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

• We present an analytical model for the space charge limited current in a thin film.
• We derive the two-dimensional version of the one-dimensional Mott–Gurney equation.
• We derive the current for an in-plane photoconductor with non-injecting contacts.
• We analyze the effects of carrier trapping on two-dimensional space charge limited currents.
• We validate the model on an in-plane organic heterojunction photoconductor.

We extend the one-dimensional space-charge limited current theory to a two-dimensional 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 equation J=98μ∊V2L3 is replaced by a similar K=2πμ∊V2L2 equation. For extraction of photo-generated carriers the one-dimensional J∼g3/4V1/2J∼g3/4V1/2 dependence is replaced by a K∼g2/3V2/3K∼g2/3V2/3 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.

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