Analysis of layer growth kinetics
During real-time RBS one is measuring the thickness of some layer, , whose value is given by integrating the rate of change in over suitable limits,
where the model of the growth of the layer is contained in the rate equation in square brackets. Eq. (1) is independent of the model, and can therefore be used for constant temperature as well as ramped temperature anneals, provided of course that an appropriate model is used. The initial conditions are expressed as and often it is taken that , but this is not necessary as long as the thickness at time is known.
The rate equation at constant temperature
Let denote the rate equation. When the temperature remains constant over time, one can model the growth with a very general model that incorporates a full description of the kinetics. However, it is not apparent which of the competing models should be used. Fortunately, one almost always finds the limiting behaviour of either diffusion controlled growth or in some rare cases linear growth for film thicknesses in the range where RBS analysis is suitable ( nm).
Diffusion controlled kinetics
In the case of diffusion controlled kinetics, the growth rate is inversely proportional to the thickness of the growing layer and is given by
where is a constant that can be related to the interdiffusion coefficient of the growing layer. Integration yields
Linear reaction kinetics
In the case of linear reaction kinetics, the growth rate is assumed to be limited by the reactions taking place at the interface, and not on the rate at which they arrive (by diffusion, for example). Hence the rate equation is quite simply:
where is a constant that can be related to the rate of the reaction. Integration yields
The rate equation in a linear ramp
During a typical experiment the temperature ramp rate, , is kept constant during the time interval , and
eq. (1) is transformed to
with an activation energy (in diffusion control it is the apparent activation energy for the interdiffusion, while in linear growth it is the activation energy of the corresponding chemical reaction), is Boltzmann’s constant and is a constant that does not depend on time or temperature.