TCAD simulation studies of novel geometries for CZT ring-drift detectors

The detector-grade CZT material resistivity exhibits values in the region of $\rho > 10^{10}-10^{11} ~ \Omega$ cm. In order to achieve such resistivity values, a combination of defects were introduced in the form of acceptors and donors of a given energy, concentration and cross section, see table 1. Previous studies [20, 21] suggest that three shallow acceptor levels representing inherent material defects and one deep level responsible for the compensation mechanism are sufficient to adequately recreate realistic material resistivity.

Table 1. List of the defects introduced in the CZT model.

Defect type	Energy (eV)	Concentration (cm−3)	Cross section (cm−2)
Acceptor	$E_v$  +  0.023	$4.0 \times 10^{7}$	$1 \times 10^{-17}$
Acceptor	$E_v$  +  0.290	$1.2 \times 10^{8}$	$3 \times 10^{-17}$
Acceptor	$E_v$  +  0.480	$1.0 \times 10^{8}$	$2 \times 10^{-14}$
Donor	Ec - 0.830	$1.0 \times 10^{9}$	$1 \times 10^{-10}$

The transport properties of the material are governed by the mobilities and life-times of the carriers. These were set such to achieve the $\mu \tau$ values for electrons and holes widely accepted in the literature for detector-grade material.

X-ray interactions were simulated in the detector based on the heavy-ion model. Out of the two models available in the Synopsys Sentaurus TCAD, namely, alpha particle and heavy ion, the latter is known to be more adequate for the simulation of x-ray interactions. The parameters for the heavy-ion model were set to generate an electron-hole cloud expected for a 25 keV x-ray photon. The characteristic depth of the interaction was chosen such to achieve a 1/e attenuation length, corresponding to $70~\mu$m in CZT. This energy was chosen so the results of simulations validated against existing experimental data [16, 17, 22].

Only a single photon interaction is allowed in the simulation package. The collected charge was calculated by integrating the electron current induced on the anode during the first 500 ns of the simulation time. The simulated values were benchmarked against the measurements. The same calculation method was later applied to compare between different geometries suggested for the new detector geometry.

The model was validated against the study of a manufactured device described in [16, 17, 22]. The device had dimensions of $10\times11\times2.3$ mm (x-y -thickness). The charge collection was studied at the Diamond synchrotron facility using a monochromatic 25 keV $10\times10$ mm beam across the centre of the detector. A 2D TCAD model of the geometrically equivalent device was created measuring $10\times1\times2.3$ mm. Because of the nature of the measurement, the 2D model was sufficient to qualitatively validate charge collection in the simulated device. Bias voltages applied to the model were replicated from the synchrotron measurements (R1  =  −500 V, R2  =  −600 V, R3  =  −700 V, cathode  =  −700 V) with the exception of the guard ring, which was biased at  −1000 V as the limitations of the TCAD would not permit correct boundary conditions at the edges of the detector model [22]. The validated model was applied to the new geometries to study the effects influencing charge collection and searching for an optimised geometrical arrangement.

Only one biasing scheme for this simulation study was chosen. The relative and absolute amplitudes of the voltages applied to the rings, cathode and guard ring were systematically studied in [17]. Their optimal voltages were chosen for this study, i.e. R1  =  −1000 V, R2  =  −1200 V, R3  =  −1400 V, cathode  =  −1400 V, guard ring  =  −2000 V. Due to the length of each simulation, it was impractical to study further whether better biasing conditions exist for the new geometries. Such a study will be carried out on a fabricated device.

The detailed simulation performed by Boothman et al in [17] concluded that increasing number of steering rings from three to four to five does not improve charge collection. However, no conclusions on the designs with one and two rings were given. The devices created in this simulation study measure $5.0 \times 5.0 \times 2.0$ mm. The collecting anode has a diameter of $250 ~\mu$m. The thickness of the rings is $125 ~\mu$m and the diameter of the starting point of the guard ring is $4200 ~\mu$m. Geometry models with one, two and three steering electrodes were simulated. In this study a comparison between a circle and squircle shape of the steering rings was performed. The geometry was developed such that only the diagonal cross section of the devices was different. This arrangement required a 3D model to be created for the comparison. A top view of the created geometries can be seen in figure 1. Charge collection was compared between the two sets of geometries. Charge was injected along diagonal X  =  Y axis at locations $0-, 250-, 500-, 750 ~\mu$m, etc.

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The intrinsic material properties used in this study are based on three acceptors representing defects and one deep donor responsible for compensation. The initial trap energy levels, cross sections and concentrations were taken from [17, 22] and references therein. The macroscopic resistivity of the material can be directly calculated from the quasi-stationary simulation results with no bias applied to the detector by using formula

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \rho = \frac{1}{q (n \mu _{n} + p \mu _{p})} , \nonumber \end{align} \tag{ 3.1 }$$

where q—unit charge, n and p —electron and hole concentrations, $\mu _{n}$ and $\mu _{p}$ electron and hole mobilities.

In the model created for this study resistivity was calculated to be $\rho = 4.3 \times 10^{10} ~ \Omega$ cm which falls within acceptable range for the detector grade material.

The mobility and life time of the carriers were also extracted directly from the simulation results based on initial values set in the model configuration. They were found to be $3.5 \times 10^{-4}$ and $5.3 \times 10^{-4}$ cm² V⁻¹ for electrons and holes respectively.

Figure 7 shows a comparison between the three ring squirkle and three ring circle configurations. The charge collection remains similar between the two configurations until approximately $900~\mu$m. After this, the three ring squirkle configuration shows improved charge collection. In this geometry, the device remains sensitive to charge injection until approximately $1200~\mu$m, while the three ring circle configurations stops collecting charge after about $1000~\mu$m. Figure 8 shows a comparison between the two ring squirkle and two ring circle geometries. The charge collection in this case remains similar between the two configurations until approximately $1100~\mu$m. After this the two ring squirkle configuration shows improved charge collection by being sensitive to charge injection until $1500~\mu$m, whilst the two ring circle configurations stops collecting charge after about $1300~\mu$m. A dip in the charge collection in the two ring squirkle can be noticed between $900~\mu$m and $1300~\mu$m. This feature is observed right below the first ring. Because such a dip is only observed prominently for the two ring squirkle geometry, it seems that the biasing conditions for this ring could be further improved to optimise charge collection in this region of the device. Figure 9 compares the one ring squirkle and one ring circle geometries. As for the three and two ring configurations, the squirkle geometry improves charge collection. The one ring circle geometry collects charge until about $900~\mu$m, while the device with one ring squirkle configuration collects charge until  ∼$1300~\mu$m. It is worth noting a small dip around $500~\mu$m for the one ring circle geometry. Because it is located between the anode and first ring, it can be attributed to the weakest point in the weighting potential. This can be improved by careful optimisation of the biasing of the first ring.

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The squircle steering ring configuration has been shown to be systematically better than the circle ring configuration. A comparison between the number of the rings in this geometry is shown in figure 10. Charge collection extends as far as  ∼$1400~\mu$m from the anode in the two ring configuration case. This gets worse for both the 3 and 1 ring geometries. The one ring configuration collects charge until $\sim$$1100~\mu$m while the three ring geometry device—until  ∼$1300~\mu$m. It is worth noting the pulse timing attributes of the geometries simulated. Figure 11 depicts three selected charge pulses from the one, two and three ring squircle geometries. Charge was injected at a distance of $x=y=1000~\mu$m from the anode. The total charge collected is marginally different between the two and three ring and $20\%$ smaller for the one ring geometries at this location. However, the times the pulses start to be collected by the anode differ. The first charge starts being seen on the anode is 50  −  , 75  −  and 85 ns for two, three and one ring configurations respectively. This indirectly confirms the more effective weighting profile for the two ring geometry. When charge is collected faster by the anode, there is less probability for recombination, which is likely to be pronounced more in a real device.

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The study of the two ring squircle configuration was extended further. A comparison was made for the device with double the thickness (4000 instead of $2000~\mu$m) and double the default bias, R1  =  −2000 V, R2  =  −2400 V, R3  =  −2800 V, cathode  =  −2800 V, guard ring  =  −4000 V. It was expected to see a change in the weighting potentials of the rings and hence variation in the charge collection.

Figure 12 shows the extension of the charge collection range for the $4000~\mu$m thick device. In this case, the charge is being collected all the way to the start of the second ring (approx. $2000~\mu$m). This is improved by $500~\mu$m compared to the nominal thinner device. The increase of charge collection when applying a factor of two increase in bias voltage is shown to be marginal, yielding only a  ∼$100~\mu$m extra.

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