The Modified Hecht Equation for Pixel Detectors: A Methodological Approach to Evaluating Mobility-Lifetime Products

In this article, we propose a modified Hecht equation tailored for pixel detectors. Building upon the Shockley-Ramo theorem and the fundamental principles of the classic Hecht equation, our approach incorporates consideration for the nonlinear weighting of the electric field. The improved Hecht equation is designed to accurately model the mobility-lifetime product of carriers in pixel detectors under both low-energy and high-energy photon irradiation. In comparison to conventional methods based on the plane Hecht relation, our approach aligns more closely with the charge collection model of the pixel detector. Notably, it eliminates the restriction that the radiation used for mobility-lifetime product fitting must be deposited solely on the detector surface. This flexibility is particularly beneficial for evaluating detector performance postpackaging and integration. We validated our approach using an 11\times11 CdZnTe pixel detector, featuring an electrode area of 22\times22 mm and a thickness of 15 mm. The mobility-lifetime product of electrons and holes was successfully determined under both 59.5- and 661.6-keV photon irradiation. Strikingly, the fitting results from different radioactive sources exhibited high consistency, affirming the reliability of the improved Hecht equation. Finally, we compared the fitting results of the carrier mobility-lifetime product obtained in this study with the results obtained using the traditional Hecht equation and other results obtained in pixel detectors and analyzed the reliability of the results.