A carrier radiation electric parameter measurement optimization model and inversion method

Abstract

This invention belongs to the field of non-contact detection technology for semiconductor materials, specifically relating to an optimized model and inversion method for measuring carrier radiation electrical parameters. This invention proposes a quasi-complete nonlinear transport model of carrier recombination mechanisms. By integrating three mechanisms—SRH recombination, radiative recombination, and Auger recombination—the dynamic characteristics of their contribution ratios are precisely incorporated into the carrier diffusion equation, constructing a physically complete nonlinear carrier recombination model. Through optimization of the numerical solution algorithm, accurate characterization of semiconductor materials under different light intensities and injection levels is achieved. The optimization of the model itself overcomes the limitations of traditional models in terms of applicability and the lack of physical mechanisms. Furthermore, through optimization of the multi-parameter fitting solution algorithm, the accuracy of electrical parameter inversion results based on photogenerated carrier detection methods can be further improved in scenarios such as industrial detection of changes in injected light intensity and differences in material properties.

Description

Technical Field

[0001] This invention belongs to the field of non-contact detection technology for semiconductor materials, specifically relating to an optimization model and inversion method for measuring charge carrier radiation electrical parameters. Background Technology

[0002] Semiconductor carrier transport parameters (equivalent recombination lifetime τ, diffusion coefficient) Front / back surface recombination rate ( ) is a core indicator that determines the material properties and the photoelectric conversion efficiency of devices: among which, The diffusion coefficient directly reflects the density and diffusion coefficient of electron / hole trap defects within the material. Characterizing carrier mobility, The levels of surface / interface defects in related materials, along with other factors, collectively determine the energy conversion efficiency of semiconductor photovoltaic cells, the switching speed of semiconductor integrated circuits, and the response sensitivity of semiconductor sensors. Therefore, achieving non-contact, high-precision in-situ detection of these parameters under different operating conditions (different light intensities, corresponding to different injection levels) is of great significance for the quality assessment and process optimization of semiconductor materials and devices.

[0003] In carrier parameter detection, photocarrier radiation (PCR) measurement technology, as a non-contact optical detection method, has become an important technical path for carrier transport parameter inversion due to its advantages of being non-destructive to samples, having a high signal-to-noise ratio, and being relatively easy to set up. Its core principle is: a periodic photocarrier density wave is generated by irradiating a semiconductor material with intensity-modulated excitation light; the near-infrared signal released by carrier radiative recombination is detected, and the amplitude-frequency response and phase-frequency response of the signal are obtained; the target electrical parameters are obtained by fitting the response curve based on a carrier transport theoretical model.

[0004] The recombination mechanism of charge carriers is the core basis for establishing the theoretical model of PCR technology. In semiconductor physics, charge carrier recombination is jointly determined by three mechanisms: SRH recombination, radiative recombination, and Auger recombination. The contribution ratio of each mechanism varies under different light injection levels. (It should be noted that the recombination lifetime formulas involved in the implementation path of this patent are all based on...) (Semiconductor hypothesis). The specific characteristics and mathematical expressions of the three different recombination mechanisms are as follows:

[0005] SRH (Shockley-Read-Hall) recombination: Primarily refers to recombination assisted by defect centers in the bandgap, dominating carrier loss during low injection. Its SRH recombination lifetime formula is: In the formula, For excess electron concentration, Doping concentration, For electronic SRH recombination lifetime, This represents the hole SRH recombination lifetime.

[0006] Radiative recombination refers to the direct interband radiative recombination of electrons and holes, emitting photons. Under medium-injection conditions, its contribution is significantly enhanced, dominating the lifetime characteristics of carriers under medium light intensity. The radiative recombination lifetime formula is: In the formula, is the radiative recombination coefficient.

[0007] Auger recombination: The energy released by electron-hole recombination is absorbed by other charge carriers instead of emitting photons. Its contribution is significantly enhanced under high injection conditions, dominating the lifetime characteristics of charge carriers under high light intensity. The Auger recombination lifetime formula is: In the formula, The Auger composite coefficient for holes. is the electronic Auger recombination coefficient.

[0008] Currently, in the application of PCR detection technologies, such as zero-difference PCR, lock-in carrier imaging (LIC), and heterodyne PCR, the theoretical parameter fitting models are all based on the assumption of low injected light intensity. The carrier diffusion model based on this assumption is as follows:

[0009] (1)

[0010] in, For excess carrier concentration, Where is the diffusion coefficient. For the body's lifespan, Let represent the carrier generation rate. Since this theoretical model is based on low injection conditions, where the excess carrier concentration is extremely low, radiative recombination and Auger recombination are ignored, and the fitting results only consider carrier losses dominated by SRH recombination.

[0011] Traditional models retain only the SRH recombination term, neglecting the contributions of radiative recombination and Auger recombination. However, the carrier recombination mechanism within semiconductors exhibits differentiated contribution ratios depending on the excitation light injection level, and intrinsic material differences lead to variations in the determination of the light intensity injection boundary. In practical semiconductor testing, when the excitation light injection conditions can be confirmed as a low injection level, i.e. The theoretical model can be simplified to reflect only the physical process of carrier SRH recombination; when the excitation light injection conditions satisfy the injection level, i.e. At this time, the radiative recombination term contributes significantly. Traditional models, neglecting this mechanism, misjudge carrier losses caused by radiative recombination as a shortened SRH lifetime, resulting in an underestimation of the lifetime measurement. When the excitation light injection conditions meet a high injection level, i.e. At this time, the Auger composite term becomes dominant. If the parameter fitting calculation is still performed according to the traditional model assumptions, the calculated effective life value will be much lower than the actual value of the material, and the error can reach an order of magnitude.

[0012] In practical industrial testing, photogenerated carrier detection technology often faces two major challenges in obtaining signals with a signal-to-noise ratio sufficient for detector recognition: first, the excitation light intensity must generate a sufficient number of photogenerated carriers to ensure detectable signals; second, for complex materials with low lifetimes or heavy doping, the rapid loss of carriers must be overcome to ensure signal detectability. At this point, the injection level within the material often exceeds the low-injection applicable conditions of traditional models. If parameter inversion is still performed using a linear fitting model applicable to low-injection conditions (a theoretical model considering only SRH recombination), it will lead to insufficient accuracy and an inability to accurately characterize the properties of complex materials in industrial applications. Summary of the Invention

[0013] To address the aforementioned issues, this invention proposes a quasi-complete nonlinear transport model for carrier recombination mechanisms. By integrating three mechanisms—SRH recombination, radiative recombination, and Auger recombination—the dynamic characteristics of the contribution ratios of these three recombination mechanisms are precisely incorporated into the carrier continuity equation, thus constructing a physically complete nonlinear carrier recombination model. Through optimization of the numerical solution algorithm, accurate characterization of semiconductor materials under arbitrary light intensities and injection levels is achieved.

[0014] The technical solution adopted in this invention is:

[0015] An optimization model and inversion method for measuring carrier radiation electrical parameters, comprising:

[0016] Constructing a quasi-complete carrier nonlinear recombination model:

[0017] (2)

[0018] In this model, Defined as a nonlinear composite law function, its expression is: .

[0019] This theoretical model considers parameter inversion under the combined effect of three mechanisms, among which:

[0020] Low injection adaptation ( The SRH compound term is automatically simplified to: ; Radiation composite term With Auger complex Related items It is minimal and naturally weakened, achieving the same accuracy as traditional models in the low injection range.

[0021] Injecting adapters ( ): Radiation composite term Automatic activation, forming a synergistic response with the SRH recombination term, accurately quantifies the competing losses of the two mechanisms, and solves the error problem caused by neglecting radiative recombination in traditional models.

[0022] High injection compatibility ( Auger compound: Significant contribution, fully covering the three composite effects, significantly reducing measurement errors under high-intensity laser excitation.

[0023] The DC and AC components of the model are solved by numerical algorithm, then the PCR radiation integral signal is calculated, and finally the convergent semiconductor electrical parameters are obtained by constructing the Nelder-Mead Simplex optimization algorithm.

[0024] The beneficial effects of this invention are as follows: By establishing a nonlinear transport model of charge carriers under a quasi-complete recombination mechanism, this invention can more fully restore the charge carrier recombination process through mathematical modeling. That is, by optimizing the model itself, it can make up for the limitations of the applicable conditions and the lack of physical mechanisms of traditional models. Furthermore, by optimizing the multi-parameter fitting solution algorithm, it can further improve the accuracy of electrical parameter inversion in scenarios such as industrial detection of changes in injected light intensity and material property differences. Attached Figure Description

[0025] Figure 1 This is a flowchart of the nonlinear transport model for fully composite carriers;

[0026] Figure 2 This is a schematic diagram of the experimental system;

[0027] Figure 3 This is a schematic diagram of the experimental procedure;

[0028] Figure 4 This is a schematic diagram of the simplex iterative optimization algorithm. Detailed Implementation

[0029] The method of the present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0030] This invention proposes a quasi-complete carrier nonlinear recombination model. By integrating three mechanisms—SRH recombination, radiative recombination, and Auger recombination—the dynamic characteristics of the contribution ratios of the three recombination mechanisms are accurately incorporated into the carrier continuity equation, thus constructing a carrier transport model with complete physical mechanisms. With the corresponding numerical solution algorithm, the electrical parameters of semiconductor materials can be accurately characterized.

[0031] like Figure 1 As shown, the specific technical solution includes the following steps:

[0032] (1) The constructed quasi-complete carrier nonlinear recombination model differs from the traditional linear model that only considers SRH recombination. This invention explicitly introduces recombination expressions for three recombination mechanisms into the carrier continuity equation, constructing a nonlinear complete governing equation independent of injection intensity:

[0033] (3)

[0034] in, The total recombination rate function comprehensively reflects the synergistic effect of the three mechanisms of SRH recombination, radiation recombination, and Auger recombination, and can be specifically expressed as:

[0035] (4)

[0036] The meanings of the parameters in the formula are as follows:

[0037] Sample spatial location Place, The excess carrier concentration at a given time (based on the P-type semiconductor assumption, where electrons are the minority carriers of interest) is a core physical quantity describing the spatiotemporal distribution of carriers.

[0038] : Bipolar diffusion coefficient of charge carriers, characterizing the ability of charge carriers to diffuse in a material;

[0039] Electron SRH recombination lifetime reflects the ability of defects in the material to capture and recombine electrons.

[0040] Hole SRH recombination lifetime reflects the ability of a material's internal defects to capture and recombine holes.

[0041] Doping concentration is an intrinsic parameter determined by both the material's doping characteristics and temperature.

[0042] : Radiation recombination coefficient, characterizing the intensity of the interband radiative recombination process between electrons and holes;

[0043] : Electron Auger recombination coefficient, characterizing the intensity of the Auger recombination process involving electrons under high injection conditions;

[0044] : Hole Auger recombination coefficient, characterizing the intensity of the Auger recombination process involving holes under high injection conditions;

[0045] Photogenerated carrier generation rate describes the spatial distribution and time-varying characteristics of carriers excited by excitation light in a material.

[0046] (2) The DC component of the numerical solution model is decomposed into the DC component of the total electron concentration. and communication weight ,Right now The DC component, generated by steady-state excitation light, forms the basis of the carrier distribution; the AC component, excited by modulated light, serves as the signal carrier for parameter inversion. The decomposition formula for the photogenerated carrier generation rate is as follows: ,in The alternating current photogenerated carrier generation rate, and the DC component Together constitute the total production rate For DC equations The solution can be obtained by spatial discretization using the finite element method, and the steps are as follows:

[0047] First, the semiconductor sample is divided into M linear units along the thickness direction, corresponding to M+1 node coordinates (including the first and last nodes). Then, a shape function ϕ_i(z) (where i ranges from 1 to M+1) corresponding to each node is introduced to interpolate and approximate the continuous concentration distribution. This enables the discretization of continuous physical fields. Specifically, it addresses the total composite law function in the equations. The residual equation is constructed using the weighted residual method. The solution is obtained using the Newton-Raphson algorithm. In each iteration, a linear correction equation is constructed based on the Taylor first-order expansion principle:

[0048]

[0049] in The two core matrix vectors in this equation are defined as the node concentration correction factor:

[0050] residual vector Its physical meaning originates from the weighted integral of the governing equations. It characterizes the current approximate solution. After substituting into the equation, the net margin that is not in equilibrium between the diffusion term, the nonlinear recombination term, and the carrier generation term is controlled by the degree to which the residual approaches zero, thereby controlling the fitting accuracy.

[0051] Stiffness matrix Defined as the derivative matrix of the residual vector with respect to the node concentration. According to the Taylor expansion derivation, the elements of this matrix not only include the contribution of the diffusion term, but also explicitly include the derivative term of the total recombination rate:

[0052]

[0053]

[0054] The concentration correction amount is obtained by solving the above linear equations. and update the solution until the residual vector The norm approaches zero, thus obtaining an accurate steady-state carrier concentration distribution. Finally, the recombination rate of the front surface was determined. and post-surface recombination rate As boundary flux conditions, the boundary terms generated by integrals of parts are coupled and superimposed onto the corresponding boundary nodes of the stiffness matrix.

[0055] (3) Numerical solution of the AC components of the model. The core equation of the fully composite model is: .Will After substituting into the equation, the steady-state equation satisfied by the DC component is used. The DC balance term can be eliminated, and the AC component equations are finally obtained: In the formula This is the recombination rate increment function, a core term representing the coupling between the AC component and the DC substrate. This increment function contains... The first, second, and third terms of the equation cause the AC equation to exhibit strong nonlinear characteristics, making it impossible to solve using conventional frequency domain analytical methods. Therefore, a hybrid strategy combining Crank-Nicolson time-domain discretization with transient Newton iteration is adopted to achieve efficient solution.

[0056] Crank-Nicolson time-domain discretization. Discretizes the time axis into... To ensure unconditional stability and achieve second-order accuracy in numerical computation, the Crank-Nicolson scheme is adopted, i.e., at time points... Establish an equilibrium equation at that point. Make the equation include the previous time step. and current time step This information effectively suppresses numerical oscillations. The central difference quotient is used. Approximate time derivative Discrete formula for the second derivative Simultaneously, both the diffusion term and the total composite term are expressed as the arithmetic mean of the old and new time steps. This leads to the construction of the discrete evolution equation:

[0057]

[0058] in, It is the identity matrix. Let be the space Laplacian operator matrix. and These represent the increments in the composite rate between the old and new time steps, respectively. Since the left-hand side of the equation includes... The quadratic and cubic terms cannot be solved directly; they need to be linearized using Newton's iteration method at each time step. Define the transient residual vector. The difference between the left and right sides of the discrete evolution equation is denoted as . Its physical meaning is to characterize the dynamic imbalance of the approximate solution at the current time step. Simultaneously, the transient Jacobian matrix is ​​defined. For the residual vector with respect to the concentration at the new time step The partial derivatives of are the core of linearizing nonlinear problems, and their analytical form is: The diagonal matrix terms are the derivatives of the composite increment function, fully preserving the nonlinear contributions under non-low injection conditions.

[0059] The goal is to find make Based on the Taylor first-order expansion, the residual vector is approximated as... ,in This is the approximate value for the current iteration. This is the concentration correction amount.

[0060] By solving linear systems It converges rapidly within each time step to obtain accurate results. Perform multi-cycle transient simulations using the above algorithm until the carrier fluctuations generated by the system enter a periodic steady state. Extract the time-domain waveform data from the last complete cycle and use Fast Fourier Transform (FFT) to extract and modulate the frequency. The corresponding fundamental frequency component is used to obtain the AC amplitude and phase at that frequency.

[0061] (4) Calculate the PCR radiation integral signal. According to the generalized Planck's law of radiation, calculate the PCR signal received by the detector. Under the corresponding injection conditions, the PCR signal intensity follows the following integral relationship:

[0062]

[0063] The amplitude of the analog signal is calculated by numerical integration. and phase .

[0064] (5) Multi-parameter global optimization fitting: Construct the Nelder-Mead Simplex optimization algorithm, using the experimentally measured amplitude. Phase Compared with simulated values , The objective function is to minimize the error between the measured parameters. As optimization variables, the optimal solution is searched in the parameter space through simplex geometric operations such as reflection, expansion, and contraction, and finally the converged semiconductor electrical parameters are obtained.

[0065] Compared with existing traditional optical carrier radiation detection techniques based on linear models (considering only SRH recombination and ignoring radiative recombination and Auger recombination terms), this invention has the following significant advantages:

[0066] This invention significantly improves the accuracy of electrical parameter measurements under non-low illumination conditions. Traditional techniques, under non-low-level injection, neglect radiative recombination and Auger recombination, incorrectly attributing this additional recombination loss to surface recombination or SRH recombination, leading to underestimating carrier lifetime values ​​and overestimating surface recombination rates. This invention considers the nonlinear characteristics of complete carrier recombination and, by coordinating the contribution changes of different recombination mechanisms, has achieved accurate inversion of electrical parameters even under illumination conditions exceeding 100 solar constants.

[0067] For low-lifetime or heavily doped semiconductor materials, traditional models cannot accurately detect them due to poor implantation adaptability. The fully recombination model of this invention can adapt to the recombination behavior of carriers under different doping concentrations, expanding the application range of carrier detection technology and enabling techniques such as PCR to be applied to a wider range of industrial detection scenarios, such as concentrated photovoltaic cell materials and power semiconductor devices.

[0068] The hybrid numerical solution method of "finite element spatial discretization + Crank-Nicolson time discretization + Newton iteration" adopted in this invention can handle arbitrarily complex non-uniform carrier generation rate distribution and nonlinear boundary conditions compared with analytical solution methods. Compared with explicit difference method, it has better numerical stability, ensures calculation convergence under high nonlinear coefficients, and guarantees the reliability of parameter fitting.

[0069] Example:

[0070] The photocarrier radiation (PCR) detection experimental system used in this embodiment is as follows: Figure 2 The main features shown include: using a semiconductor laser with a wavelength of 405nm as the pump source, and its output light intensity... Modulated by a function generator, the frequency range can be set according to detection needs; in this embodiment, the frequency covers 100Hz to 1MHz. The optical system includes a lens group (for focusing the excitation light onto the sample surface), an off-axis parabolic mirror (for efficiently collecting the infrared radiation signal emitted by the sample), and a detection unit (in this example, an InGaAs photodetector is used, with a long-pass filter added to filter out the excitation light, for receiving radiative recombination photons). Finally, a lock-in amplifier is used to demodulate the detector signal to obtain the signal amplitude and phase.

[0071] The embodiment uses a piece of thickness Using P-type silicon wafers as the detection object, the simplex optimization algorithm, combined with the aforementioned quasi-complete carrier nonlinear recombination model, is employed to achieve data fitting and parameter extraction of key electrical parameters of the silicon wafers. The specific process is as follows: Figure 3 As shown, it includes the following five core steps:

[0072] Step 1: Acquisition and preprocessing of PCR experimental data. First, the sample was scanned using the constructed photocarrier radiation detection system. The DC laser intensity was set. (Low injection) or (High injection), and load a sinusoidal modulated signal. In Select within range The frequency points increase exponentially. The amplitude and phase of the radiated signal at each frequency point are recorded using a lock-in amplifier to obtain the experimental dataset. and This data will serve as the basis for subsequent fitting and optimization.

[0073] Step 2: Parameter Space Definition and Initialization In the inversion algorithm, based on the characteristics of the physical model, the parameters are divided into two categories: "parameters to be fitted" and "fixed parameters" to reduce the degrees of freedom in optimization and improve the convergence speed.

[0074] Parameters to be fitted (Variables): The five parameters most sensitive to the signal response were selected as optimization variables, namely, electronic SRH recombination lifetime. Hole SRH composite lifetime front surface recombination rate Post-surface recombination rate and diffusion coefficient .

[0075] Constants: Set known physical constants and injection characteristic parameters for the material. Among these, the radiation recombination coefficient... Set as Electron Auger recombination coefficient Set as Hole Auger composite coefficient Set as doping concentration Set as .

[0076] To prevent the algorithm from getting stuck in a non-physically valid region, parameter boundaries are set. ,in, .

[0077] Step 3: Construct the initial simplex and use the Nelder-Mead simplex algorithm to optimize multiple parameters.

[0078] for The algorithm constructs a five-dimensional parameter space containing _____ parameters to be fitted. The initial simplex (geometry) with vertices. Each vertex (i=1,2,…,6) represents a specific set of parameter vectors. .

[0079] The algorithm invokes a quasi-complete carrier nonlinear composite model to calculate the simulated amplitudes corresponding to these six vertices. and phase Then calculate the objective function value for each vertex. The objective function is defined as the weighted mean square error between the simulation data and the experimental data. Its specific calculation formula is as follows: .

[0080] in, This represents the mean square value of the relative amplitude error. The normalized mean square value of the absolute phase error is given by the following formulas:

[0081]

[0082]

[0083] In the formula, Given the total number of frequency points in the frequency sweep test, the error value corresponding to each vertex is calculated using this formula. .

[0084] Step 4: Simplex iterative optimization loop as follows Figure 4 As shown, the algorithm enters an iterative loop, continuously approximating the minimum point of the objective function through geometric transformations. The six vertices are sorted according to the objective function value to obtain... .in This is the current optimal point (with minimal error). This is the worst-case scenario (with the largest error). First, calculate the values ​​excluding the worst-case scenario. The geometric centers of the other 5 points According to the simplex rule, try to find the worst-case scenario. Update.

[0085] Reflection operation: Regarding the center of mass Obtained by reflection The formula is: .in The reflection coefficient is usually set to 1, which controls the step size of the reflection direction. This indicates that the reflection point effect is moderate, so use... replace .

[0086] Extended operation: If the reflection point Better than the current best point This indicates that this direction has great potential, and further expansion along this direction will yield results. The formula is .in The expansion factor is usually set to 2, which controls the step size in the expansion direction.

[0087] like Then use replace Otherwise use .

[0088] Contraction operation: if the reflection point Still poor Then in and Contraction between to obtain The formula is: .in This is the contraction coefficient, typically taken as 0.5, which controls the step size in the contraction direction. If... Then use replace .

[0089] If the above operations are ineffective, then the entire simplex will be oriented to the optimal point. Compress, narrowing the search scope.

[0090] Step 5: Convergence check and result output. After each simplex update, check the convergence condition:

[0091] Function value tolerance: Whether the difference between the objective function values ​​of the best and worst vertices is less than a set threshold. ;

[0092] Parameter standard deviation: Whether the standard deviation of the parameter values ​​at each vertex is less than the set threshold. If any of the above conditions are met, the iteration is considered converged, and the algorithm terminates. At this point, the optimal vertex is found. The corresponding parameter vector These are the semiconductor electrical parameters obtained from the final inversion.

[0093] Through the above embodiments, the present invention can quickly and accurately separate more realistic transport parameters of materials from PCR signals by utilizing the global search capability of the simplex method and the physical accuracy of the quasi-complete carrier nonlinear recombination model at different injection levels.

Claims

1. An optimization model and inversion method for measuring and retrieving charge carrier radiation electrical parameters, characterized in that, include: Constructing a quasi-complete carrier nonlinear recombination model: ， in, For sample spatial location Place, The excess carrier concentration at any given time. The bipolar diffusion coefficient of charge carriers, Photogenerated carrier generation rate, It is the total composite rate function. The expression is: ， in, For electronic SRH recombination lifetime, For hole SRH recombination lifetime, Doping concentration, The radiative recombination coefficient, The Auger composite coefficient for holes. The electron Auger recombination coefficient; This model also needs to consider the boundary conditions of two surfaces: ， in These are the recombination rates of the front and rear surfaces, respectively. The thickness of the sample; The DC component of the numerical solution model is specifically: the total excess carrier concentration. Decomposed into DC component and communication weight ,Right now The DC component is generated by steady-state excitation light and serves as the basis for carrier distribution; the AC component is excited by modulated light and serves as the signal carrier for parameter inversion; the photogenerated carrier generation rate decomposition formula... ,in The alternating current photogenerated carrier generation rate, and the DC component Together constitute the total production rate For DC equations The spatial discretization solution is obtained using the finite element method. First, the semiconductor sample is divided into M linear units along the thickness direction, and M+1 node coordinates are defined accordingly. Then, shape functions corresponding to each node are introduced. The value of i ranges from 1 to M+1, and interpolation is used to approximate the continuous concentration distribution, i.e. This enables the discretization of continuous physical fields; and addresses the total composite law function in the equations. The residual equation is constructed using the weighted residual method. The solution is obtained using the Newton-Raphson algorithm. In each iteration, a linear correction equation is constructed based on the Taylor first-order expansion principle: ， in The node concentration correction amount, residual vector Characterizes the current approximate solution After substituting into the equation, the net margin that is not in equilibrium between the diffusion term, the nonlinear recombination term, and the carrier generation term is controlled by the degree to which the residual approaches zero; the stiffness matrix Defined as the derivative matrix of the residual vector with respect to the node concentration According to the Taylor expansion derivation, the elements of this matrix not only include the contribution of the diffusion term, but also explicitly include the derivative term of the total recombination rate: ， The concentration correction of the DC component is obtained by solving the linear equation system. and update the solution until the residual vector The norm approaches zero, thus obtaining an accurate steady-state carrier concentration distribution. Finally, the recombination rate of the front surface was measured. and post-surface recombination rate As a boundary flux condition, the boundary terms generated by integration by parts are coupled and superimposed onto the corresponding boundary nodes of the stiffness matrix; The numerical solution model's AC components are as follows: The core equations of the composite model are: ,Will , After substituting into the equation, the steady-state equation satisfied by the DC component is used. After eliminating the DC balance term, the AC component equations are finally obtained: In the formula The recombination rate increment function is the core term for the coupling between the AC component and the DC substrate. The increment function includes... The first, second, and third terms of the equation cause the AC equation to exhibit strong nonlinear characteristics. Therefore, a hybrid strategy combining Crank-Nicolson time-domain discretization and transient Newton iteration is adopted to achieve efficient solution. This includes: Crank-Nicolson time-domain discretization, which discretizes the time axis into... To ensure unconditional stability and achieve second-order accuracy in numerical computation, the Crank-Nicolson scheme is adopted, i.e., at time points... Establish an equilibrium equation at a point such that the equation includes the previous time step. and current time step Information using central difference quotient Approximate time derivative Discrete formula for the second derivative Simultaneously, both the diffusion term and the total composite term are expressed as the arithmetic mean of the old and new time steps, thereby constructing the discrete evolution equation: ， in, It is the identity matrix. Let be the space Laplacian operator matrix. and These represent the increments in the composite rate between the old and new time steps; since the left-hand side of the equation contains... The quadratic and cubic terms cannot be solved directly and must be linearized using Newton's iteration method at each time step; a transient residual vector is defined. This is the difference between the left and right sides of the discrete evolution equation, i.e. Its physical meaning is to characterize the dynamic imbalance of the approximate solution at the current time step; at the same time, the transient Jacobian matrix is ​​defined. For the residual vector with respect to the concentration at the new time step The partial derivatives, in analytical form, are: The diagonal matrix terms are the derivatives of the composite increment function, which fully preserves the nonlinear contributions under non-low injection conditions. The target solution has been found. make Based on the first-order Taylor expansion, the residual vector is approximated as... ,in This is the approximate value for the current iteration. This is the concentration correction amount; By solving linear systems It converges rapidly within each time step to obtain accurate results. Multi-cycle transient simulations were performed until the carrier fluctuations generated by the system entered a periodic steady state. The time-domain waveform data of the last complete cycle was captured, and the frequency was extracted and modulated using Fast Fourier Transform (FFT). The corresponding fundamental frequency component is used to obtain the AC amplitude and phase at that frequency; Calculate the PCR radiation integral signal. According to the generalized Planck's radiation law, calculate the PCR signal received by the detector. Under the corresponding injection conditions, the PCR signal intensity follows the following integral relationship: ， The amplitude of the analog signal is calculated by numerical integration. and phase ; A Nelder-Mead Simplex optimization algorithm was constructed, using experimentally measured amplitudes. Phase Compared with simulated values , Minimizing the error between them is the objective function, and the parameters to be measured are... As optimization variables, the optimal solution is searched in the parameter space through simplex geometric operations of reflection, expansion, and contraction, and finally the converged semiconductor electrical parameters are obtained.

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