JP6044926B2 - Reliability evaluation simulation program for evaluating reliability of via connection multilayer wiring, method for improving allowable current density of via connection multilayer wiring, and via connection multilayer wiring - Google Patents

Description

  The present invention relates to a reliability evaluation simulation program and the like for evaluating the reliability of a via-connected multilayer wiring in an electromigration damage process for a via-connected multilayer wiring having a reservoir structure.

  In recent years, the miniaturization of metal wiring has been advanced due to the high integration of electronic devices. On the other hand, in miniaturized wiring, damage due to electromigration (EM) is a problem due to high density current and accompanying increase in Joule heat. EM is a diffusion phenomenon of metal atoms due to high-density electron flow.

  FIG. 9 shows a multilayer wiring 50 having a reservoir structure. As shown in FIG. 9, in the multilayer wiring 50 having the via connection 56, by providing projecting portions (reservoirs) 55n and 55p outward from the wiring end portions 54n (cathode side −) and 54p (anode side +), It is known to delay disconnection due to EM damage by the reservoir effect (see Non-Patent Document 1). This effect is because the metal atoms supplied from the overhang portions (reservoirs) 55n and 55p serve as a reservoir of atoms to the metal wirings 51, 52 and 53.

On the other hand, it is known that a threshold current density j _th of EM damage exists in via connection. Conventionally, research on an evaluation method of the threshold current density j _th has been performed (see Non-Patent Document 2). The threshold current density j _th is also evaluated by numerical simulation, and the generation process of the atomic concentration distribution in the wiring is simulated. This simulation is based on the dominant parameter AFD ^* _gen of the EM damage in the polycrystalline structure wiring (see Non-Patent Document 3). This parameter can be applied to a two-dimensional wiring shape, and various evaluations have been performed (see Non-Patent Document 4). Attempts have also been made to develop an EM damage evaluation simulation considering the reservoir effect (see Non-Patent Documents 5 and 6).

  In the simulations of Non-Patent Documents 3 to 6 described above, the simulation of the EM damage process, in which voids are generated, grow, and break, has not been achieved, and a simulation technique that can evaluate the reliability of wiring while taking into account the reservoir effect is still available There was a problem that it was not developed. Therefore, an object of the present invention is to solve the above-mentioned problems. For via connection multilayer wiring having a reservoir structure, simulation of the EM damage process leading to generation of voids is performed, and the reservoir effect is taken into consideration. Another object of the present invention is to provide a reliability evaluation simulation program for evaluating the reliability of multilayer wiring by evaluating the threshold current density.

The reliability evaluation simulation program for evaluating the reliability of a via-connected multilayer wiring according to the present invention is a reliability evaluation simulation program for evaluating the reliability of a multilayered wiring in an electromigration damage process for a via-connected multilayer wiring having a reservoir structure. An element dividing step for dividing the multilayer wiring into two-dimensional elements, an initial setting step for setting an initial atomic concentration of each element divided in the element dividing step to N _0, and a two-dimensional finite element method The two-dimensional finite element method step for calculating the current density distribution and the temperature distribution in the multilayer wiring, and the current density distribution and the temperature distribution in the multilayer wiring calculated by the two-dimensional finite element method step are recorded in the recording unit. In addition, based on the physical property constants of the multilayer wiring material, Click Toro migration by how many atoms less dominant indicating whether the disappeared parameter (in the element and via the cathode and anode ends ^AFD _{* gen | end,} the other elements ^AFD _{* gen),}

Here, N: atomic concentration, D ₀ : frequency term, k: Boltzmann constant, T: absolute temperature, Q _gb : activation energy of atomic diffusion, κ: between concentration change and stress change under constraint of protective film Ω: atomic volume, σT: tensile thermal stress, N _T : atomic concentration when σT is applied, N ₀ : atomic concentration in an unstressed state, Z ^* : number of effective charges, e: unit charge, ρ : Electric resistivity, j ^* : J-direction component of current density vector, ∂N / ∂l: J-direction component of atomic concentration gradient, δ: Effective width of grain boundary, θ: Minute unit structure and x-axis Angle, d: average crystal grain size, Δφ: constant related to angle between crystal grain boundaries, Q _gb : activation energy of grain boundary diffusion, D _x = Z ^* eρj _x −κΩ / N ₀ (∂N / ∂ x), D _y = Z ^* eρj _y −κΩN ₀ (∂N / ∂y),

Here, β: angle formed by the wiring end with the x-axis, d: average crystal grain size, D _x = Z ^* eρj _x −κΩ / N ₀ (∂N / ∂x), D _y = Z ^* eρj _y −κΩN Control parameter calculation step for calculating ₀ (∂N / ∂y), based on the control parameter value calculated in the control parameter calculation step, atomic concentration N ^{* with} respect to θ, where N ^* : is the _AFD ^* gbθ the θ included in the _AFD ^* gbθ value calculated as respective values of 2π from 0, the atomic concentration N ^* calculating step of calculating the atomic concentration, of each value of θ determined from its value, the atomic concentration N ^* all average atomic concentration N atom concentration N calculation step of calculating for in the atomic concentration has each value of the computed at the computation step theta N ^* element, δN / δx, δN / δy and the like, Concentration distribution calculation step, atomic concentration A determination step for determining whether or not a steady state is reached at which no change occurs, and if it is determined in the determination step that the steady state has not been reached, a setting for repeated calculation is performed, and the control proceeds to the control parameter calculation step. It is a reliability evaluation simulation program for evaluating the reliability of a multilayer wiring for executing a repeating step of returning and repeating the calculation.

Allowable current density improved method of via connection of a multilayer wiring of the present invention, the multi-layer wiring via connections having a reservoir structure, reliability evaluation simulation program obtained from the execution result, atoms in the wiring in the steady state of the present invention Based on the evaluation of the input current density (threshold current density) when the minimum concentration value matches the critical atomic concentration value leading to void formation, a reservoir is provided only on the via side of the cathode end, and the minimum atom inside the multilayer wiring The allowable current density of the multilayer wiring is increased by increasing the concentration.

The via-connected multilayer wiring structure of the present invention is the minimum value of the atomic concentration in the wiring in the steady state obtained from the execution result of the reliability evaluation simulation program of the present invention in the via-connected multilayer wiring structure having the reservoir structure. Based on the evaluation of the input current density (threshold current density) when the value matches the critical atomic concentration value that leads to void formation, a reservoir is provided only on the via side at the cathode end to increase the minimum atomic concentration inside the multilayer wiring Thus, the allowable current density of the multilayer wiring is increased.

In the present invention, the reservoir effect in the integrated circuit wiring is evaluated by performing this numerical simulation using the EM damage control parameter. In this numerical simulation, first, a current density distribution and a temperature distribution are calculated by two-dimensional FE analysis (analysis by a finite element method. Generally, numerical analysis may also be used). The dominant parameters (AFD ^* _{gen | end} , AFD ^* _gen ) in each element are calculated from the analysis results (current density distribution and temperature distribution) and thin film characteristics (physical property constants of the wiring material) recorded on a disk or the like. Next, the atomic concentration N ^* related to θ is calculated based on the value of the dominant parameter. The atomic concentration N in each element is calculated by the average of N ^* for all values of θ. Calculating the concentration distribution of δN / δx, δN / δy, etc., determining whether the critical atom concentration or the steady state where the atomic concentration does not change has been reached, and if it has been reached, the process ends; otherwise If it is determined, the above calculation of the control parameter is repeated.

  From this numerical simulation, it was found that when a reservoir is provided on the cathode end side, the minimum atomic concentration inside the wiring increases and the allowable current density of the wiring increases. In addition, when a current exceeding the allowable value is applied to the wiring having a reservoir, the atomic concentration reaches the critical value of void generation at the cathode side via portion, and therefore, the void generation point can be evaluated as the cathode side via portion. It was. That is, it can be said that the result of this numerical simulation is appropriate because it is found that the void is generated in the via portion on the cathode side when the current exceeding the allowable value is applied, which is consistent with the experimental fact. In other words, it has been found that if the reservoir is provided only on the cathode end side, which has not been performed so far, the allowable current density of the wiring increases and it is difficult to damage. As described above, for the via connection multi-layer wiring having the reservoir structure, the numerical simulation of the EM damage process leading to the void generation is performed, and the threshold (allowable) current density is evaluated while considering the reservoir effect. It is possible to provide a simulation method for evaluating the performance.

It is a figure which shows the model of a polycrystalline structure. It is a flowchart which shows the flow of this numerical simulation or method. It is a figure which shows the change with time of the atomic concentration distribution of wiring by repetition calculation with a graph. It is a figure which shows four types of wiring structures evaluated by this numerical simulation. It is a figure which shows the relationship between the minimum value of atomic concentration N ^* in all the elements in a steady state, and the assumed current density j with a graph. Samples 1, 2 and 3 of the distribution of atomic concentration N / N ₀ along the line illustrates graphically. It is a figure which shows the atomic concentration distribution at the time of completion | finish of this numerical simulation with a graph It is a block diagram which shows the internal circuit 30 of the computer which executes the simulation program of this invention. It is a figure which shows the multilayer wiring 50 which has a reservoir structure.

  In the present invention, as a first step for developing a reliability evaluation method for a wiring having a reservoir, an evaluation of an allowable current density leading to void generation and an evaluation of a void generation location when a current exceeding the allowable value is applied. A numerical simulation program (or a numerical simulation method of the present invention for EM damage) for carrying out the following is developed. Hereinafter, each embodiment will be described in detail with reference to the drawings.

In order to construct this numerical simulation program, the dominant parameter AFD ^* _gen of EM damage is used (see Non-Patent Document 3). This parameter is given by the formulation of atomic flux due to EM damage. The EM atomic flux J considering the backflow caused by the atomic concentration gradient (stress gradient) and the effect of the stress generated in the metal wiring on the diffusivity is given by Equation 1 (see Non-Patent Document 7).

Here, N: atomic concentration, D ₀ : frequency term, k: Boltzmann constant, T: absolute temperature, Q _gb : activation energy of atomic diffusion, κ: between concentration change and stress change under constraint of protective film Ω: atomic volume, σT: tensile thermal stress, N _T : atomic concentration when σT is applied, N ₀ : atomic concentration in an unstressed state, Z ^* : number of effective charges, e: unit charge, ρ : Electric resistivity, j ^* : J direction component of current density vector, ∂N / ∂l: J direction component of atomic concentration gradient.

  In this numerical simulation, since a wide Cu wiring covered with a protective layer was assumed, grain boundary diffusion was assumed as a main diffusion mechanism. According to Non-Patent Documents 8 and 9, the crystal grain boundary becomes an EM path in wide Cu wiring (interconnetcs). The inventors have introduced a model for calculating atomic flux divergence (see Non-Patent Document 10). Therefore, the inventors used the dominant parameter of EM damage based on the model for Cu wiring.

  Considering atoms entering and exiting a minute unit structure (described later), the atomic flux divergence of the polycrystalline structure wiring is formulated as given by Equation 2.

Here, C ^* _gb represents D ₀ δ / k, and the effective width of the crystal grain boundary is represented by δ. d is the average crystal size, Δφ is a constant related to the relative angle between the grain boundaries, j _x and j _y are the x and y components of the current density vector j in the Cartesian coordinate system, and θ is the micro unit structure and the x axis. Is the angle between. FIG. 1A shows a model of a polycrystalline structure, and FIG. 1B shows a partially enlarged view (micro unit structure) of FIG. In FIG. 1 (A), a partially enlarged view of a metal line (Metal line) 10 is shown. In the enlarged view, the metal crystal grains 12 are represented by a rectangle of size d (hexagonal grain). In the fine unit structure shown in FIG. 1B, grain boundaries 1 (Arabic numerals), 2 (Arabic numerals) and 3 (Arabic numerals), an angle θ between the minute unit structure and the x-axis, A constant Δφ and the like are shown.

The expected value of only the positive value of AFD ^* _{gen in} Equation 2 represents the parameter AFD ^* _gen that governs EM damage, and Equation 3 is obtained for void formation in the polycrystalline structure wiring.

  Equation 3 means the number of atoms decreasing per unit time and unit volume.

At the end of the wiring, it is necessary to give a boundary condition regarding the atomic flow to the formation of the parameter. That is, there is no inflow of atoms at the cathode end of the wiring and no outflow of atoms at the anode end. The boundary condition can be expressed by assigning a possible zero flux for each θ range in the micro unit structure shown in FIG. Table 1 shows the boundary conditions regarding the atomic flux (see Non-Patent Document 11). In Table 1, _{1 (Arabic numerals)} , _{2 (Arabic numerals)} and _{3 (Arabic numerals)} such as J1 _{(Arabic numerals)} , J2 (Arabic numerals) and J3 (Arabic numerals) are the grain boundaries in FIG. (Grain Boundary) 1 (Arabic numeral), 2 (Arabic numeral), 2 (Arabic numeral), 2 (Arabic numeral) and 3 (Arabic numeral).

From the above, considering the entry and exit of atoms in the minute unit structure shown in FIG. 1B, the atomic flux divergence AFD ^* _{gen | end} at the end of the via connection wiring is given by Equation 4.

Where δ is the effective width of the crystal grain boundary, β is the angle between the wiring end and the x-axis (see FIG. 1B), d is the average crystal grain size, Q _{gb is} the activation energy of the grain boundary diffusion, D _x = Z ^* eρj _x− κΩ / N ₀ (∂N / ∂x), and D _y = Z ^* eρj _y −κΩN ₀ (∂N / ∂y). The atomic flux gradient AFD ^* _{gen | end} represents the amount of flux divergence at the grain boundary, and means the number of atoms decreasing per unit time and unit volume.

Using this EM damage dominant parameter, this numerical simulation of the atomic concentration distribution in the wiring was performed under conditions of several types of input current density j and substrate temperature T _s . The wiring to be evaluated was divided into two-dimensional elements, and the generation process of the atomic concentration distribution was simulated while changing the atomic concentration of each element based on the above governing parameters. The boundary condition related to temperature was given to both ends of the wiring, and the boundary condition related to current density was given to the via position. The atomic flow was interrupted around the metal wiring. The end parameter AFD ^* _{gen | end} was used for the cathode and anode end elements and vias, and AFD ^* _gen was used for the other elements.

FIG. 2 shows a flow chart of the present numerical simulation or method. As shown in FIG. 2, the initial atomic concentration of the element is N ₀ . First, the current density distribution and the temperature distribution are calculated by two-dimensional FE analysis (finite element method. Generally, numerical analysis may be used) (step S10). Based on the analysis results (current density distribution and temperature distribution) and thin film characteristics recorded on the disk 38 (described later) and the like (physical property constants of wiring materials; see Non-Patent Document 4), the above dominant parameters (AFD ^* _{gen | end} , AFD ^* _gen ) are calculated (steps S12 and S14). As the time elapses, the atomic concentration distribution in the wiring changes, and these dominant parameter values also change. Next, the atomic concentration N ^* related to θ is calculated based on the value of the dominant parameter. The atomic concentration N in each element is calculated by the average of N ^* for all values of θ. Calculation of concentration distributions such as δN / δx and δN / δy is also performed (step S16). It is determined whether or not a critical atomic concentration or a steady state where the atomic concentration does not change is reached (step S18). If it is determined that it has been reached, the process ends. If not, the process returns to step S14 to perform calculation. Repeat. When iterating, the calculation in step S14 is performed using N, δN / δx, and δN / δy as the result of step S16.

In this numerical simulation, two types of atomic concentrations (N ^* , N) are used for each element. One is to calculate the AFD ^* _gbθ value with θ (see FIG. 1) included in the AFD ^* _gbθ represented by the expression 2 as values of 0 to 2π, and the atomic concentration (N ^* ) obtained from the value. is there. Therefore, N ^* is calculated for each value of θ. The other is the atomic concentration N obtained by averaging the N ^* values for each value of θ for all of the elements. This N is used in Equation 2 as the atomic concentration of the element. From the initial value N ₀ of the atomic concentration, the number of atoms in the element before EM damage is known. AFD ^* _gen or AFD ^* _{gen | end} is a parameter for determining how many atoms are lost by EM per unit volume per unit time. From this value, how many atoms are lost from one element during one calculation step. And the atomic concentration N of the element after the elapse of time can be obtained. When the atomic concentration N of the element (and its gradient δN / δx, δN / δy, etc.) changes, the value of AFD ^* _gen or AFD ^* _{gen | end} that is a function thereof also changes. Iterative calculation is performed with the time step progressed. The iterative calculation is performed assuming a certain input current density.

FIG. 3 is a graph showing the change with time of the atomic concentration distribution of the wiring by repeated calculation. In FIG. 3, the horizontal axis represents the distance (μm from the center of the wiring (see FIG. 4), negative is the cathode side and positive is the anode side), and the vertical axis is the atomic concentration (N / N ₀ ). In the original drawing, the number of repetitions (step; the passage of time) is shown in different colors (blue: 5000 steps, red: 10,000 steps, green: 20,000 steps, purple: 30,000 steps). As shown in FIG. 3, as the number of repetitions increases (with the passage of time), the atomic concentration at the same position changes, and after a sufficient amount of time has passed, a steady state may occur where the atomic concentration does not change. Recognize. As shown in FIG. 3, as time elapses, N ^{* and} N for each element change, and then the concentration distribution becomes a steady state.

  FIG. 4 shows four types of wiring structures evaluated by this numerical simulation. In FIG. 4, the parts denoted by the same reference numerals as those in FIG. 4A shows sample 1 without a reservoir at both ends, FIG. 4B shows sample 2 in which the reservoir is arranged in the cathode via, and FIG. 4C shows sample 3 in which the reservoir is arranged in the anode via. FIG. 4D shows Sample 4 in which reservoirs are arranged in both the cathode and anode vias. Assuming a linear Cu wiring as shown in FIGS. 4A to 4D, the influence of the presence or absence of the reservoir on the magnitude of the threshold current density was evaluated. As shown in FIGS. 4A to 4D, when there is no reservoir (sample 1), a current of density j is input from the anode end, and a current of j is output from the cathode end (see arrow). The distance between vias assumed at both ends is 150 (μm), the width W of the wiring 52 is 10 (μm), and the thickness t is 410 (nm). Similarly, when there was a reservoir (samples 2 and 3), a current of density j was input / output between vias separated by 150 (μm). The lengths of the overhang portions 55n and 55P are both 25 (μm). This numerical simulation was performed for each sample assuming three types of current densities under a substrate temperature of 573 (K).

  In this numerical simulation, it was assumed that the Cu wiring had the characteristic constants shown in Table 2 (see Non-Patent Documents 12 to 16).

As the input current density, three values of 0.2, 0.4, and 0.6 (MA / cm ² ) were assumed. The environmental temperature was assumed to be 573 (K) in all samples 1-4.

When the assumed current density is small, the atomic concentration distribution in the wiring does not change without the minimum value of the atomic concentration N ^* in the wiring reaching the critical atomic concentration N ^* _min that leads to void formation. A current density at which the minimum value of the atomic concentration in the wiring is exactly equal to N ^* _min is defined as a threshold current density, and the allowable current density of the wiring is evaluated. After executing this numerical simulation at a current density smaller than the threshold current density, a steady state of atomic concentration distribution was obtained. FIG. 5A is a graph showing the relationship between the minimum value of the atomic concentration N ^* and the assumed current density j in all elements in the steady state. In FIG. 5A, the horizontal axis represents current density (MA / cm ² ), and the vertical axis represents minimum atomic concentration (N ^* / N ₀ ), which is plotted for each sample 1 to 4. Specifically, the minimum value in the wiring of the atomic concentration N ^{* in} the steady state was plotted against the assumed input current density. As shown in FIG. 5A, the threshold current density is evaluated from the intersection of the line connecting the plots of the samples 1 to 4 and the line of the critical atomic concentration (N ^* _min / N ₀ ). FIG. 5B shows the threshold current density j _th (MA) for each of the samples 1 to 4 evaluated in this way. When the atomic concentration N ^* in the wiring reaches a value of N ^* _min , voids are formed and the model is damaged. Therefore, when the steady state is reached, the minimum value of the atomic concentration N ^* in the wiring is just N ^* _min . The input current density that becomes a value is obtained from the intersection of the graph shown in FIG. 5A, and is evaluated as the threshold current density j _th .

As shown in FIGS. 5A and 5B, the threshold current densities j _th of the samples 1 and 4 are almost the same. In contrast, samples 2, threshold current density _{j th} is the threshold current density greater than _{j th} samples 1 and 4. Sample 3 The threshold current density _{j th} is the threshold current density _{j th} smaller samples 1 and 4. In the case of the sample 2 in which the reservoir is disposed on the cathode via side, the threshold current density j _th is larger than that of the other samples. This is thought to be because the atomic concentration distribution of the via-via energization portion in the steady state does not change between samples 1 and 3, but the distribution of sample 2 is higher than those and the minimum value of the atomic concentration in the wiring also increases. . In other words, since the sample 2 can flow more current than other shape samples, it can be said that the sample 2 has a shape that is not easily damaged by EM.

FIG. 6 is a graph showing the distribution of atomic concentration N / N ₀ along the wirings of Samples 1, 2, and 3. In FIG. 6, the horizontal axis is the distance (μm from the center of the wiring (see FIG. 4) (negative is the cathode side and positive is the anode side), the vertical axis is the atomic concentration (N / N ₀ ), and Sample 1 is a solid line. Sample 2 is indicated by a broken line, and sample 3 is indicated by a broken line with a short pitch. All samples have the same input current density. According to Equation 1, when the current density is equal, the driving force of the EM is equal. For this reason, in the steady state, the gradients of atomic concentrations correspond to each other. On the other hand, in the reservoir, the current density is almost zero and there is no driving force of EM. Therefore, in the steady state, the inclination in the reservoir is almost horizontal. According to the law of mass, the atomic concentration distribution in sample 2 shifts upward from the distribution in sample 1 in a wide area.

FIG. 7 is a graph showing the atomic concentration distribution at the end of the numerical simulation. In FIG. 7, the horizontal axis is the distance (μm from the center of the wiring (see FIG. 4) (negative is the cathode side and positive is the anode side), and the vertical axis is the atomic concentration N ^* (1 / μm ³ ) 2.0 (MA / cm ² ) and shown for sample 3. 7 for the distribution of N ^*, since plots they have multiple N ^* to each element, it is seen in thick line. As shown in FIG. 7, it was considered that the concentration reached N ^* _min in the via portion on the cathode side and voids were generated. This result coincides with the void generation location in the experiment (see Non-Patent Document 17).

As mentioned above, according to Example 1 of this invention, the reservoir effect in integrated circuit wiring was evaluated by implementing this numerical simulation using EM damage control parameters. In this numerical simulation, first, the current density distribution and the temperature distribution are calculated by two-dimensional FE analysis (finite element method. Generally, numerical analysis may be used). Based on the analysis results (current density distribution and temperature distribution) and thin film characteristics (physical property constants of the wiring material, see Non-Patent Document 4) recorded on the disk 38 (described later), the dominant parameter (AFD ^* _{gen | end} , AFD ^* _gen ). Next, the atomic concentration N ^* related to θ is calculated based on the value of the dominant parameter. The atomic concentration N in each element is calculated by the average of N ^* for all values of θ. It is determined whether or not a critical atomic concentration or a steady state where the atomic concentration does not change is reached. If it is determined that it has been reached, the process ends. If not, the above control parameter calculation is repeated. From this numerical simulation, it was found that when a reservoir is provided on the cathode end side, the minimum atomic concentration inside the wiring increases and the allowable current density of the wiring increases. In addition, when a current exceeding the allowable value is applied to the wiring having a reservoir, the atomic concentration reaches the critical value of void generation at the cathode side via portion, and therefore, the void generation point can be evaluated as the cathode side via portion. It was. That is, it can be said that the result of this numerical simulation is appropriate because it is found that the void is generated in the via portion on the cathode side when the current exceeding the allowable value is applied, which is consistent with the experimental fact. In other words, it has been found that if the reservoir is provided only on the cathode end side, which has not been performed so far, the allowable current density of the wiring increases and it is difficult to damage. As described above, the reliability of the wiring is evaluated by conducting this numerical simulation of the EM damage process leading to the occurrence of voids in the via-connected multilayer wiring having a reservoir structure, and evaluating the threshold current density in consideration of the reservoir effect. We were able to provide a simulation method and so on. In a via-connected multilayer wiring structure having a reservoir structure, a reservoir is provided only on the cathode end via side, and the minimum atomic concentration inside the multilayer wiring is increased, thereby increasing the allowable current density of the multilayer wiring. A wiring structure could be provided.

  FIG. 8 is a block diagram showing an internal circuit 30 of a computer such as a PC that executes the simulation program of the present invention. As shown in FIG. 8, the CPU 31, ROM 32, RAM 33, image control unit 36, controller 37, input control unit 40 and external I / F unit 42 are connected to a bus 43. In FIG. 8, the above-described simulation program of the present invention is recorded on a recording medium (including a removable recording medium) such as a ROM 32, a disk 38, a DVD or a CD-ROM 39. Thin film characteristics such as physical constants shown in Table 2 are recorded on a recording medium (recording unit) such as the disk 38. The simulation program is loaded into the RAM 33 from the ROM 32 via the bus 43 or from a recording medium such as the disk 38 or DVD or CD-ROM 39 via the controller 37 via the bus 43. The image control unit 36 sends the data of the calculation result by the simulation program to the VRAM 35. The display unit 34 displays the above data sent from the VRAM 35. The VRAM 35 is an image memory having a capacity corresponding to the data capacity of one screen of the display unit 34. The input operation unit 41 is an input device such as a mouse, a keyboard, a touch panel, and a switch for performing input, designation, and the like to the computer. The input control unit 40 is connected to the input operation unit 41 and performs input control and the like. The external I / F unit 42 has an interface function when connecting to the outside of the computer (CPU) 31.

  As described above, the computer (CPU) 31 executes the simulation program of the present invention, whereby the object of the present invention can be achieved. The simulation program can be supplied to the computer (CPU) 31 in the form of a recording medium such as a DVD or a CD-ROM 39 as described above, and a recording medium such as a DVD or a CD-ROM 39 on which the simulation program is recorded is similarly applied to the present invention. Will be configured. As a recording medium on which the simulation program is recorded, for example, a memory card, a memory stick, an optical disk, or the like can be used in addition to the recording medium described above.

  As an application example of the present invention, the present invention can be applied to improvement (increase) of allowable current density in a multilayer wiring (in particular, integrated circuit wiring) having via connection.

  10 metal wiring, 12 metal crystal grains, 30 internal circuit of computer, 31 CPU, 32 ROM, 33 RAM, 34 display unit, 35 VRAM, 36 image control unit, 37 controller, 38 disk, 39 DVD or CD-ROM, 40 Input control unit, 41 Input operation unit, 42 External I / F unit, 43 Bus, 50 Multi-layer wiring, 51, 52, 53 Metal wiring, 54n, 54P Wiring end, 55n, 55p Overhang (reservoir), 56 Via connection .

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Claims (3)

A reliability evaluation simulation program for evaluating the reliability of a via-connected multilayer wiring in a process of electromigration damage for a via-connected multilayer wiring having a reservoir structure,
An element dividing step of dividing the multilayer wiring into two-dimensional elements;
An initial setting step in which the initial atomic concentration of each element divided in the element division step is N ₀ ;
A two-dimensional finite element method step of calculating a current density distribution and a temperature distribution in the multilayer wiring by a two-dimensional finite element method;
Based on the current density distribution and temperature distribution in the multilayer wiring calculated by the two-dimensional finite element method step and the physical constants of the multilayer wiring material recorded in the recording section, what is caused by electromigration per unit volume per unit time? or are shown below governing parameters pieces atoms disappears (at the element and via the cathode and anode ends ^AFD _{* gen | end,} the other elements ^AFD _{* gen),}
Here, N: atomic concentration, D ₀ : frequency term, k: Boltzmann constant, T: absolute temperature, Q _gb : activation energy of atomic diffusion, κ: between concentration change and stress change under constraint of protective film Ω: atomic volume, σT: tensile thermal stress, N _T : atomic concentration when σT is applied, N ₀ : atomic concentration in an unstressed state, Z ^* : number of effective charges, e: unit charge, ρ : Electric resistivity, j ^* : J-direction component of current density vector, ∂N / ∂l: J-direction component of atomic concentration gradient, δ: Effective width of grain boundary, θ: Minute unit structure and x-axis Angle, d: average crystal grain size, Δφ: constant related to angle between crystal grain boundaries, Q _gb : activation energy of grain boundary diffusion, D _x = Z ^* eρj _x −κΩ / N ₀ (∂N / ∂ x), D _y = Z ^* eρj _y −κΩN ₀ (∂N / ∂y),
Here, β: angle formed by the wiring end with the x-axis, d: average crystal grain size, D _x = Z ^* eρj _x −κΩ / N ₀ (∂N / ∂x), D _y = Z ^* eρj _y −κΩN ₀ (∂N / ∂y),
Dominant parameter calculation step, which calculates
Based on the value of the dominant parameter calculated in the dominant parameter calculation step, the atomic concentration N ^* related to θ, where N ^* : θ included in AFD ^* _gbθ represented by Equation 2 is a value from 0 to 2π An atomic concentration N ^* calculation step for calculating an AFD ^* _gbθ value as, and calculating an atomic concentration for each θ value obtained from that value,
Atomic concentration N calculating step of calculating the atomic concentration N ^* calculated atomic concentration N averaged for all in the calculated atomic concentration has for each value of theta N ^* element in step,
A determination step for determining whether or not a steady state has been reached at which the atomic concentration does not change;
If the determination step determines that the steady state has not been reached, a multi-layer wiring with via connection is used to perform a repetitive step by performing settings for repetitive calculation and returning to the dominant parameter calculation step to repeat the calculation. A reliability evaluation simulation program that evaluates the reliability of a computer. For a via-connected multilayer wiring having a reservoir structure, the minimum value of the atomic concentration in the wiring in a steady state obtained from the execution result of the reliability evaluation simulation program according to claim 1 is a critical atomic concentration that leads to void formation. Based on the evaluation of the input current density (threshold current density) when the values match, the reservoir is provided only on the via side of the cathode end, and the minimum atomic concentration inside the multilayer wiring is increased, thereby increasing the allowable current density of the multilayer wiring. A method of improving an allowable current density of a via-connected multi-layer wiring, characterized by increasing the number of vias. In a via-connected multi-layer wiring structure having a reservoir structure , the critical atomic concentration at which the minimum value of the atomic concentration in the wiring in the steady state is obtained from the execution result of the reliability evaluation simulation program according to claim 1 leads to void formation Based on the evaluation of the input current density (threshold current density) when the value matches the above value, the reservoir is provided only on the via side of the cathode end, and the minimum atomic concentration inside the multilayer wiring is increased, thereby allowing the allowable current density of the multilayer wiring. Via connection multilayer wiring structure characterized by increasing