Image sensor performance prediction method based on graded doping and finite substrate thickness

Abstract

The present application relates to the technical field of image sensor, and provides a kind of image sensor performance prediction method based on gradual doping and limited substrate thickness.The method comprises: establishing the three-dimensional physical model of image sensor pixel;Three-dimensional physical model includes the continuous gradual doping distribution between epitaxial layer and substrate and the substrate of limited thickness;Under the condition of point source illumination, the three-dimensional space distribution model of photo-generated carrier in pixel is established;According to the continuous gradual doping distribution, the built-in electric field model generated by doping concentration gradient is obtained;Under the boundary condition corresponding to the substrate of limited thickness, the carrier continuity equation is solved to obtain the spatial distribution function of photo-generated carrier in pixel;According to the spatial distribution function, the pixel response function of pixel to point source illumination is obtained;According to the pixel response function, the modulation transfer function and charge collection efficiency of image sensor are obtained.The present application realizes the unified accurate prediction of the modulation transfer function and charge collection efficiency of image sensor.

Description

Technical Field

[0001] This invention relates to the field of image sensor technology, and in particular to a method for predicting the performance of image sensors based on gradient doping and finite substrate thickness. Background Technology

[0002] With the widespread application of CCD (Charge Coupled Device) and CMOS (Complementary Metal-Oxide Semiconductor) image sensors in aerospace remote sensing, astronomical observation, and high-end scientific imaging, higher demands are being placed on image sensors in terms of spatial resolution, weak signal detection capability, and photoelectric conversion efficiency. Especially under low-light and long-wavelength imaging conditions, the performance bottleneck of image sensors is becoming increasingly prominent, and accurate performance prediction models have become an indispensable technical means in device design and optimization. The modulation transfer function (MTF) characterizes the response capability of an image sensor to signals of different spatial frequencies and is an important indicator for measuring the spatial resolution of an image system. Charge collection efficiency, or internal quantum efficiency (IQE), describes the ability of photogenerated carriers generated by incident photons within the device to be effectively collected and converted into electrical signals; it is a key parameter for evaluating the photoelectric conversion performance of an image sensor. In practical applications, the MTF and charge collection efficiency often simultaneously affect image quality, and their prediction accuracy is directly related to the rational selection of device structural and process parameters.

[0003] Existing image sensor performance prediction methods are typically based on diffusion theory or simplified carrier transport models, and their modeling process generally employs one or a combination of the following assumptions: First, it assumes that the substrate thickness is sufficiently large relative to the carrier diffusion length, thus using an infinite substrate approximation; second, it assumes an abrupt doping junction between the epitaxial layer and the substrate, ignoring the gradual doping distribution commonly found in actual processes; third, in carrier transport modeling, it only considers the diffusion mechanism and ignores the electric field drift caused by doping gradients or built-in barriers; fourth, it calculates the modulation transfer function and charge collection efficiency independently using different physical models, lacking a unified physical description framework. These modeling assumptions were somewhat applicable to early image sensors with thick substrates and simple structures, but their limitations have become increasingly apparent with the continuous evolution of device structures and manufacturing processes. Particularly in back-illuminated image sensors, to improve quantum efficiency and suppress parasitic effects, the substrate thickness is usually significantly reduced; simultaneously, to improve charge transport characteristics and suppress recombination losses, a gradual doping structure is often used between the epitaxial layer and the substrate to form a built-in electric field. Such structures subject the transport process of photogenerated carriers to the combined effects of diffusion and electric field drift, and the substrate boundary conditions exert a non-negligible modulation effect on carrier distribution. Under these conditions, continuing to use traditional infinite substrate assumptions, abrupt doping models, or pure diffusion models can easily lead to significant deviations in the estimation of carrier spatial distribution under long-wavelength or weak absorption conditions, resulting in inconsistencies between predicted modulation transfer function and charge collection efficiency and actual device performance. Furthermore, existing methods that model the modulation transfer function and charge collection efficiency separately fail to reflect the intrinsic correlation between the two in the carrier transport process within the pixel, hindering the overall optimization design of the pixel structure. Summary of the Invention

[0004] This invention provides a method for predicting the performance of image sensors based on gradient doping and finite substrate thickness. It achieves unified and accurate prediction of the modulation transfer function and charge collection efficiency of image sensors, thereby overcoming the problem of insufficient applicability of existing models in next-generation image sensor structures.

[0005] In a first aspect, embodiments of the present invention provide a method for predicting the performance of an image sensor based on graded doping and finite substrate thickness, comprising: A three-dimensional physical model of the image sensor pixels is established; wherein, the three-dimensional physical model includes a continuous gradient doping distribution between the epitaxial layer and the substrate and a substrate of finite thickness; A three-dimensional spatial distribution model of photogenerated carriers within a pixel is established under point source illumination conditions. Based on the continuously gradient doping distribution, a built-in electric field model generated by the doping concentration gradient is obtained; Under the boundary conditions corresponding to the substrate of finite thickness, the carrier continuity equation is solved to obtain the spatial distribution function of photogenerated carriers within the pixel; wherein, the carrier continuity equation includes the three-dimensional spatial distribution model, the built-in electric field model, and the carrier diffusion process; The pixel response function of the pixel to point source illumination is obtained based on the spatial distribution function; The performance metrics of the image sensor are obtained based on the pixel response function; wherein, the performance metrics include the modulation transfer function and the charge collection efficiency.

[0006] In some embodiments, establishing a three-dimensional physical model of the image sensor pixels includes: Establish a three-dimensional coordinate system with the pixel plane as the horizontal direction and the incident light propagation direction as the vertical direction, and define at least one parameter among the pixel period, epitaxial layer thickness and substrate thickness; A continuously differentiable doping concentration distribution function is defined in the depth direction to describe the continuously gradient doping distribution.

[0007] In some embodiments, establishing a three-dimensional spatial distribution model of photogenerated carriers within a pixel under point source illumination conditions includes: The lateral distribution of point source illumination on the pixel plane is represented by a lateral point source distribution function; Based on the material absorption characteristics of the collection layer, the generation rate of photogenerated carriers in the depth direction is expressed as an exponential decay function; where the depth direction is the direction of incident light propagation. The three-dimensional spatial distribution model is obtained by combining the lateral point source distribution function and the exponential decay function.

[0008] In some embodiments, determining the built-in electric field model includes: Differentiating the continuously gradient doping distribution yields the doping concentration gradient; The doping concentration gradient is converted into the built-in electric field model based on the thermal equilibrium condition.

[0009] In some embodiments, obtaining the pixel response function of the pixel to point source illumination based on the spatial distribution function includes: The current density in the depth direction is obtained according to the spatial distribution function at the location of the corresponding collection layer; wherein, the depth direction is the direction of incident light propagation. Integrate the current density to obtain the pixel output current; The pixel output current is normalized to obtain the pixel response function.

[0010] In some embodiments, obtaining the modulation transfer function includes: The pixel response function is integrated along the horizontal direction of the pixel plane to obtain a one-dimensional pixel response function; Perform a Fourier transform on the one-dimensional pixel response function to obtain the modulation transfer function of the image sensor at different spatial frequencies.

[0011] In some embodiments, obtaining the charge collection efficiency includes: Integrating the pixel response function over the region where the collection layer is located yields the number of photogenerated carriers collected. The total number of photogenerated carriers generated within the pixel is obtained based on the number of photogenerated carriers collected, the material absorption characteristics of the collection layer, and the thickness of the substrate. The ratio of the number of collected photogenerated carriers to the total number of photogenerated carriers is defined as the charge collection efficiency.

[0012] Secondly, embodiments of the present invention also provide an image sensor performance prediction device based on graded doping and finite substrate thickness, comprising: A physical model building module is used to build a three-dimensional physical model of the image sensor pixels; wherein, the three-dimensional physical model includes a continuous gradient doping distribution between the epitaxial layer and the substrate and a substrate of finite thickness; The distribution model building module is used to build a three-dimensional spatial distribution model of photogenerated carriers within a pixel under point source illumination conditions. The electric field model acquisition module is used to acquire the built-in electric field model generated by the doping concentration gradient based on the continuous gradient doping distribution. The distribution function acquisition module is used to solve the carrier continuity equation under the boundary conditions corresponding to the substrate of finite thickness to obtain the spatial distribution function of photogenerated carriers within the pixel; wherein, the carrier continuity equation includes the three-dimensional spatial distribution model, the built-in electric field model, and the carrier diffusion process; The response function acquisition module is used to acquire the pixel response function of the pixel to point source illumination based on the spatial distribution function; A performance index acquisition module is used to acquire the performance index of the image sensor based on the pixel response function; wherein the performance index includes the modulation transfer function and the charge collection efficiency.

[0013] Thirdly, embodiments of the present invention also provide an image sensor, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the image sensor performance prediction method based on gradient doping and finite substrate thickness as described in the first aspect.

[0014] In some embodiments, the image sensor is a front-illuminated image sensor, a back-illuminated image sensor, or a charge-coupled device (CCD) image sensor.

[0015] This invention simultaneously considers the built-in electric field effect introduced by gradient doping and the boundary effects under finite substrate thickness conditions at the physical modeling level, making the carrier transport model more realistic. The introduction of the built-in electric field enables the electric field-assisted collection of deep carriers generated in the long wavelength range, avoiding the underestimation of carrier spatial distribution by the pure diffusion model; the finite substrate boundary condition avoids the erroneous estimation of bottom carrier behavior under the infinite substrate assumption. Therefore, a more accurate spatial distribution of photogenerated carriers can be obtained, leading to a more accurate modulation transfer function and charge collection efficiency, providing a reliable basis for device optimization. Furthermore, within a unified theoretical framework, joint prediction of the modulation transfer function and charge collection efficiency of image sensors is achieved, thereby improving the applicability and prediction accuracy of the model in next-generation image sensor structures. Attached Figure Description

[0016] To more clearly illustrate the technical solutions in this invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of this invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0017] Figure 1 This is a flowchart illustrating a method for predicting the performance of an image sensor based on gradient doping and finite substrate thickness, provided in an embodiment of the present invention. Figure 2 This is a schematic diagram of the three-dimensional structure of an image sensor pixel provided in an embodiment of the present invention; Figure 3 This is a schematic diagram of a two-dimensional distribution of the modulation transfer function of an image sensor provided in an embodiment of the present invention; Figure 4 This is a schematic diagram of the structure of an image sensor performance prediction device based on gradient doping and finite substrate thickness provided in an embodiment of the present invention; Figure 5 This is a schematic diagram of the structure of an image sensor provided in an embodiment of the present invention. Detailed Implementation

[0018] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of this invention. Based on the embodiments of this invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this invention.

[0019] Figure 1 This is a schematic flowchart illustrating an image sensor performance prediction method based on graded doping and finite substrate thickness provided in an embodiment of the present invention. The image sensor performance prediction method based on graded doping and finite substrate thickness can be executed by the image sensor performance prediction device based on graded doping and finite substrate thickness provided in this embodiment of the invention. This image sensor performance prediction device based on graded doping and finite substrate thickness can be implemented in software and / or hardware. For example... Figure 1 As shown, the image sensor performance prediction method based on graded doping and finite substrate thickness includes the following steps: S101. Establish a three-dimensional physical model of the image sensor pixels; wherein, the three-dimensional physical model includes a continuous gradient doping distribution between the epitaxial layer and the substrate, and a substrate of finite thickness.

[0020] S102. Establish a three-dimensional spatial distribution model of photogenerated carriers within a pixel under point source illumination conditions.

[0021] S103. Obtain the built-in electric field model generated by the doping concentration gradient based on the continuous gradient doping distribution.

[0022] S104. Under the boundary conditions corresponding to a substrate of finite thickness, solve the carrier continuity equation to obtain the spatial distribution function of photogenerated carriers within the pixel; wherein, the carrier continuity equation includes a three-dimensional spatial distribution model, a built-in electric field model, and a carrier diffusion process.

[0023] S105. Obtain the pixel response function of the pixel to point source illumination based on the spatial distribution function.

[0024] S106. Obtain the performance indicators of the image sensor based on the pixel response function; wherein, the performance indicators include the modulation transfer function and the charge collection efficiency.

[0025] Specifically, an image sensor pixel refers to the smallest imaging unit of an image sensor, used to convert incident light into an electrical signal. A three-dimensional physical model is a mathematical model built in a computer that describes the physical characteristics of a pixel, such as its geometry and material doping distribution. Figure 2 This is a schematic diagram of the three-dimensional structure of an image sensor pixel provided in an embodiment of the present invention. Figure 2As shown, an image sensor pixel comprises a substrate, an epitaxial layer, and a collection layer. The substrate, i.e. Figure 2 The P-type substrate shown is the bottom semiconductor material layer of the image sensor pixel, which provides support; its thickness affects the transport behavior of charge carriers. The collector layer is the top layer of the image sensor pixel, responsible for collecting photogenerated carriers to achieve image sensing. The epitaxial layer, i.e. Figure 2 The p-type epitaxial layer shown is located between the substrate and the collector layer. Light can penetrate the collector layer and reach the epitaxial layer to continue generating electrons. A continuous gradient doping distribution refers to a doping concentration along the depth direction, i.e. Figure 2 The z-direction distribution in the image is continuously varying rather than abruptly changing, determined by the manufacturing process. A finite-thickness substrate refers to a substrate with a defined, finite geometric thickness, not an assumption of infinite thickness. Point-source illumination is an idealized lighting condition where the incident light is concentrated at a single geometric point on the pixel plane, used to analyze the spatial response characteristics of a pixel. Photogenerated carriers are one type of carrier in the electron-hole pair generated after incident photons are absorbed by the semiconductor material; for example, they can be electrons.

[0026] The three-dimensional spatial distribution model is a three-dimensional distribution function describing the location of photogenerated carriers within a pixel. The doping concentration gradient is the rate of change of doping concentration in space, i.e., the derivative of concentration with respect to space. The built-in electric field model is a mathematical model of the electric field existing within the semiconductor, caused by the doping concentration gradient, which influences carrier drift. The carrier continuity equation is a partial differential equation describing the spatial and temporal changes in carrier concentration, including terms related to generation, recombination, diffusion, and drift. Boundary conditions are constraints imposed on the boundaries of the region when solving the differential equations; in this case, they are carrier behavior conditions at the bottom boundary of the substrate. The spatial distribution function is a mathematical function describing the distribution of photogenerated carrier concentration at various points within the pixel. The carrier diffusion process is the random thermal motion of carriers due to the concentration gradient. The pixel response function is a function describing the relationship between the output signal generated by the pixel in response to point source illumination and the location of the point source; it represents the spatial response characteristics of the pixel. Performance metrics are quantitative parameters used to evaluate the imaging quality of an image sensor, including the modulation transfer function and charge collection efficiency. The modulation transfer function (MTF) describes the modulation depth transfer capability of an image sensor for sinusoidal gratings of different spatial frequencies, reflecting image sharpness. Charge collection efficiency is the ratio of the number of photogenerated carriers effectively collected by a pixel to the total number of photogenerated carriers generated by the incident photons, reflecting photoelectric conversion capability.

[0027] First, a three-dimensional physical model of the image sensor pixel is established. This model must include two key elements: one is the continuous gradient doping distribution between the epitaxial layer and the substrate, which can be described by a continuously differentiable doping concentration distribution function; the other is the finite thickness of the substrate. The former reflects the smooth transition of doping concentration in actual processes, while the latter abandons the traditional infinite substrate approximation, making the model closer to the actual device structure. Second, a three-dimensional spatial distribution model of photogenerated carriers within the pixel is established under point source illumination conditions. Assuming that the incident light forms an approximate point source illumination on the pixel plane, the point source illumination, as an ideal excitation source, can excite photogenerated carriers at various locations within the pixel. This model can be used to calculate the photogenerated carrier generation rate at any location. Next, a built-in electric field model generated by the doping concentration gradient is obtained based on the continuous gradient doping distribution. Since the doping concentration changes continuously along the depth direction, a concentration gradient inevitably exists. According to the thermal equilibrium conditions of semiconductor physics, this gradient will be converted into an equivalent built-in electric field, which will have a drift effect on the carriers. This electric field can then be introduced into the carrier transport equation, i.e., the carrier continuity equation, so that the motion of photogenerated carriers is simultaneously affected by diffusion and drift.

[0028] Then, under the boundary conditions corresponding to a substrate of finite thickness, the carrier continuity equation is solved to obtain the spatial distribution function of photogenerated carriers within the pixel. This continuity equation simultaneously includes a three-dimensional spatial distribution model corresponding to the recombination term, a built-in electric field model corresponding to the drift term, and the carrier diffusion process corresponding to the diffusion term. Boundary conditions reflecting the influence of finite thickness are applied at the bottom of the substrate to obtain the steady-state concentration distribution of photogenerated carriers. Next, based on the spatial distribution function and the charge collection region, i.e., the geometric boundary of the collection layer, the pixel response function to point source illumination is obtained. This function is obtained by integrating the current density at the collection layer, reflecting the pixel's response intensity to point light sources at different locations. Finally, the image sensor's performance indicators, including the modulation transfer function and charge collection efficiency, are obtained based on the pixel response function. The charge collection efficiency can also be understood as the internal quantum efficiency. The modulation transfer function is obtained by performing a Fourier transform on the response function, and the charge collection efficiency is obtained by integrating the response function over the collection region. These two indicators evaluate the sensor's imaging quality from different perspectives.

[0029] Therefore, this invention, at the physical modeling level, simultaneously considers the built-in electric field effect introduced by gradient doping and the boundary effects under finite substrate thickness conditions, making the carrier transport model more realistic. The introduction of the built-in electric field allows for the electric field-assisted collection of deep carriers generated in the long wavelength band, avoiding the underestimation of carrier spatial distribution by the pure diffusion model; the finite substrate boundary condition avoids erroneous estimation of bottom carrier behavior under the infinite substrate assumption. Thus, a more accurate spatial distribution of photogenerated carriers can be obtained, leading to a more accurate modulation transfer function and charge collection efficiency, providing a reliable basis for device optimization. Simultaneously, joint prediction of the modulation transfer function and charge collection efficiency of image sensors is achieved within a unified theoretical framework, thereby improving the applicability and prediction accuracy of the model in next-generation image sensor structures.

[0030] In some embodiments, establishing a three-dimensional physical model of an image sensor pixel includes: establishing a three-dimensional coordinate system with the pixel plane as the lateral direction and the incident light propagation direction as the depth direction, and defining at least one parameter among the pixel period, epitaxial layer thickness, and substrate thickness; defining a continuously differentiable doping concentration distribution function in the depth direction to describe the continuously gradient doping distribution.

[0031] Specifically, the pixel plane is the lateral extension of the pixel, parallel to the chip surface, and is the plane illuminated by incident light. Figure 2 The x and y directions lie in the plane. The horizontal direction is parallel to the pixel plane and includes two orthogonal directions, namely... Figure 2 The x and y directions in the image. The incident light propagation direction is the depth direction of the light after entering the pixel, perpendicular to the pixel plane, i.e. Figure 2 The z-direction in the image. Pixel period is the distance between the centers of adjacent pixels, determining the pixel density. Epitaxial layer thickness is the geometric dimension of the epitaxial layer in the depth direction. Substrate thickness is the geometric dimension of the substrate in the depth direction. A continuously differentiable doping concentration distribution function describes the change of doping concentration with depth coordinates; this function is continuous and differentiable, thus ensuring the existence and continuity of the doping gradient.

[0032] First, a three-dimensional coordinate system is established with the pixel plane as the lateral direction and the incident light propagation direction as the depth direction. This coordinate system allows the position of any point within a pixel to be represented by coordinates (x, y, z), where (x, y) represents the lateral position and z represents the depth position. This allows lateral diffusion behavior and depth transport behavior to be described within the same coordinate system. Second, at least one parameter is defined from the pixel period, epitaxial layer thickness, and substrate thickness, where the pixel period is p, the epitaxial layer thickness is Tepi, and the substrate thickness is Tsub. These parameters are fundamental quantities describing the pixel geometry, with the pixel period affecting the lateral diffusion boundary, and the epitaxial layer and substrate thickness affecting the depth transport length. These structural parameters can be derived from device design documents, process specifications, or obtained through microscopic characterization and process inversion methods. It should be noted that the aforementioned parameters are merely examples; the three-dimensional physical model can include the geometric dimensions of all pixel structures. Finally, based on device manufacturing process data or experimental measurement results, a continuously differentiable doping concentration distribution function is defined in the depth direction to describe the continuously gradient doping distribution. This function is expressed as: Furthermore, its derivative exists, allowing for the further calculation of the doping concentration gradient and the built-in electric field. For example, the doping concentration distribution function... The following calculation formula is satisfied:

[0033] in, Initial doping concentration, in cm⁻¹ -3 , It is a continuously differentiable function used to describe the gradual doping transition relationship between the epitaxial layer and the substrate, reflecting the smooth transition characteristics of doping concentration in the depth direction in actual processes.

[0034] Therefore, this embodiment of the invention provides a unified geometric framework for subsequent carrier generation, transport, and collection by establishing a three-dimensional coordinate system and defining key geometric parameters. The introduction of a continuously differentiable doping concentration distribution function makes the mathematical description of graded doping possible and ensures the physical rationality of the built-in electric field obtained through subsequent differentiation. This step lays the structural foundation for the entire prediction method.

[0035] In some embodiments, establishing a three-dimensional spatial distribution model of photogenerated carriers within a pixel under point source illumination conditions includes: representing the lateral distribution of point source illumination on the pixel plane as a lateral point source distribution function; representing the generation rate of photogenerated carriers in the depth direction as an exponential decay function based on the material absorption characteristics of the collection layer; wherein the depth direction is the direction of incident light propagation; and combining the lateral point source distribution function and the exponential decay function to obtain a three-dimensional spatial distribution model.

[0036] Specifically, the lateral point source distribution function is a function that describes the lateral distribution of point source illumination on the pixel plane, using the Dirac function. This indicates that light energy is concentrated at a single point. The collection layer is the region within a pixel used to collect photogenerated carriers; it is the area where the photodiode is located. Material absorption characteristics refer to the ability of a semiconductor material to absorb photons of a specific wavelength, expressed as the absorption coefficient. Characterization. The exponential decay function is an exponential function describing the decrease in light intensity with penetration depth, that is, the photogenerated carrier generation rate decreases exponentially with increasing depth. The three-dimensional spatial distribution model is a three-dimensional function of the photogenerated carrier generation rate obtained by combining lateral point source and depth decay.

[0037] First, assume that the incident light forms a localized spot on the pixel plane, with a spatial scale relatively small compared to the pixel size, which can be approximated as point source illumination. The lateral distribution of point source illumination on the pixel plane is represented by a lateral point source distribution function. Ideally, point source illumination means that all the incident light energy is concentrated at a single geometric point on the pixel plane (…). , Therefore, the lateral point source distribution function Satisfy the following formula:

[0038] in,( , () represents the location of the incident source. Let be the Dirac function, used to represent the lateral distribution of an ideal point light source. Secondly, based on the material absorption characteristics of the collection layer, the generation rate of photogenerated carriers in the depth direction is expressed as an exponential decay function. Since the light intensity decays exponentially with the penetration depth z, the generation rate of photogenerated carriers in the depth direction... Satisfy the following formula:

[0039] in, The incident light wavelength, The absorption coefficient of the collection layer, Let be the incident photon flux. Finally, by combining the above transverse point source distribution function and exponential decay function, a three-dimensional spatial distribution model is obtained, i.e., the three-dimensional photogenerated carrier generation rate is obtained. Satisfy the following formula:

[0040] This function describes the point of incidence source ( , Under excitation, at any position within a pixel The photogenerated carrier generation rate at a given location, i.e., through the above modeling method, can obtain the three-dimensional generation location distribution of photogenerated electrons within the pixel, thus providing initial conditions for the analysis of the carrier transport process.

[0041] Therefore, this embodiment of the invention simplifies the modeling difficulty under complex lighting conditions by separating the lateral distribution and depth attenuation of point source illumination. This three-dimensional spatial distribution model provides accurate generation terms for subsequent solving of the carrier continuity equation, enabling a mathematical description of the initial excitation source of carrier transport, thereby allowing for accurate calculation of the contribution of carriers generated at different locations to the final signal.

[0042] In some embodiments, determining the built-in electric field model includes: taking the derivative of the continuously gradient doping distribution to obtain the doping concentration gradient; and converting the doping concentration gradient into the built-in electric field model according to the thermal equilibrium condition.

[0043] Specifically, differentiation involves performing a differential operation on a continuously differentiable function to obtain its rate of change. The doping concentration gradient is the derivative of the doping concentration with respect to the depth coordinate, representing the rate and direction of concentration change. The thermal equilibrium condition is when the semiconductor is in thermodynamic equilibrium without external excitation, at which point the Fermi level is uniform everywhere, and the carrier concentration and doping concentration satisfy a specific relationship. The built-in electric field model is an electric field expression derived from the doping concentration gradient based on the Boltzmann relation under thermal equilibrium conditions. The built-in electric field changes continuously in the depth direction and affects the direction of motion and diffusion behavior of photogenerated electrons.

[0044] First, the doping concentration gradient is obtained by differentiating the continuously gradient doping distribution. The doping concentration gradient satisfies the following calculation formula:

[0045] Due to the doping concentration distribution function It is continuously differentiable, and its derivative can be calculated. This doping concentration gradient reflects the rate at which the doping concentration changes with depth. Secondly, based on the thermal equilibrium condition, the doping concentration gradient is converted into a built-in electric field model. Under thermal equilibrium, the diffusion current and drift current of the charge carriers cancel each other out, thus deriving the following relationship between the built-in electric field and the doping gradient:

[0047] in, Boltzmann's constant, This is absolute temperature, measured in Kelvin (K). The electric field represents the elementary charge. The built-in electric field points in the direction of increasing doping concentration, causing a drift effect on majority carriers, i.e., electrons. Therefore, this embodiment of the invention converts the geometric information of the doping distribution into physical field information, enabling the subsequent carrier transport equations to include drift terms. Since the electric field is naturally formed by gradual doping in actual processes, this method requires no additional assumptions and is directly derived from the doping distribution, improving the physical consistency of the model. The introduction of the built-in electric field is key to distinguishing this method from pure diffusion models, making the prediction of long-wavelength photogenerated carrier collection more accurate.

[0048] Then, under the boundary conditions corresponding to a substrate of finite thickness, the carrier continuity equation is solved to obtain the spatial distribution function of photogenerated carriers within the pixel; wherein, the carrier continuity equation includes a three-dimensional spatial distribution model, a built-in electric field model, and a carrier diffusion process. For example, the carrier continuity equation is as follows:

[0049] That is, under steady-state conditions, the electron concentration satisfies the above equation. Wherein, Electron diffusion coefficient, in cm. 2 / s; Electron mobility, in cm 2 / (V·s); Electron concentration, unit: cm³ -3 ; Electron lifetime, measured in seconds (s); The gradient operator is then applied. Next, boundary conditions corresponding to a substrate of finite thickness are applied in the depth direction to characterize carrier reflection or low recombination conditions at the bottom of the substrate:

[0050] The finite substrate thickness boundary condition is used to describe the actual constraints on carrier transport behavior in the bottom region of the substrate, avoiding the computational errors introduced by the infinite substrate assumption and improving the model's adaptability to real device structures. Under the above boundary conditions, the carrier continuity equation is solved to obtain the spatial distribution function of photogenerated carriers within the pixel, which satisfies the following formula:

[0051] in, The electron diffusion length is given in meters (m), and r is the distance from any point (x, y, z) in space to the point of incidence. , The distance between (0, 0) in meters satisfies the following formula:

[0052] In some embodiments, obtaining the pixel response function of a pixel to point source illumination based on the spatial distribution function includes: obtaining the current density in the depth direction according to the spatial distribution function at the location of the collection layer; wherein the depth direction is the incident light propagation direction; integrating the current density to obtain the pixel output current; and normalizing the pixel output current to obtain the pixel response function.

[0053] Specifically, the collection layer is located at a specific depth within the pixel used to collect photogenerated carriers; it is either the interface between the epitaxial layer and the substrate or the junction region of a photodiode. The current density in the depth direction is the current density formed by carrier flow in the depth direction, contributed by both diffusion and drift. The pixel output current is the total current obtained by integrating the entire pixel collection area, i.e., the pixel's response intensity to point source illumination. Normalization involves dividing the output current by the total incident luminous flux or the maximum response value to normalize the range of the pixel response function to [0, 1], facilitating comparison and subsequent mathematical processing.

[0054] First, based on the spatial distribution function, the current density in the depth direction is obtained at the location of the collection layer. Satisfy the following formula:

[0055] This current density includes contributions from diffusion and drift, reflecting the carrier flux collected at that point. Next, the current density is integrated to obtain the pixel output current. It satisfies the following formula, where This indicates that double integration is performed in the pixel output current collection region:

[0056] Finally, the pixel output current is normalized to obtain the pixel response function. Satisfy the following formula:

[0057] Pixel response function represents the pixel pair located at The relative response intensity of a point source reflects the spatial response characteristics of a pixel to localized optical excitation, serving as a crucial intermediate quantity connecting the physical structure parameters of a device with its imaging performance indicators. Therefore, this embodiment of the invention transforms the abstract carrier concentration distribution into a concrete, measurable pixel output signal and establishes a mapping relationship between the point source location and the output signal, i.e., the pixel response function. This function acts as a bridge connecting the device's physical structure and imaging performance; subsequent modulation transfer function and charge collection efficiency are calculated based on this function. Normalization eliminates the influence of incident light intensity, ensuring that the response function reflects only the spatial response characteristics of the pixel itself.

[0058] In some embodiments, obtaining the modulation transfer function includes: integrating the pixel response function along the lateral direction of the pixel plane to obtain a one-dimensional pixel response function; and performing a Fourier transform on the one-dimensional pixel response function to obtain the modulation transfer function of the image sensor at different spatial frequencies.

[0059] Specifically, the one-dimensional pixel response function is a one-dimensional function obtained by integrating the two-dimensional pixel response function along a certain horizontal direction, used to analyze the spatial response of a pixel in a single direction. The Fourier transform converts the spatial domain function into a frequency domain function, used to analyze the frequency response characteristics of the system. Spatial frequency is the reciprocal of the period of brightness variation in an image in space. The modulation transfer function is a frequency domain function, defined as the ratio of the output modulation depth to the input modulation depth as a function of spatial frequency, obtained from the modulus of the Fourier transform of the pixel response function.

[0060] First, the pixel response function is integrated along the horizontal direction of the pixel plane to obtain a one-dimensional pixel response function. Satisfy the following formula:

[0061] One-dimensional pixel response function The linear diffusion characteristics of pixels in the x-direction are described. Next, a Fourier transform is performed on the one-dimensional pixel response function to obtain the modulation transfer function of the image sensor at different spatial frequencies. Satisfy the following formula:

[0062] Where f is the spatial frequency. The modulation transfer function (MTF) quantitatively characterizes a pixel's ability to transmit signals at different spatial frequencies and is an important indicator for evaluating image sensor resolution and image sharpness. Meanwhile, the diffusion-induced modulation transfer function... It can be represented as follows:

[0063] Figure 3 This is a two-dimensional distribution diagram of the modulation transfer function of an image sensor provided in an embodiment of the present invention. Figure 3 In the left-hand chart, the x-axis represents the reciprocal of the x-axis position, and the y-axis represents the reciprocal of the y-axis position. In the right-hand chart, the x-axis represents the reciprocal of the x-axis position, and the y-axis represents the modulation transfer function. Figure 3 This reflects the spatial frequency response characteristics of the pixel considering gradient doping and finite substrate thickness.

[0064] Therefore, this embodiment of the invention uses Fourier transform to convert the pixel response function in the spatial domain into a modulation transfer function in the frequency domain, thereby enabling quantitative evaluation of the sensor's imaging capability for different detail sizes. The modulation transfer function is a core indicator for evaluating image sharpness; this step realizes the transformation from a physical model to an imaging performance indicator, providing a direct optimization basis for device design.

[0065] In some embodiments, obtaining the charge collection efficiency includes: integrating the pixel response function over the region where the collection layer is located to obtain the number of photogenerated carriers collected; obtaining the total number of photogenerated carriers generated within the pixel based on the number of photogenerated carriers collected, the material absorption characteristics of the collection layer, and the thickness of the substrate; and determining the ratio of the number of photogenerated carriers collected to the total number of photogenerated carriers as the charge collection efficiency.

[0066] Specifically, the collection layer region is the spatial extent within a pixel used for collecting photogenerated carriers, including both lateral and depth regions. The number of photogenerated carriers collected is the total number of photogenerated carriers that are ultimately collected by the pixel and converted into electrical signals under point source illumination, obtained by integrating the pixel response function over the collection region. The total number of photogenerated carriers is the total number of photogenerated carriers generated within the pixel by incident photons under point source illumination, obtained by integrating the three-dimensional spatial distribution model over the entire pixel volume. The charge collection efficiency is the ratio of the number of collected photogenerated carriers to the total number of photogenerated carriers, reflecting the pixel's ability to collect photogenerated carriers.

[0067] First, the pixel response function is integrated over the region containing the collection layer to obtain the number of collected photogenerated carriers. Satisfy the following formula:

[0068] Secondly, the total number of photogenerated carriers generated within the pixel is obtained based on the number of collected photogenerated carriers, the material absorption characteristics of the collection layer, and the thickness of the substrate. Satisfy the following formula:

[0069] Finally, the ratio of the number of photogenerated carriers collected to the total number of photogenerated carriers is determined as the charge collection efficiency, or the internal quantum efficiency. Satisfy the following formula:

[0070] in, Let z be the carrier collection efficiency at depth z. Thus, this embodiment of the invention achieves a quantitative calculation of charge collection efficiency. Since this calculation is based on the same pixel response function as the modulation transfer function, the two performance indicators originate from the same physical model, ensuring inherent consistency. Charge collection efficiency is a key parameter for measuring the photoelectric conversion capability of a sensor, and is particularly important for low-light imaging. This step provides a quantitative tool for optimizing doping distribution and substrate thickness to improve collection efficiency.

[0071] In summary, compared with the prior art, the embodiments of the present invention have at least the following beneficial effects: First, it breaks through the traditional assumption of infinite substrate modeling, improving the applicability to novel image sensor structures. The present invention explicitly introduces the boundary condition of finite substrate thickness in the performance prediction process, avoiding the errors caused by the infinite substrate approximation in traditional models. This allows the model to accurately reflect the changing law of carrier transport behavior under substrate thinning conditions, thus making it applicable to back-illuminated image sensors and other image sensors using thinned substrate structures. Second, it introduces the built-in electric field effect caused by graded doping, significantly improving the prediction accuracy under long-wavelength conditions. By considering the built-in electric field formed by the continuous change of doping concentration in the carrier transport model, the present invention can simultaneously describe the diffusion and drift processes of photogenerated carriers, avoiding the problem of inaccurate estimation of carrier spatial distribution under weak absorption and long-wavelength conditions in traditional pure diffusion models, thereby improving the calculation accuracy of modulation transfer function and charge collection efficiency. Third, based on a unified pixel response function model, it achieves joint prediction of modulation transfer function and charge collection efficiency. This invention constructs a pixel response function to uniformly describe the generation, transport, and collection processes of photogenerated carriers. It simultaneously calculates the modulation transfer function and charge collection efficiency within the same physical model framework, avoiding the parameter inconsistencies and result deviations caused by using multiple independent models to calculate different performance indicators in existing technologies. This facilitates overall performance analysis and optimization of the pixel structure. Fourth, it improves the engineering usability of the model, making it suitable for applications in the design and parameter optimization stages of image sensors. Given device structural and process parameters, the method of this invention can quickly obtain key performance indicators of image sensors. Its predicted results show good consistency with experimental measurements, providing effective quantitative references for image sensor structural design, doping distribution design, and substrate thickness selection. Fifth, it has a wide range of applications and is easily extended to various types of image sensors. The method of this invention does not depend on specific material systems or pixel size assumptions and can be adjusted according to different material and structural parameters. It is applicable to CCD image sensors, CMOS image sensors, and their back-illuminated or front-illuminated structures, exhibiting good versatility and scalability.

[0072] Figure 4This is a schematic diagram of the structure of an image sensor performance prediction device based on gradient doping and finite substrate thickness, provided in an embodiment of the present invention. Figure 4 As shown, the image sensor performance prediction device based on gradient doping and finite substrate thickness includes a physical model establishment module 301 for establishing a three-dimensional physical model of the image sensor pixel; wherein, the three-dimensional physical model includes a continuous gradient doping distribution between the epitaxial layer and the substrate and a substrate of finite thickness; a distribution model establishment module 302 for establishing a three-dimensional spatial distribution model of photogenerated carriers within the pixel under point source illumination conditions; an electric field model acquisition module 303 for acquiring a built-in electric field model generated by the doping concentration gradient based on the continuous gradient doping distribution; a distribution function acquisition module 304 for solving the carrier continuity equation under the boundary conditions corresponding to the substrate of finite thickness to obtain the spatial distribution function of photogenerated carriers within the pixel; wherein, the carrier continuity equation includes a three-dimensional spatial distribution model, a built-in electric field model, and a carrier diffusion process; a response function acquisition module 305 for acquiring the pixel response function of the pixel to point source illumination based on the spatial distribution function; and a performance index acquisition module 306 for acquiring the performance index of the image sensor based on the pixel response function; wherein, the performance index includes the modulation transfer function and the charge collection efficiency.

[0073] In summary, this invention relates to the field of image sensor performance modeling and prediction technology, and in particular to a method for predicting image sensor performance considering gradient doping and finite substrate thickness. This method, targeting the pixel structure of an image sensor, introduces a continuous gradient doping distribution between the epitaxial layer and the substrate based on a three-dimensional physical model of the pixel. It constructs a built-in electric field model generated by the doping concentration gradient and establishes a photogenerated carrier generation model, i.e., a three-dimensional spatial distribution model, under point source illumination conditions. By incorporating the built-in electric field and the carrier diffusion process into the carrier continuity equation, the spatial distribution of photogenerated carriers is solved under the boundary condition of finite substrate thickness. Furthermore, the pixel response function to point source illumination is calculated based on the carrier spatial distribution, and the modulation transfer function and charge collection efficiency or internal quantum efficiency of the image sensor are obtained based on this pixel response function. Therefore, it can simultaneously consider the influence of gradient doping structure and finite substrate thickness on carrier transport behavior, achieving a unified and accurate prediction of image sensor imaging performance. This method is applicable to CMOS image sensors, CCD image sensors, and back-illuminated image sensor structures, providing an effective technical means for device design optimization and performance evaluation.

[0074] Figure 5 This is a schematic diagram of the structure of an image sensor provided in an embodiment of the present invention. Figure 5As shown, the image sensor may include: a processor 401, a communication interface 402, a memory 403, and a communication bus 404. The processor 401, communication interface 402, and memory 403 communicate with each other via the communication bus 404. The processor 401 can call logic instructions from the memory 403 to execute an image sensor performance prediction method based on graded doping and finite substrate thickness, including: A three-dimensional physical model of the image sensor pixels is established; wherein, the three-dimensional physical model includes a continuous gradient doping distribution between the epitaxial layer and the substrate and a substrate of finite thickness; A three-dimensional spatial distribution model of photogenerated carriers within a pixel is established under point source illumination conditions. The built-in electric field model generated by the doping concentration gradient is obtained based on the continuous gradient doping distribution. Under the boundary conditions corresponding to a substrate of finite thickness, the carrier continuity equation is solved to obtain the spatial distribution function of photogenerated carriers within a pixel; wherein, the carrier continuity equation includes a three-dimensional spatial distribution model, a built-in electric field model, and a carrier diffusion process; Obtain the pixel response function of the pixel to point source illumination based on the spatial distribution function; The performance metrics of an image sensor are obtained based on the pixel response function; these metrics include the modulation transfer function and charge collection efficiency.

[0075] Furthermore, the logical instructions in the aforementioned memory 403 can be implemented as software functional units and, when sold or used as independent products, can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, essentially, or the part that contributes to the prior art, or a part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods of the various embodiments of the present invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0076] In some embodiments, the image sensor is a front-illuminated image sensor, a back-illuminated image sensor, or a charge-coupled device (CCD) image sensor.

[0077] Specifically, front-illuminated image sensors are structures where light shines from the front of the chip. Back-illuminated image sensors are structures where light shines from the back of the chip, with a thinner substrate. Charge-coupled device (CCD) image sensors are image sensors that employ charge-coupled technology. Under different substrate thicknesses, doping distributions, and operating wavelengths, stable and accurate imaging performance predictions can be achieved by adjusting the corresponding structural and material parameters. Furthermore, this method can be implemented using numerical calculation software or a dedicated simulation platform. The calculation process exhibits good numerical stability and repeatability, making it suitable for performance evaluation and structural optimization during the design phase of image sensor devices.

[0078] On the other hand, the present invention also provides a computer program product, which includes a computer program that can be stored on a non-transitory computer-readable storage medium. When the computer program is executed by a processor, the computer is able to execute the above-described image sensor performance prediction method based on graded doping and finite substrate thickness, including: A three-dimensional physical model of the image sensor pixels is established; wherein, the three-dimensional physical model includes a continuous gradient doping distribution between the epitaxial layer and the substrate and a substrate of finite thickness; A three-dimensional spatial distribution model of photogenerated carriers within a pixel is established under point source illumination conditions. The built-in electric field model generated by the doping concentration gradient is obtained based on the continuous gradient doping distribution. Under the boundary conditions corresponding to a substrate of finite thickness, the carrier continuity equation is solved to obtain the spatial distribution function of photogenerated carriers within a pixel; wherein, the carrier continuity equation includes a three-dimensional spatial distribution model, a built-in electric field model, and a carrier diffusion process; Obtain the pixel response function of the pixel to point source illumination based on the spatial distribution function; The performance metrics of an image sensor are obtained based on the pixel response function; these metrics include the modulation transfer function and charge collection efficiency.

[0079] In another aspect, the present invention also provides a non-transitory computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, is implemented to perform the above-described image sensor performance prediction method based on graded doping and finite substrate thickness, comprising: A three-dimensional physical model of the image sensor pixels is established; wherein, the three-dimensional physical model includes a continuous gradient doping distribution between the epitaxial layer and the substrate and a substrate of finite thickness; A three-dimensional spatial distribution model of photogenerated carriers within a pixel is established under point source illumination conditions. The built-in electric field model generated by the doping concentration gradient is obtained based on the continuous gradient doping distribution. Under the boundary conditions corresponding to a substrate of finite thickness, the carrier continuity equation is solved to obtain the spatial distribution function of photogenerated carriers within a pixel; wherein, the carrier continuity equation includes a three-dimensional spatial distribution model, a built-in electric field model, and a carrier diffusion process; Obtain the pixel response function of the pixel to point source illumination based on the spatial distribution function; The performance metrics of an image sensor are obtained based on the pixel response function; these metrics include the modulation transfer function and charge collection efficiency.

[0080] The device embodiments described above are merely illustrative. The units described as separate components may or may not be physically separate, and the components shown as units may or may not be physical units; that is, they may be located in one place or distributed across multiple network units. Some or all of the modules can be selected to achieve the purpose of this embodiment according to actual needs. Those skilled in the art can understand and implement this without any creative effort.

[0081] Through the above description of the embodiments, those skilled in the art can clearly understand that each embodiment can be implemented by means of software plus necessary general-purpose hardware platforms, and of course, it can also be implemented by hardware. Based on this understanding, the above technical solutions, in essence or the part that contributes to the prior art, can be embodied in the form of a software product. This computer software product can be stored in a computer-readable storage medium, such as ROM / RAM, magnetic disk, optical disk, etc., including several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute the methods of various embodiments or some parts of embodiments.

[0082] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for predicting the performance of an image sensor based on graded doping and finite substrate thickness, characterized in that, include: A three-dimensional physical model of the image sensor pixels is established; wherein, the three-dimensional physical model includes a continuous gradient doping distribution between the epitaxial layer and the substrate and a substrate of finite thickness; A three-dimensional spatial distribution model of photogenerated carriers within a pixel is established under point source illumination conditions. Based on the continuously gradient doping distribution, a built-in electric field model generated by the doping concentration gradient is obtained; Under the boundary conditions corresponding to the substrate of finite thickness, the carrier continuity equation is solved to obtain the spatial distribution function of photogenerated carriers within the pixel; wherein, the carrier continuity equation includes the three-dimensional spatial distribution model, the built-in electric field model, and the carrier diffusion process; The pixel response function of the pixel to point source illumination is obtained based on the spatial distribution function; The performance metrics of the image sensor are obtained based on the pixel response function; wherein, the performance metrics include the modulation transfer function and the charge collection efficiency.

2. The image sensor performance prediction method based on graded doping and finite substrate thickness according to claim 1, characterized in that, Establish a three-dimensional physical model of the image sensor pixels, including: Establish a three-dimensional coordinate system with the pixel plane as the horizontal direction and the incident light propagation direction as the vertical direction, and define at least one parameter among the pixel period, epitaxial layer thickness and substrate thickness; A continuously differentiable doping concentration distribution function is defined in the depth direction to describe the continuously gradient doping distribution.

3. The image sensor performance prediction method based on graded doping and finite substrate thickness according to claim 1, characterized in that, A three-dimensional spatial distribution model of photogenerated carriers within a pixel is established under point source illumination conditions, including: The lateral distribution of point source illumination on the pixel plane is represented by a lateral point source distribution function; Based on the material absorption characteristics of the collection layer, the generation rate of photogenerated carriers in the depth direction is expressed as an exponential decay function; where the depth direction is the direction of incident light propagation. The three-dimensional spatial distribution model is obtained by combining the lateral point source distribution function and the exponential decay function.

4. The image sensor performance prediction method based on graded doping and finite substrate thickness according to claim 1, characterized in that, Determining the built-in electric field model includes: Differentiating the continuously gradient doping distribution yields the doping concentration gradient; The doping concentration gradient is converted into the built-in electric field model based on the thermal equilibrium condition.

5. The image sensor performance prediction method based on graded doping and finite substrate thickness according to claim 1, characterized in that, Obtaining the pixel response function of the pixel to point source illumination based on the spatial distribution function includes: The current density in the depth direction is obtained according to the spatial distribution function at the location of the corresponding collection layer; wherein, the depth direction is the direction of incident light propagation. Integrate the current density to obtain the pixel output current; The pixel output current is normalized to obtain the pixel response function.

6. The image sensor performance prediction method based on graded doping and finite substrate thickness according to claim 1, characterized in that, Obtaining the modulation transfer function includes: The pixel response function is integrated along the horizontal direction of the pixel plane to obtain a one-dimensional pixel response function; Perform a Fourier transform on the one-dimensional pixel response function to obtain the modulation transfer function of the image sensor at different spatial frequencies.

7. The image sensor performance prediction method based on graded doping and finite substrate thickness according to claim 1, characterized in that, Obtaining the charge collection efficiency includes: Integrating the pixel response function over the region where the collection layer is located yields the number of photogenerated carriers collected. The total number of photogenerated carriers generated within the pixel is obtained based on the number of photogenerated carriers collected, the material absorption characteristics of the collection layer, and the thickness of the substrate. The ratio of the number of collected photogenerated carriers to the total number of photogenerated carriers is defined as the charge collection efficiency.

8. A device for predicting the performance of an image sensor based on graded doping and finite substrate thickness, characterized in that, include: A physical model building module is used to build a three-dimensional physical model of the image sensor pixels; wherein, the three-dimensional physical model includes a continuous gradient doping distribution between the epitaxial layer and the substrate and a substrate of finite thickness; The distribution model building module is used to build a three-dimensional spatial distribution model of photogenerated carriers within a pixel under point source illumination conditions. An electric field model acquisition module is used to acquire a built-in electric field model generated by the doping concentration gradient based on the continuous gradient doping distribution. The distribution function acquisition module is used to solve the carrier continuity equation under the boundary conditions corresponding to the substrate of finite thickness to obtain the spatial distribution function of photogenerated carriers within the pixel; wherein, the carrier continuity equation includes the three-dimensional spatial distribution model, the built-in electric field model, and the carrier diffusion process; The response function acquisition module is used to acquire the pixel response function of the pixel to point source illumination based on the spatial distribution function; A performance index acquisition module is used to acquire the performance index of the image sensor based on the pixel response function; wherein the performance index includes the modulation transfer function and the charge collection efficiency.

9. An image sensor, characterized in that, The method includes a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the image sensor performance prediction method based on gradient doping and finite substrate thickness as described in any one of claims 1-7.

10. The image sensor according to claim 9, characterized in that, The image sensor is a front-illuminated image sensor, a back-illuminated image sensor, or a charge-coupled device (CCD) image sensor.

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