Influence of microstructure on the transitions between mesoscopic thin-film morphologies in ballistic-diffusive models

We study the influence of the symmetries of competing microstructures on the emergence of different mesoscopic morphologies in the growth by vapor deposition of thin solid films. We show the results of numerical simulations in $\left(1+1\right)$- and $\left(2+1\right)$-dimensional systems including different microstructures, as well as thermally activated surface diffusion in combination with a ballistic algorithm to model the deposition process. We focus on the characterization of the transitional structures that appear in the empirical structure zone model (SZM) through the evaluation of the mean packing density and the mean coordination number. We show that the maximum coordination number of the underlying microstructure classifies the statistics of the transitional morphologies at the border between zone I in the SZM, characterized by the formation of fractal-like patterns, and zone II, where pronounced faceting develops. We analyze the appearance of lattice frustration and texture competition effects in complex microstructures having mutually exclusive symmetries.

Figure 1

Microstructures in the $\left(1+1\right)$-dimensional model used in our simulations with their active positions labeled. (a) Simple square lattice with a maximum coordination number 4; a particle is buried when active position 2 is occupied. (b) Simple hexagonal lattice with a maximum coordination 6; the particle is buried when positions 2 and 3 are occupied. (c) Double square lattice with a maximum coordination 4; the particle is buried when positions 2, 3, and 4 are occupied. (d) Two-dimensional projection of the face-centered-cubic lattice (square and hexagonal lattices mixed) with a maximum coordination of 6; a particle is buried when positions 2, 3, and 4 are occupied.

Figure 2

Three-dimensional microstructures used in the $\left(2+1\right)$ simulations. (a) A close-packed hexagonal lattice with a maximum coordination number of 12. (b) A superposition of three pure simple cubic lattices. Its maximum coordination number is 6. To aid the eye, nearest-neighbor positions lying on the rotated planes are plotted in different gray tones.

Figure 3

Illustration of the ballistic deposition process in a simple hexagonal microstructure: fixed substrate (dark circles), shadowed trajectories (dashed arrows), allowed trajectory and instantaneous relaxation (solid arrows) to the closest of the existing active positions (solid dots), and new active positions created around the deposited adatom (open dots).

Figure 4

Film structures obtained at different temperatures at a fixed deposition rate of 1 monolayer per second for each given microstructure; to aid the eye, each stripe represents a height of 10 monolayers.

Figure 5

Change in the mean column width (measured as a fraction of the total width of the substrate) of dense structures with the substrate temperature.

Figure 6

Three-dimensional film structures obtained after the deposition of $50000$ particles at a fixed deposition rate $R=1\text{ML}/\text{s}$. To aid the eye, layers containing 8000 deposited particles are plotted in alternated dark-light gray tones. Top row: superposition of three simple cubic lattice microstructure $\left(3d\text{-sc}\right)$ at reduced temperatures $T/T_m$: (a) 0,20; (b) 0,26; (c) 0,32. Bottom row: close-packed hexagonal microstructure $\left(3d\text{-hex}\right)$ at reduced temperatures $T/T_m$: (d) 0,20; (e) 0,26; (f) 0,32.

Figure 7

(a) Change in the bulk mean packing density $\rho$ with the substrate temperature $T$ obtained for different microstructures and at deposition rates $R=1,100\text{ML}/\text{s}$. The location of the maxima of their statistical fluctuations (inset) determines the characteristic temperature of each curve, $T_C$. (b) Mean packing density curves rescaled with the characteristic transition temperature $T_C$. Curves corresponding to microstructures sharing the same maximum coordination number collapse onto a single master curve.

Figure 8

The same as Fig. 7a but for the $\left(2+1\right)$-dimensional system. The location of the maxima of the statistical fluctuations (inset) determines the characteristic temperature of each curve, $T_C$.

Figure 9

The same as Fig. 7b but for the $\left(2+1\right)$-dimensional system. Curves corresponding to microstructures sharing the same maximal coordination number collapse onto a single master curve.

Figure 10

Change in the transition temperatures between zones I and II, rescaled with the substrate melting temperature $T_m$, for each microstructure and at different deposition rates $R$.

Figure 11

Total and bulk mean coordination number as a function of the substrate temperature for a deposition rate $R=1\text{ML}/\text{s}$ and their statistical fluctuations (inset): (a) sq and sq-sq microstructures; (b) hex and hex-sq microstructures.

Figure 12

Statistics for the surface adatoms: (a) mean coordination number and its statistical fluctuations (inset) versus the substrate temperature at a deposition rate $R=1\text{ML}/\text{s}$ for the different microstructures; (b) total number of surface adatoms under the same conditions.

Figure 13

Fraction of bonds in the hex-sq microstructure as a function of the substrate temperature where symmetries with a maximum coordination number compete. Deposition rate: $R=1\text{ML}/\text{s}$.

Figure 14

Texture competition in sq-sq microstructure. (a) Details of single lattice domains formed at intermediate temperatures $\left(R=1\text{ML}/\text{s},T=500\text{K}\right)$. Fraction of bonds in the horizontal and the oblique bonding symmetry as a function of the substrate temperature using (b) a horizontal substrate and (c) an oblique substrate.