A carrier concentration preserving simulation calculation method based on drift-diffusion model

Abstract

The application relates to a carrier concentration preserving simulation calculation method based on a drift-diffusion model, relates to the technical field of optoelectronic component simulation, and solves the technical problems that in the simulation of a traditional semiconductor device, the carrier concentration is prone to non-physical negative values and strong numerical oscillation under high bias voltage and strong electric field conditions. The method comprises the following steps: determining a drift-diffusion physical model equation set for describing internal physical processes of a semiconductor device; performing mesh division on a simulation region; establishing a standard finite volume element bilinear form of a potential Poisson equation; establishing a bilinear form of a new upwind finite volume element method of an electron continuity equation and a hole continuity equation; using a Newton iteration method to solve a coupled nonlinear algebraic equation set; and performing non-negativity correction of electron and hole concentrations. The method can avoid numerical oscillation, meet current continuity, and improve the convergence speed to the second order, and can ensure non-negativity of carrier concentrations.

Description

Technical Field

[0001] This invention relates to the field of optoelectronic component simulation technology, and in particular to a carrier concentration-preserving simulation calculation method based on a drift-diffusion model. Background Technology

[0002] In the simulation of optoelectronic components, the core physical model describing the transport behavior of charge carriers (electrons and holes) inside semiconductors is the drift-diffusion equation. This model is a strongly coupled set of nonlinear partial differential equations, typically containing the following three interrelated equations: 1) the Poisson equation for potential: describing the spatial distribution of the potential; 2) the electron continuity equation: characterizing the evolution of electron concentration over time and space; and 3) the hole continuity equation: characterizing the evolution of hole concentration over time and space. The electron and hole continuity equations are mathematically classified as convection-diffusion equations. In many critical operating regions of optoelectronic devices, such as the depletion region formed under high reverse bias and at heteromaterial interfaces, charge carrier transport often exhibits strong convection-dominated characteristics, meaning that the drift motion of charge carriers driven by the electric field is far more effective than the diffusion motion driven by the concentration gradient. Simultaneously, these regions are usually accompanied by abrupt spatial changes in charge carrier concentration, forming steep concentration gradients.

[0003] When applying standard numerical methods to solve such strongly convection-dominated problems or regions with extremely large carrier concentration gradients, severe numerical oscillations and negative concentration solutions are prone to occur. These non-physical phenomena not only lead to serious distortion and loss of credibility in simulation results, but also directly undermine the stability of subsequent nonlinear iterative solutions, frequently causing iterative divergence and ultimately resulting in the entire simulation failing to converge or unexpectedly interrupting. Therefore, developing a stable, positively consistent, and highly accurate drift-diffusion equation solver, especially for the continuous equations dominated by strong convection, is crucial for carrier concentration simulation in optoelectronic devices.

[0004] Currently, to overcome the inherent limitations of conventional numerical methods in problems dominated by strong convection, many stabilization techniques have been developed. Among them, the Finite Volume Method (FVM), due to its inherent guarantee of flux conservation, has become the mainstream spatial discretization framework for solving drift-diffusion equations. Within the FVM framework, the mainstream stabilization techniques can be summarized into two categories: upwind strategies and the Scharfetter-Gummel (SG) scheme. However, under strong convection dominance, both the standard upwind scheme and the SG scheme... The convergence rate under the norm is first order.

[0005] In summary, the drift-diffusion equation is the core model describing the carrier transport process in the simulation of optoelectronic components. However, existing numerical simulation methods based on this model generally suffer from the following key technical problems when solving the electron and hole continuity equations:

[0006] Severe non-physical oscillations: When using standard numerical schemes (such as central difference or standard upwind scheme) to solve the problem, the calculation results are prone to severe non-physical oscillations, which seriously affect the reliability of the solution.

[0007] Non-physical negative carrier concentration: If the discrete scheme does not have positive preservation, coupled with numerical oscillation, the calculated carrier concentration will often be non-physical negative, which violates the basic physical law that the concentration must be non-negative.

[0008] The simulation process is unstable: the appearance of the above-mentioned non-physical negative concentration will seriously disrupt the stability of the nonlinear iterative solution process, causing the calculation to diverge and making it difficult to obtain a convergent solution.

[0009] Precision limitations: Existing commonly used positive or stable formats, such as the standard upwind format or SG format, have limitations in accuracy. The convergence speed under the norm is usually only first order. Summary of the Invention

[0010] This invention aims to solve the technical problems in the simulation of traditional semiconductor devices under high bias voltage and strong electric field conditions, such as the easy occurrence of non-physical negative values ​​of carrier concentration and strong numerical oscillations. It provides a carrier concentration positive-preserving simulation calculation method based on a drift-diffusion model.

[0011] To solve the above-mentioned technical problems, the technical solution of the present invention is as follows:

[0012] A carrier concentration-preserving simulation calculation method based on a drift-diffusion model includes the following steps:

[0013] Step 1: Determine the set of equations for the drift-diffusion physical model that describe the internal physical processes of semiconductor devices;

[0014] Step 2: Mesh the simulation area;

[0015] Step 3: Establish the standard finite volume element bilinear form of the potential Poisson equation;

[0016] Step 4: Combining the new upwind technology, establish the bilinear form of the new upwind finite volume element method for the electron continuity equation and the hole continuity equation;

[0017] Step 5: Solve the coupled nonlinear algebraic equations using Newton's iterative method;

[0018] Step 6: Perform nonnegative corrections for electron and hole concentrations.

[0019] In the above technical solution, step 1 uses the drift-diffusion physical model equations, including the Poisson equation for electric potential, the electron continuity equation, and the hole continuity equation, with the following expressions:

[0020]

[0021] in, Where is the dielectric constant. For electric potential, The fundamental charge, and These represent the concentrations of electrons and holes, respectively. and These represent the doping concentrations of the donor and acceptor, respectively. and These are the recombination-generation terms for electrons and holes, respectively. and These are the current densities for electrons and holes, respectively; Represents the gradient operator;

[0022] The expressions for the current density of electrons and holes are as follows:

[0023]

[0024] in, and These are the mobilities of electrons and holes, respectively. and Let be the diffusion coefficients of electrons and holes, respectively, which satisfy Einstein's relation:

[0025]

[0026]

[0027] in, Boltzmann's constant, For temperature;

[0028] Dirichlet or Neumann boundary conditions are used at the boundary of the simulation region, as defined below:

[0029]

[0030]

[0031] in, and These are the Dirichlet-type boundary and the Neumann-type boundary of the potential, respectively. and These are the Dirichlet-type boundary and Neumann-type boundary for carrier concentration, respectively. To solve for the unit outward normal vector of the region boundary, Indicates in Given the potential boundary value, Indicates in The given hole concentration boundary value, Indicates in The given electron concentration boundary value, The potential gradient is denoted as .

[0032] In the above technical solution, step 2, which involves meshing the simulation area, specifically uses the finite volume element method as spatial discretization to directly extend it to a three-dimensional spatial scene.

[0033] In the above technical solution, step 3 specifically includes: defining the trial function space. and test function space ,in:

[0034] Trial function space Let it be a bilinear polynomial space:

[0035]

[0036] in, To explore the function space The trial function in For a continuous function space, To solve the region closure, Indicates that it is defined in the reference unit The reference trial function on, For the logical symbol "arbitrary", To solve the region The original mesh partitioning, To solve the region The boundary, It is a unit square. This is a bilinear transformation used to transform a unit square... Mapped to a general quadrilateral The transformation expression is:

[0037]

[0038] in, It is a quadrilateral The x-coordinate of any point in the middle, It is a quadrilateral The x-coordinate of the midpoint. It is a quadrilateral The ordinate of any point in the middle, It is a quadrilateral The y-coordinate of the midpoint. , Reference unit The local x-coordinate in the middle, Reference unit Local ordinates in ;

[0039] In the original unit quadrilateral superior, The expression is:

[0040]

[0041] in, The function value at the node. Number the nodes. ;

[0042] Test function space Let it be a piecewise constant space on the dual unit, that is:

[0043]

[0044] in, To test the function space The test function in Let represent the space of square-integrable functions. Indicates the dual unit. Indicates dual mesh partitioning, Denotes the boundary of the solution domain Boundary nodes on, Indicates location at the boundary upper node The corresponding boundary dual element;

[0045] The bilinear form of the Poisson equation for electric potential using the finite volume element method is: [Find...] , so that:

[0046] ,

[0047] in, The bilinear operator of the finite volume element method for representing the potential Poisson equation. The numerical solution representing the electric potential. The inner product of the source terms in the Poisson equation for electric potential is represented.

[0048] Among them, the bilinear operator Inner product of source terms They are defined as follows:

[0049]

[0050]

[0051] In the formula, For dual element boundary The outer normal vector, This represents a line surface integral element on the boundary. express The length element of the direction, express The length element in the direction.

[0052] In the above technical solution, step 4 specifically includes:

[0053] The electron continuity equation and the hole continuity equation are as follows:

[0054]

[0055] The standard finite volume element bilinear forms of the diffusion terms of the electron continuity equation and the hole continuity equation are as follows:

[0056]

[0057]

[0058] in, The finite volume element method bilinear form of the diffusion term in the electron continuity equation. The finite volume element method bilinear form of the diffusion term in the hole continuity equation. This represents the gradient of the numerical solution for electron concentration. This represents the gradient of the numerical solution for hole concentration. Numerical solution representing electron concentration. Numerical solution representing hole concentration;

[0059] Define the average derivative on the boundary: assuming two adjacent primitive units and The common edge is ,but any point above average derivative at Defined as:

[0060]

[0061] For the original unit In the middle, set dual unit and For the common boundary line segment, Points on The corresponding reference plane The coordinates on are ,in ; and then define the corresponding boundary line segment upstream point for:

[0062]

[0063] in, dual unit The unit outward normal vector, Represents bilinear transformation exist The value at time, Represents bilinear transformation exist The value at time, Representing adjacent dual units and The common boundary segment, Represents the convection velocity vector;

[0064] Based on the upstream point, the electron concentration in the convection term and hole concentration The windward approach is defined as:

[0065]

[0066]

[0067] in, As the electron concentration approaches from the wind, The windward approach of hole concentration, Indicates upstream point The electron concentration value at that location, Indicates upstream point Hole concentration value at that location, Indicates the current point Upstream point The position vector, i.e. , Represents the average derivative operator;

[0068] The bilinear forms of the convection terms in the electron continuity equation and the hole continuity equation are defined as follows:

[0069]

[0070]

[0071] A new upwind finite volume element method bilinear form for the convection term of the electron continuity equation. The new upwind finite volume element method bilinear form of the convection term in the hole continuity equation. This represents the windward approximation of the numerical solution for electron concentration. This represents the windward approximation of the numerical solution for hole concentration;

[0072] The inner product of the source terms on the right-hand side of the electron continuity equation and the hole continuity equation is defined as follows:

[0073]

[0074]

[0075] in, This represents the inner product of the source terms in the electronic continuity equation. This represents the inner product of the source terms in the hole continuity equation;

[0076] The bilinear form of the new upwind finite volume element method for the electron continuity equation is: [Find...] , so that:

[0077]

[0078] The bilinear form of the new upwind finite volume element method for the hole continuity equation is: [Find...] , so that:

[0079] .

[0080] In the above technical solution, step 5 specifically includes:

[0081] Through numerical discretization in steps 3 and 4, the Poisson equation, the electron continuity equation, and the hole continuity equation are transformed into a system of coupled nonlinear algebraic equations. ,in Let be the vector to be solved, containing the electric potential of all nodes. electron concentration and hole concentration .

[0082] In the above technical solution, step 5 involves using Newton's iteration method to solve the coupled nonlinear algebraic equation system. The iteration formula is as follows:

[0083]

[0084] in, For the first Approximate solution for the next iteration For the first Approximate solution for the next iteration Let Jacobian matrix be the value at the k-th iteration. It represents the residual vector of the coupled nonlinear algebraic equation system at the k-th iteration.

[0085] In the above technical solution, step 6 specifically includes:

[0086] electron concentration at all nodes and hole concentration The process involves iterating through the data and checking each node. If the concentration value at a node is negative, it is corrected and reset to a very small positive number. After the correction is complete, the next iteration is performed to ensure that the carrier concentration meets the non-negativity requirement throughout the process.

[0087] The present invention has the following beneficial effects:

[0088] The carrier concentration positive simulation calculation method based on the drift-diffusion model of the present invention can avoid numerical oscillation, satisfy current continuity, and improve the convergence speed to the second order, while ensuring that the carrier concentration is non-negative.

[0089] The carrier concentration-preserving simulation calculation method based on the drift-diffusion model of this invention, by constructing a new upwind finite volume element method (FVEM) scheme, can effectively:

[0090] Avoid violent non-physical oscillations;

[0091] Ensure that the calculated carrier concentration result is non-negative (positive protection);

[0092] Improve the stability of the nonlinear iterative solution process and ensure simulation convergence;

[0093] exist Achieving second-order convergence speed under the norm yields higher-precision simulation results. Attached Figure Description

[0094] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0095] Figure 1 For nodes Schematic diagram of the dual unit (control volume).

[0096] Figure 2 This is a schematic diagram of a twisted quadrilateral grid.

[0097] Figure 3 For numerical solutions Distribution diagram.

[0098] Figure 4 For an exact solution Distribution diagram.

[0099] Figure 5 Error between numerical solution and exact solution Distribution diagram. Detailed Implementation

[0100] The inventive concept of this invention is as follows:

[0101] The present invention provides a carrier concentration positive-preserving simulation calculation method based on a drift-diffusion model. This method uses the finite volume element method for numerical discretization. For the electron continuity equation and the hole continuity equation, a new upwind technique is applied to the finite volume element scheme of the electron continuity equation and the hole continuity equation. Through a positive-preserving correction mechanism, it ensures that the carrier concentration always satisfies the physical non-negativity, while avoiding numerical oscillations and improving the accuracy and stability of the simulation calculation.

[0102] The carrier concentration-preserving simulation calculation method based on the drift-diffusion model of the present invention includes the following steps:

[0103] Step 1: Determine the set of equations for the drift-diffusion physical model that describe the internal physical processes of semiconductor devices;

[0104] Step 2: Mesh the simulation area;

[0105] Step 3: Establish the standard finite volume element bilinear form of the potential Poisson equation;

[0106] Step 4: Combining the new upwind technology, establish the bilinear form of the new upwind finite volume element method for the electron continuity equation and the hole continuity equation;

[0107] Step 5: Solve the coupled nonlinear algebraic equations using Newton's iterative method;

[0108] Step 6: Perform nonnegative corrections for electron and hole concentrations.

[0109] The present invention will now be described in detail with reference to the accompanying drawings.

[0110] The carrier concentration-preserving simulation calculation method based on the drift-diffusion model of the present invention is described in detail below:

[0111] Step 1: Determine the set of equations for the drift-diffusion physical model that describe the internal physical processes of semiconductor devices;

[0112] This invention considers semiconductor simulation devices under steady-state conditions. The drift-diffusion physical model used consists of the potential Poisson equation, the electron continuity equation, and the hole continuity equation, with the specific expressions as follows:

[0113]

[0114] in, Where is the dielectric constant. For electric potential, The fundamental charge, and These represent the concentrations of electrons and holes, respectively. and These represent the doping concentrations of the donor and acceptor, respectively. and These are the recombination-generation terms for electrons and holes, respectively. and These are the current densities for electrons and holes, respectively; This is the gradient operator.

[0115] The current density of electrons and holes follows the drift-diffusion theory, specifically expressed as:

[0116]

[0117] in, and These are the mobilities of electrons and holes, respectively. and Let be the diffusion coefficients of electrons and holes, respectively, which satisfy Einstein's relation:

[0118]

[0119]

[0120] in, Boltzmann's constant, For temperature.

[0121] Dirichlet or Neumann boundary conditions are used at the boundary of the simulation region, as defined below:

[0122]

[0123]

[0124] in, and These are the Dirichlet-type boundary and the Neumann-type boundary of the potential, respectively. and These are the Dirichlet-type boundary and Neumann-type boundary for carrier concentration, respectively. To solve for the unit outward normal vector of the region boundary, Indicates in Given the potential boundary value, Indicates in The given hole concentration boundary value, Indicates in The given electron concentration boundary value, The potential gradient is denoted as .

[0125] The objective of solving the entire drift-diffusion physical model equations is the electric potential. electron concentration and hole concentration Three physical quantities.

[0126] Step 2: Mesh the simulation area;

[0127] This invention takes two-dimensional spatial simulation as an example, dividing the simulation region into quadrilateral meshes and using the finite volume element method for spatial discretization. This method can be directly extended to three-dimensional spatial scenes. This method belongs to the vertex-centered finite volume method, which requires the construction of two meshing systems: the original mesh and the dual mesh.

[0128] First, construct the original mesh partitioning. The semiconductor device simulation region is divided into N convex quadrilateral units. The set of quadrilateral cell mesh nodes is denoted as All physical quantities to be solved are defined on the nodes of the original mesh, including the electric potential. electron concentration Hole concentration ,node The physical quantity at that location is abbreviated as: , , .

[0129] Secondly, construct the dual mesh partitioning. The dual mesh, i.e., the control volume of the original mesh nodes, is constructed as follows: first, the average center of each original cell is calculated. And determine the midpoint of each side of the original unit. By connecting the midpoints of adjacent sides in sequence With the mean center , forming with the original node The volume (dual element) is controlled by the polygon centered on it. For example... Figure 1 As shown, connect the points in sequence. , constitute nodes dual unit .in, All are original unit vertices. All are the average centers of the original units. All are the midpoints of the edges of the original elements. The dual mesh subdivision consisting of all dual elements is defined as follows: .

[0130] Step 3: Establish the standard finite volume element bilinear form of the potential Poisson equation;

[0131] To achieve numerical discretization of the potential Poisson equation, it is necessary to define a trial function space. and test function space This leads to the construction of a bilinear form for finite volume elements.

[0132] Trial function space Let it be a bilinear polynomial space:

[0133]

[0134] To explore the function space The trial function in For a continuous function space, To solve the region closure, Indicates that it is defined in the reference unit The reference trial function on, For the logical symbol "arbitrary", To solve the region The original mesh partitioning, To solve the region The boundary, It is a unit square. This is a bilinear transformation used to transform a unit square... Mapped to a general quadrilateral The transformation expression is:

[0135]

[0136] in, It is a quadrilateral The x-coordinate of any point in the middle, It is a quadrilateral The x-coordinate of the midpoint. It is a quadrilateral The ordinate of any point in the middle, It is a quadrilateral The ordinate of the middle vertex, ( Reference unit The local x-coordinate in the middle, Reference unit The local ordinate in ( );

[0137] In the original quadrilateral unit (Refers to a general quadrilateral) )superior, The expression is:

[0138]

[0139] in, The function value at the node. Number the nodes. .

[0140] Test function space Let it be a piecewise constant space on the dual unit, that is:

[0141]

[0142] in, To test the function space The test function in Let represent the space of square-integrable functions. Indicates the dual unit. Indicates dual mesh partitioning, Denotes the boundary of the solution domain Boundary nodes on, Indicates location at the boundary upper node The corresponding boundary dual element;

[0143] The bilinear form of the Poisson equation for electric potential using the finite volume element method is: [Find...] , so that:

[0144] ,

[0145] in, The bilinear operator of the finite volume element method for representing the potential Poisson equation. The numerical solution representing the electric potential. The inner product of the source terms in the Poisson equation for electric potential is represented.

[0146] Among them, the bilinear operator Inner product of source terms They are defined as follows:

[0147]

[0148]

[0149] In the formula, For dual element boundary The outer normal vector, This represents a line surface integral element on the boundary. express The length element of the direction, express The length element in the direction.

[0150] Step 4: Combining the new upwind technology, establish the bilinear form of the new upwind finite volume element method for the electron continuity equation and the hole continuity equation;

[0151] After simplification, both the electron continuity equation and the hole continuity equation include convection and diffusion terms. To balance numerical stability and computational accuracy, the diffusion term adopts the standard finite volume element scheme, and a new upwind technique is introduced for the convection term to avoid numerical oscillations.

[0152] The simplified electron continuity equation and hole continuity equation are as follows:

[0153]

[0154] The first term in both equations is the convection term, and the second term is the diffusion term. We first give the bilinear form of the diffusion term in the standard finite volume element. The standard finite volume element bilinear forms of the diffusion terms in the electron continuity equation and the hole continuity equation are as follows:

[0155]

[0156]

[0157] in, The finite volume element method bilinear form of the diffusion term in the electron continuity equation. The finite volume element method bilinear form of the diffusion term in the hole continuity equation. This represents the gradient of the numerical solution for electron concentration. This represents the gradient of the numerical solution for hole concentration. Numerical solution representing electron concentration. Numerical solution representing hole concentration;

[0158] To avoid numerical oscillations, we apply a novel upwind technique to the convection terms of the electron and hole continuity equations. Since the derivatives are discontinuous at the boundaries of the primitive elements, we first define the average derivative at the boundaries: assuming two adjacent primitive elements... and The common edge is ,but any point above average derivative at Defined as:

[0159]

[0160] For the original unit In the middle, set dual unit and For the common boundary line segment, Points on The corresponding reference plane The coordinates on are ,in Then, define the corresponding boundary segments. upstream point for:

[0161]

[0162] in, dual unit The unit outward normal vector, Represents bilinear transformation exist The value at time, Represents bilinear transformation exist The value at time, Representing adjacent dual units and The common boundary segment, This represents the convection velocity vector. Based on the upstream point, the electron concentration in the convection term... and hole concentration The windward approach is defined as:

[0163]

[0164]

[0165] in, As the electron concentration approaches from the wind, The windward approach of hole concentration, Indicates upstream point The electron concentration value at that location, Indicates upstream point Hole concentration value at that location, Indicates the current point Upstream point The position vector, i.e. , Represents the average derivative operator;

[0166] The bilinear forms of the convection terms in the electron continuity equation and the hole continuity equation are defined as follows:

[0167]

[0168]

[0169] in, A new upwind finite volume element method bilinear form for the convection term of the electron continuity equation. The new upwind finite volume element method bilinear form of the convection term in the hole continuity equation. This represents the windward approximation of the numerical solution for electron concentration. This represents the windward approximation of the numerical solution for hole concentration;

[0170] The inner product of the source terms on the right-hand side of the electron continuity equation and the hole continuity equation is defined as follows:

[0171]

[0172]

[0173] in, This represents the inner product of the source terms in the electronic continuity equation. This represents the inner product of the source terms in the hole continuity equation;

[0174] The bilinear form of the new upwind finite volume element method for the electron continuity equation is: [Find...] , so that:

[0175]

[0176] The bilinear form of the new upwind finite volume element method for the hole continuity equation is: [Find...] , so that:

[0177] .

[0178] Step 5: Solve the coupled nonlinear algebraic equations;

[0179] In this embodiment, Newton's iterative method is used to solve the coupled nonlinear algebraic equation system.

[0180] Through numerical discretization in steps 3 and 4, the Poisson equation, the electron continuity equation, and the hole continuity equation are transformed into a coupled set of nonlinear algebraic equations. ,in Let be the vector to be solved, containing the electric potential of all nodes. electron concentration and hole concentration This invention employs Newton's iterative method to solve a system of nonlinear algebraic equations. The iterative formula is as follows:

[0181]

[0182] in, For the first Approximate solution for the next iteration Indicates the first Approximate solution for the next iteration Let Jacobian matrix be the value at the k-th iteration. Let be the residual vector of the coupled nonlinear algebraic equations at the k-th iteration. By iteratively calculating until the convergence condition is met, the potential distribution and the concentration distribution of electrons and holes within the simulation region of the semiconductor device can be obtained.

[0183] Step 6: Perform nonnegativity correction for electron and hole concentrations;

[0184] Due to numerical discretization errors or extreme operating conditions such as strong electric fields, electron concentration... and hole concentration Negative values ​​can occur during the calculation, which violates the physical principle that concentration values ​​are non-negative and can slow down or even prevent the convergence of Newton's iterations. Therefore, after each Newton iteration, a positive-preserving correction operation needs to be performed: the electron concentration at all nodes... and hole concentration Perform a traversal check; if the concentration value at a certain node is negative, reset it to a very small positive number. (This value is much smaller than the normal concentration range and does not affect the overall calculation accuracy.) After correction, proceed to the next iteration to ensure that the carrier concentration meets the non-negativity requirement throughout the process.

[0185] In other specific embodiments of the present invention, the finite volume element method in steps 3-4 can be replaced by the element-centered finite volume method; the Newton method in step 5 can be replaced by the quasi-Newton method; the positiveity preservation correction operation in step 6 can also be replaced by other alternatives to construct a format that satisfies the M matrix; these will not be elaborated here.

[0186] To verify the convergence accuracy and stability of the method proposed in this invention, the electronic continuity equation (convection-diffusion equation) was selected as a numerical example for testing.

[0187] In the region Consider the following questions:

[0188]

[0189] The exact solution is:

[0190]

[0191] The convection velocity is taken as:

[0192]

[0193] Take the diffusion coefficient as (These correspond to diffusion-dominated, weak convection-dominated, and strong convection-dominated operating conditions, respectively.) and The expression is derived from the exact solution. The derivation is obtained by substituting into the control equation.

[0194] Exact solutions and numerical solutions The formula for calculating the modulus error is:

[0195]

[0196] Convergence order The calculation formula is:

[0197]

[0198] in, The original mesh step size, The grid step size after encryption. and Each of the corresponding grids Modulus error.

[0199] We perform calculations on a twisted quadrilateral mesh, such as Figure 2 As shown. Different diffusion coefficients. Modulus error and The convergence order is shown in Table 1. Convection-dominated operating condition ( Numerical solution for a 128×128 grid Distribution, exact solution Distribution and the error between numerical and exact solutions Distribution, see below. Figure 3-5 .

[0200] The calculation results show that, regardless of whether diffusion is dominant ( ) and convection dominance ( Under the operating conditions, the method of the present invention The modulus errors all exhibit stable second-order convergence characteristics, indicating that the method of the present invention has excellent numerical accuracy. Under the convection-dominated condition, the error distribution between the numerical solution and the exact solution is uniform, with no obvious numerical oscillation phenomenon, which verifies the effectiveness of the new upwind technique in avoiding numerical oscillation.

[0201] Table 1 Different diffusion coefficients Below Modulus error and Convergence order

[0202]

[0203] Obviously, the above embodiments are merely illustrative examples for clear explanation and are not intended to limit the implementation. Those skilled in the art will recognize that other variations or modifications can be made based on the above description. It is neither necessary nor possible to exhaustively list all possible implementations here. However, obvious variations or modifications derived therefrom are still within the scope of protection of this invention.

Claims

1. A carrier concentration-preserving simulation calculation method based on a drift-diffusion model, characterized in that, Includes the following steps: Step 1: Determine the set of equations for the drift-diffusion physical model that describe the internal physical processes of semiconductor devices; Step 2: Mesh the simulation area; Step 3: Establish the standard finite volume element bilinear form of the potential Poisson equation; Step 4: Combining the new upwind technology, establish the bilinear form of the new upwind finite volume element method for the electron continuity equation and the hole continuity equation; Step 5: Solve the coupled nonlinear algebraic equations using Newton's iterative method; Step 6: Perform nonnegativity correction for electron and hole concentrations; Step 4 is as follows: The electron continuity equation and the hole continuity equation are as follows: in, For electric potential, and These represent the concentrations of electrons and holes, respectively. and These are the recombination-generation terms for electrons and holes, respectively. Represents the gradient operator, and These are the mobilities of electrons and holes, respectively. and These are the diffusion coefficients for electrons and holes, respectively; The standard finite volume element bilinear forms of the diffusion terms of the electron continuity equation and the hole continuity equation are as follows: in, The finite volume element method bilinear form of the diffusion term in the electron continuity equation. The finite volume element method bilinear form of the diffusion term in the hole continuity equation. This represents the gradient of the numerical solution for electron concentration. This represents the gradient of the numerical solution for hole concentration. Numerical solution representing electron concentration. The numerical solution representing the hole concentration. Indicates the dual unit. Indicates dual mesh partitioning, For dual element boundary The outer normal vector, This represents a line surface integral element on the boundary. To explore the function space, To test the function space, The logical symbols are arbitrary. To test the function space The test function in; Define the average derivative on the boundary: assuming two adjacent primitive units and The common edge is ,but any point above average derivative at Defined as: in, The potential gradient; For the original unit In the middle, set dual unit and For the common boundary line segment, Points on The corresponding reference plane The coordinates on are ,in ; and then define the corresponding boundary line segment upstream point for: in, dual unit The unit outward normal vector, Represents bilinear transformation exist The value at time, Represents bilinear transformation exist The value at time, Representing adjacent dual units and The common boundary segment, Represents the convection velocity vector; Based on the upstream point, the electron concentration in the convection term and hole concentration The windward approach is defined as: in, As the electron concentration approaches from the wind, The windward approach of hole concentration, Indicates upstream point The electron concentration value at that location, Indicates upstream point Hole concentration value at that location, Indicates the current point Upstream point The position vector, i.e. , Represents the average derivative operator; The bilinear forms of the convection terms in the electron continuity equation and the hole continuity equation are defined as follows: A new upwind finite volume element method bilinear form for the convection term of the electron continuity equation. The new upwind finite volume element method bilinear form of the convection term in the hole continuity equation. This represents the windward approximation of the numerical solution for electron concentration. This represents the windward approximation of the numerical solution for hole concentration; The inner product of the source terms on the right-hand side of the electron continuity equation and the hole continuity equation is defined as follows: in, This represents the inner product of the source terms in the electronic continuity equation. This represents the inner product of the source terms in the hole continuity equation; express The length element of the direction, express The length element in the direction; The bilinear form of the new upwind finite volume element method for the electron continuity equation is: [Find...] , so that: The bilinear form of the new upwind finite volume element method for the hole continuity equation is: [Find...] , so that: 。 2. The carrier concentration-preserving simulation calculation method based on the drift-diffusion model according to claim 1, characterized in that, In step 1, the drift-diffusion physics model equations used include: the Poisson equation for electric potential, the electron continuity equation, and the hole continuity equation, with the following expressions: in, Where is the dielectric constant. The fundamental charge, and These represent the doping concentrations of the donor and acceptor, respectively. and These are the current densities for electrons and holes, respectively; The expressions for the current density of electrons and holes are as follows: The two satisfy the Einstein relation: in, Boltzmann's constant, For temperature; Dirichlet or Neumann boundary conditions are used at the boundary of the simulation region, as defined below: in, and These are the Dirichlet-type boundary and the Neumann-type boundary of the potential, respectively. and These are the Dirichlet-type boundary and Neumann-type boundary for carrier concentration, respectively. Let be the unit outward normal vector of the boundary. Indicates in Given the potential boundary value, Indicates in The given hole concentration boundary value, Indicates in The given electron concentration boundary value.

3. The carrier concentration-preserving simulation calculation method based on the drift-diffusion model according to claim 2, characterized in that, In step 2, the meshing of the simulation area is specifically performed by using the finite volume element method as spatial discretization, which is then directly extended to a three-dimensional spatial scene.

4. The carrier concentration-preserving simulation calculation method based on the drift-diffusion model according to claim 2, characterized in that, Step 3 specifically includes: defining the trial function space. and test function space ,in: Trial function space Let it be a bilinear polynomial space: in, To explore the function space The trial function in For a continuous function space, To solve the region closure, Indicates that it is defined in the reference unit The reference trial function on, To solve the region The original mesh partitioning, To solve the region The boundary, It is a unit square. This is a bilinear transformation used to transform a unit square... Mapped to a general quadrilateral The transformation expression is: in, It is a quadrilateral The x-coordinate of any point in the middle, It is a quadrilateral The x-coordinate of the midpoint. It is a quadrilateral The ordinate of any point in the middle, It is a quadrilateral The y-coordinate of the midpoint. , Reference unit The local x-coordinate in the middle, Reference unit Local ordinates in ; In the original unit quadrilateral superior, The expression is: in, The function value at the node. Number the nodes. ; Test function space Let it be a piecewise constant space on the dual unit, that is: in, Let represent the space of square-integrable functions. Denotes the boundary of the solution domain Boundary nodes on, Indicates location at the boundary upper node The corresponding boundary dual element; The bilinear form of the Poisson equation for electric potential using the finite volume element method is: [Find...] , so that: , in, The bilinear operator of the finite volume element method for representing the potential Poisson equation. The numerical solution representing the electric potential. The inner product of the source terms in the Poisson equation for electric potential is represented. Among them, the bilinear operator Inner product of source terms They are defined as follows: 。 5. The carrier concentration-preserving simulation calculation method based on the drift-diffusion model according to claim 1, characterized in that, Step 5 specifically involves: Through numerical discretization in steps 3 and 4, the Poisson equation, the electron continuity equation, and the hole continuity equation are transformed into a system of coupled nonlinear algebraic equations. ,in Let be the vector to be solved, containing the electric potential of all nodes. electron concentration and hole concentration .

6. The carrier concentration-preserving simulation calculation method based on the drift-diffusion model according to claim 1, characterized in that, In step 5: Newton's iterative method is used to solve the coupled nonlinear algebraic equation system. The iterative formula is: in, For the first Approximate solution for the next iteration For the first Approximate solution for the next iteration Let Jacobian matrix be the value at the k-th iteration. It represents the residual vector of the coupled nonlinear algebraic equation system at the k-th iteration.

7. The carrier concentration-preserving simulation calculation method based on the drift-diffusion model according to claim 6, characterized in that, Step 6 specifically involves: electron concentration at all nodes and hole concentration Perform a traversal check; if the concentration value at a certain node is negative, correct it and reset it to a very small positive number. After the correction is completed, proceed to the next iteration calculation to ensure that the carrier concentration meets the non-negativity requirement throughout the process.

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