Effects of Diffusion on Double Injection in Insulators

The theoretical treatment of double injection in insulators has been extended to include both diffusion and field terms. The effect of diffusion is shown to influence strongly both the form and the magnitude of the current-voltage characteristics, even for the case where the sample length $L$ is much greater than the ambipolar diffusion length $L_a$. The effects of diffusion showed themselves primarily in the shape of the density distribution $n\left(x\right)\mathrm{}$ and in the necessity of assuming more realistic boundary conditions. A family of numerical and approximate solutions have been found for the reduced carrier and potential distributions which depend on only one parameter, the minimum reduced carrier density ${\eta }_0$. For ${\eta }_0<1\mathrm{}$, the carrier distributions are characterized by an exponentially decreasing region from the boundaries in which $n\propto e^{\left|\frac{x}{L_a}\right|}$, merging into a more slowly varying, Lampert-like distribution. For this case ${{\eta }_0}^{-\frac{3}{2}}$ is approximately proportional to the length of the Lampert-like region measured in units of $L_a$. For ${\eta }_0>1\mathrm{}$, the carrier distributions approach rapidly that found for the pure diffusion case, $n\propto \frac{coshx}{L_a}$. The appropriate boundary conditions are given by $n\propto J{L}_a$, which allows the current-voltage characteristics to be determined with $\frac{L}{L_a}$ appearing as a parameter. It was found that $J$ rises at least as fast as $V^m$, where $m=\frac{3}{(1-\frac{10L_a}{3L})}$, compared with $J\propto V^3$ predicted by Lampert. It was also found that $J$ was increased by an approximate factor of ${\left(\frac{L}{L_{\mathrm{eff}}}\right)}^5$, where $L_{\mathrm{eff}}=L-Q\times \left(2\mathrm{}{L}_a\right)$ and $Q$ is a slowly varying function of $J$ with values normally between 1.7 and 3.8. Even in the extreme case of $\frac{L}{L_a}=100\mathrm{}$, the magnitude of $J$ is increased by a factor of ≈1.25 and $J$ is rising at least as fast as $V^{3.1}$. Experimental results described in the second paper of this series confirm most of the conclusions of this work.