervised the project. M.L., X.L., and C.L. performed all the experiments, with assistance from S.J., J.S., L.W., and Y.G. M.L. and C.L. analyzed the data. S.P., Y.C., W.Y., and H.M. participated in discussions and made valuable comments on the manuscript. M.L. and C.L. co-wrote the manuscript. All authors discussed the results and contributed to writing the manuscript.

Mei Li: Conceptualization (equal); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (equal); Project administration (equal); Validation (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Xuqiang Liu: Data curation (equal); Writing – review & editing (equal). Sheng Jiang: Data curation (supporting); Methodology (supporting); Resources (supporting). Jesse S. Smith: Data curation (equal); Resources (supporting). Lihua Wang: Data curation (supporting); Resources (supporting). Shang Peng: Data curation (supporting); Methodology (supporting). Yongjin Chen: Formal analysis (supporting); Methodology (supporting). Yu Gong: Data curation (supporting); Resources (supporting). Chuanlong Lin: Conceptualization (lead); Data curation (equal); Formal analysis (equal); Funding acquisition (lead); Project administration (lead); Resources (lead); Supervision (lead); Validation (lead); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Wenge Yang: Funding acquisition (supporting); Resources (supporting); Supervision (supporting); Validation (supporting). Ho-Kwang Mao: Funding acquisition (supporting); Resources (supporting); Supervision (supporting); Validation (supporting).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.