Exploring the Complex World of Two-Dimensional Ordering with Three Modes

Abstract

The world of two-dimensional crystals is of great significance for the design and study of structural and functional materials with novel properties. Here we examine the mechanisms governing the formation and dynamics of these crystalline or polycrystalline states and their elastic and plastic properties by constructing a generic multi-mode phase field crystal model. Our results demonstrate that a system with three competing length scales can order into all five Bravais lattices, and other more complex structures including honeycomb, kagome and other hybrid phases. In addition, nonequilibrium phase transitions are examined to illustrate the complex phase behavior described by the model. This model provides a systematic path to predict the influence of lattice symmetry on both structure and dynamics of crystalline and defected systems. . On larger length scales much progress has been made in the self-assembly of 2D crystals using particles of nano or micron size that are easier to tailor for specific functionalities and to observe. Colloidal crystals, for example, play a vital role in the study of structural properties of crystalline systems and the development of engineered, functional materials [7][8][9]. In addition, another novel technique for artificial lattice ordering is built on the trapping of ultracold atoms (e.g., 87 Rb) in optical superlattices produced by overlaying laser beams [10,11], as utilized for the study of many-body quantum physics. [7,11]. Thus it is of fundamental importance to identify the universal mechanisms underlying these distinct modes of crystallization, based on the general principle of symmetry [12]. It is also important to understand the nature of topological defects which occur frequently in such systems and are known to determine the electronic and mechanical properties of the sample [2]. Unfortunately it is very difficult to model and predict the nature of such defected states, due to multiple length and time scales involved in the non-equilibrium crystallization processes.In this work we develop a dynamic model that can be applied to the study of crystallization with a variety of ordered and defected structures. We adopt the phase field crystal (PFC) formalism [13][14][15][16], in the spirit of the Alexander-McTague analysis of crystallization based on Landau theory [12]. The advantage of this PFC approach is that one can study polycrystal formation in terms of the atomic number density on diffusive time scales that are many orders of magnitude larger than that of classical microscopic models such as molecular dynamics. One can also apply renormalization techniques [17][18][19] on the PFC equation to study problems that involve both micro and meso scales such as epitaxial growth [20] and surface patterning in ultra-thin films [21].Recently a great deal of progress has been made on generalizing the PFC formulation to include more crystal symmetries [22][23][24][25], although in 2D current PFC studies are restricted to triangular and square states. The basic idea is to incorp...

Honeycomb and triangular domain wall networks in heteroepitaxial systems

“…[27][28][29][30] This approach has been applied to the modeling of a large variety of elastic and plastic systems from nano to micron length scales. 11,12,[29][30][31][32][33][34][35][36][37][38][39] The results of this work elucidate the influence of overlayer lattice symmetry on the nature of phase transitions between various superstructured phases. Interestingly the results indicate a dramatic difference between the H-H and H-T systems.…”

“…11 For some H-T systems such as graphene or h-BN layers on metallic surfaces, typically the coupling between film and substrate is quite weak and thus the results of this work are relevant to these graphenetype materials at very small V 0 , leading to the emergence of triangular Moiré patterns as observed experimentally. 17,18 To further validate the H-T results particularly those at large V 0 , sample simulations of a three-mode PFC model for honeycomb lattice 36 were also conducted. The outcomes were consistent with the predictions described above in particular the phenomena of transitions among triangular, stripe, and commensurate phases.…”

Emergence of Chirality from Isotropic Interactions of Three Length Scales

“…Recent studies have demonstrated that up to three length scales are sufficient to produce all five two-dimensional (2D) Bravais lattices and other ordered phases [15], while interparticle interactions of two length scales can lead to 6 different 2D quasicrystalline orders [11]. One would expect this length-scale factor also plays an important role on the emergence of system chirality.…”

“…From a geometric perspective, a resonant triad of wave vectors forming a closed loop can form an equilateral, isosceles, scalene, or collinear triangle, and the following point group symmetries can be stabilized by the 3-waves coupling: C 6 , C 4 , D 2 , and C 1 . Which of them is preferred depends on the number and nature of the length scales involved [15]. For example, the chiral symmetry C 1 can emerge when N = 3, i.e., the wave-vector triplet forms a scalene triangle.…”

“…For simplicity we assume that the density waves for different length scales are of the same excitation level, i.e., b i = 0. This allows us to choose a sufficiently large range of λ (from 0.02 to 100), to tune the system elastic constants and address the chirality of both "soft" (with small λ) and "hard" crystals that are beyond the previous study [15]. The PFC dynamic equation was solved numerically via a pseudo-spectral algorithm [36] with periodic boundary conditions.…”