Operational voltage of silicon heavily irradiated strip detectors utilizing avalanche multiplication effect

Recent results on the collected charge $Q_c$ in heavily irradiated Si strip detectors developed by RD50 collaboration for the high luminosity operation of the LHC (sLHC) showed a significant $Q_c$ enhancement if the detectors were operated at a bias voltage beyond 1000 V. To explain these results, a physical model based on carrier avalanche multiplication in high electric fields near a n$^+$-p junction was developed at the Ioffe Physical-Technical Institute. This study represents an extension of the model, whose goal is finding the detector bias voltage and the other operating conditions which lead to such charge enhancement. The electric field distribution and collected charge in heavily irradiated Si strip detectors are calculated at different conditions: bias voltage, temperature, radiation fluence and strip detector geometry. It is demonstrated that: a) the minimum operational voltage, which initiates the avalanche process, depends on the fluence (being 500 V and 1500 V for LHC and sLHC fluence range, respectively), b) in avalanche multiplication mode the electric field near the n$^+$-strips is stabilized via potential redistribution inside the detector and formation of a pronounced DP electric field profile and is about 300 kV/cm, and c) the model predicts that the maximum collected charge in detectors irradiated to $(2-5) \times 10^{15}$ \rm{n}_{eq}cm^{-2}$ is larger than in a non-irradiated detector that nicely agrees with the experimental data. Another factor, which gives an enhanced avalanche multiplication at lower bias voltage, is the electric field focusing near the strips which then also leads to an enhancement of $Q_c$.