Doping Concentration

With irradiation, both donors are removed and acceptor-like defects are generated throughout the bulk. This effect leads to a decrease of the effective bulk doping concentration

$$N_{\rm bulk}=\vert N_D-N_A\vert\quad,$$ (2.25)

and eventually, there are equal numbers of donors and acceptors. The effective doping concentration is zero then and the silicon behaves as if it were intrinsic. This state is known as the inversion point. With further irradiation, the acceptors begin to dominate, and the bulk material is now effectively of p-type. This implies that the pn-junction has moved to the backplane side. As the depletion voltage scales with the bulk doping concentration (eq. ), the bias voltage has to be adjusted during the irradiation process to ensure full depletion. Initially, the depletion voltage decreases to theoretically zero at the depletion point, and then rises with the effective bulk doping concentration. The fluence needed for inversion depends on the initial doping concentration. High-resistivity sensors have a low initial donor density and reach the inversion point with less fluence than those of low resistivity.

Figure: Effective doping concentration and depletion voltage of silicon detectors vs. $1\,\rm MeV$ neutron equivalent fluence [22]. The fluence needed to reach the inversion point depends on the initial resistivity.

Fig.  shows the development of the depletion voltage for silicon detectors of various initial resistivities and manufacturers over the equivalent fluence $\Phi_{\rm eq}$ of $1\,\rm MeV$ neutrons. Various attempts have been made on flattening the $dN/d\Phi$ slope, which is a parameter describing the radiation hardness of the material. While carbon contamination was identified to have a bad influence on the radiation induced doping concentration change, oxygen enriched silicon detectors tolerate a three times higher fluence of charged hadrons compared to standard material. However, no difference was observed regarding neutrons. The comparison of the doping concentration development between standard silicon and oxygen enriched material is shown in fig. .

Figure: Dependence of the effective bulk doping concentration on the $1\,\rm MeV$ neutron equivalent fluence for standard and oxygen enriched silicon [22].

The influence of inversion on the particle induced current has been modelled with the simulation discussed in section , p. , neglecting the efficiency decrease due to radiation. At the same effective doping concentrations before and after inversion, the depletion voltages are equal, but the triangular shape of the electric field flips, since the pn-junction moves from the readout to the backplane side. Thus, the current contributions of individual carriers are quite different, but nevertheless the sum currents of both electrons and holes are the same before and after inversion. Fig.  demonstrates this amazing feature by investigating a single electron-hole pair, five pairs and the real case, a large number of charge carriers.

Figure: Particle induced current waveforms before and after inversion at the depletion voltage. The top plot shows the normalized currents of 2 single charges, which are placed in the center, moving across the bulk to the electrodes. The center plot shows the same for 5 electrons and 5 holes distributed equally along the x axis. In the bottom plot, the same is drawn for a large number of charges, as it happens in a real detector. Gradually the current curves before and after inversion merge.