Multi-Scale Simulation of Nucleation and Growth of Nanoscale SiC on Si

Abstract

The main obstacle of implementing numerical simulations for the prediction of nucleation and epitaxial growth is the variety of physical processes with a considerable difference in time and spatial scales. During the growth of nanostructures and epitaxy deposition of atoms, surface and bulk diffusion, nucleation of two-dimensional and three-dimensional clusters, transitions from two dimensional to three dimensional growth, stress relaxation occur. Thus, it is challenging to describe all of them in the framework of a single physical model. In the present work a multi-scale simulation of the epitaxial growth of silicon carbide nanostructures on silicon using three numerical methods, namely Molecular Dynamics, kinetic Monte Carlo, and the Rate Equations was implemented. Molecular Dynamics was used for the estimation of kinetic parameters of atoms and stress fields at the surface, which are input parameters for the other simulation methods. Kinetic Monte Carlo simulations allowed investigating basic nucleation processes and the transition from two dimensional nucleation to three dimensional cluster growth as well as the ordering of nanoclusters. Furthermore the influence of impurities on the nucleation of nanoscale SiC was studied. The energy barriers values obtained in Molecular Dynamics and the physical model used in the rate equation simulations was validated by Kinetic Monte Carlo. The Rate Equation simulation allowed studying the growth process at larger time scales taking into account the surface stress fields. As a result, a full time scale description spanning over a large substrate area of the morphological and structural surface evolution during SiC formation on Si was developed.

Formation and Growth of Subsurface Cobalt Clusters in Copper

“…Parameter β l ( R l ) in Eq. takes into account the efficiency of capture of Co atoms by a large cluster and depends on the radius R l ( t ), as the number of attachment places on the edge of the cluster is determined by its perimeter 2 πR l ( t ) and the area of the “feeding” zone of the cluster; i.e., the area from which the Co atoms diffuse to the cluster depends on the surface area of the cluster . The attachment rate of the atoms from region l to the clusters in the adjoining region is calculated only for the attachment of surface Co atoms to the subsurface Co clusters (process 6s).…”

“…In this case, by analogy with Refs. , it can be assumed that the binding energy of the Co atom at the cluster boundary does not depend on the number of atoms in the cluster. The binding energy for the surface cluster is equal to ≈ 1.3 eV; for the subsurface cluster, value of the binding energy is unknown.…”

Kinetic simulation of the 3D growth of subsurface impurity nanoclusters during cobalt deposition onto a copper surface

“…The sink strength for cobalt atoms in the subsurface copper layers can be estimated as [7,9,[11][12][13][14] …”

“…Since cobalt clusters are covered with only one copper layer (i.e., the amount of copper in the clusters is limited by their size), for copper atoms that diffuse along the surface we can write (7) where is the radius of capture by a cluster at cop per atomic radius R(t) (short interatomic distances, as with ) [7,9,[11][12][13][14]. Expressing the number of cobalt atoms in the first subsurface layer of a cobalt cluster via its radius, and inserting the obtained expres …”

“…Energy is an addi tional barrier to migration during the diffusion of cobalt atoms under the copper surface, which depends on depth z at which the cobalt atom is located. The expressions for attachment probabilities and are obtained with allowance for the normalization of frequencies and (11) Within the classical theory of elasticity [15,16], we can estimate energies and as (12) where = 0.085 eV is the specific elastic energy at the cobalt cluster-copper interface. Energies and , obtained with respect to the lattice geometry, cluster shape, and energies of the cobalt-cobalt, cobalt-copper, and copper-copper bonds [17], are = 0.21 eV and = 0.28 eV.…”