Semiconductor electrical characteristic efficient simulation method and system based on physical model convergence acceleration

Abstract

The invention discloses a semiconductor electrical characteristic efficient simulation method and system based on physical model convergence acceleration, and aims to effectively solve the problems that a traditional semiconductor electrical characteristic solving method is low in convergence efficiency and insufficient in stability. According to the method, firstly, based on a spectral element method, a damping Newton iteration algorithm is adopted to realize stable solution of transient electrical characteristics, and numerical oscillation is effectively inhibited through a damping factor; an Anderson acceleration method is adopted in the iteration solving process, a linear combination is constructed by means of previous m iteration results, the next iteration result is predicted, the electrical characteristics of the semiconductor are obtained through iteration, and the convergence speed is remarkably increased. According to the method, on the premise that the numerical solution stability is kept, the solution efficiency of the semiconductor physical model is improved, and an efficient and reliable numerical solution is provided for simulation of a semiconductor device.

Description

Technical Field

[0001] The present invention relates to the technical field of numerical simulation of semiconductor devices, and particularly to an efficient simulation method and system for semiconductor electrical characteristics based on physical model convergence acceleration. Background Art

[0002] In the solution of the electrical characteristics of semiconductor physical models, the damped Newton method is widely used due to its global convergence and adaptability to complex problems. However, the problem of its large number of iteration steps is particularly prominent when facing ill-conditioned problems or improper initial point selection. It is pointed out in Document 1. Zhang Jianjun, Li Chunquan, Zhang Liehui. Two important parameters affecting the convergence of the damped Newton method [J]. Pure and Applied Mathematics, 2012, 28(04): 433-439. that there is a non-linear relationship between the number of iterations of the damped Newton method and parameter selection, which further shows that it may require a large number of iteration steps to converge in practical applications. Document 2. Lu Huiping, Xi Jie, Jiang Zhixia. A damped Newton method with memory [J]. Advances in Applied Mathematics, 2024, 13(6): 2614-2626. points out that when solving some problems, the damped Newton method has a large number of iterations, especially when facing ill-conditioned problems, its convergence speed is slow. This shows that in some complex problems, the damped Newton method may require a large number of iteration steps to achieve the expected accuracy, resulting in slow prediction speed and low efficiency of the electrical characteristics of semiconductor physical models. To overcome this limitation, the present invention proposes an efficient physical model convergence acceleration method based on semiconductor device simulation to improve the convergence speed of the electrical characteristics simulation of semiconductor physical models. Summary of the Invention

[0003] The purpose of the present invention is to provide an efficient simulation method and system for semiconductor electrical characteristics based on physical model convergence acceleration, which improves the solution efficiency of semiconductor physical models while maintaining the stability of numerical solutions, and provides an efficient and reliable solution for the simulation of semiconductor devices.

[0004] The technical solution to achieve the purpose of the present invention is: An efficient simulation method for semiconductor electrical characteristics based on physical model convergence acceleration, the structure of the semiconductor physical model is composed of an AlGaN layer and a GaN layer, including:

[0005] Step 1, divide the structure of the semiconductor physical model using hexahedrons, and set different material numbers for AlGaN and GaN respectively to obtain all grid information of the semiconductor physical model;

[0006] Step 2: Normalize the current continuity equation, current density equation, Poisson equation, and their corresponding boundary conditions and continuity conditions based on the grid information, and discretize the current continuity equation. The normalized and discretized current continuity equation, current density equation, Poisson equation, and their corresponding boundary conditions and continuity conditions form a drift-diffusion model;

[0007] Step 3: Use the damped Newton iteration method, introduce a damping factor, solve the equations in the drift-diffusion model. The Anderson acceleration method is used in the iterative solution process, and the linear combination is constructed using the results of the previous m iterations to predict the next iteration result, and the semiconductor electrical characteristics are obtained through iteration.

[0008] Furthermore, it includes: using the damped Newton iteration method, introducing a damping factor, and obtaining the iterative coupling equation of the drift-diffusion model as:

[0009]

[0010] where λ is the damping factor, φ n , φ p , respectively represent the electron quasi-Fermi potential, hole quasi-Fermi potential, and electric potential, respectively represent the electron quasi-Fermi potential, hole quasi-Fermi potential, and electric potential at the current l-th moment, F n , F p and respectively represent the electron continuity equation, hole continuity equation, and Poisson equation.

[0011] Furthermore, the damping factor satisfies:

[0012]

[0013] where ‖·‖ represents any norm of a vector or matrix, the variable p = -[f′(x l )] -1 ·f(x l ), f(x l ) is the equation to be solved, and the coefficient μ is a set value.

[0014] Furthermore, λ = 0.6, μ ∈ (0, 1).

[0015] Furthermore, the Anderson acceleration form is:

[0016]

[0017] In the formula, is the increment matrix, γ (l) is the acceleration parameter, g(x l) is the function value.

[0018] Furthermore, during the Anderson acceleration process, the acceleration parameters of the electron quasi-Fermi potential, hole quasi-Fermi potential, and electric potential are calculated independently. If convergence is not achieved after continuously exceeding a preset specific number of steps during the iteration process, the Anderson acceleration is terminated and the damped Newton iteration method is restored.

[0019] Furthermore, the calculation method of the acceleration parameter is as follows:

[0020]

[0021] where f l represents the residual vector, represents the increment matrix composed of the residual vectors, γ (l) represents the optimal parameter vector of the l-th iteration, γ represents the parameter vector to be optimized, which respectively represent different dimensions of the Q matrix and R matrix, α represents the number of unknowns, and m represents the set memory depth;

[0022] Furthermore, the m = 2.

[0023] An efficient semiconductor electrical property simulation system based on physical model convergence acceleration, comprising:

[0024] A semiconductor physical model meshing unit that meshes the semiconductor physical model structure using hexahedrons and sets different material numbers for AlGaN and GaN respectively to obtain all grid information of the semiconductor physical model;

[0025] A drift-diffusion model construction unit that normalizes the current continuity equation, current density equation, Poisson equation, and corresponding boundary conditions and continuity conditions based on the grid information, and discretizes the current continuity equation. The normalized and discretized current continuity equation, current density equation, Poisson equation, and corresponding boundary conditions and continuity conditions form a drift-diffusion model;

[0026] A semiconductor electrical property solving unit that uses the damped Newton iteration method, introduces a damping factor, solves the equations in the drift-diffusion model, and uses the Anderson acceleration method in the iterative solving process. It constructs a linear combination using the results of the previous m iterations to predict the next iteration result, and obtains the semiconductor electrical properties through iteration.

[0027] Compared with the prior art, the present invention has the following remarkable advantages: The method of the present invention introduces the damped Newton iteration method and the Anderson acceleration method. The damped Newton iteration method introduces a smaller damping factor to make the iteration step smaller, effectively reducing the overshoot and oscillation when approaching the optimal solution and enhancing the algorithm stability. On this basis, the Anderson acceleration method is introduced, which constructs a linear combination using the results of previous iterations to predict the next iteration result, accelerating the convergence and improving the solution efficiency of the semiconductor physical model, providing an efficient and reliable solution for the simulation of semiconductor devices. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] Figure 1 is the block diagram of the adaptive Anderson acceleration program.

[0029] Figure 2 is the schematic structural diagram of the semiconductor physical model.

[0030] Figure 3 is the bias voltage applied to the drain of the semiconductor physical model and the solved drain current response.

[0031] Figure 4 is the potential distribution diagram of the semiconductor physical model at 100 ps. Fig. (a) is the solution result of the spectral element method of the present patent method, and Fig. (b) is the solution result of the commercial software COMSOL.

[0032] Figure 5 is the potential distribution diagram of the semiconductor physical model at 200 ps. Fig. (a) is the solution result of the spectral element method of the present patent method, and Fig. (b) is the solution result of the commercial software COMSOL.

[0033] Figure 6 is the convergence curve of the hole quasi-Fermi potential with different memory depths when the source voltage of the semiconductor physical model is 4V. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0034] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0035] The present invention proposes an efficient physical model convergence acceleration method based on semiconductor device simulation. The efficient physical model convergence acceleration method based on semiconductor device simulation particularly focuses on the damped Newton iteration optimization of the drift-diffusion equations under the grid discretization framework, and realizes the improvement of the convergence speed through the construction of a linear combination prediction mechanism of historical iteration solutions. Next, in conjunction with the accompanying drawings, taking Figure 2Taking the semiconductor physical model shown as an example, the geometric dimensions of the model are supplemented as follows: the distance between the gate and the drain is 0.1um, and the distance between the gate and the source is 0.1um. The specific operation steps are as follows:

[0036] The first step is to use a hexahedron Figure 2 The semiconductor physical model structure shown is divided, and different material numbers are set for AlGaN and GaN respectively, to obtain all mesh information of the semiconductor physical model;

[0037] The second step is to normalize the current continuity equation, current density equation, Poisson equation and corresponding boundary conditions and continuity conditions based on the grid information, specifically:

[0038] The normalized form of the current continuity equation is

[0039]

[0040] The current density equation includes the electron current density equation and the hole current density equation, and their normalized forms are:

[0041]

[0042] The normalized form of Poisson's equation is

[0043]

[0044] In formulas (1)-(5), φ n ,φ p , They represent the electron quasi-Fermi potential, hole quasi-Fermi potential and electric potential respectively, M represents the ratio of the intrinsic carrier concentration of the normalized material to the intrinsic carrier concentration of the normalized material, and J n , J p are the electron current density and hole current density, μ n and μ p are the electron mobility and hole mobility of the material, Γ is the net doping concentration, ε and ε f is the dielectric constant of the raw material and the dielectric constant of the normalized material; the continuity condition of thermal electron emission ensures the continuity of the normal current density of the heterojunction surface, and the normalized form is

[0045]

[0046] Introducing the surface charge density ρ into the Poisson equation s To characterize the effect of the two-dimensional electron gas, the potential continuity condition is introduced, and after normalization, it is:

[0047]

[0048] The normal current density and electric potential are specified by the Schottky contact boundary at the gate of the GaN HEMT, and after normalization, they are as follows:

[0049]

[0050] The electric potential is specified by the Ohmic contact boundary at the source and drain of the GaN HEMT, and after normalization, they are as follows:

[0051]

[0052] At other boundaries of the GaN HEMT, the floating boundary condition is satisfied, and after normalization, they are as follows:

[0053]

[0054] In equations (6)-(14), n ⊥ is the normal vector, and the subscripts 1 and 2 represent different cell labels on both sides of the heterojunction interface. v n , v p are the electron recombination rate and hole recombination rate respectively. E c , E v , E g , E f are the conduction band energy level, valence band energy level, energy band gap, and intrinsic energy level respectively. χ is the electron affinity. N has the same meaning as M. n0 and p0 are the electron concentration and hole concentration in the quasi-equilibrium state. Φ m is the work function of the metal. ΔE f is the difference between the intrinsic energy levels of the metal and the semiconductor. V0 represents the applied voltage. N c , N v are the effective density of states of the conduction band and valence band respectively.

[0055] In the third step, the current continuity equation is discretized. The normalized and discretized current continuity equation, current density equation, Poisson equation, and the corresponding boundary conditions and continuity conditions form a drift-diffusion model. The current continuity equation and Poisson equation are tested based on the spectral element method, and the form is

[0056]

[0057] On the Schottky surface of the gate, A is equal to 1, and is equal to 0 in other regions. On the heterojunction surface, B is equal to 1, and is equal to 0 in other regions. C is the dielectric constant ratio on the heterojunction surface and 0 in other regions. D is the dielectric constant ratio. WXF is the weight to ensure the continuity of the electric potential, and its value is the ratio of the dielectric constant to the diffusion wavelength. N i is the test basis function. For the unknowns in (15)-(17), the electron quasi-Fermi potential φ n , hole quasi-Fermi potential φ p , and electric potential Expand and solve using the Newton iteration method. The coupling form is as follows:

[0058]

[0059] Step 4: Introduce damped Newton iteration.

[0060] The form of damped Newton iteration is:

[0061]

[0062] In the formula, λ is the damping factor. Based on this, change formula (18) to the following formula.

[0063]

[0064] The selection principle of λ in the formula is to make it satisfy the following formula

[0065]

[0066] where μ ∈ (0, 1) is a certain fixed value, ||·|| represents any norm of a vector or matrix, p = -[f′(x l )] -1 ·f(x l ). It should be noted that when λ = 1, it means that the algorithm degenerates into Newton iteration. In this paper, λ = 0.6 is selected.

[0067] Step 5: Introduce Anderson acceleration.

[0068] In the process of numerically simulating the semiconductor physical model, a key challenge is the huge difference in the order of magnitude between the electron quasi-Fermi potential φ n , the hole quasi-Fermi potential φ p and the electric potential , which may lead to the inability to converge to the predetermined relative error range during the calculation process. To address this challenge, this paper adopts an adaptive strategy, that is, independently calculate the coefficients γ of φ n , φ p and respectively, and introduce a judgment mechanism on this basis. Specifically, if the convergence is not achieved after continuously exceeding the preset specific number of steps during the iteration process, terminate the Anderson acceleration process and restore to the traditional damped Newton iteration method. This strategy aims to ensure numerical stability while making full use of the advantages of Anderson acceleration to maintain computational efficiency and avoid instability problems caused by the order of magnitude difference. Through this comprehensive method, it is possible to effectively control the oscillation and non-convergence phenomena during the calculation process while ensuring the accuracy of the numerical simulation. The program flowchart is as Figure 1 shown

[0069] The essence of the nonlinear problem is:

[0070]

[0071] Equation (21) can be equivalently expressed as a fixed - point problem:

[0072]

[0073] Based on the fixed - point iteration (22), the residual is defined as:

[0074] f(x) = g(x) - x (23)

[0075] Anderson acceleration is based on the data of the previous m steps and finds the optimal iteration direction by analyzing historical rules. This method shows a higher level of intelligence compared to traditional algorithms. According to the historical information of the iteration, the matrix is continuously updated

[0076] where Δf l = f l+1 - f l 、Δg l = g l+1 - g l .

[0077] Based on the unconstrained problem, a new form of the solution can be defined as:

[0078]

[0079] For the solution of γ, this paper uses the QR decomposition method with better stability to solve it.

[0080]

[0081] where

[0082] Through the above - mentioned iterative method, the electrical characteristics of semiconductor devices are solved, and the simulation of the physical model of semiconductor devices is realized.

[0083] Table 1 Efficiency comparison

[0084]

[0085] As shown in Table 1, when m = 0, only the standard damped Newton iteration method is used. In contrast, the unconstrained Anderson acceleration algorithm shows significant advantages in improving the calculation efficiency of the electrical characteristics of semiconductor devices. Specifically, when m = 2 and the damping factor is 0.6, the Anderson acceleration algorithm reduces the total number of iteration steps by 75% and shortens the calculation time by 40% compared to the damped Newton iteration. As Figure 6As shown, for the iterative calculation of each voltage point, the number of iterative steps is reduced from 15 steps to 6 steps, and the acceleration effect is remarkable.

[0086] This embodiment also provides a high-efficiency semiconductor electrical property simulation system based on physical model convergence acceleration, including:

[0087] A semiconductor physical model meshing unit that meshes the semiconductor physical model structure using hexahedrons and sets different material numbers for AlGaN and GaN respectively to obtain all grid information of the semiconductor physical model;

[0088] A drift-diffusion model construction unit that normalizes the current continuity equation, current density equation, Poisson equation, and corresponding boundary conditions and continuity conditions based on the grid information, and discretizes the current continuity equation. The normalized and discretized current continuity equation, current density equation, Poisson equation, and corresponding boundary conditions and continuity conditions form a drift-diffusion model;

[0089] A semiconductor electrical property solving unit that uses the damped Newton iteration method, introduces a damping factor, solves the equations in the drift-diffusion model, and uses the Anderson acceleration method in the iterative solving process. It constructs a linear combination using the results of the previous m iterations to predict the next iteration result, and obtains the semiconductor electrical properties through iteration.

[0090] This embodiment also provides a computer storage medium that stores an executable program, and the executable program is executed by a processor to implement the steps of the high-efficiency semiconductor electrical property simulation method described above.

[0091] The present invention proposes a high-efficiency physical model convergence acceleration method based on semiconductor device simulation. Taking the spectral element method as the simulation platform, it processes the complex physical fields with large electric field gradients and drastic changes in carrier concentration in semiconductor devices through its high-precision discretization ability. Combining the damped Newton iteration algorithm with the introduction of a damping factor effectively suppresses the oscillation divergence of the traditional Newton method in the high-field strength nonlinear region and improves the stability of transient electrical property solving. To further optimize the efficiency, an unconstrained Anderson acceleration algorithm is adopted, and by dynamically adjusting the memory depth of historical residual information, the number of iterative steps is reduced while avoiding the increase in memory overhead. This method combines algorithmic innovation to balance the convergence speed while ensuring accuracy, and is especially suitable for high-frequency and high-power electrothermal coupling simulations of semiconductor physical models, providing an efficient solution for multi-physical field coupling analysis and device design automation.

[0092] The above shows and describes the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the above embodiments do not limit the present invention in any form. Any technical solutions obtained by using equivalent replacements or equivalent transformations fall within the protection scope of the present invention.

Claims

1. An efficient simulation method for semiconductor electrical characteristics based on physical model convergence acceleration, characterized in that Including: Step 1: Use a hexahedron to divide the semiconductor physical model structure, and set different material numbers for AlGaN and GaN respectively to obtain all the grid information of the semiconductor physical model; Step 2: Based on the grid information, normalize the current continuity equation, current density equation, Poisson equation, and the corresponding boundary conditions and continuity conditions, and discretize the current continuity equation. The normalized and discretized current continuity equation, current density equation, Poisson equation, and the corresponding boundary conditions and continuity conditions constitute a drift-diffusion model; Step 3: Use the damped Newton iteration method, introduce a damping factor, solve the equations in the drift-diffusion model. The Anderson acceleration method is used in the iterative solution process. A linear combination is constructed using the results of the previous m iterations to predict the next iteration result, and the semiconductor electrical characteristics are obtained through iteration.

2. The high - efficiency simulation method for semiconductor electrical characteristics based on physical model convergence acceleration according to claim 1, wherein Including: Use the damped Newton iteration method, introduce a damping factor, and the iterative coupling equation of the drift-diffusion model is: where λ is the damping factor, φ n , φ p , represent the electron quasi-Fermi potential, hole quasi-Fermi potential, and electric potential respectively, represent the electron quasi-Fermi potential, hole quasi-Fermi potential, and electric potential at the current l-th moment respectively, F n , F p and represent the electron continuity equation, hole continuity equation, and Poisson equation respectively.

3. The high-efficiency simulation method for semiconductor electrical characteristics based on physical model convergence acceleration according to claim 2, wherein The damping factor satisfies: Among them, ||·|| represents any norm of a vector or a matrix, the variable p = -[f′(x l )] -1 ·f(x l ), f(x l ) is the equation to be solved, and the coefficient μ is a set value.

4. An efficient semiconductor electrical property simulation method based on physical model convergence acceleration according to claim 3, characterized in that Where λ = 0.6 and μ ∈ (0, 1).

5. An efficient simulation method for semiconductor electrical characteristics based on physical model convergence acceleration according to claim 1, characterized in that, The Anderson acceleration form is: In the formula, is the increment matrix, and γ (l) is the acceleration parameter, and g(x l ) is the function value.

6. An efficient simulation method for semiconductor electrical characteristics based on physical model convergence acceleration according to claim 5, characterized in that, During the Anderson acceleration process, the acceleration parameters of the electron quasi-Fermi potential, hole quasi-Fermi potential, and electric potential are calculated independently. If convergence is not achieved after continuously exceeding a preset specific number of steps during the iteration process, the Anderson acceleration is terminated and the damped Newton iteration method is restored.

7. An efficient simulation method for semiconductor electrical characteristics based on physical model convergence acceleration according to claim 6, characterized in that, The calculation method of the acceleration parameter is: Among them, f l represents the residual vector, represents the increment matrix composed of residual vectors, γ (l) represents the optimal parameter vector of the l-th iteration, γ represents the parameter vector to be optimized, respectively representing different dimensions of the Q matrix and the R matrix, α represents the number of unknowns, and m represents the set memory depth; 8. An efficient semiconductor electrical property simulation method based on physical model convergence acceleration according to claim 1, characterized in that, Where m = 2.

9. A high-efficiency semiconductor electrical characteristic simulation system for implementing the method according to any one of claims 1-8, characterized in that, Including: A semiconductor physical model division unit that uses a hexahedron to divide the semiconductor physical model structure, and sets different material numbers for AlGaN and GaN respectively to obtain all the grid information of the semiconductor physical model; A drift-diffusion model construction unit that normalizes the current continuity equation, current density equation, Poisson equation, and the corresponding boundary conditions and continuity conditions based on the grid information, and discretizes the current continuity equation. The normalized and discretized current continuity equation, current density equation, Poisson equation, and the corresponding boundary conditions and continuity conditions constitute a drift-diffusion model; A semiconductor electrical characteristic solution unit that uses the damped Newton iteration method, introduces a damping factor, solves the equations in the drift-diffusion model. The Anderson acceleration method is used in the iterative solution process. A linear combination is constructed using the results of the previous m iterations to predict the next iteration result, and the semiconductor electrical characteristics are obtained through iteration.

10. A computer storage medium, characterized in that, The computer storage medium stores an executable program, and the executable program is executed by a processor to implement the steps of the semiconductor electrical characteristic high-efficiency simulation method according to any one of claims 1-8.

Images