Adaptive encryption method based on resonant tunneling diode current-voltage simulation

Abstract

The application discloses an adaptive encryption method based on resonance tunneling diode current-voltage simulation, which comprises the following steps: solving a Schrodinger-Poisson model by utilizing a fourth-order compact difference format coupling on an initial grid; calculating an encryption indicator of each subinterval on an existing grid, and encrypting the grid according to an encryption strategy until a predetermined grid encryption requirement is met, so that a final calculation grid is obtained; solving the Schrodinger-Poisson model by utilizing the compact difference format coupling on the obtained local encryption grid, and then obtaining convergent electron density and potential energy function; and calculating a current density according to a Tsu-Esaki formula within a specified bias range, constructing a current-voltage characteristic curve through a numerical analysis tool, and realizing simulation of a negative differential resistance effect. While ensuring the solution accuracy of physical quantities such as electron density and current density, the application effectively controls the size of the grid, significantly reduces the calculation amount, and greatly improves the calculation efficiency.

Description

Technical Field

[0001] This invention relates to the field of semiconductor device current-voltage characteristic simulation, and specifically to an adaptive encryption method based on resonant tunneling diode current-voltage simulation. Technical Background

[0002] Resonant tunneling diodes (RTDs) have shown broad application prospects in high-speed electronic devices, terahertz communication, and quantum computing due to their unique quantum tunneling effect and negative differential resistance characteristics. Their core structure utilizes a combination of quantum wells and barriers to achieve current transport through the resonant tunneling of charge carriers between quantum energy levels. Their current-voltage characteristics exhibit a nonlinear negative resistance region, a key feature that distinguishes RTDs from traditional semiconductor devices.

[0003] However, accurately simulating the current-voltage characteristics of resonant tunneling diodes faces numerous challenges. From a physical mechanism perspective, the tunneling process of resonant tunneling diodes involves complex quantum effects, including quantum tunneling probability, band bending, and carrier scattering and recombination. These effects are closely related to the microscopic parameters of the material (such as barrier height, well width, and doping concentration) and exhibit strong nonlinear coupling. Traditional classical physical models, such as the drift-diffusion model, neglect quantum effects and struggle to accurately describe the tunneling current of resonant tunneling diodes, leading to significant discrepancies between simulation results and actual measurements.

[0004] From a computational perspective, accurately capturing the quantum tunneling phenomenon in resonant tunneling diodes requires quantum mechanical models, such as the Schrödinger equation and the non-equilibrium Green's function method. However, these quantum models have high computational complexity; the computational load increases exponentially with the increase in computational degrees of freedom, making the simulation of large-scale devices extremely difficult. Currently, commonly used numerical methods for solving the Schrödinger equation include the finite difference method and the finite element method, as exemplified by the research of foreign scholars Jan-Frederik Mennemann et al. (Jan-Frederik Mennemann, Ansgar Jüngel, Hans Kosina, Transient...). Simulations of a high-frequency resonant tunneling diode oscillator[J]. Journal of Computational Physics, 2013, 239: 187-205.), using a second-order central difference scheme to numerically solve the Schrödinger-Poisson coupling model, successfully simulated the current-voltage characteristics of the resonant tunneling diode. However, this algorithm suffers from insufficient accuracy. To obtain convergent numerical results, it is necessary to increase the number of iterations or refine the mesh, which undoubtedly further increases the computational load. Olivier Pinaud used the Runge-Kutta method to numerically solve the Schrödinger equation (Olivier Pinaud, Transient simulations of aresonant tunneling diode[J], Journal of Applied Physics, 2002, 92: 1987-1994.). Although this improved the computational accuracy to some extent, the computational load was still relatively large. Moreover, the Runge-Kutta algorithm lacked an automatic step size control mechanism, requiring manual selection of the step size to balance accuracy and efficiency, which increased the complexity and uncertainty of the operation in actual calculations.

[0005] Furthermore, since the physical quantities (such as electron density and electric potential) in the resonant tunneling diode exhibit drastic changes in spatial distribution, the traditional uniform mesh generation method is difficult to adapt to this complex distribution. It cannot effectively reduce the amount of computation while ensuring the accuracy of the calculation results, thus affecting the simulation efficiency. Summary of the Invention

[0006] This invention proposes a high-precision adaptive refinement method based on the current-voltage simulation of resonant tunneling diodes (RTDs) to improve the efficiency of current-voltage characteristic simulation. First, a Schrödinger-Poisson model is solved on a relatively coarse initial grid, typically a uniform grid with a step size of 1 nm. Then, the refinement indicator for each sub-interval is calculated. Combined with a refinement strategy, a new locally refined grid is obtained. The Schrödinger-Poisson model is solved on this grid. This process is repeated until the grid is sufficiently refined, yielding convergent current densities, etc. This high-precision simulation method based on adaptive refinement can significantly shorten the simulation time, providing a feasible path for accurate and efficient simulation of the current-voltage characteristics of RTDs.

[0007] The objective of this invention is achieved by at least one of the following technical solutions.

[0008] An adaptive encryption method based on resonant tunneling diode current-voltage simulation includes the following steps:

[0009] S1. Based on the relevant physical parameters of the resonant tunneling diode, select an initial grid and use a tight difference scheme to solve the Schrödinger-Poisson model on the initial grid. The electron density and potential energy function under bias voltage are obtained through the coupling operation.

[0010] S2. Based on the potential energy function, calculate the encryption indicator for each sub-interval on the current grid. According to the selected encryption strategy, the newly added grid points that meet the encryption conditions form a set. If the set is not empty, construct a new grid. If the set is empty, it means that the current grid is already fine enough and no further encryption is needed. Proceed directly to step S4.

[0011] S3. On the new grid in step S2, the Schrödinger-Poisson model is solved by coupling the tight difference scheme on the non-uniform grid. Through coupling operations, the electron density and potential energy function under the set bias voltage are obtained. Then, return to step S2 and iterate to optimize the solution accuracy.

[0012] S4. Calculate the current density and construct the current-voltage characteristic curve using numerical analysis tools for comparison with theoretical results.

[0013] Furthermore, in step S1, the specific coupling form of the Schrödinger-Poisson model is as follows: substituting the potential energy function into the following Schrödinger equation and transmission boundary conditions with the spatial position coordinates x∈[0,L] as independent variables:

[0014]

[0015] ψ x (0)+ik1ψ(0)=2ik1

[0016] ψ x (L)-ik2ψ(L)=0

[0017] Where L represents the length of the domain of the problem to be solved. and m * Let represent the reduced Planck constant and the effective mass of the electron, respectively, and ψ(x) represent the electron wavefunction. xx V(x) represents the second derivative of the wave function, V(x) represents the potential energy function of the electron, E is the energy of the electron, and ψ x (0) and ψ x (L) represent the first derivative values ​​of the wavefunction at 0 and L, respectively, where i is the imaginary unit, satisfying i 2 =-1, k1 and k2 are wave vectors; calculate the electron density based on the wave function at different energies, and substitute the electron density into the Poisson equation to solve for the potential energy function; repeat the above calculation until the numerical error meets the accuracy requirements, and obtain the convergent electron density and potential energy function, and obtain the curve of the electron transmission coefficient as a function of energy.

[0018] Further, in step S1, the initial grid is selected as a uniform grid. To construct the fourth-order compact difference schemes of the Schrödinger equation and the Poisson equation, Taylor expansions are performed on the wave function and potential energy function at the node to be solved and at two adjacent nodes, respectively. Based on the requirement of fourth-order truncation error, the corresponding coefficients are determined, and the function values ​​at the node to be solved and two adjacent nodes are linearly combined according to these coefficients. The linear combination constitutes the fourth-order discrete scheme of the second derivative value of the function at the node to be solved. Substituting the fourth-order discrete scheme of the second derivative into the Schrödinger equation and the Poisson equation, the fourth-order discrete schemes of these two equations can be obtained.

[0019] Furthermore, the transmission boundary conditions of the Schrödinger equation have two discretization schemes on a uniform grid:

[0020] (I) Discrete fourth-order transmission boundary conditions D4TBCs;

[0021] (II) Analyze the discrete transmission boundary conditions aDTBCs.

[0022] Furthermore, by assembling the Poisson equation with von Neumann boundary conditions, a Taylor expansion of the potential function is performed at the boundary nodes, adjacent grid points of the boundary nodes, and virtual points. Based on the coefficients of the fourth-order truncation error, the function values ​​at the boundary nodes, adjacent grid points of the boundary nodes, and virtual points are linearly combined to construct a fourth-order discretization scheme for the first derivative of the function at the boundary nodes. Substituting this fourth-order discretization scheme into the Poisson equation yields the fourth-order discretization scheme for the von Neumann boundary conditions.

[0023] Furthermore, in step S2, the first and second derivatives of the potential energy function are numerically integrated over the interval to be judged, and the numerical result of the integration is used as the encryption indicator.

[0024] Further, in step S2, the grid encryption of the encryption strategy is as follows: the encryption indicator value of each interval is compared with the threshold. If the indicator value of the interval is greater than the threshold, it indicates that the interval needs to be encrypted. A point is inserted at the midpoint of the interval and added to the set of new grid points. If the encryption indicator value is less than or equal to the threshold, it indicates that the interval is already fine enough and does not need to be encrypted.

[0025] Furthermore, the encryption strategy includes selecting one indicator or considering two indicators simultaneously;

[0026] When selecting only one indicator, there are two encryption strategies:

[0027] (iv) Encryption strategy SPRS1: Use the numerical integration result of the first derivative of the potential energy function on the interval to be judged as the judgment indicator. If the value of the indicator of the interval to be judged is greater than the average value of the indicators on all intervals, the interval needs to be encrypted.

[0028] (V) Encryption Strategy SPRS2: The numerical integral of the second derivative of the potential energy function over the interval to be judged is used as the judgment indicator. If the value of the indicator of the interval to be judged is greater than the average value of the indicators over all intervals, the interval needs to be encrypted.

[0029] (vi) When two encryption indicators are used at the same time, there is a third strategy denoted as SPRS3: the numerical integrals of the first and second derivatives of the potential energy function on the interval to be determined are used as two encryption indicators respectively. If the value of at least one of these two indicators is greater than the average value of the indicator in all intervals, then the interval to be determined needs to be encrypted.

[0030] Further, in step S3, if the refined mesh is a non-uniform mesh, Taylor expansions of the wave function and potential energy function are performed at the node to be solved and at two adjacent nodes. Based on the principle of minimizing the truncation error, the corresponding coefficients are determined, and the function values ​​at the node to be solved and the two adjacent nodes are linearly combined according to these coefficients. The linear combination constitutes a higher-order compact difference scheme of the second derivative value of the function at the node to be solved. Under the condition of non-uniform mesh, the higher-order compact difference scheme of the second derivative value has third-order accuracy. When the mesh is uniform, the convergence order of the higher-order compact difference scheme of the second derivative value will be increased to the fourth order. Substituting the higher-order compact difference scheme of the second derivative value of the node to be solved into the Schrödinger equation and the Poisson equation respectively, the higher-order numerical discretization schemes of these two equations on arbitrary meshes can be derived.

[0031] Analytical discrete transmission boundary conditions aDTBCs are applied to a non-uniform grid:

[0032]

[0033] Where, ψ -j ,ψ0, These represent the wavefunction at the -j, 0, and Nth digits, respectively. x N x+j The value of N grid points x It is the total number of grid points in the current grid, x is the spatial coordinate, and i is the imaginary unit, satisfying i 2 =-1, k1 and k2 represent wave vectors, It is the grid spacing between the -j-th grid point and the 0th grid point. It is the Nth x The grid point and the Nth grid point x+j Grid spacing between grid points and It can be calculated using Euler's formula:

[0034] Further, in step S4, the current density is calculated according to the Tsu-Esaki formula.

[0035] A computer device according to the present invention includes: a memory and a processor, and a computer program stored in the memory, which, when executed on the processor, implements the method.

[0036] Compared with the prior art, the present invention has the following advantages and technical effects:

[0037] This invention combines a high-precision compact differential scheme with an adaptive encryption method. By solving the Schrödinger-Poisson coupling model, the electron density and potential energy function are obtained. The current density is then calculated using the Tsu-Esaki formula, thus yielding the current-voltage characteristic curve of the resonant tunneling diode. This method ensures the accuracy of the calculation results while reducing the computational load, significantly improving computational efficiency. It provides a practical and feasible solution for simulating the current-voltage characteristics of resonant tunneling diodes. Attached Figure Description

[0038] Figure 1 This is a schematic diagram of the resonant tunneling diode in Embodiments 1 and 2 of the present invention;

[0039] Figure 2 In Embodiment 1 of the present invention, in V b A schematic diagram of the electron density and total potential function distribution obtained by the fourth-order compact difference scheme when =0(V);

[0040] Figure 3 In Embodiment 1 of the present invention, in V b A schematic diagram of the electron density and total potential function distribution obtained by the fourth-order compact difference scheme at V = 0.26.

[0041] Figure 4 In Embodiment 1 of the present invention, in V b A schematic diagram of the electron density and total potential function distribution obtained by the fourth-order compact difference scheme at V = 0.28.

[0042] Figure 5 This is a graph showing the change in transmission coefficient with energy obtained using a fourth-order compact difference scheme in Embodiment 1 of the present invention.

[0043] Figure 6 This is a current-voltage characteristic curve of a resonant tunneling diode simulated using a fourth-order compact differential scheme in Embodiment 1 of the present invention.

[0044] Figure 7 This is a schematic diagram of the electron density and potential energy function distribution on the initial grid of the adaptive encryption method in Embodiment 2 of the present invention;

[0045] Figure 8 In Embodiment 2 of the present invention, in V b A schematic diagram of the distribution of electron density, potential energy function and grid density obtained by the adaptive encryption method using three encryption strategies when = 0 (V);

[0046] Figure 9 In Embodiment 2 of the present invention, in V b A schematic diagram showing the distribution of electron density, potential energy function and grid density obtained by the adaptive encryption method using three encryption strategies when the voltage is 0.26 (V).

[0047] Figure 10 This is a schematic diagram showing the numerical error of electron density and potential energy function compared with the reference solution on the initial grid and the grids obtained by the three encryption algorithms in Embodiment 2 of the present invention.

[0048] Figure 11 This is a current-voltage characteristic curve of a resonant tunneling diode simulated using an adaptive encryption method based on an encryption strategy in Embodiment 2 of the present invention.

[0049] Figure 12 This is a flowchart of the adaptive encryption method based on resonant tunneling diode current-voltage simulation of the present invention; Detailed Implementation

[0050] The following specific examples further illustrate the usage method and process of the present invention, which are only preferred embodiments of the present invention, but the implementation and protection scope of the present invention are not limited thereto.

[0051] Example 1

[0052] In this embodiment, a classic resonant tunneling diode model is used as an example to simulate the current-voltage characteristics. In this embodiment, the current-voltage characteristics of the device are simulated only on a uniform grid, and the implementation of the present invention on a uniform grid is explained.

[0053] like Figure 12 As shown, the adaptive encryption method based on resonant tunneling diode current-voltage simulation includes the following steps:

[0054] S1. Based on the relevant physical parameters of the resonant tunneling diode, select an initial grid and solve the Schrödinger-Poisson model on the initial grid using a compact difference scheme. The bias voltage V is obtained through coupling calculation. b Electron density n(x) and potential energy function V s(x), where the independent variable x represents the spatial coordinates of the electron, and the bias voltage satisfies V b >0.

[0055] As a specific embodiment, the relevant physical parameters of the resonant tunneling diode in this embodiment are as follows: Figure 1 As shown, the overall material of the device is gallium arsenide (GaAs), with two sections of aluminum gallium arsenide (GaAs). 1-x Al x The heterogeneous region of As implies that the device has two intrinsic potential barriers, each with a height of 0.3 eV. The total potential function V(x) is the sum of the intrinsic potential barriers and the potential energy function V. s The sum of (x) and the bias voltage across the device is V. b The doping function varies in the range of 0 (V) to 0.4 (V), and is a piecewise constant, which is more pronounced in heavily doped (n) doped (n) doping ... + The GaAs region is 1×10 18 (cm -3 In other areas, take 5×10 15 (cm -3 The physical constants involved in the embodiments: the effective mass m of the electron. * =0.067m0, where m0 is the electron rest mass; dielectric constant ε = 11.44ε0, where ε0 is the vacuum dielectric constant; simulated temperature is T. L =300(K); the total length of the device is L=135(nm), and the independent variable x represents the spatial coordinates of the electron.

[0056] Under the premise of ballistic transport, the boundary condition of the Schrödinger equation is a transmission boundary condition, while the contact at both ends of the resonant tunneling diode simulated in this invention satisfies the condition of electrical neutrality, so the Poisson equation is equipped with the Neumann boundary condition.

[0057] The initial grid is selected as a uniform grid with a step size Δx = 0.5 (nm). The j-th grid point x in the current grid... j =jΔx, j = 0, 1, ..., N x j represents the grid point number, where N x =270, and the Schrödinger-Poisson model is solved using a tight difference scheme coupled on this grid;

[0058] On a uniform grid, the function to be solved is u(x) (in solving the Schrödinger equation and the Poisson equation, u(x) represents the wave function ψ(x) and the potential energy function V, respectively). s The second derivative u of (x)) xx (x) at grid point x j The fourth-order compact difference discretization scheme at point is:

[0059]

[0060] Among them, u j-1 u j u j+1 These represent u(x) at grid point x, respectively. j-1 x j x j+1 The value at (u) xx ) j-1 、(u xx ) j 、(u xx ) j+1 They represent u respectively xx (x) at grid point x j-1 x j x j+1 The value at that location.

[0061] The transmission boundary conditions of the Schrödinger equation can be discretized in two ways on a uniform grid:

[0062] (I) Discrete Fourth-Order Transmission Boundary Conditions (D4TBCs):

[0063] ψ -1 -αψ0=α -1 -α,

[0064]

[0065] Introduce two virtual points: x -1 =-Δx, L represents the length of the domain of the problem to be solved. In the above expression, ψ -1 ,ψ0, Let ψ(x) represent the wave function ψ(x) at x. -1 x0 The values ​​of α and β are parameters calculated based on the potential energy function and the grid step size.

[0066] (II) Analytical Discrete Transmission Boundary Conditions (aDTBCs):

[0067]

[0068] Where i is the imaginary unit, satisfying i 2 =-1, k1 and k2 are wave vectors, It can be calculated using Euler's formula:

[0069] Poisson equation with Neumann boundary conditions, potential energy function V s The first derivative of (x) V s ′ (x) in x j The fourth-order compact difference discretization scheme at point is:

[0070]

[0071] Among them, (V) s ) j+1 and (V) s ) j-1 Representing the potential energy function V s (x) in x j+1 x j-1 The value at (V) s ") j+1 and (V) s ") j-1 Representing the potential energy function V s The second derivative V of (x) s "(x) in x j+1 x j-1 The value at that location.

[0072] The specific coupling form of the Schrödinger model is: the potential energy function V s Substitute (x) into the Schrödinger equation to solve for the electron wave function ψ(x) (where x represents the electron's spatial coordinates). Calculate the electron density n(x) based on the wave function ψ(x) at different energies. Substitute the electron density n(x) into the Poisson equation to solve for the potential energy function V. s (x), repeat the above calculation until the numerical error meets the accuracy requirements, and obtain the result in V. b The electron density n(x) and total potential function V(x) at V = 0, 1, 0.26, and 0.28 are respectively as follows: Figure 2 , Figure 3 , Figure 4 As shown, the curve of electron transmission coefficient T(E) as a function of energy E is obtained as follows. Figure 5 As shown. Among them Figure 2 (a) and (b) respectively represent V b =0(V), schematic diagram of the distribution of electron density n(x) and total potential function V(x) obtained by combining the two boundary condition discretization schemes D4TBCs and aDTBCs with the compact difference scheme. Figure 3 (a) and (b) respectively represent V b When V = 0.26, the electron density n(x) and total potential function V(x) distributions are obtained by combining the two boundary condition discretization schemes, D4TBCs and aDTBCs, with the compact difference scheme. Figure 4 (a) and (b) respectively represent V b When V = 0.28, the electron density n(x) and total potential function V(x) distributions are obtained by combining the two boundary condition discretization schemes, D4TBCs and aDTBCs, with the compact difference scheme. Figure 5(a) and (b) represent the curves of the transmission coefficient T(E) as a function of energy E, obtained by combining the two boundary condition discretization schemes D4TBCs and aDTBCs with the compact difference scheme under different bias voltages.

[0073] S2. The current mesh is the reference mesh selected in this embodiment, which is already fine enough and does not need further encryption. Proceed directly to step S4.

[0074] S3, on the local encrypted mesh constructed in step S2.

[0075] S4. Calculate the current density using the Tsu-Esaki formula and present it as a current-voltage characteristic curve to simulate and evaluate the current-voltage characteristics of the resonant tunneling diode.

[0076] The current density I(V) was calculated according to the Tsu-Esaki formula. b ), current density I(V b ) Calculated by the following formula:

[0077]

[0078] Where V b q represents the bias voltage across the device. e and m * k represents the charge and effective mass of an electron, respectively. B and T represents the Boltzmann constant and the reduced Planck constant. L The experimental simulation temperature is represented by T(E), which is the electron transmission coefficient, and E represents the steady-state electron energy. F This indicates the Fermi level at the left contact of the device.

[0079] Finally, the current-voltage characteristic curve is constructed using numerical analysis tools, as shown below. Figure 6 As shown, the negative differential resistance effect is simulated, which facilitates comparison with theoretical results.

[0080] Example 2

[0081] The resonant tunneling diode model in this embodiment is the same as that in Embodiment 1. The difference is that this embodiment fully implements steps S1 to S4, and further details the implementation of the present invention.

[0082] S1. The physical parameters of the resonant tunneling diode in this embodiment are the same as those in Embodiment 1, such as... Figure 1 As shown; the initial grid is selected as a uniform grid with a step size of 1 nm, i.e., N x =135. The Schrödinger-Poisson model is solved on this grid using a tight difference scheme. The solution steps and difference scheme are the same as in Example 1. The results are as follows: Figure 7 As shown, Figure 7 For V b A schematic diagram of the electron density and potential energy function distribution on the initial grid of the adaptive encryption method when V = 0.

[0083] S2. On the current grid, calculate each subinterval [x] j ,x j+1 The encryption indicator, based on the selected encryption strategy, forms a set S of newly added grid points that meet the encryption conditions. k If S k If S is not empty, then construct a new grid. k If the set is empty, it means that the current mesh is already fine enough and no further refinement is needed. Proceed directly to step S4.

[0084] Here, we choose the potential energy function V. s The first and second derivatives of (x) are used as encryption indicators, and their mathematical expressions are as follows:

[0085] I 1,j =|(V s ) j+1 -(V s ) j |,

[0086] and

[0087]

[0088] Among them, I 1,j and I 2,j They respectively represent the use of the potential energy function V s The first and second derivatives of (x) serve as indicators of the sub-time, and the subinterval [x] j ,x j+1 The value of the corresponding indicator, x j This represents the j-th grid point in the current grid, (V s ) j+1 and (V) s ) j Representing the potential energy function V s (x) in x j+1 x j The value at (V) s ") j+1 and (V) s ") j Representing the potential energy function V s The second derivative V of (x) s "(x) in x j+1 x j The value at h j+1 =x j+1 -x jRepresents the subinterval [x] j ,x j+1 The interval length of ].

[0089] The specific steps of the grid encryption strategy are as follows: compare the encryption indicator value of each interval with the threshold. If the indicator value of an interval is greater than the threshold, it indicates that the interval needs to be encrypted. Insert a point at the midpoint of the interval and add it to the set S. k If the encryption indicator is less than or equal to the threshold, it indicates that the range is already fine enough and does not need to be encrypted.

[0090] When constructing an encryption strategy, choose one indicator or consider two indicators simultaneously;

[0091] When selecting only one indicator, there are two encryption strategies:

[0092] (1), indicator I 1,j The corresponding encryption strategy SPRS1

[0093] When indicator I 1,j satisfy:

[0094]

[0095] When, the interval [x j ,x j+1 The encryption needs to be performed, where θ1≥1 is the set encryption coefficient;

[0096] (2), indicator I 2,j The corresponding encryption strategy SPRS2

[0097] When indicator I 2,j satisfy:

[0098]

[0099] When, the interval [x j ,x j+1 The encryption needs to be performed, where θ2≥1 is the set encryption coefficient;

[0100] When considering both encryption indicators, there is a third strategy, SPRS3, which is when...

[0101] or

[0102] When it was established, [x] j ,x j+1 Encryption is required.

[0103] In this embodiment, each subinterval [x] is calculated. j ,x j+1 Encryption indicator I1,j and I 2,j By using three encryption strategies, SPRS1, SPRS2, and SPRS3, to encrypt once each, with encryption coefficients θ1 = θ2 = 1, a new locally encrypted grid is obtained. When V b When V = 0 (V), the new grid points obtained by encrypting once using encryption strategies SPRS1, SPRS2, and SPRS3 are 172, 172, and 176, respectively; when V b When the voltage is 0.26 (V), i.e. the tunneling voltage, the new grid points are 176, 173 and 191 respectively, obtained by encrypting once using encryption strategies SPRS1, SPRS2 and SPRS3.

[0104] S3. On the locally refined mesh constructed in step S2, similar to step S1, the Schrödinger-Poisson model is solved using a tightly differential scheme on the non-uniform mesh. Through coupled computation, the model is obtained under a given bias voltage V. b The electron density and potential energy function are determined, and then the process is repeated in step S2 to optimize the solution accuracy.

[0105] The encrypted mesh is not uniform; assume the current mesh has N... x +1 grid point, denoted as: The function to be solved is u(x) (in solving the Schrödinger equation and the Poisson equation, u(x) represents the wave function ψ(x) and the potential energy function V, respectively). s The second derivative u of (x)) xx (x) in x j (x j The expression representing the compact difference scheme at the j-th grid point is:

[0106]

[0107] Among them, u j-1 u j u j+1 They represent u(x) at x respectively j-1 x j x j+1 The value at (u) xx ) j-1 、(u xx ) j 、(u xx ) j+1 They represent u respectively xx (x) in x j-1 x j x j+1 The value at the grid spacing h j =x j -x j-1 α jβ j and γ j All are related to h j and h j+1 The relevant parameters are obtained through Taylor expansion according to the required accuracy;

[0108] Analytical discrete transmission boundary conditions aDTBCs are applied to a non-uniform grid:

[0109]

[0110] Where i is the imaginary unit, satisfying i 2 =-1, k1 and k2 represent wave vectors, x -j and These are virtual points introduced based on the numerical format: and It is the set grid spacing, ψ -j ,ψ0, Let ψ(x) represent the wave function ψ(x) at x. -j x0 The value, and It can be calculated using Euler's formula:

[0111] In this embodiment, the distribution curves of electron density and potential energy function are calculated on the grid obtained in step S2 according to the above difference scheme, in V b When = 0 (V), the electron density n(x) and potential energy function V s The distribution curves of (x) and grid density ρ(x) are as follows: Figure 8 As shown, where Figure 8 In (a), (c), and (e), the potential energy functions V are obtained using the SPRS1, SPRS2, and SPRS3 encryption strategies, respectively. s Distribution curves of (x) and grid density ρ(x), Figure 8 Figures (b), (d), and (f) show the distribution curves of electron density n(x) and grid density ρ(x) obtained using SPRS1, SPRS2, and SPRS3 encryption strategies, respectively; in V b When V = 0.26, the electron density n(x) and potential energy function V are given. s The distribution curves of (x) and grid density ρ(x) are as follows: Figure 9 As shown, where Figure 9 In (a), (c), and (e), the potential energy functions V are obtained using the SPRS1, SPRS2, and SPRS3 encryption strategies, respectively. s Distribution curves of (x) and grid density ρ(x), Figure 8Figures (b), (d), and (f) show the distribution curves of electron density n(x) and grid density ρ(x) obtained using SPRS1, SPRS2, and SPRS3 encryption strategies, respectively; N obtained in Example 1 x The result at 270° is used as a reference solution. Comparisons are made with the results on the initial grid and the three refined grids to obtain error curves for the electron density and potential energy function, as shown below. Figure 10 As shown.

[0112] S4. The current density is calculated using the Tsu-Esaki formula and presented as an IV curve, thereby simulating and evaluating the current-voltage characteristics of the resonant tunneling diode.

[0113] The current density I(V) was calculated according to the Tsu-Esaki formula. b ), current density I(V b ) Calculated by the following formula:

[0114]

[0115]

[0116] Where V b q represents the bias voltage across the device. e and m * k represents the charge and effective mass of an electron, respectively. B and T represents the Boltzmann constant and the reduced Planck constant. L The experimental simulation temperature is represented by T(E), which is the electron transmission coefficient, and E represents the steady-state electron energy. F This indicates the Fermi level at the left contact of the device.

[0117] In this embodiment, in order to select the most suitable encryption strategy, V is calculated according to the Tsu-Esaki formula. b The current densities corresponding to the reference grid, initial grid, and three types of refined grids at a voltage of 0.26 (V) are shown in Table 1.

[0118] Table 1 shows the current densities calculated on the reference grid, the initial grid, and the three types of refined grids.

[0119] <![CDATA[N x +1]]> <![CDATA[I(V b )(10 -4 A·nm -2 )]]> error 271 (Reference) 1.39979e-6 * 136 (Starting) 1.43095e-6 3.11532e-8 176(SPRS1) 1.40109e-6 1.29412e-9 173(SPRS2) 1.46274e-6 6.29487e-8 191(SPRS3) 1.39972e-6 7.01079e-11

[0120] By comparison Figure 10 Compared with the errors in Table 1, the calculation results on the mesh obtained after one encryption by the SPRS3 encryption strategy are more accurate, so the SPRS3 encryption strategy is selected; the bias voltage V is calculated using the adaptive encryption method. bThe current density from 0 (V) to 0.4 (V), and the calculation time at some bias points are shown in Table 2. Table 2 also lists the reference grid N. x With a computation time of 270, it can be seen that the computation time of the adaptive encryption method combined with SPRS3 is 1 / 4 to 1 / 3 of that of the reference grid, which greatly improves the computational efficiency.

[0121] Table 2 shows the calculation time at some bias points (unit: seconds).

[0122] <![CDATA[V b (V)]]> 0 0.10 0.26 0.32 Adaptive Mesh 20.3 41.2 646.1 715.5 Reference grid 77.2 115.3 2694.3 2887.9

[0123] Finally, using numerical analysis tools, current-voltage characteristic curves are constructed, such as... Figure 11 As shown, the simulation of the negative differential resistance phenomenon can be successfully achieved. As a specific embodiment, the numerical analysis tool can be MATLAB, Beitai Tianyuan, or similar tools.

[0124] The present invention provides a high-precision adaptive encryption method based on current-voltage simulation of resonant tunneling diodes. This simulation method can accurately simulate the quantum tunneling effect of resonant tunneling diodes and efficiently calculate their IV characteristics. It has important practical significance for promoting the application and development of resonant tunneling diodes in related fields.

[0125] The preferred embodiments of the present invention disclosed above are merely illustrative of the invention. These preferred embodiments do not describe all details exhaustively, nor do they limit the invention to the specific implementations described.

Claims

1. An adaptive encryption method based on resonant tunneling diode current-voltage simulation, characterized in that, Includes the following steps: S1. Based on the relevant physical parameters of the resonant tunneling diode, select an initial grid and use a tight difference scheme to solve the Schrödinger-Poisson model on the initial grid. The electron density and potential energy function under bias voltage are obtained through the coupling operation. S2. Based on the potential energy function, calculate the encryption indicator for each sub-interval on the current grid. According to the selected encryption strategy, the newly added grid points that meet the encryption conditions form a set. If the set is not empty, construct a new grid. If the set is empty, it means that the current grid is already fine enough and no further encryption is needed. Proceed directly to step S4. S3. On the new grid in step S2, the Schrödinger-Poisson model is solved by coupling the tight difference scheme on the non-uniform grid. Through coupling operations, the electron density and potential energy function under the set bias voltage are obtained. Then, return to step S2 and iterate to optimize the solution accuracy. S4. Calculate the current density and construct the current-voltage characteristic curve using numerical analysis tools for comparison with theoretical results.

2. The adaptive encryption method based on resonant tunneling diode current-voltage simulation according to claim 1, characterized in that, In step S1, the specific coupling form of the Schrödinger-Poisson model is as follows: Substitute the potential energy function into the following Schrödinger equation and transmission boundary conditions with the spatial position coordinates x∈[0,L] as independent variables: ψ x (0)+ik1ψ(0)=2ik1 ψ x (L)-ik2ψ(L)=0 Where L represents the length of the domain of the problem to be solved. and m * Let represent the reduced Planck constant and the effective mass of the electron, respectively, and ψ(x) represent the electron wavefunction. xx V(x) represents the second derivative of the wave function, V(x) represents the potential energy function of the electron, E is the energy of the electron, and ψ x (0) and ψ x (L) represent the first derivative values ​​of the wavefunction at 0 and L, respectively, where i is the imaginary unit, satisfying i 2 =-1, k1 and k2 are wave vectors; calculate the electron density based on the wave function at different energies, and substitute the electron density into the Poisson equation to solve for the potential energy function; repeat the above calculation until the numerical error meets the accuracy requirements, and obtain the convergent electron density and potential energy function, and obtain the curve of the electron transmission coefficient as a function of energy.

3. The adaptive encryption method based on resonant tunneling diode current-voltage simulation according to claim 1, characterized in that, In step S1, a uniform grid is selected as the initial grid. To construct the fourth-order compact difference schemes for the Schrödinger and Poisson equations, Taylor expansions are performed on the wave function and potential energy function at the node to be solved and at two adjacent nodes, respectively. Based on the requirement of fourth-order truncation error, the corresponding coefficients are determined, and the function values ​​at the node to be solved and two adjacent nodes are linearly combined according to these coefficients. The linear combination constitutes the fourth-order discrete scheme of the second derivative value of the function at the node to be solved. Substituting the fourth-order discrete scheme of the second derivative into the Schrödinger and Poisson equations, the fourth-order discrete schemes of these two equations can be obtained.

4. The adaptive encryption method based on resonant tunneling diode current-voltage simulation according to claim 2, characterized in that, The transmission boundary conditions of the Schrödinger equation can be discretized in two ways on a uniform grid: (I) Discrete fourth-order transmission boundary conditions D4TBCs; (II) Analyze the discrete transmission boundary conditions aDTBCs.

5. The adaptive encryption method based on resonant tunneling diode current-voltage simulation according to claim 2, characterized in that, By assembling the Poisson equation with von Neumann boundary conditions, a Taylor expansion of the potential function is performed at the boundary nodes, adjacent grid points, and virtual points. Based on the coefficients of the fourth-order truncation error, the function values ​​at the boundary nodes, adjacent grid points, and virtual points are linearly combined to construct a fourth-order discretization scheme for the first derivative of the function at the boundary nodes. Substituting this fourth-order discretization scheme into the Poisson equation yields the fourth-order discretization scheme for the von Neumann boundary conditions.

6. The adaptive encryption method based on resonant tunneling diode current-voltage simulation according to claim 1, characterized in that, In step S2, the first and second derivatives of the potential energy function are numerically integrated over the interval to be judged, and the numerical result of the integration is used as the encryption indicator.

7. The adaptive encryption method based on resonant tunneling diode current-voltage simulation according to claim 1, characterized in that, In step S2, the grid encryption of the encryption strategy is as follows: the encryption indicator value of each interval is compared with the threshold. If the indicator value of the interval is greater than the threshold, it indicates that the interval needs to be encrypted. A point is inserted at the midpoint of the interval and added to the set of new grid points. If the encryption indicator value is less than or equal to the threshold, it indicates that the interval is already fine enough and does not need to be encrypted.

8. The adaptive encryption method based on resonant tunneling diode current-voltage simulation according to claim 1, characterized in that, Encryption strategies include selecting one indicator or considering two indicators simultaneously; When selecting only one indicator, there are two encryption strategies: (I) Encryption Strategy SPRS1: Use the numerical integration result of the first derivative of the potential energy function on the interval to be judged as the judgment indicator. If the value of the indicator of the interval to be judged is greater than the average value of the indicators on all intervals, the interval needs to be encrypted. (II) Encryption Strategy SPRS2: The numerical integral of the second derivative of the potential energy function over the interval to be judged is used as the judgment indicator. If the value of the indicator of the interval to be judged is greater than the average value of the indicators over all intervals, the interval needs to be encrypted. (III) When two encryption indicators are used at the same time, there is a third strategy denoted as SPRS3: the numerical integrals of the first and second derivatives of the potential energy function on the interval to be determined are used as two encryption indicators respectively. If the value of at least one of these two indicators is greater than the average value of the indicator in all intervals, then the interval to be determined needs to be encrypted.

9. The adaptive encryption method based on resonant tunneling diode current-voltage simulation according to claim 1, characterized in that, In step S3, if the refined mesh is non-uniform, Taylor expansions of the wave function and potential energy function are performed at the node to be solved and at two adjacent nodes. Based on the principle of minimizing truncation error, the corresponding coefficients are determined, and the function values ​​at the node to be solved and two adjacent nodes are linearly combined according to these coefficients. The linear combination constitutes a higher-order compact difference scheme of the second derivative value of the function at the node to be solved. Under non-uniform mesh conditions, the higher-order compact difference scheme of the second derivative value has third-order accuracy. When the mesh is uniform, the convergence order of the higher-order compact difference scheme of the second derivative value will be increased to fourth-order. Substituting the higher-order compact difference scheme of the second derivative value of the node to be solved into the Schrödinger equation and the Poisson equation respectively, the higher-order numerical discretization schemes of these two equations on arbitrary meshes can be derived. Analytical discrete transmission boundary conditions aDTBCs are applied to a non-uniform grid: Where, ψ -j ,ψ0, These represent the wavefunction at the -j, 0, and Nth digits, respectively. x , x+j The value of N grid points x It is the total number of grid points in the current grid, x is the spatial coordinate, and i is the imaginary unit, satisfying i 2 =-1, k1 and k2 represent wave vectors, It is the grid spacing between the -j-th grid point and the 0th grid point. It is the Nth x The grid point and the Nth grid point x+j Grid spacing between grid points and It can be calculated using Euler's formula: .

10. The adaptive encryption method based on resonant tunneling diode current-voltage simulation according to claim 1, characterized in that, In step S4, the current density is calculated according to the Tsu-Esaki formula.

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