A solution method for the drift-diffusion model of high-power semiconductor devices based on stabilized high-order mixed discontinuous Galerkin

Abstract

The present invention discloses a method for solving the drift-diffusion model of high-power semiconductor devices based on a stabilized high-order mixed discontinuous Galerkin method. During the calculation process of the steady-state semiconductor drift-diffusion equation, the strong convection conditions caused by the high field effect cause the numerical solution of the equation to oscillate violently, resulting in calculation errors. This method constructs a stable high-order numerical framework to avoid calculation errors caused by the amplification of oscillation errors in the iterative process, thereby ensuring the high precision, strong robustness and fast convergence advantages of the numerical framework. Compared with traditional methods, 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 mixed discontinuous Galerkin method, without introducing additional computational costs. This method is the first to extend the high-order mixed discontinuous Galerkin method to the carrier drift-diffusion equation dominated by strong convection, and can provide a high-precision and strong robust numerical framework for solving high-power semiconductor devices.

Description

Technical Field

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

[0002] During the design and manufacturing stages of semiconductor devices, high-precision simulation of the quasi-static electric field-carrier coupling system within the device is required to obtain the device's AC / DC effects and predict device degradation performance. This challenge lies in the fact that the drift-diffusion equation describing the carrier distribution is subject to convection-dominated effects, exhibiting highly nonlinear and singular distributions. The finite volume method (FVM) and Scharfetter-Gummel (SG) method, which respectively adhere to vector conservation and upwind schemes, enable the FVM-SG method to handle carrier drift-diffusion models with drastic magnitude variations.

[0003] However, with the increasing sophistication of semiconductor device structures and complexity of materials, solving carrier transport models using the FVM-SG scheme has become extremely challenging. Firstly, deep submicron structures and wide bandgap materials introduce complex quantum and high-field effects, making the FVM-SG scheme, which only exhibits first-order convergence, difficult to achieve. Secondly, the constant current assumption in the SG scheme fails, and its computational accuracy deteriorates in high-power semiconductor devices. Specifically, the FVM-SG scheme only satisfies the upwind scheme in the tangential direction of the cell boundary, while the drift-diffusion equation in the normal direction exhibits "air leakage."

[0004] In recent years, the hybrid discontinuous Galerkin method (HDG) has attracted attention in diverse fields, including computational mathematics, fluid dynamics, and computational electromagnetics. As an improved finite element method, HDG satisfies high-order convergence and vector conservation, and overcomes the high computational cost of the discontinuous Galerkin method (DG) through static projection, making it a promising candidate to replace the FVM-SG scheme. However, the HDG method does not conform to the upwind scheme, leading to numerical instabilities when discretizing drift-diffusion systems with convective terms, resulting in unphysical oscillations in the numerical solution. Existing HDG semiconductor solvers focus on proving mathematically optimal convergence, but these proofs often rely on the assumption of smooth numerical solutions, circumventing the instabilities caused by convective terms. While these solvers have advanced the mathematical understanding of the HDG method, carrier drift-diffusion models with convective terms are relatively lacking. Therefore, developing HDG solvers that combine high accuracy and robustness for high-power semiconductor devices is crucial.

[0005] Artificial diffusion terms are a common method for addressing numerical instabilities and are widely used in FEM, FVM, and DG methods. However, in most frameworks, they depend on the mesh size, basis function order, and physical field distribution characteristics, requiring tedious parameter tuning. Furthermore, these parameters depend on the specific problem scenario and require readjustment whenever the model, equations, and boundary conditions are changed.

[0006] In the calculation of high-power semiconductor devices dominated by convection, high-order HDG methods exhibit weak stability, causing strong unphysical oscillations. Taking into account the characteristics of this problem, the present invention proposes a hybrid discontinuous Galerkin stabilization technique for the coupled calculation of nonlinear drift-diffusion systems and develops a corresponding preprocessing solution framework: at each update of the boundary conditions / time step, a trial calculation is performed and the corresponding numerical error distribution is estimated, providing 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 oscillation 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] During the calculation of nonlinear drift-diffusion systems of carriers in high-power semiconductor devices, the convective occupation effect caused by strong fields can lead to unphysical oscillations in the numerical solution. To address this issue, this paper proposes a solution for the drift-diffusion model of high-power semiconductor devices based on a stabilized high-order hybrid discontinuous Galerkin (HDG) algorithm. This approach employs a stabilization technique using an adaptive artificial diffusion term to ensure the robustness of the high-order HDG numerical framework.

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

[0009] A method for solving a high-power semiconductor device drift-diffusion model based on a stabilized high-order mixed discontinuous Galerkin method comprises: first, tentatively solving a carrier drift-diffusion system under an updated applied voltage and estimating a numerical error; then, constructing an initial artificial diffusion term based on the numerical error, discarding the tentative solution, and restarting a stable iteration; in each subsequent iteration, if there is a numerical error greater than a preset threshold, updating the artificial diffusion term until the nonlinear system converges under the current voltage.

[0010] The above technical solution further specifically includes the following steps:

[0011] Step 1: Establish a multi-physics coupled system geometric model of the semiconductor device and perform initial meshing to obtain the mesh topology, i.e., shape function information, unit adjacency, and 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. Use the hybrid discontinuous Galerkin method to perform a weak formalization of the carrier drift-diffusion equation in steady-state semiconductors. Then, use a high-order discontinuous polynomial space for discretization and assemble the linear system matrix of the drift-diffusion model.

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

[0015] Step 5: Traverse all grid cells and estimate the numerical error distribution using the numerical jump amount;

[0016] Step 6: Determine whether the relative error between the two iterations is less than the preset error tolerance. If so, proceed to step 7; otherwise, proceed 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 grid cell whose numerical error is greater than the preset threshold. If so, proceed to step 9; otherwise, proceed to step 10.

[0019] Step 9: Keep the boundary conditions unchanged, iterate and solve, and go to step 5;

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

[0021] Furthermore, the truncation error of the numerical solution is estimated by the redundant degrees of freedom of the hybrid discontinuous Galerkin method, and the strong convection unit in the carrier convection-diffusion system is located. 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:

[0022]

[0023] Thus, the diffusion error and the oscillation error are adaptively balanced based on the error estimate, where is the exact solution of the system without introducing artificial diffusion terms, In order to introduce the accurate solution of the artificial diffusion equation, is the operator that projects the infinite-dimensional exact solution into the finite element space, is the numerical solution of the equation with the artificial diffusion term. The first error on the right-hand side comes from the modification of the physical equation by the artificial diffusion term. The second error comes from the truncation caused by projecting the infinite-dimensional function onto the finite-dimensional function space. The third error comes from the accuracy and stability of the numerical framework itself.

[0024] Furthermore, the solution variables of the carrier drift-diffusion model in semiconductor devices are 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.

[0025] Furthermore, in step 5, the optimization process of the artificial diffusion term is combined with the iterative process of the nonlinear problem to construct an a posteriori error indicator to estimate the numerical error. The redundant degrees of freedom that overlap in the grid cells have different values ​​and 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 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, without solving the adjoint system or constructing other problems. The a posteriori error indicator constructed to estimate the numerical error is:

[0026]

[0027] in, is the estimated numerical error, For any grid cell, For unit The border, is the local physical field defined within the cell, is the global physics field defined on the skeleton, For unit Feature size, For the definition Modules in space.

[0028] Furthermore, the convergence criterion in step 6 is the solution of two adjacent iterations 、 The modulus of the difference between The ratio does not exceed the preset error tolerance ,Right now:

[0029]

[0030] in, 、 are the solutions of two adjacent iterations, and the preset error tolerance for .

[0031] Furthermore, in step 8, it is determined whether there is a unit The posterior error indicator value in Greater than or equal to the preset error threshold:

[0032]

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

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

[0035] Update the artificial diffusion term as a function of the numerical error:

[0036]

[0037] in, is the maximum value of the artificial diffusion term defined by the user, The artificial diffusion term parameter defined by the user determines the sensitivity of the diffusion term value to the value jump. In the iterative process of the nonlinear system, the artificial diffusion term needs to be updated with the accumulated value of the error rather than the true value, that is, Pick , and there is:

[0038]

[0039] in, For unit The estimated mean error is The value in the iteration, For unit The estimated mean error is The value in iterations.

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

[0041] The present invention adopts a stabilized high-order mixed discontinuous Galerkin method to solve the carrier drift-diffusion model of high-power semiconductor devices. In the calculation process of the steady-state semiconductor drift-diffusion equation, this method effectively avoids the strong convection conditions caused by high field effects, which causes calculation errors caused by violent oscillations in the numerical solution of the equation, thereby ensuring the high precision, strong robustness and fast convergence advantages of the numerical framework. Compared with traditional methods, 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 mixed discontinuous Galerkin method, without introducing additional computational costs. This method is the first to extend the high-order mixed discontinuous Galerkin method to the carrier drift-diffusion equation dominated by strong convection, and can provide a high-precision and strong robust numerical framework for solving high-power semiconductor devices. BRIEF DESCRIPTION OF THE DRAWINGS

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

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

[0044] Figure 3 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 used for calculation, and (c) is the structured two-dimensional triangular mesh used for calculation;

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

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

[0047] Figure 6 Comparing the effects of introducing the stabilization strategy of the present invention on stability;

[0048] Figure 7 It is the comparison of the convergence speed between this method and the traditional method;

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

[0050] Figure 9 This is a comparison chart of electrical characteristics calculated by this method and traditional methods;

[0051] Figure 10 The hole quasi-Fermi potential gradient distribution at a drain voltage of 22 V is shown in the figure below and is compared between this method and the traditional method. DETAILED DESCRIPTION

[0052] The technical solution of the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0053] To address the nonlinearities in carrier drift-diffusion systems in high-power semiconductor devices, an iterative solution is required until the electric field, electron, and hole distributions reach a steady state. This paper proposes a stabilization method for a high-order hybrid discontinuous Galerkin framework. Combined with adaptive artificial diffusion terms, this novel solution addresses the unphysical oscillations caused by the dominance of convection terms in the carrier transport equation. This method effectively controls numerical oscillations and suppresses oscillation errors during the iterative process. This method utilizes the redundant degrees of freedom of the hybrid discontinuous Galerkin method to construct a highly reliable a posteriori error indicator and adaptive artificial diffusion terms, eliminating the need for complex parameter tuning. Figure 1 This is a flow chart of the present invention's method for solving the drift-diffusion model for high-power semiconductor devices based on a stabilized high-order mixed discontinuous Galerkin. The method includes: first, after updating boundary conditions / time steps, performing a trial solution without a stabilization strategy and estimating the numerical error, providing 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 based on the numerical error until the error falls below a preset threshold. The specific steps are as follows:

[0054] The first step is to establish a multi-physics coupled system geometric model of the device and perform initial meshing to obtain the mesh topology, namely, shape function information, unit adjacency, and boundary conditions.

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

[0056] Step 3: Load the grid topology and semiconductor physics model. For the carrier drift-diffusion equation in a steady-state semiconductor, a hybrid discontinuous Galerkin method is used to perform a weak formalization. This is then discretized using a 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, Nabla operator , is the dielectric constant of the material, is the elementary charge, is the doping concentration, is the hole concentration, is the electron concentration, is the artificial diffusion term introduced in this method, 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, For hot pressing, is the Boltzmann constant, is the lattice temperature, is the carrier recombination rate, the superscript Indicates the The corresponding physical quantity of the step iteration.

[0060] In the above method, the drift-diffusion equation for carrier transport needs to be discretized using the HDG method, which requires the selection of a pair of scalar-vector unknowns and the realization of global coupling based on the conservation of flux between units. The key to the problem is to construct a suitable scalar-vector field so that it can satisfy the high-order polynomial basis function space as much as possible. The present invention converts the unknown quantity from Change to , and an equivalent transformation of the quasi-Fermi potential transport equation is performed, still reducing it to the nonlinear drift-diffusion equation. The dimensions of the three scalar unknowns in the system are unified, and the functional form approximately satisfies polynomial basis functions. Without considering the excitation term, all three physical fields satisfy the discrete extreme value theorem: the values ​​vary monotonically, and extreme values ​​occur only at the boundaries.

[0061] The derivation results show that the artificial diffusion term has no effect on the gradient information involved in the k-th step, and therefore does not reduce the convergence rate. The hybrid discontinuous Galerkin method is used to obtain the weak form, which is then discretized using a high-order discontinuous polynomial space. To simplify the notation, in the aforementioned linearized control equations, the scalar unknown is marked as , the trace of the scalar unknown on the skeleton is marked as , the vector unknowns are marked as , the convection vector field is marked as , the corresponding discrete form of the equation is:

[0062]

[0063] Among them, the inner product is a function , The product of the unit The integral in is a function , The product of the unit Integral on the boundary, It is defined in the unit The outward normal unit vector of the surface, It is defined in the unit scalar polynomial basis functions in , It is defined in the unit The vector polynomial basis functions in , It is defined in the unit Scalar polynomial basis functions on the boundaries. is a flux term defined on the skeleton, defined as:

[0064]

[0065] in, is the local stability parameter introduced by the HDG framework itself, which is respectively:

[0066]

[0067] Combining the above equations, we can obtain the linear system matrix of the drift-diffusion model:

[0068]

[0069] Among them, in addition to and is a vector, and other bold capital letters represent matrices. Their elements are defined as follows:

[0070]

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

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

[0073]

[0074] The first error on the right side of the above equation originates from the modification of the physical equations by the artificial diffusion term. The second error originates from the truncation caused by projecting the infinite-dimensional function onto the finite-dimensional function space. The third error originates from the accuracy and stability of the numerical framework itself. As the strength of the artificial diffusion term increases, the numerical errors of the latter two terms decrease monotonically, while the first diffusion error increases monotonically. The total error shows a trend of first decreasing and then increasing. Therefore, the setting of the artificial diffusion term can be summarized as the following optimization problem:

[0075]

[0076] However, calculating the optimal artificial diffusion term is quite expensive. In fact, semiconductor device calculations do not require a globally optimal artificial diffusion term distribution; instead, it is sufficient to ensure that oscillation errors are not amplified by the iterative process, thereby avoiding calculation errors.

[0077] The present invention combines the optimization process of the artificial diffusion term with the iterative process of the nonlinear problem without introducing additional computational costs. Figure 2 As shown, the redundant degrees of freedom overlapped in the triangular element have different values ​​and can be used to characterize the oscillation error of the numerical solution. Figure 2 (a) is based on the skeleton solution The redundant degrees of freedom of Calculate the global solution error:

[0078] ,

[0079] Here For unit mid-node, is the number of unit nodes. Figure 2 (b) is based on skeleton solution and local solution The redundant degrees of freedom between the calculation unit local error:

[0080] .

[0081] The characterization of the error only requires the local field and skeleton field naturally generated by the HDG method. There is no 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 as 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 the third step;

[0087] Step 8: Determine whether a unit exists The posterior error indicator value in is greater than the preset error threshold:

[0088]

[0089] The error threshold here is taken to be smaller than the order of magnitude of the thermal pressure, that is, .

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

[0091] Step 9: Keep the boundary conditions unchanged, iterate and solve, and go to step 5;

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

[0093]

[0094] During the iteration process of the nonlinear system, the artificial diffusion term needs to be updated with the accumulated value of the error rather than the true value:

[0095]

[0096] If the error estimate is used directly to update the artificial diffusion term, the iterative process will fail to converge.

[0097] The method of the present invention constructs a stable, high-order numerical framework. Conventional methods for calculating the steady-state semiconductor drift-diffusion equation suffer from strong convection conditions caused by high fields, which can lead to violent oscillations in the numerical solution and computational errors. The method of the present invention avoids computational errors caused by amplification of oscillation errors during the iterative process, thereby ensuring the high precision, strong robustness, and rapid convergence of the numerical framework.

[0098] by Figure 3 The PN junction model in is used as an example. The stabilized high-order mixed discontinuous Galerkin method proposed in this invention is used 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 used for calculation, and (c) is the structured two-dimensional triangular mesh used for calculation. Figure 4 The estimated values ​​of the posterior error indicators are given. In the data calculated using four groups of grids, the estimated error values ​​are in good agreement with the true error values.

[0099] Figure 5 The authors present the trends of the numerical and true errors in the PN junction model as the strength of the maximum artificial diffusion term increases. At each voltage, the estimated error decreases monotonically with the artificial diffusion term, but the overall error shows a general trend of first decreasing and then increasing, consistent with the discussion of the error form in the previous method.

[0100] Figure 6 The electron concentration distribution calculation results for a PN junction model at applied voltages of 2V and 20V are presented, which agree well with the results from the commercial simulation software COMSOL Multiphysics and Sentaurus TCAD. At high voltages, the numerical algorithm of the present invention exhibits improved numerical stability, while COMSOL's first-order FEM solver exhibits severe numerical oscillations, demonstrating the robustness of the proposed method.

[0101] Figure 7The present invention compares the error convergence rates of the stabilized hybrid discontinuous Galerkin method, the FEM method, and the FVM method. The relative error convergence rates of the three methods with respect to mesh size are 4.5858, 1.6622, and 1.3037, respectively. This invention provides stability to the HDG framework while retaining 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 grid to fail. The stabilized mixed discontinuous Galerkin method proposed in this paper can solve this problem. Figure 8 The high-power LDMOS device shown is an example, where (a) is the device doping distribution and (b) is the unstructured two-dimensional triangular mesh used in the calculation. Keep the Gate voltage constant and continuously increase the Drain voltage. Figure 9 As shown in the figure, COMSOL's first-order and second-order solvers have calculation errors at 3.9 V and 21.8 V respectively. The S-HDG method proposed in this invention is consistent with FVM and can complete the complete calculation of the Drain terminal from 0V to 40V. The difference is that the calculation accuracy achieved by this invention is higher than that of the FVM method. Figure 10 As shown in the figure, the S-HDG method ensures both computational accuracy and computational stability. The FVM method, as a low-order stable method, has poor computational accuracy. The FEM method, as an unstable method, loses computational accuracy and cannot guarantee computational robustness.

[0103] The embodiments described above are merely some preferred embodiments of the present invention and are not intended to limit the present invention. Persons skilled in the art may make various changes and modifications without departing from the spirit and scope of the present invention. Therefore, any technical solution obtained by equivalent substitution or equivalent transformation falls within the scope of protection 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 applied voltage and estimating the numerical error; then, constructing an 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; specifically, the steps include: Step 1: Establish a multi-physics coupled system geometric model of the semiconductor device and perform initial meshing to obtain the mesh topology, i.e., shape function information, unit adjacency, and boundary conditions. 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; Step 3: Update the grid topology, semiconductor physical model, and boundary conditions in the simulation software. A hybrid discontinuous Galerkin method is used to perform a weak formalization of the carrier drift-diffusion equation in steady-state semiconductors. This is then discretized using a high-order discontinuous polynomial space to assemble the linear system matrix of the drift-diffusion model. The redundant degrees of freedom of the hybrid discontinuous Galerkin method are used to estimate the truncation error of the numerical solution, locate the strong convection units in the carrier convection-diffusion system, and derive 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 the requirement to not exceed the sum of the following three components: the error due to the artificial diffusion term modifying the physical equation, the truncation error due to projecting the infinite-dimensional function onto the finite-dimensional function space, and the accuracy and stability errors of the numerical framework itself. Thus, the diffusion error and oscillation error are adaptively balanced based on the error estimate. 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 the 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, specifically: 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, The artificial diffusion term parameter defined by the user 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 rather than the true value, that is, Pick , and there are 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 the iteration.

2. 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 1, 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.

3. 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 1, characterized in that: The numerical jump amount is used in the fifth step to estimate the numerical error distribution specifically 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 unit 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 unit 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.

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 1, characterized in that: The convergence criterion in step 6 is: the solution of two adjacent iterations 、 The modulus of the difference between The ratio does not exceed the preset error tolerance .

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 1, characterized in that: In step 8, determine whether a unit exists The posterior error indicator value in Greater than or equal to a preset error threshold, the preset error threshold is taken as an order of magnitude less than the thermal pressure, that is, .

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