Shape of the entire low energy tail

A similar phenomenon - splitting of the electron cloud between silicon and silicon dioxide - should obviously take place for any cloud which originates within the distance R from the Si-SiO₂ interface, where R is the cloud radius. Since the number of the electron-hole pairs generated per eV of the incoming photon energy is significantly higher in silicon than in silicon dioxide, the amplitude of such events will change gradually from the primary peak down to the low energy peak, as the center of the cloud moves from silicon into oxide. This is schematically shown in Fig. 4.13, which explains how the low energy tail of the spectral redistribution function is formed.

  

Figure 4.13: Scheme of forming low energy tail from the electron clouds generated close to Si-SiO₂ interface.

Fig. 4.14 provides evidence that the entire low energy tail comes from the photons interacting within a small distance from the silicon interface. This figure shows the fraction of the events in the tail as a function of the characteristic absorption length $\lambda_{si}$ in silicon. In order to produce this plot events in the fluorescent and in the escape peaks were ignored, as well as the low energy (oxide) peak events discussed above. Each point in this plot corresponds to a measurement at a particular energy, and the points are connected sequentially in energy ascending order, starting at the lowest energy point labeled 1487 eV.

  

Figure 4.14: Fraction of single pixel events in tail as a function of characteristic absorption length in silicon.

The plot demonstrates that for widely separated energies (for instance, 1487, 1836, and 3600 eV) with the similar valaues of $\lambda_{si}$ the fraction of events in the tail is similar. Moreover, immediately above the silicon edge, at very small $\lambda_{si}$, the tail intensity goes up sharply. If one makes a very crude assumption that the cloud radius R is constant (or slowly changing function of energy in the range of interest), then an approximately $1/\lambda_{si}$dependence is an indication that the flat part of the tail results from the the photons interacting in a shallow region near the surface, whose thickness 2R is small compared to $\lambda_{si}$.

If all the events in the flat part of the low energy tail come from the electron clouds formed within a distance R from the Si-SiO₂ interface, the fraction of the events in the flat part of the tail can be used to determine the electron cloud size. The results of such calculations are shown on Fig. 4.15. The values of cloud sizes are significantly lower than 150 - 200 nm [Scholze and Ulm1994,Lechner and Struder1995]. They also are much smaller than cloud sizes in the bulk silicon extracted from our mesh experiments [Pivovaroff et al.1998]. We believe the reason for this large discrepancy is the presence of a potential barrier for electrons at the Si - SiO₂ interface. Electron cloud generation starts with emission of relatively high energy primary photo- and Auger electrons which dissipate their energy in an ionization cascade. The range of electrons in the initial stages of the cascade is roughly consistent with the cloud radius in our model. It is the low energy, nearly thermalized electrons with sharply increased mean free path (of an order of 100 nm), that are responsible for the large cloud sizes quoted in Scholze and Ulm (1994) and Lechner and Struder (1995). In the MOS structure analyzed here a ``potential wall'' at the Si - SiO₂ interface prevents the low energy electrons from penetrating into the SiO₂ for the clouds centered at a distance greater than R from this interface. Only hot electrons in the initial stages of cascading (for which the range is very small) can participate in the cloud splitting between the silicon and the silicon dioxide. This feature of the buried channel MOS X-ray CCDs is very beneficial for the spectrometric properties of the frontside illuminated CCDs, since it results in larger fraction of counts going into the main peak instead of the tail, and, hence, better energy resolution and quantum efficiency.

  

Figure 4.15: Cloud radius in silicon as a function of energy.

The precise shape of the tail depends on the density distribution of charge in the cloud and this distribution in principle can be extracted. We have not accomplished this task yet, because it requires much higher number of counts than we were able to obtain during the limited time at the synchrotron facility.

We have developed a model based on the scheme shown at Fig. 4.13. The basic assumption is that each electron cloud is a sphere and when the sphere crosses the silicon interface, the number of electrons produced in each material is proportional to the volume of the corresponding segment of the sphere. Each material is assumed to have different electron-hole creation energy. This is equivalent to assuming that electrons have the same mean free path in both materials. This may be not such a bad approximation, especially for high energy electrons in the original stages of cascading. As discussed above, electrons liberated in the oxide contribute to the total charge collected. From the low energy peak position (see Fig. 4.9) we deduce an effective electron-hole pair creation energy w_ox in SiO₂ of approximately 52 eV. This value includes recombination losses and hence differs significantly from the reported value of 17 eV/pair. We have made no attempt here to decouple the true value of w_ox and the losses. This matter is clearly worth pursuing, because it will allow to produce a more accurate model of the device response.

In Fig. 4.16 - 4.19 are shown the results of the best fit of the model to the data at several energies.

  

Figure 4.16: Response of the CCD to 1700 eV photons and the model prediction.

  

Figure 4.17: Response of the CCD to 1870 eV photons and the model prediction.

  

Figure 4.18: Response of the CCD to 2015 eV photons and the model prediction.

  

Figure 4.19: Response of the CCD to 4510 eV photons and the model prediction.

The results of this section are used as the basis for the model of the CCD described in section 4.14 to produce a response matrix for the flight devices.