Analysis of layer growth kinetics

Introduction

During real-time RBS one is measuring the thickness of some layer, x, whose value is given by integrating the rate of change in x over suitable limits,

(1)   \begin{equation*} x=\int_{t_0}^t \left[\frac{\mathrm{d}x}{\mathrm{d}t}\right] \mathrm{d}t \end{equation*}

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 x(t_0)=x_0 and often it is taken that x_0\approx 0, but this is not necessary as long as the thickness at time t_0 is known.

The rate equation at constant temperature

Let R(t) = \left[\frac{\mathrm{d}x}{\mathrm{d}t}\right] 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 (5\ldots 1000 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

(2)   \begin{equation*} R(t) = R_0/x \end{equation*}


where R_0 is a constant that can be related to the interdiffusion coefficient of the growing layer. Integration yields

(3)   \begin{equation*} x^2-x_0^2 = 2R_0t \end{equation*}

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:

(4)   \begin{equation*} R(t) = R_0 \end{equation*}


where R_0 is a constant that can be related to the rate of the reaction. Integration yields

(5)   \begin{equation*} x-x_0 = R_0t \end{equation*}

The rate equation in a linear ramp

During a typical experiment the temperature ramp rate, \rho, is kept constant during the time interval (t_0,t), and

(6)   \begin{eqnarray*} T(t)&=&T_0+\rho t \\ \rho &=& \frac{\mathrm{d}T}{\mathrm{d}t} \end{eqnarray*}

Since,

(7)   \begin{equation*} \frac{\mathrm{d}x}{\mathrm{d}t} = \frac{\mathrm{d}x}{\mathrm{d}T} \cdot \frac{\mathrm{d}T}{\mathrm{d}t} \end{equation*}

eq. (1) is transformed to

(8)   \begin{equation*} x=\rho\int_{T_0}^T \left[\frac{\mathrm{d}x}{\mathrm{d}T}\right] \mathrm{d}T \end{equation*}

The rate constants, R_0, used in diffusion controlled growth (2) and linear growth (4), at constant temperature, usually have a temperature activated dependence, and generally

(9)   \begin{equation*} R_0 = R_0^{\prime} \exp\left( \frac{E_a}{k_BT}\right) \end{equation*}

with E_a 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), k_B is Boltzmann’s constant and R_0^{\prime} is a constant that does not depend on time or temperature.

The introduction of the temperature activated term into the rate equation in eq. (8) allows its solutions to be determined in terms of the exponential integral, \mathrm{Ei}.

Exponential integrals

The exponential integral, \mathrm{Ei} with real z>0 is defined as:

(10)   \begin{equation*} \mathrm{Ei}(1,z) = \int_1^\infty\frac{\exp(-\theta z)}{\theta} \mathrm{d}\theta \end{equation*}

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