Characterization of Crystal Properties and Defects in CdZnTe Radiation Detectors

Abstract

CdZnTe-based detectors are highly valued because of their high spectral resolution, which is an essential feature for nuclear medical imaging. However, this resolution is compromised when there are substantial defects in the CdZnTe crystals. In this study, we present a learning-based approach to determine the spatially dependent bulk properties and defects in semiconductor detectors. This characterization allows us to mitigate and compensate for the undesired effects caused by crystal impurities. We tested our model with computer-generated noise-free input data, where it showed excellent accuracy, achieving an average RMSE of 0.43% between the predicted and the ground truth crystal properties. In addition, a sensitivity analysis was performed to determine the effect of noisy data on the accuracy of the model.

1. Introduction

Single-photon detectors are essential in nuclear medical imaging techniques, where high spectral resolution is critical for tissue differentiation. In particular, detector technologies based on cadmium zinc telluride (CdZnTe) are increasingly favored in modalities that operate with relatively low photon flux, such as single photon emission computed tomography (SPECT) [1]. Thick CdZnTe crystals can provide high stopping power, enabling the absorption of most incoming photons and their energy classification across a wide spectral range [2]. Additionally, this semiconductor possesses a wide and tunable band gap of approximately 1.6 eV [3], with the exact value depending on the crystal stoichiometry and preparation conditions, allowing detector operation at room temperature [4]. These photon-counting detectors (PCDs) offer a rapid temporal response (less than 10 ns) and can be configured with a pixelated geometry, enabling sub-millimeter spatial resolution. However, the energy resolution of these detectors is often compromised due to the high defect density typically found in the thick CdZnTe crystals [5,6,7]. These defects include point defects, dislocations, and grain boundaries [8], which adversely influence the optoelectronic properties of the crystal.

In this study, we employ a reverse-engineering approach to determine the material properties and defects in the CdZnTe crystals used for radiation detectors. Building upon prior physics-based learning models developed by our team [9,10,11,12,13,14], our novel model accurately identifies fundamental spatially dependent properties, such as charge mobility and the lifetimes for charge recombination, trapping, and de-trapping. By precisely characterizing these features, the study seeks to mitigate the detrimental effects of crystal impurities. The model takes as input the charge-induced signals and the charge concentrations after a particular photon-detector event interaction. We initially validate our model using computer-generated noise-free data to establish a baseline for performance, consistently with our previous work. Subsequently, we now assess for the first time the impact of noise in the input data on the accuracy of this characterization. This approach bridges the gap between purely simulated signals and the potential application of our learning-based characterization for real detectors in the future.

2. Detector Simulator

We have built a physics-driven model that simulates the functioning of CdZnTe photon-counting detectors. Various detector geometries are employed in CZT-based photon-counting devices, including basic planar configurations (with one cathode and one anode), pixelated designs, and Frisch grid structures (coplanar, parallel stripe, or capacitive) [15,16,17,18,19]. It is important to emphasize that our proposed methodology is generalizable to different configurations and dimensions. In our simulations, we consider a representative detector with reasonable dimensions of $1\times 1\times 1$ cm³ [20,21,22], using a standard pixelated configuration with a single cathode on top and nine anodes on the bottom, as seen in Figure 1.

Pixelated detectors take advantage of the small-pixel effect [19,23], which enhances signal resolution by focusing on the contributions from electrons, which have higher mobility and less trapping compared to holes. This effect becomes more pronounced as the anode pixel size decreases relative to the cathode area, while pixelated detectors can include $10\times 10$ anode pixels or more [24], it is sufficient to focus the analysis on the main pixel where the charge injection occurs and the neighbor pixels, and thus the common $3\times 3$ analysis [25,26], since these receive enough signal strength to provide useful information for detector characterization.

When a $\gamma$-ray penetrates the cathode and interacts with the CZT crystal, several electron–hole (e–h) pairs are generated after absorption. The number of e–h pairs depends on the energy of the $\gamma$-ray and the ionization energy of the CZT crystal, which is approximately 4.6 eV, depending on the stoichiometry and crystal growth conditions [27]. Typical SPECT applications employ radionuclides with $\gamma$-ray energies ranging from 30–250 keV [28], potentially leading to the generation of approximately 6500 to 54,000 e–h pairs per photon. Without loss of generality, we have employed an deposited energy of 140.5 keV for these particular simulations, considering the commonly employed Technetium-99m source. At these photon energies, other dominant interactions within the detector include the photoelectric effect and Compton scattering, both of which may also contribute to electron–hole pair generation. Simulation tools such as GEANT4 [29] or MCNP [30] are commonly used to simulate the interaction of $\gamma$-rays with matter, including the statistical modeling of energy deposition and the spatial distribution of interactions within the detector material. These simulations account for various physical processes such as the pair production, photoelectric absorption, and Compton scattering, which are critical in determining where and how energy is deposited in the CZT crystal.

Once the energy deposition and interaction locations are defined, our model takes these as inputs and focuses on the subsequent stages: the transport and the fluctuation of the charges. Indeed, the charge carriers move through the crystal drifted by an external electric field E, therefore producing charge-induced signals at the nearby electrodes [31,32].

Following a delta-like photon-detector event $\delta (x_0,y_0,z_0,t_0)$ occurring at time $t_0$ and specific location $(x_0,y_0,z_0)\in {[0,1]}^3$, the electron concentration $n_{\mathrm{e}}(x,y,z,t)$ can be modeled with a system of partial differential equations (PDEs) [33,34]:

$$\begin{array}{cc}\hfill {\partial }_t{n}_{\mathrm{e}}-\nabla ·\left({\mu }_{\mathrm{e}}E{n}_{\mathrm{e}}\right)& =-{\displaystyle \frac{1}{{\tau }_{\mathrm{eR}}}}{n}_{\mathrm{e}}-{\displaystyle \frac{1}{{\tau }_{\mathrm{Te}}}}{n}_{\mathrm{e}}+{\displaystyle \frac{1}{{\tau }_{\mathrm{eD}}}}{\tilde{n}}_{\mathrm{e}}+\delta \hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill {\partial }_t{\tilde{n}}_{\mathrm{e}}& ={\displaystyle \frac{1}{{\tau }_{\mathrm{eT}}}}{n}_{\mathrm{e}}-{\displaystyle \frac{1}{{\tau }_{\mathrm{eD}}}}{\tilde{n}}_{\mathrm{e}}\hfill \end{array}$$

(2)

Note that an equivalent system and notation can also be applied to model the dynamics of holes, where the concentration is denoted as $n_{\mathrm{h}}$. Equations (1) and (2) accounts for various dynamic processes: charge drift, charge generation–recombination, and trapping–de-trapping effects. The variable ${\tilde{n}}_{\mathrm{e}}$ represents the concentration of electrons within the trapping energy levels, which includes shallow and deep traps. These traps are conceptualized as an infinite well that attracts (and eventually expels) a percentage of the charges, as described by the Shockley–Read–Hall theory [35].

The mean lifetimes for electron charge trapping, de-trapping, and recombination are denoted by ${\tau }_{\mathrm{eT}}$, ${\tau }_{\mathrm{eD}}$, and ${\tau }_{\mathrm{eR}}$, respectively. The electron mobility is given by ${\mu }_{\mathrm{e}}$, with its drift velocity being $v_{\mathrm{e}}=-{\mu }_{\mathrm{e}}E$. In the simulations, we considered a vertical operative field with an average reasonable magnitude $E=850$ V/cm [36]. Our model simplifies the charge dynamics by assuming a purely vertical trajectory for the charges along the z-axis, omitting only the minor deviations in the $(x,y)$-directions that could arise near the interpixel gaps [37] or from particular polarization effects [38]. Additionally, we incorporate an empirical relationship between the mobilities of electrons and holes, estimated as ${\mu }_{\mathrm{h}}\approx {\mu }_{\mathrm{e}}/10$, based on the findings reported in [9]. Our model neglects the weak effects of diffusion and Coulomb repulsion, which is reasonable under a high electric field strength [33,39,40].

Delving into the mathematical model presented in Equations (1) and (2), we can break down the behavior of each equation. In Equation (1), the term ${\partial }_t{n}_{\mathrm{e}}$ represents the rate of change in the free charge concentration over time. The first term $\nabla ·\left(v_{\mathrm{e}}{n}_{\mathrm{e}}\right)$ then models the drift of the charges under the influence of the velocity field. This term captures the transport of the charges through the medium: At each time step, the charges then move toward their corresponding electrode. Next, there is a decrease in the free charge concentration due to recombination and trapping processes, expressed by the combined term $(1/{\tau }_{\mathrm{eR}}+1/{\tau }_{\mathrm{eT}})n_{\mathrm{e}}$. These processes remove free charges from the conduction band back to the valence band or to certain intermediate trapping energy levels. Additionally, there is a regeneration of free charges from the de-trapped population, described by the term $1/{\tau }_{\mathrm{eD}}{\tilde{n}}_{\mathrm{e}}$.

In contrast, Equation (2) models the evolution of the trapped charge concentration, and shows that trapped charges remain stationary in space and do not contribute to transport or signal generation. The term $1/{\tau }_{\mathrm{eT}}{n}_{\mathrm{e}}$ represents the rate at which free charges become trapped, while $1/{\tau }_{\mathrm{eD}}{\tilde{n}}_{\mathrm{e}}$ models the rate at which trapped charges are released (de-trapped) back into the conduction band. Once de-trapped, these charges are free to move under the influence of the electric field and contribute to signal production once again.

Equations (1) and (2) can be efficiently solved numerically using the explicit finite-difference method [41]. We defined a spatial step of $\Delta z=0.01$ cm and a time step of $\Delta t=10$ ns, capturing the high temporal response of the real CdZnTe-PCDs. Note that the crystal properties (charge mobility and lifetimes) can be equivalently reformulated using dimensionless computational parameters, as detailed in

Table 1. Common material properties of CdZnTe crystals [41], along with the corresponding dimensionless computational parameters used in the simulations. The table also shows the NRMSE between the predicted parameters and the ground-truth ones when employing noise-free and noisy input data.

Material Properties	Symbol	Value	Parameter	Value	NRMSE (%)
					Noise-Free	Noisy
Charge mobility [cm2/Vs]	${\mu }_{\mathrm{e}}$	1120	$R_{\mathrm{e}}=\left({\mu }_{\mathrm{e}}E\right)\Delta t/\Delta z$	0.95	<${10}^{-2}$	0.22
	${\mu }_{\mathrm{h}}\approx {\mu }_{\mathrm{e}}/10$	112	$R_{\mathrm{h}}=\left({\mu }_{\mathrm{h}}E\right)\Delta t/\Delta z$	0.095	<${10}^{-2}$	0.22
Recombination lifetime [$\mathsf{\mu }$s]	${\tau }_{\mathrm{eR}}$	10	$P_{\mathrm{eR}}=\Delta t/{\tau }_{\mathrm{eR}}$	0.001	2.81	23.81
	${\tau }_{\mathrm{hR}}$	1	$P_{\mathrm{hR}}=\Delta t/{\tau }_{\mathrm{hR}}$	0.01	<${10}^{-2}$	2.86
Trapping lifetime [$\mathsf{\mu }$s]	${\tau }_{\mathrm{eT}}$	10	$P_{\mathrm{eT}}=\Delta t/{\tau }_{\mathrm{eT}}$	0.001	0.18	3.46
	${\tau }_{\mathrm{hT}}$	0.067	$P_{\mathrm{hT}}=\Delta t/{\tau }_{\mathrm{hT}}$	0.15	<${10}^{-2}$	0.74
De-trapping lifetime [$\mathsf{\mu }$s]	${\tau }_{\mathrm{eD}}$	0.4	$P_{\mathrm{eD}}=\Delta t/{\tau }_{\mathrm{eD}}$	0.025	0.01	4.52
	${\tau }_{\mathrm{hD}}$	0.067	$P_{\mathrm{hD}}=\Delta t/{\tau }_{\mathrm{hD}}$	0.15	<${10}^{-2}$	0.88

. These parameters naturally emerge during the discretization of the PDE system, as further explained in [11]. To account for the spatial variability of material properties, we define a stratified medium consisting of $N=100$ layers sequentially stacked from cathode to anode, with each layer assigned distinct parameter values. The choice for the number of layers N is motivated by the time resolution limitations of CZT detectors (which is 10 ns for our representative and common scenario). During each time step, electrons move approximately 0.01 cm, given their velocity $v_{\mathrm{e}}={\mu }_{\mathrm{e}}E={10}^6$ cm/s, while holes travel a shorter distance, around 0.001 cm. This resolution constraint supports the use of $N=100$ or fewer layers, as this level of stratification aligns with the system ability to resolve charge transport, as further detailed in [9,12].

After numerically solving the system from Equations (1) and (2) for both electron and hole concentrations, we obtain the charge densities as functions of time and space. The current signals for each electrode, indexed by $k\in \{1,2,\dots ,10\}$, are subsequently calculated using the Shockley–Ramo theorem [31]:

$$i^{\left(k\right)}\left(t\right)={\int }_{\mathsf{\Omega }}q{n}_{\mathrm{e}}(t,x,y,z)v_{\mathrm{e}}\left(z\right)E_{\mathrm{w}}^{\left(k\right)}(x,y,z)d\mathsf{\Omega }$$

(3)

In this equation, $\mathsf{\Omega }={[0,1]}^3$ cm³ is the crystal space. represents the crystal volume. The induced current $i^{\left(k\right)}$ at time t computed by integrating over the three-dimensional charge density $q{n}_{\mathrm{e}}(t,x,y,z)$, where q is the elementary charge. A higher concentration of charges induces a larger current. Additionally, the electron drift velocity $v_{\mathrm{e}}={\mu }_{\mathrm{e}}E$ contributes to the induced current. Since the velocity depends on the spatially varying mobility and electric field in the stratified medium, the current is also depth-dependent within the material. This explains why holes, moving approximately ten times slower than electrons, generate much weaker signals.

Another crucial factor in the Shockley–Ramo theorem is the weighting electric field $E_{\mathrm{w}}^{\left(k\right)}$ which accounts for how close charges are to the collecting electrode. Charges that are nearer to the collection electrode induce a higher current. According to the theorem, the weighting potential ${\varphi }_{\mathrm{w}}^{\left(k\right)}$ is determined by solving the Poisson equation:

$${\nabla }^2{\varphi }_{\mathrm{w}}^{\left(k\right)}(x,y,z)=0$$

(4)

This is subject to the boundary conditions ${\varphi }_{\mathrm{w}}^{\left(k\right)}=1$ at the electrode of interest k, and ${\varphi }_{\mathrm{w}}^{\left(k\right)}=0$ at all other electrodes. This requires solving a separate partial differential equation (PDE) for each electrode, which in our case involves ten electrodes. Figure 2 illustrates the calculated weighting potential for an anode located at the edge of the bottom pixel array, obtained using the finite element method (FEM) [42] via the FEniCS package [43] in Python.

It is worth noting that the commonly measured charge-induced signals are simply calculated as $Q^{\left(k\right)}\left(t\right)=\int i^{\left(k\right)}\left(t\right)dt$.

3. Detector Characterization

Given the detector signals and charge concentrations, we can now construct an inverse model that provides us with the material properties and defects of the CdZnTe crystal at different locations, as outlined in Figure 3. As proposed in [11], we aim to solve the following optimization problem:

$$\begin{array}{cc}\hfill minC\left(\mathsf{\Theta }\right)=& \parallel n_{\mathrm{e}}^{\mathrm{sim}}(t,z;\mathsf{\Theta })-n_{\mathrm{e}}^{\mathrm{given}}{(t,z)\parallel }_{\mathrm{F}}^2\hfill \\ \hfill & +\parallel n_{\mathrm{h}}^{\mathrm{sim}}(t,z;\mathsf{\Theta })-n_{\mathrm{h}}^{\mathrm{given}}{(t,z)\parallel }_{\mathrm{F}}^2\hfill \\ \hfill & +\parallel {\tilde{n}}_{\mathrm{e}}^{\mathrm{sim}}(t,z;\mathsf{\Theta })-{\tilde{n}}_{\mathrm{e}}^{\mathrm{given}}{(t,z)\parallel }_{\mathrm{F}}^2\hfill \\ \hfill & +\parallel {\tilde{n}}_{\mathrm{h}}^{\mathrm{sim}}(t,z;\mathsf{\Theta })-{\tilde{n}}_{\mathrm{h}}^{\mathrm{given}}{(t,z)\parallel }_{\mathrm{F}}^2\hfill \\ \hfill & +\lambda \sum _{i=1}^{10}{\parallel Q_i^{\mathrm{sim}}(t;\mathsf{\Theta })-Q_i^{\mathrm{given}}\left(t\right)\parallel }_{\mathrm{F}}^2\hfill \end{array}$$

(5)

The 2D array $n_{\mathrm{e}}^{\mathrm{sim}}(t,z;\mathsf{\Theta })$ depicts the simulated concentration of electrons at discrete times and positions, and ${\parallel ·\parallel }_{\mathrm{F}}$ the Frobenius matrix norm. The simulated concentration and signals depend on the computational parameters, which are encapsulated by $\mathsf{\Theta }\in {\mathbb{R}}^{7\times 100}$, seven parameters over the $N=100$ voxels in depth. Equation (5) describes a problem in which we try to fit the simulator’s output to the provided data. It is crucial to note that the data used in this study were generated in a computational manner, not from experiments. This means that we have prior knowledge of the ground-truth parameters, enabling a straightforward evaluation of our inverse solver algorithm. The errors between the calculated and given charge-induced signals $Q^{\left(k\right)}$ at each electrode $k\in \{1,2,\dots ,10\}$ are incorporated into Equation (5) as a regularization term with a relatively low coefficient ($\lambda =0.1$), as proposed in [11].

The non-linear fitting problem from Equation (5) displays a cost function $C\left(\mathsf{\Theta }\right)$ with several local minima, and therefore describes a complicated global optimization problem. Due to the curse of dimensionality [44], conventional global optimizers have a relatively high computational cost and are less suitable for our problem. To address this challenge, we employ the Adam optimizer [45], a momentum-based gradient descent method designed to handle non-convex optimization problems efficiently while avoiding stagnation at local minima. For that optimizer, we set the hyperparameters with a learning rate of $5\times {10}^{-4}$, a first moment ${\beta }_1=0.9$, and a second moment ${\beta }_2=0.999$.

Please observe that we used automatic differentiation to accelerate gradient computations with the PyTorch library. We also utilized an NVIDIA Tesla GPU to run our program, which is specifically designed to accelerate AI operations. We are able to perform 20,000 iterations, until convergence, in approximately 3 h. It should be noted that the convergence time is approximately the same when analyzing signals with different noise level.

To evaluate the model outcomes, we adopt the normalized root mean square error (NRMSE) of each parameter within a specified region of interest (ROI). For an event at the center of the detector (depth voxel 50), the ROI for the electron properties spans voxels 50–100, as most of the generated free electrons reach the anode location (voxel 100). In contrast, due to the relative lower drift velocity of holes, they predominantly undergo recombination or trapping when traveling through the first 10 voxels, being their ROI between voxels 40–50. It is important to note that we enforce box constraints using the projected gradient method [46]. The NRMSE could range ±25% relative to the values of the ground-truth parameters.

4. Results

We first want to extract detector characteristics and defects considering ideally noise-free computer-generated input data. Figure 4 a shows both the predicted and ground-truth computational parameter $R_{\mathrm{e}}$ (see Table 1) in each depth voxel, indicative of charge mobility. Consequently, we are using a single mobility parameter for both electrons (with ROI highlighted in blue) and holes (ROI in red). Figure 4b offers a fitting result for another representative parameter (associated with the electron trapping lifetime). Please find in Table 1 all the NRMSEs found for the multiple computational parameters. The average error was only 0.43% for all the parameters, indicating the correct convergence.

We will now analyze the case in which an additive Gaussian noise is introduced to the computer-generated input datasets. The standard deviation of the Gaussian noise is 0.1% of the maximum signal value. It is crucial to note that the cost function C should not reach zero in these noisy scenarios: A cost below the threshold given by the ground truth parameters, $C\left({\mathsf{\Theta }}^{\mathrm{gt}}\right)$, would indicate an overfit, suggesting that the model has learned the noise. The regularizer in Equation (5) prevents such overfitting, as it inhibits the algorithm from adjusting both the noise in the concentrations and the signals simultaneously. Figure 4c,d show a comparison between the predicted parameters $R_{\mathrm{e}}$ and $P_{\mathrm{eT}}$ and their ground-truth values within the regions of interest for this case with noisy data. The average NRMSE of the parameters determined became 4.89%. Although $R_{\mathrm{e}}$ maintained a relatively low error of 0.22%, the error for $P_{\mathrm{eT}}$ increased to 3.46% (see Figure 4). The last column of Table 1 presents the results for all other parameters, in this case of noisy input data. In particular, we can see that the electron recombination parameter, $P_{\mathrm{eR}}$, exhibits a significantly larger error. Its precise prediction during training becomes less critical because small variations in this parameter do not significantly impact the overall concentrations and signals [41].

Figure 5a indicates the correlation between the increase in the standard deviation in the input data and the increase in the average error in the predicted parameters. With a standard deviation ranging from 0% to 2%, the highest observed NRMSE was 16.19%. Interestingly, the random nature of the added noise and the ADAM optimizer itself led to the peak at 1.5% and not 2.0%, as one might initially anticipate. In contrast, the NRMSE for the mobility ratio increased almost linearly with the added noise level and found a maximum value of only 3.01%. In Figure 5b, one sees the ideal ground-truth signals compared to the signals generated by the predicted parameters. Even with an input noise level that has a standard deviation of 2%, the predicted signals align closely with the ground truth, resulting in an error of 0.25%.

5. Discussions

Although our inverse model demonstrated high accuracy with noise-free input data, the introduction of additive Gaussian noise to the charge concentrations and signals led to more pronounced errors. It is important to emphasize that the model still yields valuable insights even under high noise levels, as explained in the next two arguments.

First, the initial parameters deviate by 20% from the ground truth values, and the limit of the box constraint was up to $\pm 25\%$. In contrast, the average error of our model remained well below 16.2% for all scenarios (see Figure 5a). These results demonstrate that even under pathologically noisy conditions, the algorithm still refines the initial estimates.

Second, throughout all noise conditions, the mobility ratio maintained a relatively low error. Slight variations in this parameter induce large changes in the signal production, and therefore in the cost function value [41]. Therefore, this high sensitivity enables the correct characterization of the charge mobility.

However, our approach is not without limitations. On the one hand, we recognize that our approach involves the so-called “inverse crime”, since we employ the same simulator for the generation of input data (with or without noise) and to solve the optimization problem. Future studies should validate the model using real-world experimental data. On the other hand, it must be noted that, while our detector simulator [41] is efficient, it operates under the assumption of a uniformly directed external E-field from the anodes to the cathode, resulting in one-dimensional charge trajectories, in the z-direction. Therefore, it also neglects the charge cloud expansion in the 3D space due to the second-order effects of charge diffusion and Coulomb repulsion.

6. Conclusions

This study introduces a reverse-engineering model to infer spatially varying material properties and defects in CdZnTe detectors using charge concentrations and signals. Our developed program shows a relatively high accuracy (with an average error of 0.43%) when using noise-free input data. However, the error increased to 4.89% when processing data inputs affected by the additive Gaussian noise, with a standard deviation of 0.1% of the signal peak. Despite the performance degradation with higher noise levels, the derived parameters still provide a refined understanding compared to the initial assumptions. Furthermore, our model accurately predicted the mobility ratio even under conditions of very high noise levels. It is important to highlight that the parameters related to the lifetimes of recombination, trapping, and de-trapping are highly susceptible to noise disturbances in the input data. Finally, it should be emphasized that this work bridges the gap between pure signal analysis and the handling of noisy signals, with the aim of applying our learning-based crystal characterization technique to the analysis of real experimental semiconductor detectors in future studies.