In mathematics and physics, surface growth refers to models used in the dynamical study of the growth of a surface, usually by means of a stochastic differential equation of a field.
Examples Popular growth models include:
KPZ equation Dimer model Eden growth model SOS model Self-avoiding walk Abelian sandpile model Kuramoto–Sivashinsky equation (or the flame equation, for studying the surface of a flame front) They are studied for their fractal properties, scaling behavior, critical exponents, universality classes, and relations to chaos theory, dynamical system, non-equilibrium / disordered / complex systems. Popular tools include statistical mechanics, renormalization group, rough path theory, etc.
Kinetic Monte Carlo surface growth model
Kinetic Monte Carlo (KMC) is a form of computer simulation in which atoms and molecules are allowed to interact at given rate that could be controlled based on known physics. This simulation method is typically used in the micro-electrical industry to study crystal surface growth, and it can provide accurate models surface morphology in different growth conditions on a time scales typically ranging from micro-seconds to hours. Experimental methods such as scanning electron microscopy (SEM), X-ray diffraction, and transmission electron microscopy (TEM), and other computer simulation methods such as molecular dynamics (MD), and Monte Carlo simulation (MC) are widely used.
How KMC surface growth works
1. Absorption process First, the model tries to predict where an atom would land on a surface and its rate at particular environmental conditions, such as temperature and vapor pressure. In order to land on a surface, atoms have to overcome the so-called activation energy barrier. The frequency of passing through the activation barrier can by calculated by the Arrhenius equation:
A i n = A 0 , i n exp ( − E a , i n k T ) {\displaystyle A_{in}=A_{0,in}\exp \left(-{\frac {E_{a,in}}{kT}}\right)}
where A is the thermal frequency of molecular vibration, E a {\displaystyle E_{a}} is the activation energy, k is the Boltzmann constant and T is the absolute temperature.
2. Desorption process When atoms land on a surface, there are two possibilities. First, they would diffuse on the surface and find other atoms to make a cluster, which will be discussed below. Second, they could come off of the surface or so-called desorption process. The desorption is described exactly as in the absorption process, with the exception of a different activation energy barrier.
A o u t = A 0 , o u t exp ( − E a , o u t k T ) {\displaystyle A_{out}=A_{0,out}\exp \left(-{\frac {E_{a,out}}{kT}}\right)}
For example, if all positions on the surface of the crystal are energy equivalent, the rate of growth can be calculated from Turnbull formula:
V c = h C 0 ( A o u t − A 0 , o u t ) = h C 0 exp ( − E a , i n k T ) ⋅ ( 1 − exp ( − Δ G k T ) ) {\displaystyle V_{c}=hC_{0}(A_{out}-A_{0,out})=hC_{0}\exp \left(-{\frac {E_{a,in}}{kT}}\right)\cdot \left(1-\exp \left(-{\frac {\Delta G}{kT}}\right)\right)}
where V c {\displaystyle V_{c}} is the rate of growth, ∆G = Ein – Eout, Aout, A0 out are frequencies to go in or out of crystal for any given molecule on the surface, h is the height of the molecule in the growth direction and C0 the concentration of the molecules in direct distance from the surface.
3. Diffusion process on surface Diffusion process can also be calculated with Arrhenius equation:
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