Rate equations and scaling in pulsed laser deposition

We study a simplified model for pulsed laser deposition [Hinnemann et al., Phys. Rev. Lett. 87, 135701 (2001)] by rate equations. We consider a set of equations where islands are assumed to be pointlike, as well as an improved one that takes the size of the islands into account. The first set of equations is solved exactly but its predictive power is restricted to a few pulses. The improved set of equations is integrated numerically, is in excellent agreement with simulations, and fully accounts for the crossover from continuous to pulsed deposition. Moreover, we analyze the scaling of the nucleation density and show numerical results indicating that a previously observed logarithmic scaling does not apply.

Figure 1

(Color online) Nucleation density $n\left(t\right)$ and island density $N\left(t\right)$ obtained from simulations in the PLD limit for $I=0.001$.

Figure 2

(Color online) Island density in the PLD limit as a function of $I$ for the first three pulses $l=1,2,3$ obtained from simulations (points) and from relations (12, 28, 29) (lines).

Figure 3

(Color online) Comparison of the nucleation density obtained by simulations (points) and the rate equations (15, 16) (lines) with $I=0.001$ and, in the MBE limit, $R={10}^4$. Note that for comparing the two cases for given $I$, $R$ has to be chosen such that $R⪡I^{1/\left(2\gamma -1\right)}$.

Figure 4

(Color online) Data collapse to verify Eq. (6) obtained from numerical integration of Eqs. (36, 37) with $\alpha =1$ at the coverage $\theta =0.1$.

Figure 5

(Color online) Comparison between simulations (s) and numerical integration of the rate equations (36, 37) (r) of the nucleation density (upper panel) and mean island size (lower panel) in the PLD limit for the pulse intensities $I=0.0001$ (a), $I=0.0002$ (b), $I=0.0005$ (c), and $I=0.001$ (d).

Figure 6

(Color online) The probability distribution function (7) for $I=0.0001$ in the PLD limit obtained from simulations (s) and from a numerical integration of the rate equations (36, 37) (r).

Figure 7

(Color online) The probability distribution $p_s$, obtained from simulations, multiplied by mean island size $⟨s⟩$ as a function of $s/⟨s⟩$ for $I=0.0001$ and different values of $\theta$ (left); for $\theta =0.1$ and different values of $I$ (right).

Figure 8

(Color online) Data collapse of the normalized nucleation density, coming from simulations data, with the logarithmic scaling form. The inset shows the interval $0.2<\text{ln}\theta /\text{ln}I<0.25$ where the scaling function increases with the pulse intensity $I$.

Figure 9

(Color online) Nucleation density, obtained from simulations, as a function of $I$ for different values of $\theta$. The graph suggests that the exponent $\alpha$ increases with $\theta$.