A simulation method for micro-hole evolution in electric field-induced interconnection wire based on phase field method
By constructing a phase-field method for simulating micropores in interconnects induced by electric field, the problem of dynamic simulation of real-time evolution of micropores in interconnects of integrated circuits is solved. This method enables accurate description and reliability analysis of the micropore process and provides a theoretical basis for thermal erosion trenches at polycrystalline grain boundaries.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- NANJING COLLEGE OF INFORMATION TECH
- Filing Date
- 2026-04-01
- Publication Date
- 2026-07-07
AI Technical Summary
Existing technologies struggle to dynamically and continuously simulate the real-time evolution of micropores inside interconnects in integrated circuits under the influence of an electric field. In particular, the effects of anisotropic surface diffusion and grain boundary diffusion are not fully considered in polycrystalline interconnects.
A simulation method for electric field-induced micropores in interconnecting wires based on the phase field method is constructed. This method includes constructing a phase field model, introducing conservative and non-conservative phase field variables, electric field free energy function, surface diffusion function, and grain boundary diffusion mobility, and simulating the evolution process of micropores by coupling the phase field and electrostatic field control equations.
The dynamic simulation of micropores under the action of an electric field was realized, describing the drift, segmentation, merging and decomposition of micropores, providing a quantitative analysis tool for the reliability of interconnects in integrated circuits, and verifying the reliability and theoretical basis of the model.
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Abstract
Description
Technical Field
[0001] This invention relates to the field of electric field-induced micro-damage in interconnecting wires, and more particularly to a simulation method for the evolution of micropores in interconnecting wires induced by electric field based on the phase field method. Background Technology
[0002] With the continuous development of modern microelectronics technology and the constant reduction in the width of interconnects, the linewidth of interconnects has approached or is smaller than the average grain diameter of thin-film interconnects, exhibiting a structure resembling a "bamboo joint." Inevitably, some irregular defects exist within these interconnects, such as micropores, microcracks, or second-phase particles. These increase the resistance of the interconnects and may even lead to open-circuit failure. As the interconnect width decreases, transgranular micropore evolution has become the main damage form induced by electromigration in interconnects. Due to the significant reduction in the number of grain boundaries, the lifetime of "bamboo joint" interconnects has been significantly increased.
[0003] Currently, based on the phase-field method, the migration and growth behavior of micropores in single-crystal interconnects under the influence of electric and stress fields, mediated by surface and bulk diffusion mechanisms, has been studied in detail. However, the relationship between pore migration and grain boundaries under surface diffusion in polycrystalline interconnects, while considering the influence of grain boundary diffusion, is relatively rare. Current research based on anisotropic surface mobility largely focuses on the evolution of intracrystalline micropores, with few studies investigating the evolution of microstructures in bamboo-like interconnects under anisotropic surface diffusion mobility and grain boundary diffusion. Summary of the Invention
[0004] The technical problem to be solved by this invention is that the existing technology is difficult to dynamically and continuously simulate the real-time evolution process of micropores inside interconnects of integrated circuits under the action of an electric field, which leads to simulation distortion. This invention mainly explores the evolution process of circular micropores under anisotropic surface diffusion, and focuses on analyzing the influence of initial deflection angle, electric field strength and anisotropic intensity factor on the micropore change law.
[0005] 1. A simulation method for the evolution of micropores in interconnecting wires induced by electric field based on the phase-field method, characterized by comprising the following steps: Step 1: Construct a phase-field model of micropores in transcrystalline interconnects; the phase-field model of micropores in electric field-induced transcrystalline interconnects includes a matrix phase, a grain boundary phase, and a micropore phase. Step 2: Construct the conservative and non-conservative field phase variables related to time and space. , Spatial coordinates are The time coordinate is t ; Step 3: Construct an electric field free energy function that varies continuously with spatial coordinates. The potential smoothly transitions from the matrix phase to the porous phase through a diffusion interface layer; Step 4, the potential containing phase field variables Substituting into the Laplace equation, we obtain the potential distribution of the entire matrix; Step 5, introduce the total free energy function of the system. F This includes local free energy, conservative field and non-conservative field gradient energy, and electric field energy; Step 6: Introduce the surface diffusion function and grain boundary diffusion mobility in the {110} crystal direction. ; Step 7: Introduce the mass flux of surface atoms, and obtain the chemical potential by taking the variational function of the free energy of the whole system, and then obtain the mass flux of surface atoms. ; Step 8 yields the modified Cahn-Hilliard equation and Allen-Cahn equation, which serve as the phase field control equations. Step 9: Given the basic parameters of the conservative field, non-conservative field, and electrostatic field control equations, construct boundary conditions for the phase field and electrostatic field control equations, and complete the simulation of micropores in transcrystalline interconnects by coupling the conservative field, non-conservative phase field, and electrostatic field.
[0006] The beneficial effects of this invention application are as follows: (1) Based on the anisotropic surface diffusion and grain boundary diffusion mechanism, this invention constructs the governing equations of the phase containing crystal phase, grain boundary phase and micropore under the action of electric field. The total free energy of the model includes local free energy, gradient energy and electrostatic energy, which describes the evolution process of transcrystalline micropores.
[0007] (2) This invention verifies the reliability of the model by observing the evolution of grain boundary thermal erosion grooves under the surface diffusion mechanism, and also provides a theoretical basis for the analysis of polycrystalline grain boundary thermal erosion grooves.
[0008] (3) The present invention can intuitively describe the transcrystalline process of micropores, including the phenomena of micropore drift, segmentation, merging and re-decomposition, providing direct and quantitative theoretical basis for experimental analysis. Attached Figure Description
[0009] Figure 1 This is a flowchart illustrating the evolution of micropores in a metallic interconnect.
[0010] Figure 2 This is a flowchart of the solution process for the adaptive finite element method of the phase field model.
[0011] Figure 3 This is a schematic diagram of the phase field model of the internal connecting wires of a bicrystalline material under an electric field.
[0012] Figure 4 This is a comparison chart of the calculated results and theoretical solutions from the phase-field model.
[0013] Figure 5 for A schematic diagram illustrating the evolution of circular micropores.
[0014] Figure 6 for A schematic diagram illustrating the evolution of circular micropores.
[0015] Figure 7 for A schematic diagram illustrating the evolution of the lower circular micropores.
[0016] Figure 8 The diagram shows the number of segments under different angles and electric field intensities.
[0017] Figure 9 A diagram showing the number of segments for different angles and different anisotropic intensity factors.
[0018] Figure 10 This is a schematic diagram illustrating the change in the critical initial deflection angle with the electric field strength.
[0019] Figure 11 Critical anisotropy intensity factor A schematic diagram illustrating the variation of the initial angle. Detailed Implementation
[0020] The present invention will be further described below with reference to the accompanying drawings and examples.
[0021] This invention relates to a simulation method for the evolution of micropores in electric field-induced interconnects based on the phase-field method. The method includes constructing a phase-field model for the evolution of micropores in electric field-induced {110}-oriented "bamboo-like" interconnects in face-centered cubic metals based on coupled Cahn-Hilliard and Allen-Cahn equations; constructing phase-field variables for micropores and grain boundaries; constructing distribution functions for conductivity and atomic mobility varying with spatial coordinates; and constructing the system free energy function. F By introducing the free energy function of the fourth-order double potential well function to modify the Cahn-Hilliard equation and the Allen-Cahn equation, the phase field control equation of the system is obtained. The conservative phase field order parameter, the non-conservative phase field order parameter and the electrostatic field parameter are determined. By coupling the phase field and the electrostatic field, the evolution of micropores in the transcrystalline interconnect is obtained.
[0022] The model constructed in this invention can not only accurately simulate the evolution of micropores in transcrystalline interconnects under anisotropic surface diffusion, but also present the dynamic processes of segmentation, merging, and re-segmentation of micropores under different external fields, and obtain the critical anisotropy factor under different electric fields. and critical initial deflection angle This provides a direct quantification tool for the reliability of interconnects in integrated circuits.
[0023] Figure 1 This is a flowchart of the method of the present invention, by Figure 1 As can be seen, the simulation method for the evolution of micropores in interconnecting wires induced by electric field based on the phase field method proposed in this invention includes the following steps: Step 1: Construct a phase-field model of micropores in transcrystalline interconnects; the phase-field model of micropores in transcrystalline interconnects induced by electric field includes a matrix phase, a grain boundary phase, and a micropore phase.
[0024] In this embodiment, in the phase-field model, through order parameters The entire area is divided into inner conductor zones. and damaged area In the interface region between the hole and the wire, the order parameter It changes continuously from 0 to 1. This is achieved by introducing two additional distinct order parameters. and These represent different grains. Indicates the first The order parameter takes a value of 1 within the crystal and a value of 0 in other crystals. At grain boundaries, the order parameter... It changes continuously from 0 to 1 and satisfies the following conditions throughout the region. .
[0025] Step 2: Construct the conservative and non-conservative field phase variables related to time and space. , Spatial coordinates are The time coordinate is t .
[0026] Step 3: Construct an electric field free energy function that varies continuously with spatial coordinates. The potential smoothly transitions from the matrix phase to the porous phase through a diffusion interface layer.
[0027] In this embodiment, the phase field variable expression is:
[0028] The first term is the fourth-order double-well potential function of a general conservative field, satisfying that the conservative field order parameter is 1 inside the connecting wire. The second term indicates that when... When inside the connecting wire, in different There exists a minimum value at a certain point, meaning that for different grain boundary directions, at... , , The first term indicates that the minimum value is obtained at the point where the damage defect is located; the third term indicates that it needs to be satisfied within the damage defect, i.e. At that time, there is a unique minimum value, that is, at It reaches a minimum value at that point;
[0029] electric field free energy function ;in For the amount of electric charge, For the effective dielectric constant, For electric potential, This represents the atomic volume.
[0030] Step 4, the potential containing phase field variables Substituting into the Laplace equation, we obtain the potential distribution throughout the matrix. The conductivity satisfies... in ;in Represents the Laplace operator. Represents the potential gradient and resistivity , and These are the electrical conductivity of the inner conductor substrate and the hole, respectively.
[0031] Step 5, introduce the total free energy function of the system. F This includes local free energy, gradient energy between conservative and non-conservative fields, and electric field energy. The total free energy function of the system is...
[0032] in The above equation represents the surface energy per unit area, and the second and third terms are the conservative field order parameters, respectively. Nonconservative field order parameters The gradient energy term; The gradient energy coefficient is related to the thickness of the conservative field interface layer. It represents a micro-volume.
[0033] Step 6: Introduce the surface diffusion function and grain boundary diffusion mobility in the {110} crystal direction. The surface mobility is
[0034] in The diffusion function of an anisotropic surface is expressed as follows: Diffusion on anisotropic surfaces is determined by three unquantized parameters, namely... , , ;in The number of characteristic crystal symmetry directions characterizes the direction of rapid atomic diffusion. In face-centered cubic crystals, the symmetry axis corresponding to the {110} direction has only one symmetry axis, therefore... , For the diffusion intensity of anisotropic surfaces, The angle between the tangent to the surface of the hole and the applied electric field in the clockwise direction; Let be the initial deflection angle between the fastest diffusion direction and the electric field direction, where , conservative field variables Partial derivatives with respect to coordinates.
[0035] Step 7: Introduce the mass flux of surface atoms, and obtain the chemical potential by taking the variational function of the free energy of the whole system, and then obtain the mass flux of surface atoms. .
[0036] Mass flux of surface atoms With chemical potential The gradients are linearly related, that is... in , , is the surface diffusion coefficient, For the surface layer thickness, Boltzmann's constant, Absolute temperature Let π be the mathematical constant. For the conservative field governing equations, the chemical potential... free energy function For conservative field order parameters The variation, i.e. ; For the governing equations of a nonconservative field, the chemical potential For free energy function pairs The variation, i.e. .
[0037] Step 8 yields the modified Cahn-Hilliard and Allen-Cahn equations, which serve as the phase-field control equations; the Cahn-Hilliard diffusion equation, derived from the law of conservation of mass, is... ; According to the laws of dynamics, the Allen-Cahn equation is expressed as follows: .
[0038] For ease of description, this invention selects... Characteristic size, spatial dimensionlessization according to , The dimensionless Hamiltonian operator is The dimensionless potential is Dimensionless conductivity Dimensionless time of conservative field order parameter Dimensionless time of nonconservative field order parameters , , Substituting the dimensionless expression above into the governing equation, we get... ; ; ; in It represents the relative magnitude of electric potential energy and surface energy. The length of the model, The potential at infinity is the magnitude of the electric potential.
[0039] As described in step 9, given the basic parameters of the conservative field, non-conservative field, and electrostatic field control equations, boundary conditions are constructed for the phase field and electrostatic field control equations. The simulation of micropores in transcrystalline interconnects is completed by the mutual coupling of the conservative field, non-conservative phase field, and electrostatic field.
[0040] To realize the discretization process of finite element method, the weighted residual method is used to construct weak solution forms of equivalent integrals for the CH equation and AC equation respectively; firstly, the equivalent integral form of the dimensionless governing equation CH is constructed. ; In the formula, This is the weight function corresponding to the CH equation. Next, we construct the equivalent integral form of the dimensionless governing equations, the AC equation; ; In the formula Let be the weight function corresponding to the AC equation; The modified CH equation, by introducing auxiliary variables The fourth-order phase-field governing equations are decomposed into two second-order differential equations, and then the order is reduced using the divergence theorem. ; ; The modified second-order Allen-Cahn equation can be solved directly, i.e. , in , , Let represent the residuals of chemical potential, conservative field order parameter, and non-conservative order parameter, respectively. This represents the volume integral of the inner product. It represents the surface integral of the inner product.
[0041] Figure 2 The flowchart shows the solution process of the adaptive finite element method for the phase field model. Figure 3 This is a schematic diagram of the phase field model of the connecting wires in a bicrystalline material under an electric field, where... a and b These are the major and minor axes of the ellipse, respectively.
[0042] Figure 4 This figure shows the variation of the thermal erosion trench depth with respect to the 1 / 4 power of time under the surface diffusion mechanism without external field action in this embodiment of the invention. As can be seen from the figure, the numerical simulation results agree very well with the theoretical solution, therefore the reliability of the program is considered to meet the requirements. Furthermore, the calculated equilibrium dihedral angle is 139°, and the simulation results are basically consistent with the theoretical value, thus verifying the reliability of this method.
[0043] Figures 5-7 for , , The evolution of circular micropores was investigated. Comparing the evolution patterns of circular micropores under different initial deflection angles, electric field intensities, and anisotropy factors, three main categories were identified: 1) stable drift along the electric field direction with a quasi-triangular or wedge-shaped morphology; 2) stable oscillating drift along the electric field direction; and 3) different segmentation phenomena occurring with increasing initial deflection angle, electric field intensity, and anisotropy factor. Furthermore, when the electric field intensity, initial deflection angle, or anisotropy factor is large, the circular micropores will segment due to the drastic evolution of the micropore surface, with the number of segments increasing accordingly.
[0044] Figures 8-9 The diagram shows the number of sections under different angles, different electric field intensities, and different anisotropy intensity factors.
[0045] As shown in the figure, when the electric field strength or the initial deflection angle is large, the circular micropores will segment due to drastic morphological evolution. The number of segments increases with the increase of the electric field strength and the initial deflection angle, and when the initial angle is large... The segmentation phenomenon of circular micropores becomes more pronounced with increasing initial deflection angle and anisotropy intensity factor. The maximum number of pores tends to stabilize as the initial deflection angle and anisotropy intensity factor increase. As the initial deflection angle and anisotropy intensity factor increase, circular micropores will segment due to the drastic evolution of the micropore surface, and the number of segmentations increases with increasing anisotropy intensity factor and initial deflection angle.
[0046] Figure 10 The critical initial deflection angle As the electric field strength changes The value decreases as the electric field strength increases, indicating that increasing the electric field strength promotes segmentation. When the electric field strength is relatively large, The curve becomes flatter, indicating that when the electric field strength is large, the initial deflection angle has a limited impact on segmentation. When the electric field strength is constant, the critical initial deflection angle... With anisotropy intensity factor The rapid decrease following the increase indicates that the increase... This will facilitate segmentation; Figure 11 Critical anisotropy intensity factor With initial angle The changing pattern is shown in the figure. As can be seen from the initial angle... When smaller, Approaching infinity, this indicates The smaller the value, the more stable the circular micropores. However, when the initial angle... When the angle exceeds 30°, the circular micropores become very unstable and small. This can induce segmentation.
[0047] An apparatus includes a memory, a processor, and computer software (Moose) stored in the memory and executable on the processor, wherein the processor, when executing the computer program, implements the steps of the present invention's simulation method for electric field-induced micropore evolution in interconnecting wires based on the phase-field method.
[0048] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Although the present invention has been shown above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make some modifications or alterations to the above-disclosed technical content to create equivalent embodiments without departing from the scope of the present invention. Any simple modifications, equivalent changes, and alterations made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the scope of the present invention.
Claims
1. A simulation method for the evolution of micropores in interconnecting wires induced by electric field based on the phase-field method, characterized in that, Includes the following steps: Step 1: Construct a phase-field model of micropores in transcrystalline interconnects; the phase-field model of micropores in electric field-induced transcrystalline interconnects includes a matrix phase, a grain boundary phase, and a micropore phase. Step 2: Construct the conservative and non-conservative field phase variables related to time and space. , Spatial coordinates are The time coordinate is t ; Step 3: Construct an electric field free energy function that varies continuously with spatial coordinates. The potential smoothly transitions from the matrix phase to the porous phase through a diffusion interface layer; Step 4, the potential containing phase field variables Substituting into the Laplace equation, we obtain the potential distribution of the entire matrix; Step 5, introduce the total free energy function of the system. F This includes local free energy, conservative field and non-conservative field gradient energy, and electric field energy; Step 6: Introduce the surface diffusion function and grain boundary diffusion mobility in the {110} crystal direction. ; Step 7: Introduce the mass flux of surface atoms, and obtain the chemical potential by taking the variational function of the free energy of the whole system, and then obtain the mass flux of surface atoms. ; Step 8 yields the modified Cahn-Hilliard equation and Allen-Cahn equation, which serve as the phase field control equations. Step 9: Given the basic parameters of the conservative field, non-conservative field, and electrostatic field control equations, construct boundary conditions for the phase field and electrostatic field control equations, and complete the simulation of micropores in transcrystalline interconnects by coupling the conservative field, non-conservative phase field, and electrostatic field.
2. The simulation method for the evolution of micropores in interconnecting wires induced by electric field based on the phase-field method according to claim 1, wherein in step 1, the phase-field model of micropores in transcrystalline interconnecting wires is obtained through order parameters. The entire region is divided into an inner conductor region and a damage region. At the interface between the hole and the conductor, the order parameter... It changes continuously from 0 to 1; by introducing two additional distinct order parameters. and These represent different grains; among them, Indicates the first The order parameter takes a value of 1 within the crystal and a value of 0 in other crystals; at grain boundaries, the order parameter... It changes continuously from 0 to 1 and satisfies the following conditions throughout the region. .
3. The simulation method for the evolution of micropores in interconnecting wires induced by electric field based on the phase field method according to claim 2, wherein the expression for the phase field variables is as follows: ; The first term is the fourth-order double-well potential function of a general conservative field, satisfying that the conservative field order parameter is 1 inside the connecting wire. The second term indicates that when... When inside the connecting wire, in different There exists a minimum value at a certain point, meaning that for different grain boundary directions, at... , , The first term indicates that the minimum value is obtained at the point where the damage defect is located; the third term indicates that it needs to be satisfied within the damage defect, i.e. At that time, there is a unique minimum value, that is, at It reaches a minimum value at that point; 。 4. The simulation method for the evolution of micropores in interconnecting wires induced by electric field based on the phase field method according to claim 3, wherein step 3, the electric field free energy function... ;in For the amount of electric charge, For the effective dielectric constant, For electric potential, This is the atomic volume.
5. The simulation method for the evolution of micropores in interconnecting wires induced by electric field based on the phase field method according to claim 4, wherein in step 4, the conductivity satisfies ,in Represents the Laplace operator. Represents the potential gradient and resistivity , and These are the electrical conductivity of the inner conductor substrate and the hole, respectively.
6. The simulation method for the evolution of micropores in interconnecting wires induced by electric field based on the phase field method according to claim 5, wherein in step 5, the total free energy function of the system is: ,in The above equation represents the surface energy per unit area, and the second and third terms are the conservative field order parameters, respectively. Nonconservative field order parameters The gradient energy term, The gradient energy coefficient is related to the thickness of the conservative field interface layer. It represents a micro-volume.
7. The simulation method for the evolution of micropores in interconnecting wires induced by electric field based on the phase field method according to claim 6, wherein in step 6, the surface mobility is, , , These represent the grain boundary diffusion coefficient and the surface diffusion coefficient, respectively. in The diffusion function of an anisotropic surface is expressed as follows: Diffusion on anisotropic surfaces is determined by three unquantized parameters, namely... , , ;in The characteristic crystal symmetry direction number is used to characterize the direction of rapid atomic diffusion; in a face-centered cubic crystal, the symmetry axis corresponds to the {110} direction, and there is only one symmetry axis. , For the diffusion intensity of anisotropic surfaces, The angle between the tangent to the surface of the hole and the applied electric field in the clockwise direction; Let be the initial deflection angle between the fastest diffusion direction and the electric field direction, where , conservative field variables Partial derivatives with respect to coordinates.
8. The simulation method for the evolution of micropores in interconnecting wires induced by electric field based on the phase field method according to claim 7, wherein step 7 refers to the mass flux of surface atoms. With chemical potential The gradients are linearly related, that is... in , , is the surface diffusion coefficient, For the surface layer thickness, Boltzmann's constant, Absolute temperature Pi; For the conservative field control equations, the chemical potential free energy function For conservative field order parameters variation , Represents the local free energy function with respect to the conservative field order parameter The differential.
9. The simulation method for the evolution of micropores in interconnecting wires induced by electric field based on the phase-field method according to claim 8, wherein in step 8, the Cahn-Hilliard diffusion equation can be obtained from the law of conservation of mass as follows: ； According to the laws of dynamics, the Allen-Cahn equation is expressed as follows: , The relaxation coefficient is a non-conservative field order parameter, representing the rate of interface movement.
10. A device, characterized in that, The system includes a memory, a processor, and computer software stored in the memory and capable of running on the processor. When the processor executes the computer program, it implements the steps of the simulation method for the electric field-induced evolution of micropores in interconnecting wires based on the phase field method as described in any one of claims 1 to 9.