2.3 The MINIMOS-NT AVC Models

The generic device simulator MINIMOS-NT was enhanced to be capable of simulating AVC measurements. Appropriate models were added to the simulator to account for the injection of electrons by the electron beam and the electron-hole pair generation caused by the incident high energy electrons.

For modeling the injection rate of primary electrons n_inj and the generation rate of secondary carriers n_sec and p_sec in lateral and vertical direction Gaussian distributions were assumed which are good approximations for a wide range of primary electron energies [14].

$${\frac{n_{0}}{2\cdot\pi\cdot\sigma_{\mathrm{l}}\cdot\sigma_{\mathrm{v}}}} \left(\vphantom{\frac{(l - \mu_{\mathrm{l}})^{2}}{2\cdot\sigma_{\mathrm{l}}^{2}} + \frac{(v - \mu_{\mathrm{v}})^{2}}{2\cdot\sigma_{\mathrm{v}}^{2}}}\right. {\frac{(l - \mu_{\mathrm{l}})^{2}}{2\cdot\sigma_{\mathrm{l}}^{2}}} {\frac{(v - \mu_{\mathrm{v}})^{2}}{2\cdot\sigma_{\mathrm{v}}^{2}}} \left.\vphantom{\frac{(l - \mu_{\mathrm{l}})^{2}}{2\cdot\sigma_{\mathrm{l}}^{2}} + \frac{(v - \mu_{\mathrm{v}})^{2}}{2\cdot\sigma_{\mathrm{v}}^{2}}}\right)$$ (2.60)

Here l and v are the lateral and vertical coordinates, respectively. $\mu_{\mathrm{l}}^{}$ is the lateral position of the center of the primary electron beam. $\mu_{\mathrm{v}}^{}$ is the vertical distance of the center of the distribution from the semiconductor surface. $\sigma_{\mathrm{l}}^{}$ and $\sigma_{\mathrm{v}}^{}$ are the lateral and vertical standard deviations, respectively. The vertical distance of the maximum of the distribution function is of the same order of magnitude as the standard deviation in vertical direction. Therefore a considerable part of the distribution function where the function value is significant lies outside the semiconductor and the total number of electrons injected per second is less than n₀.

$\mu_{\mathrm{v}}^{}$, $\sigma_{\mathrm{l}}^{}$, and $\sigma_{\mathrm{v}}^{}$ are functions of the energy of the incident electrons E_prim. Estimates for these parameters were taken from Monte Carlo simulations performed by the program SESAME [15]. For typical primary electron energies of several keV the distance of the maximum from the surface and the corresponding lateral and vertical standard deviations are of the order of 100 nm.

The parameters of the distribution function of the generated secondary electrons n_sec and holes p_sec can be approximated by the parameters for the injected electrons n_inj because the electron-hole pairs are generated mainly by the injected primary electrons and not by secondary carriers. This is equivalent with the assumption that the average kinetic energy of the primary electrons is much higher than the average kinetic energy of the secondary carriers.

$$\gg$$ (2.61)

For typical devices the size of the generation region is small compared to the minority carrier diffusion length L_$\nu$ and compared to the dimensions of the device.

$$\nu \sqrt{D_{\nu}\cdot\tau_{\nu}}$$ (2.62)

In this equation $\tau_{\nu}^{}$ is the minority carrier lifetime and D_$\nu$ is the minority carrier diffusivity which can be calculated from the carrier mobility $\mu_{\nu}^{}$ by the Einstein relation.

$${\frac{D_{\nu}}{\mu_{\nu}}} {\frac{\ensuremath{\mathrm{k_{B}}}\cdot T}{\mathrm{q}}}$$ (2.63)

Typical values for the mobility and the lifetime for electrons in silicon are 1360 cm²s^{- 1} and 10^{- 6} s, respectively. This results in a typical diffusion length of approximately 60 $\mu$m at room temperature.

Therefore it is sufficient to use rough approximations for the lateral and vertical standard deviations. The most important parameter of the model for the secondary electron-hole pairs is the average number of pairs generated per incident electron. For Si in the average an energy of 3.65 eV is necessary to generate an electron-hole pair.

The diameter of electron beams used for measurements is in the order of 100 nm.

The effects caused by the surface potential barrier are neglected in the simulations because they are not relevant for the beam currents typically used for measurements.