Crossover from BKT-rough to KPZ-rough surfaces for interface-limited crystal growth/recession

The crossover from a Berezinskii–Kosterlitz–Thouless (BKT) rough surface to a Kardar–Parisi–Zhang (KPZ) rough surface on a vicinal surface is studied using the Monte Carlo method in the non-equilibrium steady state in order to address discrepancies between theoretical results and experiments. The model used is a restricted solid-on-solid model with a discrete Hamiltonian without surface or volume diffusion (interface limited growth/recession). The temperature, driving force for growth, system size, and surface slope dependences of the surface width are calculated for vicinal surfaces tilted between the (001) and (111) surfaces. The surface velocity, kinetic coefficient of the surface, and mean height of the locally merged steps are also calculated. In contrast to the accepted theory for (2 + 1) surfaces, we found that the crossover point from a BKT (logarithmic) rough surface to a KPZ (algebraic) rough surface is different from the kinetic roughening point for the (001) surface. The driving force for crystal growth was found to be a relevant parameter for determining whether the system is in the BKT class or the KPZ class. It was also determined that ad-atoms, ad-holes, islands, and negative-islands block surface fluctuations, which contributes to making a BKT-rough surface.


S1 Model
The model that the Monte Carlo treatment in this study uses is based on the RSOS model on a square lattice. Here, "restricted" means that the surface height difference between nearest neighbor sites is restricted to 0 or ±1. The surface Hamiltonian for the RSOS model is given by the following equation: where h(m, n) is the height of the surface at a site (n, m), ϵ is the microscopic ledge energy, N is the total number of the unit cells on the (001) surface, and E surf is the surface energy per unit cell. The RSOS condition is required implicitly. Here, ∆µ is introduced such that ∆µ = µ ambient − µ crys , where µ ambient and µ crys are the bulk chemical potential of the ambient and crystal phases, respectively. At equilibrium, ∆µ = 0; for ∆µ > 0, the crystal grows; whereas ∆µ < 0, the crystal recedes.
For the first-principles quantum mechanical calculations, E surf corresponds to the surface free energy, which includes entropy originating from lattice vibrations and distortions. Hence, E surf or ϵ slightly decreases as the temperature increases. However, E surf and ϵ are assumed to be constant throughout the work because we concentrate on the crossover phenomena of the surface roughness.

S2 Monte Carlo method
Vicinal surfaces of a (001) surface tilted towards the [111] direction are considered using the Monte Carlo method for non-conserved systems with the Metropolis algorithm. Here, "non-conserved" indicates that the number of crystals is not conserved. Atoms on the crystal surface can escape to the ambient phase or atoms in the ambient phase can be captured. The external parameters are temperature T , ∆µ, number of steps N step , and the linear size of the system L. The surface slope p = N step a/L = tan θ, where θ is the tilt angle from the ⟨001⟩ direction.
Initially, N step steps run in the mean directionỹ = ⟨110⟩, for which a periodic boundary condition is required. Thex direction is assigned to the ⟨110⟩ direction. The configurations on the lower height side (right side of the top-down view of the surface) are connected to the upper height side (left side of the top-down view of the surface) by adding N step a steps. We considered two types of initial surface configuration: a surface with a macrostep where all the steps are combined , and a surface with a train of steps. After 2 × 10 8 Monte Carlo steps per site (MCS/site), the results from both initial conditions agree well.
The lattice sites on the surface at which an event occurs are selected randomly and a "capture" or "escape" event for the lattice site is selected randomly with a probability of 1/2. The energy change between the states before and after an event is calculated using Eq. (1). If the energy change is negative, the surface configuration is updated with a probability of 1. If the energy change is positive, the surface configuration is updated with a probability of exp[−(E after −E before )/k B T ]. Here, E after and E before are the surface energies calculated using Eq. (1) after and before the surface configuration update, respectively.
The surface diffusion of atoms or volume diffusion are not taken into consideration. The advance and recession of an elementary step are respectively caused by the capture and escape of atoms at kinks on the step edges. In the following, snapshots of the top-down and side views of the simulated surfaces are shown.