Equivalent path and electricity-heat-force multi-physical path modeling method for simulating electro-thermal coupling effect of integrated circuit

Abstract

The invention discloses an equivalent path and electricity-heat-force multi-physical path modeling method for simulating an electro-thermal coupling effect of an integrated circuit. The method comprises the following steps: (1) establishing a circuit model, an equivalent thermal circuit model and an equivalent circuit model according to a to-be-simulated integrated circuit structure; (2) calculating values of electrical elements in the circuit, the equivalent thermal circuit and the equivalent circuit according to actual working conditions and formulas; (3) constructing boundary conditions of an electric-thermal-force equivalent multi-physical circuit; and (4) solving the constructed equivalent multi-physical circuit. According to the equivalent multi-physical circuit modeling method developed by the invention, the electro-thermal-mechanical coupling effect in an integrated circuit can be simulated, and support is provided for circuit design.

Description

Technical Field

[0001] This invention belongs to the field of electronic design automation (EDA) and relates to an equivalent circuit and an electro-thermal-mechanical multi-physics circuit modeling method for simulating the electro-thermal-mechanical coupling effect of integrated circuits. Background Technology

[0002] As the integration density of 3D heterogeneous integrated circuits (3D HICs) continues to increase, multiphysics effects in the circuits are becoming increasingly significant and complex. Multiphysics effects can severely impact material and device performance, thereby affecting the performance and reliability of 3D HICs. Therefore, accurately and effectively modeling and analyzing multiphysics effects has become one of the most challenging problems in 3D HIC design.

[0003] Electrothermal coupling significantly degrades the signal transmission characteristics of integrated circuits, such as delay jitter and noise margin, and can cause device reliability issues such as transistor threshold voltage drift. For 3D HICs, the equivalent thermal circuit method has become the core method for realizing electrothermal co-simulation due to its fast computation speed and seamless compatibility with the SPICE platform, and can effectively support the multiphysics joint optimization design of 3D HICs.

[0004] Thermal stress analysis in 3D HIC is also crucial for maintaining system stability and reliability, as thermal stress can lead to device performance degradation, device failure, and even microcracks. Previous studies have introduced spring-mass-damped networks integrated with equivalent thermal circuits to simulate electro-thermo-mechanical interactions (M. Garci, J.-B. Kammerer and L. Hebrard, "Towards electro-thermo-mechanical simulation of integrated circuits in standard CAD environment," Microelectronics Journal, 2015). However, there is currently no equivalent multiphysics circuit framework available for simulating electro-thermo-mechanical interactions.

[0005] Based on this, the present invention proposes an equivalent circuit and an electro-thermal-mechanical multi-physics circuit modeling method for simulating the electro-thermal-mechanical coupling effect of integrated circuits, which can simulate the electro-thermal-mechanical interaction of integrated circuits based on the equivalent circuit. Summary of the Invention

[0006] To address the shortcomings of existing technologies, this invention provides an equivalent path and an electro-thermal-mechanical multi-physics path modeling method for simulating the electro-thermal-mechanical coupling effect of integrated circuits.

[0007] The technical solution adopted in this invention is as follows:

[0008] An equivalent path and electro-thermal-mechanical multi-physics path modeling method for simulating electro-thermal-mechanical coupling effects in integrated circuits includes the following steps:

[0009] 1) Based on the integrated circuit structure to be simulated, establish the circuit, equivalent thermal path, and equivalent mechanical path to form an electro-thermal-mechanical equivalent multi-physics circuit;

[0010] 2) Set the values ​​of electrical components in the circuit according to the actual working conditions, and calculate the values ​​of electrical components in the equivalent thermal circuit and the equivalent thermal circuit.

[0011] 3) Based on the actual working conditions of the integrated circuit to be simulated, construct the boundary conditions of the electro-thermal-mechanical equivalent multi-physics circuit;

[0012] 4) Solve for the constructed equivalent multi-physics circuit.

[0013] In the above technical solution, further, in step 1), an equivalent circuit is established according to the integrated circuit structure to be simulated, specifically following the following steps: (1) Set circuit nodes at the desired location in the structure; (2) Connect adjacent nodes and set resistors between nodes; (3) Add DC current sources to the circuit nodes.

[0014] Furthermore, in step 1), the equivalent thermal circuit and the equivalent thermal circuit are coupled based on the equivalent current source in the equivalent thermal circuit.

[0015] Furthermore, the equivalent circuit electrical element described in 2) has the following expression: For the equivalent circuit used to calculate the displacement u in the x-direction, we have:

[0016] The resistance in the x-direction is:

[0017]

[0018] The resistance in the y-direction is:

[0019]

[0020] The resistance in the z-direction is:

[0021]

[0022] The current source is:

[0023]

[0024] Where: Δx is the distance between the two nodes of the resistor in the x-direction, Δy is the distance between the two nodes of the resistor in the y-direction, Δz is the distance between the two nodes of the resistor in the z-direction, λ and μ are Lamé coefficients, v represents the displacement in the y-direction, w represents the displacement in the z-direction, T represents the temperature, and α... xx Indicates the coefficient of thermal expansion;

[0025] The same applies to the equivalent path used to calculate the displacement v in the y direction and the equivalent path used to calculate the displacement w in the z direction.

[0026] Furthermore, in step 3), the equivalent circuit boundary conditions are constructed according to the actual working conditions of the integrated circuit to be simulated. Specifically: for fixed boundary conditions, the nodes in the corresponding equivalent circuit are set to ground to construct fixed boundary conditions; for equivalent circuit structures with periodic boundaries, the displacements at the two nodes can satisfy the periodic boundary by connecting the two boundary nodes with wires; for free boundary conditions, the resistance of the absorption layer is constructed in the equivalent circuit to construct the free boundary.

[0027] Furthermore, the method for solving the constructed equivalent path in step 4) includes the following steps:

[0028] Step 1: Obtain the temperature based on the equivalent thermal path method;

[0029] Step 2: Guess the initial displacement;

[0030] Step 3: For displacements in different directions, solve for the corresponding equivalent path and obtain the displacement results;

[0031] Step 4: If the displacement change is less than the preset tolerance, the simulation ends; otherwise, proceed to the next step.

[0032] Step 5: Update the current source in the equivalent circuit, and then proceed to step 3.

[0033] Compared with the prior art, the present invention has the following beneficial effects:

[0034] 1. This invention provides an equivalent circuit and an electro-thermal-mechanical multi-physics circuit modeling method for simulating electro-thermal-mechanical coupling effects in integrated circuits. This method can simulate electro-thermal-mechanical coupling effects in various complex structures.

[0035] 2. The equivalent thermal circuit method provided by this invention can be deployed on the SPICE platform and is compatible with the equivalent thermal circuit method, thus greatly improving simulation efficiency. Attached Figure Description

[0036] Figure 1 (a) The cube structure to be simulated; (b) The equivalent path for calculating displacement u; (c) The equivalent path for calculating displacement v; (d) The equivalent path for calculating displacement w.

[0037] Figure 2 (a) Periodic boundary conditions; (b) Free boundary conditions.

[0038] Figure 3The diagram shows (a) the current density spectrum distribution of the FinFET, (b) the temperature distribution calculated by FEM, (c) the thermal stress distribution calculated by FEM, (d) the circuit diagram, (e) the equivalent thermal circuit diagram, and (f) the equivalent thermal circuit diagram.

[0039] Figure 4 The displacement distribution results calculated based on multi-physics equivalent circuit and FEM numerical simulation are compared.

[0040] Figure 5 The conductivity distribution results calculated based on multi-physics equivalent circuit and FEM numerical simulation are compared. Detailed Implementation

[0041] The present invention will be described in detail below with reference to specific embodiments. The following examples will help those skilled in the art to further understand the present invention, but are not intended to limit the present invention in any form. It should be noted that, for those skilled in the art, several changes and improvements can be made without departing from the scope of the present invention. These all fall within the scope of protection of the present invention.

[0042] In the following examples, a typical three-dimensional nanoscale semiconductor electronic device is selected, taking the p-type fin field-effect transistor (FinFET) as an example, to describe in detail the quantum transport-thermal-thermal stress coupling simulation of the three-dimensional nanoscale device of the present invention. However, the equivalent circuit and electro-thermal-mechanical multi-physics circuit modeling method that is compatible with SPICE proposed in this invention is not only applicable to fin field-effect transistors, but also applicable to the modeling and analysis of electro-thermal-mechanical coupling effects in most structures.

[0043] According to a specific embodiment of the present invention, the method for modeling the equivalent circuit and the electro-thermal-mechanical multi-physics circuit of the above structure specifically includes the following:

[0044] 1. Construct circuits, equivalent thermal circuits, and equivalent mechanical circuits to form an electro-thermal-mechanical equivalent multi-physics circuit:

[0045] by Figure 1 The cubic structure in (a) is used as an example to illustrate the method of constructing equivalent multi-physical paths. Since the methods of constructing circuits and equivalent thermal paths are well known, they will not be described again; only the method of constructing equivalent thermal paths will be detailed.

[0046] The method for constructing an equivalent circuit should follow these steps: 1) Set up circuit nodes at the desired location in the structure; 2) Connect adjacent nodes and set resistors between nodes; 3) Add DC current sources to the circuit nodes.

[0047] 2. Expressions for resistance and current source in equivalent circuit:

[0048] use Figure 1When calculating the displacement u in the x-direction using the equivalent circuit in (b), the resistance in the x-direction is calculated using the following expression.

[0049]

[0050] Where Δx is the distance between the two nodes of the resistor in the x-direction, Δy is the distance between the two nodes of the resistor in the y-direction, and Δz is the distance between the two nodes of the resistor in the z-direction. λ and μ are the Lamé coefficients. The resistance in the y-direction is calculated using the following expression:

[0051]

[0052] The resistance in the z-direction is calculated using the following expression.

[0053]

[0054] The current source is calculated using the following expression:

[0055]

[0056] Where v represents displacement in the y-direction, w represents displacement in the z-direction, and T represents temperature. The numerical values ​​of the variable subscripts correspond to... Figure 1 The node numbers in (a). α xx This represents the coefficient of thermal expansion.

[0057] use Figure 1 In (c), when calculating the displacement v in the y-direction using the equivalent path, the resistance in the x-direction is calculated using the following expression.

[0058]

[0059] The resistance in the y-direction is calculated using the following expression.

[0060]

[0061] The resistance in the z-direction is calculated using the following expression.

[0062]

[0063] The current source is calculated using the following expression:

[0064]

[0065] use Figure 1 When calculating the z-direction displacement w in the equivalent path in (d), the x-direction resistance is calculated using the following expression.

[0066]

[0067] The resistance in the y-direction is calculated using the following expression.

[0068]

[0069] The resistance in the z-direction is calculated using the following expression.

[0070]

[0071] The current source is calculated using the following expression:

[0072]

[0073] Among them, the equivalent current source in the equivalent thermal circuit can couple the equivalent thermal circuit with the equivalent thermal circuit, as shown in formulas (7), (14) and (21).

[0074] 3. Boundary conditions of equivalent paths

[0075] For fixed boundary conditions, the fixed boundary conditions can be constructed by setting the nodes in the corresponding equivalent path to ground.

[0076] Equivalent road structures with periodic boundaries, such as Figure 2 As shown in (a), by connecting node 1 and node 2 with a wire, the displacements at node 1 and node 2 can satisfy the periodic boundary.

[0077] For free boundary conditions, Figure 2 (b) shows a cube with free boundaries on its top and bottom surfaces, and a cube with a low Young's modulus absorbing layer whose top and bottom surfaces are fixed. The low Young's modulus absorbing layer is used to construct the free boundaries. Therefore, the free boundaries can be constructed by the resistor that forms the low Young's modulus absorbing layer in the equivalent circuit.

[0078] 4. The solution algorithm for the equivalent path is shown in Table I.

[0079] Table I. Equivalent Path Solution Algorithm

[0080]

[0081] In the finite element method (FEM) simulation process, temperature and thermal stress are calculated using FEM, with the FinFET power calculated by the non-equilibrium Green's function (NEGF) set as the heat source. For the temperature boundary, the outer surface of the STI is set to 300K. For the thermal stress boundary, the upper / lower edges of the FinFET are set to fixed, and the left / right edges are set to periodic. The current density spectrum of the FinFET, the temperature distribution calculated by FEM, and the thermal stress distribution calculated by FEM are shown below. Figure 3As shown in (a)-(c), the 3DFinFET structure has been equivalently represented as a 2D dual-gate structure. For circuit simulation, the circuit diagram, equivalent thermal diagram, and equivalent thermal diagram are shown below. Figure 3 As shown in (d)-(f), the values ​​of the voltage-controlled current source (VCCS) are calculated from the current density spectrum.

[0082] The displacement v distribution calculated based on the multiphysics equivalent circuit and FEM are as follows: Figure 4 As shown in (a1) and (a2), where the tolerance is set to 10. -6 nm. The displacement u distribution calculated based on the multi-physics equivalent circuit and finite element method is as follows: Figure 4 As shown in (b1) and (b2). The displacement v along the y-direction x = 5 / 10 / 15 nm line calculated based on the multiphysics equivalent circuit and FEM is as follows. Figure 4 As shown in (a3). The displacement u along the x-direction of the line y = 11 / 10 / 7.5nm, calculated based on the multiphysics equivalent circuit and FEM, is as follows. Figure 4 As shown in (b3). The calculated results of the multi-physics equivalent circuit agree well with the numerical results.

[0083] σ calculated from multi-physics equivalent circuit and FEM xx The distributions are as follows: Figure 5 As shown in (a1) and (a2). σ is calculated based on the multi-physics equivalent circuit and FEM. yy Distribution as Figure 5 As shown in (b1) and (b2). σ is calculated based on the multi-physics equivalent circuit and FEM. xx The line along the x-direction with y = 2.5 / 7.5 nm is as follows: Figure 5 As shown in (a3). σ is calculated based on the multi-physics equivalent circuit and FEM. yy The line along the x-direction with y = 2.5 / 7.5 nm is as follows: Figure 5 As shown in (b3). The results verify the correctness of the multi-physics equivalent circuit method.

[0084] The above describes specific embodiments of the present invention. It should be understood that the present invention is not limited to the specific embodiments described above, and those skilled in the art may make various changes or modifications within the scope of the claims, which do not affect the essence of the present invention. The embodiments of this application and the features in the embodiments may be combined with each other in any manner unless there is a conflict.

Claims

1. A method for modeling equivalent circuits and electro-thermal-mechanical multi-physics circuits for simulating electro-thermal-mechanical coupling effects in integrated circuits, characterized in that, Includes the following steps: 1) Based on the integrated circuit structure to be simulated, establish the circuit, equivalent thermal path, and equivalent mechanical path to form an electro-thermal-mechanical equivalent multi-physics circuit; 2) Set the values ​​of electrical components in the circuit according to the actual working conditions, and calculate the values ​​of electrical components in the equivalent thermal circuit and the equivalent thermal circuit. 3) Based on the actual working conditions of the integrated circuit to be simulated, construct the boundary conditions of the electro-thermal-mechanical equivalent multi-physics circuit; 4) Solve for the constructed equivalent multi-physics circuit.

2. The equivalent path and electro-thermal-mechanical multi-physics path modeling method for simulating electro-thermal coupling effects in integrated circuits according to claim 1, characterized in that, In step 1), an equivalent circuit is established based on the integrated circuit structure to be simulated. Specifically, the following steps are followed: (1) Set circuit nodes at the locations to be simulated in the structure; (2) Connect adjacent nodes and set resistors between nodes; (3) Add DC current sources to the circuit nodes.

3. The equivalent path and electro-thermal-mechanical multi-physics path modeling method for simulating electro-thermal coupling effects in integrated circuits according to claim 1, characterized in that, In step 1), the equivalent current source in the equivalent thermal circuit enables coupling between the equivalent thermal circuit and the equivalent thermal circuit.

4. The equivalent path and electro-thermal-mechanical multi-physics path modeling method for simulating electro-thermal coupling effects in integrated circuits according to claim 1, characterized in that, The equivalent circuit electrical element described in 2) has the following expression: For the equivalent circuit used to calculate the displacement u in the x-direction, we have: The resistance in the x-direction is: The resistance in the y-direction is: The resistance in the z-direction is: The current source is: Where: Δx is the distance between the two nodes of the resistor in the x-direction, Δy is the distance between the two nodes of the resistor in the y-direction, Δz is the distance between the two nodes of the resistor in the z-direction, λ and μ are Lamé coefficients, v represents the displacement in the y-direction, w represents the displacement in the z-direction, T represents the temperature, and α... xx Indicates the coefficient of thermal expansion; The same applies to the equivalent path used to calculate the displacement v in the y direction and the equivalent path used to calculate the displacement w in the z direction.

5. The equivalent path and electro-thermal-mechanical multi-physics path modeling method for simulating electro-thermal coupling effects in integrated circuits according to claim 1, characterized in that, In step 3), the equivalent circuit boundary conditions are constructed according to the actual working conditions of the integrated circuit to be simulated. Specifically: for fixed boundary conditions, the nodes in the corresponding equivalent circuit are set to ground to construct fixed boundary conditions; for equivalent circuit structures with periodic boundaries, the displacements at the two nodes can be made to satisfy the periodic boundary by connecting the two boundary nodes with wires; for free boundary conditions, the free boundary can be constructed by constructing the resistance of the absorption layer in the equivalent circuit.

6. The equivalent path and electro-thermal-mechanical multi-physics path modeling method for simulating electro-thermal coupling effects in integrated circuits according to claim 1, characterized in that, The method for solving the constructed equivalent path in step 4) includes the following steps: Step 1: Obtain the temperature based on the equivalent thermal path method; Step 2: Guess the initial displacement; Step 3: For displacements in different directions, solve for the corresponding equivalent path and obtain the displacement results; Step 4: If the displacement change is less than the preset tolerance, the simulation ends; otherwise, proceed to the next step. Step 5: Update the current source in the equivalent circuit, and then proceed to step 3.

Images