An electromigration process simulation method and system combining atomic evolution information and an electronic device

Abstract

The application discloses a kind of electromigration process simulation method, system and electronic equipment combined with atomic evolution information, the present application is aimed at the defects that traditional simulation method is difficult to describe the phenomenon of cavity growth and expansion in electromigration (EM) in the service process of electronic packaging device, combined with finite element simulation method (FEM), by identifying temperature and density information file, atomic stress information is calculated using iterative algorithm, and atomic evolution information after iterative solution is obtained using visualization software.The present application gets rid of the relatively cumbersome restrictions on the mathematical method of solving atomic concentration by mass flux differential equation, and provides a theoretical basis for studying the formation and expansion of cavities in the electromigration process at the microscale by simulating atomic evolution in the electromigration process using molecular dynamics (MD).

Description

Technical Field

[0001] This invention belongs to the field of molecular dynamics simulation, specifically relating to a simulation method and electronic device that can combine atomic evolution information to describe electromigration under multi-physics coupling. Background Technology

[0002] Electromigration (EM) is a mass transfer phenomenon commonly occurring in electronic circuits, induced by high current density. Collisions between electrons in a conductor and metal atoms / ions generate momentum transfer, causing atoms to migrate and adjust their atomic arrangement. This leads to void nucleation on the cathode side and mound formation on the anode side, potentially causing short circuits in electrical devices. With the shrinking feature size of integrated circuits, electromigration significantly impacts their lifespan. Therefore, electromigration and its impact on integrated circuits remain an important topic in microelectronics reliability research. Since Gerardin observed electromigration in 1861, numerous experiments have been conducted to study its formation mechanism, resulting in a wealth of empirical theories. According to Fiks and Huntington, the driving force of an "electron wind" from the current induces electromigration; this technique may be applied to indirectly determine the activation energy of diffusion. Furthermore, the migration of metal atoms is often influenced by factors such as temperature, stress, and concentration.

[0003] Based on the characteristics of electromigration, many theoretical models have been used to reveal the phenomenon. In 1993, Korhonen et al. established an electromigration model to analyze the coupling relationship between vacancy migration and stress evolution, but neglected the influence of vacancy self-diffusion on the electromigration process. In 2004, Sukharev et al. developed a multiphysics model and implemented it in general-purpose commercial finite element software. In recent years, with the continuous deepening of research, a general coupling model (GCM) for electromigration has been established. This model considers all physical fields and fully couples them. However, the fully coupled multiphysics theory has a unique set of differential equations that cannot be directly simulated by standard finite element software such as ANSYS. In response, in 2022, Cui et al. used the weak-form PDE modulus in COMSOL software to obtain the weak-form governing equations for three-dimensional finite element modeling of electromigration. In the experiment, the one-dimensional finite element solution and the analytical solution of the conductor under full constraint showed good consistency. However, the above models are difficult to incorporate the formation process of pores. The formation, development, and migration of pores affect the distribution of current density and stress. Moreover, its migration is a dynamic process. In 2023, Zhang et al. proposed a modified fully coupled peridynamics (PD) model based on this, which includes the coupling relationship between electromigration, thermal strain and mechanical strain. The model numerically simulated the nucleation and growth of voids in metal wires. The results obtained by this method are consistent with previous experimental data, which verifies the accuracy of the PD model.

[0004] Current research on the electromagnetic phenomenon mostly uses the finite element method (FEM) numerical simulation, focusing on optimizing the mathematical solution process of the coupled equations. However, some details are still not well explained. This is because many factors influence the electromagnetic process, such as multiple competing driving forces, grain boundary structure, interface structure, and metal composition. In the electromagnetic process, atoms can diffuse through the bulk, grain boundaries, or surface, which involves dynamic mechanisms. Simulating only through constitutive equations and element birth and death judgment methods within the finite element method cannot fully describe the atomic motion, energy distribution, and stress distribution information. Summary of the Invention

[0005] In view of the above-mentioned shortcomings of the prior art, the purpose of this invention is to provide a simulation method, system, and electronic device capable of describing electromigration under multiphysics coupling by incorporating atomic evolution information. This method focuses on analyzing the atomic concentration distribution during electromigration and studies the evolutionary mechanism of void formation and expansion under electrical loading based on molecular dynamics.

[0006] To achieve the above objectives, the technical solution of the method of the present invention is as follows:

[0007] An electromigration simulation method incorporating atomic evolution information includes the following steps:

[0008] S1: Initialize the settings in the finite element software and set the time step ΔT and total duration T for the iterative calculation of the electric field intensity;

[0009] S2: Input the atomic configuration file into the molecular dynamics calculation software to calculate the temperature information file, atomic density information file, and new atomic configuration file;

[0010] S3: Read the temperature information file, atomic density information file and new atomic configuration file output in step S2, calculate and output the atomic force and temperature information after the completion of this iteration, update the velocity information of each atom and write a new atomic configuration file;

[0011] S4: Repeat steps S2-S3 once for the new atomic configuration file output in step S3. To complete one iteration, a total of T / ΔT iterations are required.

[0012] S5: Based on the atomic configuration files output from steps S2-S4, analyze the evolution mechanism of void formation and expansion using visualization software.

[0013] Further, step S1 specifically involves setting the mesh generation method for the discontinuous medium system in the discrete system finite element software, and setting the initial temperature field, current boundary conditions, time step ΔT and total duration T of the iterative calculation process.

[0014] Further, step S2 includes the following sub-steps:

[0015] S2.1 In molecular dynamics calculation software, read the atomic configuration file, including the type, coordinates, velocity and mass information of each atom;

[0016] S2.2 divides the system into multiple atomic blocks, each containing a certain number of atoms;

[0017] S2.3 For each atomic block, at specified step intervals, calculate the atomic number density information and store it in the density information file;

[0018] S2.4 For each atomic block, the velocity of the atoms within that block is statistically analyzed, and the average kinetic energy of the atoms within that block is calculated. Based on the theory of statistical physics, the kinetic energy is converted into temperature, and the temperature value is calculated using the average kinetic energy.

[0019] S2.5 Iterate through all atomic blocks, record the temperature of each atomic block and store it in a temperature information file.

[0020] Furthermore, step S3 includes the following specific steps:

[0021] S3.1 reads the temperature information file and the atomic density information file, calculates and outputs the electric field strength matrix and temperature matrix, and then calculates the atomic force information;

[0022] S3.2 reads the initial velocity and temperature of each atom from the new atomic configuration file provided by S2, performs velocity scaling, and updates the velocity information of each atom;

[0023] S3.3 Based on the calculated atomic force information, solve the iterative atomic velocity for each atom according to Newton's laws of motion, and write a new atomic configuration file based on the atomic velocity information.

[0024] Furthermore, the formula for calculating the atomic force information in step S3.1 is as follows:

[0025] E = -gardV

[0026] F = Z * eE

[0027] Where E represents the electric field strength, V represents the electric potential, Gard represents the gradient, F represents the force on the atom, and Z represents the electric field strength. * This represents the effective charge number of the material at the current temperature, and e represents the charge constant.

[0028] Furthermore, in step S3.2, the initial velocity value of each atom is read from the new atomic configuration file provided in S2. Based on the relationship between atomic velocity and temperature:

[0029]

[0030] Calculate the initial temperature T corresponding to the atoms in the original atomic configuration file. ori Where KE is the overall kinetic energy of N atoms, dim is the corresponding system dimension, and k B This is the Boltzmann constant, and T is the temperature corresponding to KE at this point. The atomic temperature is calculated as T. new After that, T new / T ori As a scaling factor, based on the relation Perform velocity scalar scaling to calculate the updated atomic velocities.

[0031] Furthermore, in step S3.3, based on the calculated atomic force information, each atom is subjected to the following formula according to Newton's laws of motion:

[0032] F = ma

[0033] Where F represents the force acting on the atom, m is the atomic mass, and a is the acceleration. Integrating over a time interval ΔT yields the atomic velocity information for this iteration. For the iterated atomic velocities, write a new atomic configuration file based on the atomic velocity information.

[0034] Furthermore, molecular dynamics software was used to observe the motion of metal atoms after iterative calculations, and visualization software was used to analyze the evolution mechanism of void formation and expansion during electromigration.

[0035] Secondly, the present invention provides an electromigration simulation system incorporating atomic evolution information, comprising:

[0036] Module 1: It is used to perform initialization settings in the finite element software and to set the time step ΔT and total duration T of the iterative calculation process of the electric field intensity;

[0037] Module 2: It is used to input atomic configuration files into molecular dynamics calculation software to calculate temperature information files, atomic density information files, and new atomic configuration files;

[0038] Module 3: It is used to read the output temperature information file, atomic density information file and new atomic configuration file, calculate and output the atomic force and temperature information after the completion of this iteration, update the velocity information of each atom and write a new atomic configuration file;

[0039] Module 4: This module is used to generate a new atomic configuration file from the output of the steps. It repeats the steps of Module 2 and Module 3. To complete one iteration, a total of T / ΔT iterations are required.

[0040] Module 5: It is used to analyze the evolution mechanism of void formation and expansion through visualization software based on the atomic configuration files output by Modules 2 to 4;

[0041] The electromigration simulation incorporating atomic evolution information is used to execute the steps in the electromigration simulation method incorporating atomic evolution information described above.

[0042] Thirdly, the present invention provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement the electromigration simulation method incorporating atomic evolution information as claimed in any one of claims 1 to 7.

[0043] Compared with the prior art, the present invention has the following beneficial effects:

[0044] (1) The simulation method described in this invention can overcome the distortion problem caused by the inability to identify vacuum regions in traditional models and the influence of non-uniform and discontinuous medium systems.

[0045] (2) The simulation method described in this invention can provide specific atomic-level evolution information, realize real-time simulation of the atomic diffusion process of a large number of particles in electromigration, and simulate void growth.

[0046] (3) The simulation method described in this invention provides a theoretical basis for studying the formation and expansion of voids in the electromigration process at the microscale. Attached Figure Description

[0047] Figure 1 This is a flowchart of the steps in a method for simulating electromigration processes that incorporates atomic evolution information, as described in this invention.

[0048] Figure 2 This is an initial model diagram according to an embodiment of the present invention.

[0049] Figure 3 This is the initial atomic distribution according to an embodiment of the present invention.

[0050] Figure 4 This refers to the atomic flux at different times according to an embodiment of the present invention.

[0051] Figure 5 This is information on the specific evolution of atoms at different times according to an embodiment of the present invention. Detailed Implementation

[0052] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, 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 some, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0053] Example 1

[0054] like Figure 1 As shown in the figure, this embodiment provides a simulation method for electromigration processes that incorporates atomic evolution information. The specific steps are as follows:

[0055] S1: Initialize the settings in the finite element software and set the time step ΔT and total duration T for the iterative calculation of the electric field intensity;

[0056] S2: Input the atomic configuration file into the molecular dynamics calculation software to calculate the temperature information file, atomic density information file, and new atomic configuration file;

[0057] S3: Read the temperature information file, atomic density information file and new atomic configuration file output in step S2, calculate and output the atomic force and temperature information after the completion of this iteration, update the velocity information of each atom and write a new atomic configuration file;

[0058] S4: Repeat steps S2-S3 once for the new atomic configuration file output in step S3. To complete one iteration, a total of T / ΔT iterations are required.

[0059] S5: Based on the atomic configuration files output from steps S2-S4, analyze the evolution mechanism of void formation and expansion using visualization software.

[0060] The technical solution of the present invention will be further described below with reference to specific embodiments, focusing on the electromigration process in a three-dimensional copper conductor. The constructed model has dimensions of 2×0.2×0.6μm and is made of copper.

[0061] In step S1, the mesh generation method for the system is first set as shown in Table 1:

[0062] The system settings used in Table 1

[0063]

[0064] Initially, the temperature of all grids was set to 300K (room temperature), the anode potential was set to 3V, the cathode was grounded, and a fixed constraint was applied to the bottom surface. Figure 2 As shown. The time step ΔT of the calculation process is set to 0.0001 μs, and the total duration is 5 μs. Step S1 ends.

[0065] Proceed to step S2, run the molecular dynamics calculation software (LAMMPS in this embodiment), and select "metal" as the unit system in the input file. Calculate the interatomic interaction forces using the eam / alloy potential function, using the Mendelev-Cu2-2012.eam potential function file as the input. The molecular dynamics calculation software outputs the calculation results, including: atomic configuration file, temperature information file, and atomic density information file. Step S2 is now complete.

[0066] In step S3, the calculation program reads the initial state of the system from the input atomic configuration file, such as... Figure 3 As shown, following the mesh generation method of the molecular dynamics calculation software in step S2, atomic blocks are divided within the mesh, and the density of each divided atomic block is further calculated. Subsequently, the calculation results are averaged over time and summarized, and output to a density information file (density-profile.txt), with the density unit being atoms / ... Meanwhile, using the information from each atomic block as defined above, the temperature distribution is calculated every specified number of steps and output to the temperature information file (temperature-profile.txt).

[0067] The relevant physical parameters used in the electromigration iterative calculation process are shown in Table 2:

[0068] Table 2 Physical parameters used in the calculation process

[0069]

[0070] In this embodiment, the time step size during the iterative calculation process is set to 0.0001 μs, and the spatial step size after mesh generation is... Based on the aforementioned parameter settings and calculation equations, run the calculation program.

[0071] Furthermore, the computational program reads the atomic configuration file output in step S2, performs scalar scaling on the velocity, updates the velocity information of each atom, and writes a new atomic configuration file according to the format required by the molecular dynamics computational software. The atomic configuration file is then input into the molecular dynamics computational software, and the above steps are repeated to begin iterative iteration. Step S3 concludes.

[0072] For the atomic configuration file output in step S3, repeat steps S2-S3 once. To complete one iteration, a total of 50,000 iterations are required before ending the iteration loop. Organize all output result files and analyze the final calculated data.

[0073] According to Fick's law, diffusion flux is related to the concentration gradient. Fick's first law states:

[0074]

[0075] J C Let D be the diffusion flux, C be the diffusion coefficient, and D be the concentration. The atomic diffusion flux divergence is div(J). C If the value is positive, it promotes the loss of atoms and the formation of voids; otherwise, it inhibits the loss.

[0076] like Figure 4 The left side shows the atomic flux distribution at t=0. The black arrows indicate the direction of current flow, showing that copper atoms have not yet migrated in the wire. Figure 4The right side shows the atomic flux distribution at t = 5 μs. The calculation results show that the atomic diffusion flux divergence caused by electromigration is almost zero in straight interconnects, but larger at the bends of interconnects. The direction of current is opposite to the direction of electron movement. At the bend, the atomic flux divergence is negative in the direction of current inflow, and atomic deposition occurs; at the bend, the atomic flux divergence is positive in the direction of current outflow, and atomic migration occurs.

[0077] further, Figure 5 (a), (c), and (e) show the specific evolution information of copper atoms at the initial, intermediate, and final moments on the anode side, respectively. Figure 5 (b), (d), and (f) show the evolution of copper atoms at three different time points on the cathode side. It can be seen that initially, in (a) and (b), the atomic distribution is uniform; after a period of energization, in (c), a small amount of atoms accumulate at the anode-side corner; in (d), some atoms migrate at the cathode-side corner, beginning to form voids; as the electromigration process continues, in (e), a large number of atoms deposit at the anode-side corner, with obvious nucleation; and in (f), the voids gradually enlarge. This is consistent with previous research findings.

[0078] Example 2

[0079] This embodiment provides module one, which is used to perform initialization settings in finite element software and set the time step ΔT and total duration T of the iterative calculation process of electric field strength;

[0080] Module 2 is used to input atomic configuration files into molecular dynamics calculation software to calculate temperature information files, atomic density information files, and new atomic configuration files;

[0081] Module 3: It is used to read the output temperature information file, atomic density information file and new atomic configuration file, calculate and output the atomic force and temperature information after the completion of this iteration, update the velocity information of each atom and write a new atomic configuration file;

[0082] Module 4: This module is used to generate a new atomic configuration file from the output of the steps. It repeats the steps of Module 2 and Module 3. To complete one iteration, a total of T / ΔT iterations are required.

[0083] Module 5: It is used to analyze the evolution mechanism of void formation and expansion through visualization software based on the atomic configuration files output by Modules 2 to 4.

[0084] Example 3

[0085] An electronic device includes a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the program, it implements the electromigration process simulation method incorporating atomic evolution information as described in Example 1.

[0086] The above description is merely a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any modifications, equivalent substitutions, and improvements made by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the invention. Although preferred embodiments of the present invention have been described, those skilled in the art, once they understand the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments as well as all changes and modifications falling within the scope of the present invention.

[0087] Obviously, those skilled in the art can make various modifications and variations to the embodiments of the present invention without departing from the spirit and scope of the embodiments of the present invention. Thus, if these modifications and variations to the embodiments of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention also intends to include these modifications and variations.

[0088] All other parts not described in detail are existing technologies.

Claims

1. An electromigration simulation method incorporating atomic evolution information, characterized in that, Includes the following steps: S1: Initialize the settings in the finite element software and set the time step ΔT and total duration T for the iterative calculation of the electric field intensity; S2: Input the atomic configuration file into the molecular dynamics calculation software to calculate the temperature information file, atomic density information file, and new atomic configuration file; S3: Read the temperature information file, atomic density information file and new atomic configuration file output in step S2, calculate and output the atomic force and temperature information after the completion of this iteration, update the velocity information of each atom and write a new atomic configuration file; The specific steps include the following: S3.1 reads the temperature information file and the atomic density information file, calculates and outputs the electric field strength matrix and temperature matrix, and then calculates the atomic force information; S3.2 reads the initial velocity and temperature of each atom from the new atomic configuration file provided by S2, performs velocity scaling, and updates the velocity information of each atom; S3.3 Based on the calculated atomic force information, solve the iterative atomic velocity for each atom according to Newton's laws of motion, and write a new atomic configuration file based on the atomic velocity information; S4: Repeat steps S2-S3 once for the new atomic configuration file output in step S3. To complete one iteration, a total of T / ΔT iterations are required. S5: Based on the atomic configuration files output from steps S2-S4, analyze the evolution mechanism of void formation and expansion using visualization software.

2. The electromigration simulation method incorporating atomic evolution information according to claim 1, characterized in that, Step S1 specifically involves setting the mesh generation method for the discontinuous medium system in the discrete system finite element software, and setting the initial temperature field, current boundary conditions, time step ΔT and total duration T of the iterative calculation process.

3. The electromigration simulation method incorporating atomic evolution information according to claim 1, characterized in that, Step S2 includes the following sub-steps: S2.1 In molecular dynamics calculation software, read the atomic configuration file, including the type, coordinates, velocity and mass information of each atom; S2.2 divides the system into multiple atomic blocks, each containing a certain number of atoms; S2.3 For each atomic block, at specified step intervals, calculate the atomic number density information and store it in the density information file; S2.4 For each atomic block, the velocity of the atoms within that block is statistically analyzed, and the average kinetic energy of the atoms within that block is calculated. Based on the theory of statistical physics, the kinetic energy is converted into temperature, and the temperature value is calculated using the average kinetic energy. S2.5 Iterate through all atomic blocks, record the temperature of each atomic block and store it in a temperature information file.

4. The electromigration simulation method incorporating atomic evolution information according to claim 1, characterized in that, The formula for calculating the force information of atoms in step S3.1 is as follows: in, Indicates electric field strength. Represents electric potential, Represents the gradient. This indicates that the atom is subjected to force. This indicates the effective charge number of the material at the current temperature. This represents the charge constant.

5. The electromigration simulation method incorporating atomic evolution information according to claim 1, characterized in that, In step S3.2, the initial velocity value of each atom is read from the new atomic configuration file provided in S2. Based on the relationship between atomic velocity and temperature: Calculate the initial temperature corresponding to the atoms in the original atomic configuration file. ,in yes The overall kinetic energy of an individual atom It corresponds to the system dimension. It is the Boltzmann constant. It is at this moment The corresponding temperature magnitude, the atomic temperature is calculated as follows: Afterwards, As a scaling factor, based on the relation Perform scalar scaling on the velocity to calculate the updated atomic velocities. .

6. The electromigration simulation method incorporating atomic evolution information according to claim 5, characterized in that, In step S3.3, based on the calculated force information of the atoms, each atom is subjected to the following formula according to Newton's laws of motion: in For atoms to experience forces, atomic mass For acceleration, the integral is performed over time ΔT to obtain the atomic velocity information for this iteration. , For the iterated atomic velocities, write a new atomic configuration file based on the atomic velocity information.

7. The electromigration simulation method incorporating atomic evolution information according to claim 1, characterized in that, The motion of metal atoms after iterative calculations was observed using molecular dynamics software, and the evolution mechanism of void formation and expansion during electromigration was analyzed using visualization software.

8. An electromigration simulation system incorporating atomic evolution information, characterized in that: include: Module 1: It is used to perform initialization settings in the finite element software and to set the time step ΔT and total duration T of the iterative calculation process of the electric field intensity; Module 2: It is used to input atomic configuration files into molecular dynamics calculation software to calculate temperature information files, atomic density information files, and new atomic configuration files; Module 3: It is used to read the output temperature information file, atomic density information file and new atomic configuration file, calculate and output the atomic force and temperature information after the completion of this iteration, update the velocity information of each atom and write a new atomic configuration file; Module 4: This module is used to generate a new atomic configuration file from the output of the steps. It repeats the steps of Module 2 and Module 3. To complete one iteration, a total of T / ΔT iterations are required. Module 5: It is used to analyze the evolution mechanism of void formation and expansion through visualization software based on the atomic configuration files output by Modules 2 to 4; The electromigration simulation incorporating atomic evolution information is used to perform the steps in the electromigration simulation method incorporating atomic evolution information as described in any one of claims 1-6.

9. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the program, it implements the electromigration simulation method incorporating atomic evolution information as described in any one of claims 1 to 6.

Images