Murray A. Lampert

An ohmic contact between a metal and an insulator facilitates the injection of electrons into the insulator. Subsequent flow of the electrons is space-charge limited. In real insulators the trapping of electrons in localized states in the forbidden gap profoundly influences the current flow. The interesting features of the current density-voltage ($J\ensuremath{-}V$) characteristic are confined within a "triangle" in the $logJ\ensuremath{-}logV$ plane bounded by three limiting curves: Ohm's law, Child's law for solids ($J\ensuremath{\propto}{V}^{2}$) and a traps-filled-limit curve which has a voltage threshold and an enormously steep current rise. Simple inequalities relating the true field at the anode to the ohmic field facilitate qualitative discussion of the $J\ensuremath{-}V$ characteristic. Exact solutions have been obtained for an insulator with a single, discrete trap level in a simplified theory which idealizes the ohmic contact and neglects the diffusive contribution to the current. The discrete trap level produces the same type of nonlinearity discovered by Smith and Rose and attributed by them to traps distributed in energy.

Received 14 November 1958DOI:https://doi.org/10.1103/PhysRevLett.1.450©1958 American Physical Society

Current injection in insulators provides a valuable technique for obtaining such information about defect states in insulators as their density, their energetic location in the forbidden gap, and their cross sections for capture of free electrons and holes. It also enables determination of free carrier drift mobilities. The physical principles underlying one-carrier space-charge-limited current injection, both steady-state and transient, and steady-state, two-carrier, largely neutralized current injection are discussed. Experimental verifications of the different types of current injection are also discussed briefly.

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.