## Morphology of growing surfaces

Molecular Beam Epitaxy is a technique for growing metal and semiconductor structures allowing a control of the thickness on the atomic scale.

The main goal of the theoretical investigation of the growth process is to understand how the morphology on mesoscopic scale depends on microscopic processes (deposition, surface diffusion, nucleation, evaporation).

#### Submonolayer growth

Once deposited on the substrate, atoms diffuse, until they are incorporated by preexisting steps or they meet other diffusing atoms giving rise to islands, that act as aggregation centers for diffusing atoms. The typical distance between islands in the submonolayer regime is the diffusion length

where D is the diffusion coefficient, F is the deposition flux and the exponent depends on the size of the stable island nucleus.

If the substrate is not exactly oriented along a high-symmetry direction, it contains steps and atoms attached to them, producing a step-flow growth, typical of vicinal surfaces.

#### Multilayer regime

It is often desirable to obtain smooth surfaces, but this is not possible in many cases, due to instabilities intrinsic to the growth process. There are several types of instabilities, distinguished in two broad classes depending on their thermodynamic or kinetic origin.

- Thermodynamic instabilities are due to the difference of lattice constant (lattice mismatch) between the substrate and the adsorbate. They are exploited to create quantum dots.
- Kinetic instabilities are due to the nonequilibrium nature of the growing surface. The most prominent example is the instability due to the reduced interlayer diffusion (Ehrlich-Schwoebel barrier) that gives rise to the structures shown in the figure for Pt(111).

A theoretical approach to the Ehrlich-Schwoebel instability is done via a Langevin equation for the dynamics of the surface

where represents the atom current along the surface. The Ehrlich-Schwoebel effect determines the presence of an **unstable** contribution to the current

A more microscopic approach to the growth process must include the analysis of the nucleation process. This is important to understand both the island morphology of the submonolayer regime, and the mounded profile of the multilayer regime.

#### Our publications

- Claudio Castellano and Paolo Politi,

*“Spatio-temporal distribution of nucleation events during crystal growth.”*

Phys. Rev. Lett.**87**056102, (2001).

- Paolo Politi and Claudio Castellano,

*“The process of irreversible nucleation in multilayer growth. II. Failure of the mean-field approach.”*

Phys. Rev. E**66**, 031605 (2002).

- Paolo Politi and Claudio Castellano,

*“The process of irreversible nucleation in multilayer growth. II. Exact results in one and two dimensions.”*

Phys. Rev. E**66**, 031606 (2002).

- Paolo Politi and Claudio Castellano,

*“Irreversible nucleation in Molecular Beam Epitaxy: From theory to experiments.”*

Phys. Rev. B**67**, 075408 (2003).

- Daniele Vilone, Claudio Castellano and Paolo Politi,

*“Nucleation and step-edge barriers always destabilize step-flow growth of a vicinal surface.”*

Surf. Sci. Lett.**588**, L227 (2005).

- Daniele Vilone, Claudio Castellano and Paolo Politi,

*“Breakdown of metastable step-flow growth of vicinal surfaces induced by nucleation.”*

Phys. Rev. B (in press), cond-mat/0508149.