# On the weighting field of irradiated silicon detectors

## TL;DR

This paper investigates how the weighting field in irradiated silicon detectors evolves over time, deriving a formula linking it to resistivity and showing that at high fluences, the weighting field becomes effectively time-independent, simplifying simulations.

## Contribution

It introduces a simple formula relating the time constant of the weighting field to the resistivity and low-field region extension in irradiated silicon sensors, and demonstrates the transition to a time-independent weighting field at high fluences.

## Key findings

- Time constant increases with resistivity and low-field region size.
- Weighting field becomes time-independent at high fluences.
- Time-independent approximation is valid for highly irradiated sensors.

## Abstract

The understanding of the weighting field of irradiated silicon sensors is essential for calculating the response of silicon detectors in the radiation environment at accelerators like at the CERN LHC. Using 1-D calculations of non-irradiated pad sensors and 1-D TCAD simulations of pad sensors before and after irradiation, it is shown that the time-dependence of the weighting field is related to the resistivity of low field regions with ohmic behaviour in the sensor. A simple formula is derived, which relates the time constant of the time-dependent weighting field, $\tau $, with the resistivity and the extension of the low-field region for pad detectors. As the resistivity of irradiated silicon increases with fluence and finally reaches the intrinsic resistivity, $\tau $ becomes much larger than the charge-collection time and the weighting field becomes essentially independent of time. The TCAD simulations show that the transition from a time-dependent to a time-independent weighting field occurs at a neutron-equivalent fluence of $ \approx 5 \times 10^{12}$ cm$^{-2}$ for a 200 $\mu$m thick pad diode operated at 40 V and $- 20^\circ$C. It is therefore concluded that the use of a time-independent weighting field calculated with the same method as for a fully-depleted non-irradiated sensor is also appropriate for the simulation of highly irradiated silicon sensors.