Numerical analysis of quantum dots on off-normal incidence ion sputtered surfaces

We implement substrate rotation in a $2+1$-dimensional solid-on-solid model of ion-beam sputtering of solid surfaces. With this extension of the model, we study the effect of concurrent rotation, as the surface is sputtered, on possible topographic regions of surface patterns. In particular, we perform a detailed numerical analysis of the time evolution of dots obtained from our Monte Carlo simulations at off-normal-incidence sputter erosion. We found the same power-law scaling exponents of the dot characteristics for two different sets of ion-material combinations, without and with substrate rotation.

Figure 1

Possible topographies of the model, within the experimental constraints considered in Ref. 10. $t=3$, $a=6$, and $\theta =50°$. From left to right columns, $\sigma =1$, 3, and 5, respectively. From bottom row to top row, $\mu =0.5$, 1, 1.5, 2, and 5, respectively.

Figure 2

Profiles obtained from simultaneous sputtering and rotation using the same parameters as in Fig. 1. $t=3$, $a=6$, and $\theta =50°$. From left to right columns, $\sigma =1$, 3, and 5, respectively. From bottom row to top row, $\mu =0.5$, 1, 1.5, 2, and 5, respectively.

Figure 3

Structure factor of the lettered surface profiles in Fig. 2. $\mathbf{S}$ (relatively smooth), $\mathbf{H}$ (hole), $\mathbf{N}$ (nonoriented structures), and $\mathbf{D}$ (dot). $\mathbf{k}=0$ at the center and $\mathbf{k}=\frac{2\pi }{8}(\pm 1,\pm 1)$ at the corners.

Figure 4

Surface roughness $W$ with dots excluded. Main plot: $W$ as a function of $\mu$ for $\sigma =5$. Inset: $W$ as a function of $\sigma$ for $\mu =5$. $t=3$ ions/atom.

Figure 5

The coefficients, ${\nu }_{av}$ and ${\chi }_{av}$, of the isotropic version of Eq. (2) (Ref. 12) for the rotated case as functions of $\mu$. (a) $\sigma =1.0$, (b) $\sigma =3.0$, and (c) $\sigma =5.0$.

Figure 6

Time evolution of the $\mathbf{S}$ topography of Fig. 1 with rotation. $\sigma =3$, and $\mu =0.5$. [(a)–(d)] $t=3$, 40, 90, and 150, respectively.

Figure 7

Topography changes as a result of changing the angle of incidence $\theta$ for $\sigma =1$, $\mu =2$ (top row), and 5 (bottom row). Top row, from left to right $\theta =10°$, 30°, 40°, 50° and 80°. Bottom row, from left to right $\theta =5°$, 10°, 30°, 50°, and 80°.

Figure 8

(Color online) Surface roughness $W$ as a function of $\theta$ for profiles shown in Fig. 7 for $\sigma =1$ and $t=3$ ions/atom. Error bars are much smaller than symbol size except at $\theta =10°$ $(\mu =5)$. For this case for about half of the runs, the structure had already evolved to a nonoriented structure pattern with a considerable roughness, while the other half exhibited still a rather smooth surface. Hence, there seems to be a sharp transition in time from smooth to rough, where the transition time of finite samples fluctuates strongly.

Figure 9

Sample surface profiles for the unrotated case (top) and the rotated case (bottom) at $t=3$ ions/atom. As a visual check of our clustering algorithm, we show on the second column, the clusters (as defined in the text) formed from the corresponding profiles on the left. The scales indicate the surface height measured from the lowest height.

Figure 10

(Color online) Time evolution of (a) the average number $N_c$ and (b) the average height $H_c$ (both in lattice units) of the sampled dots with time (in ions/atom). Symbols: open and closed circle (triangle) symbols denote data obtained with and without substrate rotation, respectively, for Ar-GaAs (Ne-Si).

Figure 11

(Color online) Time evolution of the average area of crosssection $A_c$ of the sampled dots as a function of time, with and without sample rotation. Symbols: open and closed circle (triangle) symbols denote data obtained with and without substrate rotation, respectively, for Ar-GaAs (Ne-Si).