On the correctness of mathematical models of diffusion and cathodoluminescence

Mathematical models of diffusion and cathodoluminescence of nonequilibrium minority charge carriers generated by a wide electron beam in homogeneous and multilayer semiconductor materials are considered. The use of wide electron beams makes it possible to reduce these problems to one–dimensional ones and to describe these mathematical models by ordinary differential equations. For the collective movement model, the corresponding mathematical model is:
\begin{equation*} D\frac{{d^2 \Delta p\left( z \right)}}{{dz^2 }} - \frac{{\Delta p\left( z \right)}}{\tau } = - \rho \left( z \right) \end{equation*}
\noindent with boundary conditions
\begin{equation*} D\;\frac{{d\Delta p\left( z \right)}}{{dz}}\left| \begin{array}{l} \\ {\kern 1pt} z = 0 \\ \end{array} \right. = \upsilon _{s} \Delta p\left( 0 \right), \quad \Delta p\left( \infty \right)=0. \end{equation*}

\noindent Here the function $\rho \left( z \right)$ is the dependence on the coordinate $z$ of the density of minority charge carriers generated by an electron beam in a semiconductor target prior to their diffusion, $\Delta p\left( z \right)$ is the sought distribution of minority charge carriers after their diffusion, the remaining parameters for homogeneous materials are constants.
For the model of independent sources, the corresponding mathematical model is:
\begin{equation*} D\,\frac{{d^2 \Delta p\left( {z,\;z_0 } \right)}}{{dz^2 }} - \frac{{\Delta p\left( {z,\;z_0 } \right)}}{\tau } = - \rho \left( z \right)\delta \left( {z - z_0 } \right) \end{equation*}
\noindent with boundary conditions
\begin{equation*} \label{Dif. ur.-Gran. usl.-Independent} D\,\left. {\frac{{d\Delta p\left( {z,\;z_0 } \right)}}{{dz}}} \right|_{z = 0} = \upsilon _s \Delta p\left( {0,\;z_0 } \right), \quad \Delta p\left( {\infty ,\;z_0 } \right) = 0. \end{equation*}

\noindent Here the function $\Delta p\left( {z,z_0 } \right)$ describes the distribution over the depth of the minority charge carriers generated by a plane infinitely thin source located at a depth $z_0,$ $z_0 \in \left[ {0,\infty } \right).$ The distribution of nonequilibrium charge carriers $\Delta p(z)$ in this case is found as
\begin{equation*} \Delta p(z) = \int\limits_0^\infty {\Delta p\left( {z,\;z_0 } \right)dz_0 }. \end{equation*}

For both models, the intensity of cathodoluminescence $I$ taking into account absorption at a fixed radiation wavelength $\lambda$ was calculated as
\begin{equation*} I \sim \int\limits_{l_s}^\infty \Delta p\left( z \right) \exp \left[ -\alpha (\lambda) z \right] dz. \end{equation*}

The study of the considered models is carried out, including the proof of the uniqueness of solutions and the continuous dependence of solutions on the data of the problem. Estimates are obtained for solving the problems under consideration, which make it possible to use them in electron probe technologies.
In the case of one–dimensional diffusion into the n–layer final semiconductor structure $\left( {z \in \left[ {0,\;l} \right]} \right)$ the depth distribution of the minority charge carrier is found as a solution to the differential equations
\begin{equation*} D^{(i)} \left( z \right)\frac{{{\rm{d}}^2 \Delta p^{(i)} \left( z \right)}}{{{\rm{d}}z^2 }} - \frac{{\Delta p^{(i)} \left( z \right)}}{{\tau ^{(i)} \left( z \right)}} = - {\rm{\rho }}^{(i)} \left( z \right),{\rm{ }}i = \overline {1,n} \end{equation*}

\noindent with boundary conditions
\begin{equation*} D^{(1)} \,\left. {\frac{{{\rm{d}}\Delta p^{(1)} \left( z \right)}}{{{\rm{d}}z}}} \right|_{z = 0} = \nu _s ^{(1)} \Delta p^{(1)} \left( 0 \right), \quad D^{(n)} \,\left. {\frac{{{\rm{d}}\Delta p^{(n)} \left( z \right)}}{{{\rm{d}}z}}} \right|_{z = l} = - \nu _s ^{(n)} \Delta p^{(n)} \left( l \right). \end{equation*}
\noindent The superscript in parentheses indicates the layer number.
The possibilities of using this approach for multilayer structures with an arbitrary number of layers are discussed.