Equilibrium Crystal Shape Research Articles

銀/硫化銀(固柚/固相)界面平衡条件における硫化銀の平衡形

銀/硫化銀(固柚/固相)界面平衡条件における硫化銀の平衡形

Equilibrium crystal shape of the Potts model at the first-order transition point

We investigate the Q-state two-dimensional Potts model. The anisotropic interfacial tension is related by duality to the anisotropic correlation length. For Q > 4 we calculate exactly the anisotropic correlation length at the first-order transition point. From the calculated anisotropic correlation length, the equilibrium crystal shape (ECS) is derived via the Wulff construction. The ECS is expressed by means of a simple algebraic curve. Regarding Q as a temperature scale, we show that the Potts model has the same ECS as the eight-vertex model. We discuss a connection between the ECS and the q-deformed pseudo-Euclidian algebra.

The free energy singularity of the asymmetric six-vertex model and the excitations of the asymmetric XXZ chain

We consider the asymmetric six-vertex model, widely used to describe the equilibrium shape of crystals, and the relevant asymmetric XXZ chain. By means of the Bethe-ansatz solution we determine the free energy singularity, as function of the external field, at two special points on the phase boundary. We confirm the exponent 3 2 (checked experimentally, as the antiferroelectric ordered phase is reached from the incommensurate phase normally to this boundary, and we determine a new singularity along the tangential direction. Both singularities describe the rounding off of the crystal near a facet. At this point the hole excitations of the spin chain show dispersion relations ΔE ∼ (ΔP) 1 2 at small momenta, leading to a finite-size scaling ΔE ∼ N − 1 2 for the low-lying excited states, N being the chain size. We discuss the nature of the phase transition and the behavior of arrow-arrow correlation lengths in the ordered phase.

GaAs equilibrium crystal shape from first principles.

Surface energies for different GaAs surface orientations have been calculated as a function of the chemical potential. We use an energy density formalism within the first-principles pseudopotential density-functional approach. The equilibrium crystal shape (ECS) has been derived from the surface energies for the (110), (100), (111), and (-1-1-1) orientations. Under As-rich conditions all four considered surface orientations exist in thermodynamic equilibrium, in agreement with experimental observations. Moreover, our calculations allow us to decide on previous contradictory theoretical values for the surface energies of the (111) and (-1-1-1) facets.

Crystal shapes and phase equilibria: A common mathematical basis

Geometrical constructions, such as the tangent construction on the molar free energy for determining whether a particular composition of a solution is stable, are related to similar tangent constructions on the orientation-dependent interfacial energy for determining stable interface orientations and on the orientation dependence of the crystal growth rate which tests whether a particular orientation appears on a growing crystal. Subtle differences in the geometric constructions for the three fields arise from the choice of a metric (unit of measure). Using results from studies of extensive and convex functions, we demonstrate that there is a common mathematical structure for these three disparate topics and use this to find new uses for well-known graphical methods for all three topics. Thus, the use of chemical potentials for solution thermodynamics is very similar to known vector formulations for surface thermodynamics and the method of characteristics which tracks the interfaces of growing crystals; the Gibbs-Duhem equation is analogous to the Cahn-Hoffman equation. The Wulff construction for equilibrium crystal shapes can be modified to construct a “phase shape” from solution free energies that is a potentially useful method of numerical calculations of phase diagrams from known thermodynamical data.

Quasi-Reentrant Interface Behavior of Two-Dimensional Frustrated Ising Models

For a two-dimensional frustrated Union Jack Ising model, we give exact calculations of anisotropic interface tension, equilibrium crystal shape (ECS), squared scaled interface width and interface entropy by an extended Feynman-Vdovichenko's random walk method. In addi- tion to the region where the reentrant phase transition occurs, we find a wide area of the quasi-reentrant region where an interface entropy is negative and ECS extends with increasing temperature.

Chemical potential dependence of silver atom on equilibrium crystal shape for high temperature phases of silver selenide and silver sulfide

Chemical potential dependence of silver atom on equilibrium crystal shape for high temperature phases of silver selenide and silver sulfide

Curvature Jump at the Facet Edge of a Crystal for Arbitrary Surface Orientation

The equilibrium crystal shape (ECS) for the body-centered solid-on-solid (BCSOS) model in the low-temperature phase is studied. Based on the equivalence between the BCSOS model and the asymmetric six-vertex model, the surface free energy f (ρ) (ρ: step density) is calculated for general mean surface tilt angle. In the expansion, f (ρ)= f (0)+α ρ+βρ 3 + O (ρ 4 ), the universal relation between a and b is verified. Accordingly, the Gaussian curvature of the ECS is shown to assume a universal finite jump at any position of the facet edge of the crystal.

Equilibrium crystal shapes and their application to nucleation in solids

A major aim of theoretical calculations of nucleation rates, for diffusional phase transformations involving solids, is to obtain the equilibrium crystal shape (ECS) of the critical nucleus of the precipitating phase in the surrounding medium (matrix). The importance of the ECS arises from the fact that, in phase transformations involving negligible strain energy, the homogeneous nucleation rate is strongly dependent on the anisotropic interfacial energy of the precipitate-matrix interphase interface. The ECS is determined by this interfacial energy which, in turn, depends on the concentration profile at the interface. Equations obtained on the basis of the discrete lattice plane (DLP) model for the composition profile at a structurally sharp, compositionally diffuse interface between two crystal structures in a binary system, are presented in this paper. The results of application of the DLP model to determine the ECS of an h.c.p. precipitate in an f.c.c. matrix are also described. Good agreement is obtained on comparing the ECS obtained by this model with experiments which reveal the ECS. These results, as well as the computed temperature dependence of the ECS are discussed.

Numerical Examination of the Universality of the Equilibrium Crystal Shape in Some SOS Models

By means of the Monte Carlo twist method, we examine the universality of the following critical properties of the equilibrium crystal shape in the discrete Gaussian (DG) model and restricted solid-on-solid (RSOS) model: c1) a finite curvature jump at the roughening temperature T R , c2) the profile of a facet with a singularlity at the facet edge for T < T R , and c3) the radius of the facet near T R proportional to \(\exp(-a/ \sqrt{1-T/T_{R}})\). We obtain the following results with the help of some results obtained analytically. The DG and RSOS models have T R =1.44 ±0.02 and 1.60 ±0.05, respectively. c2) is universal for these models. For the DG model c1) well agrees with the exact result of the body-centered SOS model, whereas for the RSOS model it is almost equal to twice the latter. Finite-size-scaling for c3) is excellent for the DG model, while it is consistent but inconclusive for the RSOS model.

Quantum effects on the step interaction energy and the equilibrium shape of crystals

This paper shows that, due to quantum effects associated with surface states forming standing waves at steps surfaces, the leading term of the surface steps interaction energy near to a facet goes like p 2, where p is the surface slope. This term dominates over the p 3 contribution of the existing theoretical model predictions and therefore the equilibrium shape of the crystal near the flat surface of size x 0 is not singular but goes like ( x − x 0) 2.

Phase separation of crystal surfaces: A lattice gas approach.

We consider both equilibrium and kinetic aspects of the phase separation (``thermal faceting") of thermodynamically unstable crystal surfaces into a hill--valley structure. The model we study is an Ising lattice gas for a simple cubic crystal with nearest--neighbor attractive interactions and weak next--nearest--neighbor repulsive interactions. It is likely applicable to alkali halides with the sodium chloride structure. Emphasis is placed on the fact that the equilibrium crystal shape can be interpreted as a phase diagram and that the details of its structure tell us into which surface orientations an unstable surface will decompose. We find that, depending on the temperature and growth conditions, a number of interesting behaviors are expected. For a crystal in equilibrium with its vapor, these include a low temperature regime with logarithmically--slow separation into three symmetrically--equivalent facets, and a higher temperature regime where separation proceeds as a power law in time into an entire one--parameter family of surface orientations. For a crystal slightly out of equilibrium with its vapor (slow crystal growth or etching), power--law growth should be the rule at late enough times. However, in the low temperature regime, the rate of separation rapidly decreases as the chemical potential difference between crystal and vapor phases goes to zero.

Surface tension, step free energy, and facets in the equilibrium crystal

Some aspects of the microscopic theory of interfaces in classical lattice systems are developed. The problem of the appearance of facets in the (Wulff) equilibrium crystal shape is discussed, together with its relation to the discontinuities of the derivatives of the surface tension τ(n) (with respect to the components of the surface normaln) and the role of the step free energy τstep(m) (associated with a step orthogonal tom on a rigid interface). Among the results are, in the case of the Ising model at low enough temperatures, the existence of τstep(m) in the thermodynamic limit, the expression of this quantity by means of a convergent cluster expansion, and the fact that 2τstep(m) is equal to the value of the jump of the derivative ∂τ/∂δ (when δ varies) at the point δ=0 [withn=(m 1 sin δ,m 2 sin δ, cos δ)]. Finally, using this fact, it is shown that the facet shape is determined by the function τstep(m).

Ambiguity of Anisotropic Interface Tension for Complex Crystals

For two-dimensional (2D) lattice gas which consists of more than one sublattices, we show that microscopic interface energy cannot be determined uniquely. Accordingly, the interface tension has unavoidable ambiguity in choice of the origin (Wulff point) of the Wulff plot. On the other hand, interface stiffness, equilibrium crystal shape and interface entropy are free from the ambiguity. For anisotropic systems, it has been known that the interface tension in various “physical” formulas is replaced by the interface stiffness, which can be naturally understood from the ambiguity argument. As a demonstration, we consider a step (regarded as a 1D interface) on the (10\(\overline{1}\)0) surface of the hexagonal ZnS-type crystal, and calculate interface (step) quantities at finite temperatures. Critical curve is successfully obtained from zeros of the interface stiffness.

Thermal roughness of a Si(331) surface

The roughness of a high-indexed semiconductor surface, Si(331), was studied in a synchrotron X-ray scattering experiment. Below the transition temperature, the surface is macroscopically flat with mean-squared roughness close to that of an ideally terminated (331) facet. The height fluctuations of the surface above the transition temperature diverge logarithmically. The equilibrium crystal shape near the 〈331〉 direction is composed of the (331) facet and round regions connected by a sharp corner whose angle vanishes at the transition temperature.

Crossover Scaling Functions in One Dimensional Dynamic Growth Models.

The crossover from Edwards-Wilkinson ($s=0$) to KPZ ($s>0$) type growth is studied for the BCSOS model. We calculate the exact numerical values for the $k=0$ and $2\pi/N$ massgap for $N\leq 18$ using the master equation. We predict the structure of the crossover scaling function and confirm numerically that $m_0\simeq 4 (\pi/N)^2 [1+3u^2(s) N/(2\pi^2)]^{0.5}$ and $m_1\simeq 2 (\pi/N)^2 [1+ u^2(s) N/\pi^2]^{0.5}$, with $u(1)=1.03596967$. KPZ type growth is equivalent to a phase transition in meso-scopic metallic rings where attractive interactions destroy the persistent current; and to endpoints of facet-ridges in equilibrium crystal shapes.

Model for the shapes of islands and pits on (111) surfaces of fcc metals.

It is experimentally observed that adsorbate atoms and vacancies on (111) surfaces of fcc metals cluster into islands which are approximately hexagonal, but which on closer inspection turn out to have equilibrium facets that alternate in length ABABAB around the six sides of the island. By contrast, previous theoretical models for island faceting predict a rotating sequence of three lengths ABCABC around the island. We propose a model for the observed shapes, whose physical basis is the variation of the local arrangements of substrate atoms seen by an adsorbate atom. We map our model onto a generalized form of the two-dimensional Ising model having three- as well as two-spin interactions, and estimate using atom-embedding calculations the strengths of these interactions for Cu adsorbed on a Cu(111) surface. We then describe a highly efficient Monte Carlo technique for calculating the equilibrium crystal shapes of general Ising-type models in two or three dimensions, and apply it to the model in hand. Our results do indeed show alternating facet lengths very similar to those seen in experiments.

A geometric model for anisotropic crystal growth

Equilibrium crystal shapes are defined uniquely by the Wulff construction. The classical kinematic theory of crystal growth, due mainly to Frank and Chernov, provides a mathematically equivalent prescription for the limiting growth shape. To connect these two well studied states, we derive a local geometric growth model and examine the transient shape evolution of an equilibrium form containing both facets and rough regions. Our model is appropriate to the weakly driven growth of a two-dimensional single crystal with n-gonal symmetry and arbitrary closed initial shape. The model links disparate kinetic processes determined by the local interfacial structure to the isotropic growth drive, and reproduces the experimentally observed transition from a partly rounded equilibrium shape to a highly faceted crystal which we term 'global kinetic faceting'. We solve for the transient shape dynamics globally, and locally, and in the latter case present a curvature evolution equation valid for any local growth law. Both approaches show that, during kinetic faceting, rough orientations grow out of existence with decreasing curvature.

First-principles determination of equilibrium crystal shapes for metals at T=0.

We propose a simple method to evaluate the energies of ideal metal surfaces as a function of orientation based on cluster energy expansion. By symmetry only clusters with even-number corners will be present. It is found that the energy expansion converges rapidly and in most cases can be truncated at the pair interaction level. The parameters can be determined from a limited number of low-index surface energies obtained from first-principles calculations. The equilibrium crystal shape at T=0 is then predicted and the step energy on major facets is derived for some fcc metals.

Equilibrium shapes of crystals attached to walls

We discuss equilibrium shapes of crystals attached to walls. Optimal shapes for different configurations of walls are found and the minimality of the overall surface tension is proven with the help of a simple geometrical argument based on the isoperimetric inequality and monotonicity. Stability results in the form of Bonnesen inequalities are obtained in the two-dimensional case.