Density gradient quantum correction carrier transport simulation method for reducing grid quality dependence

Abstract

The invention discloses a density gradient quantum correction carrier transport simulation method for reducing grid quality dependence. According to the method, starting from a density gradient model in quantum correction drift diffusion transport, reasons for errors generated when obtuse triangle grids are used for solving in a finite volume method in carrier transport simulation are studied, and performance indexes such as device carrier distribution and current-voltage relation are evaluated. In the process of solving the transport equation of the density gradient model in a full-coupling manner, the control volume is determined by the mass center of the grid, and the carrier current of the control volume is calculated by adopting vector interpolation, so that errors generated by using obtuse triangle grid simulation in a finite volume method are avoided; the method is a method with low grid quality dependence and has important application value in the field of advanced nano semiconductor devices.

Description

Technical Field

[0001] The invention belongs to the field of quantum effect modeling and simulation of advanced nano-semiconductor devices, and relates to a density gradient quantum corrected carrier transport simulation method for reducing grid quality dependence. Background Art

[0002] As transistor size decreases, the impact of quantum confinement effect on carrier transport simulation becomes more and more significant. For example, in FinFET and GAA nanosheets, quantum confinement effect will lead to deactivation of dopants and increase of band gap, thus affecting the characteristics of the device. However, the classic drift-diffusion model cannot reflect the impact of quantum confinement effect on carrier transport simulation, resulting in inaccurate simulation results.

[0003] In order to consider the influence of quantum confinement effects on the transport simulation of advanced nanoscale devices, quantum transport simulation methods and quantum correction simulation methods are constantly being developed. Among them, the non-equilibrium Green's function method can rigorously describe quantum transport and is widely used in the simulation of advanced nanodevices. In addition, by making quantum corrections to the classical drift-diffusion model, it can also be applied to the transport simulation of nanoscale devices. For example, in the density gradient model, the influence of the density gradient of the electron gas on its energy is introduced. After calibration, accurate simulation results and fast simulation speed can be taken into account in the simulation of high quantum confinement areas, so it is adopted by many TCAD software.

[0004] In recent years, there have been many methods to improve the convergence and accuracy of the density gradient model in carrier transport simulation. In order to solve the convergence problem caused by the exponential correlation between the quantum potential and the quasi-Fermi potential in the density gradient equation, Taylor expansion is adopted to reduce the numerical overflow of the exponential term (A. Wettstein, O. Penzin, and E. Lyumkis, "Integration of the density gradient model into a general purpose devicesimulator," VLSI Design, 2002.). In addition, since the simulation results of the density gradient model in nanodevices are affected by discrete charges, the second-order quantum correction term is introduced into the density gradient equation to analyze the carrier distribution in a narrow area (S. Rhee, D. Kim, K. Kim, S. Choi, BGPark, and YJPark, "Extension of the DG Model to the Second-Order Quantum Correction for Analysis of the Single-Charge Effect in Sub-10-nm MOS Devices," IEEE Journal of the Electron Devices Society, 2020.). In addition, different density gradient equation discretization methods have been proposed to increase the accuracy and convergence of density gradient model simulation, including the exponential fitting scheme based on Scharfetter-Gummel discretization (MG Ancona, "Finite-differences schemes for the density-gradient equations," J. Comput. Electron, 2002) and the second-order differential scheme (AJ Garcia-Loureiro et al., "Implementation of the Density Gradient Quantum Corrections for 3-DSimulations of Multigate Nanoscaled Transistors," IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 2011).

[0005] In particular, the quality of the grid has a significant impact on the simulation results of the finite volume method for solving density gradient models, such as the aspect ratio, minimum / maximum angle and asymmetry of the grid. The finite volume method uses the circumcenter of the grid to determine the control volume. In an obtuse triangle grid, the circumcenter is located outside the obtuse triangle, which leads to an overestimation of the control volume, thereby increasing the simulation error and even producing non-physically meaningful solutions. Based on this, the present invention proposes an improved method to solve the problem of overestimation of the control volume when the finite volume method uses an obtuse triangle grid in solving the density gradient model. The method is applicable to any orthogonal grid and non-orthogonal grid. Summary of the invention

[0006] In view of the shortcomings of the prior art, the present invention provides a density gradient quantum corrected carrier transport simulation method that reduces grid quality dependence.

[0007] The technical solution adopted by the present invention is as follows:

[0008] A density gradient quantum corrected carrier transport simulation method with reduced grid quality dependence includes the following:

[0009] First, meshing is performed according to the device structure.

[0010] Then, the initial values ​​of the entire device structure are guessed, including the initial potential, the initial quasi-Fermi level and the initial quantum potential;

[0011] Then, based on the determined initial value and grid structure, the transport equation of the fully coupled density gradient model is carried out on the device, that is, the Poisson equation, the current continuity equation and the density gradient equation are solved to obtain the potential, carrier distribution and current-voltage characteristic curve of the entire device. In this solution process, the control volume is determined by the grid centroid.

[0012] Furthermore, since the boundary of the control volume is no longer perpendicular to the edge of the grid when the center of mass is used to determine the control volume, vector interpolation in the vector finite element method is used to determine the carrier flow at the grid boundary.

[0013] Furthermore, the solution specifically includes the following:

[0014] (1) Determine the control volume of each grid node by connecting the grid centroid and the midpoint of the grid edge;

[0015] (2) Integrate the current continuity equation and density gradient equation on the control volume of each grid node, and convert the integral into the integral of the carrier flow on the boundary through the divergence theorem;

[0016] (3) The carrier flow on the grid edge is calculated by using the central difference through the potential, quantum potential and electron / hole concentration at the grid vertex. The carrier flow on the boundary of each control volume is further obtained by vector interpolation, thereby obtaining the matrix form of the control equation of the density gradient model.

[0017] (4) The fully coupled Newton-Raphson method is used to solve the matrix form of the density gradient model control equation until convergence, and the carrier distribution and current-voltage characteristic curve are obtained.

[0018] The present invention compares the results of carrier transport simulation using the proposed improved method and the traditional finite volume method on grids of different quality, and further evaluates the effectiveness of the proposed improved algorithm when simulating using low-quality grids.

[0019] Compared with the prior art, the present invention has the following beneficial effects:

[0020] 1. The present invention provides a density gradient quantum corrected carrier transport simulation method that reduces the dependence on grid quality. Compared with the traditional finite volume method, this method can reduce the dependence on grid quality when solving the density gradient quantum corrected drift diffusion model, which has guiding significance for further improving the accuracy and convergence of carrier transport simulation of advanced nano devices, and has important application value in the field of advanced nano semiconductor devices.

[0021] 2. The present invention uses the centroid of the grid instead of the circumcenter to determine the control volume, thereby avoiding the problem of overestimation of the control volume when the finite volume method uses a low-quality grid. In addition, the present invention uses vector interpolation in the vector finite element to determine the exact value of the carrier flow at the grid boundary, thereby improving the convergence and accuracy of the simulation. BRIEF DESCRIPTION OF THE DRAWINGS

[0022] Other features, objectives and advantages of the present invention will become more apparent from a reading of the detailed description of non-limiting embodiments made with reference to the following accompanying drawings.

[0023] Figure 1 The difference between the improved method proposed in the present invention and the finite volume method in terms of control volume;

[0024] Figure 2 A simulation flow chart of a density gradient quantum corrected carrier transport simulation method for reducing grid quality dependence provided by the present invention;

[0025] Figure 3 The present invention provides a device overall structure of a dual-gate metal-oxide-metal field effect transistor;

[0026] Figure 4(a) a high-quality grid map and (b) a low-quality grid map used in the embodiments of the present invention;

[0027] Figure 5 The hole density distribution obtained by simulating the improved method and the finite volume method on high / low quality grids respectively in the embodiment of the present invention;

[0028] Figure 6 : The hole distribution in (a) the carrier transport direction (y=0 nm) and (b) perpendicular to the transport direction (x=0 nm) in the embodiment of the present invention;

[0029] Figure 7 The hole density distribution is obtained by solving the current continuity equation and the density gradient equation in a low-quality grid using the improved method and the finite volume method respectively in the embodiment of the present invention;

[0030] Figure 8 The hole density distribution is obtained by solving the current continuity equation and the density gradient equation in the y-axis direction using the improved method and the finite volume method in the low-mass grid in the embodiment of the present invention.

[0031] Fig. 9 The current-voltage characteristic curves are obtained by solving the current continuity equation and the density gradient equation in a low-quality grid using the improved method and the finite volume method respectively in the embodiments of the present invention.

[0032] Fig.10 The capacitance-voltage characteristic curves are obtained by solving the current continuity equation and the density gradient equation in a low-quality grid using the improved method and the finite volume method respectively in the embodiments of the present invention. DETAILED DESCRIPTION

[0033] The present invention is described in detail below in conjunction with specific embodiments. The following embodiments will help those skilled in the art to further understand the present invention, but are not intended to limit the present invention in any form. It should be noted that, for those of ordinary skill in the art, several changes and improvements can also be made without departing from the concept of the present invention. These all belong to the protection scope of the present invention.

[0034] A typical nano-semiconductor device is selected, and a dual-gate metal-oxide-metal field-effect transistor is taken as an example to comprehensively and detailedly describe the process of solving the density gradient model by the improved method proposed in the present invention. The improved method is not only applicable to two-dimensional triangular grids, but also to most two-dimensional and three-dimensional orthogonal grids and non-orthogonal grids.

[0035] According to a specific embodiment of the present invention, a density gradient quantum corrected carrier transport simulation method for reducing grid quality dependence comprises the following steps:

[0036] Step 1: Guess the initial values ​​of the electric potential, quasi-Fermi potential, and quantum potential equations.

[0037] Step 2: The density gradient model is solved by fully coupling the improved method and the finite volume method, respectively. The initial values ​​of each variable are derived from step 1. First, the control volume of each grid vertex is determined by the grid centroid and the center point of the grid edge. The control equation of the density gradient model is integrated on the control volume of each node, and the divergence theorem is applied to convert the integral into the integral of the carrier flow on the boundary. Then, the central difference is used through the potential, quantum potential and electron / hole concentration at the grid vertex to calculate the carrier flow on the grid edge. The carrier flow on the boundary of each control volume is further obtained by vector interpolation, thereby obtaining the matrix form of the control equation of the density gradient model. The control equations of the solution process are listed in Table 1.

[0038] Step 3: Use the fully coupled Newton-Raphson method to solve the matrix form of the density gradient model control equation; since the nonlinear equations obtained in step 2 are highly nonlinear and cannot be solved directly by matrix inversion, Newton-Raphson iteration is used to solve the nonlinear equations obtained in step 2, calculate the Jacobian matrix and residual matrix of the Poisson equation, current continuity equation and density gradient equation, and continuously iterate until the electric potential, quasi-Fermi potential and quantum potential converge. The model equations are listed in Table 1.

[0039] Step 4: According to step 3, the converged carrier distribution and current-voltage characteristic curve are obtained, and the results of the improved method of the present invention and the finite volume method on high / low quality grids are compared.

[0040] Table 1 Control equations in the model

[0041]

[0042]

[0043] The symbols and terms used in this paper are shown in Table 2.

[0044] Table 2 Symbols and terms used in this paper

[0045]

[0046]

[0047] like Figure 1The finite volume method shown uses the circumcenter to divide the control volume, which will lead to the problem of overestimation of the control volume in the obtuse triangle, thereby introducing errors in the carrier transport simulation. The improved method proposed in the present invention uses the grid centroid to divide the control volume, which solves the problem of overestimation of the control volume in the finite volume method. Since the boundary of the control volume is no longer perpendicular to the edge of the grid when the grid is divided using the centroid, it is necessary to use vector interpolation in the vector finite element to calculate the carrier flow at the boundary of the control volume.

[0048] like Figure 2 As shown, a flow chart of a density gradient quantum corrected carrier transport simulation method that reduces grid quality dependence is provided. First, the grid is divided according to the device structure; then the Newton method is used to fully couple the Poisson equation (Equation 1), the current continuity equation (Equation 2) and the density gradient equation (Equation 3); finally, the carrier distribution and the current-voltage characteristic curve are obtained based on the converged results.

[0049] like Figure 3 As shown in Figure 3, a dual-gate metal-oxide-metal field-effect transistor is used as an example to demonstrate the superiority of the improved algorithm. The relevant parameters are listed in Table 3.

[0050] Table 3 Device related parameters

[0051] symbol parameter value <![CDATA[L channel ]]> Channel length 10nm <![CDATA[L Source ]]> Source length 10nm <![CDATA[L Drain ]]> Drain length 10nm <![CDATA[W channel ]]> Channel width 3nm <![CDATA[T oxide ]]> Gate oxide thickness 1nm

[0052] like Figure 4 Shown is a high / low quality mesh diagram. The high quality mesh is composed of sharp triangles, while the low quality mesh has obtuse triangles introduced in the center. Figure 5 The carrier concentration distribution diagrams obtained by simulation using the improved method and the finite volume method respectively. Figure 6 is the hole distribution in the carrier transport direction (a) and perpendicular to the transport direction (b). It can be seen that in low-quality grids, the improved method proposed in the present invention can still maintain high accuracy, while the simulation results using the finite volume method produce obvious errors. Figure 7 The electron density distribution is obtained by solving the current continuity equation and the density gradient equation in a low-quality grid using the improved method and the finite volume method respectively. The specific method description is listed in Table 4.

[0053] Table 4 Methods for solving density gradient model

[0054] Method Abbreviation Current Continuity Equation Density Gradient Equation Method I Finite Volume Method Finite Volume Method Method II Finite Volume Method Improved Algorithm Method III Improved Algorithm Finite Volume Method Method IV Improved Algorithm Improved Algorithm

[0055] Figure 8 , Fig. 9 and Fig.10The hole density distribution in the y-axis direction, the current-voltage characteristic curve, and the capacitance-voltage characteristic curve are obtained by solving the current continuity equation and the density gradient equation in a low-quality grid using the improved method and the finite volume method, respectively.

[0056] Combined with the density gradient quantum corrected drift diffusion model carrier transport simulation results, it can be seen that in the carrier transport simulation using the density gradient model, the finite volume method will overestimate the carrier concentration and current. This is because the finite volume method overestimates the control volume, resulting in errors in the calculation of the quantum potential and the metal contact boundary, which further affects the calculation results of the carrier concentration and current. Therefore, in low-quality grids, compared with the simulation results of the density gradient model solved using the finite volume method, the improved method proposed in the present invention has a very significant optimization in accuracy. An improved method for solving quantum corrected drift diffusion transport based on vector finite elements provided by the present invention has important application value in the field of advanced nano-semiconductor devices.

[0057] The above describes the specific embodiments of the present invention. It should be understood that the present invention is not limited to the above specific embodiments, and those skilled in the art can make various changes or modifications within the scope of the claims, which does not affect the essence of the present invention. In the absence of conflict, the embodiments of the present application and the features in the embodiments can be combined with each other arbitrarily.

Claims

1. A density gradient quantum corrected carrier transport simulation method with reduced grid quality dependence, characterized in that: include: Meshing the device structure; Analyze the device structure and determine the initial values, including the initial potential, initial quasi-Fermi level, and initial quantum potential; Based on the initial value, the transport equation of the density gradient model is fully coupled to the segmented device, and the control volume is determined by the centroid of the grid during the solution process.

2. The density gradient quantum corrected carrier transport simulation method with reduced grid quality dependence according to claim 1, characterized in that: Vector interpolation in vector finite element is used in the solution process to calculate the carrier flow at the boundary of the control volume.

3. The density gradient quantum corrected carrier transport simulation method with reduced grid quality dependence according to claim 1, characterized in that: The fully coupled solution of the transport equation of the density gradient model for the segmented device specifically includes the following: (1) Determine the control volume of each grid node by connecting the grid centroid and the midpoint of the grid edge; (2) Integrate the current continuity equation and density gradient equation on the control volume of each grid node, and convert the integral into the integral of the carrier flow on the boundary through the divergence theorem; (3) The carrier flow on the grid edge is calculated by using the central difference through the potential, quantum potential and electron / hole concentration at the grid vertex. The carrier flow on the boundary of each control volume is further obtained by vector interpolation, thereby obtaining the matrix form of the control equation of the density gradient model. (4) The fully coupled Newton-Raphson method is used to solve the matrix form of the density gradient model control equation until convergence, and the carrier distribution and current-voltage characteristic curve are obtained.

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