2.5.1 Simulation of a p⁺n Diode

In the first example AVC scans of a p⁺n diode are simulated for beam currents varying from 10 pA up to 1 nA. The AVC potential $\varphi_{\mathrm{AVC}}^{}$ extracted from these simulations is shown in Fig. 2.11. For comparison the built-in potential is shown in this plot too. The equilibrium potential is nearly identical to the potential for a beam current of 10 pA or 100 pA.

For increasing beam currents the number of electron-hole pairs generated by the injected electrons becomes much higher than the equilibrium carrier concentration on the lower doped n-side of the diode. The secondary electrons and holes are separated very quickly by the strong electric field in the vicinity of the pn-junction. Recombination in the depletion region can be neglected because the extension of the depletion region is below 1 $\mu$m and the average carrier velocities are of the order of 10⁶ m s^{- 1}. This gives a time constant of approximately 1 ps which is much smaller than the average recombination time for Si which is approximately 1 $\mu$s. The holes which are the minorities on the lower doped n-side are pulled across the junction by the built-in electric field leaving back the electrons. This causes a reduction of the potential difference across the pn-junction and of the slope at the pn-junction.

On the higher doped side of the pn-junction the perturbation of the majority carrier concentration is negligible. Therefore the potential shows only a small change.

Figure 2.11: AVC potential of a p⁺n diode.

The second derivative of the AVC potential is shown in Fig. 2.12. From this figure it can be seen that the location where the second derivative equals zero shifts towards the lower doped side with increasing beam current. Fig. 2.13 shows the shift of the location where the second derivative of the AVC potential equals zero as function of the beam current.

Figure 2.12: Second derivative of the AVC potential of a p⁺n diode. The metallurgical junction is located at x = 0.4 $\mu$m.

Figure 2.13: Shift of the location where the second derivative of the AVC potential of a p⁺n diode equals zero as function of the electron beam current.