Method for solving drift-diffusion model of high-power semiconductor device based on stabilized high-order mixed discontinuous Galerkin

Abstract

The invention discloses a method for solving a drift-diffusion model of a high-power semiconductor device based on stabilized high-order mixed discontinuous Galerkin. In the calculation process of a steady-state semiconductor drift-diffusion equation, severe convection conditions caused by a high field effect enable an equation numerical solution to generate severe oscillation, and calculation errors are caused. According to the method, a stable high-order numerical framework is constructed, and calculation errors caused by amplification of oscillation errors in an iteration process are avoided, so that the advantages of high precision, high robustness and rapid convergence of the numerical framework are ensured. Compared with a traditional method, the method has the advantages that the high-reliability posterior error indicator and the self-adaptive artificial diffusion term are constructed based on the redundancy freedom degree of the mixed discontinuous Galerkin method, and extra calculation cost does not need to be introduced. According to the method, the high-order mixed discontinuous Galerkin method is popularized to a current carrier drift-diffusion equation under severe convection leading for the first time, and a high-precision and high-robustness numerical framework can be provided for solving a high-power semiconductor device.

Description

Technical Field

[0001] The present invention belongs to the field of numerical calculation of multi-physical fields in electronic devices, and particularly relates to a method for solving the drift-diffusion model of high-power semiconductor devices based on stabilized high-order hybrid discontinuous Galerkin. Background Art

[0002] In the design and manufacturing stages of semiconductor devices, it is necessary to perform high-precision simulations on the quasi-static electric field-carrier coupling system in semiconductor devices to obtain the AC / DC effects of the devices and predict the degradation performance of the devices, etc. The difficulty lies in that the drift-diffusion equation describing the carrier distribution is subject to the convection-dominated effect and exhibits a highly nonlinear singular distribution. The finite volume method (FVM) and the Scharfetter-Gummel (SG) method satisfy vector conservation and upwind scheme respectively, enabling FVM-SG to handle the carrier drift-diffusion model with a drastic change in order of magnitude.

[0003] However, with the refinement of semiconductor device structures and the complexity of material types, it is extremely challenging to solve the carrier transport model using the FVM-SG format. On the one hand, complex quantum effects and high-field effects are introduced in deep submicron structures and wide bandgap materials, and the FVM-SG format with only first-order convergence is difficult to converge; on the other hand, the constant current assumption in the SG format fails, and its calculation accuracy deteriorates in high-power semiconductor devices. Specifically, the FVM-SG format only satisfies the upwind scheme in the tangential direction of the cell boundary, and there is a "leakage" effect in the normal drift-diffusion equation.

[0004] In recent years, the hybrid discontinuous Galerkin method (HDG) has received attention in different fields such as computational mathematics, fluid mechanics, and computational electromagnetics. As an improved finite element method, HDG satisfies high-order convergence and vector conservation, and overcomes the problem of high computational cost of the discontinuous Galerkin method (DG) through static projection, and can be a potentially powerful candidate to replace the FVM-SG format. However, the HDG method does not satisfy the upwind scheme, resulting in numerical instability and non-physical oscillations in the numerical solution when discretizing the drift-diffusion system occupied by the convective term. Existing HDG semiconductor solvers focus on proving their optimal convergence in mathematics, and these proofs often rely on the assumption of smooth numerical solutions, avoiding the instability problem caused by the occupation of the convective term. Although they improve people's mathematical understanding of the HDG method, there is a relative lack of research on the carrier drift-diffusion model with convection dominance. Therefore, it is crucial to develop an HDG solver with both high precision and high robustness for high-power semiconductor devices.

[0005] Artificial diffusion terms are a general method for solving numerical instability and have been widely used in FEM, FVM, and DG methods. However, in most frameworks, it depends on the mesh size, the order of basis functions, and the characteristics of the physical field distribution, involving a cumbersome parameter tuning process. What's more troublesome is that these parameters depend on the specific problem scenario, and whenever the model, equation, and boundary conditions are changed, they need to be readjusted.

[0006] In the calculation of high-power semiconductor devices dominated by convection, the high-order HDG method exhibits weak stability, causing strong non-physical oscillations. Considering the characteristics of this problem, the present invention proposes a hybrid discontinuous Galerkin stabilization technique for the coupled calculation of non-linear drift-diffusion systems, and develops a corresponding preconditioned solution framework: at each update of the boundary conditions / time stepping, a trial calculation is performed, and the corresponding numerical error distribution is estimated to provide a reference distribution of the artificial diffusion term for subsequent iterative calculations. In the iterative calculation, the artificial diffusion term is set as a power function of the estimated numerical error, thereby controlling the oscillatory error within a preset threshold. The proposed method retains the high-order convergence of the HDG method and provides iterative robustness at a very low cost. Summary of the Invention

[0007] In the calculation process of the carrier non-linear drift-diffusion system in high-power semiconductor devices, the convection-dominated effect caused by the strong field leads to non-physical oscillations in the numerical solution. In view of this problem, the present invention proposes a method for solving the drift-diffusion model of high-power semiconductor devices based on stabilized high-order hybrid discontinuous Galerkin, and adopts a stabilization technique with an adaptive artificial diffusion term to ensure the robustness of the high-order HDG numerical framework.

[0008] The technical solution adopted by the present invention is as follows:

[0009] A method for solving the drift-diffusion model of high-power semiconductor devices based on stabilized high-order hybrid discontinuous Galerkin, including: First, trial-solve the carrier drift-diffusion system under the updated applied voltage and estimate the numerical error; then, construct an initial artificial diffusion term based on the numerical error and discard the trial solution, and restart the stable iteration. In each subsequent iteration, if the numerical error is greater than the preset threshold, update the artificial diffusion term until the non-linear system converges under the current voltage.

[0010] In the above technical solution, further, it specifically includes the following steps:

[0011] Step 1, establish a geometric model of the multi-physical field coupling system of the semiconductor device, perform an initial mesh dissection on it to obtain the mesh topology, that is, the shape function information, the element adjacency relationship, and the boundary conditions;

[0012] Step 2: Establish a physical model of the semiconductor device, including doping material properties, doping concentration distribution, mobility degradation model, and carrier impact ionization model;

[0013] Step 3: Update the grid topology, semiconductor physical model, and boundary conditions in the simulation software. For the carrier drift-diffusion equation in a steady-state semiconductor, use the mixed discontinuous Galerkin method for weak formulation, and then discretize it using a high-order discontinuous polynomial space to assemble the linear system matrix of the drift-diffusion model;

[0014] Step 4: Set the artificial diffusion term to zero and find the trial solution;

[0015] Step 5: Traverse all grid cells and use numerical jumps to estimate the numerical error distribution;

[0016] Step 6: Determine whether the relative error between two iterations is less than the preset error tolerance. If so, go to Step 7; otherwise, go to Step 8;

[0017] Step 7: Determine whether the preset voltage value is reached. If so, end the algorithm; otherwise, go to Step 3;

[0018] Step 8: Determine whether there is a numerical error in any grid cell greater than the preset threshold. If so, go to Step 9; otherwise, go to Step 10;

[0019] Step 9: Keep the boundary conditions unchanged, perform iterative solution, and go to Step 5;

[0020] Step 10: Update the artificial diffusion term according to the error distribution.

[0021] Furthermore, estimate the truncation error of the numerical solution through the redundant degrees of freedom of the mixed discontinuous Galerkin method, locate the strong convection cells in the carrier convection-diffusion system, and obtain the distribution of the artificial diffusion term based on the distribution of the numerical error system and the calculation results. The total numerical error distribution satisfies:

[0022]

[0023] Thus, adaptively balance the diffusion error and oscillation error based on the error estimate value, where, is the exact solution of the system without introducing the artificial diffusion term, is the exact solution of the equation with the artificial diffusion term introduced, is the operator that projects the infinite-dimensional exact solution onto the finite element space, is the numerical solution of the equation with the artificial diffusion term introduced. The first error term on the right side stems from the change of the physical equation by the artificial diffusion term, the second error term stems from the truncation generated by projecting the infinite-dimensional function onto the finite-dimensional function space, and the third error term stems from the accuracy and stability of the numerical framework itself.

[0024] Further, the solution variables of the carrier drift-diffusion model in the semiconductor device are replaced by the carrier quasi-Fermi potential , and based on the equivalent transformation, the control equation describing the carrier transport is reduced to a nonlinear drift-diffusion equation, where is the electrostatic potential, is the electron concentration, is the hole concentration, is the electron quasi-Fermi potential, is the hole quasi-Fermi potential.

[0025] Further, in step 5, the optimization process of the artificial diffusion term is combined with the iterative process of the nonlinear problem to construct a posteriori error indicators to estimate the numerical error. The redundant degrees of freedom overlapping in the grid cells have different numerical values, which are used to characterize the oscillation error of the numerical solution. The global solution error is calculated based on the redundant degrees of freedom of the skeleton solution, and the local element error is calculated based on the redundant degrees of freedom between the skeleton solution and the local solution. Thus, the characterization of the error only requires the generated local fields and the skeleton fields, without solving the adjoint system or constructing other problems. The constructed a posteriori error indicator for estimating the numerical error is:

[0026]

[0027] where is the estimated numerical error, is any grid cell, is the boundary of the cell , is the local physical field defined within the cell, is the global physical field defined on the skeleton, is the cell characteristic size, is defined on the norm in the space.

[0028] Further, in step 6, the convergence criterion is that the ratio of the norm of the difference between the solutions , in two adjacent iterations to does not exceed the preset error tolerance , that is:

[0029]

[0030] where , are the solutions in two adjacent iterations respectively, and the preset error tolerance is .

[0031] Further, in step 8, it is judged whether there is a cell the numerical value of the posterior error indicator in is greater than or equal to a preset error threshold:

[0032]

[0033] the preset error threshold is taken to be less than the order of magnitude of the thermal pressure, i.e., .

[0034] Furthermore, the updating of the artificial diffusion term is specifically as follows:

[0035] The artificial diffusion term is updated as a function of the numerical error:

[0036]

[0037] where is the maximum value of the user-defined artificial diffusion term, is the user-defined artificial diffusion term parameter, which determines the sensitivity of the diffusion term value to the numerical jump; during the iteration of the nonlinear system, the artificial diffusion term needs to be updated with the cumulative value of the error instead of the true value, i.e., in the above formula takes , and there is:

[0038]

[0039] where is the numerical value of the error estimate in element at the -th iteration, is the numerical value of the error estimate in element at the -th iteration.

[0040] The beneficial effects of the present invention are as follows:

[0041] The present invention uses a stabilized high-order hybrid discontinuous Galerkin method to solve the carrier drift-diffusion model of high-power semiconductor devices. During the calculation of the steady-state semiconductor drift-diffusion equation, this method effectively avoids the calculation errors caused by the severe oscillation of the numerical solution of the equation due to the strong convection condition caused by the high-field effect, and ensures the high-precision, strong robustness and fast convergence advantages of the numerical framework. Compared with the traditional method, this method constructs a highly reliable posterior error indicator and an adaptive artificial diffusion term based on the redundant degrees of freedom of the hybrid discontinuous Galerkin method, without the need to introduce additional computational costs. This method first extends the high-order hybrid discontinuous Galerkin method to the carrier drift-diffusion equation dominated by strong convection, and can provide a high-precision and strong-robustness numerical framework for the solution of high-power semiconductor devices. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Figure 1 It is a schematic diagram of the specific process for solving the carrier drift-diffusion model in semiconductor devices by the stabilized mixed discontinuous Galerkin method described in the present invention;

[0043] Figure 2 It is a schematic diagram of the posterior error indicator, where (a) calculates the global solution error based on the redundant degrees of freedom of the skeleton solution, and (b) calculates the local element error based on the redundant degrees of freedom between the skeleton solution and the local solution;

[0044] Figure 3 It is a schematic diagram of the steady-state diode structure and doping, where (a) is the doping concentration distribution, (b) is the unstructured two-dimensional triangular mesh for calculation, and (c) is the structured two-dimensional triangular mesh for calculation;

[0045] Figure 4 It is the test result of the posterior error estimate value described in the present invention in the steady-state diode;

[0046] Figure 5 It is the test result of the variation trend of the numerical error with the artificial diffusion term in the diode;

[0047] Figure 6 Compare the influence on stability with or without introducing the stabilization strategy of the present invention;

[0048] Figure 7 It is a comparison of the convergence rates between the present method and the traditional method;

[0049] Figure 8 It is a schematic diagram of the structure, doping and mesh of a high-power LDMOSFET device, where (a) is the device doping distribution and (b) is the unstructured two-dimensional triangular mesh used for calculation;

[0050] Figure 9 It is a comparison diagram of the electrical characteristics calculated by the present method and the traditional method;

[0051] Figure 10 It is the hole quasi-Fermi potential gradient distribution of the present method and the traditional method under a drain voltage of 22 V. Specific embodiments

[0052] The technical solution of the present invention will be further described in detail below in conjunction with the accompanying drawings and specific embodiments.

[0053] In order to deal with the nonlinearity in the carrier drift-diffusion system in high-power semiconductor devices, it is necessary to iterate the solution until the electric field, electron and hole distribution reach a steady state. This paper proposes a stabilization method for a high-order hybrid discontinuous Galerkin framework, combined with the adaptive artificial diffusion term technology, to form a new solution method for solving the non-physical oscillation problem caused by the occupation of the convection term in the carrier transport equation; it can effectively control the oscillation of numerical calculations and suppress the oscillation error during the iteration process. This method constructs a highly reliable a posteriori error indicator and an adaptive artificial diffusion term based on the redundant degrees of freedom of the hybrid discontinuous Galerkin method, without involving a complex parameter adjustment process. Figure 1 The present invention is a flow chart of a method for solving the drift-diffusion model of a high-power semiconductor device based on a stabilized high-order mixed discontinuous Galerkin. The method comprises: first, after updating the boundary conditions / time step, a trial solution is performed without a stabilization strategy, and its numerical error is estimated to provide a reference value for the initial artificial diffusion term distribution for subsequent iterative solutions. During the iteration process, the artificial diffusion term distribution is updated according to the numerical error until the numerical error is lower than a preset threshold. The specific steps are as follows:

[0054] The first step is to establish the multi-physics field coupling system geometric model of the device, perform initial meshing on it, and obtain the mesh topology, that is, shape function information, unit adjacency relationship, and boundary conditions;

[0055] The second step is to establish a physical model of semiconductor devices, including doping material properties, doping concentration distribution, mobility degradation model, carrier impact ionization model, etc.

[0056] Step 3: Load the grid topology and semiconductor physical model. For the carrier drift-diffusion equation in the steady-state semiconductor, the mixed discontinuous Galerkin method is used for weak formalization, and then discretized using high-order discontinuous polynomial space to obtain the linear system matrix of the drift-diffusion model;

[0057] The governing equation describing semiconductor carrier transport is nonlinear and is usually linearized and solved iteratively using the Newton-Raphson method. The linearized governing equation is:

[0058]

[0059] in, is the Nabla operator , is the material dielectric constant, is the elementary charge, is the doping concentration, is the hole concentration, is the electron concentration, The artificial diffusion term introduced in this method is is the electron quasi-Fermi potential, is the hole quasi-Fermi potential, is the electric field strength, is the electron mobility, is the hole mobility, is the thermal pressure, is the Boltzmann constant, is the lattice temperature, is the carrier recombination rate, and the superscript represents the corresponding physical quantity at the th iteration step.

[0060] In the above method, it is necessary to discretize the drift-diffusion equation describing carrier transport using the HDG method, and the HDG method requires selecting a pair of scalar-vector unknowns and achieving global coupling based on the conservation of fluxes between elements. The key to the problem lies in constructing a suitable scalar-vector field to make it as close as possible to satisfying the high-order polynomial basis function space. The present invention changes the unknown quantity from to , and performs an equivalent transformation on the hole quasi-Fermi potential transport equation, still reducing it to a non-linear drift-diffusion equation. The dimensions of the three scalar unknowns in the system are unified, and the functional form approximately satisfies the polynomial basis function. Without considering the excitation term, the three physical fields all satisfy the discrete extreme value theorem: the numerical value changes monotonically, and the extreme value only appears on the boundary.

[0061] The derivation result shows that the artificial diffusion term has no influence on the gradient information related to the kth step, and thus will not reduce the convergence rate. Using the mixed discontinuous Galerkin method, the weak form is obtained, and then it is discretized using the high-order discontinuous polynomial space. To simplify the symbolic expression, in the aforementioned linearized control equation set, the scalar unknown is labeled as , the trace of the scalar unknown on the skeleton is labeled as , the vector unknown is labeled as , the convective vector field is labeled as , and the corresponding discrete form of the equation is:

[0062]

[0063] Among them, the inner product is the integral of the product of the functions , in the element , is the integral of the product of the functions , on the boundary of the element , is the unit normal vector pointing outwards on the surface of the element , is the scalar polynomial basis function defined in the element ​ is the vector polynomial basis function defined on the element and is the scalar polynomial basis function defined on the boundary of the element . is the flux term defined on the skeleton and is defined as:

[0064]

[0065] where is the local stability parameter introduced by the HDG framework itself and is respectively in the three governing equations:

[0066]

[0067] Combining the above equations, the linear system matrix of the drift-diffusion model is obtained:

[0068]

[0069] where, except that and are vectors, other bold uppercase letters represent matrices, and their element definitions are as follows:

[0070]

[0071] Step 4: Set the artificial diffusion term to zero and find the trial solution. The trial solution provides a reference value for the initial artificial diffusion term and does not participate in the iteration process.

[0072] Step 5: Traverse all grid elements and estimate the numerical error distribution. Generally speaking, the artificial diffusion term alleviates the convection domination problem and also introduces new errors. The total error satisfies the following form:

[0073]

[0074] In the above formula, the first error term on the right side comes from the change of the physical equation by the artificial diffusion term, the second error term comes from the truncation generated by projecting the infinite-dimensional function onto the finite-dimensional function space, and the third error term comes from the accuracy and stability of the numerical framework itself. As the strength of the artificial diffusion term increases, the latter two numerical errors decrease monotonically, the first diffusion error increases monotonically, and the total error shows a trend of first decreasing and then increasing. Therefore, the setting of the artificial diffusion term can be reduced to the following optimization problem:

[0075]

[0076] However, the cost of calculating the optimal artificial diffusion term is quite high. In fact, for semiconductor device calculations, a globally optimal artificial diffusion term distribution is not required, and it is only necessary to ensure that the oscillation error is not amplified during the iteration process to avoid causing calculation errors.

[0077] The optimization process of the artificial diffusion term is combined with the iterative process of the nonlinear problem in the present invention, without the need to introduce additional computational costs. The schematic diagram of constructing the posterior error indicator is as shown in Figure 2 Figure (a) in [reference] shows the calculation of the global solution error based on the redundant degrees of freedom of the skeleton solution Figure 2 in (a) is based on the skeleton solution to calculate the global solution error using the redundant degrees of freedom:

[0078] ,

[0079] where is the node in element , and is the number of nodes in the element. While Figure 2 in (b) is to calculate the local error of the element based on the redundant degrees of freedom between the skeleton solution and the local solution :

[0080] .

[0081] The characterization of the error only requires the local field and the skeleton field naturally generated by the HDG method, without the need to solve the adjoint system or construct other problems, and the computational cost is negligible.

[0082] Step 6: Determine whether the relative error between two iterations is less than the preset convergence criterion:

[0083]

[0084] The convergence criterion here is set to . If yes, go to Step 7; otherwise, go to Step 8;

[0085] Step 7: Determine whether the current voltage value reaches the preset voltage value.

[0086] If yes, end the algorithm; otherwise, go to Step 3;

[0087] Step 8: Determine whether there is an element in which the value of the posterior error indicator is greater than the preset error threshold:

[0088]

[0089] The error threshold here is taken to be less than the order of magnitude of the hot pressing, i.e., .

[0090] If yes, go to Step 9; otherwise, go to Step 10;

[0091] Step 9: Keep the boundary conditions unchanged, perform iterative solution, and go to Step 5;

[0092] Step 10: Update the artificial diffusion term as a function of the numerical error:

[0093]

[0094] During the iterative process of the nonlinear system, the artificial diffusion term needs to be updated with the cumulative value of the error instead of the true value:

[0095]

[0096] If the artificial diffusion term is directly updated with the error estimate value, the iterative process will not converge.

[0097] The method of the present invention constructs a stable high-order numerical framework. In the calculation process of the steady-state semiconductor drift-diffusion equation by traditional methods, the strong convection condition caused by the high-field effect makes the numerical solution of the equation show severe oscillations, resulting in calculation errors. Using the method of the present invention can avoid the calculation errors caused by the amplification of oscillation errors during the iterative process, thus ensuring the high precision, strong robustness, and fast convergence advantages of the numerical framework.

[0098] Taking Figure 3 the PN junction model in Figure 4 as an example, use the stabilized high-order hybrid discontinuous Galerkin method proposed by the present invention to solve the nonlinear drift-diffusion model of carriers in semiconductor devices, where (a) is the doping concentration distribution, (b) is the unstructured two-dimensional triangular mesh for calculation, and (c) is the structured two-dimensional triangular mesh for calculation.

[0099] Figure 5 The estimated values of the posterior error indicators are given. Among the data calculated using four groups of meshes, the estimated error values and the true error values are in good agreement.

[0100] Figure 6 The calculation results of the electron concentration distribution in the PN junction model under the applied voltages of 2V and 20V are given, which are in good agreement with the results of the commercial simulation software COMSOL Multiphysics and Sentaurus TCAD. At high voltages, the numerical algorithm of the present invention shows better numerical stability, while the first-order FEM solver of COMSOL has shown severe numerical oscillations, proving the strong robustness of the present method.

[0101] Figure 7The stabilized hybrid discontinuous Galerkin method of the present invention, the error convergence rates of the FEM method and the FVM method are compared. The convergence rates of the relative errors of the three methods with respect to the mesh size are 4.5858, 1.6622, and 1.3037 respectively. While providing stability for the HDG framework, the present invention retains high-order convergence.

[0102] In high-power semiconductor devices, as the applied voltage increases, the region dominated by strong convection moves significantly, causing the initial mesh to fail. The stabilized hybrid discontinuous Galerkin method proposed by the present invention can solve this problem. Taking Figure 8 the high-power LDMOS device shown as an example, where (a) is the device doping profile and (b) is the unstructured two-dimensional triangular mesh used for the calculation. Keeping the Gate voltage constant, the Drain voltage is continuously increased. As Figure 9 shown, the first-order and second-order solvers of COMSOL encounter calculation errors at 3.9 V and 21.8 V respectively. The S-HDG method proposed by the present invention is consistent with the FVM and can complete the full calculation of the Drain voltage from 0 V to 40 V. The difference is that the calculation accuracy achieved by the present invention is higher than that of the FVM method. As Figure 10 shown, the S-HDG method ensures both calculation accuracy and calculation stability; the FVM method, as a low-order stable method, has poor calculation accuracy; the FEM, as an unstable method, loses calculation accuracy and cannot guarantee calculation robustness.

[0103] The above-described embodiments are only some of the better solutions of the present invention, but they are not intended to limit the present invention. Those of ordinary skill in the relevant technical fields can make various changes and modifications without departing from the spirit and scope of the present invention. Therefore, all technical solutions obtained by means of equivalent replacement or equivalent transformation fall within the protection scope of the present invention.

Claims

1. A method for solving the drift-diffusion model of high-power semiconductor devices based on stabilized high-order mixed discontinuous Galerkin, characterized in that: The method comprises: first, tentatively solving the carrier drift-diffusion system under the updated external voltage and estimating the numerical error; then, constructing the initial artificial diffusion term based on the numerical error and discarding the tentative solution, restarting the stable iteration, and in each subsequent iteration, if there is a numerical error greater than a preset threshold, updating the artificial diffusion term until the nonlinear system under the current voltage converges.

2. The method for solving the drift-diffusion model of a high-power semiconductor device based on a stabilized high-order mixed discontinuous Galerkin according to claim 1 is characterized in that: The specific steps include: Step 1: Establish a multi-physics coupling system geometric model of semiconductor devices, perform initial meshing, and obtain mesh topology, i.e., shape function information, unit adjacency, and boundary conditions. Step 2: Establish a physical model of semiconductor devices, including doping material properties, doping concentration distribution, mobility degradation model, and carrier impact ionization model; Step 3: Update the grid topology, semiconductor physical model and boundary conditions in the simulation software. For the carrier drift-diffusion equation in the steady-state semiconductor, use the mixed discontinuous Galerkin method for weak formalization, and then use the high-order discontinuous polynomial space for discretization to assemble the linear system matrix of the drift-diffusion model. Step 4: Set the artificial diffusion term to zero and find a trial solution; Step 5: Traverse all grid cells and estimate the numerical error distribution using the numerical jump amount; Step 6: Determine whether the relative error between two iterations is less than the preset error tolerance. If so, proceed to step 7; otherwise, proceed to step 8. Step 7: Determine whether the preset voltage value is reached. If so, end the algorithm, otherwise go to step 3; Step 8: Determine whether there is a grid cell whose numerical error is greater than the preset threshold. If so, proceed to step 9, otherwise proceed to step 10; Step 9: Keep the boundary conditions unchanged, iterate and solve, and go to step 5; Step 10: Update the artificial diffusion term according to the error distribution.

3. The method for solving the drift-diffusion model of a high-power semiconductor device based on a stabilized high-order mixed discontinuous Galerkin according to claim 2, characterized in that: The truncation error of the numerical solution is estimated by the redundant degrees of freedom of the hybrid discontinuous Galerkin method, the strong convection unit in the carrier convection-diffusion system is located, and the distribution of the artificial diffusion term is obtained based on the distribution of the numerical error system and the calculation results. The total numerical error distribution satisfies that it does not exceed the sum of the following three parts: the error caused by the artificial diffusion term changing the physical equation, the truncation error caused by projecting the infinite-dimensional function into the finite-dimensional function space, and the accuracy and stability error caused by the numerical framework itself; Thus, the diffusion error and the oscillation error are adaptively balanced based on the error estimate.

4. The method for solving the drift-diffusion model of a high-power semiconductor device based on a stabilized high-order mixed discontinuous Galerkin method according to claim 2, characterized in that: The solution variables of the carrier drift-diffusion model in semiconductor devices Replaced by carrier quasi-Fermi potential , and based on the equivalent transformation, the governing equation describing carrier transport is reduced to the nonlinear drift diffusion equation, where is the electrostatic potential, is the electron concentration, is the hole concentration, is the electron quasi-Fermi potential, is the hole quasi-Fermi potential.

5. The method for solving the drift-diffusion model of a high-power semiconductor device based on a stabilized high-order mixed discontinuous Galerkin method according to claim 2, characterized in that: The fifth step uses numerical jumps to estimate the numerical error distribution as follows: the optimization process of the artificial diffusion term is combined with the iterative process of the nonlinear problem, and a posterior error indicator is constructed to estimate the numerical error. The redundant degrees of freedom overlapping in the grid cells have different values, which are used to characterize the oscillation error of the numerical solution. The global solution error is calculated based on the redundant degrees of freedom of the skeleton solution. The local error of the cell is calculated based on the redundant degrees of freedom between the skeleton solution and the local solution. Therefore, the characterization of the error only requires the generated local field and skeleton field, and there is no need to solve the adjoint system or construct other problems.

6. The method for solving the drift-diffusion model of a high-power semiconductor device based on a stabilized high-order mixed discontinuous Galerkin method according to claim 2, characterized in that: The convergence criterion in step 6 is: the solutions of two consecutive iterations , The modulus of the difference between The ratio does not exceed the preset error tolerance .

7. The method for solving the drift-diffusion model of a high-power semiconductor device based on a stabilized high-order mixed discontinuous Galerkin according to claim 2, characterized in that: In step 8, determine whether there is a unit The posterior error indicator value in is greater than or equal to a preset error threshold, wherein the preset error threshold is an order of magnitude less than the thermal pressure, that is, .

8. The method for solving the drift-diffusion model of a high-power semiconductor device based on a stabilized high-order mixed discontinuous Galerkin according to claim 2, characterized in that: The updating of artificial diffusion items is specifically as follows: The function of updating the artificial diffusion term as the numerical error is ,in, is the maximum value of the artificial diffusion term defined by the user, is the user-defined artificial diffusion term parameter, which determines the sensitivity of the diffusion term value to the value jump. In the nonlinear system iteration process, the artificial diffusion term needs to be updated with the accumulated value of the error instead of the true value, that is, Pick , and there is for and The larger of For unit The estimated mean error is The value in the iteration, For unit The estimated mean error is The value in iterations.

Images