Charge collection in semi-insulator radiation detectors in the presence of a linear decreasing electric field

In 1932 Hecht obtained his famous equation concerning the charge induced on the plates of a planar radiation detector in the presence of a uniform electric field. It is well known that in many cases, due to non-ohmic contacts or, in any case, in the presence of spatial charge, the internal electric field is no longer constant, so this formula could lead to wrong conclusions. In this article the authors examine the common case of an electric field decreasing linearly along the detector thickness. This is a very interesting case because this shape of field is fairly widespread in the presence of diffused spatial charge and the functional dependence of the collected charge on varying the applied bias, in some cases, is similar to the Hecht equation. The authors believe that this model could be an important instrument for interpreting the data arising from pulsed photocurrent measures. Starting from Ramo-Shockley theorem and under the same Hecht's hypotheses (except for the uniform field), we calculate the new relation between the collected charge and the applied bias.