Space-charge limited surface currents between two semi-infinite planar electrodes embedded in a uniform dielectric medium

Abstract

We extend the one-dimensional space-charge limited current theory to a twodimensional geometry where current flows in a thin layer between two coplanar semi-infinite electrodes. It is shown that the surface charge density in the gap between the electrodes is the finite Hilbert transform of the in-plane component of the electric field. This enables us to derive analytical expressions for the field and charge density for single carrier injection and for photo-carrier extraction by solving a non-linear integral equation for the field. The analytical expressions have been verified by numerical calculations. For the in-plane geometry, the one-dimensional Mott-Gurney equationFor extraction of photo-generated carriers the one-dimensional J ⇠ g 3/4V 1/2 dependence is replaced by a K ⇠ g 2/3 V 2/3 dependence, where g is the generation rate of photo-carriers. We also extend these results to take into account trapping. We show experimental evidence obtained with an organic photoconductor confirming the predicted voltage, width and generation dependencies.

Geometry dependence of 2-dimensional space-charge-limited currents

“…Similar behavior has been observed in a planar two-dimensional (2D) setting in organic thin films [2,3,4,5] and more recently in several types of monolayers [6,7]. In [8] we derived the following 2D version of eq. ( 1) for an infinitesimally thin layer between two semi-infinite co-planar electrodes…”

“…We note that the specific variation of the field components in eq. ( 53), has been confirmed by a more realistic numerical model for symmetric infinite "strip" electrodes [8].…”

“…There exist some experimental evidence for the voltage and length dependence in (3): see [8] for experiments with an organic photoconductor and [6] for experiments on a hexagonal-BN monolayer. Unfortunately since in these experiments the mobility usually is not known it's not possible to extract a value for the prefactor.…”

“…is a constant within the gap −1 < x < 1 and zero elsewhere. This type of equation can be solved analytically [8,11] but to obtain α the explicit solution is not needed (in section 5 we explain how the field components can be obtained). It suffices to integrate (8) over the gap after removing possible singularities.…”

“…This result was found by solving analytically a boundary value problem for the square of the complex electric field. In our method [8] the problem is reduced to solving a non-linear integral equation with a known solution, which was published by Peters [11]. We will also show how these two methods are related.…”