# Charge Transportation

The detector is biased to collect the charges generated by x-ray interaction. The charges will drift along the electric field, *that is*, electrons drifting to the anodes and holes moving toward the cathode. The drift velocity of electrons and holes is proportional to the electric field strength, £„,

where *ц _{еП},* is the electron/hole mobility. For CZT, the mobility is typically about 1000 cm

^{2}/ Vs for electrons and 80 cm

^{2}/ Vs for holes. If a 900-V bias voltage is applied to a 2-mm CZT

**FIGURE **13.3 Radius covering 95% of charge, the solid triangle is derived from charge cloud generated by 10.000 electrons and solid curve shows *ftR _{K0},* where

*fi*equals to 0.42 and

*R*is the КО radius (Eq. 13.2).

_{KO}crystal, it will generate an electric field with a strength of 450 V/mm. It takes around 44 ns for electrons to travel from the cathode to the anode pixel. The holes move around ten times slower than electrons.

When drifting along the electric field, random thermal motions of electrons (holes) w ill expand the charge cloud size. If we assume the initial charge cloud has a Gaussian distribution, the charge cloud w ill evolve as a Gaussian distribution with a standard deviation given by,

where *D* is the diffusion coefficient, w^{f}hich is calculated by the *Einstein relation*,

where *D* is proportional to the electron/hole mobility, *ц _{е}ц„* the Boltzmann constant

*(K),*and the absolute temperature,

*T,*in Kelvin; and

*e*is the charge of an electron. At room temperature (25°C),

*Щ-*equals to 2.57 mV.

During charge transportation, crystal defects could trap the charge and prevent them from being collected by the electrodes. The duration of an electron and hole from its generation to being trapped is defined as the lifetime, *x _{e},_{h}.* For a hole in the CZT, the typical value is around 1 ^s, and for the electrons, the value is around 3 //s. Since the electron lifetime is significantly larger than its drift time, electron trapping can be ignored. However, the hole drift time has a similar order of its lifetime, and it prevents the holes from being fully collected by the cathode. Consequently, it will degrade the detector energy resolving performance.

In the above, we discussed that the charge cloud distribution is affected by the applied electric field, the thermal diffusion, and charge trapping. The charge cloud size also is affected by self repulsion between electrons/holes. All of these factors are coupled together. For this case, there is no

**TABLE 13.2**

**Materials Properties of CZT [6]**

Electron Mobility (/',.) |
Electron Lifetime (r,,) |
Hole Mobility (/<,,) |
Hole Lifetime (r |
Permittivity (e) |

1000 cm |
3 / |
80 cm |
1 |
9.4x 10“ |

simple solution, like Eq. 13.6, anymore. To accurately model these effects, we have to solve charge transportation partial differential equations. For the electron cloud, we have,

where the electron cloud (*q _{e}(r,t*)) includes a moving part (

*q*)) and a trapped part (

_{em}(r,t*q*)). The moving part will be affected by three terms. The first one is diffusion,

_{es}(r,t*that is, Dy*The second term is a convectional term,

^{2}q_{em}(r,t).*that is, jj*It models the charge convection caused by the electric field (£,,). The last term is the trapping term,

_{<}V (q_{<}.,„(r,t)E_{e}).*that is,*ч™<

^{г}-

^{п}

*.*Typical material properties of CZT are shown in Table 13.2.

The term, *E _{e},* in Eq. 13.8 includes the applied electric field

*(E„)*generated by the high-voltage bias and the electric field generated by the electron cloud itself, wfiich models the charge repulsion effect. Here we did not take into account interaction between holes and electrons, because electrons and holes will be separated in a 0.1 ns time scale and after that the interaction between holes and electrons will be reduced rapidly by the inverse of the distance squared.

*E*= V0(r,f), wfiich is given by

_{c}where

For the hole cloud, we have,

where the notations of Eqs. 13.11 and 13.12 are the same as Eqs. 13.8 and 13.9, except the subscript *“e”* is for the electron cloud and “A” for the hole cloud.

Figure 13.4A shows a 2D profile as the charge cloud drifts toward the anode, and corresponding radius (*R*_{g5}) covering 95% of the charge cloud is shown in Figure 13.4B. The initial size of the charge cloud has a radius around 6.5 //in. The thermal diffusion and charge repulsion expand the radius to around 45.0 //m when the cloud reaches the anode side. The overall size (2/?_{95}, *i.e.,* 90 /лп) is comparable to a small anode pitch, like 250 //m. Figure 13.4B also show's that the model, without considering the repulsion effect, significantly underestimates the charge cloud size by around 20%. When the charge repulsion is not taken into account, only thermal diffusion will expand the charge cloud size, the solution of PDE in Eq. 13.8 is the same as the w'idely used empirical formula in Eq. 13.6. The empirical formula significantly underestimates the charge cloud size change over time by around 20%.

**FIGURE 13.4 **Illustration of electron cloud distribution change over time when the cloud drifts toward the cathode. (A) is the 2D profile; (B) shows radius covering 95% of the charge cloud. The solid red curve considers both repulsion and diffusion effect, while the solid black curve only incorporates the thermal diffusion effect.

Figure 13.5 shows the charge cloud size evolution for 30 keV, 60 keV, and 120 keV events. The corresponding initial sizes of these charge clouds have a radius around 2.1 /лп, 6.5 //m. and 21.0 //m, respectively. The thermal diffusion and charge repulsion expand the radius to 41.8 /mi. 45.0 /im, and 51.3 /mi as the cloud reaches the anode side. As expected, the cloud size differences are reduced over time.