Phase-field-crystal dynamics for binary systems: Derivation from dynamical density functional theory, amplitude equation formalism, and applications to alloy heterostructures

Abstract

The dynamics of phase field crystal (PFC) modeling is derived from dynamical density functional theory (DDFT), for both single-component and binary systems. The derivation is based on a truncation up to the three-point direct correlation functions in DDFT, and the lowest order approximation using scale analysis. The complete amplitude equation formalism for binary PFC is developed to describe the coupled dynamics of slowly varying complex amplitudes of structural profile, zerothmode average atomic density, and system concentration field. Effects of noise (corresponding to stochastic amplitude equations) and species-dependent atomic mobilities are also incorporated in this formalism. Results of a sample application to the study of surface segregation and interface intermixing in alloy heterostructures and strained layer growth are presented, showing the effects of different atomic sizes and mobilities of alloy components. A phenomenon of composition overshooting at the interface is found, which can be connected to the surface segregation and enrichment of one of the atomic components observed in recent experiments of alloying heterostructures.

Modeling the structure of liquids and crystals using one- and two-component modified phase-field crystal models

“…Thus, when q = 1 the phase transition is of second order for r > −9/38 and of first order for r < −9/38. As already mentioned, this result is exact in the sense that it is unchanged if more modes are included in the ansatz (11) and improves on the prediction r = −1/4 obtained for the PFC model using a one-mode approximation [2]. Note that the vacancy term (5) does not affect the transition because φ is positive everywhere.…”

“…This interpretation is bolstered by the fact that it has been shown that the PFC model [Eqs. (1) and (2)] may be derived from the density functional theory of freezing [9] and the dynamical density functional theory for colloidal particles [10,11] with certain approximations. The free energy is minimized by either periodic structures or by a homogeneous flat profile, depending on the values of q, r, andφ =…”

Patterning of Heteroepitaxial Overlayers from Nano to Micron Scales

“…To overcome the spatial limitation of the standard PFC, in this work we will use an amplitude expansion of the density field [25][26][27][28][29]. This methodology allows the study of 2D films adsorbed on a rigid substrate with spatial dimensions of several hundred nanometers, yet retains atomistic resolution.…”

“…This periodic variation limits the applicability of the model as the grid spacing must be less than the atomic spacing. To overcome this difficulty, an amplitude expansion was developed by Goldenfeld et al [25][26][27][28][29]. In this formulation the density is expanded around the basis set as the substrate, i.e., n ¼ P j e iG s j Ár þ c:c:, where j is a complex amplitude that varies on length scales much larger than 2=jG …”

“…Neither of these effects were included in the amplitude expansions discussed in Refs. [25][26][27][28][29] and are essential for describing the phenomena of interest here. While the amplitudes can be used to reconstruct the atomic density n and thus incorporate atomistic details such as dislocations, it should be noted that this description is missing the subtle coupling of the two length scales (a and the variation of j ) that leads to Peierls barriers [36] and faceting.…”