Electric fields, weighting fields, signals and charge diffusion in detectors including resistive materials

In this report we discuss static and time dependent electric fields in detector geometries with an arbitrary number of parallel layers of a given permittivity and weak conductivity. We derive the Green's functions i.e. the field of a point charge, as well as the weighting fields for readout pads and readout strips in these geometries. The effect of 'bulk' resistivity on electric fields and signals is investigated. The spreading of charge on thin resistive layers is also discussed in detail, and the conditions for allowing the effect to be described by the diffusion equation is discussed. We apply the results to derive fields and induced signals in Resistive Plate Chambers, Micromega detectors including resistive layers for charge spreading and discharge protection as well as detectors using resistive charge division readout like the MicroCAT detector. We also discuss in detail how resistive layers affect signal shapes and increase crosstalk between readout electrodes

Passive quenching, signal shapes, and space charge effects in SPADs and SiPMs

In this report we study the dynamics of passive quenching in a single-photon avalanche diode. Our discussion is based on a microscopic description of the electron-hole avalanche coupled to the equivalent circuit of the device, consisting of the quench resistor and the junction capacitance. Analytic expressions for the resulting signal shape are derived from this model for simple electric field configurations, and efficient numerical prescriptions are given for realistic device geometries. Space charge effects are included using simulations. They are shown to distort the signal shape, but alter neither its basic characteristics nor the underlying quenching mechanism

The statistics of electron-hole avalanches

Charge multiplication through avalanche processes is commonly employed in the detection of single photons or charged particles in high-energy physics and beyond. In this report, we provide a detailed discussion of the properties of avalanches driven by two species of charge carriers, e.g. electrons and holes in a semiconductor exposed to an electric field. We derive equations that describe the general case of avalanches developing in inhomogeneous electric fields and give their analytical solutions for constant fields. We discuss consequences for the time resolution achievable with detectors that operate above the breakdown limit, e.g. single-photon avalanche diodes (SPADs) and silicon photomultipliers (SiPMs). Our results also describe avalanches that achieve finite gain and are important for avalanche photodiodes (APDs) and low-gain avalanche detectors (LGADs)

An extension of the Gluckstern formulas for multiple scattering: analytic expressions for track parameter resolution using optimum weights

Momentum, track angle and impact parameter resolution are key performance parameters that tracking detectors are optimised for. This report presents analytic expressions for the resolution of these parameters for equal and equidistant tracking layers. The expressions for the contribution from position resolution are based on the Gluckstern formulas and are well established. The expressions for the contribution from multiple scattering using optimum weights are discussed in detail

Signals induced on electrodes by moving charges, a general theorem for Maxwell's equations based on Lorentz-reciprocity

We discuss a signal theorem for charged particle detectors where the finite propagation time of the electromagnetic waves produced by a moving charge cannot be neglected. While the original Ramo-Shockley theorem and related extensions are all based on electrostatic or quasi-electrostatic approximations, the theorem presented in this report is based on the full extent of Maxwell's equations and does account for all electrodynamic effects. It is therefore applicable to all devices that detect fields and radiation from charged particles