UNIVERSIT . . AT BONN Physikalisches Institut - Prof. Dr. Norbert ...

14 2. X-ray imaging
component and a smaller, longer lasting hole component. The difference in the electron
and hole components of the signal can be explained by Ramo’s theorem (2.11). The high
mobility of the electrons causes them to quickly travel through the detector inducing a
rapidly changing charge at the anode. As the holes progress more slowly, the hole signal
lasts longer but at the same time has a smaller amplitude. Overall, the charge collection
is complete once the carriers arrive at the respective electrodes. Note that this example
treats only the signals at the anode. Still, it is evident that a negative charge, which moves
away from the cathode, is equal to a positive charge moving closer to said electrode. Hence,
the signal can be observed at both sides of the detector with the sole difference of inverted
polarities. However, this is only true if no charge is lost inside the sensor and if the charges
are created right in the middle of the sensor. In case the electron-hole pairs are not created
in the middle of the sensor volume, the electron and hole components contribute to the
constant total collected charge according to their distance of travel.
Pixel detector
In contrast to this simple example, a special situation arises for pixel detectors in which
the pixel dimensions are small compared to the sensor thickness. Fig. 2.8(b) illustrates
the so-called small-pixel effect in one dimension. The weighting potential is given by the
following formula [29]:
Φw = 1
π arctan
�
sin(πy)sinh(π a
2 )
cosh(πx) − cos(πy)cosh(π a
2 )
�
(2.14)
The x coordinate gives the position in the anode plane and the y coordinate parameterizes
the distance from the anode. The thickness of the sensor is set to one and the width of
the anode pixel electrodes is given by the constant a. In this example the starting point
of the charge carrier movement is chosen close to the cathode, so that only the electrons
have a significant contribution to the signals. Holes reach their collecting electrode almost
instantly and thus induce only a negligible signal on the anode. Fig. 2.8(b) shows the
induced signals for two different paths through the detector. One electron (A) is generated
directly above the central read-out pixel and the second (B) starts its translation above
the neighboring electrode. At the starting point of both movements the weighting field
of the central electrode is weak. Therefore the induced charge and the current signals
increase only slowly as the carriers move towards the anode. At the end of the charge
carrier movement, when electron A approaches the central pixel, the steep increase in the
weighting potential causes a strong coupling of the movement to this particular electrode.
The figure also illustrates that the weighting potential of the central pixel extends into the
space above the neighboring pixels. This means that a charge carrier movement towards
a neighboring electrode (electron B) also induces a current signal on the central electrode.
As shown, the signal from electron B does not differ from that of electron A during the
initial phase of the charge carrier movement. Nevertheless, once electron B approaches its
collecting pixel the sign of the current signal is inverted and the induced charge on the
central electrode returns to zero.
In summary this means that due to the small pixel effect the largest part of the signal
is generated when the charge carriers are close to the pixel electrodes, e.g. at a 250 µm
electrode on a 3 mm thick sensor 90 % of the charge is collected in only 40 % of the
charge carrier’s transit time. Furthermore, only charges that are collected at the read-out
electrode yield a net charge signal.