Understanding the Crystallisation Dynamics in Phase-Change Materials Using the Master Rate Equation & Multi-Scale Modelling

Abstract

The increasing demand for low-power, high-speed data storage technologies and shift in computing paradigms from conventional computing (with separate processing and memory elements) to in-memory computing, led to intensive research efforts in non-volatile memory materials and technologies. Phase-change materials (chalcogenide-based) present one the most promising solutions for the development of high-speed, low-power solid-state, optical and photonic memories and processors with excellent scaling capability, high cyclability, and thermal stability for data retention. Due to their extraordinary technological potential, phase-change materials were considered for non-volatile random-access memories (Phase-change RAM), optical data storage, neuromorphic computing and functional photonic and optical devices (switches, absorbers, and modulators and beam steerers). Phase-change materials (PCM) exhibit a transition from a disordered amorphous phase to an ordered crystalline phase, and vice versa, when subjected to certain annealing conditions achieved by electrical Joule heating or optical absorption. This reversible switching process is accompanied by modifications in the electrical and optical properties of the phase-change material or heated area, with sufficient contrast to enable practicable applications including memory functions (storage of 1s and 0s), and optical absorption and transmission. Switching from the crystalline to amorphous phase during switching is achieved through melting the phase-change material followed by fast cooling and quenching. This process can occur over very short time scales (nanoseconds or sub-nanoseconds). The crystallisation process, however, is more complex (involves nucleation of critical nuclei followed by their growth) and limits the overall switching speed in phase-change materials and devices. Moreover, the crystallisation mechanism, the switching rates and data retention are also dependent on the class of phase-change material used; crystallisation is dominated by nucleation in nucleation dominated materials and dominated by growth in growth dominated materials. Understanding the crystallisation kinetics – the rate at which crystallisation occurs and the stability of the amorphous phase – is therefore critical to enhance the data rates and writing and erasure speeds of phase-change memories and associated technologies. Thus, it is important to accurately model and simulate the crystallisation process in phase-change materials to develop high performing phase change materials and devices.

A detailed literature review of crystallisation models identified the Master rate equation method as the most practicable and physically realistic method for modelling crystallisation in phase-change materials. This method traces the transient attachment and detachment of molecules (monomers), driven by the thermodynamic and kinetic parameters of the material and temperature history, to realistically simulate the transient nucleation and growth processes during crystallisation. This approach is relatively fast compared to atomistic simulations and more accurate than classical analytical nucleation and growth theories and ensemble kinetic theories.

A numerical algorithm based on the discrete form of the Master rate equation was developed and solved in MATLAB using the ode15s solver to primarily study and understand the crystallisation process in the AgInSbTe (AIST) growth dominated material phase-change material, which is the first objective of this research project. Previous research (both theoretical and experimental) indicated that crystallisation is interface driven in growth dominated materials and proceeds primarily through growth from interfaces, and thus requires modelling approaches that include spatially dependent, moving crystallisation and thermal fronts to accurately model crystallisation. Recently published ultra-fast, high-temperature calorimetry measurements, however, indicated that the kinetics of crystallisation in growth dominated materials are rather controlled by the strong-to-fragile transition in material viscosity at increasing temperatures beyond the glass transition temperatures. Thus, the first objective of this project was to address the question of whether the Master rate equation (cluster density based) can be used to model crystallisation in the AIST growth dominated phase-change material. The physically realistic Mauro-Yue-Ellison-Gupta-Allan (MYEGA) model for the viscosity dependence on temperature, incorporating the measured viscosity parameters for AIST (including the fragility index and glass transition temperature) was integrated in the discrete Master rate equation model. Transient simulations carried out at different heating rates using ramped annealing were carried and the calculated crystalline volume fractions and nucleation and growth rates (as functions of temperature) were in qualitative and quantitative agreement with published experimental measurements. Furthermore, simulations of calorimetry measurements using the Master rate equation for AIST demonstrated very good agreement with measured Kissinger plots over a wide range of heating rates and temperatures. These simulations and agreement with reported experimental work therefore provide strong evidence of the feasibility of the Master rate equation in modelling crystallisation in growth dominated materials and other classes of phase-change materials.

The crystallisation model developed in this work, and previous work on modelling crystallisation in phase-change materials using the discrete Master rate equation relied on solving a relatively small system of equations corresponding to a limited number of monomers (in tens or hundreds), and hence cluster sizes, to achieve convergence over practical computation times. Realistic phase-change material structures in practical devices consist of tens of thousands or even millions of monomers over their volumes, which results in computationally demanding large-scale simulations. The question of the accuracy and applicability of the solution to the discrete Master rate equation to model realistic phase-change materials and devices with large number of interacting monomers and clusters, and the impact of this interaction on the crystallisation dynamics in phase-change materials has not been previously investigated and was the second question addressed in this research. To address the second research question, a multi-scale method, incorporating the discrete Master rate equation for small clusters and the continuous form of the Master rate equation for larger clusters, was developed to model the crystallisation in large and more realistic phase-change material systems. The Chang-Cooper algorithm was implemented numerically to partition the cluster space of the continuous Master rate equation using finite-differences and transform the partial differential problem into a system of ordinary differential equations. The continuity of cluster density flux condition was used to link the two forms of the Master rate equation, and the complete system was solved in MATLAB using the ode15s solver and validated. Multi-scale simulations of crystallisation in Ge2Sb2Te5 (GST) phase change material (nucleation dominated) and AIST (growth dominated) were carried out using 9 x 104 monomers (corresponding to the number of monomers in a phase-change film structure used in photonic devices). Comparisons to solutions of the discrete Master rate equation with a limited number of monomers (400 monomers) revealed an increase in the crystallisation temperatures with multi-scale simulations. This is attributed to the dependence of the transient nucleation and growth rates in the Master rate equation on cluster size, which cannot be fully considered using discrete simulations only. Moreover, multi-scale simulations of GST and AIST materials at different heating rates indicated higher crystallisation temperatures for AIST compared to GST, controlled by the strong-to-fragile transition in viscosity in AIST which is consistent with the higher amorphous phase stability and improved data retention observed in this class of phase change material. Thus, the multi-scale method provides more accurate simulations of crystallisation compared to the discrete model, and in qualitative and quantitative agreement with reported measurements and simulations for different classes of phase-change materials (nucleation and growth dominated).

The developed numerical algorithms and understanding gained from this research would contribute to the design and development of advanced non-volatile memory and processor technologies. Moreover, the numerical tools developed would be invaluable for the interpretation of experimental measurements (for material or device characterisation) and can be used for the extraction of pertinent material parameters from scanning calorimetry and other measurements methods for the characterisation of kinetics in the different classes of phase-change materials.