Supercooled Liquid Silicon Research Articles

Kinetic reconstruction of free energies as a function of multiple order parameters.

A vast array of phenomena, ranging from chemical reactions to phase transformations, are analyzed in terms of a free energy surface defined with respect to a single or multiple order parameters. Enhanced sampling methods are typically used, especially in the presence of large free energy barriers, to estimate free energies using biasing protocols and sampling of transition paths. Kinetic reconstructions of free energy barriers of intermediate height have been performed, with respect to a single order parameter, employing the steady state properties of unconstrained simulation trajectories when barrier crossing is achievable with reasonable computational effort. Considering such cases, we describe a method to estimate free energy surfaces with respect to multiple order parameters from a steady state ensemble of trajectories. The approach applies to cases where the transition rates between pairs of order parameter values considered is not affected by the presence of an absorbing boundary, whereas the macroscopic fluxes and sampling probabilities are. We demonstrate the applicability of our prescription on different test cases of random walkers executing Brownian motion in order parameter space with an underlying (free) energy landscape and discuss strategies to improve numerical estimates of the fluxes and sampling. We next use this approach to reconstruct the free energy surface for supercooled liquid silicon with respect to the degree of crystallinity and density, from unconstrained molecular dynamics simulations, and obtain results quantitatively consistent with earlier results from umbrella sampling.

Liquid-liquid phase transition in deeply supercooled Stillinger-Weber silicon.

The existence of a phase transition between two distinct liquid phases in single-component network-forming liquids (e.g. water, silica, silicon) has elicited considerable scientific interest. The challenge, both for experiments and simulations, is that the liquid-liquid phase transition (LLPT) occurs under deeply supercooled conditions, where crystallization occurs very rapidly. Thus, early evidence from numerical equationof state studies was challenged with the argument that slow spontaneous crystallization had been misinterpreted as evidence of a second liquid state. Rigorous free-energy calculations have subsequently confirmed the existence of a LLPT in some models of water, and exciting new experimental evidence has since supported these computational results. Similar results have so far not been found for silicon. Here, we present results from free-energy calculations performed for silicon modeled with the classical, empirical Stillinger-Weber-potential. Through a careful study employing state-of-the-art constrained simulation protocols and numerous checks for thermodynamic consistency, we find that there are two distinct metastable liquid states and a phase transition. Our results resolve a long-standing debate concerning the existence of a liquid-liquid transition in supercooled liquid silicon and address key questions regarding the nature of the phase transition and the associated critical point.

Thermodynamics and kinetics of crystallization in deeply supercooled Stillinger-Weber silicon.

We study the kinetics of crystallization in deeply supercooled liquid silicon employing computer simulations and the Stillinger-Weber three-body potential. The free energy barriers to crystallization are computed using umbrella sampling Monte Carlo simulations and from unconstrained molecular dynamics simulations using a mean first passage time formulation. We focus on state points that have been described in earlier work [S. Sastry and C. A. Angell, Nat. Mater. 2, 739 (2003)] as straddling a liquid-liquid phase transition (LLPT) between two metastable liquid states. It was argued subsequently [Ricci et al., Mol. Phys. 117, 3254 (2019)] that the apparent transition is due to the loss of metastability of the liquid state with respect to the crystalline state. The presence of a barrier to crystallization for these state points is therefore of importance to ascertain, which we investigate, with due attention to ambiguities that may arise from the choice of order parameters. We find a well-defined free energy barrier to crystallization and demonstrate that both umbrella sampling and mean first passage time methods yield results that agree quantitatively. Our results thus provide strong evidence against the possibility that the liquids at state points close to the reported LLPT exhibit slow, spontaneous crystallization, but they do not address the existence of a LLPT (or lack thereof). We also compute the free energy barriers to crystallization at other state points over a broad range of temperatures and pressures and discuss the effect of changes in the microscopic structure of the metastable liquid on the free energy barrier heights.

Phase relation between supercooled liquid and amorphous silicon

The phase relation between supercooled liquid silicon (l-Si) and amorphous silicon (a-Si) is discussed based on experimental results. Electrostatically levitated l-Si samples were supercooled down to low temperatures, 300 K below the melting temperature (Tcl: 1683 K), and solidified accompanied by the release of latent heat. It was found that solidified Si samples melted again at 1480 K caused by the latent heat. Also, it was found that the Si samples that rapidly quenched near the solidification temperature contained a large amount of a-Si with tetrahedral coordination. These two findings show that the supercooled l-Si samples solidified into a-Si and a-Si melted, confirming the idea of a first-order phase transition between two metastable phases proposed by Turnbull et al. [Metall. Mater. Trans. A 29, 1825 (1998)].

Polymorphism in glassy silicon: inherited from liquid-liquid phase transition in supercooled liquid.

Combining molecular dynamics (MD) simulation and Voronoi polyhedral analyses, we discussed the microstructure evolution in liquid and glassy silicon during cooling by focusing on the fraction of various clusters. Liquid-liquid phase transition (LLPT) is detected in supercooled liquid silicon However, freezing the high-density liquid (HDL) to the glassy state is not achieved as the quenching rate goes up to 1014 K/s. The polyamorphism in glassy silicon is found to be mainly associated with low-density liquid (LDL).

Dynamic transitions in molecular dynamics simulations of supercooled silicon

Two dynamic transitions or crossovers, one at a low temperature (T${}^{*}$ \ensuremath{\approx} 1006 K) and the other at a high temperature (T${}^{0}$ \ensuremath{\approx} 1384 K), are shown to emerge in supercooled liquid silicon using molecular dynamics simulations. The high-temperature transition (T${}^{0}$) marks the decoupling of stress, density, and energy relaxation mechanisms. At the low-temperature transition (T${}^{*}$), depending on the cooling rate, supercooled silicon can either undergo a high-density-liquid to low-density-liquid (HDL-LDL) phase transition or experience an HDL-HDL crossover. Dynamically heterogeneous domains that emerge with supercooling become prominent across the HDL-HDL transition at 1006 K, with well-separated mobile and immobile regions. Interestingly, across the HDL-LDL transition, the most mobile atoms form large prominent aggregates while the least mobile atoms get spatially dispersed akin to that in a crystalline state. The attendant partial return to spatial uniformity with the HDL-LDL phase transition indicates a dynamic mechanism for relieving the frustration in supercooled states.

Ornstein–Zernike direct correlation function from diffraction experiments in supercooled liquid silicon and in disordered cobalt

Using experimental data for the structure factor S(q) of supercooled liquid silicon, and amorphous and supercooled liquid cobalt, we have calculated and compared the Fourier transform c(q) of the Ornstein–Zernike direct correlation function for these two systems. Differences of c(q) between Si and Co are ascribed to the different atomic coordination and packing, high in Co due to typical metallic bonding, and lower in Si due to more directional bonding. Also, in contrast to previous expectations, differences are found between the direct correlation function of amorphous and supercooled liquid cobalt.

Pressure effects on the transitions between disordered phases in supercooled liquid silicon

We investigate the pressure effects on the transitions between the disordered phases in supercooled liquid silicon through Monte Carlo simulations and efficient methods to compute free energies. Our calculations, using an environment dependent interatomic potential for Si, indicate that at zero pressure the liquid-liquid phase transition, between the high density liquid and the low density liquid, occurs at a temperature 325K below melting. We found that the liquid-liquid transition temperature decreases with increasing pressure, following the liquid-solid coexistence curve. As pressure increases, the liquid-liquid coexistence curve approaches the region where the glass transition between the low density liquid and the low density amorphous takes place. Above 5 GPa, our calculations show that the liquid-liquid transition is suppressed by the glassy dynamics of the system. We also found that above 5 GPa, the glass transition temperature is lower than that at lower pressures, suggesting that under these conditions the glass transition occurs between the high density liquid and the high density amorphous.

First-principles coexistence simulations of supercooled liquid silicon

We perform first-principles coexistence simulations of the low-density and the high-density phases of supercooled liquid silicon and find a negative slope for the coexisting line in the temperature–pressure plane. Electron density maps and electron-localization function plots of the two phases of silicon show marked differences. The calculated differences suggest more localized electrons in the low-density liquid compared to the high-density liquid, coming from an increased population of covalent bonds, which further explain the calculated negative slope in the two phase coexistence regime. This is consistent with the presence of a pseudo-gap in low-density liquid silicon, absent in the high-density liquid which shows a metallic behavior.

Noncontact modulated laser calorimetry in a dc magnetic field for stable and supercooled liquid silicon

Thermal conductivity of liquid silicon is necessary for numerical process modeling. It is also of scientific interest. However, measuring thermal conductivity is a difficult task because of convections in the liquid and contamination from contact materials. To overcome these experimental difficulties, we have developed noncontact modulated laser calorimetry in a dc magnetic field to measure heat capacity, thermal conductivity and emissivity of high-temperature liquid metals. In this study, through improvement in temperature measurements, we considerably reduced the experimental uncertainty of measurements. Furthermore, the thermal conductivity and heat capacity of supercooled liquid silicon were measured. Thermal conductivity of liquid silicon agrees with the values calculated assuming the Wiedemann–Franz law near the melting point. This result suggests that free electrons play a dominant role in the thermal transport process in liquid silicon.

Nucleation of tetrahedral solids: A molecular dynamics study of supercooled liquid silicon

The early stages of crystallization of tetrahedral systems remain largely unknown, due to experimental limitations in spatial and temporal resolutions. Computer simulations, when combined with advanced sampling techniques, can provide valuable details about nucleation at the atomistic level. Here we describe a computational approach that combines the forward flux sampling method with molecular dynamics, and we apply it to the study of nucleation in supercooled liquid silicon. We investigated different supercooling temperatures, namely, 0.79, 0.86, and 0.95 of the equilibrium melting point T(m). Our results show the calculated nucleation rates decrease from 5.52+/-1.75x10(28) to 4.77+/-3.26x10(11) m(-3) s(-1) at 0.79 and 0.86 T(m), respectively. A comparison between simulation results and those of classical nucleation theory shows that the free energy of the liquid solid interface gamma(ls) inferred from our computations differ by about 28% from that obtained for bulk liquid solid interfaces. However the computed values of gamma(ls) appear to be rather insensitive to supercooling temperature variations. Our simulations also yield atomistic details of the nucleation process, including the atomic structure of critical nuclei and lifetime distributions of subcritical nuclei.

Density and Structure of Supercooled Liquid Silicon—Does Exist Density Maximum in Liquid Silicon—

Density and Structure of Supercooled Liquid Silicon—Does Exist Density Maximum in Liquid Silicon—

Dynamic aspects of the liquid-liquid phase transformation in silicon

First principles molecular dynamics simulations are used to characterize dynamic properties of supercooled liquid silicon across the liquid-liquid phase transition. Our results evidence the presence of a boson peak in the terahertz frequency range for the low-density liquid and support the scenario of a fragile-to-strong transition. The analysis of the collective excitations shows that its origin is due to a coupling of the longitudinal and transverse acoustic modes localized on the nanometer scale and can be attributed to collective vibrations of connected tetrahedra.

Structural and dynamical heterogeneity in deeply supercooled liquid silicon

We report on a first-principles molecular-dynamics study of structural and dynamical heterogeneity in supercooled liquid silicon. We find that highly tetrahedral configurations are intermittently formed and that spatially heterogeneous dynamics is concurrently induced in the deeply supercooled state (1000 K). This heterogeneity is responsible for the anomalous structural relaxation characterized by the stretched-exponential function. The temporal structural fluctuation is found to give rise to the 1/f dependence in the corresponding power spectral density. In a moderately supercooled state (1600 K), the structural and dynamical heterogeneity is quite weak, in contrast to the deeply supercooled state. The applicability of the Stillinger-Weber potential to the deeply supercooled state is also discussed.

Structural study of supercooled liquid silicon

For many years, theoretical studies using model and ab initio potentials have predicated the existence of a liquid/liquid phase transition in silicon, based on a continuous change of the liquid A5 structure to A4. In contrast, we report here a quantitative analysis of data from high-energy X-ray diffraction measurements of containerlessly-processed supercooled liquid silicon that demonstrates that the fractions of regions with A5 and A4 order instead remain essentially constant with supercooling, but that the coherence length of the A5 order increases.

The structures of normal and supercooled liquid silicon metal and SiGe alloy

The atomic structures of liquid Si metal and SiGe alloy in the normal and supercooled liquid states have been measured by X-ray diffraction from aerodynamically levitated and laser heated liquid droplets. The measurements covered a wide range of scattering vector Q (0.8–20A−1) and a wide temperature range that extended into the supercooled region. The present results on liquid Si confirm earlier measurements that showed a decrease in first-shell coordination with increased supercooling. For liquid SiGe, the shoulder on the high-Q side of the first peak in the structure factor S(Q) sharpens with increased supercooling, suggesting an increase in covalent bonding and development of chemical short-range order.

Does supercooled liquid Si have a density maximum?

We have performed precise measurements of the density of supercooled liquid silicon (l-Si) in the temperature range of 1530-1800 K using an electromagnetic levitation (EML) technique with static magnetic fields. We also performed first-principles molecular dynamics simulation (FPMD) of supercooled l-Si. The observed density of the supercooled l-Si and the FPMD results show good agreement in the temperature range of 1530-1800 K. The structure of the supercooled l-Si also showed good agreement between the experimental measurements and FPMD simulations. Based on these results, we discuss nucleation in supercooled l-Si and also the existence of a density maximum in the supercooled l-Si, which is well known in water at 4 degrees C.

Elastic modulus of supercooled liquid and hot solid silicon measured by inelastic X-ray scattering

The dynamical structure factors of supercooled-liquid and hot-solid silicon are measured by inelastic X-ray scattering at the same temperature, 1620 K. Two significant changes in the averaged longitudinal sound velocities and in the longitudinal modulus are observed. First, we observe a different longitudinal modulus in the polycrystalline hot-solid silicon compared to the extrapolated value obtained from the single-crystal measurement. This reduction of the modulus may be a precursor of the semiconductor-to-metal transition. Second, the increase in the longitudinal modulus in the liquid upon supercooling is consistent with an increase in the degree of the directional bonding.

Anomalous diffusivity in supercooled liquid silicon under pressure

We perform isothermal-isobaric first-principles molecular-dynamics simulations to investigate the dynamics of liquid silicon (l-Si) under pressure. We find that the self-diffusion coefficient increases with increasing pressure in the deeply supercooled state. This anomalous diffusivity is attributed to the formation of locally tetrahedral configurations which on average reduces the diffusivity at low pressures. Densification hinders the formation of the tetrahedral configurations, thus the diffusivity increases with increasing pressure. The tetrahedral configurations frequently formed at low pressures may be viewed as fragments of the low-density form of l-Si . It is therefore conceivable that transformations between two distinct liquids, low- and high-density liquids, locally occur in deeply supercooled l-Si . The present findings indicate the profound generality of the dynamics in liquids with a tetrahedral network such as water.

Transitions between disordered phases in supercooled liquid silicon.

We have investigated the transitions between disordered phases in supercooled liquid silicon using computer simulations. The thermodynamic properties were directly obtained from the free energy, which was computed using the recently proposed reversible scaling method. The calculated free energies of the crystalline and liquid phases of silicon at zero pressure, obtained using the environment dependent interatomic potential, are in excellent agreement with the available experimental data. The results show that, at zero pressure, a weak first-order liquid-liquid transition occurs at 1135 K and a continuous liquid-amorphous transition takes place at 843 K. These results are consistent with the existence of a second critical point for the liquid-liquid transition at a negative pressure.