In recent years，planar single-photon avalanche diodes （SPADs），sometimes referred to as planar Geiger-mode avalanche photodiodes （GM-APDs），have been extensively utilized in several disciplines，including quantum key distribution （QKD），quantum imaging，and three-dimensional laser detection and ranging （LADAR） ^［1-5］. SPADs that utilize the InGaAs/InP material system exhibit exceptional sensitivity at the single-photon level within the near-infrared （NIR） range of 900-1700 nm. Consequently，they are regarded as very promising detectors for single-photon detection within the short-wave infrared （SWIR） spectrum ^［6-11］. SPADs function at voltages over the threshold for avalanche breakdown，operating in Geiger mode. The disparity between the operating voltage and the breakdown voltage in Geiger mode is denoted as the excess bias. Dark counting rate （DCR），photon detection efficiency （PDE），and afterpulse probability are crucial metrics that characterize the operation of SPADs. The width of the multiplication region has a substantial impact on PDE，DCR，and afterpulse probability ^［13-17］. Conventional InP-based SPADs that are fabricated utilizing double diffusion techniques successfully lower the electric field at the edges of the InP p-region. This helps to prevent premature breakdown in the multiplication area and enhances the likelihood of avalanche events occurring at the center of the multiplication region ^［12］. The extent of Zn diffusion in the double diffusion process has a direct impact on the multiplication width （MW），whereas shallow diffusion is essential for minimizing the electric field at the edges. Hence，achieving accurate manipulation of the dual diffusion behavior is crucial in the production of high-performance devices.

The prevailing configuration of InGaAs/InP SPADs is the Separate Absorption Grading Charge Multiplication （SAGCM） structure，which is distinguished by the distinct segregation of the multiplication zone and the absorption region. The P+ active area in the InP cap layer is created via diffusion. The double diffusion structure is initially created with a shallow depth using the first mask，and subsequently，a deep diffusion structure is formed with a greater depth using the second mask. The double Zn diffusion structure enhances the concentration of the intense electric field needed for avalanche breakdown in the multiplication zone，facilitating the detection of large-scale signals and the efficient identification of photo-generated carriers. A 2013 study conducted by Politecnico di Milano analyzed the influence of multiplier layers on the efficiency of InGaAs/InP SPAD devices ^［14］. The study demonstrated that diminishing the MW led to a decline in the breakdown voltage. Furthermore，PDE and DCR both showed a progressive rise as the MW dropped. In 2016，the University of Science and Technology of China did an analysis on how the multiplier layer affects the performance of InGaAs/InP SPAD devices ^［18］. The investigation demonstrated a robust association between the MW and the performance of PDE and DCR. While numerous theoretical studies have been published on the impact of the MW ^［13-18］，there is a scarcity of papers about the prediction of the multiplier layer's width.

The diffusion process becomes increasingly intricate when the SPADs undergo two sequential diffusion stages，with deep diffusion following shallow diffusion. As a result，the fitting of the two diffusion depths is distinct from single-step diffusion.

This paper employs numerical simulations to forecast the diffusion patterns that occur in two distinct steps. The characterization of diffusion depth in experiments is achieved by the utilization of SEM and SIMS. By integrating simulation and experimental data，a very accurate predictive model for profound diffusion depth has been established. The functionality of devices with various dual diffusion architectures was examined and studied in both linear and Geiger modes. Precise regulation of the depth of diffusion is crucial for attaining superior performance in InGaAs/InP SPADs.

Fig. 1（a） displays the cross-sectional schematic diagram of the double diffusion InGaAs/InP SAGCM SPAD structure. The device structure，starting from the bottom and moving upwards，comprises a substrate made of n-type InP with metallization，followed by an n-type InP buffer layer，an unintentionally-doped （uid） absorption layer made of InGaAs，an n-type InGaAsP grading layer，a moderately doped n-type InP charge layer，and finally an uid InP cap layer that includes a multiplication layer. The experiment involved the design of four sets of devices，each with fixed shallow diffusion times and different deep diffusion times. Fig. 1（b） displays the schematic design of the first shallow diffusion profile. In this diagram，X_j1 denotes the distance of the shallow diffusion from the InP/SiNx surface. The schematic diagram of the structure after double diffusion is shown in Fig. 1（a）. In this context，X_j2 denotes the shallow diffusion distance following deep diffusion，while X_j3 indicates the deep diffusion distance. The value of X_j4 is determined by subtracting X_j1 from X_j2 in order to precisely define the re-diffusion process. The multiplication layer is situated underneath the deep diffusion region and functions as the major zone for impact ionization. The cap layer thickness is subtracted from the deep diffusion distance to obtain MW. The correlation between the depth of diffusion and the width of the multiplier is clearly apparent. In shallow diffusion，a p-type floating guard ring is added near the boundary to further reduce the strength of the electric field at the edge of the device.

This work utilized a Two-Dimensional Model ^［19-20］ for simulating the device diffusion process. This model implies a unidirectional link between defects and dopants，where the diffusion of dopants is greatly influenced by point defects，while the diffusion of point defects is thought to have no connection to the dopant diffusion process. During this diffusion process，it is presumed that there is an excess of point defects，suggesting that the diffusion process is not in a state of thermal equilibrium. The equation governing the continuation of dopants is as follows：

Where CI and CV represent the actual concentrations of interstitials and vacancies，while CI* and CV* represent the equilibrium concentrations. The fx is an empirical defect factor. Excess point defects will affect the dopant diffusion coefficient through a simple proportionality factor.

RB represents the bulk recombination rate，RT represents trap capture or release by interstitials，and R311 represents interstitial aggregation.

During the simulation procedure，the initial step involved etching the bigger mask to generate a diffusion structure with a relatively shallow depth. Afterwards，the smaller mask was etched to generate a profound diffusion structure with increased depth. Fig. 2 below illustrates the 2D double-diffused SPAD structure used for device simulation.

In Fig. 3，the diffusion depth is defined as the distance at which the doping concentration is equal to the background concentration. The deep diffusion distance X_j3，which corresponds to various deep diffusion durations，was determined using fitting analysis，as depicted in Fig. 4. It was observed that the fitting relationship followed the formula X=kt+c. In this context，the variable k indicates the diffusion coefficient，whereas t represents the entire diffusion time. Deep diffusion is distinct from typical single-step diffusion because it is rooted in shallow diffusion. Thus，a constant c was included to calibrate the deep diffusion process，taking into account its deviation from single-step diffusion.

Upon comparing the diffusion state contours depicted in Fig. 2，it becomes evident that there is a substantial increase in the depth of the guard ring. However，the diffusion mask of the guard ring lacks a diffusion source during the deep diffusion phase. The increased depth of the guard ring diffusion is a result of re-diffusion caused by temperature. After conducting an analysis of the guard ring's surface Zn concentration following both shallow and deep diffusion，it was determined that the surface concentration after shallow diffusion is 10¹⁹ cm^-3，whereas after deep diffusion，it is around 6×10¹⁸ cm^-3. This verifies that the increase in the depth of the guard ring is truly a result of temperature. In addition，a surface concentration analysis was performed to examine the enhanced depth of shallow diffusion. The trend in the change in surface concentration aligns with that of the guard ring. Therefore，the increased depth of shallow diffusion can be attributed to temperature-induced re-diffusion.

When the shallow diffusion time t1 is sufficiently short，shallow diffusion can be neglected，and instead it serves as a diffusion source for the second diffusion，with a solubility point of approximately 10¹⁸ cm^-3. Therefore，a fraction of the material （with negligible diffusion） that diffused during t1 is considered as the source of diffusion for the second diffusion. The second diffusion depth is denoted as X_j4，and the corresponding time is t2. The equation that accurately describes this process is x=kt+c. Considering it as a solitary diffusion process ^［23］，the behavior of x can be described by the equation x=kt. However，it should be noted that this formula alone does not provide complete accuracy. Since the first diffusion serves as the diffusion source and eliminates any surface conditions，there is no need to take interface circumstances into account. In order to offset this effect，a constant c is used to modify the formula. Consequently，shallow diffusion may be conceptualized as consisting of two distinct diffusion processes：the initial diffusion depth can be accurately estimated using the usual formula x=kt，while the subsequent diffusion is a result of temperature-induced re-diffusion，which can be anticipated using the aforementioned fitting formula.

The experimental portion involved the fabrication of devices with varying diffusion times using a two-step sealed ampoule zinc diffusion technique. The experiment entailed creating four sets of devices with consistent shallow diffusion times and different deep diffusion times. The SEM and SIMS methods were employed to characterize and assess the diffusion profile and depth.

The experiments rigorously adhered to the same approach as the simulation. Initially，the first mask created a diffusion structure that was relatively shallow. Following that，a more profound diffusion structure was formed utilizing the second mask. The shallow diffusion time was set at a constant duration of 7.5 minutes，whereas four distinct durations were employed for deep diffusion：15 minutes，22 minutes，29 minutes，and 36 minutes. SEM and SIMS tests were performed on the manufactured devices，and the outcomes of the testing are displayed in Fig. 5.

Securely affix the sample to a sample stage using conductive adhesive. Subsequently，insert it into the scanning electron microscope （SEM） system for the purpose of conducting tests. Modify the contrast and brightness of the image，as exemplified in the picture. The region exhibiting the greatest luminosity corresponds to the double diffusion area that encompasses the Zn element. The SEM studies revealed diffusion depths of 2.010 μm，2.689 μm，3.231 μm，and 3.638 μm.

By harnessing the heightened sensitivity of sims ions，it is feasible to accurately measure the exact depth of the Zn element in the diffused sample. This enables the measurement of the extent of profound diffusion within the device. The SIMS measurements resulted in diffusion depths of 1.83 μm，2.46 μm，2.96 μm，and 3.24 μm.

The equation shown below was constructed in order to enhance the accuracy of the fitting method for simulating deep diffusion depth：The equation X=kt-t0+c illustrates the relationship between the diffusion distance X，the diffusion coefficient k，the overall diffusion time t，and the time required for the temperature to rise during the diffusion process t0. An adjustment is required due to the temperature increase period involved in the two-step sealed ampoule zinc diffusion method. Throughout this time frame，the diffusion coefficient exhibits variations in accordance with temperature. Hence，the time interval during which the diffusion coefficient remains unchanged should not include the heating period t0. The constant c is utilized to modify the diffusion distance throughout the heating procedure. The revised equation was utilized to accurately model the experimental data collected from both SEM and SIMS experiments. The results depicted in Fig. 6 indicate that the formula successfully matches the experimental data.

In order to simulate the electrical performance of the device，we utilized carrier transport equations that are based on drift-diffusion. The models primarily consist of mobility models，Shockley-Read-Hall （SRH） recombination models，Auger recombination models，and impact ionization models. In this work，the Selberherr model，which relies on temperature and electric field，was utilized for the impact ionization model. The model presents the expressions for the electron impact ionization coefficient αn and the hole impact ionization coefficient αp as follows. The impact ionization parameters for the InP cap layer and InGaAs absorption layer are chosen according to the values presented in Table 1. The material parameters used for simulation were derived from the prior study ^［21］.

Where E represents the electric field at a specific location in the structure，indicating the direction of current flow.

The ray tracing approach is utilized to simulate the photoelectric response of devices. The simulation was performed by inputting data such as incident angle，wavelength，intensity，and reflection into the luminous module，enabling the calculation of light ray behavior.

The simulation examined the correlation between breakdown voltage and MW. Initially，the discrepancy between shallow and deep diffusion was rectified at a measurement of 1.6 μm. Subsequently，devices with different MW were simulated to analyze the distribution of the electric field，as illustrated in Fig. 7. The strong electric field is predominantly situated beneath the core deep diffusion InP region，as anticipated. The shallow diffusion region has moderate electric field intensities in its surrounding locations. The lateral electric field remains balanced with the gradient profile due to the inclusion of the guard ring. However，if the MW is excessively large，the high electric field region becomes concentrated at the margins of shallow and deep diffusion，hence increasing the likelihood of experiencing edge breakdown occurrences. Moreover，the electric field within the InGaAs absorption layer remains within the low-field regime. After doing a thorough analysis of the two-dimensional electric fields，it is evident that the electric field within the InP multiplication layer diminishes progressively as the MW increases. The device with the MW of 0.365 μm exhibits a significantly greater electric field in its core region compared to the device with the MW of 1.359 μm，with a difference of approximately 2x10⁵ V/cm.

The devices were evaluated for their photodetector performance within the linear range. Current-voltage （I-V） characteristics were measured for devices with varying MW under both lit and dark conditions at a temperature of 300 K. The results are presented in Fig. 8. To achieve illumination，a light source emitting at a wavelength of 1550 nm and an intensity of 0.1 W/cm² was employed. InP exhibits no light absorption within this specific range of wavelengths. Hence，the production of photocurrent signifies a reduction in the InGaAs layer. The graph illustrates the correlation between breakdown voltage，punch-through voltage （the voltage at the device's ends when the depleted region extends to the absorption layer），and MW. It is evident that the breakdown voltage （V_BR） does not exhibit a linear relationship with MW at all intervals. When the MW is insufficiently narrow，the depletion zone penetrates the InGaAs layer prematurely. This leads to a reduction in the partial voltage of the multiplier layer and a drop in the electric field，thus resulting in an increase in the breakdown voltage ^［15］. Nevertheless，the punch-through voltage （V_P） exhibits a progressive rise as the MW increases.

MW were determined using the data obtained from the standard error of the mean （SEM） tests. Fig. 9 displays a comparison between the simulated and measured dark current （IV） curves of four distinct MW. The photodetector characteristic curves derived from IV tests are displayed. Throughout the simulation，the second diffusion depth was meticulously regulated to align precisely with the results obtained from the SEM test. When simulating the electrical performance of devices，the focus is primarily on simulating their breakdown behavior. However，factors like defects and stress in the material are not considered，leading to an overly idealized simulated dark current that deviates significantly from the measured dark current. Based on the IV test results，it can be noted that the V_BR （the voltage at which a current of 10 μA is reached） of the device climbs progressively when the MW is incremented，reaching values of 48.9 V，59 V，76.2 V，and 94.5 V，respectively. This is because an increase in MW will result in a reduction of its electric field strength. A linear correlation is established between MW and V_BR within a specific range of multiplication width，as illustrated in Fig. 10.

Geiger-mode measurements were conducted using a custom-built gating measuring system. Two devices with MW values were chosen for Geiger-mode performance evaluation，as depicted in Fig. 11. It is evident that the DCR reduces significantly as the temperature decreases. At a temperature of 300 K and with an excess bias voltage of 3.5 volts，the DCR is roughly 2×10⁶ counts per second. Nevertheless，when the sample was exposed to a temperature of 253 K，the DCR increased to 2×10⁴ counts per second. The decrease in dark counts is a result of the elimination of thermally produced dark counts in InGaAs and tunneling dark counts in the InP multiplication layer at lower temperatures ^［22］. It is worth mentioning that devices with thinner multiplication layers demonstrate elevated dark count rates at identical temperatures. At the identical temperature and excess bias voltage of 3 volts，the device with the MW of 0.8 μm exhibits a DCR that is 5×10⁴ counts per second greater than the device with the MW of 1.5 μm. This phenomenon is ascribed to the intensified electric field that arises from the decrease in MW.

The primary objective of this study was to concentrate on the design of the device structure and the numerical simulation of InGaAs/InP single-photon avalanche diodes （SPADs） using both shallow and deep diffusion processes. Models were created to predict the depths of diffusion in both cases. Experiments were carried out to analyze the performance of the device in both linear and Geiger modes. The research findings suggest that the prediction model for shallow diffusion depth can be regarded as a composite of an initial diffusion and a subsequent diffusion resulting from temperature. The formula X=at-t0+c can be used to fit the predictive model for deep diffusion depth. The width of the multiplication layer has a substantial impact on the electric field within the multiplication zone. The breakdown voltage exhibits an initial fall followed by an increase as the multiplication width grows. In addition，the study examined how the thickness of the multiplication layer and temperature affect the performance of Geiger mode. It was noted that decreasing the temperature significantly decreases the rate at which dark counts occur，while decreasing the thickness of the multiplication layer increases the rate at which dark counts occur. These findings are crucial reference points for developing high-performance InP near-infrared SPAD arrays，which facilitate advanced applications like high-speed quantum communication and high-resolution 3D laser imaging.