Simulation method for semiconductor device, electronic device, and storage medium

Abstract

Example embodiments of the present disclosure relate to a simulation method for a semiconductor device, an electronic device and a storage medium. The method comprises: discretizing a barrier-related parameter based on a predetermined range and a step length of the predetermined barrier-related parameter to generate a plurality of first data points; and determining a transmission coefficient corresponding to each of the plurality of first data points, wherein the transmission coefficient comprises a first transmission coefficient corresponding to a first barrier unit having a constant barrier height and a second transmission coefficient corresponding to a second barrier unit having a linearly varying barrier height. Embodiments of the present disclosure can efficiently and accurately calculate the barrier transmission coefficient for any shape barrier, improve the calculation efficiency of the tunneling current, and thus reduce the calculation amount and save costs.

Description

Technical Field

[0001] The embodiments of this disclosure primarily relate to simulation techniques, and more specifically, to simulation methods for semiconductor devices, electronic devices, and storage media. Background Technology

[0002] With technological advancements, people have placed higher demands on the performance of semiconductor devices, including faster computing, more intelligent processing, longer battery life, lower latency, and smaller size. These increasing demands have driven the miniaturization of semiconductor devices.

[0003] With the continuous shrinking of semiconductor device dimensions, the thickness of the gate dielectric layer has approached the physical limit of 1 nm, leading to a significant enhancement of direct tunneling and Fowler-Nordheim tunneling effects. These quantum effects not only affect the subthreshold characteristics of the device, such as subthreshold swing (SS), but also induce significant drain-induced barrier lowering (DIBL) and static power consumption issues, increasing the difficulty of calculating tunneling current. Summary of the Invention

[0004] According to exemplary embodiments of this disclosure, a simulation scheme for semiconductor devices is provided to at least partially overcome the above or other potential defects.

[0005] In a first aspect of this disclosure, a simulation method for a semiconductor device is provided. The method includes: discretizing the barrier-related parameters based on a predetermined range and a predetermined step size to generate a plurality of first data points; and determining a transmission coefficient corresponding to each of the plurality of first data points, wherein the transmission coefficient includes a first transmission coefficient corresponding to a first barrier cell having a constant barrier height and a second transmission coefficient corresponding to a second barrier cell having a linearly varying barrier height.

[0006] In some embodiments, determining the transmission coefficient corresponding to each of the plurality of first data points includes: determining the first transmission coefficient corresponding to each of the plurality of first data points using a first analytical expression for a first barrier unit; and determining the second transmission coefficient corresponding to each of the plurality of first data points using a second analytical expression for a second barrier unit.

[0007] In some embodiments, the method further includes: generating a pre-interpolation table for determining the transmission coefficient of a barrier of a semiconductor device based on the correspondence between each of a plurality of first data points and its corresponding first transmission coefficient and its corresponding second transmission coefficient.

[0008] In some embodiments, the method further includes: in response to the barrier-related parameters of the semiconductor device exceeding the range of predetermined barrier-related parameters, expanding the pre-interpolation table to generate an extrapolation pre-interpolation table.

[0009] In some embodiments, in response to a barrier-related parameter of a semiconductor device exceeding a predetermined range of barrier-related parameters, expanding the pre-interpolation table to generate an extrapolation pre-interpolation table includes: determining a first transmission coefficient variation trend relative to the barrier-related parameter and a second transmission coefficient variation trend relative to the barrier-related parameter for a plurality of first data points adjacent to the barrier-related parameter of the barrier under test within the predetermined range of the barrier-related parameter; discretizing the barrier-related parameter based on the portion of the barrier-related parameter of the barrier under test exceeding the predetermined range of the predetermined barrier-related parameter and a predetermined step size to generate a plurality of second data points; and determining an expanded transmission coefficient corresponding to each of the plurality of second data points based on the determined first transmission coefficient variation trend relative to the barrier-related parameter and the second transmission coefficient variation trend relative to the barrier-related parameter, wherein the expanded transmission coefficient includes a first expanded transmission coefficient corresponding to a first barrier unit having a constant barrier height and a second expanded transmission coefficient corresponding to a second barrier unit having a linearly varying barrier height.

[0010] In some embodiments, expanding the pre-interpolation table to generate an extrapolation pre-interpolation table in response to the barrier-related parameters of the barrier under test exceeding a predetermined range of barrier-related parameters includes: discretizing the barrier-related parameters based on the portion of the barrier-related parameters of the barrier under test that exceeds a predetermined range of predetermined barrier-related parameters and a predetermined step size to generate a plurality of second data points; and determining an extended transmission coefficient corresponding to each of the plurality of second data points, wherein the extended transmission coefficient includes a first extended transmission coefficient corresponding to a first barrier unit having a constant barrier height and a second extended transmission coefficient corresponding to a second barrier unit having a linearly varying barrier height.

[0011] In some embodiments, determining the extended transmission coefficient corresponding to each of the plurality of second data points includes: determining a first extended transmission coefficient corresponding to each of the plurality of second data points using a first analytical expression for a first barrier unit; and determining a second extended transmission coefficient corresponding to each of the plurality of second data points using a second analytical expression for a second barrier unit.

[0012] In some embodiments, in response to the barrier-related parameters of the semiconductor device exceeding the range of the predetermined barrier-related parameters, expanding the pre-interpolation table to generate the extrapolation pre-interpolation table includes: expanding the pre-interpolation table to generate the extrapolation pre-interpolation table based on the correspondence between each of the plurality of second data points and the corresponding first extended transmission coefficient and the corresponding second extended transmission coefficient.

[0013] In some embodiments, barrier-related parameters include barrier length, barrier width, and electron energy.

[0014] In a second aspect of this disclosure, a method for processing the transmission coefficient of a semiconductor device is provided. The method includes: dividing a potential barrier of the semiconductor device along the electron tunneling direction into a plurality of barrier units, wherein the plurality of barrier units include one or more first barrier units having a constant barrier height and one or more second barrier units having a linearly varying barrier height; determining a first transmission coefficient of each of the one or more first barrier units and a second transmission coefficient of each of the one or more second barrier units based on a predetermined correspondence between barrier-related parameters and corresponding transmission coefficients; and determining the total transmission coefficient of the semiconductor device's barrier based on the first and second transmission coefficients.

[0015] In some embodiments, the predetermined correspondence is a pre-interpolation table, which stores transmission coefficients corresponding to predetermined barrier-related parameters.

[0016] In some embodiments, the potential barrier of a semiconductor device along the electron tunneling direction is divided into multiple potential barrier units, including dividing the potential barrier along the electron tunneling direction by a dichotomy method, dividing the potential barrier into multiple first potential barrier units and second potential barrier units.

[0017] In some embodiments, dividing the potential barrier along the electron tunneling direction into multiple first barrier units and second barrier units by a dichotomy method includes: determining a first slope of the line connecting the left endpoint and the midpoint of the barrier and a second slope of the line connecting the right endpoint and the midpoint; and in response to the first slope not being equal to the second slope, dividing the left endpoint and the midpoint, and the midpoint and the right endpoint by a dichotomy until the first slope and the second slope are equal.

[0018] In some embodiments, the method further includes: in response to determining that the first slope and the second slope of the barrier cell in the segmented barrier are equal, randomly selecting a random point between the two endpoints of the barrier cell to verify whether the first slope of the line connecting the random point to the left endpoint of the barrier cell and the second slope of the line connecting the random point to the right endpoint of the barrier cell are equal; and in response to determining that the first slope and the second slope are not equal, continuing to divide the barrier cell into two segments until the first slope and the second slope are equal or a predetermined number of divisions is reached.

[0019] In some embodiments, determining the first transmission coefficient of each first barrier unit in one or more first barrier units and the second transmission coefficient of each second barrier unit in one or more second barrier units based on a predetermined correspondence between barrier-related parameters and corresponding transmission coefficients includes: in response to the barrier-related parameters of each barrier unit in the plurality of barrier units being within a predetermined range of predetermined barrier-related parameters, determining the transmission coefficients corresponding to barrier-related parameters adjacent to the corresponding barrier-related parameters from a pre-interpolation table based on the barrier-related parameters corresponding to each barrier unit; and determining the first transmission coefficient and the second transmission coefficient by an interpolation algorithm based on the transmission coefficients corresponding to each barrier-related parameter.

[0020] In some embodiments, determining the first transmission coefficient of each first barrier unit in one or more first barrier units and the second transmission coefficient of each second barrier unit in one or more second barrier units based on a predetermined correspondence between barrier-related parameters and corresponding transmission coefficients includes: in response to a barrier-related parameter in one or more barrier units exceeding the range of a predetermined barrier-related parameter, expanding a pre-interpolation table to generate an extrapolation pre-interpolation table; determining the transmission coefficient corresponding to the barrier-related parameter adjacent to the corresponding barrier-related parameter from the extrapolation pre-interpolation table based on the barrier-related parameter corresponding to each barrier unit; and determining the first transmission coefficient and the second transmission coefficient using an interpolation algorithm based on the transmission coefficient corresponding to each barrier-related parameter.

[0021] In some embodiments, in response to a barrier-related parameter of a semiconductor device exceeding a predetermined range of barrier-related parameters, expanding the pre-interpolation table to generate an extrapolation pre-interpolation table includes: determining a first transmission coefficient variation trend relative to the barrier-related parameter and a second transmission coefficient variation trend relative to the barrier-related parameter for a plurality of first data points adjacent to the barrier-related parameter of the barrier under test within the predetermined range of the barrier-related parameter; discretizing the barrier-related parameter based on the portion of the barrier-related parameter of the barrier under test exceeding the predetermined range of the predetermined barrier-related parameter and a predetermined step size to generate a plurality of second data points; and determining an expanded transmission coefficient corresponding to each of the plurality of second data points based on the determined first transmission coefficient variation trend relative to the barrier-related parameter and the second transmission coefficient variation trend relative to the barrier-related parameter, wherein the expanded transmission coefficient includes a first expanded transmission coefficient corresponding to a first barrier unit having a constant barrier height and a second expanded transmission coefficient corresponding to a second barrier unit having a linearly varying barrier height.

[0022] In some embodiments, in response to a barrier-related parameter of the barrier under test of a semiconductor device exceeding a predetermined range of barrier-related parameters, expanding the pre-interpolation table to generate an extrapolation pre-interpolation table includes: discretizing the barrier-related parameter based on a portion of the barrier-related parameter of the barrier under test that exceeds a predetermined range of the predetermined barrier-related parameter and a predetermined step size to generate a plurality of second data points; determining a third extended transmission coefficient corresponding to each of the plurality of second data points using a first analytical expression for a first barrier cell; and determining a fourth extended transmission coefficient corresponding to each of the plurality of second data points using a second analytical expression for a second barrier cell, wherein the extended transmission coefficient includes a first extended transmission coefficient corresponding to a first barrier cell having a constant barrier height and a second extended transmission coefficient corresponding to a second barrier cell having a linearly varying barrier height.

[0023] In some embodiments, in response to the barrier-related parameters of one or more barrier cells in a plurality of barrier cells exceeding the range of predetermined barrier-related parameters, expanding the pre-interpolation table to generate an extrapolation pre-interpolation table includes: expanding the pre-interpolation table to generate an extrapolation pre-interpolation table based on the correspondence between each of the plurality of second data points and the corresponding first extended transmission coefficient and the corresponding second extended transmission coefficient.

[0024] In some embodiments, the interpolation algorithm includes a trilinear interpolation algorithm and a cubic spline interpolation algorithm.

[0025] In some embodiments, determining the total transmission coefficient of the barrier of a semiconductor device based on a first transmission coefficient and a second transmission coefficient includes multiplying the corresponding first transmission coefficient and second transmission coefficient to generate the total transmission coefficient.

[0026] In some embodiments, the predetermined correspondence includes a pre-interpolation table generated according to the method of the first aspect.

[0027] In some embodiments, barrier-related parameters include barrier length, barrier width, and electron energy.

[0028] In a third aspect of this disclosure, an electronic device is provided. The electronic device includes a processor; and a memory coupled to the processor, the memory having instructions stored therein, the instructions causing the device to perform an action when executed by the processor, the action including: discretizing the barrier-related parameters based on a predetermined range and a predetermined step size of predetermined barrier-related parameters to generate a plurality of first data points; and determining a transmission coefficient corresponding to each of the plurality of first data points, wherein the transmission coefficient includes a first transmission coefficient corresponding to a first barrier unit having a constant barrier height and a second transmission coefficient corresponding to a second barrier unit having a linearly varying barrier height.

[0029] In some embodiments, the electronic device is capable of performing the methods of either the first or second aspect.

[0030] In a fourth aspect of this disclosure, a computer-readable storage medium is provided. A computer program is stored on this computer-readable storage medium, which, when executed by a processor, implements the methods of the first and second aspects of this disclosure.

[0031] The solution of this disclosure divides the barrier of arbitrary shape into smaller barrier units to efficiently and accurately calculate the transmission coefficient of the barrier of arbitrary shape, thereby improving the calculation efficiency of tunneling current, reducing the amount of calculation and saving costs.

[0032] It should be understood that the description in the Summary of the Invention is not intended to limit the key or essential features of the embodiments of this disclosure, nor is it intended to restrict the scope of this disclosure. Other features of this disclosure will become readily apparent from the following description. Attached Figure Description

[0033] The above and other features, advantages, and aspects of the embodiments of this disclosure will become more apparent from the accompanying drawings and the following detailed description. In the drawings, the same or similar reference numerals denote the same or similar elements, wherein:

[0034] Figure 1 A schematic diagram of an example environment in which embodiments of the present disclosure can be implemented is shown;

[0035] Figure 2 A flowchart of a simulation method for a semiconductor device according to some embodiments of the present disclosure is shown;

[0036] Figure 3 A flowchart of a method for processing the transmission coefficient of a semiconductor device according to some embodiments of the present disclosure is shown;

[0037] Figure 4 A barrier diagram along the electron tunneling direction is shown according to some embodiments of the present disclosure;

[0038] Figure 5 Barrier diagrams along the electron tunneling direction with rectangular barrier cells are shown according to some embodiments of the present disclosure.

[0039] Figure 6 Barrier diagrams with triangular barrier cells along the electron tunneling direction are shown according to some embodiments of the present disclosure;

[0040] Figure 7 A barrier diagram along the electron tunneling direction with random points set according to some embodiments of the present disclosure is shown;

[0041] Figure 8 A block diagram of a computing device capable of implementing several embodiments of the present disclosure is shown. Detailed Implementation

[0042] Embodiments of this disclosure will now be described in more detail with reference to the accompanying drawings. While some embodiments of this disclosure are shown in the drawings, it should be understood that this disclosure can be implemented in various forms and should not be construed as limited to the embodiments set forth herein. Rather, these embodiments are provided to provide a more thorough and complete understanding of this disclosure. It should be understood that the accompanying drawings and embodiments of this disclosure are for illustrative purposes only and are not intended to limit the scope of protection of this disclosure.

[0043] In the description of embodiments of this disclosure, the term "comprising" and similar terms should be understood as open-ended inclusion, i.e., "including but not limited to". The term "based on" should be understood as "at least partially based on". The term "one embodiment" or "the embodiment" should be understood as "at least one embodiment". The terms "first", "second", etc., may refer to different or the same objects. The terms "equal" and "unequal" refer to equality or inequality within a specific tolerance range. The specific tolerance may be a conventional value in the art or may be set by those skilled in the art according to specific implementation methods, and this disclosure does not limit it. Other explicit and implicit definitions may also be included below.

[0044] As mentioned earlier, with the miniaturization of semiconductor devices, quantum effects become increasingly significant, directly impacting the functionality and energy consumption of these devices and making it difficult to accurately calculate tunneling current. Efficient and accurate calculation of tunneling current is crucial for device simulation in semiconductor device design and development. Therefore, it is necessary to accurately calculate the tunneling current of the semiconductor device's potential barrier during device simulation.

[0045] In traditional calculation methods, the tunneling current I can be calculated using the following formula (1):

[0046] (1)

[0048] Where e is the electron charge, m For the effective mass of electrons in dielectric materials, To reduce Planck's constant, f L (E) is the electron energy distribution function on the left (at the barrier entrance), f R (E) is the electron energy distribution function on the right side (at the barrier exit), and T(E) is the total transmission coefficient of the barrier. The electron energy distribution function f(E) is usually a Fermi-Dirac distribution and is calculated using the following formula (2):

[0049] (2)

[0050] Among them, E F For the Fermi level, k B is Boltzmann's constant, and T is the absolute temperature (K).

[0051] Based on formulas (1) and (2), it can be seen that the conventional method for calculating tunneling current requires integrating the transmission coefficient with electron energy E as a variable. To ensure calculation accuracy, a very small integration step size is required. This results in a large amount of computation and low simulation efficiency.

[0052] In view of this, this disclosure provides an improved solution.

[0053] According to some embodiments of this disclosure, a simulation method for a semiconductor device is provided. The method includes: discretizing the barrier-related parameters based on a predetermined range and a predetermined step size to generate a plurality of first data points; and determining a transmission coefficient corresponding to each of the plurality of first data points. The transmission coefficient includes a first transmission coefficient corresponding to a first barrier cell having a constant barrier height and a second transmission coefficient corresponding to a second barrier cell having a linearly varying barrier height.

[0054] This simulation method does not require integrating the transmission coefficient to calculate the tunneling current of the device, thus overcoming the problems of low efficiency and high computational cost of known schemes, and providing strong support for the design and optimization of semiconductor devices.

[0055] The embodiments of this disclosure will now be described in detail with reference to the accompanying drawings. First, refer to... Figure 1 , Figure 1 A schematic diagram of an example environment 100 that can be implemented according to embodiments of the present disclosure is shown. Figure 1 As shown, the example environment 100 includes a computing device 110 and a client 120.

[0056] In some embodiments, the computing device 110 can interact with the client 120. For example, the computing device 110 can receive input messages from the client 120 and output feedback messages to the client 120. In some embodiments, the input messages from the client 120 can be relevant data information of semiconductor devices, such as barrier-related parameters. The computing device 110 can perform corresponding mathematical operations on the aforementioned relevant data information and output the corresponding operation results to the client 120.

[0057] In some embodiments, the computing device 110 may include, but is not limited to, a personal computer, a server computer, a handheld or laptop device, a mobile device (such as a mobile phone, a personal digital assistant PDA, a media player, etc.), a consumer electronics product, a minicomputer, a mainframe computer, cloud computing resources, etc.

[0058] It should be understood that the description of the structure and functionality of example environment 100 for illustrative purposes only is not intended to limit the scope of the subject matter described herein. The subject matter described herein may be implemented in different structures and / or functionalities. This environment is merely illustrative and is not intended to limit the application environment of the embodiments of this disclosure.

[0059] To more clearly explain the principles of this disclosure, reference will be made below. Figures 2 to 8 Let's describe it in more detail.

[0060] Figure 2 A flowchart of a simulation method 200 for a semiconductor device according to some embodiments of the present disclosure is shown.

[0061] At step 202, the barrier-related parameters are discretized based on a predetermined range and a predetermined step size to generate a plurality of first data points. In some embodiments, the semiconductor device may include a planar metal-oxide-semiconductor field-effect transistor (MOSFET), a fin field-effect transistor (FinFET), a gate-around transistor (GAA), and a complementary field-effect transistor (CFET), etc. The semiconductor device in this application may also be any other electronic device having a barrier and a corresponding tunneling effect. These electronic devices may be existing or future developments, and this disclosure does not limit them. In some embodiments, the barrier-related parameters include barrier length, barrier width, and electron energy. In practice, the predetermined range and predetermined step size of the barrier-related parameters can be determined according to actual needs. As an example, the predetermined range of the barrier height V0 can be [0.0, 10.0] eV, with a predetermined step size ΔV0 = 0.01 eV; the predetermined range of the barrier width L can be [0.0, 50.0] nm, with a predetermined step size ΔL = 0.1 nm; and the predetermined range of the electron energy E can be [0.0, 10.0] eV, with a predetermined step size ΔE = 0.01 eV. The first data point can be represented as [V0, L, E]. In other embodiments, the predetermined step size for the same barrier-related parameter can be different within different ranges of the predetermined range. The smaller the predetermined step size, the higher the resolution and accuracy, but the corresponding computational load and data volume are also larger. Those skilled in the art can set the predetermined step size according to specific needs.

[0062] In some embodiments, the parameter range and step size of the pre-interpolation table can be determined based on a balance between actual device requirements and computational efficiency. For example, the barrier height V0 ranges from [0,10] eV with a step size of 0.01 eV to cover typical barrier values ​​of the gate dielectric of semiconductor devices while maintaining accuracy; the barrier width L ranges from [0,50] nm with a step size of 0.1 nm to adapt to the gate dielectric layer thickness in the process node and avoid excessive memory usage caused by data storage; the electron energy E ranges from [0,10] eV with a step size of 0.01 eV to ensure high-precision calculations near the Fermi level. The dynamic expansion mechanism supplements the data exceeding the range through extrapolation and real-time calculation, and gradually expands the table with a fixed step size (e.g., 0.01 eV) to balance storage and accuracy requirements.

[0063] At step 204, a transmission coefficient corresponding to each of the plurality of first data points is determined. The transmission coefficient includes a first transmission coefficient corresponding to a first barrier unit having a constant barrier height and a second transmission coefficient corresponding to a second barrier unit having a linearly varying barrier height. That is, two transmission coefficients can be determined for each first data point for subsequent selective use. In some embodiments, the transmission coefficient corresponding to each of the plurality of first data points can be determined by the following steps: using a first analytical expression for the first barrier unit, determining the first transmission coefficient corresponding to each of the plurality of first data points; and using a second analytical expression for the second barrier unit, determining the second transmission coefficient corresponding to each of the plurality of first data points.

[0064] In some embodiments, the first barrier cell with a constant barrier height may be referred to as a rectangular barrier cell. The first transmission coefficient T corresponding to the rectangular barrier cell can be determined by the following formula (3). rect (E):

[0065] (3)

[0066] Where e is the electron charge and m is the electron mass. Here, E is the reduced Planck constant, L is the barrier width, and V0 is the barrier height.

[0067] In some embodiments, the second barrier unit with a linearly varying barrier height may be referred to as a triangular barrier unit. The second transmission coefficient T corresponding to the triangular barrier unit can be determined by the following formula (4). tri (E):

[0068] (4)

[0069] Where e is the electron charge and m is the electron mass. To reduce Planck's constant, E is the electron energy, L is the barrier width, and V0 is the barrier height. Therefore, through the above process, for each first data point [V0, L, E], a corresponding first transmission coefficient T can be obtained. rect (E) and a second transmission coefficient T tri (E).

[0070] In some embodiments of this disclosure, rectangular and triangular barrier cells are used as examples for illustration. It should be understood that the embodiments of this disclosure are not limited thereto. In some embodiments, other barrier cells may be used as needed.

[0071] In some embodiments, the simulation method 200 may further include: generating a pre-interpolation table for determining the transmission coefficient of a potential barrier of a semiconductor device based on the correspondence between each of the plurality of first data points and its corresponding first transmission coefficient and corresponding second transmission coefficient. As mentioned above, for each first data point [V0, L, E], a corresponding first transmission coefficient T can be obtained. rect (E) and a second transmission coefficient T tri (E). Based on this correspondence, a pre-interpolation table is established, which facilitates the subsequent determination of the first or second transmission coefficient of the corresponding data point based on the pre-interpolation table, saving tedious calculations, improving the calculation efficiency of the transmission coefficient, and thus improving the calculation efficiency of the tunneling current and the simulation efficiency of the corresponding device.

[0072] When constructing the pre-interpolation table, it can be understood as building a three-dimensional data table using the barrier height V0, barrier width L, and electron energy E as a three-dimensional space. When querying the transmission coefficient, the location of the point to be queried can be determined first, and the data of the eight adjacent points can be found. Interpolation calculations are then performed on these data to obtain the transmission coefficient of that point. It should be understood that the eight adjacent points mentioned here are merely an example of this disclosure, and the scope of this disclosure is not limited thereto and can be changed according to actual needs.

[0073] For ease of understanding, let's assume we've constructed a simple two-column data interpolation table as shown below. The corresponding coefficients can be obtained using the analytical formula described above, for example:

[0074] {0, 0, 0} -> Rectangular transmission coefficient 0, triangular transmission coefficient 0;

[0075] {10, 10, 10} -> Rectangular transmission coefficient 1, Triangular transmission coefficient 2.

[0076] In other words, for the parameter {0, 0, 0}, the corresponding rectangular transmission coefficient is 0, and the triangular transmission coefficient is 0; for the parameter {10, 10, 10}, the corresponding rectangular transmission coefficient is 1, and the triangular transmission coefficient is 2. It should be understood that this is merely illustrative for ease of understanding, and the embodiments of this disclosure are not limited thereto. When using this interpolation table subsequently, the appropriate coefficients can be selected based on whether the specific barrier unit corresponds to a rectangular or triangular barrier.

[0077] In some embodiments, where the barrier-related parameters of the barrier under test of a semiconductor device exceed the range of barrier-related parameters in an established pre-interpolation table, this disclosure also provides a method for expanding the pre-interpolation table to update the pre-interpolation table and expand its range to cover the range of barrier-related parameters of the semiconductor device under test.

[0078] In some embodiments, the simulation method 200 may further include: expanding the pre-interpolation table when the barrier-related parameters of the barrier to be measured in the semiconductor device exceed the range of predetermined barrier-related parameters. In some embodiments, the pre-interpolation table may be extrapolated based on the trend of the transmission coefficient of first data points within the range of the pre-interpolation table adjacent to the barrier-related parameters to generate an extrapolated pre-interpolation table. Specifically, the process of expanding the pre-interpolation table may include: determining the trend of the first transmission coefficient relative to the barrier-related parameters and the trend of the second transmission coefficient relative to the barrier-related parameters for a plurality of first data points within the range of predetermined barrier-related parameters adjacent to the barrier-related parameters of the barrier to be measured. In some embodiments, the trend may include linear or exponential changes. This disclosure is not limited in this respect, and the trend may also be of other types, as long as the first transmission coefficient or the second transmission coefficient corresponding to the subsequent barrier-related parameters can be determined based on the trend. As an example, the transmission coefficient (including the first transmission coefficient and the second transmission coefficient) of the target data point (V0, L, E) is determined based on the trend.

[0079] In some embodiments, whether it is linear can be determined in the following ways.

[0080] Data fitting: In the pre-interpolation table, select multiple adjacent data points (e.g., V0=9.98, 9.99 eV, 10.00 eV) close to the boundary (e.g., the maximum value of V0) and calculate the trend of the transmission coefficient T with V0.

[0081] Linear Regression: Perform linear regression analysis (least squares method) on these data points and calculate the correlation coefficient R of the fitted line. 2 If R 2 Close to 1 (e.g., R) 2 If the value is greater than 0.99, then the relationship between T and V0 is considered to be approximately linear.

[0082] As an example, the barrier height V0 of the target data point exceeds the maximum value V of the barrier height in the pre-interpolation table. max The barrier width L and electron energy E are less than the maximum values ​​of the barrier height and electron energy in the pre-interpolation table, respectively. When the transmission coefficient changes linearly with the barrier height, the transmission coefficient T of the target data point can be calculated using the following formulas (5) and (6). extrap :

[0083] (5)

[0084] (6)

[0085] Among them, T max For the data point with the largest barrier height (V) in the pre-interpolation table maxThe transmission coefficients of L and E are given, K is the linear slope at the boundary point, and ΔV0 represents the step size of the barrier height. It should be understood that the transmission coefficient T of the target data point... extrap The first transmission coefficient corresponding to the rectangular barrier unit and the second transmission coefficient corresponding to the triangular barrier unit are calculated based on the first and second transmission coefficients of the boundary points using the above formulas (5) and (6), respectively.

[0086] In some embodiments, it can be determined whether it is exponential decay in the following ways.

[0087] Logarithmic transformation: Take the natural logarithm of T (ln T) and observe its relationship with V0. If ln T has a linear relationship with V0 (as mentioned above in the linearity judgment), then the original T has an exponentially decreasing relationship with V0.

[0088] Exponential fitting: using the exponential function Fit the data points and calculate the goodness of fit (e.g., R²). 2 Or the sum of squared residuals). If the goodness of fit is high (e.g., R0), then the goodness of fit is high (e.g., R0). 2 If the value is greater than 0.98, it is considered exponential decay.

[0089] As an example, when the transmission coefficient decreases exponentially with the barrier height, the transmission coefficient T of the target data point can be calculated using the following formulas (7) and (8). extrap :

[0090] (7)

[0091] (8)

[0092] Among them, T max For the data point with the largest barrier height (V) in the pre-interpolation table max The transmission coefficients of L and E are given by , k is the exponential decay coefficient at the boundary point, and ΔV0 represents the step size of the barrier height. Similarly, the transmission coefficient T of the target data point is given by . extrap The first transmission coefficient corresponding to the rectangular barrier unit and the second transmission coefficient corresponding to the triangular barrier unit are calculated based on the first and second transmission coefficients of the boundary points using the above formulas (7) and (8), respectively.

[0093] The above embodiments are described using examples where the barrier height exceeds a predetermined range. If the barrier width or electron energy exceeds a predetermined range, a similar process can be used to extrapolate the pre-interpolation table, which will not be elaborated here.

[0094] To improve the accuracy of the extrapolation pre-interpolation table generated by the above extension method, the transmission coefficient of the target data points can be calculated using formula (9). The real-time transmission coefficient T determined by formulas (3) and (4) real Relative error RE:

[0095] (9)

[0096] When the relative error RE exceeds the predetermined allowable error, embodiments of this disclosure can switch to real-time calculation, i.e., recalculate the first and second transmission coefficients for each second data point according to (3) and (4).

[0097] The newly calculated parameter combination {V0, L, E, T} can be stored in a pre-interpolation table, expanding its range. For portions of V0 that exceed the range, the pre-interpolation table is expanded incrementally, for example, by a step size ΔV0 = 0.01 eV (e.g., from [0,10] eV to [0,12] eV). A similar strategy is used to expand portions of L or E that exceed the range.

[0098] Therefore, embodiments of this disclosure also provide another process for extrapolating a pre-interpolation table, which may further include: discretizing the barrier-related parameters based on a portion of the barrier-related parameters that exceeds a predetermined range and a predetermined step size to generate a plurality of second data points. In some embodiments, the range exceeding the predetermined range is discretized with the same step size as the first data points to generate the second data points. However, this disclosure is not limited thereto. As mentioned above, the step size can be set according to specific actual needs, and different step sizes can be set for the same barrier-related parameter within the range exceeding the predetermined range.

[0099] In some embodiments, the process of expanding the pre-interpolation table may further include: determining an expanded transmission coefficient corresponding to each of a plurality of second data points based on the determined variation trends of the first transmission coefficient with respect to the barrier-related parameters and the second transmission coefficient with respect to the barrier-related parameters. The expanded transmission coefficient includes a first expanded transmission coefficient corresponding to a first barrier cell with a constant barrier height and a second expanded transmission coefficient corresponding to a second barrier cell with a linearly varying barrier height. As previously mentioned, the first barrier cell includes a rectangular barrier cell, and the second barrier cell includes a triangular barrier cell. Therefore, the second data points also include rectangular barrier cells and triangular barrier cells. Similar to the first data points, for each second data point [V0, L, E], a corresponding first expanded transmission coefficient T can be obtained. rect (E) and a second extended transmission coefficient T tri (E), which will not be elaborated upon here.

[0100] In some embodiments, the process of expanding the pre-interpolation table to generate an extrapolation pre-interpolation table may further include: expanding the pre-interpolation table based on the correspondence between each of the plurality of second data points and its corresponding first extended transmission coefficient and corresponding second extended transmission coefficient to generate an extrapolation pre-interpolation table. This method can update the pre-interpolation table, avoid real-time calculations, and reduce computational load while maintaining accuracy, thereby improving the calculation efficiency of tunneling current and the simulation efficiency of the device.

[0101] In embodiments where the extrapolation method described above has significant errors, a new method is needed to dynamically update the pre-interpolation table. Therefore, in addition to the method described above for extrapolating the pre-interpolation table based on the changing trend of the transmission coefficients relative to the barrier-related parameters within adjacent pre-interpolation table ranges, embodiments of this disclosure also propose another method for extrapolating the pre-interpolation table. This method recalculates the first and second extended transmission coefficients corresponding to the second data points based on analytical expressions. Specifically: the barrier-related parameters are discretized based on the portion of the barrier-related parameters of the barrier under test that exceeds a predetermined range and a predetermined step size to generate multiple second data points; and the extended transmission coefficient corresponding to each of the multiple second data points is determined, the extended transmission coefficient including a first extended transmission coefficient corresponding to a first barrier unit with a constant barrier height and a second extended transmission coefficient corresponding to a second barrier unit with a linearly varying barrier height. In some embodiments, the transmission coefficient corresponding to each of the plurality of second data points can be determined by the following steps: determining a first extended transmission coefficient corresponding to each of the plurality of second data points using a first analytical expression for a first barrier cell; and determining a second extended transmission coefficient corresponding to each of the plurality of second data points using a second analytical expression for a second barrier cell.

[0102] The process of extrapolating the pre-interpolation table is similar to the previous reference. Figure 2 The process of establishing the pre-interpolation table is similar and will not be repeated here. It should be noted that, as mentioned earlier, in some embodiments, the range exceeding the predetermined range is discretized with the same step size as the first data point to generate the second data point. However, this disclosure does not limit this. The step size can be set according to specific actual needs, and different step sizes can be set for the same barrier-related parameter within the range exceeding the predetermined range.

[0103] In some embodiments, taking barrier interpolation as an example, a pre-interpolation table is used for 0-10 eV, and an extrapolation method is used for 10-12 eV. The extrapolation method utilizes the pre-interpolation table for numerical extrapolation, enabling fast calculation while maintaining a certain level of accuracy and reducing memory usage (no table is needed for 10-12 eV). When a case of 11 eV occurs, according to section 3.2.3 extrapolation error control, if the extrapolation error is large, it indicates that the original pre-interpolation table and extrapolation cannot cover this case, and a 10-12 eV pre-interpolation table needs to be further constructed.

[0104] It should be noted that although some embodiments of this disclosure use extrapolation interpolation tables to calculate the transmission coefficient for data points exceeding a predetermined range, those skilled in the art will understand that the transmission coefficient for a target data point can be obtained directly based on the trend of change or calculated in real time. This disclosure does not impose any limitations in this regard.

[0105] The pre-interpolation table generated through the above process can be used to calculate the total transmission coefficient of the barrier under test of the semiconductor device and, based on this, calculate the tunneling current and perform simulation. The following will combine... Figures 3 to 7 Describe it.

[0106] Figure 3 A flowchart of a transmission coefficient processing method 300 for a semiconductor device according to some embodiments of the present disclosure is shown.

[0107] At step 302, the potential barrier of the semiconductor device along the electron tunneling direction is divided into multiple barrier units. The multiple barrier units include one or more first barrier units with a constant barrier height and one or more second barrier units with a linearly varying barrier height. As previously mentioned, barrier-related parameters include barrier length, barrier width, and electron energy. The first barrier unit may be referred to as a rectangular barrier unit, and the second barrier unit may be referred to as a triangular barrier unit. In some embodiments, the potential barrier of the semiconductor device along the electron tunneling direction can be divided into multiple barrier units by: dividing the barrier along the electron tunneling direction using a bisection method, dividing the barrier into multiple first barrier units and second barrier units. Specifically, a first slope of the line connecting the left endpoint and the midpoint of the barrier and a second slope of the line connecting the right endpoint and the midpoint are determined; and if the first slope is not equal to the second slope, the left endpoint and the midpoint, and the midpoint and the right endpoint are bisected until the first slope and the second slope are equal.

[0108] In some embodiments, the bisection method may include: determining a first slope of the line connecting the left endpoint and the midpoint of the barrier and a second slope of the line connecting the right endpoint and the midpoint; and, if the first slope is not equal to the second slope, bisecting the left endpoint and the midpoint, and the midpoint and the right endpoint, until the first slope and the second slope are equal. Additionally, to ensure the accuracy of the barrier unit division, the bisection method further includes: if the first slope and the second slope of the barrier unit in the divided barrier are determined to be equal, randomly selecting a random point between the two endpoints of the barrier unit and continuing to bisect the barrier unit to verify whether the first slope of the line connecting the random point to the left endpoint of the barrier unit and the second slope of the line connecting the random point to the right endpoint of the barrier unit are equal; and if the first slope and the second slope are determined to be unequal, continuing to bisect the barrier unit until the first slope and the second slope are equal or a predetermined number of divisions is reached. This allows for the accurate division of arbitrarily shaped barriers into rectangular and triangular barrier cells, providing a foundation for determining the total transmission coefficient of the barrier based on a pre-interpolation table and ensuring the accuracy of tunneling current calculations and device simulations based on the transmission coefficient. It is understood that those skilled in the art can set the specific maximum number of divisions according to the accuracy requirements and computational capabilities of the simulation in a particular implementation. This step will be described below with reference to... Figures 4 to 7 Provide a detailed description.

[0109] At step 304, based on a predetermined correspondence between the barrier-related parameters and the corresponding transmission coefficients, a first transmission coefficient of each of the one or more first barrier cells and a second transmission coefficient of each of the one or more second barrier cells are determined. In some embodiments, the predetermined relationship includes the pre-interpolation table generated above. This pre-interpolation table stores transmission coefficients corresponding to the predetermined barrier-related parameters.

[0110] In some embodiments, the creation of the pre-interpolation table typically does not require device information, as long as the maximum value of the table can cover the possible maximum barrier values. A device may need to calculate barrier shapes in multiple directions and regions; therefore, any barrier shape can be decomposed into the smallest barrier unit and the transmission coefficient can be obtained by looking up the table. For this purpose, the pre-interpolation table can be generated in advance, for example, by setting a predetermined range of barrier-related parameters, using the corresponding analytical expression (as described above) to determine the transmission coefficient corresponding to each data point within the set range, and storing it in the table.

[0111] In some embodiments, the pre-interpolation table can also be generated when determining the tunneling current for the barrier of the device under test, for example, by reading the barrier-related parameters of the device and constructing a corresponding pre-interpolation table for the range of the read barrier-related parameters.

[0112] In some embodiments, when the barrier-related parameters of each barrier unit in a plurality of barrier units are within a predetermined range of predetermined barrier-related parameters, the transmission coefficients corresponding to the barrier-related parameters adjacent to the corresponding barrier-related parameters are determined from a pre-interpolation table based on the barrier-related parameters corresponding to each barrier unit; and the first transmission coefficient and the second transmission coefficient are determined by an interpolation algorithm based on the transmission coefficients corresponding to each barrier-related parameter. In some embodiments, the interpolation algorithm includes a trilinear interpolation algorithm and a cubic spline interpolation algorithm. As an example, by using the trilinear interpolation algorithm, in a three-dimensional network, the transmission coefficient T(x, y, z) of the test data point (x, y, z) is calculated based on a linear combination of 8 adjacent points using the following formula (10):

[0113] (10)

[0114] Among them, w ijk For corresponding to (x) i , y j , z k The weighting coefficients of the test data point (x, y, z) are inversely proportional to the distance between the test data point (x, y, z) and its neighboring points. x represents the barrier height, y represents the barrier width, z represents the electron energy, and i, j, and k are indices used to traverse the 8 neighboring points, which are 0 or 1 respectively.

[0115] As an example, for ease of understanding, suppose we decompose the directional barrier of a device, ultimately obtaining multiple small triangular and rectangular barriers t1, t2, t3, t4. We then interpolate the transmission coefficients of these triangular and rectangular barriers using an interpolation table, for example:

[0116] Triangle t1 {0.1, 0.1, 0.1}: 0.2;

[0117] t2 rectangle {0.1, 0.1, 0.1}: 0.1;

[0118] t3 triangle {0.5, 0.5, 0.5}: 1.0;

[0119] t4 rectangle {0.5, 0.5, 0.5}: 0.5.

[0120] This table can be used for interpolation as long as the value of the small barrier is within {10, 10, 10}.

[0121] Alternatively, in some other embodiments, if the barrier-related parameters of one or more barrier units in a plurality of barrier units exceed the range of predetermined barrier-related parameters, the pre-interpolation table is expanded to generate an extrapolation pre-interpolation table; based on the barrier-related parameters corresponding to each barrier unit, the transmission coefficient corresponding to the barrier-related parameter adjacent to the corresponding barrier-related parameter is determined from the extrapolation pre-interpolation table; based on the transmission coefficients corresponding to each barrier-related parameter, a first transmission coefficient and a second transmission coefficient are determined by an interpolation algorithm. (Refer to the above...) Figure 2 In conjunction with the descriptions in formulas (4) to (8), some embodiments of this disclosure provide two alternative processes for expanding the pre-interpolation table. When the error between the transmission coefficient determined by the method of extrapolating the pre-interpolation table based on the changing trend and the transmission coefficient calculated in real time is greater than a predetermined threshold, the method of extrapolating the pre-interpolation table by real-time calculation can be selected. For distinction, the transmission coefficient of the rectangular barrier unit determined by real-time calculation can be referred to as the third extended transmission coefficient, and the transmission coefficient of the triangular barrier unit determined by real-time calculation can be referred to as the fourth extended coefficient. Specifically, instead of determining the extended transmission coefficient corresponding to each of the plurality of second data points to generate the extrapolation pre-interpolation table based on the changing trend of the determined first transmission coefficient relative to the barrier-related parameters and the changing trend of the second transmission coefficient relative to the barrier-related parameters, the method of determining the extended transmission coefficient corresponding to each of the plurality of second data points to generate the extrapolation pre-interpolation table is: using the first analytical expression for the first barrier unit to determine the third extended transmission coefficient corresponding to each of the plurality of second data points; and using the second analytical expression for the second barrier unit to determine the fourth extended transmission coefficient corresponding to each of the plurality of second data points. The extended transmission coefficient includes a third extended transmission coefficient corresponding to a first barrier cell with a constant barrier height and a fourth extended transmission coefficient corresponding to a second barrier cell with a linearly varying barrier height.

[0122] In some embodiments, the pre-interpolation table can be expanded by real-time calculation if the error between the first extended transmission coefficient and the third extended transmission coefficient, and / or the error between the second extended transmission coefficient and the fourth extended transmission coefficient, of any second data point among a plurality of second data points is greater than a predetermined error. The predetermined error can be selected by a person skilled in the art based on actual conditions. Real-time calculation improves accuracy compared to trend-based calculation. In some embodiments, the predetermined error can be represented by a relative error. Specifically, the pre-interpolation table can be expanded by real-time calculation when the relative error between the first extended transmission coefficient and the third extended transmission coefficient, and / or the relative error between the second extended transmission coefficient and the fourth extended transmission coefficient, of any second data point among a plurality of second data points is greater than a predetermined relative error. In other words, for data points outside a predetermined range, if the difference between the transmission coefficient calculated based on the trend and the real-time calculated transmission coefficient for any data point is greater than a predetermined error, or the relative error between the two is greater than a predetermined relative error, the pre-interpolation table is extrapolated by real-time calculation. Those skilled in the art can set the predetermined error or the predetermined relative error according to actual needs, for example, a predetermined relative error of 5%. Furthermore, it is understood that those skilled in the art can, according to actual needs, for example, choose to expand the pre-interpolation table using real-time calculation methods to improve accuracy, or choose to expand the pre-interpolation table using trend-based calculation methods to reduce computational load, or combine trend-based calculation methods with real-time calculation methods across different potential barrier-related parameters to achieve a trade-off between accuracy and computational load. This disclosure does not impose any limitations in this regard.

[0123] Additionally, in some embodiments, the method for expanding the pre-interpolation table based on the changing trend further includes: expanding the pre-interpolation table based on the correspondence between each of the plurality of second data points and its corresponding first extended transmission coefficient and its corresponding second extended transmission coefficient to generate an extrapolation pre-interpolation table. Alternatively, in some embodiments, the method for expanding the pre-interpolation table in real time may further include: expanding the pre-interpolation table based on the correspondence between each of the plurality of second data points and its corresponding third extended transmission coefficient and its corresponding fourth extended transmission coefficient to generate an extrapolation pre-interpolation table.

[0124] At step 306, the total transmission coefficient of the semiconductor device barrier is determined based on the first and second transmission coefficients. In some embodiments, the corresponding first and second transmission coefficients are multiplied to generate the total transmission coefficient. Specifically, the total transmission coefficient T... total (E) can be calculated using the following formula (11):

[0125] (11)

[0126] Where N is a positive integer greater than or equal to 1, representing the total number of barrier segments, and T i (E) is the transmission coefficient of the i-th barrier unit, where i is a positive integer.

[0127] In some embodiments, similar to method 200, method 300 may also update the pre-interpolation table. Specifically, method 300 may further include: expanding the pre-interpolation table based on the correspondence between each of the plurality of second data points and its corresponding first extended transmission coefficient and corresponding second extended transmission coefficient, to generate an extrapolation pre-interpolation table. The updated extrapolation pre-interpolation table can avoid real-time calculations, reducing the computational load while ensuring accuracy, thereby improving the calculation efficiency of tunneling current and the simulation efficiency of the device.

[0128] Figure 4 A barrier diagram along the electron tunneling direction during a partitioning process according to some embodiments of the present disclosure is shown. The barrier diagram includes five points A0, B0, A1, B1, and A2, where point A1 is the midpoint between A0 and A2, point B0 is the midpoint between A0 and A1, and point B1 is the midpoint between A1 and A2. The x-axis represents the electron tunneling direction, and the y-axis represents the barrier height.

[0129] like Figure 4 As shown, the horizontal axis X represents the barrier width (L), the vertical axis Y represents the barrier height (V), and the first slope K1 between the midpoint A1 and the corresponding left endpoint A0 is (V). A1 -V A0 ) / (X A1 -X A0 The second slope K2 of the midpoint A1 and the corresponding right endpoint A2 is (V A2 -V A1 ) / (X A2 -X A1 As shown in the figure, K1 is not equal to K2. Continuing to bisect the barrier unit, the first slope between the reset midpoint B0 and the corresponding left endpoint A0 is K1' = (V... B0 -V A0 ) / (X B0 -X A0 The second slope K2' between the reset midpoint B0 and the corresponding right endpoint A0 is K2' = (V A1 -V B0 ) / (X A1 -X B0 The first slope K1” between the reset midpoint B1 and the corresponding left endpoint A1 is (V B1 -V A1 ) / (X B1 -X A1), the second slope K2” between the reset midpoint B1 and the corresponding right endpoint A2 is = (V A2 -V B1 ) / (X A2 -X B1 ).like Figure 4 As shown, K1' is not equal to K2', and K1' is not equal to K2. Therefore, it is necessary to continue performing the bisection as shown above on the cells composed of points A0 and B0, the cells composed of points B0 and A1, the cells composed of points A1 and B1, and the cells composed of points B1 and A1. This continues until the first slope relative to any midpoint equals the second slope.

[0130] Figure 5 A barrier diagram with rectangular barrier cells along the electron tunneling direction is shown according to some embodiments of the present disclosure. The contents of its x and y axes are as follows: Figure 4 Same. The barrier diagram includes three points: A0, B0, and A1. (Similar to...) Figure 4 Similarly, point B0 is the midpoint between points A0 and A1, the x-axis represents the barrier width in the electron tunneling direction, and the y-axis represents the barrier height. For example... Figure 5 As shown, the first slope between the midpoint B0 and the left endpoint A0 is K1 = (V0 - V0) / (X). B0 -X A0 The second slope between the midpoint B0 and the corresponding right endpoint A1 is K2 = (V0 - V0) / (X). A1 -X B0 The first slope K1 is equal to the second slope K2, and both are equal to 0. In other words, the barrier unit formed by points A0 and A1 is a barrier unit with a constant barrier height, i.e., a rectangular barrier unit.

[0131] Figure 6 A barrier diagram with triangular barrier cells along the electron tunneling direction is shown according to some embodiments of the present disclosure. The contents of its x and y axes are related to... Figure 4 Same. The barrier diagram includes three points: A0, B0, and A1. (Similar to...) Figure 4 Similarly, point B0 is the midpoint between points A0 and A1, the x-axis represents the electron tunneling direction, and the y-axis represents the potential barrier height. For example... Figure 6 As shown, the first slope K1 between the midpoint B0 and the left endpoint A0 is (V B0 -V A0 ) / (X B0 -X A0 The second slope K2 between the midpoint B0 and the corresponding right endpoint A1 is (V A1 -V B0 ) / (X A1 -X B0The first slope K1 is equal to the second slope K2, and is not equal to 0. In other words, the barrier unit composed of points A0 and A1 is a barrier unit with a linearly changing barrier height, i.e., a triangular barrier unit.

[0132] To ensure the correctness of barrier partitioning and avoid erroneous partitioning (e.g., the midpoint satisfies the condition that the first slope equals the second slope, but the barrier cell is neither a rectangular nor a triangular barrier cell), further verification is performed after partitioning into multiple barrier cells. For example, method 300 further includes: if it is determined that the first slope and the second slope of the barrier cell in the partitioned barrier are equal, randomly selecting a random point between the two endpoints of the barrier cell to verify whether the first slope of the line connecting the random point to the left endpoint of the barrier cell and the second slope of the line connecting the random point to the right endpoint of the barrier cell are equal; and if it is determined that the first slope and the second slope are not equal, continuing to divide the barrier cell into two parts until the first slope and the second slope are equal or a predetermined number of partitions is reached. In some embodiments, the random point can be selected using a Monte Carlo random sampling method. However, this disclosure does not limit this, and other sampling methods can also be used. The random point can be one or more; the more random points selected, the higher the accuracy of cell partitioning, but the corresponding computational load is also increased. Therefore, those skilled in the art can choose according to specific implementation methods. This step will be referred to below. Figure 7 Provide a detailed description.

[0133] Figure 7 A potential barrier diagram with random points along the electron tunneling direction is shown according to some embodiments of the present disclosure. Its x and y axes represent the following... Figure 4 The same. The potential barrier diagram includes three points: A0, B0, A1, C0, and C1. Point B0 is the midpoint between points A0 and A1, C0 is a random point between points A0 and B0, and C1 is a random point between points B0 and A1. The x-axis represents the direction of electron tunneling, and the y-axis represents the height of the potential barrier.

[0134] like Figure 7 As shown, the first slope between the midpoint B0 and the left endpoint A0 is K1 = (V0 - V0) / (X). B0 -X A0 =0, the second slope K2 between the midpoint B0 and the corresponding right endpoint A1 is K2 = (V0 - V0) / (X) A1 -X B0 =0. As mentioned earlier, the barrier cell formed by points A0 and A1 can be determined as a barrier cell with a constant barrier height, i.e., a rectangular barrier cell. To verify the accuracy of this determination, random points C0 and C1 were set respectively. The first slope K1' between random point C0 and the left endpoint A0 is = (V C0 -V A0) / (X C0 -X A0 The second slope K2' between random points C0 and B0 is (V B0 -V C0 ) / (X B0 -X C0 As shown in the figure, K1' is not equal to K2', and is not equal to 0. The first slope between another random point C1 and B0 is K1” = (V C1 -V B0 ) / (X C1 -X B0 ), the second slope K2” between another random point C1 and A1 = (V A1 -V C1 ) / (X A1 -X C1 As shown in the figure, K1” is not equal to K2” and is not equal to 0. Therefore, it is necessary to continue to divide the barrier unit composed of points A0 and B0 and the barrier unit composed of points A1 and B0 into rectangular barrier units and triangular barrier units, or to reach the maximum number of divisions.

[0135] In some embodiments of this disclosure, the potential barrier is divided into rectangular potential barriers and triangular potential barriers by using a barrier segmentation method and a random sampling method (mainly used to exclude erroneous segmentation cases).

[0136] In some embodiments of this disclosure, a pre-interpolation table for transmittance coefficients within a predetermined range is determined. The transmittance coefficients within the predetermined range are retrieved using an interpolation method based on the pre-interpolation table. Specifically, within a predetermined error threshold range, the interpolation table is updated using linear or exponential extrapolation. This method avoids real-time integration when calculating transmittance coefficients, significantly reducing computational complexity. Furthermore, through error control and dynamic update mechanisms, accuracy is maintained while efficiency is improved.

[0137] In some embodiments of this disclosure, the trade-off between efficiency and accuracy in traditional transmission coefficient integration methods is resolved through barrier shape segmentation and transmission coefficient pre-interpolation tables, significantly improving the efficiency of tunneling current calculation. Barrier decomposition allows for calculation of barriers of arbitrary shapes by dividing the barrier into multiple barrier units based on the barrier slope, avoiding the use of extremely small integration step sizes in traditional methods. Acceleration is achieved through pre-interpolation tables; for example, by pre-calculating the transmission coefficient and constructing a pre-interpolation table, real-time integration is avoided, significantly reducing the computational load. Error control and dynamic update mechanisms ensure accuracy while improving efficiency.

[0138] Figure 8 A schematic block diagram of an example device 800 that can be used to implement embodiments of the present disclosure is shown. For example, Figure 2 The simulation method 200 and / or for semiconductor devices shownFigure 3 The transmission coefficient processing method 300 for semiconductor devices shown can be implemented by device 800. As shown, device 800 includes a central processing unit (CPU) 810, which can perform various appropriate actions and processes according to computer program instructions stored in read-only memory (ROM) 820 or loaded from storage unit 880 into random access memory (RAM) 830. Various programs and data required for the operation of device 800 can also be stored in RAM 830. CPU 810, ROM 820, and RAM 830 are interconnected via bus 840. Input / output (I / O) interface 850 is also connected to bus 840.

[0139] Multiple components in device 800 are connected to I / O interface 850, including: input unit 860, such as keyboard, mouse, etc.; output unit 870, such as various types of monitors, speakers, etc.; storage unit 880, such as disk, optical disk, etc.; and communication unit 890, such as network card, modem, wireless transceiver, etc. Communication unit 890 allows device 800 to exchange information / data with other devices through computer networks such as the Internet and / or various telecommunications networks.

[0140] Processing unit 810 executes the various methods and processes described above, such as method 200 and / or method 300. For example, in some embodiments, method 200 and / or method 300 may be implemented as computer software programs tangibly contained in a machine-readable medium, such as storage unit 880. In some embodiments, part or all of the computer program may be loaded and / or installed on device 800 via ROM 820 and / or communication unit 890. When the computer program is loaded into RAM 830 and executed by CPU 810, one or more steps of method 200 and / or method 300 described above may be performed. Alternatively, in other embodiments, CPU 810 may be configured to execute method 200 and / or method 300 by any other suitable means (e.g., by means of firmware).

[0141] The functions described above in this document can be performed at least in part by one or more hardware logic components. For example, exemplary types of hardware logic components that can be used, without limitation, include: field programmable gate arrays (FPGAs), application-specific integrated circuits (ASICs), application-specific standard products (ASSPs), systems-on-a-chip (SoCs), payload programmable logic devices (CPLDs), and so on.

[0142] The program code used to implement the methods of this disclosure may be written in any combination of one or more programming languages. This program code may be provided to a processor or controller of a general-purpose computer, special-purpose computer, or other programmable data processing apparatus, such that when executed by the processor or controller, the program code causes the functions / operations specified in the flowcharts and / or block diagrams to be implemented. The program code may be executed entirely on a machine, partially on a machine, as a standalone software package partially on a machine and partially on a remote machine, or entirely on a remote machine or server.

[0143] In the context of this disclosure, a machine-readable medium can be a tangible medium that may contain or store a program for use by or in conjunction with an instruction execution system, apparatus, or device. A machine-readable medium can be a machine-readable signal medium or a machine-readable storage medium. A machine-readable medium can be, but is not limited to, electronic, magnetic, optical, electromagnetic, infrared, or semiconductor systems, apparatus, or devices, or any suitable combination of the foregoing. More specific examples of machine-readable storage media include electrical connections based on one or more wires, portable computer disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), optical fiber, portable compact disk read-only memory (CD-ROM), optical storage devices, magnetic storage devices, or any suitable combination of the foregoing.

[0144] Furthermore, although the operations are described in a specific order, this should be understood as requiring that such operations be performed in the specific order shown or in sequential order, or requiring that all illustrated operations be performed to achieve the desired result. In certain environments, multitasking and parallel processing may be advantageous. Similarly, although several specific implementation details are included in the above discussion, these should not be construed as limiting the scope of this disclosure. Certain features described in the context of individual embodiments may also be implemented in combination in a single implementation. Conversely, various features described in the context of a single implementation may also be implemented individually or in any suitable sub-combination in multiple implementations.

[0145] Although the subject matter has been described using language specific to structural features and / or methodological logic, it should be understood that the subject matter defined in the appended claims is not necessarily limited to the specific features or actions described above. Rather, the specific features and actions described above are merely illustrative examples of implementing the claims.

Claims

1. A simulation method for semiconductor devices, comprising: The barrier-related parameters are discretized based on a predetermined range and a predetermined step size to generate multiple first data points. Determine the transmission coefficient corresponding to each of the plurality of first data points, wherein the transmission coefficient includes a first transmission coefficient corresponding to a first barrier unit having a constant barrier height and a second transmission coefficient corresponding to a second barrier unit having a linearly varying barrier height; and A pre-interpolation table of transmission coefficients is generated based on the first and second transmission coefficients to determine the potential barrier of a semiconductor device.

2. The simulation method according to claim 1, wherein determining the transmission coefficient corresponding to each of the plurality of first data points comprises: Using a first analytical expression for the first barrier unit, a first transmission coefficient corresponding to each of the plurality of first data points is determined, wherein the first analytical expression is a transmission coefficient analytical expression for a rectangular barrier. as well as Using a second analytical expression for the second barrier unit, a second transmission coefficient corresponding to each of the plurality of first data points is determined, wherein the second analytical expression is a transmission coefficient analytical expression for a triangular barrier.

3. The simulation method according to claim 2 further includes: Based on the correspondence between each of the plurality of first data points and its corresponding first transmission coefficient and corresponding second transmission coefficient, a pre-interpolation table for determining the transmission coefficient of the potential barrier of a semiconductor device is generated.

4. The simulation method according to claim 3 further includes: In response to the barrier-related parameters of the semiconductor device exceeding the range of the predetermined barrier-related parameters, the pre-interpolation table is expanded to generate an extrapolation pre-interpolation table.

5. The simulation method according to claim 4, wherein expanding the pre-interpolation table to generate the extrapolation pre-interpolation table in response to the barrier-related parameters of the semiconductor device exceeding the range of the predetermined barrier-related parameters comprises: Determine the variation trends of the first transmission coefficient and the second transmission coefficient relative to the barrier correlation parameter for multiple first data points adjacent to the barrier correlation parameter of the barrier to be measured within the range of the predetermined barrier correlation parameter. Based on the portion of the barrier-related parameters of the barrier under test that exceeds the predetermined range of the predetermined barrier-related parameters and the predetermined step size, the barrier-related parameters are discretized to generate multiple second data points. as well as Based on the determined trends of the first transmission coefficient with respect to the barrier-related parameters and the second transmission coefficient with respect to the barrier-related parameters, an extended transmission coefficient corresponding to each of the plurality of second data points is determined, wherein the extended transmission coefficient includes a first extended transmission coefficient corresponding to the first barrier unit with a constant barrier height and a second extended transmission coefficient corresponding to the second barrier unit with a linearly varying barrier height.

6. The simulation method of claim 4, wherein expanding the pre-interpolation table to generate an extrapolation pre-interpolation table in response to a barrier-related parameter of the semiconductor device exceeding the range of the predetermined barrier-related parameter comprises: Based on the portion of the barrier-related parameters of the barrier to be measured that exceeds the predetermined range of the predetermined barrier-related parameters and the predetermined step size, the barrier-related parameters are discretized to generate multiple second data points. as well as Determine the extended transmission coefficient corresponding to each of the plurality of second data points, wherein the extended transmission coefficient includes a first extended transmission coefficient corresponding to the first barrier cell having a constant barrier height and a second extended transmission coefficient corresponding to the second barrier cell having a linearly varying barrier height.

7. The simulation method of claim 6, wherein determining the extended transmission coefficient corresponding to each of the plurality of second data points comprises: Using the first analytical expression for the first barrier unit, a first extended transmission coefficient corresponding to each of the plurality of second data points is determined; as well as Using the second analytical expression for the second barrier unit, a second extended transmission coefficient corresponding to each of the plurality of second data points is determined.

8. The simulation method according to any one of claims 5 to 7, wherein, In response to the barrier-related parameters of the semiconductor device exceeding the range of the predetermined barrier-related parameters, expanding the pre-interpolation table to generate the extrapolation pre-interpolation table includes: Based on the correspondence between each of the plurality of second data points and its corresponding first extended transmission coefficient and its corresponding second extended transmission coefficient, the pre-interpolation table is expanded to generate the extrapolation pre-interpolation table.

9. The simulation method according to claim 1, wherein the barrier-related parameters include barrier length, barrier width, and electron energy.

10. A method for processing the transmission coefficient of a semiconductor device, comprising: The potential barrier of a semiconductor device along the electron tunneling direction is divided into multiple barrier units, wherein the multiple barrier units include one or more first barrier units with a constant barrier height and one or more second barrier units with a linearly varying barrier height. Based on the predetermined correspondence between the barrier-related parameters and the corresponding transmission coefficients, the first transmission coefficient of each first barrier unit in the one or more first barrier units and the second transmission coefficient of each second barrier unit in the one or more second barrier units are determined respectively. as well as The total transmission coefficient of the barrier of the semiconductor device is determined based on the first transmission coefficient and the second transmission coefficient.

11. The method of claim 10, wherein the predetermined correspondence is a pre-interpolation table, the pre-interpolation table storing transmission coefficients corresponding to predetermined barrier-related parameters.

12. The method of claim 10, wherein dividing the potential barrier of the semiconductor device along the electron tunneling direction into the plurality of barrier units comprises: The potential barrier along the electron tunneling direction is divided by a dichotomy method, and the potential barrier is divided into multiple first potential barrier units and second potential barrier units.

13. The method of claim 12, wherein dividing the potential barrier along the electron tunneling direction into a plurality of first barrier units and second barrier units by a dichotomy comprises: Determine the first slope of the line connecting the left endpoint and the midpoint of the barrier, and the second slope of the line connecting the right endpoint and the midpoint; as well as In response to the first slope not being equal to the second slope, the left endpoint and the midpoint, and the midpoint and the right endpoint are bisected until the first slope and the second slope are equal.

14. The method of claim 13, further comprising: In response to the determination that the first slope and the second slope of the barrier cell in the segmented barrier are equal, a random point is selected between the two endpoints of the barrier cell to verify whether the first slope of the line connecting the random point to the left endpoint of the barrier cell and the second slope of the line connecting the random point to the right endpoint of the barrier cell are equal; and In response to determining that the first slope and the second slope are not equal, the barrier unit is further divided into two parts until the first slope and the second slope are equal or a predetermined number of divisions is reached.

15. The method of claim 11, wherein determining the first transmission coefficient of each of the one or more first barrier units and the second transmission coefficient of each of the one or more second barrier units based on a predetermined correspondence between barrier-related parameters and corresponding transmission coefficients comprises: In response to the fact that the barrier-related parameters of each of the plurality of barrier units are within a predetermined range of the predetermined barrier-related parameters, the transmission coefficient corresponding to the barrier-related parameter adjacent to the corresponding barrier-related parameter is determined from the pre-interpolation table based on the barrier-related parameters corresponding to each barrier unit. as well as Based on the transmission coefficients corresponding to each of the barrier-related parameters, the first transmission coefficient and the second transmission coefficient are determined by an interpolation algorithm.

16. The method of claim 11, wherein determining the first transmission coefficient of each of the one or more first barrier units and the second transmission coefficient of each of the one or more second barrier units based on a predetermined correspondence between the barrier-related parameters and the corresponding transmission coefficients comprises: In response to the barrier-related parameters of one or more of the plurality of barrier units exceeding the range of the predetermined barrier-related parameters, the pre-interpolation table is expanded to generate an extrapolation pre-interpolation table. Based on the barrier-related parameters corresponding to each barrier unit, the transmission coefficients corresponding to the barrier-related parameters adjacent to the corresponding barrier-related parameters are determined from the extrapolation pre-interpolation table; and Based on the transmission coefficients corresponding to each of the barrier-related parameters, the first transmission coefficient and the second transmission coefficient are determined by an interpolation algorithm.

17. The method of claim 16, wherein expanding the pre-interpolation table to generate the extrapolation pre-interpolation table in response to a barrier-related parameter of the semiconductor device exceeding the range of the predetermined barrier-related parameter comprises: Determine the variation trends of the first transmission coefficient and the second transmission coefficient relative to the barrier correlation parameter for multiple first data points adjacent to the barrier correlation parameter of the barrier to be measured within the range of the predetermined barrier correlation parameter. Based on the portion of the barrier-related parameters of the barrier under test that exceeds the predetermined range of the predetermined barrier-related parameters and the predetermined step size, the barrier-related parameters are discretized to generate multiple second data points. as well as Based on the determined trends of the first transmission coefficient with respect to the barrier-related parameters and the second transmission coefficient with respect to the barrier-related parameters, an extended transmission coefficient corresponding to each of the plurality of second data points is determined, wherein the extended transmission coefficient includes a first extended transmission coefficient corresponding to the first barrier unit with a constant barrier height and a second extended transmission coefficient corresponding to the second barrier unit with a linearly varying barrier height.

18. The method of claim 16, wherein expanding the pre-interpolation table to generate the extrapolation pre-interpolation table in response to a barrier-related parameter of the semiconductor device exceeding the range of the predetermined barrier-related parameter comprises: Based on the portion of the barrier-related parameters of the barrier under test that exceeds the predetermined range of the predetermined barrier-related parameters and the predetermined step size, the barrier-related parameters are discretized to generate multiple second data points. Using a first analytical expression for the first barrier unit, a third extended transmission coefficient corresponding to each of the plurality of second data points is determined, wherein the first analytical expression is an analytical expression for the transmission coefficient of a rectangular barrier. as well as Using a second analytical expression for the second barrier unit, a fourth extended transmission coefficient corresponding to each of the plurality of second data points is determined, wherein the second analytical expression is an analytical expression for the transmission coefficient of a triangular barrier, and the extended transmission coefficient includes a first extended transmission coefficient corresponding to the first barrier unit having a constant barrier height and a second extended transmission coefficient corresponding to the second barrier unit having a linearly varying barrier height.

19. The method of claim 17 or 18, wherein expanding the pre-interpolation table to generate the extrapolation pre-interpolation table in response to one or more of the plurality of barrier cells having barrier-related parameters exceeding the range of the predetermined barrier-related parameters further comprises: Based on the correspondence between each of the plurality of second data points and its corresponding first extended transmission coefficient and its corresponding second extended transmission coefficient, the pre-interpolation table is expanded to generate the extrapolation pre-interpolation table.

20. The method according to claim 16, wherein the interpolation algorithm includes a trilinear interpolation algorithm and a cubic spline interpolation algorithm.

21. The method of claim 10, wherein determining the total transmission coefficient of the barrier of the semiconductor device based on the first transmission coefficient and the second transmission coefficient comprises: The corresponding first transmission coefficient and second transmission coefficient are multiplied to generate the total transmission coefficient.

22. The method of claim 10, wherein the predetermined correspondence comprises the pre-interpolation table generated by the simulation method of any one of claims 3 to 8.

23. The method of claim 10, wherein the barrier-related parameters include barrier length, barrier width, and electron energy.

24. An electronic device comprising: processor; as well as A memory coupled to the processor, the memory having instructions stored therein, the instructions causing the device to perform actions when executed by the processor, the actions including: The barrier-related parameters are discretized based on a predetermined range and a predetermined step size to generate multiple first data points. Determine the transmission coefficient corresponding to each of the plurality of first data points, wherein the transmission coefficient includes a first transmission coefficient corresponding to a first barrier unit having a constant barrier height and a second transmission coefficient corresponding to a second barrier unit having a linearly varying barrier height; and A pre-interpolation table of transmission coefficients is generated based on the first and second transmission coefficients to determine the potential barrier of a semiconductor device.

25. A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the method according to any one of claims 1-23.

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