Description of the technique

This technique makes use of the following idea. Let us assume that CCD is illuminated with a monochromatic source of X-ray photons whose

  

Figure 4.51: Schematic of the CCD depleted layers.

characteristic absorption length $\lambda$ is smaller than, but comparable to the depletion depth. If an interaction of the photon with Si occurres in the depleted region of the CCD, the created electron cloud will be drawn into the potential wells by the electric field and all the charge will be collected (see Fig. 4.51).

Depending on where relative to pixel boundaries the photon landed the resulting event can be either a single-pixel or a split one (can be either horizontally or vertically split, or grade 6), but the sum of the amplitudes of the pixels containing a signal charge should account for all the charge generated by the photon, and, hence, be part of the main peak in the response histogram. On the other hand, if an interaction occurred in an undepleted bulk of the semiconductor, the charge cloud diffuses more or less uniformly in all directions. Part of the cloud drifting towards the back side of the device will be lost. Electrons moving towards the front surface enter the depletion region and being pulled by an electric field end up in the potential wells of the CCD. Due to initial diffusion of charge the registered event in this case is a widespread multipixel cloud. According to the ASCA grading scheme this is a grade 7 event and, since part of the initial charge is lost to the backside junction, the amplitude is lower than the peak energy even if all the pixels of the event are summed together.

If we denote N_d the number of interactions in the depleted region and N_und the number of events in the undepleted bulk, then, due to exponential distribution of the of the number of interacting photons as as a function of depth, a simple equation holds (see Fig. 4.52 for a better understanding):  

$$N_{und}=(N_d+N_{und}) \exp(- \frac {d_d}{\lambda})$$ (30)

where d_d is the depletion depth of the device.

  

Figure 4.52: Exponential distribution of the number of interacting photons as a function of distance from the surface.

This implies the following algorithm of the depletion depth calculation.

1. Sum the the intensities of the grades 0,1,2,3,4,6 and determine what is the number of events N_d in the main peak. Find the peak center E_c and width $\sigma$,and then count how many events are within the $\pm 3\sigma$ interval from the peak center.

2. Count what is the total number of events N_und in the grade 7 below $E_c-3\sigma$.

3. Calculate the depletion depth d_d according to the formula derived from (4.19):  

$$d_d = \lambda \ln(\frac{N_d}{N_{und}} + 1)$$ (31)

The logarithmic function in the above equation makes the result very insensitive to small changes in both N_d and N_und caused by statistical uncertainties or, say, choosing different set of grades for the peak counts calculation. At the same time, small changes of the depletion depth cause changes in the N_d and N_und to go in the opposite directions (because the sum N_d + N_und = const, see Fig. 4.52), thus amplifying the change in the result. Sharp sensitivity of the result to the changes of the depletion depth has been confirmed by experiment where the gate voltage was stepped from 0 to +10 Volts.