Possible mechanism for the onset of step-bunching instabilities during the epitaxy of single-species crystalline films

A thermodynamically consistent continuum theory for single-species, step-flow epitaxy that extends the classical Burton-Cabrera-Frank (BCF) framework is derived from basic considerations. In particular, an expression for the step chemical potential is obtained that contains two energetic contributions—one from the adjacent terraces in the form of the jump in the adatom grand canonical potential and the other from the monolayer of crystallized adatoms that underlies the upper terrace in the form of the nominal bulk chemical potential—thus generalizing the classical Gibbs-Thomson relation to the dynamic, dissipative setting of step-flow growth. The linear stability analysis of the resulting quasistatic free-boundary problem for an infinite train of equidistant rectilinear steps yields explicit—i.e., analytical—criteria for the onset of step bunching in terms of the basic physical and geometric parameters of the theory. It is found that, in contrast with the predictions of the classical BCF model, both in the absence as well as in the presence of desorption, a growth regime exists for which step bunching occurs, except possibly in the dilute limit where the train is always stable to step bunching. In the present framework, the onset of one-dimensional instabilities is directly attributed to the energetic influence on the migrating steps of the adjacent terraces. Hence the theory provides a “minimalist” alternative to existing theories of step bunching and should be relevant to, e.g., molecular beam epitaxy of GaAs where the equilibrium adatom density is shown by Tersoff, Johnson, and Orr [Phys. Rev. B 78, 282 (1997)] to be extremely high.

Figure 1

(Color online) Three-dimensional schematic of a single monatomic step separating two adjacent terraces upon which adsorption-desorption occurs, accompanied by adatom diffusion.

Figure 2

A monatomic step idealized as a smoothly evolving curve $\mathcal{S}=\mathcal{S}\left(t\right)$ in the plane, separating the trace of the lower terrace ${\Omega }_{+}$ from its upper counterpart ${\Omega }_{-}$. Here, the step velocity is assumed purely normal, $\mathbf{v}=V\mathbf{n}$, and ${\mathcal{R}}_{+}$ $\left({\mathcal{R}}_{-}\right)$ is the intersection of the lower (upper) terrace with an arbitrary two-dimensional domain $\mathcal{R}$.

Figure 3

One-dimensional schematic view of a section of a two-terrace periodic array of parallel rectilinear steps.

Figure 4

The critical value ${\Theta }_{\mathrm{c}}$ of $\Theta =\frac{{\varrho }_{\mathrm{eq}}}{a{\varrho }^b}$, beyond which step pairing occurs, as a function of the dimensionless period $\Lambda$ for fixed $K_{+}$ and varying $K_{-}$. Note that ${\Theta }_c$ increases with $\delta =K_{+}-K_{-}=\frac{1}{\tau \sqrt{M}}(C_{+}-C_{-})$, i.e., the barrier for the onset of step bunching is higher the more pronounced the normal Ehrlich-Schwoebel effect is.

Figure 5

One-dimensional simulation of the evolution of a three-periodic train of steps without desorption, Eq. (37), with $L=1$, $C_{+}=1$, $C_{-}=0.8$, $F=0.1$, $M=2$, $a{\varrho }^b=0.1$, and $\Theta =0.15$. The initial configuration corresponds to three nonequidistant steps per unit length, extended by periodicity outside the unit interval. Here $\Theta >{\Theta }_c=0.05$ and step pairing occurs in finite time.

Figure 6

One-dimensional simulation of the evolution of a three-periodic train of steps without desorption, see Eq. (37). The parameter values and initial conditions are as in Fig. 5. Here $\Theta =0.03<{\Theta }_c=0.05$ and the steps tend to become equidistant in finite time.