Learning-based physical models of room-temperature semiconductor detectors with reduced data

Room-temperature semiconductor radiation detectors (RTSD) have broad applications in medical imaging, homeland security, astrophysics and others. RTSDs such as CdZnTe, CdTe are often pixelated, and characterization of these detectors at micron level can benefit 3-D event reconstruction at sub-pixel level. Material defects alongwith electron and hole charge transport properties need to be characterized which requires several experimental setups and is labor intensive. The current state-of-art approaches characterize each detector pixel, considering the detector in bulk. In this article, we propose a new microscopic learning-based physical models of RTSD based on limited data compared to what is dictated by the physical equations. Our learning models uses a physical charge transport considering trapping centers. Our models learn these material properties in an indirect manner from the measurable signals at the electrodes and/or free and/or trapped charges distributed in the RTSD for electron–hole charge pair injections in the material. Based on the amount of data used during training our physical model, our algorithm characterizes the detector for charge drifts, trapping, detrapping and recombination coefficients considering multiple trapping centers or as a single equivalent trapping center. The RTSD is segmented into voxels spatially, and in each voxel, the material properties are modeled as learnable parameters. Depending on the amount of data, our models can characterize the RTSD either completely or in an equivalent manner.


Methods
In this section, we present the classical approach for detector modeling, learning-based full model of the detector and our learning-based physical model of detectors with reduced data. Our approach of developing the learning-based physical models based on reduced data has been progressively developed with reduced amount of data during training the models, thereby reducing the amount of learnable parameters (and hence the material properties) in the model as shown in the subsequent sections of the paper.
Classical approach for detector modeling. Electrons and holes transport properties play a vital role in selecting the RTSD. Shockley-Read-Hall (SRH) Theory 36,37 governs the trapping, detrapping and recombination during charge transport. Rodrigues et al. 38 measured the detailed properties of these detectors using the charge transport and charge continuity equations with multiple electron and hole defect levels-coupled with Poisson's equation [39][40][41][42][43] . In the RTSD, the electron-hole pairs are created when high energy photons interact with the material. Subsequently in the RTSD, the free charges drift towards the respective electrodes, along with trapping, de-trapping in the defect levels and recombination of free charges in the bulk of the material 44 . The temporal dynamics of free electrons and holes following the SRH model is affected by the trapping energy states in the bandgap 45 . For free hole concentration (p), with trapping and detrapping lifetimes τ Ti and τ Di respectively for ith trap, and p ti as the concentration of holes in the ith trap, the following equation can be written 45 , The trapping and detrapping lifetimes dictate whether the defects induce short term or long term trapping of charges in the detector. Considering the low probability of transition of charges between the trapping centers, the occurrence of such process 46 is neglected in Eq. (1). The equations governing the transport, trapping and detrapping, and diffusion of electrons and holes is detailed 35 . Signals collected at the electrodes arise due to the movement of charges [47][48][49][50] . The detector setup is shown in Fig. 1 with the 9 grid electrodes on the anode side-north-west (NW), north (N), north-east (NE), west (W), center (C), east (E), south-west (SW), south (S), south-east (SE), and, 1 single large cathode electrode (CAT).
The simulated data for training the proposed learning model has been generated using the classical physical equations 34,35 . A MATLAB code was developed for describing the charge transport equations in the detector, by defining the transport, trapping, de-trapping and lifetimes of electrons and holes which are µ e , µ h , τ eT , τ eD , τ hT1 , τ hD1 , τ hT2 , τ hD2 , τ e , and τ h respectively as the fixed pre-defined parameters, with electric field along the material 35 . As in our previous work 35 , the classical model was created for a discretized (voxelized) RTSD, with charge input at any voxel. At each time step, the signals are collected at the cathode and pixelated anodes. The free and trapped charges in different voxels are also computed at each time step. The time steps and total number of time steps is defined a priori. The input for training the learning-based model consists of electron-hole pairs injected at known voxel positions. The signals, alongwith free and trapped charges in different voxels of the classical model over different time steps are the output of the learning-based model 35 . Learning-based full model of detector. We subdivide the RTSD into N voxels, (N = number of subdivisions in the material in any dimension). In each of the discretized voxels, the material properties such as µ e,h,i , τ eT,i , τ eD,i , τ hT 1 ,i , τ hD 1 ,i , τ hT 2 ,i , τ hD 2 ,i , τ e,i , and τ h,i are defined, which refer to the drift coefficients for electrons and (1)   35 . The model therefore allows the determination of the unknown material properties with higher spatial resolution than the bulk of the RTSD. For each of these coefficients (referred here as τ in general), we compute the number of charged particles (electrons or holes) remaining in that particular level as N left = N 0 e −t/τ , where N 0 and N left are the number of charged particles at a particular level at t = 0 and at time t respectively 35 . We use these charges in our model. For a particular material property τ , we can find out the fraction of charges remaining in that energy level. A voxelized representation of the detector in 1D is shown in Fig. 2a with the electrodes at either end -anode on the right and cathode on the left. The high energy radiation (for example Gamma rays or X-rays) can interact in any position of the RTSD. The learning-based model is a recurrent network structure over time with the input to the model as the injection positions of the electron-hole pairs with the magnitude of the injected charges normalized to 1 as shown in Fig. 2b. Each voxel consists of discretized material properties as the trainable weights. As charges drift under the influence of Electric Field, the electrons moves towards the anode (blue arrows in Fig. 2b) and the holes moves towards the cathode (red arrows in Fig. 2b). The operations in each voxel can be either that of the full model as shown in Fig. 3a or as an equivalent model as in Fig. 4. Each voxel consists of free and trapped electron and hole charges at each time instant. The movement of charges between the   www.nature.com/scientificreports/ different voxels gives rise to signals at the electrodes. The output of the model are the signals from the electrodes, free and trapped electron and hole charges in the voxels over time. Based on the electron-hole pair input to this model, the outputs (signals, free and trapped charges) are computed over time 35 . Figure Fig. 3a. This approach is as per our previous work 35 . Figure 3b shows the model with the detector discretized into 5 voxels (for illustration). The high energy rays are incident on voxel 3 creating electron-hole pairs in that voxel. The electrons drift towards the anode (right of voxel 5), while the holes drift towards the cathode (left of voxel 1). While drifting from one voxel to another, the electron charges are multiplied by difference of induced potentials, to generate electrical current from movement of electrons ( signal electrons ) as per the Schokley-Ramo Theorem 48 . Similar phenomena occur to generate electrical current from movement of holes ( signal holes ). The induced potential differences due to the motion of charges are fixed based on the geometry and pre-computed in this model 35 .
Additionally, we consider the applied bias voltage to the electrodes to be fixed at V i and V f . In general, the voltage can vary in any manner within the RTSD depending on the defects within the material. Here we consider the voltage at each voxel of the detector to be a linearly increasing function from cathode to the anode. An error term error voltage , is formulated as the difference between the voltages at a particular voxel obtained from successive linear segments, as detailed in our previous work 35 .
The learning-based model is trained with the data simulated in MATLAB. The input is the voxel position and magnitude of the injected electron-hole pair. The signals obtained at the electrodes along with the electron and hole charges (free and trapped) in each of the voxels over time are the output of this learning-based model. The overall loss function is computed as the sum of the squared errors between the signals and charges in the voxels compared to the ground truth signals along with the error 2 voltage . We consider CZT detector with 2 trapping centers for holes and 1 trapping center for electrons 14,34 as shown in Eq. (2). In the loss function, the subscript gt refers to the ground truth data for the particular parameter generated in MATLAB, while the subscript L refers to the data generated by the learning based model as detailed in our previous paper 35 . Learning-based physical models of detector with reduced data. The learning-based full model of the detector uses a loss function taking into consideration the complete data as required by the classical physical equations-signals, voltage distribution in the material, free and trapped charges in the different trapping centers for both electrons and holes. In the real world, each of these data must be obtained from experimental hardware setups with the detector, which not only requires costly equipments, but also skilled manpower and time. In order to address this issue, we propose learning-based models to learn from fewer data than in the full model (which is dictated by the classical physical model). We explore the models by training with fewer data than what is dictated by the classical physical equations, step-wise removing small fraction of relevant data from the full www.nature.com/scientificreports/ learning-model (such as charge trapped and trapping centers) and evaluate the performance of the learningmodel. In the end, we remove a significant portion of the data from the full model and use only the signals for training the physical model. Our models have been developed keeping in mind what can be measured with the hardware setups and labor required to generate these data.
Physical Model-1. The Physical Model-1 has been developed using signals at the electrodes, voltage distribution along the detector, free electron charges and, free and trapped hole charges in one trap center as shown in Eq. (3). We use hole charges corresponding to one trapping center and no electron charges for its trapping center for CZT detector with 2 trapping centers for holes and 1 trapping center for electrons. Our model as shown in Figs. 2 and 3 can be trained using charges corresponding to any one of the hole trapping center. However, for illustration purposes, we have used data from hole trapping center 1. For any other material with NT h trapping centers for holes and NT e trapping centers for electrons, we would use the data corresponding to NT e − 1 trapped charges for electrons and NT h − 1 trapped charges for holes.
The learning-based model uses the same hyperparameters k, l, n in the loss function in Eq. (2) which was derived through optimization in our earlier work 35 .
Physical Model-2. In Physical Model-1, we observed from our simulation experiments that the hyperparameters k, l, n in the loss function are biased heavily towards the free and trapped charges, and hence the hyperparameters l, n are much higher than k. Thus, the Physical Model-2 has been developed considering only the free electron charges and, free and trapped hole charges, corresponding to hole charges in only one trapping center-in this case using hole charges in hole trapping center-1 (for illustration purposes). The loss function ( LF 2 ) is then defined in Eq. (4), The model does not use the signals and the voltage distribution in bulk of the detector, as well as the hole charges in the trapping center 2 of the CZT detector to characterize the material. In general, for NT h trapping centers for holes and NT e trapping centers for electrons, we can use the free hole and electron charges, as well as, electron and hole trapped charges for NT e − 1 and NT h − 1 trapping centers respectively in training the model. Physical Model-3. In the Physical Model-3, we further reduce the dependency on any of the trapped hole charges which were used in the Physical Model-2. This results in a model which can characterize the trapping centers in an equivalent manner. The equivalent trapping and detrapping lifetimes are the equivalent contribution of several trapping and detrapping lifetimes in the detector which contributes to the dynamics of charge motion in the detector as shown in Eq. (1). The properties of the physical detector can be attributed as defect-free properties in addition to equivalent defects in the material. The detector has inherent properties such as transport of charges (electrons and holes) in bulk of the material alongwith recombination of charges which form the defect free model. On the other hand, the defects (equivalent) in the model are contributed due to trapping and detrapping of charges at the trapping centers within the detector. The presence of multiple trapping and detrapping levels can be converted to equivalent trapping and detrapping levels. In such a scenario, for 2 hole trapping levels of CZT with trapping lifetimes τ 1 and τ 2 , the equivalent trapping lifetime is given in Eq. (5), Considering the probability of trapping 1 level as p τ 1 and detrapping 1 lifetime as τ dt 1 , alongwith probability of trapping 2 level as p τ 2 and detrapping 2 lifetime as τ dt 2 , we can calculate the equivalent detrapping lifetime as in Eq. (6), The physical model-3 is designed as a combination of defect-free model and model with equivalent defects. The equivalent computations in a voxel i is shown in Fig. 4. The equivalent trapping and detrapping weights are w hT eq ,i and w hD eq ,i for holes, and similarly for electrons, the corresponding trapping and detrapping weights are w eT eq ,i and w eD eq ,i . The charges in equivalent trap center is q h eq ,i and q e eq ,i for holes and electrons respectively. The loss function consists of only the free electron and hole charges is used to train the model, as shown in Eq. (7), Physical Model-4. In Physical Model-4, we use only the signals generated at the cathode and anodes due to motion of the charged particles to train. Typically, the signals are generated at the electrodes by the superposition of signals generated individually due to transport of electrons and transport of holes. However, in Physical Model-4, we separate out the signals generated due to the transport of electrons from the signals generated due www.nature.com/scientificreports/ to the transport of holes and use them in training the model. The loss function for training the physical model-4 is shown in Eq. (8), It is also observed from our simulation experiments that by using the loss function in Eq. (8) leads to the trained model weights which fail to converge to the ground truth solution. The solution converges to a local minimum which is different from global minimum, and hence the trained weights differ from the ground truth weights. We use a Total Variation (T.V.) regularization on the different weights of the model corresponding to the trapping, detrapping and recombination of electrons and holes to converge the learned solution to the global minimum solution as observed in our simulation experiments. Similar to Physical Model-3, the multiple trapping and detrapping hole coefficients in actual material is learned in the Physical Model-4 as a single equivalent trapping and detrapping weights as well. The loss function gets modified to LF 4,m as shown in Eq. (9), The T.V. regularization ensures smoothness in the trained weights of the model. The optimal values of 1 and 2 are determined through simulation experiments by finding the minimum error between the ground truth weights and the trained weights.
Implementation details. The learning-based physical models of RTSD with reduced data is trained with synthetic data by considering the classical model in MATLAB as in 35 . The scarcity of experimental data in literature for signals and charge distribution in bulk of the RTSDs is the major reason for developing an algorithm for generating simulated data for training and testing the different learning-based physical models. In these learning-based physical models, the input to the model are the injection positions and magnitude of the generated electron-hole pair charges. The magnitude of the injected charges are normalized to 1. The output from the model are the free and trapped charges in the voxels, alongwith the signals at the electrodes. The complete data as dictated by the physical equations for the RTSD is first generated using all the known phenomena such as drift, trapping, detrapping and recombination of electrons and holes. Subsequently, limited data is chosen out of these complete data in order to train the learning based models with complete physical properties for multiple trapping centers or a single equivalent trapping center, depending on the data.
The model weights are initialized during the start of the training process to its initial values. The models are trained over several epochs by computing the loss function based on the output corresponding to each input injections for the different reduced models. The model is a recurrent network structure over time, and hence Backpropagation through Time (BPTT) 51,52 is used to compute the gradients of the loss with respect to the trainable weights in the model. The weights are updated based on a stochastic gradient descent method -ADAM optimization 53 . The learning rate of 5 × 10 −4 is used alongwith 2 momentum terms set as β 1 = 0.9 and β 2 = 0.999 . The optimization reduces the loss function over epochs and the weights are trained. Once the model is trained, the weights of the model converges to the ground truth detector parameters used to generate the data in MATLAB. Our model has been developed using the popular machine learning Tensorflow library 54 in Python in eager execution mode.
The trained weights for the different Physical Models has been evaluated by computing an error metric for each of the weights. For example, for electron trapping weight ( w eT ) with w eT,gt and w eT,lr as the ground truth and trained weights respectively, the error is expressed as, The error is computed over the injection positions of the electrons/holes and the number of voxels over which the weight gets trained over the epochs. The difference between the trained weights and ground truth weights for the trained region is normalized by the ground truth weights, to take into account the different ranges of weights in the model and put equal emphasis on the different model weights. For multiple injections of electron-hole pair, when the model weights are not trained in a contiguous manner, only the voxels where the weights are trained are taken into account in order to compute this error metric. For characterization purposes in this paper, more emphasis is placed on the RTSD properties in the bulk of the material than at the ends. The mean error (Err(Mean)) is computed as the arithmetic mean of these individual weights. The relative error, expressed as for each of the trained weight are also shown for electrons and holes for the four different physical models. For electron and hole weights, the mean of the different electron and hole weights are computed as the relative error metric. (9) LF 4,m =[(sg e,gt − sg e,L ) 2 ] + 1 �∇w eT eq + ∇w eD eq + ∇w e,Rec � 2 + [(sg h,gt − sg h,L ) 2 ] + 2 �∇w hT eq + ∇w hD eq + ∇w h,Rec � 2 .

Results
In this section, we present the simulation experimental results from the different physical models which have been presented in the previous section.  Table 1 shows the different error values of the learned model properties. The error for the electron coefficients are computed from Voxels 9 to 99 since the electrons move towards anode and the coefficients in those voxels gets trained, while for the hole coefficients, the error is computed from Voxels 1 to 59, where the hole move towards cathode. For k = 1 , l = 10 4 and n = 10 3 , the error is minimum, as shown in Table 1. The electron trapping, detrapping and recombination coefficients fit closely to the ground truth values as shown in Fig. 5b. The drift coefficients follow the ground truth which is uniform in the material as shown in Fig. 5a. Here we consider the fact that the hole drift coefficients is one-tenth that of electron drift coefficients (µ h = 0.1µ e ) . The hole trapping, detrapping coefficients for the 2 trapping centers alongwith the recombination coefficients are shown in Fig. 5c. The holes travel from the point of injection towards the cathode which is Voxel 0 in our model. The holes get trapped inside the material due to its lower drift coefficient ( µ h ), and thus the hole coefficients are trained only in those voxels where the holes propagate.

Numerical experiments with Physical Model-2. Unit charge in terms of electron-hole pair injections
at voxel positions 9 with stride of 5 voxels until voxel 59 are fed into the model in order to train the model. The trained weights of the model-drift coefficients, electron coefficients and hole coefficients are shown in Fig. 6a-c, respectively. In the weighted loss function of Eq. (4), we use the weights l = 10 , n = 1 , is based on the same ratio for the weights in the loss function of Eq. (3) used in Physical Model-1 which gives marginally smaller mean error. For the drift coefficients ( µ e ), the trained weights follow the ground truth as shown in Fig. 6a. The learned trapping, detrapping and recombination coefficients for the electrons match closely to the ground truth parameters as in Fig. 6b. Similarly, for holes, the trapping, detrapping coefficients for trapping centers 1 and 2, alongwith the recombination coefficients match closely to the ground truth parameters as in Fig. 6c. For the trained weights as shown in Fig. 6, the error values of drift coefficients ( µ e ), trapping ( w eT ), detrapping ( w eD ), recombination coefficients ( w eRec ) for electrons are 1.6108 × 10 −4 , 0.1241, 0.0240, 0.1673 respectively which are computed for voxels 9 to 99. Similarly, the error values of the trapping 1 ( w hT,1 ), detrapping 1 ( w hD,1 ), trapping 2 ( w hT,2 ), detrapping 2 ( w hD,2 ) and recombination coefficients ( w hRec ) for holes are 0.0768, 0.0447, 0.1005, 0.0640,    Fig. 7a-c respectively. Equation (7) is used as the loss function with weights l = 10 and n = 1 based on the same ratio for the weights in the free and trapped electron and hole charges as in simulation experiments with Physical Model-1 and Physical Model-2. Clearly, the electron drift, trapping, detrapping and recombination coefficients follow the ground truth values. The learned recombination coefficients for the holes follow the ground truth values as well. For multiple trapping centers for holes (2 in this case), the learning based model finds the equivalent trapping center, with equivalent trapping and detrapping lifetimes following Eqs. (5) and (6) respectively. The ground truth values of trapping 1 lifetime is 0.195 µ s and trapping 2 lifetime is 0.094 µ s. This corresponds to the probability of trapping holes in trap center 1 and 2 to be 0.05 and 0.10 respectively. The ground truth equivalent trapping lifetime ( τ eq ) is calculated using Eq. (5) to be 0.0634 µ s. The fraction of holes remaining as free holes after getting trapped in the equivalent trapping center is N after = N before e −dt/τ eq . Considering time step dt = 10 ns, N after = 0.8541N before . Thus, the fraction of holes getting trapped in the equivalent trapping center is 1 − 0.8541 = 0.1459 . Similarly, in the ground truth simulation data, we considered fraction of charge getting detrapped from trap center 1 and 2 are 0.10 and 0.05 respectively. Thus, after detrapping, the fraction of charges remaining in trapping center 1 and 2 would be 0.90 and 0.95 respectively. Considering N after = N before e −dt/τ eq , with the same time step of dt = 10 ns, the detrapping 1 and 2 lifetimes are 94.9122 ns and 194.9573 ns respectively. Considering Eq. (6), and equivalent trapping probability as 0.1459, we compute the equivalent detrapping lifetime τ dt,eq as 145.9 ns. The fraction of holes remaining after detrapping from the equivalent trapping  www.nature.com/scientificreports/ level would be 0.9338. Thus, the fraction of charges getting detrapped from the equivalent trapping center is 1 − 0.9338 = 0.0662 . From our simulation experiments, we find that our learning based model is able to correctly identify the equivalent trapping and detrapping probabilities as shown in Fig. 7c. For the trained weights as shown in Fig. 7, the error values of drift coefficients ( µ e ), trapping ( w eT ), detrapping ( w eD ), recombination coefficients ( w eRec ) for electrons are 1.32 × 10 −5 , 0.0412, 0.0316, 0.0277 respectively which are computed for voxels 9 to 99. Similarly, the error values of the equivalent trapping ( w hT,eq ), equivalent detrapping ( w hD,eq ), and recombination coefficients ( w hRec ) for holes are 0.0954, 0.1957 and 0.3378 respectively which are computed for voxels 1 to 59. The arithmetic mean of the error of these material properties is 0.1042.

Numerical experiments with Physical
Numerical experiments with Physical Model-4. The physical model-4 is trained with signals generated from motion due to electrons and holes separately. We show the convergence of trained hole coefficients and electrons coefficients. For holes, the physical model-4 finds out the equivalent trapping and detrapping coefficients similar to physical model-3. Figure 8 shows the hole trapping, detrapping and recombination coefficients due to electron-hole pair injection at voxel positions 24, 27 and 30 for different 2 in Eq. (9). During training, the trapping hole and detrapping hole weights were bounded in [0.04, 0.07], and [0.15, 0.30] respectively which are close to the actual ground truth weights. The initialization of trapping, detrapping and recombination coefficients for holes are done uniformly at 0.05, 0.2 and 0.005 respectively. This is represented as 'bound' in Table 3. It is seen that for 2 = 0 , without T.V. regularization, the hole trapping, detrapping and recombination coefficients does not converge to the ground truth hole coefficients. On the other hand using the T.V. regularization improves the convergence of these coefficients to actual ground truth values. The different error values, computed for hole coefficients from Voxels 13 to 30 for varying 2 values are shown in Table 2. For 2 = 0.001 and 0.01, the hole trapping and detrapping coefficients are closer to the ground truth coefficients than for 2 = 0.1 and hence smaller the error. However, for recombination coefficients, the trained weights for 2 = 0.1, 0.01 and 0.001 are better than for 2 = 0 . Thus, it is seen that the weights 2 = 0.01, 0.001 in the loss function of Eq. (9) provides better convergence for the hole coefficients.
Additional simulation experiments has been done with 2 = {0.001, 0.01} without bounds on trapping and detrapping coefficients and electron-hole pair injections at Voxels 24, 26 and 28. All the initial weights of trapping and detrapping over the voxels has been uniformly initialized as {0.005, 0.005}, {0.05, 0.2}, {0.07, 0.3} which corresponds to 'far' , 'ip1' and 'ip2' respectively in Figs. 9 and 10. Table 3 shows the different error values, computed for hole coefficients from Voxels 13 to 28. It is observed that bounds on the trapping and detrapping weights do not have any influence on the final trained weights of the holes. However, for 2 = 0.001 , the hole trapping, detrapping and recombination coefficients converge more closely to the ground truth parameters. Additionally, initializing the trapping and detrapping hole weights with 'ip2' converges the trained weights closer to the actual       Table 5. It is seen that for 'grtr' case, the mean error has the value of 0.1159 which is minimum of these three cases. Overall, the case with 'bound' provides us with the minimum error for the electron coefficients. Figure 11 shows the electron drift, trapping, detrapping and recombination coefficients for = 0.1 with 'bound' and 'grtr' cases. The material properties for the electrons converges very closely to the corresponding ground truth values.
Comparison of the different Physical Models. The performance of the different physical models has also been evaluated with the Relative Error in %, is shown in Table 6. The mean relative error due to the electron coefficients (Err2(electrons)) are separated from that of holes (Err2(holes)) and the mean of Err2(electrons) and Err2(holes) are then computed as Err2(Total) in Table 6   www.nature.com/scientificreports/ refers to the relative error result for the model trained for electron coefficients only. The electron injections are at voxel 81, 84 and 87 with 1 = 0.1 with 'bound' condition as described in Table 5. On the other hand, Err2(holes) only refer to the relative error result for the model trained for hole coefficients only. The hole injections are at voxel 24, 26 and 28 with 1 = 0.001 with 'ip2' condition as described in Table 3. It is seen that for Physical Models 1 and 2, the Err2(Total) value is small. On the other hand, for Physical Model 3, the Err2(holes) are maximum. This is because in the learned model, the hole coefficients (equivalent in this case) tend to oscillate around the ground truth value. Overall, the Err2(Total) is less than 9% , which shows good convergence of the RTSD material parameters to the ground truth values.

Discussion
The learning-based approach for obtaining the detector parameters is novel for RTSD. We have developed four physical models using different amount of data for modeling the RTSD. In practice, experimentally generating data for building these learning models, such as free and trapped charges (holes and electrons) in the bulk of the material requires several experimental setups, expert know how and time. Physical learning based model have been developed step-wise using fewer data which directly relates to fewer experimental setups. Physical Model-1 uses most information regarding the system, while Physical Model-4 utilizes just signals in the learning model, which is obtained directly from the electrodes without any additional hardware requirements. We also observe that progressively using reduced or limited data in developing these models affect the characterization properties of the RTSD. While in Physical Models-1 and 2, we characterize all the properties of the RTSD, in Physical Models-3 and 4, the characterization of the RTSD is done in a single equivalent manner for multiple trapping centers (holes in this case). Thus, the Physical Model-3 and 4 is agnostic to the actual (ground truth) number of trapping centers in the RTSD. In this work, we have shown our results of a model with a 100 voxel, but our approach can be extended to models with other voxel as well. Our learning approach learns the properties of the RTSD in a fast and efficient way and can identify defects in the detector spatially and their variations over time.
In experimental results with Physical Model-3, for the hole coefficients in Fig. 7c, we observe slight fluctuations in the trained parameters around the converged value. We observe that these fluctuations gradually diminish if we continue training over additional several epochs. Additionally, in Physical Model-4, we observe that the addition of T.V. regularization to the loss function in Eq. (8), improves the solution drastically and converges the hole and electron coefficients to the ground truth parameters. However, from our numerical experiments with Physical Model-4, we observe that different weights on the T.V. regularization is required for electrons compared to the holes.
In this work, a 1D learning model of the detector trained with different amount of data is presented. The 3D learning model of the detector will follow the same principles. Moreover, in this work, the ground truth data has been simulated using a classical model in MATLAB as in our previous work 35 . In actual practice, the simulation results must be validated with actual experimental data. This experimental data can be obtained using thermoelectric emission spectroscopy, thermally stimulated current measurements, laser induced techniques and others. However, this work reduces the burden on generating experimental data and can still characterize the RTSD at higher resolution (in order of microns) than any other technique in the literature. The impact of additional noise (such as electronic noise) in the data needs to be addressed as well. Nevertheless, extending this model to work with actual experimental data in 3D detector systems is one of the future directions of work.

Conclusion
The paper introduces novel learning-based physical models of the radiation detector using limited data, which characterizes the RTSD. The limited data dictates the requirement of fewer experimental setups and less information to train these models, which is hugely beneficial for practical wide-scale implementation. Four physical models have been demonstrated which progressively utilize fewer data and characterize the material. Based on the amount of data, the models either characterize the detector completely or in an equivalent manner. The model shows promising results which could lead the way for future developments in characterization of the RTSD with fewer experimental setups and data.

Data availability
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.