Violation of neutrality and occurrence of $S$-shaped current-voltage characteristic for doped semiconductors under double injection

It is shown that violation of quasi-neutrality and its subsequent recovery (with an increase in the current density) may occur in doped $n$ layers of $p^+$–$n$–$n^+$ structures under double injection at a high injection level and for a certain combination of electrical parameters. The violation of quasi-neutrality leads to a significant increase in the voltage across the base and subsequent recovery of neutrality gives rise to sharp decrease in voltage drop, as a result of which an $S$-shaped current-voltage characteristic is formed. The characteristic threshold current density for this effect is proportional to the base doping level $N_d$.