Volume-Controlled, Two-Carrier Currents in Solids: The Injected Plasma Case

TLDR

Double injection into semiconductors and insulators is examined under conditions where the injected carriers form a free plasma, the current is volume‑controlled by distributed space charge, and the transport is field‑driven with negligible diffusion. The study finds that in extrinsic semiconductors the current follows a J∝V² law over an extended voltage range and is depressed at fixed voltage by increased thermal minority carriers (J∝|n_T−p_T|), a behavior driven by recombination kinetics, while in insulators the current scales as J∝V³, and a rigorous constant‑lifetime solution—also applicable to variable‑lifetime cases such as bimolecular recombination—provides a good approximation for both materials.

Abstract

Double injection into semiconductors and insulators is studied under conditions where the injected electrons and holes are free (injected plasma), the current is volume-controlled, i.e., determined by distributed space charge, and the current is field-driven (diffusion negligible). The major results are, assuming a one-dimensional geometry and carrier lifetime independent of injection level, for extrinsic semiconductors, (i) an extended voltage region over which $J\ensuremath{\propto}{V}^{2}$ ($J$ current density and $V$ voltage), and (ii) depression of the current, at fixed voltage, in the square-law region through increase in the number of thermal minority carriers, $J\ensuremath{\propto}|{n}_{T}\ensuremath{-}{p}_{T}|$, with ${n}_{T}$, ${p}_{T}$ the thermal-equilibrium densities of electrons and holes, respectively. This unusual behavior is shown to be a direct consequence of recombination kinetic requirements. For insulators, assuming trapping is negligible, $J\ensuremath{\propto}{V}^{3}$. A rigorous solution is obtained for the constant-lifetime problem, valid for both semiconductors and trap-free insulators. This solution furnishes a good approximation also for variable-lifetime cases, e.g., bimolecular recombination kinetics.