Response Estimation and Evaluation of Direct-Conversion Dual-Layer Perovskite X-Ray Detectors: A Numerical Study with a Cascaded Signal Model

Han Cui¹, Yu-Hang Tan², Xin Zhang², Hao-Di Wu², Ting Su², Jiong-Tao Zhu³, Hai-Rong Zheng⁴,⁵, Dong Liang⁶,⁴,⁵, Xiang-Ming Sun⁷,†, and Yong-Shuai Ge²,⁴,⁵,⁸,†

¹College of Physics Science and Technology, Guangxi Normal University, Guilin 541004, China
²Research Center for Advanced Detection Materials and Medical Imaging Devices, Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, Shenzhen, Guangdong 518055, China
³College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China
⁴Paul C Lauterbur Research Center for Biomedical Imaging, Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, Shenzhen, Guangdong 518055, China
⁵Key Laboratory of Biomedical Imaging Science and System, Chinese Academy of Sciences
⁶Research Center for Medical Artificial Intelligence, Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, Shenzhen, Guangdong 518055, China
⁷Key Laboratory of Quark and Lepton Physics, Central China Normal University, Wuhan, Hubei 430079, China
⁸National Innovation Center for Advanced Medical Devices, Shenzhen, Guangdong 518131, China

This study aims to numerically investigate the responses of a perovskite-based direct-conversion dual-layer flat-panel detector (DL-FPD). The X-ray sensitivity, spatial resolution quantified by the modulation transfer function (MTF), and detective quantum efficiency (DQE) of the DL-FPD are evaluated numerically using a linear cascade model. Both single-crystal (SC) and polycrystalline (PC) structures of MAPbI₃ are investigated, along with various key parameters such as material thickness, electric field strength, X-ray beam spectrum, and electronic readout noise. The results demonstrate that SC perovskite consistently exhibits better performance than PC perovskite owing to fewer material defects.

Increasing the layer thickness may decrease the MTF, but can also enhance the sensitivity and DQE. Moreover, appropriately increasing the external electric field within the material can improve the sensitivity, MTF, and DQE. Finally, reducing the electronic readout noise can significantly enhance the DQE for low-dose imaging. This study demonstrates the potential of high-quality dual-energy X-ray imaging using direct-conversion perovskite DL-FPDs.

Keywords: X-ray imaging, Dual-layer flat-panel detector, Perovskite X-ray detector

INTRODUCTION

In recent years, metal halide perovskites (MHPs), denoted as ABX₃ (e.g., A: Cs⁺, MA⁺, B: Pb²⁺, Bi²⁺, and X: Br⁻, I⁻) [1], as illustrated in Fig. 1(a) [FIGURE:1], have been considered promising alternatives to traditional semiconductors such as Si, α-Se, CdTe, CdZnTe, diamond, HgI₂, and Ga₂O₃ [2–7] due to their high X-ray absorption capabilities, high charge carrier mobilities µ, and long carrier lifetimes τ. MHPs can be grown in both single-crystal (SC) and polycrystalline (PC) forms and can be integrated with various readout circuits. Owing to their orientation-dependent transport behavior and low defect concentrations, SC MHPs exhibit excellent potential for X-ray detection. For instance, flat-panel detectors (FPDs) fabricated from hybrid organic-inorganic SC MAPbBr₃ [8, 9] and MAPbI₃ [10, 11] achieved significantly higher X-ray detection sensitivity than commercial α-Se detectors. Despite their superior X-ray detection performance, fabricating SC MHPs with scalable dimensions for large-area X-ray detectors is challenging and expensive. Fortunately, experiments have demonstrated that a conversion layer made of PC MHPs can be more easily scaled up to cover large areas at much lower cost [12, 13]. The first PC MAPbI₃-based FPD prototype was demonstrated in 2015, enabling direct detection of soft X-ray photons [14]. In 2017, Kim et al. reported the possibility of direct hard-X-ray imaging using a thick CH₃NH₃PbI₃ film on a thin-film transistor (TFT) array for the first time [12]. In 2021, Deumel et al. developed a new procedure for manufacturing a direct X-ray detector by soft-sintering CH₃NH₃PbI₃ on a hydrogenated amorphous silicon TFT array, employing a grid structure to mechanically adhere the thick perovskite film [15].

In 2022, Xia et al. prepared a TFT array via soft pressing and in situ polymerization of a multifunctional binder (TMTA) for a CH₃NH₃PbI₃ film [16]. In 2024, Liu et al. developed a novel complementary metal oxide semiconductor (CMOS) array-based dynamic perovskite X-ray detector using a CsPbBr₃ film [17]. Although many studies have demonstrated the excellent potential of FPDs composed of MHP materials for X-ray imaging, as illustrated in Fig. 1(c), few studies have extended this research to spectral imaging, a technique that captures X-ray attenuation information at multiple energy levels. In this manner, X-ray spectral imaging allows distinction between different materials, enhancing image contrast and specificity, making it valuable for medical, industrial, and security imaging applications [18, 19]. In 2022, Pang et al. proposed a perovskite X-ray detector with a vertical structure that offers opportunities for improved multi-energy X-ray imaging [20]. However, this innovative vertical design presents challenges for fabricating detectors with large surface areas. Therefore, alternative solutions for utilizing MHP materials in spectral imaging require further investigation.

In biomedical X-ray imaging applications, the dual-layer FPD (DL-FPD) has emerged as a promising tool for quantitative dual-energy X-ray imaging [21–23]. The DL-FPD comprises two stacked FPD layers with varying thicknesses. In this configuration, lower-energy X-ray photons are detected by the top layer, whereas higher-energy photons are detected by the bottom layer. Studies have also explored the use of DL-FPDs in dual-energy computed tomography imaging [24–27]. However, current DL-FPDs primarily employ scintillator materials, which pose a significant challenge: the imaging performance of the bottom detector, including its spatial resolution, sensitivity, and detective quantum efficiency (DQE), is inferior to that of the top detector, consequently limiting dual-energy X-ray imaging results. Alternatively, the DL-FPD can be constructed using MHP materials, as shown in Fig. 1(d). As opposed to scintillator-based indirect X-ray detectors, MHP-based X-ray detectors directly convert X-ray photons into electric charges, which are then driven by the applied electric field and subsequently collected by electrodes. DL-FPDs made of MHP materials offer improved imaging performance compared with scintillator-based detectors. For instance, because the applied electric field can be well confined, signal spreading in direct-conversion X-ray FPDs is less severe than in indirect-conversion X-ray FPDs, where X-ray-converted visible light photons may spread across several neighboring detector elements. Consequently, the spatial resolution of the bottom layer of MHP-based DL-FPDs can be enhanced. Despite the potential for improved imaging performance of MHP-based DL-FPDs, comprehensive investigations to evaluate the imaging performance of direct X-ray DL-FPDs have not yet been conducted.

This study focuses on estimating and evaluating the responses of a novel direct-conversion X-ray DL-FPD made of MHP material. Specifically, the imaging performance, including X-ray sensitivity, modulation transfer function (MTF), and DQE, of the MHP-based direct X-ray DL-FPD is explored through numerical calculations based on a linear cascaded model (see workflow illustrated in Fig. 1(e)). Moreover, the dependency of overall imaging performance on various factors such as MHP material layer thickness, X-ray beam spectra, and external electric fields is investigated. Additionally, the electronic readout noise effect is analyzed by considering three different readout arrays: hydrogenated amorphous silicon TFT (α-Si: TFT), metal oxide TFTs based on indium gallium zinc oxide (IGZO-TFT), and CMOS, as shown in Fig. 1(b). To date, α-Si: TFT arrays have been the leading choice for perovskite X-ray FPDs owing to their simple structure, low cost, and suitability for large-area applications. CMOS technology [29] combines both p-type and n-type metal-oxide-semiconductor field-effect transistors to achieve low power consumption and high noise immunity, offering advantages such as lower readout noise and higher readout speeds, making them promising alternatives. IGZO-TFTs [30] have a structure similar to α-Si: TFT, with the primary difference being the use of an IGZO layer instead of amorphous silicon, resulting in less readout noise and increased readout speed; these characteristics are intermediate between α-Si: TFT and CMOS, positioning IGZO-TFT as a competitive option for fabricating X-ray detectors.

The remainder of this paper is organized as follows. Section II reviews the linear cascade model and derives calculations for sensitivity, MTF, and DQE. Section III introduces the main parameters considered in this study and presents details of the numerical calculations. Section IV presents the DL-FPD response results obtained using different parameters. Section V provides discussion and a brief conclusion.

II. DETECTOR RESPONSE MODEL

A. Linear Cascade Model

The seven dominant stages of the linear cascade model [31–34], which describe the propagation of signals and noise from X-ray photons to electric signals in a direct FPD, are summarized as follows:

Stage 0: X-ray input. (X-ray quanta $\bar{\Phi}_0(E)$, fluctuation $\sigma_0(E)$. See Eqs. 8 and 9.)

Stage 1: Absorption of X-ray photons. (Absorption efficiency $g_1(E, x)$, variance $\sigma_1(E, x)$. See Eqs. 10 and 11.)

Stage 2: Electron-hole cloud effect. ($\text{MTF}_m(E; f)$ owing to charge cloud. See Eq. 12.)

Stage 3: Electron-hole conversion. (Charge conversion multiplication $g_3(E)$, variance $\sigma_3(E)$. See Eqs. 15 and 16.)

Stage 4: Electron-hole collection. (Charge collection efficiency $g_4(E, x)$, variance $\sigma_4(E, x)$. See Eqs. 17 and 18.)

Stage 5: Electron-hole blurring by charged traps. ($\text{MTF}_{tr}(E; f)$ owing to charge trapping. See Eq. 19.)

Stage 6: Electron-hole blurring by pixel aperture ($\text{MTF}_a(f)$ owing to aperture collection. See Eq. 23.)

Stage 7: Signal output. (Additive noise power spectrum owing to dark current shot noise $\sigma_{shot}$ and electronic readout noise $\sigma_{readout}$. See Eq. 26.)

The variables $E$, $x$, and $f$ denote X-ray photon energy, vertical position in the detector, and spatial frequency, respectively. Essentially, these seven stages can be divided into three steps: the gain step (including stages 1, 2, and 4), the stochastic blurring step (including stages 3 and 5), and the deterministic blurring step (including stage 6). Further details of the cascaded model can be found in Appendix V.

For a given gain stage $n$, the propagations of signal quantum $\Phi_n$ and noise power spectrum $\text{NPS}_n$ are expressed as follows [31, 33]:

$$
\bar{\Phi}n(E; f) = \bar{g}_n(E) \bar{\Phi}(E; f)
$$

$$
\text{NPS}n(E; f) = \bar{g}_n^2(E) \text{NPS}}(E; f) + \bar{\sigman^2(E) \bar{\Phi}(E; f)
$$

For a stochastic blurring stage $n$, the propagations of $\Phi_n$ and $\text{NPS}_n$ can be expressed as...

B. Response Evaluation

To evaluate the responses of the perovskite DL-FPD, the sensitivity, MTF, and DQE are calculated separately for the top and bottom layers. Specifically, the sensitivity $S_i$ represents the ability to convert X-ray photons into electric charges for the $i$-th layer, where $i = t$ or $b$ denotes the top or bottom layer in a DL-FPD, respectively. Mathematically:

$$
S_i = \frac{\int \bar{\Phi}i^0(E) S}}(E) \int_0^{L_i} \bar{g}_1(E, x) \bar{g}_4(E, x) \, dx \, dE}{\int \bar{\Phi}_0(E) \, dE
$$

where $\bar{\Phi}i^0$ and $L_i$ denote the received X-ray quanta and thickness of the $i$-th layer, respectively. In particular, $S(E)$, which represents the maximum sensitivity, is expressed as [35]:}

$$
S_{\text{max}}(E) = 1.14 \times 10^8 \cdot \frac{5.45 \times 10^{13} e (\alpha(E)/\rho){\text{air}}}{W \alpha}}(E)
$$

where $e$ (C) is the elementary charge, $W$ (eV) is the average energy for electron-hole pair creation, $(\alpha/\rho){\text{air}}$ (cm²/g) is the mass-energy absorption coefficient of air, and $\alpha$ and $\alpha$ (cm⁻¹) are the energy absorption and absorption coefficients of the detector material, respectively. The factor $1.14 \times 10^2$ converts the inverse exposure units from R⁻¹ into Gy⁻¹ [36], resulting in calculated sensitivity in units of µC·Gy⁻¹·cm⁻². Note that the X-ray quanta $\Phi_t^0(E)$ received on the top layer differ from $\Phi_b^0(E)$ received on the bottom layer, considering that X-rays are filtered by the top layer. $\Phi_i^0(E)$ can be expressed as follows:}

$$
\bar{\Phi}_t^0(E) = \bar{\Phi}_0(E)
$$

$$
\bar{\Phi}_b^0(E) = \bar{\Phi}_0(E) \left[1 - \int_t \bar{g}_1(E, x) \, dx\right]
$$

The MTF quantifies the contrast of an object at various spatial frequencies $f$ (in units of line pairs per millimeter, lp/mm) and is often used to quantify the spatial resolution of X-ray detectors. In particular, the $\text{MTF}_i$ of each detector layer is expressed as:

$$
\text{MTF}_i(f) = \frac{\int \bar{\Phi}_i^0(E) \text{MTF}_i^m(E; f) \text{MTF}_i^{\text{tr}}(E; f) \, dE}{\int \bar{\Phi}_i^0(E) \, dE} \cdot \text{MTF}_i^a(f)
$$

In addition, the DQE describes the spatial frequency-dependent signal-to-noise ratio (SNR) propagation efficiency of the X-ray detector. The $\text{DQE}_i$ of each detector layer is defined as the square ratio of the corresponding output $\text{SNR}_i$ to the top-layer input $\text{SNR}_t$:

$$
\text{DQE}_i(f) = \frac{(\text{SNR}_i^{\text{out}})^2}{(\text{SNR}_t^{\text{in}})^2} = \frac{(\bar{\Phi}_7^i(f))^2}{\bar{\Phi}_0 \cdot \text{NPS}_7^i(f)}
$$

Note that both the top and bottom layers utilize the same input SNR as the input X-ray quantum.

III. METHODS

A. Detector Configuration

In this study, a metal-semiconductor-metal (MSM) structure with ohmic contacts is assumed for the perovskite detector, ignoring internal photoconductive gain. Key parameters that may affect DL-FPD imaging performance, such as input beam spectra, layer thickness, electric field, dark current, readout noise, and material structure, are considered during numerical calculations.

1. Material Structure

Calculations are performed for both SC and PC MAPbI₃. The mobility-lifetime product for electrons $(\mu\tau)_e$ and holes $(\mu\tau)_h$ is assumed to be equal, set to $1 \times 10^{-3}$ cm²·V⁻¹ for SC MAPbI₃ and $1 \times 10^{-4}$ cm²·V⁻¹ for PC MAPbI₃ [10–12, 15].

2. Beam Spectra

Three X-ray beam spectra with different radiation qualities (RQA) are simulated according to International Electrotechnical Commission (IEC) 61267:1994 guidelines: (a) RQA5, (b) RQA7, and (c) RQA9 (see Table 1 [TABLE:1] and Fig. 2(a) [FIGURE:2]).

Table 1. Key parameters for different X-ray beam settings.

3. Layer Thickness

The total thickness $L$ of the top and bottom MHP layers is fixed at 1.0 mm to ensure sufficient (≥80%) X-ray attenuation. As shown in Fig. 2(b) [FIGURE:2], the attenuation efficiencies of the 1.0 mm MAPbI₃ layer are 82%, 88%, and 96% for RQA5, RQA7, and RQA9 beams, respectively. The thickness of the top layer, denoted by $L_t$, increases from 100 µm to 500 µm at intervals of 100 µm, corresponding to a top-layer thickness occupation $L_t/L$ of 10%, 20%, 30%, 40%, and 50%. The sensitivities, MTF($f$), and DQE($f$) of the top and bottom layers are calculated for different $L_t/L$ values.

4. Electric Field

The electric field inside each MHP layer, as shown in Fig. 1(c), is calculated as the ratio of applied bias voltage $U_i$ to layer thickness $L_i$, that is, $F_i = U_i/L_i$ ($i = t$ or $b$). For sensitivity estimations, the electric field is explored within the range of 0.01 to 0.6 V/µm in increments of 0.01 V/µm. The MTF and DQE responses are investigated under electric fields of 0.01, 0.05, 0.1, 0.5, and 1.0 V/µm.

5. Dark Current

As discussed in Section V.8, dark current impacts DQE by introducing additive noise. Assuming ohmic-type contacts, the dark current density can be modeled as $J_d = \sigma_d F$, where $\sigma_d$ denotes the dark conductivity of the material. According to literature [11, 12, 14, 15, 37], the $\sigma_d$ of PC MAPbI₃ can achieve values comparable to SC MAPbI₃, with both being approximately $1 \times 10^{-10}$ (Ω·cm)⁻¹. Therefore, a $\sigma_d$ value of $1 \times 10^{-10}$ (Ω·cm)⁻¹ is assigned to both SC and PC MAPbI₃. Consequently, the dark current density depends on the electric field applied inside the MHP layers, and its impact on DQE is included in the electric field analysis and not discussed independently.

6. Readout Noise

Similarly, electronic readout noise $\sigma_{\text{readout}}$ affects DQE by introducing additive noise, as discussed in Section V.8. Three types of pixelated readout circuits are compared: CMOS, IGZO-TFT, and traditional α-Si: TFT, with typical readout noise levels of 200 e⁻, 700 e⁻, and 2000 e⁻, respectively [30].

B. Numerical Study

The RQA5, RQA7, and RQA9 beam spectra are generated using SpekCal [38]. The pixel size is set to 100 µm. The mass density $\rho_{\text{mass}}$ of MAPbI₃ material is 4.16 g/cm³. X-ray absorption coefficients are obtained from the National Institute of Standards and Technology (NIST) database. The electron-hole pair creation energy $W_{\pm}$ is 4.7 eV [12, 15]. The signal integration period $\Delta t$ is set to 10.0 ms. Incident X-ray exposure is examined at 1.0 and 0.1 µGy to investigate electric field and readout noise effects, whereas it is fixed at 1 µGy to investigate other parameters. Detailed settings are presented in Table 2 [TABLE:2]. All numerical calculations are performed using Python (version 3.10) on a desktop (Dell XPS, Intel i7-13700, 16 GB DDR5 RAM).

Table 2. Key parameters used for numerical simulations.

IV. RESULTS

A. Dependence on Material Thickness

The calculated sensitivity responses are presented in Fig. 3 [FIGURE:3] for five different $L_t/L$ ratios: 10%, 20%, 30%, 40%, and 50%. Solid and dashed lines represent sensitivity responses of the top and bottom layers, respectively, with the input beam spectrum set to RQA7. Sensitivity increases as electric field increases and reaches maximum when the external electric field exceeds a certain threshold. In general, SC MAPbI₃ achieves maximum sensitivity at lower electric field than PC MAPbI₃, indicating that SC MAPbI₃ can achieve sufficient charge collection efficiency at lower electric field. Moreover, sensitivity responses of the top and bottom detector layers exhibit a competitive relationship as $L_t/L$ varies. Specifically, top detector layer sensitivity increases as $L_t/L$ increases, whereas bottom layer sensitivity decreases because more X-ray photons are absorbed by the top layer, reducing the number collected by the bottom layer. Finally, the ratio $L_t/L$ is crucial for determining relative sensitivities of the top and bottom layers. As observed, top layer sensitivity exceeds bottom layer sensitivity when $L_t/L \geq 30\%$, whereas it becomes lower when $L_t/L \leq 20\%$. Therefore, a ratio $L_t/L$ between 20% and 30% is preferred to achieve similar sensitivities for both layers.

Estimated MTF($f$) curves obtained from the five different $L_t/L$ ratios are plotted in Fig. 4 [FIGURE:4]. The input spectrum is set to RQA7 and external electric field to 0.1 V/µm for both layers. The ideal MTF, determined by pixel aperture collection blurring, is indicated by cyan lines. For SC MAPbI₃, generated MTF($f$) curves are almost identical across different $L_t/L$ values and closely resemble the ideal MTF, as shown in Fig. 4(a), indicating minimal impact on spatial resolution. However, under the same conditions, MTF($f$) of detectors made of PC MAPbI₃ varies dramatically with different $L_t/L$ ratios, with most values lower than ideal MTF except for very thin top layers when $L_t/L = 10\%$ (see Fig. 4(b)). Furthermore, MTF($f$) of the top detector layer exhibits a competitive relationship with that of the bottom layer as $L_t/L$ varies. For very thin top layers such as $L_t/L = 10\%$, top MTF($f$) is notably higher than bottom MTF($f$), as indicated by solid and dashed blue lines in Fig. 4(b). As $L_t/L$ increases, top detector layer MTF($f$) decreases while bottom detector layer MTF($f$) increases due to more severe charge trapping with increasing material thickness. When material thicknesses become equal ($L_t/L = 50\%$), top detector layer MTF($f$) approaches that of the bottom layer.

Estimated DQE($f$) curves obtained from different $L_t/L$ values are plotted in Fig. 5 [FIGURE:5]. The input spectrum is set to RQA7, electric field to 0.1 V/µm for both detector layers, entrance dose to 1 µGy, and CMOS readout noise is assumed. For both SC and PC MAPbI₃, top detector layer DQE($f$) increases with increasing $L_t/L$, whereas bottom detector layer DQE($f$) decreases. This occurs because more X-ray photons are absorbed by the top layer as it becomes thicker, while fewer are absorbed by the bottom layer. Additionally, DQE($f$) values of detectors made from PC MAPbI₃ decrease more rapidly in the high spatial frequency range compared with SC MAPbI₃, especially for the bottom layer (see Fig. 5(b)), attributed to more severe signal blurring from charge trapping in PC materials. Similar to sensitivity responses, top detector layer DQE($f$) is lower than bottom layer when $L_t/L \leq 20\%$ but exceeds bottom layer when $L_t/L \geq 30\%$. Thus, an $L_t/L$ ratio between 20% and 30% can help achieve similar DQE($f$) values for both layers.

Overall, sensitivity, MTF($f$), and DQE($f$) of the top layer exhibit a competitive relationship with those of the bottom layer for various $L_t/L$ values. In subsequent studies, we concentrate on scenarios where $L_t/L = 30\%$.

B. Dependence on Beam Spectrum

Estimated sensitivities with respect to three different input beam spectra (RQA5, RQA7, and RQA9) are shown in Fig. 6 [FIGURE:6]. For the top detector layer, sensitivity responses for RQA5 and RQA7 are identical but higher than RQA9 (see solid lines in Fig. 6), jointly determined by X-ray absorption efficiency and charge-conversion multiplication. For the bottom layer, sensitivity response is highest for RQA9 and lowest for RQA9 (see dashed lines in Fig. 6) because the bottom layer is sufficiently thick to generate similar absorption efficiencies for X-ray photons of different energies, resulting in higher sensitivity for higher-energy beam spectra. Interestingly, sensitivities of top and bottom detector layers are similar for RQA9, indicating that an appropriately low $L_t/L$ should be selected for low-energy spectra to generate similar sensitivity responses.

Obtained MTF($f$) curves for different input spectra are plotted in Fig. 7 [FIGURE:7]. The electric field is fixed at 0.1 V/µm for both layers. MTF($f$) curves are almost identical and closely resemble ideal MTF for SC MAPbI₃ material (see Fig. 7(a)). However, MTF($f$) curves are consistently lower than ideal MTF for PC MAPbI₃ (see Fig. 7(b)) due to charge-trapping-induced signal blurring. Specifically, high-energy spectra tend to generate slightly better MTF($f$) than low-energy spectra for the bottom layer.

DQE($f$) responses are plotted in Fig. 8 [FIGURE:8]. The entrance dose is set to 1 µGy and CMOS readout noise is assumed. For the top layer, DQE($f$) is highest for RQA5 and lowest for RQA9 (see solid lines in Fig. 8). Conversely, DQE($f$) is highest for RQA9 and lowest for RQA5 in the bottom layer (see dashed lines in Fig. 8). Moreover, high-energy spectrum RQA9 exhibits a narrower DQE($f$) gap between top and bottom layers than RQA5 and RQA7 spectra. Consequently, a lower $L_t/L$ ratio is preferable to achieve similar DQE($f$) responses between layers.

C. Dependence on Electric Field

MTF($f$) responses with respect to five different electric field strengths (0.01, 0.05, 0.1, 0.5, and 1.0 V/µm) are plotted in Fig. 9 [FIGURE:9]. The input spectrum is set to RQA7 and $L_t/L$ to 30%. For SC MAPbI₃, MTF($f$) exhibits less dependence on electric field except for the lower bottom detector layer with ultra-small electric field of 0.01 V/µm (see Fig. 9(a)). Conversely, for detectors made of PC MAPbI₃, MTF($f$) of the bottom detector layer depends heavily on electric field. Increasing electric field can significantly enhance MTF($f$) response, as shown in Fig. 9(b). At high electric fields, MTF($f$) of the bottom detector layer can become comparable to or even outperform that of the top detector layer. For example, the bottom layer with 0.5 V/µm exhibits similar MTF($f$) to the top layer with 0.05 V/µm. However, increasing electric field leads to higher dark current density, potentially compromising detection limit and DQE($f$) response. Hence, electric field must be optimized to generate comparable MTF($f$) in both layers.

DQE($f$) responses with respect to different electric fields are shown in Fig. 10 [FIGURE:10] for 1 µGy and 0.1 µGy. An RQA7 input spectrum and CMOS readout noise are assumed. For SC MAPbI₃, electric field has negligible impact on DQE($f$) except for ultra-low electric field of 0.01 V/µm (see Figs. 10(a) and 10(c)). Conversely, reducing electric field dramatically degrades DQE($f$) of detectors made from PC MAPbI₃ (see Figs. 10(b) and 10(d)) due to insufficient charge collection efficiency. Consequently, DL-FPDs made from PC MAPbI₃ can achieve similar DQE($f$) to SC MAPbI₃ if higher electric field is applied. Increasing electric field does not always enhance DQE($f$) because charge collection efficiency saturates beyond certain field strength. For example, the DQE($f$) curve obtained using 0.5 V/µm is similar to that using 1 V/µm, as shown in Fig. 10. Under these conditions, continuously increasing electric field increases dark current but does not improve DQE($f$).

D. Dependence on Readout Noise

DQE($f$) responses with respect to three different electronic readout noise levels are plotted in Fig. 11 [FIGURE:11] for CMOS, IGZO-TFT, and α-Si: TFT circuits. The electric field is set to 0.5 V/µm to ensure sufficient charge collection efficiency for both layers, and only PC MAPbI₃ material is investigated. DQE($f$) responses of IGZO-TFT are comparable to CMOS and higher than α-Si: TFT for 1 µGy dose level (see Fig. 11(a)). For 0.1 µGy dose level, DQE($f$) responses of CMOS are slightly higher than IGZO-TFT but significantly higher than α-Si: TFT (see Fig. 11(b)). Therefore, back-plate readout noise has more pronounced impact on low-dose imaging tasks, suggesting that a back-plate with lower readout pixel noise is recommended for low-dose imaging scenarios.

V. DISCUSSION AND CONCLUSION

In this study, responses of a direct-conversion perovskite DL-FPD were investigated numerically based on a linear cascade signal model under various configurations. Specifically, sensitivity, MTF($f$), and DQE($f$) were evaluated and compared across settings including beam spectrum, material structure, material thickness, electric field, and readout noise. Although some material parameters such as attenuation coefficient and mobility-lifetime product ($\mu\tau$) were tailored to MAPbI₃ material, the methods and results can be easily generalized to analyze DL-FPDs made from other MHP materials such as CsPbBr₃.

Material thickness strongly impacts sensitivity, MTF($f$), and DQE($f$) responses of top and bottom detector layers. Sensitivity and DQE($f$) of the top detector layer improve as $L_t/L$ ratio increases, whereas those of the bottom layer decrease. In contrast, MTF($f$) of the top layer degrades with increasing $L_t/L$, whereas that of the bottom layer is enhanced. Consequently, thickness should be optimized to ensure good low-energy imaging performance, especially for the top layer. X-ray beam spectra may significantly affect sensitivity and DQE($f$) responses while having minimal impact on MTF($f$). In high-energy imaging scenarios, sensitivity and DQE($f$) of the top layer may decrease while those of the bottom layer may increase.

Increasing electric field can improve sensitivity, MTF($f$), and DQE($f$), potentially bridging performance gaps between PC and SC materials. Nonetheless, sensitivity may saturate when exceeding certain threshold. Moreover, increasing electric field may consistently increase dark current and potentially degrade DQE response. Therefore, appropriate electric field is required to achieve satisfactory imaging performance. Additionally, a pixel array with lower readout noise is necessary to achieve high DQE performance for low-dose imaging applications.

This study has several limitations. First, a fairly ideal detector setting was considered without accounting for non-uniform material responses, which can potentially impact detector imaging performance. Second, we assumed the detector functions as a photoresistor with no photoconductive gain. However, perovskite-based detectors may be photodiodes with p-i-n structure, which could have much higher sensitivity due to internal photoconductive gain [47, 48]. Additionally, traps that cause signal blurring may enhance sensitivity and SNR [49], making it important to balance signal blurring with SNR. Third, $\mu\tau$ values were assumed equal for electrons and holes in MAPbI₃ [50]; however, $\mu\tau_h$ may be slightly greater than $\mu\tau_e$, potentially affecting MAPbI₃ detector performance when electrodes are interchanged. Fourth, we assumed uniform electric field in the region directly opposite electrodes with zero electric field at pixel boundaries. In reality, weak electric field exists at boundaries, which may cause signal loss or charge sharing, degrading MTF and DQE. These issues can be mitigated via advanced designs such as guard rings and nodal separation on CMOS chips [51]. Fifth, only one popular MHP material, MAPbI₃, was investigated; other MHP materials require detailed exploration for detector development. Finally, imaging performance of MHP DL-FPD was only investigated for total material thickness of 1.0 mm; analysis should be repeated for thicker MHP detectors. No experiments were performed to verify numerical predictions due to lack of DL-FPD prototype. Future studies should conduct detailed investigations to validate these findings.

In conclusion, direct-conversion DL-FPD made of MHP material has potential to achieve superior sensitivity, consistent spatial resolution, and high detection efficiency compared with traditional indirect-conversion DL-FPD made of scintillator materials. Future high-quality dual-energy imaging using novel direct-conversion perovskite DL-FPDs will be very useful.

1. X-ray Input

The mean input quantum $\bar{\Phi}_0$ of X-ray photons per unit exposure of radiation and per unit area can be calculated as follows [33, 39]:

$$
\bar{\Phi}0 = \int \frac{5.45 \times 10^{13} p(E)}{E \cdot (\alpha(E)/\rho) \, dE}}
$$

where $p(E)$ is the X-ray probability density at energy $E$. $\bar{\Phi}_0$ has units of cm⁻²·R⁻¹ from Eq. 8 and can be converted to cm⁻²·Gy⁻¹ by multiplying by a factor of $(8.76 \times 10^{-3})^{-1}$ R/Gy [36]. The input X-ray quantum obeys Poisson distribution with variance:

$$
\bar{\sigma}_0 = \sqrt{\bar{\Phi}_0}
$$

2. X-ray Absorption

X-ray photons absorbed by perovskite material with attenuation coefficient $\alpha(E)$ have mean absorption probability $\bar{g}_1$ at any position $x'$ expressed as:

$$
\bar{g}_1(E, x) = e^{-x/\Delta(E)}
$$

where $x$ is normalized distance of thickness $L$ ($x = x'/L$, $0 < x \leq 1$) and $\Delta$ is normalized attenuation coefficient:

$$
\Delta(E) = 1/[\alpha(E)L]
$$

Because interaction between X-ray and material is a binary selection process, $g_1(x)$ has variance [33]:

$$
\bar{\sigma}_1(x) = \bar{g}_1(x)(1 - \bar{g}_1(x))
$$

3. EHP Cloud Blurring

The primary photoelectron deposits its energy and creates electron-hole pairs (EHPs) within a distance in any direction. This distance depends on photoelectron energy and material properties. Therefore, an X-ray photon creates an EHP cloud that causes stochastic blurring and degrades intrinsic spatial resolution. The process contribution to MTF can be expressed as [40]:

$$
\text{MTF}_m(E; f) \approx \exp(-\pi^2 \sigma(E)^2 f^2)
$$

where $\sigma$ is proportional to the maximum range of the primary photoelectron $R_{\text{max}}$. An empirical expression for $R_{\text{max}}$ can be obtained from [41]:

$$
R_{\text{max}}(E) = \frac{2.761 \times 10^{-6} \times \text{Mat} \cdot E^{5/3}}{\rho Z^{8/9} (1 + 0.978 \times 10^{-6}E)^{5/3} (1 + 1.957 \times 10^{-6}E)^{4/3}}
$$

where Mat is atomic mass and $Z$ is atomic number. According to [40], $\sigma$ can be approximated as $\sigma = R_{\text{max}}/2.5$.

4. EHP Conversion

An X-ray photon with energy $E$ absorbed by the material emits a primary photoelectron that creates thousands of charge carriers (EHPs). The mean quantity of carriers created by an X-ray photon can be calculated as [33]:

$$
\bar{g}3(E) = \frac{E \alpha}}(E)}{\alpha W_{\pm}
$$

where $\alpha_{\text{en}}$ is the energy absorption coefficient of the material and $W_{\pm}$ is the EHP creation energy. This process obeys Poisson distribution with variance:

$$
\bar{\sigma}_3 = \sqrt{\bar{g}_3(E)}
$$

5. Charge Collection

Charge carriers are collected by electrodes with efficiency $g_4$. Assuming electrons are collected by bottom electrodes, the mean efficiency $\bar{g}_4$ at normalized position $x$ is described by Ramo's theorem [42] and expressed as [43]:

$$
\bar{g}_4(x) = \chi_h(1 - e^{-x/\chi_h}) + \chi_e(1 - e^{-(1-x)/\chi_e})
$$

where $\chi_{e(h)}$ is the electron (hole) normalized schubweg given by:

$$
\chi_{e(h)} = \frac{\mu_{e(h)} \tau_{e(h)} F}{L}
$$

and $F$ is the electric field in the detector bulk. The corresponding variance of $g_4$ is calculated as [43]:

$$
\bar{\sigma}_4(x) = \chi_e^2 + \chi_h^2 - \chi_h^2 e^{-2x/\chi_h} - \chi_e^2 e^{-2(1-x)/\chi_e} - 2\chi_h x e^{-x/\chi_h} - 2\chi_e(1-x)e^{-(1-x)/\chi_e}
$$

6. Trap Blurring

Trapped charge carriers in the detector bulk also blur signals stochastically. The MTF due to charge-trapping effect is denoted as $\text{MTF}_{\text{tr}}$, modeled by Fourier transformation of the line spread function of induced charge caused by trapped charges in the bulk [44]:

$$
\text{MTF}_{\text{tr}} = G(f)/G(0)
$$

where $G(f)$ and $G(0)$ are given by complex expressions involving $\Delta$, $\chi_e$, $\chi_h$, $\omega = 2\pi f$, and $\eta$ (mean absorption efficiency calculated in combination with Eq. 10).

7. Aperture Collection Blurring

Electric signals are collected and averaged by pixels, resulting in deterministic blurring expressed as:

$$
\text{MTF}_a = |\text{sinc}(af)|
$$

where $a$ denotes pixel dimension.

8. Additive Noise

Additive noise is mainly composed of dark current noise $\sigma_d$ and electronic readout noise $\sigma_{\text{readout}}$. Dark current $I_d$ exists in perovskite materials, and its fluctuations create noise [34, 45]. Accumulated dark charge quantum $Q_d$ contributes to shot noise and can be calculated by integrating $I_d$ over interval time $\Delta t$. It follows Poisson distribution, yielding $\sigma_{\text{shot}} = \sqrt{Q_d}$. In addition to shot noise, $1/f$ noise is a dominant fluctuation source [46], but can only be determined experimentally and is often unavailable for materials of interest; therefore, it is not discussed in this study. Dark current contribution to NPS can be modeled as:

$$
\text{NPS}_d = \sqrt{\frac{\bar{I}_d \Delta t}{e}}
$$

where $e$ denotes elementary charge (C). Electronic noise includes readout noise $\sigma_{\text{readout}}$ contribution, which is strongly dependent on readout technology, with NPS contribution:

$$
\text{NPS}{\text{readout}} = \sigma^2}
$$

Overall NPS for additive electronic noise is:

$$
\text{NPS}{\text{add}} = \text{NPS}_d + \text{NPS}}
$$

AUTHOR CONTRIBUTIONS

All authors contributed to study conception and design. Material preparation, data collection, and analysis were performed by Han Cui. The first draft was written by Han Cui and all authors commented on previous versions. All authors read and approved the final manuscript.

DATA AVAILABILITY

The data supporting this study's findings are openly available at https://cstr.cn/31253.11.sciencedb.j00186.00816 and https://www.doi.org/10.57760/sciencedb.j00186.00816.

CONFLICT OF INTEREST

The authors declare no competing interests.

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