Analytical expressions for the calculation of pixel detector capacitances

In this paper, analytical expressions for calculating the capacitance of an isolated pixel diode as well as a pixel diode in an array are presented. These expressions fit the three-dimensional numerical solution of the Laplace equation better than 10% as previously calculated for these diodes, covering most practical cases. The total pixel capacitance C_p is obtained by a simple substitution of parameters in the analytical expressions. In addition, a simple method to obtain the real diode capacitance, the parasitic capacitance, and the substrate concentration is proposed on the basis of the analytical equation for the capacitance for an isolated diode. The coincidence of measured and calculated data for the capacitance is shown. The procedures presented permit a very precise description of the real C-V curve for an isolated diode as well as for diodes in a pixel array, allowing the design of the best coupling between each pixel and its corresponding readout preamplifier with a minimum calculation effort