Semiconductor physical model analysis method and system with strong initial value robustness

Abstract

The invention discloses a semiconductor physical model analysis method and system with strong initial value robustness. The method comprises the following steps of: subdividing a semiconductor physical model structure by adopting a hexahedron, calculating boundary conditions by utilizing semiconductor doping concentration based on subdivision grids, dividing regions through the boundary conditions, and calculating different initial values; substituting the initial value distribution into the normalized current continuity equation, current density equation and Poisson equation to generate a group of remainders; a current continuity equation, a current density equation and a Poisson equation are stably solved by introducing a variable parameter which is added before a remainder and gradually changes from 1 to 0, and a steady-state solution of a semiconductor physical model under a specific voltage bias condition is obtained. The method aims at effectively solving the problems that a traditional initial value solving method is difficult to converge and easy to diverge in a strong nonlinear semiconductor physical model.

Description

Technical Field

[0001] The present invention relates to the technical field of semiconductor physical model simulation, and in particular to a semiconductor physical model analysis method and system with strong initial value robustness. Background Art

[0002] In the numerical solution process of semiconductor physics models, the divergence problem is a long-standing challenge. Due to the complex physical characteristics of semiconductor physics models, their physical models usually contain highly nonlinear equations, which are extremely sensitive to the initial values ​​when solving. For example, reference 1.Z. JM González-Medina, et al. Quasi-fermi-based charge transport scheme for device simulation incryogenic, Wide Bandgap, and High-Voltage Applications [J]. IEEE Transactions on Electron Devices, 2023, 70(2): 708-713 pointed out that when dealing with complex nonlinear semiconductor models, traditional numerical methods are very sensitive to initial conditions and parameter changes, which can easily diverge during the iteration process, resulting in insufficient robustness of the numerical solution and unreliable analysis results. To overcome this problem, the present invention proposes a semiconductor physical model analysis method with strong initial value robustness. Summary of the invention

[0003] The purpose of the present invention is to provide a semiconductor physical model analysis method and system with strong initial value robustness, thereby ensuring the stability and reliability of the semiconductor physical model solution process.

[0004] The technical solution to achieve the purpose of the present invention is:

[0005] A semiconductor physical model analysis method with strong initial value robustness comprises the following steps:

[0006] Step 1, using hexahedron to divide the semiconductor physical model structure, based on the meshing, using semiconductor doping concentration to calculate boundary conditions, dividing the region by boundary conditions and calculating different initial values;

[0007] Step 2, substituting the initial value distribution into the normalized current continuity equation, current density equation, and Poisson equation to generate a set of remainders;

[0008] Step 3: By introducing a variable parameter that gradually changes from 1 to 0 before the remainder, the parametric Newton iteration method is used to stably solve the current continuity equation, current density equation, and Poisson equation, and the steady-state solution of the semiconductor physical model under specific voltage bias conditions is obtained.

[0009] Further, for the semiconductor AlGaN layer, Gaussian doping is performed at the source and drain, with a doping concentration of 5×10 18 cm -3 , a junction depth of 0.015 μm, and a background doping concentration of Gaussian doping of 10 16 cm -3 . The GaN layer is uniformly doped with N type, with a doping concentration of 10 16 cm -3 , and the surface charge density of the heterojunction is 0.016·1e -4 [C / cm 2 .

[0010] Further, the boundary conditions are as follows: The source, drain, and base of the semiconductor physical model are set to Ohmic contact conditions, and the gate is set to Schottky contact conditions.

[0011] Further, the Ohmic contact condition is:

[0012]

[0013] In the formula, n is the electron concentration, p is the hole concentration, is the electric potential, is the net doping concentration, n i is the intrinsic carrier concentration, V0 is the source-drain voltage, k is the Boltzmann constant, T is the temperature, q is the elementary charge, N c , N v are the effective density of states in the conduction band and valence band respectively, E f is the Fermi level, E g is the bandgap width, and χ is the electron affinity.

[0014] Further, the Schottky contact condition is:

[0015]

[0016] In the formula, 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 is the voltage value, and q is the elementary charge.

[0017] Furthermore, dividing the region by boundary conditions and calculating different initial values ​​specifically include: in the initial value setting of the semiconductor physical model, the AlGaN region where the source, gate and drain are located is first treated uniformly as a whole, and the Ohmic contact boundary conditions are adopted from the source to the gate to the heterojunction region, and from the drain to the gate to the heterojunction region; the Schottky contact boundary conditions are adopted for the gate region, and for the remaining parts, the Ohmic contact boundary conditions are set according to the condition that the substrate voltage is 0V; the initial value is calculated through the corresponding boundary conditions.

[0018] Furthermore, by introducing a variable parameter that gradually changes from 1 to 0 before the remainder, the parameter Newton iteration method is used to stably solve the current continuity equation, current density equation, and Poisson equation, which are:

[0019]

[0020] In the formula, φ n ,φ p , denote the electron quasi-Fermi potential, hole quasi-Fermi potential and electric potential, respectively. They represent the electron quasi-Fermi potential, hole quasi-Fermi potential and electric potential at the current time k, respectively. n 、F p and They represent the electron continuity equation, the hole continuity equation and the Poisson equation, respectively. represents the remainder produced in step 2, and Δξ is a variable parameter.

[0021] Furthermore, the Δξ is 0.05, which is set to 0.05, meaning that 20 systems are required to restore the parameter ξ from 1 to 0.

[0022] A semiconductor physical model analysis system with strong initial value robustness, comprising:

[0023] The initial value calculation unit uses hexahedrons to divide the semiconductor physical model structure, calculates the boundary conditions based on the meshing, uses the semiconductor doping concentration, divides the area by the boundary conditions and calculates different initial values;

[0024] The remainder generating unit substitutes the normalized current continuity equation, current density equation and Poisson equation according to the initial value distribution to generate a set of remainders;

[0025] The solving unit introduces a variable parameter that gradually changes from 1 to 0 before the remainder, stably solves the current continuity equation, the current density equation, and the Poisson equation, and obtains the steady-state solution of the semiconductor physical model under specific voltage bias conditions.

[0026] A computer storage medium stores an executable program, which, when executed by a processor, implements the steps of the semiconductor physical model analysis method with strong initial value robustness as described above.

[0027] Compared with the prior art, the present invention has significant advantages: through precisely designed boundary conditions and reasonable doping distributions, the present invention can obtain a set of definite solutions for the drift-diffusion equations, which fully considers the complexity of the semiconductor physical model and ensures the stability and reliability of the solution process; in the solution process, the present invention introduces a gradually decreasing parameter, the introduction of which not only effectively expands the convergence domain of the numerical solution but also enables the initial value to be gradually guided to the steady-state solution under specific voltage bias conditions; in this way, the present invention significantly improves the robustness of the solution process, effectively avoiding the divergence phenomenon that may occur in the traditional Newton iteration method, and thus showing higher stability and efficiency in the initial value solution of the strongly nonlinear semiconductor physical model. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] Figure 1 is a schematic diagram of the semiconductor physical model structure.

[0029] Figure 2 is a schematic diagram of the doping distribution.

[0030] Figure 3 is a schematic diagram of the initial value distribution.

[0031] Figure 4 is a distribution diagram of the initial potential value of the semiconductor physical model.

[0032] Figure 5 is a potential distribution diagram when the source voltage of the semiconductor physical model is 0.5V, Figure 5 where (a) in is the solution result of the spectral element method of the present patent method, Figure 5 and (b) in is the solution result of the commercial software COMSOL.

[0033] Figure 6 is a diagram of the potential change near the gate when the source voltage of the semiconductor physical model is 0.5V. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0034] The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of them. 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 a semiconductor physical model analysis method with strong initial value robustness. Below, with reference to the accompanying drawings, takingFigure 1 Taking the semiconductor physical model shown as an example, the supplementary description of the model geometric dimensions is as follows: The distance between the gate and the drain is 0.1 μm, and the distance between the gate and the source is 0.1 μm. The specific operation steps are as follows:

[0036] First step, use hexahedrons to Figure 1 mesh the semiconductor physical model structure shown to obtain all the grid information of the semiconductor physical model;

[0037] Second step, dope the semiconductor physical model. The doping distribution is Gaussian doping at the source and drain of the AlGaN layer, with a doping concentration of 5×10 18 cm -3 , and the junction depth is 0.015 μm. The background doping concentration of Gaussian doping is 10 16 cm -3 ; The doping method of the GaN layer is N-type uniform doping, with a doping concentration of 10 16 cm -3 , and the surface charge density of the heterojunction is 0.016·1e -4 [C / cm 2 . The final distribution is as Figure 2 shown;

[0038] Third step, set the boundary conditions. Set the source, drain, and base of the semiconductor physical model as Ohmic contact conditions, and set the gate as a Schottky contact condition. The Ohmic contact boundary condition formula is as follows:

[0039]

[0040] In the formula, n is the electron concentration, p is the hole concentration, is the electric potential, is the net doping concentration, n i is the intrinsic carrier concentration, V0 is the source-drain voltage, k is the Boltzmann constant, T is the temperature, q is the elementary charge, N c , N v are the effective density of states in the conduction band and valence band respectively, E f is the Fermi level, E g is the band gap width, and X is the electron affinity.

[0041] The Schottky contact condition formula is as follows:

[0042]

[0043] In the formula, 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, and V0 is the voltage value.

[0044] Step 4: Assign the obtained boundary values according to the following regions. For example, Figure 3 As shown, the AlGaN regions where the source, gate, and drain are located are treated as a whole. In terms of boundary condition setting, from the source to the gate extending to the heterojunction region, and from the drain to the gate extending to the heterojunction region, Ohmic contact boundary conditions are adopted; while for the gate region, Schottky contact boundary conditions are used. For the remaining parts, according to the condition that the substrate voltage is 0V, it is set as Ohmic contact boundary conditions to calculate the initial values.

[0045] Step 5: Normalize the current continuity equation, current density equation, Poisson equation, and the corresponding boundary conditions and continuity conditions based on the grid information. Specifically:

[0046] The normalized form of the current continuity equation is

[0047]

[0048] The current density equation includes the electron current density equation and the hole current density equation, and their respectively normalized forms are

[0049]

[0050] The normalized form of the Poisson equation is

[0051]

[0052] In equations (5)-(9), φ n , φ p , respectively represent the electron quasi-Fermi potential, hole quasi-Fermi potential, and electric potential. M represents the ratio of the intrinsic carrier concentration of the material to be normalized to the intrinsic carrier concentration of the normalized material. J n , J p are the electron current density and hole current density respectively. μ n and μ p are the electron mobility and hole mobility of the material respectively. Γ is the net doping concentration. ε and ε f are the dielectric constant of the raw material and the dielectric constant of the normalized material;

[0053] Step 6: The drift-diffusion model consists of the current continuity equation, current density equation, and Poisson equation. Based on the spectral element method, the current continuity equation and Poisson equation are tested, and the form is

[0054]

[0055]

[0056] On the gate Schottky surface, A equals 1, and equals 0 in other regions. On the heterojunction surface, B equals 1, and equals 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 potential continuity, with a value of the ratio of the dielectric constant to the diffusion wavelength, N i is the test basis function.

[0057] In the seventh step, substitute the initial value distribution into the nonlinear system, and let each equation generate a remainder. For the newly generated nonlinear system, the arbitrarily selected initial iteration value is a set of its solutions. Since the current system with the added remainder is not the nonlinear system to be solved at the beginning, in order to restore the system, a coefficient ξ is added in front of the remainder on this basis. When ξ is 0, the solutions of the newly generated system and the original system are the same. That is, when the added coefficient changes from 1 to 0, the new system is also continuously approaching the original system to be solved, and this changing coefficient is Δξ. For the unknowns electron quasi-Fermi potential φ n and hole quasi-Fermi potential φ p and electric potential in (10)-(12), expand and solve using the Newton iteration method. The coupling form is as follows:

[0058]

[0059] where is the initial value substituted into the nonlinear system, each equation generates a remainder, Δξ is a variable parameter, and in this patent, Δξ = 0.05, which means it takes 20 systems for the parameter ξ to recover from 1 to 0.

[0060] Through the above method, the steady-state electrical characteristics of the semiconductor physics model are solved, and the initial value convergence of the semiconductor physics model is achieved.

[0061] Table 1 Iteration steps for different parameters ξ

[0062]

[0063]

[0064] As shown in Table 1, for the system with ξ = 0.95, compared with the original system, it only changes by 0.05L j , and the system converges to a value very close to the initial value, which has far reduced the large fluctuations generated in the first step, and only small changes are produced. Compared with directly using the traditional Newton iteration method for the initial system, the Jacobi matrix will directly report matrix singularity or near singularity in the first step of the iteration, which has obvious advantages.

[0065] According to the method described in the present invention, forFigure 1 A detailed simulation study was carried out on the semiconductor physical model shown. During the simulation, according to Figure 2 the doping distribution and fixed boundary conditions shown, the initial values were set according to Figure 3 the initial value distribution shown. The specific simulation conditions were as follows: the drain voltage was set to 0.5 V, while the gate, source, and the voltage at the bottom of the semiconductor physical model were all maintained at 0 V. Through the method of the present invention, the final obtained initial value distribution was as shown in Figure 4 . Substituting the initial value distribution into the drift-diffusion model formed a set of remainders to obtain a unique solution for the equation. To solve the steady-state electrical distribution of the physical model, a set of variable parameters Δξ was introduced before the remainders to skip the divergent system. Finally, when ξ = 0, the required steady-state electrical distribution of the physical model could be solved.

[0066] The simulation results showed that when the drain voltage was 0.5 V, the electric potential near the source was significantly higher than that in the drain region (see Figure 5 and Figure 6 ). This phenomenon indicated that the increase in the drain voltage had a significant impact on the electric potential distribution, not only changing the spatial distribution of the electric potential but also causing the relative increase in the electric potential in the source region. This result was highly consistent with the physical characteristics of the actual semiconductor physical model, verifying the accuracy and effectiveness of the method of the present invention in simulating the behavior of complex semiconductor physical models.

[0067] During the numerical solution process of the semiconductor physical model, the traditional Newton iteration method often suffered from divergence and non-convergence problems due to improper selection of the initial value or the high non-linearity of the model, which severely limited its application in the simulation of complex semiconductor physical models. To overcome this problem, the present invention proposes an analysis method for semiconductor physical models with strong initial value robustness. By combining different continuity conditions and boundary conditions, the discrete forms of the drift-diffusion equation and the Poisson equation are derived, and the initial values are assigned to the model according to the boundary conditions and doping distribution under fixed bias voltages, while introducing the parametric Newton iteration method. This method effectively solves the problem of initial value divergence in the traditional method and significantly improves the stability and reliability of the calculation. Compared with the prior art, the present invention has significant advantages: First, by dynamically adjusting the iteration parameters, the convergence domain of the numerical solution is significantly expanded, effectively avoiding the divergence problem caused by improper selection of the initial value, and being able to stably solve even under complex non-linear conditions, showing high robustness; in addition, this method is applicable to various semiconductor physical model structures, including AlGaN / GaN heterojunction physical models, wide-bandgap semiconductor physical models, etc., and has broad engineering application prospects; finally, the present invention not only has innovation in theory but also verifies its effectiveness and reliability under complex working conditions through actual simulations, providing a powerful tool for the design and optimization of semiconductor physical models.

[0068] The present invention also provides a semiconductor physical model analysis system with strong initial value robustness, including:

[0069] An initial value calculation unit, which divides the structure of the semiconductor physical model using hexahedrons, calculates boundary conditions based on the divided grids using the semiconductor doping concentration, divides regions through the boundary conditions, and calculates different initial values;

[0070] A remainder generation unit, which substitutes the initial value distribution into the normalized current continuity equation, current density equation, and Poisson equation to generate a set of remainders;

[0071] A solution unit, which stably solves the current continuity equation, current density equation, and Poisson equation by introducing a variable parameter that gradually changes from 1 to 0 before the remainder, and obtains the steady-state solution of the semiconductor physical model under specific voltage bias conditions.

[0072] The present invention also provides a computer storage medium, which stores an executable program. When the program is executed by a processor, the steps of the semiconductor physical model analysis method with strong initial value robustness are implemented.

[0073] The method of the present invention uses the spectral element method as the simulation calculation basic platform, combines fixed boundary conditions and given doping distributions, and sets reasonable initial values for the model; subsequently, the parametric Newton iteration method is adopted, and a variable parameter is set to solve the steady-state solution under specific voltage bias conditions; through the variable parameter, the problem of initial value convergence is effectively overcome, divergence is avoided, and the calculation stability and reliability are significantly improved. The method of the present invention is concise and clear, easy to be programmed and implemented, and has extremely high practical engineering application value.

[0074] 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 substitution or equivalent transformation fall within the protection scope of the present invention.

Claims

1. A method for analyzing a semiconductor physical model with strong initial value robustness, characterized in that include: Step 1, using hexahedron to divide the semiconductor physical model structure, based on the meshing, using semiconductor doping concentration to calculate boundary conditions, dividing the region by boundary conditions and calculating different initial values; Step 2, substituting the initial value distribution into the normalized current continuity equation, current density equation, and Poisson equation to generate a set of remainders; Step 3, by introducing a variable parameter that gradually changes from 1 to 0 before the remainder, the parameter Newton iteration method is used to stably solve the current continuity equation, the current density equation, and the Poisson equation, and the steady-state solution of the semiconductor physical model under specific voltage bias conditions is obtained.

2. The semiconductor physical model analysis method with strong initial value robustness according to claim 1, characterized in that For the semiconductor AlGaN layer, Gaussian doping is performed at the source and drain, with a doping concentration of 5×10 18 cm -3 , the junction depth is 0.015 μm, and the background doping concentration of Gaussian doping is 10 16 cm -3 . The GaN layer is uniformly doped with N-type, with a doping concentration of 10 16 cm -3 , and the surface charge density of the heterojunction is 0.016·1e -4 [C / cm 2 .

3. A semiconductor physical model analysis method with strong initial value robustness according to claim 1, characterized in that, The boundary conditions are: the source, drain and base of the semiconductor physical model are set to ohmic contact conditions, and the gate is set to Schottky contact conditions.

4. The semiconductor physical model analysis method with strong initial value robustness according to claim 3, characterized in that, The ohmic contact conditions are: Where n is the electron concentration, p is the hole concentration, is the electric potential, is the net doping concentration, n i is the intrinsic carrier concentration, V0 is the source-drain voltage, k is the Boltzmann constant, T is the temperature, q is the elementary charge, N c , N v are the effective density of states in the conduction band and valence band respectively, E f is the Fermi level, E g is the band gap width, and χ is the electron affinity.

5. A method for analyzing a semiconductor physical model with strong initial value robustness according to claim 1, characterized in that, Schottky contact conditions are: where 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 is the voltage value, and q is the elementary charge.

6. The semiconductor physical model analysis method with strong initial value robustness according to claim 1, wherein The method of dividing the region by boundary conditions and calculating different initial values ​​specifically includes: in the initial value setting of the semiconductor physical model, the AlGaN region where the source, gate and drain are located is first treated as a whole, and the Ohmic contact boundary conditions are used from the source to the gate to the heterojunction region, and from the drain to the gate to the heterojunction region; the Schottky contact boundary conditions are used in the gate region, and for the remaining parts, the Ohmic contact boundary conditions are set according to the condition that the substrate voltage is 0V; the initial value is calculated through the corresponding boundary conditions.

7. A method for analyzing a semiconductor physical model with strong initial value robustness according to claim 1, characterized in that By introducing a variable parameter that gradually changes from 1 to 0 before the remainder, the parameter Newton iteration method is used to stably solve the current continuity equation, current density equation, and Poisson equation: where φ n , φ p , represent the electron quasi-Fermi potential, the hole quasi-Fermi potential, and the electric potential respectively, represent the electron quasi-Fermi potential, the hole quasi-Fermi potential, and the electric potential at the current k-th moment respectively, F n , F p and represent the electron continuity equation, the hole continuity equation, and the Poisson equation respectively, j = 1, 2, 3 represents the remainder generated in step 2, and Δξ is a variable parameter.

8. A method for analyzing a semiconductor physical model with strong initial value robustness according to claim 7, characterized in that The Δξ is 0.

05.

9. A semiconductor physical model analysis system with strong initial value robustness for implementing the method according to any one of claims 1-8, characterized in that, include: The initial value calculation unit uses hexahedrons to divide the semiconductor physical model structure, calculates the boundary conditions based on the meshing, uses the semiconductor doping concentration, divides the area by the boundary conditions and calculates different initial values; The remainder generating unit substitutes the normalized current continuity equation, current density equation and Poisson equation according to the initial value distribution to generate a set of remainders; The solving unit introduces a variable parameter that gradually changes from 1 to 0 before the remainder, stably solves the current continuity equation, the current density equation, and the Poisson equation, and obtains the steady-state solution of the semiconductor physical model under specific voltage bias conditions.

10. A computer storage medium, characterized in that, The computer storage medium stores an executable program, which, when executed by a processor, implements the steps of the semiconductor physical model analysis method with strong initial value robustness as described in any one of claims 1 to 8.

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