Crystal silicon is a good material used for tracking detectors. It has a small band gap and is a semiconductor which can therefore be depleted. Hence, it is suitable for detecting traversing charged particles as explained in the following. The sensor is made up of two differently doped areas. An n-doped area has implemented atoms with one more electron than silicon which are called donors. In case of silicon a donor would be phosphorus or arsenic. In contrast a p-doped area has an implemented atom with an electron less which is called acceptor. In case of silicon an acceptor would be boron. To establish an equilibrium state the free charge carriers drift to the oppositely doped area and built a space charge region, see Figure 3.3. In addition, an electric field opposite to the field between the doped areas arises. This zone is also called depletion zone because no free charge carriers are present.
If an external reverse voltage is applied to this pn-junction the space charge region expands.
Is the reverse bias high enough, the doped areas are fully depleted and do not contain any free
3.3 Silicon as sensitive detector material
holes electrons
p-doped n-doped
carrier concentration [log]
depletion zone without reverse bias
X
E- Field
a)
depletion zone with reverse bias
p-doped n-doped
E- Field
- +
b)
Figure 3.3: a) The pn-junction without a reverse bias voltage. The coloured areas indicate the particular doped area in a non depleted state. They grey area is the depletion zone and contains no free charge carriers. b) The pn-junction with reverse bias voltage and increased depletion zone.
charge carriers. This bias voltage is called depletion voltage. The n- and p-doped areas together with an applied external electric field behave similar to a reverse-biased diode.
If a charged particle traverses this depletion area electron-hole pairs are created along the path of the incident particle and the generated free charge carriers can be measured. If no electric field is applied the electron-hole pairs would recombine immediately and thus do not contribute to the detector signal.
To determine the average number of electron-hole pairs N the absorbed energy E has to be divided by the average energy needed to create one electron-hole pair w which is 3.62 eV in silicon1. This is much higher than the band gap of silicon which is 1.12 eV. The excess of energy is transformed in lattice oscillations or phonons inside the silicon material and heats up the sensor. The fluctuation of the number of generated electron-hole pairs N is described by Formula 3.3. An electron or hole generated this way can be assumed to be a free charge carrier which can travel through the medium due to different effects. Diffusion occurs if there is a local excess of charge carriers inside the silicon material. This disequilibrium is compensated by moving charge carriers from one to the other excess region.
1The creation of one electron-hole pair inside a gaseous detector is in the order of 20 eV.
3 Particle interaction with matter
More important for the detection of signals in silicon is the effect of charge drift. An external electric field accelerates the electrons and holes along the direction of the electric field E~ and leads to a current per unit area:
J~n,drif t=−enµnE ,~ J~h,drif t=enµhE .~ (3.12) µn andµh are the mobilities for electrons and holes, respectively. nis the carrier concentration and e the electron charge. The mobility depends on the temperature and the doping and impurity concentration of the material. Inside a high electric field the free charge carriers are accelerated until a saturation velocity is reached. The latter just depends on the mean free path or the mobility µ inside the material. The applied voltage Ubias for particle detectors depends on the doping concentration and the sensor thickness and is slightly higher than the depletion voltage Udepl. This is done to establish a non-zero electric field at the electrodes which then can be reached by the drifting charge carriers. A common value for unirradiated sensors is Ubias= Udepl+ Ua≈150 V, in which Ua is the increase relative to the depletion voltage of about 30-50 V. While the charges are drifting due to the external electric field simultaneous diffusion causes a spread of the charges which is perpendicular to the direction of the electric field. This leads to charge sharing in which signal charges do not induce a charge on one pixel segment but on several ones.
In the presence of a magnetic field the carriers are deflected according to the Lorentz force described in Equation 3.9. The resulting deviation angle, or Lorentz angle θL, can then be described via the Hall mobility µHall:
tan θL,n=µHall,nB⊥ , tan θL,h =µHall,hB⊥ . (3.13) B⊥ is the magnetic field perpendicular to the velocity of the charge carriers. The Hall mobility can be described with the before mentioned mobility µHall = rµ, with r being the Lorentz factor2. Due to the Lorentz angle, the charge sharing effect is increased. Thus, the Lorentz angle has an impact on the spatial resolution and the cluster size [31].
To realise a particle detector, n+-doped pixel implants are placed in the n-type bulk of the sensor, see Figure 5.3. As a result, an incident charged particle creates electron-hole pairs with the electrons being accelerated in the direction of the higher n+-doped implants on top of the sensor. The implants act as electrodes and measure an electrically induced signal according to Ramo’s law while the electrons are moving towards them [32]:
i=Evev . (3.14)
iis the current measured at the implant due to a single electron,eis the electron charge,v the velocity of the electron and Ev is the weighting field. A sensor with an n-bulk and n+-doped implants is called n+-in-n sensor and is used as sensitive material for the Pixel Detector discussed in this thesis. n+-doped implants provide a still n-doped area while the original n-doped bulk changed to a p-doped bulk at a certain radiation dose, see Section 3.4. A possibility to isolate the n+- and n-type zones from each other is the implementation of a p-doped zone called p-spray which is brought up between the two n-type zones [33]. Thus the implants are isolated in case the reverse bias is applied. Using such implants the granularity of a pixel detector can be very high. This is especially important for b-tagging as described in Chapter 5. Taking Equation 3.8 into account the resulting small distances, which are in the order of several µm, between the n+-doped implants yield a very good spatial resolution of charged particle’s hits.
2The Lorentz factorr is measured to be 1.15 for electrons and 0.72 for holes at a temperature of 0◦C.