Simulation construction method of physical analysis model of nc-gaafet

Abstract

The application discloses a simulation construction method of a physical analysis model of an NC-GAAFET, and comprises the following steps: S1, constructing an electrostatic potential model of a short-channel NC-GAAFET; S2, obtaining an electric potential distribution function in a silicon channel by model analysis; S3, calculating stress tensor components in a ferroelectric layer and proving relevant spatial gradients; S4, coupling a conventional GAAFET model and a constitutive relation of the ferroelectric layer; S5, introducing a flexoelectric effect in the constitutive relation, and correcting a voltage-charge relation of the ferroelectric layer; S6, deducing and calculating a drain-to-barrier lowering effect parameter, and calculating electrical characteristics of the NC-GAAFET; and S7, adjusting specified parameters, and repeating steps S2 to S6 to simulate and analyze influences of each parameter on electrical performance of the NC-GAAFET. The application overcomes prediction deviation caused by ignoring nanometer scale surface effects in an existing model, and significantly improves accuracy of the model in a subthreshold region.

Description

Technical Field

[0001] This invention relates to the field of numerical simulation technology for the performance of negative capacitance transistors, specifically to a physical analytical model of a short-channel negative capacitance gate-all-around field-effect transistor (NC-GAAFET) that considers the surface tension-induced flexoelectric effect, as well as a method, apparatus, and storage medium for simulating the model. Background Technology

[0002] The combination of negative capacitance field-effect transistor (NC-FET) and gate-all-around field-effect transistor (GAA-FET) is considered to be one of the core technology routes for realizing next-generation ultra-low power chips. Currently, various analytical models have been established for the research of NC-GAAFET to simulate its electrical characteristics.

[0003] However, as device dimensions enter the nanoscale, short-channel effects (such as drain-induced barrier reduction (DIBL)) cause traditional models based on long-channel assumptions to become severely ineffective. Therefore, these existing models generally share a common defect: they all ignore the surface tension effect, which is significantly enhanced at the nanoscale. This effect generates a considerable strain gradient in the ferroelectric layer, which in turn modulates the ferroelectric polarization state and negative capacitance characteristics through flexoelectric effect. That is, for negative capacitance gate-around transistors with metal-ferroelectric-insulator-semiconductor (MFIS) structures, existing models fail to fully consider the flexoelectric effect induced by the significant mechanical strain gradient at the nanoscale. This effect generates a non-negligible built-in electric field in the ferroelectric layer, which significantly affects the threshold voltage and transport characteristics of the device.

[0004] The lack of this physical mechanism prevents existing models from accurately predicting the true performance of nanoscale NC-GAAFETs. In particular, it fails to explain and utilize the potential increase in drive current brought about by the flexoelectric effect, limiting further optimization of device performance. Therefore, a physical model that can accurately describe the coupling effect between strain gradient and negative capacitance provides an accurate theoretical tool for the design of high-performance, low-power NC-GAAFETs.

[0005] To address this, this application proposes a simulation method for constructing a physical analytical model of NC-GAAFET. By constructing a new physical model that can accurately describe the coupling effect between strain gradient and negative capacitance, it can provide accurate theoretical tools for the design of high-performance, low-power NC-GAAFET, fill the theoretical gap caused by the lack of physical mechanisms in conventional schemes, and solve the aforementioned technical problems. Summary of the Invention

[0006] The main objective of this invention is to provide a simulation method for constructing a physical analytical model of an NC-GAAFET. Starting from the electrostatic potential solution, the method gradually couples the ferroelectric negative capacitance and flexural effect to finally simulate the electrical characteristics of the device, thereby solving the technical problems mentioned in the background art.

[0007] The present invention solves the above-mentioned technical problems by adopting the following technical solutions:

[0008] A simulation construction method for a physical analytical model of NC-GAAFET specifically includes the following steps:

[0009] S1. An electrostatic potential model of a short-channel NC-GAAFET is constructed using the MFIS structure. The model uses the MFIS structure instead of the MFMIS structure because the presence of the metal floating gate may suppress the negative capacitance effect when the ferroelectric layer leakage current of the MFMIS structure is large. The charge trapping and detrapping process in the metal layer will cause the threshold voltage to fluctuate.

[0010] S2. Model analysis of conventional short-channel GAAFET: By solving the two-dimensional Poisson equation in cylindrical coordinates, the potential distribution function in the silicon channel is obtained, and then the boundary conditions of the source and drain are determined.

[0011] S3. Based on the mechanical equilibrium conditions of the nanocylindrical structure, calculate the stress tensor components inside the ferroelectric layer induced by the surface tension coefficient μ, and prove that it has a spatial gradient related to the radius of curvature.

[0012] S4. Based on the aforementioned model, the constitutive relationship between the model of a conventional GAAFET and the ferroelectric layer is achieved;

[0013] S5. In the constitutive relation of the ferroelectric layer, a flexural electric effect induced by the mechanical strain gradient is introduced, wherein the strain gradient corrects the voltage-charge relationship of the ferroelectric layer in the form of an equivalent internal electric field;

[0014] S6. Based on the boundary conditions of the source and drain, the source and drain current model of conventional GAAFET is derived and the drain-to-barrier reduction effect parameter is calculated. Based on the corrected ferroelectric layer constitutive relation and gate voltage balance condition, the electrical characteristics of NC-GAAFET, including source and drain current and subthreshold swing, are calculated.

[0015] S7. Adjust at least one of the following parameters: surface tension coefficient, flexural coefficient, ferroelectric layer thickness, and channel radius. Repeat steps S2 to S6 to simulate and analyze the effect of each parameter on the electrical performance of the NC-GAAFET.

[0016] Preferably, S1-S2 all adopt the same model construction method as the numerical simulation method for a short-channel negative capacitance gate-around-the-field-effect transistor as described in the prior art application number CN202410243471.9.

[0017] Preferably, the specific calculation process for solving the two-dimensional Poisson equation in cylindrical coordinates to obtain the potential distribution function in the silicon channel in step S2 includes:

[0018] For a uniformly doped cylindrical silicon channel, its two-dimensional Poisson equation in the subthreshold region is: Where r is the radial distance from the center of the Si channel in the nanowire, and z is the distance along the channel. , , It represents the channel doping concentration, and q represents the electron charge. It is the dielectric constant of silicon;

[0019] The potential distribution function at the center of the channel is calculated;

[0020] The potential distribution function of the channel surface is calculated by using the boundary conditions near the source and drain regions.

[0021] Preferably, the calculation process for the channel center potential distribution function includes:

[0022] Using the parabolic approximation method, assuming the electric potential follows a quadratic distribution along the radial direction, a solution to the two-dimensional Poisson equation exists in the form of: ;

[0023] The function terms A(r,z) and B(r,z) are strongly correlated with the variable z and weakly correlated with the variable r. Therefore, A(r,z) and B(r,z) can be approximated as terms and The solution formula for the two-dimensional Poisson equation exists as follows: ;

[0024] According to Gauss's law, the channel surface should satisfy the following boundary conditions: ,in It is the gate-insulator capacitance per unit area. R is the dielectric constant of the insulator, and R is the channel radius. For the thickness of the insulator, Defined as the gate voltage of a GAA-FET. This represents the difference in work function between the gate electrode and the semiconductor. It is the surface potential;

[0025] Calculation obtained about Functions: Then, substituting the function into the two-dimensional Poisson equation and the solution formula, we obtain information about the variables r and Functions: AND function

[0026] At the center of the silicon channel (r=0), let... = Substitute the above Solving the equation yields the function. The following relationship should be satisfied: ;

[0027] The calculation can be obtained ,in , and Let r be a constant, and substitute it into the equation with respect to the variables r and r. The approximate function for calculating the potential distribution at the center of the channel is given by the function. .

[0028] Preferably, the calculation process for the channel surface potential distribution function includes:

[0029] Obtain the boundary conditions near the source and drain regions:

[0030]

[0031]

[0032] in, This represents the inherent potential difference between the channel and the source / drain regions. It is the drain-source voltage, and L is the channel length;

[0033] Based on an approximate function of the potential distribution at the center of the channel, the coefficients are calculated and obtained. and The calculation formula is as follows:

[0034]

[0035]

[0036] A function is set at the interface between the channel and the gate insulator. ,but Existence condition equation: ,in Solving this conditional equation yields the following formula:

[0037]

[0038] Substitute this formula into the variables r and The function of the channel surface potential distribution function can be obtained as follows:

[0039] Where the coefficient and Based on the boundary conditions near the source region and the drain region and Calculation yields: , ,in It is represented as a set of hyperbolic sine functions for calculation.

[0040] Preferably, the formula for calculating the internal stress tensor components of the ferroelectric layer induced by the surface tension coefficient μ in step S3 is as follows:

[0041] Using the parabolic approximation method, the two-dimensional potential distribution function in the radial (r) direction is expressed as: ;

[0042] Let represent the electrostatic potential at any point (r, z) in the channel. This represents the electric potential at the centerline r=0 of the channel, also known as the center potential. The potential at the channel surface at r=R is also called the surface potential, where r represents the radial distance from the center of the channel, R represents the radius of the channel, and z represents the longitudinal distance along the channel direction.

[0043] Determine the stress distribution within the ferroelectric layer caused by surface tension, and calculate the stress components. With effective surface pressure and The component caused by the relevant uniform radial pressure, where μ is the surface tension.

[0044] Force coefficient, This represents the outer radius of the ferroelectric layer, that is, the radius from the center of the cylindrical tube to its outermost surface. The inner radius of the ferroelectric layer, i.e., the radius from the center of the cylindrical tube to the inner surface of its hollow portion, is represented in cylindrical coordinates. Mechanical equilibrium conditions of the surface of a medium cylinder It has the following form: , Represents the surface normal vector. Represents the stress tensor. It is compressive stress. Let the radial axis radiate outward from the center of the cylinder. Let z be the tangential axis along the tangential direction around the cylinder, and z be the axis along the length of the cylinder from the source to the drain.

[0045] By using the mechanical equilibrium conditions in the block body, the two stress components in the Cartesian coordinates caused by surface tension are calculated and obtained.

[0046] Preferably, the calculation process for the stress components includes:

[0047] The mechanical equilibrium conditions in the block body are obtained as follows The formulas for satisfying boundary conditions and equilibrium conditions are as follows: ;

[0048] It acts on the outer surface of the cylindrical tube. Effective radial pressure at the location, It acts on the inner surface of the cylindrical tube. Effective radial pressure at the location;

[0049] The tensor components in Cartesian coordinates are calculated using the following formula:

[0050]

[0051]

[0052] Reducing coordinates to the Cartesian coordinate system Plane, i.e. There are stress components It becomes: ;

[0053] The non-uniformity of the two stress components caused by surface stretching results in a stress gradient that leads to an internal bias flexural electric field, used to describe the flexural effect. Its expression is as follows: , and All are flexural electric field coefficients.

[0054] Preferably, the specific operation procedure for correcting the voltage-charge relationship of the ferroelectric layer through the strain gradient in step S5 in the form of an equivalent internal electric field includes:

[0055] Based on the introduction of a torsion effect term and a reverse torsion effect term into the conventional free energy expression, the expression for the free energy G exists as follows:

[0056]

[0057] in and It is the Landau coefficient. and Both are electrostriction coefficients. , These are two sets of stress components. Represented as polarization, and All are elastic coefficients. and All are positive flexural coefficients. and All are inverse flexural coefficients, and K represents the sum of all higher-order terms containing polarization gradients;

[0058] Because of the surface stretching of the nanotube ferroelectric layer, a stress gradient will be generated along the radial direction. Therefore, the stress gradient that varies with the radius will lead to the generation of a torsion signal field.

[0059] The equilibrium solution is obtained by minimizing the order parameter. Differentiating, we have:

[0060]

[0061] in, and The stress-related flexocoupler coefficient, and The inverse torsion coefficient is given, and the formula can be converted to:

[0062]

[0063]

[0064] Finally, ignore higher-order terms. There is a voltage across the ferroelectric layer. for: ;

[0065] Since the Poisson equation in cylindrical coordinates is expressed as... ,in It is the radial component of the displacement vector. There are no moving charges in the insulating layer, therefore, according to Gauss's law... In the intermediate insulating layer With ferroelectric layer At the boundary, there exists a proportional relationship as follows: ,in, In position The displacement vector at that point is equal to ,and In the insulating layer and ferroelectric layer ( Displacement vector at the interface;

[0066] Calculation of ferroelectric layer The values ​​in the range are: ;

[0067] In the ferroelectric layer, the electric field and polarization The following relationships exist: ,in It is an electric displacement field, where the polarization displacement field is approximated as: ;

[0068] Therefore, it exists. and The formula for the relationship is:

[0069]

[0070] The coefficients a, b, and c are related to the Landau parameters of the ferroelectric material of the cylindrical gate. and Related.

[0071] Preferably, the formula for calculating the Landau parameter is:

[0072]

[0073]

[0074]

[0075] in , , It is expressed as the thickness of the ferroelectric layer.

[0076] Preferably, the derivation of the source-drain current model of a conventional GAAFET and the calculation process of the drain-to-barrier reduction effect parameters in step S6 include:

[0077] Substituting the approximate potential distribution function at the channel center and the potential distribution function at the channel surface into the two-dimensional potential distribution function in the radial r direction, the two-dimensional potential distribution function is calculated, revealing the presence of source and leakage currents. The calculation formula is: ;

[0078] in, It is absolute temperature. It is electron mobility. Represented as a set of Boltzmann constants;

[0079] The formula for calculating the leakage barrier reduction effect parameter DIBL is as follows: .

[0080] Preferably, the method for calculating the electrical characteristics of the NC-GAAFET, including source-drain current and subthreshold swing, in step S6 includes:

[0081] Coupled with the conventional GAAFET model, there exists a gate surface charge density. The model is ;

[0082] in, It is the length from the gate to the drain;

[0083] According to the LK equation, using the moving charge density express The expression is:

[0084]

[0085] Where the coefficient and It is related to the Landau parameters of ferroelectric materials. It is the flexural coefficient;

[0086] The voltage balance condition for a short-channel NC-GAAFET considering the flexural effect is: ;

[0087] in, and All Landau parameters;

[0088] When the gate voltage of NC-GAAFET At that time, the effective gate voltage of the corresponding GAAFET Corresponding source-drain current The expression is:

[0089]

[0090] in, It is a set of Boltzmann constants;

[0091] The subthreshold swing SS of the NC-GAAFET is:

[0092]

[0093] in, Indicates drain-source current, It is the subthreshold swing of GAA-FET. This is represented as gate voltage.

[0094] In another aspect, the present invention also discloses a computer-readable storage medium storing a computer program, which, when executed by a processor, causes the processor to perform the steps of the method described above.

[0095] In another aspect, the present invention also discloses a computer device, including a memory and a processor, wherein the memory stores a computer program, and when the computer program is executed by the processor, the processor performs the steps of the method described above.

[0096] As can be seen from the above technical solution, the present invention provides a simulation construction method for the physical analytical model of NC-GAAFET. Compared with the prior art, the present invention has the following advantages:

[0097] 1. This invention overcomes the prediction bias caused by neglecting nanoscale surface effects in existing models, significantly improves the accuracy of the model in the subthreshold region, and can increase the conduction current by 1-2 orders of magnitude, providing an effective theoretical tool for stress engineering optimization of nanoscale negative capacitance transistors.

[0098] 2. This invention introduces a complete electromechanical coupling path of "surface tension-strain gradient-flexoelectric effect" into the NC-GAAFET analytical model for the first time, filling the gap in existing models for modeling nanoscale surface effects.

[0099] 3. By clearly defining key design parameters, including surface tension coefficient, flexural coefficient, ferroelectric layer thickness, and channel radius, this invention provides a specific technical path for optimizing device performance through "stress engineering." It also facilitates the establishment of a complete theoretical framework from mechanical equilibrium to electrical characteristics, achieving seamless coupling of electromechanical multi-physics fields.

[0100] It should be understood that the description in this section is not intended to identify key or essential features of embodiments of the invention, nor is it intended to limit the scope of the invention. Other features of the invention will become readily apparent from the following description. Of course, implementing any product of the invention does not necessarily require achieving all of the advantages described above simultaneously. Attached Figure Description

[0101] The accompanying drawings, which form part of this application, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an undue limitation of the invention. In the drawings:

[0102] Figure 1 This is a three-dimensional schematic diagram of the structure of the MFIS short-channel negative capacitance gate-around-the-field-effect transistor of the present invention;

[0103] Figure 2 This is a schematic cross-sectional view of the structure of the MFIS short-channel negative capacitance gate-around-the-field-effect transistor of the present invention.

[0104] Figure 3 This is a schematic diagram of the electrostatic potential model structure of the MFIS short-channel negative capacitance gate-around-the-gate field-effect transistor of the present invention.

[0105] Figure 4 The charge density-ferroelectric layer voltage of the present invention ( ) with surface tension coefficient A schematic diagram of the change curve, in which the S-shape The curve exhibits a negative capacitance region (i.e., a negative slope characteristic);

[0106] Figure 5This is a schematic diagram of the drain current variation curve of the NC-GAAFET of the present invention under the positive flexural coefficient, when the surface tension coefficient μ is different;

[0107] Figure 6 Different μ values ​​of the NC-GAAFET of the present invention under positive flexural coefficient Characteristic curve diagram;

[0108] Figure 7 The present invention illustrates the relationship between SS and gate voltage for NC-GAAFET (solid line) and GAAFET (dashed line) at different surface tension coefficients μ when the positive flexural coefficient is achieved. Detailed Implementation

[0109] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. Unless otherwise specified, the embodiments and features in the embodiments of this application can be combined with each other. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0110] For details in the embodiments, please refer to Figures 1 to 7 .

[0111] The simulation construction method for the physical analytical model of NC-GAAFET proposed in this embodiment of the invention can, based on the core model of undoped cylindrical metal-ferroelectric-insulator-semiconductor (MFIS) short-channel GAA-NCFET, add the influence of flexural electricity caused by surface tension on the short-channel NC-GAAFET. In one embodiment, it specifically includes the following steps:

[0112] S1. Constructed using an MFIS structure, as follows Figure 1 value Figure 3 The model shown is a set of electrostatic potential models for short-channel NC-GAAFETs. The model uses an MFIS structure instead of an MFMIS structure because the presence of a metal floating gate may suppress the negative capacitance effect when the ferroelectric layer leakage current of the MFMIS structure is large. The charge trapping and detrapping process in the metal layer will cause threshold voltage fluctuations.

[0113] S2. Model analysis of conventional short-channel GAAFET: By solving the two-dimensional Poisson equation in cylindrical coordinates, the potential distribution function in the silicon channel is obtained.

[0114] Specifically, it includes:

[0115] First, the electrostatic potential is obtained by solving the two-dimensional Poisson equation for a cylindrical channel. For a uniformly doped cylindrical silicon channel, the two-dimensional Poisson equation in the subthreshold region is:

[0116] (1)

[0117] Where r is the radial distance from the center of the Si channel in the nanowire, and z is the distance along the channel. , , It represents the channel doping concentration, and q represents the electron charge. It is the dielectric constant of silicon. Using the parabolic approximation, assuming the potential follows a quadratic distribution along the radial direction, the solution takes the form:

[0118] (2)

[0119] A(r,z) and B(r,z) are strongly correlated with variable z and weakly correlated with variable r. Therefore, A(r,z) and B(r,z) can be approximated as and This can be obtained by solving the two-dimensional Poisson equation under specific boundary conditions.

[0120] At this time, in order to obtain and Substituting the solution into the two-dimensional Poisson equation, we have:

[0121] (3)

[0122] According to Gauss's law, the channel surface should satisfy the following boundary conditions:

[0123] (4)

[0124] in It is the gate-insulator capacitance per unit area. Defined as the gate voltage of a GAA-FET.

[0125] Furthermore, in order to simplify the research, (=0) indicates the difference in work function between the gate electrode and the semiconductor. It is the surface potential.

[0126] At this point, based on the boundary conditions satisfied by the channel surface, we can obtain... about The function is:

[0127] (5)

[0128] Re-evaluating the form of the solution to the two-dimensional Poisson equation (by substituting equation (5) into equation (2)), we can obtain the relationship between the variables r and... The function is:

[0129] (6)

[0130] Then the above about Substituting the function into equation (3), we can obtain:

[0131] (7)

[0132] At the center of the silicon channel (r=0), let... = Substituting into equation (7), we obtain the function. The following relationships should be satisfied:

[0133] (8)

[0134] Solving equation (8) above yields:

[0135] (9)

[0136] in , and The constant can be obtained by solving the boundary conditions along the source-drain direction. Substituting equation (9) into the variables r and In the function, the potential distribution function at the center of the channel is approximately:

[0137] (10)

[0138] Furthermore, based on the boundary conditions near the source and drain regions:

[0139] (11)

[0140] (12)

[0141] in, This represents the inherent potential difference between the channel and the source / drain regions. It is the drain-source voltage.

[0142] Therefore, we can obtain and The two coefficients are:

[0143] (13)

[0144] (14)

[0145] At the interface between the channel and the gate insulator (r=R), let the function... ,but The following equation should be satisfied:

[0146] (15)

[0147] in Equation (15) can be solved to obtain the following form:

[0148] (16)

[0149] Substituting equation (16) into equation (6), we can obtain the channel surface potential distribution function:

[0150] (17)

[0151] Then, based on the boundary conditions near the source and drain regions:

[0152] (18)

[0153] (19)

[0154] Therefore, the coefficients can be calculated. and for:

[0155] (20)

[0156] (twenty one)

[0157] In summary, we can obtain the electrostatic potential model of the short-channel NC-GAAFET and the potential distribution function in the silicon channel.

[0158] S3. Based on the mechanical equilibrium conditions of the nanocylindrical structure, calculate the stress tensor components inside the ferroelectric layer induced by the surface tension coefficient μ, and prove that it has a spatial gradient related to the radius of curvature.

[0159] Based on the above potential distribution function, the specific operation steps include the following:

[0160] Using the parabolic approximation method, the two-dimensional potential distribution function in the radial (r) direction can be expressed as:

[0161] (twenty two)

[0162] Then, the stress distribution within the ferroelectric layer caused by surface tension is determined. First, we calculate the stress distribution caused by surface tension. With effective surface pressure and (μ is the surface tension coefficient) Component caused by uniform radial pressure. Cylindrical coordinate system. Mechanical equilibrium conditions of the surface of a medium cylinder It has the following form:

[0163] (twenty three)

[0164] The mechanical equilibrium conditions in the block body are as follows:

[0165] (twenty four)

[0166] It can be seen that the boundary conditions and equilibrium conditions, equations (23) and (24), can be satisfied:

[0167] (25)

[0168] The tensor components in Cartesian coordinates (x, y, z) can be obtained using the following relationship:

[0169] (26)

[0170] From equations (25) and (26), we can obtain:

[0171] (27)

[0172] To simplify the problem, we can consider the Cartesian coordinate system. Plane, i.e. At this point, the stress components It becomes:

[0173] (28)

[0174] Clearly, the two stress components caused by surface stretching are non-uniform. Therefore, the stress gradient will result in an internal bias field (i.e., the so-called flexural electric field), which describes the flexural effect and is expressed as:

[0175] (29)

[0176] In summary, the stress tensor components inside the ferroelectric layer can be obtained.

[0177] S4. Based on the aforementioned model, this is achieved by coupling the model of a conventional GAAFET with the constitutive relationship of the ferroelectric layer.

[0178] S5. In the constitutive relation of the ferroelectric layer, a flexural electric effect induced by a mechanical strain gradient is introduced, wherein the strain gradient corrects the voltage-charge relationship of the ferroelectric layer in the form of an equivalent internal electric field.

[0179] Based on the above-mentioned potential distribution function and stress tensor components, the specific steps include:

[0180] By introducing a torsion effect term and a reverse torsion effect term into the conventional free energy expression, the free energy expression becomes:

[0181] (30)

[0182] in and It is the Landau coefficient. And... It is temperature-dependent. Due to surface stretching of the nanotube ferroelectric layer, a stress gradient is generated along the radial direction. This radius-varying stress gradient leads to the generation of a torsion signal field. The equilibrium solution is obtained by minimizing the order parameter. For The derivative of the derivative yields:

[0183] (31)

[0184] in and The stress-related flexocoupler coefficient, and Let be the reverse torsion coefficient. Equation (31) can be transformed into:

[0185] (32)

[0186] (33)

[0187] Finally, higher-order terms This can be ignored; at this point, calculate the voltage across the ferroelectric layer. for:

[0188] (34)

[0189] Poisson's equation in cylindrical coordinates is expressed as:

[0190] (35)

[0191] in It is the radial component of the displacement vector.

[0192] Since there are no moving charges in the insulating layer, Gauss's law leads to the conclusion that: In the intermediate insulating layer ( ) and ferroelectric layer ( At the boundary of ), the proportional relationship is:

[0193] (36)

[0194] in, In position The displacement vector at that point is equal to ,and In the insulating layer and ferroelectric layer ( The displacement vector at the interface can be used to calculate the ferroelectric layer using equation (42). The value in the middle is obtained as follows:

[0195] (37)

[0196] In the ferroelectric layer, the electric field and polarization Related:

[0197] (38)

[0198] in It is an electric displacement field. For the small voltage range considered in this embodiment, polarization can be well approximated by a displacement field:

[0199] (39)

[0200] This approximation holds true because at low voltages... The term can be ignored. Therefore, equation (34) becomes:

[0201] (40)

[0203] By substituting equation (37) into equation (40), we can further obtain and Relationship:

[0204] (41)

[0205] The coefficients a, b, and c are related to the Landau parameters of the ferroelectric material of the cylindrical gate. and Related to ), and they are represented as:

[0206] (42)

[0207] (43)

[0208] (44)

[0209] in , .

[0210] S6. Based on the boundary conditions of the source and drain terminals, the source and drain current model of a conventional GAAFET is derived and the drain-to-barrier reduction effect parameters are calculated. Based on the corrected ferroelectric layer constitutive relation and gate voltage balance condition, the electrical characteristics of the NC-GAAFET, including the source and drain current and subthreshold swing, are calculated.

[0211] Based on the above steps S2 to S5, step S6 specifically includes:

[0212] Substituting equations (10) and (17) into equation (22), we obtain the two-dimensional potential distribution function. The source-drain current Ids can be calculated as follows:

[0213] (45)

[0214] The leakage barrier lowering effect parameter DIBL can be calculated as follows:

[0215] (46)

[0216] By coupling the conventional GAAFET model, the gate surface charge density The model can be used to calculate:

[0217] (47)

[0218] According to the LK equation The moving charge density can be used Represented as:

[0219] (48)

[0220] Where the coefficient and It is related to the Landau parameters of ferroelectric materials. It is the flexural coefficient.

[0221] The voltage balance condition for a short-channel NC-GAAFET considering the flexural effect is:

[0222] (49)

[0223] Gate voltage of NC-GAAFET At that time, the effective gate voltage of the corresponding GAAFET Corresponding source-drain current It can be obtained as follows:

[0224] (50)

[0225] The subthreshold swing SS of NC-GAAFET is:

[0226] (51)

[0227] in It is the subthreshold swing of GAA-FET.

[0228] S7. Adjust at least one of the following parameters: surface tension coefficient, flexural coefficient, ferroelectric layer thickness, and channel radius. Repeat steps S2 to S6 to simulate and analyze the effect of each parameter on the electrical performance of the NC-GAAFET.

[0229] In this embodiment, the actual parameter value comparison table is as follows:

[0230]

[0231] For example, in one specific embodiment, by inputting specified data and using the above model, the surface tension coefficient μ under different conditions is obtained. Curves Figure 4 As shown, the S-shaped polarization curves of the ferroelectric layer are displayed for different μ values. The curve shifts to the left as μ increases, indicating that the flexural voltage alters the polarization state of the ferroelectric layer. Further analysis is conducted for different surface tension coefficients μ. Curves Figure 5 As shown, the drain current increases significantly with increasing μ, especially in the subthreshold region. This illustrates the effect of different μ values ​​at the positive flexural coefficient. Curves Figure 6 As shown, under a positive flexural coefficient, a larger μ corresponds to a higher saturation current, indicating that the flexural effect enhances the driving capability. This contrasts with the value of μ under a different positive flexural coefficient. Curves Figure 7 As shown, the minimum value of SS remains basically unchanged, but the curve shifts to the left as μ increases, indicating that the flexural effect does not change the switching steepness, but changes the position of the threshold voltage.

[0232] In summary, by clearly defining key design parameters including surface tension coefficient, flexural coefficient, ferroelectric layer thickness, and channel radius, this application provides a specific technical path for optimizing device performance through "stress engineering." It also facilitates the establishment of a complete theoretical framework from mechanical equilibrium to electrical characteristics, achieving seamless coupling of electromechanical multiphysics fields. For the first time, a complete electromechanical coupling path of "surface tension-strain gradient-flexural effect" is introduced into the NC-GAAFET analytical model, filling the gap in existing models for modeling nanoscale surface effects. Furthermore, it overcomes the prediction bias caused by neglecting nanoscale surface effects in existing models, significantly improving the model's accuracy in the subthreshold region and enabling a 1-2 order of magnitude increase in conduction current. This provides an effective theoretical tool for stress engineering optimization of nanoscale negative capacitance transistors.

[0233] In another aspect, the present invention also discloses a computer-readable storage medium storing a computer program, which, when executed by a processor, causes the processor to perform the steps of the method described above.

[0234] In another aspect, the present invention also discloses a computer device, including a memory and a processor, wherein the memory stores a computer program, and when the computer program is executed by the processor, the processor performs the steps of the method described above.

[0235] In another embodiment provided in this application, a computer program product containing instructions is also provided, which, when run on a computer, causes the computer to execute the simulation construction method of the physical analysis model of any of the NC-GAAFETs in the above embodiments.

[0236] It is understood that the system provided in the embodiments of the present invention corresponds to the method provided in the embodiments of the present invention, and the explanation, examples and beneficial effects of the relevant content can be referred to the corresponding parts of the above methods.

[0237] This application also provides an electronic device, including a processor, a communication interface, a memory, and a communication bus, wherein the processor, communication interface, and memory communicate with each other via the communication bus.

[0238] Memory, used to store computer programs;

[0239] A simulation construction method for implementing the physical analytical model of the NC-GAAFET described above when the processor executes the program stored in memory.

[0240] The communication bus mentioned in the above-mentioned electronic devices can be a standard bus for interconnecting peripheral components or an extended industrial standard structure bus, etc. This communication bus can be divided into address bus, data bus, control bus, etc.

[0241] The communication interface is used for communication between the aforementioned electronic devices and other devices.

[0242] The memory may include random access memory or non-volatile memory, such as at least one disk storage device. Optionally, the memory may also be at least one storage device located remotely from the aforementioned processor.

[0243] The processors mentioned above can be general-purpose processors, including central processing units, network processors, etc.; they can also be digital signal processors, application-specific integrated circuits, field-programmable gate arrays or other programmable logic devices, discrete gate or transistor logic devices, or discrete hardware components.

[0244] In the above embodiments, implementation can be achieved entirely or partially through software, hardware, firmware, or any combination thereof. When implemented using software, it can be implemented entirely or partially in the form of a computer program product. The computer program product includes one or more computer instructions. When the computer program instructions are loaded and executed on a computer, all or part of the processes or functions described in the embodiments of this application are generated. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device. The computer instructions can be stored in a computer-readable storage medium or transmitted from one computer-readable storage medium to another. For example, the computer instructions can be transmitted from one website, computer, server, or data center to another website, computer, server, or data center via wired or wireless means. The computer-readable storage medium can be any available medium accessible to a computer or a data storage device such as a server or data center that integrates one or more available media. The available medium can be a magnetic medium, an optical medium, or a semiconductor medium, etc.

[0245] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

[0246] Furthermore, it should be noted that if any directional indication (such as up, down, left, right, front, back, etc.) is involved in the embodiments of the present invention, the directional indication is only used to explain the relative positional relationship and movement of each component in a specific posture. If the specific posture changes, the directional indication will also change accordingly.

[0247] Furthermore, if the embodiments of this invention involve descriptions such as "first" or "second," these descriptions are for descriptive purposes only and should not be construed as indicating or implying their relative importance or implicitly specifying the number of technical features indicated. Therefore, a feature defined with "first" or "second" may explicitly or implicitly include at least one of those features. Additionally, the meaning of "and / or" throughout the text includes three parallel solutions; for example, "A and / or B" includes solution A, solution B, or a solution where both A and B are satisfied simultaneously. Furthermore, in the embodiments of this invention, "multiple" refers to two or more. Moreover, the technical solutions of the various embodiments can be combined with each other, but this must be based on the ability of those skilled in the art to implement them. When the combination of technical solutions is contradictory or impossible to implement, it should be considered that such a combination of technical solutions does not exist and is not within the scope of protection claimed by this invention.

Claims

1. A method for simulating and constructing a physical analytical model of an NC-GAAFET, characterized in that, include: S1. An electrostatic potential model for a short-channel NC-GAAFET is constructed using an MFIS structure; S2. Model analysis of conventional short-channel GAAFET: By solving the two-dimensional Poisson equation in cylindrical coordinates, the potential distribution function in the silicon channel is obtained. S3. Based on the mechanical equilibrium conditions of the nanocylindrical structure, calculate the stress tensor components inside the ferroelectric layer induced by the surface tension coefficient μ, and prove that it has a spatial gradient related to the radius of curvature. S4. Based on the aforementioned model, the constitutive relationship between the model of a conventional GAAFET and the ferroelectric layer is achieved; S5. In the constitutive relation of the ferroelectric layer, the flexural electric effect induced by the mechanical strain gradient is introduced to modify the voltage-charge relationship of the ferroelectric layer; S6. The source-drain current model of conventional GAAFET is derived and the drain-to-barrier reduction effect parameters are calculated. The electrical characteristics of NC-GAAFET, including source-drain current and subthreshold swing, are also calculated. S7. Adjust the specified parameters and repeat steps S2 to S6 to simulate and analyze the influence of each parameter on the electrical performance of the NC-GAAFET. The specific operational procedure for correcting the voltage-charge relationship of the ferroelectric layer through the strain gradient in step S5, in the form of an equivalent internal electric field, includes: Based on the conventional free energy expression, a torsion effect term is introduced, along with a reverse torsion effect term; The equilibrium solution is obtained by minimizing the order parameter, and the polarization transformation is performed. Finally, the higher-order terms are ignored to obtain the voltage across the ferroelectric layer. According to Gauss's law, the values ​​in the ferroelectric layer are calculated. Combining this with the correlation between the electric field and polarization in the ferroelectric layer, the voltage-charge relationship is obtained as follows: The coefficients a, b, and c are related to the Landau parameters of the ferroelectric material of the cylindrical gate. and Related, and Both are flexural electric field coefficients. It acts on the outer surface of the cylindrical tube. Effective radial pressure at the location, It acts on the inner surface of the cylindrical tube. Effective radial pressure at the location, Let the radial axis radiate outward from the center of the cylinder. This represents the outer radius of the ferroelectric layer, that is, the radius from the center of the cylindrical tube to its outermost surface. It represents the inner radius of the ferroelectric layer, that is, the radius from the center of the cylindrical tube to the inner surface of its hollow part.

2. The simulation construction method for the physical analytical model of NC-GAAFET as described in claim 1, characterized in that, The specific calculation process for solving the two-dimensional Poisson equation in cylindrical coordinates to obtain the potential distribution function in the silicon channel in step S2 includes: For a uniformly doped cylindrical silicon channel, using the parabolic approximation method and assuming a quadratic potential distribution along the radial direction, the solution to the two-dimensional Poisson equation in the subthreshold region takes the form: The function terms A(r,z) and B(r,z) are strongly correlated with the variable z and weakly correlated with the variable r. Therefore, A(r,z) and B(r,z) can be approximated as terms and Where r is the radial distance from the center of the Si channel in the nanowire, and z is the distance along the channel; By combining the boundary conditions satisfied by Gauss's theorem on the channel surface, the following can be calculated: about The function is then substituted into the two-dimensional Poisson equation and the solution formula to obtain information about the variables r and The function; At the center of the silicon channel (r=0), let... = Through function By satisfying the relationship, an approximate function of the potential distribution at the center of the channel can be calculated; By using the boundary conditions near the source and drain regions, combined with an approximate function of the potential distribution at the center of the channel, the potential distribution function on the channel surface can be calculated.

3. The simulation construction method for the physical analytical model of NC-GAAFET as described in claim 2, characterized in that, The formula for calculating the internal stress tensor components of the ferroelectric layer induced by the surface tension coefficient μ in step S3 is as follows: Using the parabolic approximation method, the two-dimensional potential distribution function in the radial (r) direction is expressed as: ; Let represent the electrostatic potential at any point (r, z) in the channel. The electric potential at the centerline r=0 of the channel is called the center potential. The potential at the surface of the channel at a point r = R is called the surface potential, where r represents the radial distance from the center of the channel, R represents the radius of the channel, and z represents the longitudinal distance along the channel direction. Determine the stress distribution within the ferroelectric layer caused by surface tension, and calculate the stress components. With effective surface pressure and The component caused by the relevant uniform radial pressure, where μ is the surface tension coefficient. This represents the outer radius of the ferroelectric layer, that is, the radius from the center of the cylindrical tube to its outermost surface. The inner radius of the channel, i.e., the radius from the center of the cylindrical tube to the inner surface of its hollow portion, is expressed in cylindrical coordinates. Mechanical equilibrium conditions of the surface of a medium cylinder It has the following form: , Represents the surface normal vector. Represents the stress tensor. It is compressive stress. Let the radial axis radiate outward from the center of the cylinder. Let z be the tangential axis along the tangential direction around the cylinder, and z be the axis along the length of the cylinder from the source to the drain. By using the mechanical equilibrium conditions in the block body, the two stress components in the Cartesian coordinates caused by surface tension are calculated and obtained.

4. The simulation construction method for the physical analytical model of NC-GAAFET as described in claim 3, characterized in that, The calculation process for the stress components includes: The mechanical equilibrium conditions in the block body are obtained as follows The formulas for satisfying boundary conditions and equilibrium conditions are as follows: ; It acts on the outer surface of the cylindrical tube. Effective radial pressure at the location, It acts on the inner surface of the cylindrical tube. Effective radial pressure at the location; The tensor components in Cartesian coordinates are calculated using the following formula: Reducing coordinates to the Cartesian coordinate system Plane, i.e. There are stress components It becomes: ; The non-uniformity of the two stress components caused by surface stretching results in a stress gradient that leads to an internal bias flexural electric field, used to describe the flexural effect. Its expression is as follows: , and All are flexural electric field coefficients.

5. The simulation construction method for the physical analytical model of NC-GAAFET as described in claim 4, characterized in that, The specific operational procedure for correcting the voltage-charge relationship of the ferroelectric layer through the strain gradient in step S5, in the form of an equivalent internal electric field, includes: Based on the introduction of a torsion effect term and a reverse torsion effect term into the conventional free energy expression, the expression for the free energy G is as follows: in and It is the Landau coefficient. and Both are electrostriction coefficients. , These are two sets of stress components. Represented as polarization, and All are elastic coefficients. and All are positive flexural coefficients. and All are inverse flexural coefficients, and K represents the sum of all higher-order terms containing polarization gradients; Because of the surface stretching of the nanotube ferroelectric layer, a stress gradient will be generated along the radial direction. Therefore, the stress gradient that varies with the radius will lead to the generation of a torsion signal field. The equilibrium solution is obtained by minimizing the order parameter. Differentiating, we have: in, and The stress-related flexocoupler coefficient, and The inverse torsion coefficient is given, and the formula can be converted to: Finally, ignore higher-order terms. There is a voltage across the ferroelectric layer. for: ; Since the Poisson equation in cylindrical coordinates is expressed as... ,in It is the radial component of the displacement vector. There are no moving charges in the insulating layer, therefore, according to Gauss's law... In the intermediate insulating layer With ferroelectric layer At the boundary, there exists a proportional relationship as follows: ,in, In position The displacement vector at that point is equal to ,and In the insulating layer and ferroelectric layer ( The displacement vector at the interface, where R is the channel radius; Calculation of ferroelectric layer The values ​​in the range are: ; In the ferroelectric layer, the electric field and polarization The following relationships exist: ,in It is an electric displacement field, where the polarization displacement field is approximated as: ; Therefore, it exists. With charge The formula for the relationship is: The coefficients a, b, and c are related to the Landau parameters of the ferroelectric material of the cylindrical gate. and Related.

6. The simulation construction method for the physical analytical model of NC-GAAFET as described in claim 5, characterized in that, The formula for calculating the Landau parameter is as follows: in , , It is expressed as the thickness of the ferroelectric layer.

7. The simulation construction method for the physical analytical model of NC-GAAFET as described in claim 6, characterized in that, The derivation of the source-drain current model of a conventional GAAFET and the calculation process of the drain-to-barrier reduction effect parameters in step S6 include: Substituting the approximate potential distribution function at the channel center and the potential distribution function at the channel surface into the two-dimensional potential distribution function in the radial r direction, the two-dimensional potential distribution function is calculated, revealing the presence of source and leakage currents. The calculation formula is: ; in, It is absolute temperature. q is electron mobility, and q is electron charge. It is the drain-source voltage. Represented as a set of Boltzmann constants; The formula for calculating the leakage barrier reduction effect parameter DIBL is as follows: .

8. The simulation construction method for the physical analytical model of NC-GAAFET as described in claim 7, characterized in that, The calculation method for the electrical characteristics of the NC-GAAFET, including source-drain current and subthreshold swing, in step S6 includes: Coupled with the conventional GAAFET model, there exists a gate surface charge density. The model is ; in, It is the length from the gate to the drain. and For function The specified coefficient, and For function The specified coefficient; According to the LK equation, using the moving charge density express The expression is: Where the coefficient and It is related to the Landau parameters of ferroelectric materials. It is the flexural coefficient; The voltage balance condition for a short-channel NC-GAAFET considering the flexural effect is: ; in, and All Landau parameters; When the gate voltage of NC-GAAFET At that time, the effective gate voltage of the corresponding GAAFET Corresponding source-drain current The expression is: in, It is a set of Boltzmann constants; The subthreshold swing SS of the NC-GAAFET is: in, Indicates drain-source current, It is the subthreshold swing of GAA-FET. Represented as gate voltage, Defined as the gate voltage of a GAA-FET.

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