# An application of extensions of the Ramo-Shockley theorem to signals in silicon sensors

## TL;DR

This paper extends the Ramo-Shockley theorem to accurately calculate signals in silicon sensors with non-linear materials, finite conductivity, and space-charge, facilitating improved semiconductor sensor analysis.

## Contribution

It introduces a generalized formulation of the Ramo-Shockley theorem for complex semiconductor sensor environments, including non-linear and resistive materials.

## Key findings

- Derived a new method for calculating signals with time-dependent weighting fields.
- Provided an analytic example for un-depleted silicon sensors.
- Enhanced the practical application of the theorem in semiconductor device simulations.

## Abstract

We discuss an extension of the Ramo-Shockley theorem that allows the calculation of signals in detectors that contain non-linear materials of arbitrary permittivity and finite conductivity (volume resistivity) as well as a static space-charge. The readout-electrodes can be connected by an arbitrary impedance network. This formulation is useful for the treatment of semiconductor sensors where the finite volume resistivity in the sensitive detector volume cannot be neglected. The signals are calculated by means of time dependent weighting fields and weighting vectors. These are calculated by adding voltage or current signals to the electrodes in question, which has a very practical application when using semiconductor device simulation programs. An analytic example for an un-depleted silicon sensor is given.