Double Injection in Insulators

Abstract

Double injection in insulators is analyzed taking into account that the lifetimes for the injected electrons and holes are different and vary with injection level. Assuming charge neutrality, a detailed solution is obtained for the simple model of an insulator with a single set of recombination centers filled with electrons in thermal equilibrium. The major results are: (i) There is a threshold voltage ${V}_{\mathrm{th}}$ below which the double-injection current is negligible and at which this current rises steeply with voltage. At this threshold voltage the hole transit time, ${t}_{p,\mathrm{th}}=\frac{{L}^{2}}{{\ensuremath{\mu}}_{p}{V}_{\mathrm{th}}}$, is comparable to the hole lifetime ${\ensuremath{\tau}}_{p,\mathrm{low}}$. The subscript "low" refers to the lifetime at low injection levels. The lifetime at high injection levels in this model will generally be longer. (ii) For an electron capture cross section ${\ensuremath{\sigma}}_{n}$ much smaller than the hole capture cross section ${\ensuremath{\sigma}}_{p}$, there is a negative resistance between ${V}_{\mathrm{th}}$ and ${V}_{M}\ensuremath{\approx}(\frac{{\ensuremath{\sigma}}_{n}}{{\ensuremath{\sigma}}_{p}}){V}_{\mathrm{th}}$. With increasing current, the voltage decreases from ${V}_{\mathrm{th}}$ to ${V}_{M}$. This negative resistance has its origin in an increasing hole lifetime with increasing injection level, owing to electron depopulation of the recombination centers by hole capture. (iii) At still higher currents the double-injection current-voltage characteristic is similar to that for a semiconductor at high injection levels. At sufficiently low currents the neutrality-based, double-injection solution is no longer self-consistent with respect to the neglect of space charge, and the true current is a one-carrier SCL (space-charge-limited) current which, for the simple model analyzed, is the electron SCL current for a trap-free insulator. In real insulators the one-carrier SCL current may mask the voltage threshold effect, (i) above, depending on the physical parameters of the crystal. On the other hand, under the condition specified in (ii) above, the negative resistance will always be observed. Experimentally, the negative resistance should produce either current oscillations or a hysteresis in the voltage vs current for dc applied voltages. Both effects have been widely observed in insulators and high-resistivity semiconductors. It is shown how the theory can be extended to more complicated models.