A neural network-based multi-subband boltzmann transport equation solving method

Abstract

The application belongs to the field of electronic design automation, and discloses a solving method of a multi-subband Boltzmann transport equation based on a neural network. First, a Boltzmann-Schrodinger-Poisson system is established for a transistor electron transport problem in nanometer scale; a grid division is performed on a simulation device in time, position space and wave vector space; a training data set is generated by self-consistently solving the multi-subband Boltzmann transport equation through a deterministic method; the Boltzmann transport equation is rewritten into the form of a neural partial differential equation to embed the neural network therein, so as to form a neural Boltzmann equation; a difference between an equilibrium electron distribution function and a current electron distribution function is taken as an input of the neural network, and an output is taken as an approximated part of a right end scattering term of the neural Boltzmann equation; the neural network is trained based on a time series modeling strategy; and the trained neural network can make a solution of the neural Boltzmann transport equation as an accurate approximate solution of the electron distribution, so as to greatly improve the calculation efficiency while ensuring the accuracy.

Description

Technical Field

[0001] This invention relates to a method for solving the Boltzmann transport equation for channel electron distribution in nanoscale transistors. By embedding a neural network method into the solution of the differential equation, it can efficiently and accurately calculate the electron transport problem of nanoscale transistors such as FinFET, GAAFET, and MBCFET, and belongs to the field of electronic design automation. Background Technology

[0002] As semiconductor dimensions enter the deep submicron range, short-channel effects alter the threshold voltage and electron drift characteristics within the device channel. To overcome these effects, nanoscale transistors such as FinFETs, Gate-All-Around FETs (GAAFETs), and Multi-Bridge-Channel FETs (MBCFETs) have emerged. These transistors evolved from traditional planar field-effect transistors, using thin silicon (Si) channels and mitigating short-channel effects by enhancing the contact surface between the gate and the channel. Semiclassical transport equations can still be used for devices with channel lengths greater than the electron mean free path in the transport direction. However, as the silicon bulk thickness decreases, quantum confinement leads to energy quantization, the impact of which on electron transport cannot be ignored. Currently, fully quantum transport methods, such as non-equilibrium Green's function method, density matrix method, and Wigner transport equation, are not yet mature enough for devices with channel lengths greater than the electron mean free path. The method based on the Boltzmann Transport Equation (BTE) for multiple subbands represents a compromise between the fully quantum transport method and the semiclassical transport method, because it takes into account the quantization effect caused by confinement, while semiclassically handling the transport in each subband.

[0003] The Boltzmann transport equation is an evolution equation of a time-dependent distribution function. While Monte Carlo simulations, statistical methods, and many deterministic solvers for Boltzmann-Poisson (BTE) systems exist, all methods have limitations in accuracy or efficiency, and few are universally accurate over a wide range of lengths. Furthermore, because the microscopic scattering process couples the distribution functions in different subbands and energies, solving BTEs requires significant computational resources, and 3D modeling of nanoscale transistors such as FinFETs, GAAFETs, and MBCFETs typically demands even more computational resources.

[0004] Neural networks possess strong adaptability, learning ability, fault tolerance, and nonlinear mapping capabilities. Parameters are obtained through network training, making them suitable for solving complex nonlinear problems with complex causal relationships. Recently, methods attempting to combine the advantages of differential equations and neural networks have emerged, such as Physics-Informed Neural Networks (PINNs), which directly incorporate the structure of differential equations into the cost function and train solutions based on the neural network. However, the Boltzmann transport equation is an integrative differential equation, with its right-hand side being a multiple integral in phase space. Due to the lack of a directly differentiable structure, this convolutional scattering operator poses significant challenges to the application of PINNs. Therefore, for nanoscale transistors such as FinFET, GAAFET, and MBCFET, this invention proposes a method that integrates the profound physical insights of multi-subband Boltzmann transport equation modeling with the ideal computational efficiency of neural networks by embedding a neural network into the Boltzmann transport equation. This method then utilizes neural network technology to solve the Boltzmann transport equation, achieving both high-precision calculation results and high computational efficiency. Summary of the Invention

[0005] The purpose of this invention is to use neural network technology to accelerate the calculation of the Boltzmann transport equation in the multiple subband, thereby obtaining the electron distribution in the multiple subband of nanoscale transistors such as FinFET, GAAFET, and MBCFET.

[0006] The technical solution of this invention is: a method for solving the multi-subband Boltzmann transport equation based on neural networks, the steps of which are as follows:

[0007] (1) A Boltzmann-Schrödinger-Poisson system was established to address the electron transport problem in nanoscale transistors such as FinFET, GAAFET, and MBCFET;

[0008] First, the symbols used are defined as follows:

[0009] r = (x, y, z) represents the position vector of the electron in 3Dr-space;

[0010] k=(k x ,k y ,k z ) represents the wave vector of the electron in 3Dk-space;

[0011] v represents the subband number;

[0012] N v Indicates the total number of sub-bands;

[0013] m x,v m y,v m z,vThese represent the effective mass of the electron in the subband v in the x, y, and z directions, respectively.

[0014] V represents electric potential;

[0015] f v This is the electron distribution function on subband v;

[0016] v x,v This represents the velocity of the electrons in the subband v in the x direction;

[0017] This represents the eigenfunction on subband v;

[0018] E v This represents the eigenenergy on subband v;

[0019] This represents the force exerted on an electron in the subband v along the x-direction;

[0020] Q v Represents the scattering term;

[0021] This represents the electron density on subband v;

[0022] n = ∑ v n v Indicates the total electron density;

[0023] (h is Planck's constant);

[0024] ∈ represents the dielectric constant of the material;

[0025] q e It is a positron.

[0026] The transistor type for simulation calculation is determined, and a Boltzmann-Schrödinger-Poisson system is established based on its structural characteristics and the electron transport problem, as follows:

[0027] ① Model the channel electron transport problem of a FinFET with a high aspect ratio channel, as shown in Figure 2(a). A Cartesian coordinate system is established for the source, drain, and channel regions as shown in Figure 2(d). Electron transport in the x-direction (from source to drain) is considered, and the 1D quantum confinement effect in the y-direction is considered at each x-slice (channel cross-section). The following 1D Boltzmann-1D Schrödinger-2D Poisson system is established to self-consistently solve its electron transport problem:

[0028] The 1D multi-subband semiclassical BTE with the transport direction in the x direction is:

[0029]

[0030] The 1D Schrödinger equation for each x-slice (channel section) is:

[0031]

[0032] The corresponding 2D Poisson equation is:

[0033]

[0034] ② Model the channel electron transport problem of the nanopillar-type GAAFET shown in Figure 2(b). A Cartesian coordinate system is established for the source, drain, and channel regions as shown in Figure 2(e). Considering electron transport in the x-direction, and the 2D quantum confinement effect in the y and z directions at each x-slice, the following 1D Boltzmann-2D Schrödinger-3D Poisson system is established to self-consistently solve its electron transport problem:

[0035] The 1D multi-subband semiclassical BTE in the x-direction of transport is:

[0036]

[0037] The 2D Schrödinger equation for each x-slice (channel section) is:

[0038]

[0039] The corresponding 3D Poisson equation is:

[0040]

[0041] ③ Model the channel electron transport problem of the nanosheet-type MBCFET shown in Figure 2(c). A Cartesian coordinate system is established for the source, drain, and channel regions as shown in Figure 2(d). Considering electron transport in the x-direction and the 1D quantum confinement effect in the z-direction at each x-slice, the following 1D Boltzmann-1D Schrödinger-2D Poisson system is established to self-consistently solve its electron transport problem:

[0042] The 1D multi-subband semiclassical BTE in the x-direction of transport is:

[0043]

[0044] The 2D Schrödinger equation for each x-slice (channel section) is:

[0045]

[0046] The corresponding 2D Poisson equation is:

[0047]

[0048] (2) Perform grid division in time, location space and wave vector space;

[0049] The transistor selected for simulation in step (1) is meshed in time, location space, and wave vector space. Let the source length be L. s The drain length is L d The length of the channel is L g Width is W, height is H, and oxide layer thickness is T. ox Considering electron transport in the x-direction, let the intervals Δt, Δx = Δy = Δz = Δr, and Δk be given. x =Δk is divided into the following grid points:

[0050] {(t i ,x j1 ,y j2 ,z j3 ,k k )}∈Ω,

[0051] Ω=[0,T]×[0,L s +L g +L d ]×[-T ox W+T ox ]×[0,H+T ox ]×[-K,K],

[0052] 0≤i≤N t ,0≤j1≤N x ,0≤j2≤N y ,0≤j3≤N z ,0≤i≤N k In the formula,

[0053] (3) Use a deterministic solver to solve the time-varying multi-subband Boltzmann transport equation to generate a training dataset;

[0054] ① Considering the scattering mechanism of the crystal lattice, determine the scattering term Q in the Boltzmann transport equation. v Optionally, consider three types of lattice-automatic scattering: intravalley acoustic phonon-electron scattering, intravalley optical phonon-electron scattering, and intervalley optical phonon-electron scattering. The right-hand side Q of the Boltzmann transport equation... v It can be written as the sum of different scattering terms:

[0055]

[0056] In the formula, the exponent χ represents different types of scattering. One type of scattering... It can be written as:

[0057]

[0058] In the formula, Because it has energy E χ The scattering rate of the electronic transition from state (k′,i′) to state (k,i) caused by the absorption of phonons. Because the energy is E χ The scattering rate of the electron transition from state (k,i) to state (k′,i′) caused by phonon emission.

[0059] ② After determining the scattering term of the transistor selected in step (1), the corresponding Boltzmann-Schrödinger-Poisson system is self-consistently solved using the deterministic solver of the Boltzmann transport equation, resulting in the grid points (x) divided in step (2). j1 ,k k Discrete electron distribution time series on sub-band v {f v (t i )} as a reference value f ref This constitutes the training dataset.

[0060] like Figure 3 As shown, the process of generating the training dataset through self-consistent solution of the Boltzmann-Schrödinger-Poisson system is as follows:

[0061] Step 1: Solve the Poisson equation to obtain time t. i The potential V under i ;

[0062] Step 2: Solve the Schrödinger equation at each x-slice to obtain time t. i Sub-intrinsic energy and eigenfunctions

[0063] Step 3: Solve the time-varying Boltzmann equations for the many-subzones to obtain time t. i+1 The electron distribution function f under the following conditions v (t i+1 ), and thus obtain the electron density n i+1 ;

[0064] Step 4: Return to the solution time t from Step 1 i+1 The potential V under i+1 If the potential converges or t i+1 If the value is greater than T, the loop ends; otherwise, continue with steps two and three.

[0065] (4) Construct the neural Boltzmann transport equation. Rewrite the Boltzmann transport equation into the form of a neural partial differential equation, embed the neural network into the Boltzmann transport equation, and determine the structure of the neural network.

[0066] ① For the transistor selected in step (1) for simulation calculation, the neural network is embedded into the Boltzmann transport equation by rewriting its Boltzmann transport equation into the form of a neural partial differential equation, that is, the scattering term Q on the right side of the equation is... v The solution is divided into a mechanical part and a part to be approximated. The scattering term in the simplified relaxation time approximation model of the Boltzmann transport equation is taken as the mechanical part, while the difference is left as the part to be approximated by a neural network. Using a simplified relaxation approximation model avoids the complex calculation of multiple integrals of the scattering term, effectively improving computational efficiency. Furthermore, using a neural network to approximate the difference compensates for the roughness of the relaxation model, ensuring the accuracy of the solution.

[0067] The simplified relaxation time approximation model is as follows:

[0068]

[0069] In the formula, f + Let τ be the equilibrium electron distribution function. e This is the electronic relaxation time.

[0070] The Boltzmann transport equation embedded in a neural network can be written in the form of a neural partial differential equation as follows:

[0071]

[0072] In the formula, NN θ For a specific neural network, f + -f v This is the input to the neural network.

[0073] ② Select the neural network type and determine the network structure. Optionally, use NN... θ For example Figure 4 The following explanation uses a multi-input, multi-output multilayer perceptron neural network as an example. Its input is the current time t of all sub-bands. i Each grid point (x j1 ,k k The difference f between the equilibrium electron distribution function at position ) and the current electron distribution function + -f v (t i ,x j1 ,k k Therefore, the number of input layer neurons obtained by the grid partitioning in step (2) is N. v ×N x ×N k The number of neurons in the output layer is the same as that in the input layer, which is N. v ×N x ×N k The number of neurons in the hidden layer is N. v ×N x×N k ×M (M≥1), layers are connected by a weight matrix W. The input layer is defined as layer 0, the number of hidden layers is N-1, and the output layer is layer N. The input of layer l (1≤l≤N) is defined as a. (l-1) The weight between layer l-1 and layer l is W. (l) The threshold between layer l-1 and layer l is b. (l) Therefore, the output of the neural network is:

[0074]

[0075] layer l (a (l-1) ) = W (l) ×a (l-1) +b (l) ,

[0076] In the formula, This represents the activation function, which can be such as sigmoid, tanh, etc.

[0077] ③ The solution method for the neuro-Boltzmann transport equation is as follows:

[0078] Step 1: Combining the Schrödinger-Poisson equation corresponding to the selected transistor in step (1), a coarse electron distribution f is obtained by self-consistently solving the relaxation approximation model using a deterministic method. v ;

[0079] Step 2: Then calculate the difference f + -f v Substituting into the neural network yields the output NN θ (f + -f v );

[0080] Step 3: Transfer NN θ (f + -f v Substituting the neural Boltzmann transport equation into the Schrödinger-Poisson equation corresponding to the selected transistor, the neural Boltzmann transport equation is solved self-consistently using a deterministic method to obtain the electron distribution f. θ .

[0081] (5) Train the neural network constructed in step (4) based on the time series modeling strategy;

[0082] ① The output value f of the neural network is measured by calculating the cost function C(θ). θ Compared with reference value f ref The error between the predicted and actual values ​​is used for backpropagation, and the weights and thresholds in the network are continuously adjusted using gradient descent to gradually reduce the gap between the predicted and actual values. Optionally, the cost function C(θ) can be defined as:

[0083]

[0084] Alternatively, the cost function C(θ) can be defined as

[0085]

[0086] ② Considering that the Boltzmann transport equation is time-varying and the training dataset has time series characteristics, a time series-based modeling strategy is used to train the neural network established in step (4). The steps are as follows:

[0087] Step 1: Initialize the neural network parameters (weights W and threshold b), and set the convergence accuracy ε of the cost function;

[0088] Step 2: Initialize the time controller t i (i = 0);

[0089] Step 3: Using t i The electron distribution at time t is obtained by solving the relaxation time approximation model. i+1 Coarse electron distribution at time f i+1 ;

[0090] Step 4: Calculate the difference f + -f i+1 As input to the neural network, the output NN is obtained. θ (f + -f i+1 );

[0091] Step 5: NN θ (f + -f i+1 Substituting into the neuro-Boltzmann transport equation, we obtain t obtained from the solution. i+1 Electron distribution at time

[0092] Step 6: If i+1 <N t Then use t i+1 Electron distribution at time Return to step three; otherwise, proceed to the next step.

[0093] Step 7: Calculate the cost function C(θ). If C(θ) < ε, training ends; otherwise, update the network parameters and proceed to step 2.

[0094] (6) Change the initial conditions and use the trained neural network to solve the neural Boltzmann equation constructed in step (4) in a self-consistent manner to obtain the predicted value of the electron distribution at the grid point at each time step.

[0095] Compared with the prior art, the advantages of the present invention are as follows:

[0096] This invention applies neural network technology to solving the multi-subband Boltzmann transport equation for nanoscale transistors such as FinFET, GAAFET, and MBCFET. The scattering term on the right-hand side of the Boltzmann transport equation is divided into a mechanical part and an approximation part. The mechanical part adopts a simplified relaxation approximation model, which avoids the complex calculation of multiple integrals and can effectively improve the computational efficiency. The neural network is used to approximate the difference, which can compensate for the roughness of the relaxation model and ensure the accuracy of the solution. Within the allowable error range, the solution speed is further accelerated, which has the advantages of high precision and high efficiency. Attached Figure Description

[0097] Figure 1 This is a flowchart of the operation of the present invention;

[0098] Figure 2 shows schematic diagrams and models of three types of nanoscale transistors;

[0099] Figure 3 This is a schematic diagram of a self-consistent solver for the Boltzmann-Schrödinger-Poisson system;

[0100] Figure 4 This is a schematic diagram of a multilayer perceptron neural network;

[0101] Figure 5 This is a schematic diagram of a neural network training strategy based on time series modeling. Detailed Implementation

[0102] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0103] This invention relates to a method for solving the Boltzmann transport equation of the channel electron distribution in nanoscale transistors. The method embeds a neural network into the solution of the differential equation, with the aim of efficiently and accurately calculating the electron transport problem of nanoscale transistors such as FinFET, GAAFET, and MBCFET.

[0104] like Figure 1As shown: First, a Boltzmann-Schrödinger-Poisson system is established for the nanoscale transistor electron transport problem; the simulation device is meshed in time, position space, and wave vector space; a training dataset is generated by self-consistently solving the multi-subband Boltzmann transport equation using a deterministic method; the Boltzmann transport equation is rewritten into a neural partial differential equation and a neural network is embedded in it to form the neural Boltzmann equation; the difference between the equilibrium electron distribution function and the current electron distribution function is used as the input of the neural network, and the output is used as the approximation part of the scattering term on the right side of the neural Boltzmann equation; the neural network is trained based on a time series modeling strategy. The trained neural network can make the solution of the neural Boltzmann transport equation an accurate approximation of the electron distribution, greatly improving computational efficiency while ensuring accuracy.

[0105] 1. For the FinFET with a high aspect ratio channel as shown in Figure 2(a), neural network technology is used to accelerate the solution of its multi-subband Boltzmann equation. The specific implementation steps are as follows:

[0106] ① Establish a Boltzmann-Schrödinger-Poisson system for the electronic transport problem;

[0107] A Cartesian coordinate system is established for the source, drain, and channel regions of the FinFET with a high aspect ratio channel shown in Figure 2(a), as shown in Figure 2(d). Considering the electron transport in the x-direction (from the source to the drain), and considering the 1D quantum confinement effect in the y-direction at each x-slice (channel cross-section), the following 1D Boltzmann-1D Schrödinger-2D Poisson system is established to self-consistently solve its electron transport problem:

[0108] The time-varying 1D multi-subband semiclassical BTE with the transport direction in the x-direction is:

[0109]

[0110] The 1D Schrödinger equation for each x-slice (channel section) is:

[0111]

[0112] The corresponding 2D Poisson equation is:

[0113]

[0114] ② Perform grid division in time, location space, and wave vector space;

[0115] As shown in Figure 2(d), let the source length be L. s The drain length is L d The length of the channel is L g Width is W, height is H, and oxide layer thickness is T. oxConsidering electron transport in the x-direction, let the intervals Δt, Δx = Δy = Δz = Δr, and Δk be given. x =Δk is divided into the following grid points:

[0116] {(t i ,x j1 ,y j2 ,z j3 ,k k )}∈Ω,

[0117] Ω=[0,T]×[0,L s +L g +L d ]×[-T ox W+T ox ]×[0,H+T ox ]×[-K,K],

[0118] 0≤i≤N t ,0≤j1≤N x ,0≤j2≤N y ,0≤j3≤N z ,0≤i≤N k ,

[0119] In the formula,

[0120] ③ Use a deterministic solver to solve the time-varying multi-subband Boltzmann transport equations to generate a training dataset;

[0121] Considering the scattering mechanism of the crystal lattice, determine the scattering term Q in the Boltzmann transport equation. v Optionally, consider three types of lattice-automatic scattering: intravalley acoustic phonon-electron scattering, intravalley optical phonon-electron scattering, and intervalley optical phonon-electron scattering. The right-hand side Q of the Boltzmann transport equation... v It can be written as the sum of different scattering terms:

[0122]

[0123] In the formula, the exponent χ represents different types of scattering. One type of scattering... It can be written as:

[0124]

[0125] In the formula, Because it has energy E χ The scattering rate of the electronic transition from state (k′,i′) to state (k,i) caused by the absorption of phonons. Because the energy is E χThe scattering rate of the electron transition from state (k,i) to state (k′,i′) caused by phonon emission.

[0126] After determining the scattering term, the corresponding Boltzmann-Schrödinger-Poisson system is self-consistently solved using a deterministic solver for the Boltzmann transport equation, yielding the grid points (x) defined in step ②. j ,k k Discrete electron distribution time series on sub-band v {f v (t i )} as a reference value f ref This constitutes the training dataset. For example... Figure 3 As shown, the process of generating the training dataset through self-consistent solution of the Boltzmann-Schrödinger-Poisson system is as follows:

[0127] Step 1: Solve the Poisson equation to obtain time t. i The potential V under i ;

[0128] Step 2: Solve the Schrödinger equation at each x-slice to obtain time t. i Sub-intrinsic energy and eigenfunctions

[0129] Step 3: Solve the time-varying Boltzmann equations for the many-subzones to obtain time t. i+1 The electron distribution function f under the following conditions v (t i+1 ), and thus obtain the electron density n i+1 ;

[0130] Step 4: Return to the solution time t from Step 1 i+1 The potential V under i+1 If the potential converges or t i+1 If the value is greater than T, the loop ends; otherwise, continue with steps two and three.

[0131] ④ Construct the neural Boltzmann transport equation. Rewrite the Boltzmann transport equation into the form of neural partial differential equations, embed the neural network into the Boltzmann transport equation, and determine the structure of the neural network.

[0132] By rewriting the Boltzmann transport equation into the form of a neural partial differential equation, a neural network is embedded into the Boltzmann transport equation, that is, the scattering term Q on the right-hand side of the equation... v It is divided into a mechanical part and a part to be approximated. The scattering term in the simplified relaxation time approximation model of the Boltzmann transport equation is taken as the mechanical part, and the difference is left as the part to be approximated by the neural network.

[0133] The simplified relaxation time approximation model is as follows:

[0134]

[0135] In the formula, f + Let τ be the equilibrium electron distribution function. e f is the electronic relaxation time. + Alternatively, the Fermi-Dirac distribution function or the Maxwell distribution function can be used.

[0136] The Boltzmann transport equation embedded in a neural network can be written in the form of a neural partial differential equation as follows:

[0137]

[0138] In the formula, NN θ For a specific neural network, f + -f v This is the input to the neural network.

[0139] Select the neural network type and determine the network structure. Optionally, use a neural network... θ For example Figure 4 The following explanation uses a multi-input, multi-output multilayer perceptron neural network as an example. Its input is the current time t of all sub-bands. i Each grid point (x j1 ,k k The difference f between the equilibrium electron distribution function at position ) and the current electron distribution function + -f v (t i ,x j1 ,k k Therefore, the number of input layer neurons obtained from the grid partitioning in step ② is N. v ×N x ×N k The number of neurons in the output layer is the same as that in the input layer, which is N. v ×N x ×N k The number of neurons in the hidden layer is N. v ×N x ×N k ×M (M≥1), layers are connected by a weight matrix W. The input layer is defined as layer 0, the number of hidden layers is N-1, and the output layer is layer N. The input of layer l (1≤l≤N) is defined as a. (l-1) The weight between layer l-1 and layer l is W. (l) The threshold between layer l-1 and layer l is b. (l) Therefore, the output of the neural network is:

[0140]

[0141] layerl (a (l-1) ) = W (l) ×a (l-1) +b (l) ,

[0142] In the formula, This represents the activation function, which can be such as sigmoid, tanh, etc.

[0143] ⑤ The neural network constructed in step ④ is trained based on the time series modeling strategy;

[0144] The output value f of the neural network is measured by calculating the cost function C(θ). θ Compared with reference value f ref The error between the predicted and actual values ​​is used for backpropagation, and the weights and thresholds in the network are continuously adjusted using gradient descent to gradually reduce the gap between the predicted and actual values. Optionally, the cost function C(θ) can be defined as:

[0145]

[0146] Alternatively, the cost function C(θ) can be defined as

[0147]

[0148] Considering that the Boltzmann transport equation is time-varying and the training dataset has time series characteristics, a time series-based modeling strategy is used to train the neural network established in step (4), and the steps are as follows:

[0149] Step 1: Initialize the neural network parameters (weights W and threshold b), and set the convergence accuracy ε of the cost function;

[0150] Step 2: Initialize the time controller t i (i = 0);

[0151] Step 3: Using t i The electron distribution at time t is obtained by solving the relaxation time approximation model using a deterministic method. i+1 Coarse electron distribution at time f i+1 ;

[0152] Step 4: Calculate the difference f + -f i+1 As input to the neural network, the output NNθ(f) is obtained. + -fi +1 );

[0153] Step 5: NNNθ(f) + -fi +1) is substituted into the neural Boltzmann transport equation, and the deterministic method is used to self-consistently solve the neural Boltzmann transport equation to obtain ti +1 electron distribution at the moment

[0154] Step 6: If i+1<Nt, use ti +1 the electron distribution at the moment to return to Step 3; otherwise, proceed to the next step;

[0155] Step 7: Calculate the cost function C(θ). If C(θ)<ε, the training ends; otherwise, update the network parameters and go to Step 2.

[0156] ⑥ Change the initial conditions, and use the neural network trained in step ⑤ to self-consistently solve the neural Boltzmann equation constructed in step ④ to obtain the predicted value of the electron distribution at the grid point at each time step.

[0157] 2. For the nanopillar-type GAAFET shown in Figure 2(b), neural network technology is used to accelerate the solution of its multi-subband Boltzmann equation. The specific implementation steps are as follows:

[0158] ① Establish a Boltzmann-Schrödinger-Poisson system for the electron transport problem;

[0159] A rectangular coordinate system is established for the source, drain and channel regions of the GAAFET with the nanopillar-type channel shown in Figure 2(b), as shown in Figure 2(e). Considering the electron transport in the x direction (from the source to the drain), and considering the 2D quantum confinement effect in the y direction and z direction at each x slice (channel cross-section), the following 1D Boltzmann-2D Schrödinger-3D Poisson system is established to self-consistently solve its electron transport problem:

[0160] The 1D multi-subband semiclassical BTE in the transport direction (x direction) is:

[0161]

[0162] The 2D Schrödinger equation at each x slice (channel cross-section) is:

[0163]

[0164] The corresponding 3D Poisson equation is:

[0165]

[0166] ② Perform grid division in time, position space and wave vector space;

[0167] As shown in Figure 2(b), a spatial rectangular coordinate system is established. Let the source length be Ls, the drain length be Ld, the channel length be Lg, the width be W, the height be H, and the oxide layer thickness be Tox. Considering electron transport in the x-direction, the grid is divided with intervals Δt, Δx = Δy = Δz = Δr, and Δkx = Δk as follows:

[0168] {(ti,xj1,yj2,zj3,kk)}∈Ω,

[0169] Ω=[0,T]×[0,Ls+Lg+Ld]×[-Tox,W+Tox]×[0,H+Tox]×[-K,K],

[0170] 0≤i≤Nt, 0≤j1≤Nx, 0≤j2≤Ny, 0≤j3≤Nz, 0≤i≤Nk,

[0171] In the formula,

[0172] ③ Use a deterministic solver to solve the time-varying multi-subband Boltzmann transport equations to generate a training dataset;

[0173] Considering the scattering mechanism of the crystal lattice, determine the scattering term Q in the Boltzmann transport equation. v Optionally, consider three types of lattice-automatic scattering: intravalley acoustic phonon-electron scattering, intravalley optical phonon-electron scattering, and intervalley optical phonon-electron scattering. The right-hand side Q of the Boltzmann transport equation... v It can be written as the sum of different scattering terms:

[0174]

[0175] In the formula, the exponent χ represents different types of scattering. One type of scattering... It can be written as:

[0176]

[0177] In the formula, Because it has energy E χ The scattering rate of the electronic transition from state (k′,i′) to state (k,i) caused by the absorption of phonons. Because the energy is E χ The scattering rate of the electron transition from state (k,i) to state (k′,i′) caused by phonon emission.

[0178] After determining the scattering term, the corresponding Boltzmann-Schrödinger-Poisson system is self-consistently solved using a deterministic solver for the Boltzmann transport equation, yielding the grid points (x) defined in step ②. j ,k kDiscrete electron distribution time series on sub-band v {f v (t i )} as a reference value f ref This constitutes the training dataset. For example... Figure 3 As shown, the process of generating the training dataset through self-consistent solution of the Boltzmann-Schrödinger-Poisson system is as follows:

[0179] Step 1: Solve the Poisson equation to obtain time t. i The potential V under i ;

[0180] Step 2: Solve the Schrödinger equation at each x-slice to obtain time t. i Sub-intrinsic energy and eigenfunctions

[0181] Step 3: Solve the time-varying Boltzmann equations for the many-subzones to obtain time t. i+1 The electron distribution function f under the following conditions v (t i+1 ), and thus obtain the electron density n i+1 ;

[0182] Step 4: Return to the solution time t from Step 1 i+1 The potential V under i+1 If the potential converges or t i+1 If the value is greater than T, the loop ends; otherwise, continue with steps two and three.

[0183] ④ Construct the neural Boltzmann transport equation. Rewrite the Boltzmann transport equation into the form of neural partial differential equations, embed the neural network into the Boltzmann transport equation, and determine the structure of the neural network.

[0184] By rewriting the Boltzmann transport equation into the form of a neural partial differential equation, a neural network is embedded into the Boltzmann transport equation, that is, the scattering term Q on the right-hand side of the equation... v It is divided into a mechanical part and a part to be approximated. The scattering term in the simplified relaxation time approximation model of the Boltzmann transport equation is taken as the mechanical part, and the difference is left as the part to be approximated by the neural network.

[0185] The simplified relaxation time approximation model is as follows:

[0186]

[0187] In the formula, f + Let τ be the equilibrium electron distribution function. e f is the electronic relaxation time. + Alternatively, the Fermi-Dirac distribution function or the Maxwell distribution function can be used.

[0188] The Boltzmann transport equation embedded in a neural network can be written in the form of a neural partial differential equation as follows:

[0189]

[0190] In the formula, NN θ For a specific neural network, f + -f v This is the input to the neural network.

[0191] Select the neural network type and determine the network structure. Optionally, use a neural network... θ For example Figure 4 The following explanation uses a multi-input, multi-output multilayer perceptron neural network as an example. Its input is the current time t of all sub-bands. i Each grid point (x j1 ,k k The difference f between the equilibrium electron distribution function at position ) and the current electron distribution function + -f v (t i ,x j1 ,k k Therefore, the number of input layer neurons obtained from the grid partitioning in step ② is N. v ×N x ×N k The number of neurons in the output layer is the same as that in the input layer, which is N. v ×N x ×N k The number of neurons in the hidden layer is N. v ×N k ×N k ×M (M≥1), layers are connected by a weight matrix W. The input layer is defined as layer 0, the number of hidden layers is N-1, and the output layer is layer N. The input of layer l (1≤l≤N) is defined as a. (l-1) The weight between layer l-1 and layer l is W. (l) The threshold between layer l-1 and layer l is b. (l) Therefore, the output of the neural network is:

[0192]

[0193] layer l (a (l-1) ) = W (l) ×a (l-1) +b (l) ,

[0194] In the formula, This represents the activation function, which can be such as sigmoid, tanh, etc.

[0195] ⑤ The neural network constructed in step ④ is trained based on the time series modeling strategy;

[0196] The output value f of the neural network is measured by calculating the cost function C(θ). θ Compared with reference value f ref The error between the predicted and actual values ​​is used for backpropagation, and the weights and thresholds in the network are continuously adjusted using gradient descent to gradually reduce the gap between the predicted and actual values. Optionally, the cost function C(θ) can be defined as:

[0197]

[0198] Alternatively, the cost function C(θ) can be defined as

[0199]

[0200] Considering that the Boltzmann transport equation is time-varying and the training dataset has time-series characteristics, a time-series-based modeling strategy is used to train the neural network established in step ④, and the steps are as follows:

[0201] Step 1: Initialize the neural network parameters (weights W and threshold b), and set the convergence accuracy ε of the cost function;

[0202] Step 2: Initialize the time controller t i (i = 0);

[0203] Step 3: Using t i The electron distribution at time t is obtained by solving the relaxation time approximation model using a deterministic method. i+1 Coarse electron distribution at time f i+1 ;

[0204] Step 4: Calculate the difference f + -f i+1 As input to the neural network, the output NN is obtained. θ (f + -f i+1 );

[0205] Step 5: NN θ (f + -f i+1 Substituting into the neural Boltzmann transport equation, and using a deterministic method to self-consistently solve the neural Boltzmann transport equation, we obtain t. i+1 Electron distribution at time

[0206] Step 6: If i+1 <N t Then use t i+1 Electron distribution at time Return to step three; otherwise, proceed to the next step.

[0207] Step 7: Calculate the cost function C(θ). If C(θ) < ε, training ends; otherwise, update the network parameters and proceed to step 2.

[0208] ⑥ Change the initial conditions, use the neural network trained in step ⑤, and solve the neural Boltzmann equation constructed in step ④ in a self-consistent manner to obtain the predicted value of electron distribution at grid points at each time step.

[0209] 3. For the MBCFET with a nanosheet-type channel as shown in Figure 2(c), neural network technology is used to accelerate the solution of its multi-subband Boltzmann equation. The specific implementation steps are as follows:

[0210] ① Establish a Boltzmann-Schrödinger-Poisson system for the electronic transport problem;

[0211] A Cartesian coordinate system is established for the source, drain, and channel region of the nanosheet-type MBCFET shown in Figure 2(c), as shown in Figure 2(f). Considering electron transport in the x-direction and the 1D quantum confinement effect in the z-direction at each x-slice, the following 1D Boltzmann-1D Schrödinger-2D Poisson system is established to self-consistently solve its electron transport problem:

[0212] The 1D multi-subband semiclassical BTE in the x-direction of transport is:

[0213]

[0214] The 2D Schrödinger equation for each x-slice (channel section) is:

[0215]

[0216] The corresponding 2D Poisson equation is:

[0217]

[0218] ② Perform grid division in time, location space, and wave vector space;

[0219] As shown in Figure 2(f), let the source length be L. s The drain length is L d The length of the channel is L g Width is W, height is H, and oxide layer thickness is T. ox Considering electron transport in the x-direction, let the intervals Δt, Δx = Δy = Δz = Δr, and Δk be given. x =Δk is divided into the following grid points:

[0220] {(t i ,x j1 ,y j2 ,zj3 ,k k )}∈Ω,

[0221] Ω=[0,T]×[0,L s +L g +L d ]×[-T ox W+T ox ]×[0,H+T ox ]×[-K,K],

[0222] 0≤i≤N t ,0≤j1≤N x ,0≤j2≤N y ,0≤j3≤N z ,0≤i≤N k ,

[0223] In the formula,

[0224] ④ Use a deterministic solver to solve the time-varying multi-subband Boltzmann transport equations to generate a training dataset;

[0225] Considering the scattering mechanism of the crystal lattice, determine the scattering term Q in the Boltzmann transport equation. v Optionally, consider three types of lattice-automatic scattering: intravalley acoustic phonon-electron scattering, intravalley optical phonon-electron scattering, and intervalley optical phonon-electron scattering. The right-hand side Q of the Boltzmann transport equation... v It can be written as the sum of different scattering terms:

[0226]

[0227] In the formula, the exponent χ represents different types of scattering. One type of scattering... It can be written as:

[0228]

[0229] In the formula, Because it has energy E χ The scattering rate of the electronic transition from state (k′,i′) to state (k,i) caused by the absorption of phonons. Because the energy is E χ The scattering rate of the electron transition from state (k,i) to state (k′,i′) caused by phonon emission.

[0230] After determining the scattering term, the corresponding Boltzmann-Schrödinger-Poisson system is self-consistently solved using a deterministic solver for the Boltzmann transport equation, yielding the grid points (x) defined in step ②. j ,k kDiscrete electron distribution time series on sub-band v {f v (t i )} as a reference value f ref This constitutes the training dataset. For example... Figure 3 As shown, the process of generating the training dataset through self-consistent solution of the Boltzmann-Schrödinger-Poisson system is as follows:

[0231] Step 1: Solve the Poisson equation to obtain time t. i The potential V under i ;

[0232] Step 2: Solve the Schrödinger equation at each x-slice to obtain time t. i Sub-intrinsic energy and eigenfunctions

[0233] Step 3: Solve the time-varying Boltzmann equations for the many-subzones to obtain time t. i+1 The electron distribution function f under the following conditions v (t i+1 ), and thus obtain the electron density n i+1 ;

[0234] Step 4: Return to the solution time t from Step 1 i+1 The potential V under i+1 If the potential converges or t i+1 If the value is greater than T, the loop ends; otherwise, continue with steps two and three.

[0235] ④ Construct the neural Boltzmann transport equation, rewrite the Boltzmann transport equation into the form of neural partial differential equations, embed the neural network into the Boltzmann transport equation, and determine the structure of the neural network.

[0236] By rewriting the Boltzmann transport equation into the form of a neural partial differential equation, a neural network is embedded into the Boltzmann transport equation, that is, the scattering term Q on the right-hand side of the equation... v It is divided into a mechanical part and a part to be approximated. The scattering term in the simplified relaxation time approximation model of the Boltzmann transport equation is taken as the mechanical part, and the difference is left as the part to be approximated by the neural network.

[0237] The simplified relaxation time approximation model is as follows:

[0238]

[0239] In the formula, f + Let τ be the equilibrium electron distribution function. e f is the electronic relaxation time. + Alternatively, the Fermi-Dirac distribution function or the Maxwell distribution function can be used.

[0240] The Boltzmann transport equation embedded in a neural network can be written in the form of a neural partial differential equation as follows:

[0241]

[0242] In the formula, NN θ For a specific neural network, f + -f v This is the input to the neural network.

[0243] Select the neural network type and determine the network structure. Optionally, use a neural network... θ For example Figure 4 The following explanation uses a multi-input, multi-output multilayer perceptron neural network as an example. Its input is the current time t of all sub-bands. i Each grid point (x j1 ,k k The difference f between the equilibrium electron distribution function at position ) and the current electron distribution function + -f v (t i ,x j1 ,k k Therefore, the number of input layer neurons obtained from the grid partitioning in step ② is N. v ×N x ×N k The number of neurons in the output layer is the same as that in the input layer, which is N. v ×N x ×N k The number of neurons in the hidden layer is N. v ×N x ×N k ×M (M≥1), layers are connected by a weight matrix W. The input layer is defined as layer 0, the number of hidden layers is N-1, and the output layer is layer N. The input of layer l (1≤l≤N) is defined as a. (l-1) The weight between layer l-1 and layer l is W. (l) The threshold between layer l-1 and layer l is b. (l) Therefore, the output of the neural network is:

[0244]

[0245] layer l (a (l-1) ) = W (l) ×a (l-1) +b (l) ,

[0246] In the formula, This represents the activation function, which can be such as sigmoid, tanh, etc.

[0247] ⑤ The neural network constructed in step ④ is trained based on the time series modeling strategy;

[0248] The output value f of the neural network is measured by calculating the cost function C(θ). θ Compared with reference value f ref The error between the predicted and actual values ​​is used for backpropagation, and the weights and thresholds in the network are continuously adjusted using gradient descent to gradually reduce the gap between the predicted and actual values. Optionally, the cost function C(θ) can be defined as:

[0249]

[0250] Alternatively, the cost function C(fref) can be defined as

[0251]

[0252] Considering that the Boltzmann transport equation is time-varying and the training dataset has time-series characteristics, a time-series-based modeling strategy is used to train the neural network established in step ④, and the steps are as follows:

[0253] Step 1: Initialize the neural network parameters (weights W and threshold b), and set the convergence accuracy ε of the cost function;

[0254] Step 2: Initialize the time controller t i (i = 0);

[0255] Step 3: Using t i The electron distribution at time t is obtained by solving the relaxation time approximation model using a deterministic method. i+1 Coarse electron distribution at time f i+1 ;

[0256] Step 4: Calculate the difference f + -f i+1 As input to the neural network, the output NN is obtained. θ (f + -f i+1 );

[0257] Step 5: NN θ (f + -f i+1 Substituting into the neural Boltzmann transport equation, and using a deterministic method to self-consistently solve the neural Boltzmann transport equation, we obtain t. i+1 Electron distribution at time

[0258] Step 6: If i+1 <N t Then use t i+1 Electron distribution at time Return to step three; otherwise, proceed to the next step.

[0259] Step 7: Calculate the cost function C(θ). If C(θ) < ε, training ends; otherwise, update the network parameters and proceed to step 2.

[0260] ⑥ Change the initial conditions, use the neural network trained in step ⑤, and solve the neural Boltzmann equation constructed in step ④ in a self-consistent manner to obtain the predicted value of electron distribution at grid points at each time step.

[0261] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit the scope of protection of the present invention. Referring to the description of these embodiments, those skilled in the art should be able to understand and make relevant modifications or substitutions to the technical solutions of the present invention without departing from the spirit and scope of the present invention.

Claims

1. A method for solving the multi-subband Boltzmann transport equation based on neural networks, characterized in that, Includes the following steps: (1) Establish a Boltzmann-Schrödinger-Poisson system for the electron transport problem of nanoscale transistors; (2) The electron transport problem is meshed in time, location space and wave vector space; (3) Use a deterministic solver to solve the time-varying multi-subband Boltzmann transport equation to generate a training dataset; (4) Construct the neural Boltzmann transport equation, wherein, for the transistor selected in step (1) for simulation calculation, its Boltzmann transport equation is rewritten as a neural partial differential equation, and the scattering term on the right side of the equation is... It is divided into a mechanical part and a part to be approximated; the scattering term in the simplified relaxation time approximation model is taken as the mechanical part, and the difference between the equilibrium electron distribution function and the current electron distribution function is used. As a neural network The input is the neural network output. ( Approximating the part to be approximated, the Boltzmann transport equation embedded in the neural network becomes: In the formula, Let be the equilibrium electron distribution function. For electronic relaxation time, For neural networks; (5) Use time series-based modeling strategies to train the neural network constructed in step (4); (6) Change the initial conditions and use the neural network trained in step (5) to solve the neural Boltzmann transport equation in step (4) in a self-consistent manner to obtain the predicted value of electron distribution at grid points at each time step.

2. The method for solving the multi-subband Boltzmann transport equation based on neural networks as described in claim 1, characterized in that, In step (1), a Boltzmann-Schrödinger-Poisson system is established for nanoscale transistors such as FinFET, GAAFET, and MBCFET regarding the electron transport problem; First, the symbols used are defined as follows: Representing electrons in 3D - A position vector in space; Representing electrons in 3D - Wave vector in space; Indicates the number of subbands; Indicates the total number of sub-bands; These are respectively represented as sub-bands Shanghai Electronics Effective mass in the direction; Represents electric potential; For children The electron distribution function on; Subband Electrons on Velocity in the direction; Subband Upper eigenwave function; Subband The intrinsic energy on; = Subband electron edge The direction of the force; Represents the scattering term; Subband Electron density on; Indicates the total electron density; , It is Planck's constant; The dielectric constant of the material; Positrons; The transistor type for simulation calculation is determined, and a Boltzmann-Schrödinger-Poisson system is established based on its structural characteristics and the electron transport problem, as follows: ① Model the channel electron transport problem of FinFETs with high aspect ratio channels; establish a Cartesian coordinate system for the source, drain, and channel regions, and consider the transport from the source to the drain. Electron transport in the direction of each channel cross-section Consider the section The directional 1D quantum confinement effect leads to the following self-consistent solution of the electron transport problem for the 1D Boltzmann-1D Schrödinger-2D Poisson system: The direction of transport is The 1D multi-subband semiclassical BTE in the direction is: Each channel cross-section The 1D Schrödinger equation at the slice is: The corresponding 2D Poisson equation is: ② Model the channel electron transport problem of the nanopillar-type GAAFET; establish a Cartesian coordinate system for the source, drain, and channel regions, and consider... Electron transport in each direction Consider the section direction and The 2D quantum confinement effect in the direction of electron transport is used to establish a self-consistent solution for the electron transport problem of the 1D Boltzmann-2D Schrödinger-3D Poisson system: Transport direction The 1D multi-subband semiclassical BTE in the direction is: Each channel cross-section The 2D Schrödinger equation at the slice is: The corresponding 3D Poisson equation is: ③ Model the channel electron transport problem of nanosheet-type MBCFET; establish a Cartesian coordinate system for the source, drain, and channel regions, and consider... Electron transport in each direction Consider the section The directional 1D quantum confinement effect leads to the following self-consistent solution of the electron transport problem for the 1D Boltzmann-1D Schrödinger-2D Poisson system: Transport direction The 1D multi-subband semiclassical BTE in the direction is: Each channel cross-section The 2D Schrödinger equation at the slice is: The corresponding 2D Poisson equation is:

3. The method for solving the multi-subband Boltzmann transport equation based on neural networks as described in claim 1, characterized in that, In step (2), the transistor selected for simulation calculation in step (1) is meshed in time, location space, and wave vector space; let the source length be... The drain length is The length of the channel is Width is Height is The oxide layer thickness is ;consider Electron transport in the direction of the interval , , The grid points are divided as follows: In the formula, , , , , .

4. The method for solving the multi-subband Boltzmann transport equation based on neural networks as described in claim 1, characterized in that, In step (3), after determining the scattering term of the transistor selected in step (1), the corresponding Boltzmann-Schrödinger-Poisson system is solved by combining the deterministic solver of the Boltzmann transport equation, and the grid points divided in step (2) are obtained. virgin belt Discrete electron distribution time series As a reference value This constitutes the training dataset; The process of generating the training dataset through self-consistent solution of the Boltzmann-Schrödinger-Poisson system is as follows: Step 1: Solve the Poisson equation to obtain the time interval. The potential under ; Step 2: Solve for each The Schrödinger equation at the slice location yields the time... Sub-intrinsic energy and eigenfunctions ; Step 3: Solve the time-varying Boltzmann equations for the many-subzones to obtain the time intervals. electron distribution function under Thus, the electron density is obtained. ; Step 4: Return to the solution time in Step 1 The potential under If the potential converges or If the condition is met, the loop ends; otherwise, continue with steps two and three.

5. The method for solving the multi-subband Boltzmann transport equation based on neural networks as described in claim 1, characterized in that, In step (4), for the transistor selected in step (1) for simulation calculation, the neural network is embedded into the Boltzmann transport equation by rewriting its Boltzmann transport equation into the form of a neural partial differential equation, that is, the scattering term on the right side of the equation is... It is divided into a mechanical part and a part to be approximated; the scattering term in the simplified relaxation time approximation model of the Boltzmann transport equation is taken as the mechanical part, and the difference is left as the part to be approximated by the neural network. The simplified relaxation time approximation model is as follows: In the formula, Let be the equilibrium electron distribution function. For electronic relaxation time; The Boltzmann transport equation embedded in a neural network can be written in the form of a neural partial differential equation as follows: , In the formula, For specific neural networks, This serves as the input to the neural network; Select the neural network type and determine the network structure; the neural network This is a multi-input, multi-output multilayer perceptron neural network, where the input is the current time step of all sub-bands. Each grid point The difference between the equilibrium electron distribution function and the current electron distribution function at a given point Therefore, the number of input layer neurons obtained from the grid partitioning in step (2) is... The number of neurons in the output layer is the same as that in the input layer. The number of neurons in the hidden layer is ,in 1. Layers are connected via weight matrices. To connect; define the input layer as layer 0, the number of hidden layers as N-1, and the output layer as layer N; define The input of the layer is ,in , Layers and The weights between layers are , Layers and The threshold between layers is Therefore, the output of the neural network is: In the formula, This represents the activation function, which can be either sigmoid or tanh. The solution method for the neuro-Boltzmann transport equation is as follows: Step 1: Combining the Schrödinger-Poisson equation corresponding to the selected transistor in step (1), a coarse electron distribution is obtained by self-consistently solving the relaxation approximation model using a deterministic method. ; Step 2: Then calculate the difference. Inputting into the neural network yields the output. ); Step 3: Substituting the neural Boltzmann transport equation into the equation and combining it with the Schrödinger-Poisson equation corresponding to the selected transistor, the neural Boltzmann transport equation is solved self-consistently using a deterministic method to obtain the electron distribution. .

6. The method for solving the multi-subband Boltzmann transport equation based on neural networks as described in claim 1, characterized in that, In step (5), the cost function is calculated. To measure the output value of a neural network Compared with reference value The error between the predicted and actual values ​​is propagated in reverse, and the weights and thresholds in the network are continuously adjusted using gradient descent, thereby reducing the gap between the predicted and actual values; cost function Defined as: Considering that the Boltzmann transport equation is time-varying and the training dataset has time series characteristics, a time series-based modeling strategy is used to train the neural network established in step (4), and the steps are as follows: Step 1: Initialize the neural network parameters, i.e., the weights. and threshold Set the convergence precision of the cost function ; Step 2: Initialize the time controller and order ; Step 3: Utilize The electron distribution at time t is obtained by solving a self-consistent relaxation time approximation model using a deterministic method. Rough electron distribution at any time ; Step 4: Difference As input to the neural network, the output is obtained. ; Step 5: Substituting the neural Boltzmann transport equation into the equation, and using a deterministic method to self-consistently solve the neural Boltzmann transport equation, we obtain... Electron distribution at time ; Step 6: If Then utilize Electron distribution at time Go back to step three; Otherwise, proceed to the next step; Step 7: Calculate the cost function ;like If the training ends, the network parameters are updated, and the process proceeds to the second step.

7. The method for solving the multi-subband Boltzmann transport equation based on neural networks as described in claim 1, characterized in that, In step (6), the initial conditions are changed, and the neural network trained in step (5) is used to solve the neural Boltzmann transport equation in step (4) in a self-consistent manner to obtain the predicted value of the electron distribution at the grid point at each time step.

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