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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.
2.5. Direct converting sensor materials 15 2.5 Direct converting sensor materials The selection of the direct converting sensor material plays an important role on the detector performance. Therefore this section will introduce three different sensor materials, which were used in this work. A special focus will be placed on the discussion of material related effects. 2.5.1 Silicon Silicon is a widely used and well understood elementary semiconductor material. This and the lack of unwanted material effects make it a very good reference for the testing of hybrid pixel detector concepts. Concerning its physical properties, Si is a tetravalent metalloid with an atomic number of 14, i.e. it is situated in the carbon group of the periodic table of elements. In its crystalline form Si is arranged in a face-centered cubic lattice with the base � (0, 0, 0), ( 1 /4, 1 /4, 1 /4) � . This structure is also called a diamond lattice. The relevant material properties of Si are given in Tab. 2.4. At room temperature the free charge carrier density in high purity Si is approximately 10 10 cm 3 . Compared to the other sensor materials in Tab. 2.4 this constitutes a relatively low resistance, which is also the reason why Si sensors have to be doped. In the doping process elements with three (p-dopant e.g. In) and five (n-dopant e.g. As) valence electrons are implanted into the Si crystal, resulting in additional acceptor (p-dopant) and donator (n-dopant) energy levels in the band gap. The combination of such p- and n-doped zones then establishes a p-n junction whose schematic representation is shown in Fig. 2.9. The major feature of this junction is the depletion zone, which is created by the diffusion of majority carriers from one zone into the oppositely doped region. In this context, the term majority carriers refers to electrons in the n-doped zone and holes in the p-doped zone. The diffusion of the charge carriers causes the build-up of a positive space charge inside the n-type Si and a negative space charge in the p-type section. This introduces an electric field, which counteracts the diffusion process and thus causes a thermal equilibrium across the junction. Fig. 2.9(b) gives a schematic view of the space charge distribution with NA (p-type) and ND (n-type) describing the acceptor and Element Si CdTe CZT Atomic number Z 14 48,52 48,30,52 Average Z 14 50 49.1 Density [g/cm 3 ] 2.33 5.85 5.78 Band gap [eV] 1.12 1.50 1.57 e − - h + creation energy [eV] 3.61 4.43 4.64 Typ. resistance [Ωcm] 2.3 · 10 5 10 9 >10 10 Radiation length X0 [cm] 9.36 1.52 1.52 Dielectric constant [As/Vm] 11.7 11 10.9 Mobility e − [cm 2 /Vs] 1400 1000 1000-1300 Mobility h + [cm 2 /Vs] 480 100 50-80 Average lifetime e − [s] 10 −3 3 · 10 −6 3 · 10 −6 Average lifetime h + [s] 2 · 10 −3 2 · 10 −6 10 −6 µτ e − [cm 2 /V] 1.4 3.3 · 10 −3 (3 − 5) · 10 −3 µτ h + [cm 2 /V] 1 2 · 10 −4 5 · 10 −5 Tab. 2.4: Material constants of Si, CdTe and CdZnTe.
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Bibliography [1] E. Kraft, Counting
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Bibliography 137 [32] A. E. Bolotni
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Acknowledgements Looking back at my
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UNIVERSIT . . AT BONN Physikalisches Institut - Prof. Dr. Norbert ...

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