Characterizing equilibrium in epitaxial growth

Using a kinetic model of epitaxial growth, we describe how geometry controls kinetic pathways through which external deposition influences the state of a vicinal surface. The state of the surface is determined by three key, adjustable parameters: the local step angle θ, the Péclet number P , and the single-bond detachment rate b̆. By scaling arguments in P , we find three steady-state regimes. In one regime, detailed flux balance approximately holds, so that the system is near equilibrium. In the other two regimes, geometric effects compete with deposition as the system is driven progressively out of equilibrium. Our analytical results are in excellent agreement with those of kinetic Monte Carlo simulations. Copyright c © EPLA, 2012 Epitaxial growth involves a competition between certain atomistic processes that disrupt equilibrium and others that tend to restore it [1]. It is often unclear whether such growth phenomena can be understood in terms of equilibrium principles; nonetheless, concepts such as the free energy and local chemical potential are often invoked in descriptions of epitaxial systems [2–7]. Kinetic models that are valid in and out of equilibrium provide an alternate perspective of surfaces. This can yield insight into the conditions necessary for the use of thermodynamic concepts. For crystal surfaces, an important question (in the context of a kinetic model) is therefore: when is an epitaxial system close to equilibrium, and how can experimental parameters be used to control the state of the system? The rates of deposition and kinetic processes at step edges are usually seen as the key factors controlling growth [2–6]. Changing the local geometry at a step edge can also favor certain kinetic processes [3], so that the microstructure of a step should play a critical role in determining the state of the system. (a)E-mail: [email protected] Our goal in this letter is to provide a criterion, in the context of a tractable model, that indicates when an epitaxial system is near equilibrium, as opposed to other kinetic steady states. We address two tasks. First, we define the state of the system by means of the kink density (number of atomic defects per unit length of a step). Second, we show how experimentally adjustable parameters, e.g., the local step angle θ and the Péclet number P ∝ F/De, can control what state the system is in (F is the external deposition rate, and De is a diffusivity associated with atomic motion at a step). In particular, we show how increasing θ favors a return to equilibrium by creating additional kinks for adatoms to attach to. Our work is motivated by the issue of how accurately experimental surface systems can be described by nearequilibrium theories based on the celebrated BurtonCabrera-Frank (BCF) model [3–6]. By starting with a more general kinetic model, which contains information about kinks, we aim to provide some insight into the conditions necessary for the validity of BCF-type theories. A broader goal of our work is to describe qualitative features of surfaces under high-growth conditions.