JP4889040B2 - Metal wiring reliability evaluation apparatus and method, and recording medium storing program for metal wiring reliability evaluation - Google Patents

Description

  The present invention relates to a technique for predicting electromigration (EM) damage, which is a cause of failure of metal wiring, and evaluating reliability.

Technical background

  Along with the high integration of semiconductor devices, a decrease in reliability associated with the miniaturization of metal wiring connecting fine elements is an important issue. Electromigration (EM) can be cited as a main cause of failure of metal wiring. EM is a phenomenon in which metal atoms constituting the wiring move due to high-density electron flow. In a place where the atomic flux due to EM is not uniform, local loss or accumulation of atoms occurs, which is called atomic flux divergence. Voids (voids) and hillocks (metal atomic blocks) are generated by the loss or accumulation of atoms, respectively. When voids grow and the cross-sectional area of the wiring decreases with energization, the current density increases, and accordingly the temperature rises locally, leading to fusing. In order to ensure the reliability of wiring, it is important to accurately predict EM damage such as formation of voids or hillocks and disconnection.

  So far, the effects of two-dimensional current density and temperature distribution on void formation have been studied, and the current density, current density gradient, temperature, temperature gradient, and physical properties of the materials constituting the wiring influence the void formation. It has been shown to be a factor. The key to predicting EM damage is how to obtain the divergence of atomic flux with high accuracy and ease. Until now, the divergence of atomic flux has been formulated in consideration of the one-dimensional temperature distribution, and based on this, the disconnection prediction has been performed for the linear polycrystalline wiring. However, since the crystal grain structure of the wiring, that is, the wiring structure is not taken into consideration, the application thereof is limited. In addition, regarding bamboo wiring, which is a continuous single crystal, atomic flux divergence considering the two-dimensional distribution of current density and temperature has not yet been studied.

  The life prediction of the wiring is performed by extrapolating the disconnection test result under the EM acceleration condition under the practical condition. Empirical formulas are currently used for this extrapolation, but there is a problem that the prediction results differ depending on the selection of accelerated test conditions and the wiring shape, and the development of a universal life prediction method has been awaited.

An object of the present invention is to make predictions regarding EM damage such as formation of voids or hillocks, and disconnection failure of wiring due to the damage, by atomic flux divergence (AFD _gen ), which is an EM damage dominant parameter of bamboo wiring.

導出した原子流束発散（ＡＦＤ_gen）の式を用い、前記電流密度および温度分布と配線材料の物性定数とにより、前記分割した各要素の原子流束発散を計算し、計算した原子流束発散から、各要素の体積減少を求め、各要素の厚さの変化を求めとともに、各動作を繰り返すことにより、厚さを貫通する要素が配線幅を占める状態、或いは厚さを貫通する要素または温度が材料の融点を超える要素が配線幅を占める状態となるまで処理を行い、配線寿命および断線箇所を予測することを特徴とする。
上記の処理を実行する装置および上記の処理をコンピュータに実行させるプログラムを格納した記録媒体も本発明である。 In order to achieve the above object, the present invention divides a bamboo wiring into elements, obtains a current density and a temperature distribution by a numerical analysis method, obtains a divergence type AFD _lat of atomic flux in a crystal grain, An equation for atomic flux divergence (AFD _gen ) in the wiring is derived in advance as follows:

Using the derived formula of atomic flux divergence (AFD _gen ), the atomic flux divergence of each divided element is calculated from the current density and temperature distribution and the physical constant of the wiring material, and the calculated atomic flux divergence is calculated. From determining the volume reduction of each element, determining the change in thickness of each element, and repeating each operation, the element that penetrates the thickness occupies the wiring width, or the element or temperature that penetrates the thickness The process is performed until the element exceeding the melting point of the material occupies the wiring width, and the wiring life and the disconnection location are predicted.
An apparatus for executing the above processing and a recording medium storing a program for causing a computer to execute the above processing are also the present invention.

  As described above, the simulation of the present invention makes it possible to accurately predict the lifetime and failure location of the metal wiring.

Embodiments of the present invention will be described in detail with reference to the drawings. First, an outline of derivation of the EM damage control parameter (AFD _gen ) will be described.

ここに、Ｊは原子流束ベクトル、Ｎは原子密度、ｋはボルツマン定数、Ｔは絶対温度、Ｚ*は有効電荷数、ｅは単位電荷、およびρは電気抵抗率であり、Ｄは次式で表される拡散係数である。 (Basic formula)
The EM in the metal thin film wiring also occurs in the crystal grains as lattice diffusion along the crystal grain boundaries. Therefore, the total AFD _gen of atomic flux divergence in the wiring is expressed as the sum of atomic flux divergence in the grain boundary and in the crystal grain, and is defined by the following equation.
[Equation 2]
AFD _gen = AFD _gb + AFD _lat (1)
Here, AFD _gb and AFD _lat are the divergence of the atomic flux in the grain boundary and in the crystal grain, respectively. Equation (1) holds for both polycrystalline wiring and bamboo wiring. AFD _gb and AFD _lat are derived in consideration of the wiring structures of the polycrystalline wiring and the bamboo wiring. Here, the movement of atoms in the crystal grain boundary and in the crystal grain are both given by the following equations.

Where J is the atomic flux vector, N is the atomic density, k is the Boltzmann constant, T is the absolute temperature, Z * is the number of effective charges, e is the unit charge, and ρ is the electrical resistivity, and D is Is a diffusion coefficient.

ここに、Ｄ₀は振動数項であり、Ｑは活性化エネルギである。またｊ^*はＪ方向の電流密度であり、結晶粒界においては原子が粒界にそって移動することから、ｊ^*は電流密度ベクトルｊの結晶粒界方向成分である。一方、結晶粒内においては、Ｊとｊの方向が一致することから、ｊ^*は｜ｊ｜と等しくなる。図１に示す様に、ｊと結晶粒界とのなす角をφとすれば、

Ｑは、結晶粒界または結晶粒内における値をとり、それぞれＱ_gb、Ｑ_latと表す。

Here, D ₀ is a frequency term, and Q is activation energy. Further, j ^* is a current density in the J direction, and atoms move along the grain boundary at the grain boundary, so j ^* is a grain boundary direction component of the current density vector j. On the other hand, in the crystal grain, since J and j are in the same direction, j ^* is equal to | j |. As shown in FIG. 1, if the angle between j and the grain boundary is φ,

Q takes a value in the crystal grain boundary or in the crystal grain, and is expressed as Q _gb and Q _lat , respectively.

(Atomic flux divergence at grain boundaries)
First, let us consider the divergence of atomic flux at the grain boundaries. First, grain boundaries in polycrystalline wiring are handled. In order to consider the grain boundaries in the polycrystalline wiring, a model of the grain boundary structure is introduced. As shown in FIG. 2, it is assumed that a quadrilateral element having a unit thickness including only one triple point formed by three grain boundaries having a length that is √3 / 6 times the average grain size b is assumed. To do. Area of the element is √3b ^2/4. The grain boundaries II and III are symmetrical with respect to the grain boundary I, and the included angle is close to 120 °, but there is a slight deviation 2Δφ.

上述の電流密度ベクトルの成分および温度を式（２）および（３）に代入することにより、それぞれの結晶粒界端における結晶粒界にそった原子流束が得られる。ここに要素から外に出る方向を正と定義する。結晶粒界Ｉ，ＩＩおよびＩＩＩの端部における原子流束に粒界の幅δおよび単位厚さを乗じることにより、それぞれの粒界端における単位時間あたりの原子の移動数を得、それらを各々加える。そして微小項を無視し、さらに電流保存則を用いることで式を簡単化し、要素の体積√３ｂ²／４で除す。このようにして、結晶粒界Ｉとｘ軸とのなす角がθなる場合の結晶粒界拡散における単位時間、単位体積あたりの減少原子数、すなわち原子流束の発散ＡＦＤ_gbθが次のように与えられる。

ここに、Ｃ＝ＮＤ₀Ｚ^*ｅ／ｋ、ρは温度に依存した局部的な抵抗率、およびＱ_gbは結晶粒界における活性化エネルギである。式（５）の右辺におけるかぎ括弧内の第一項は結晶粒界三重点での原子流束の発散に関係する項であり、第二項および第三項は結晶粒界自身での原子流束の発散に関係した項である。また、ＡＦＤ_gbθが正の値をとる場合はボイド（空隙部）の形成を、負の場合はヒロック（金属原子塊）の形成を意味する。 If the x-direction component and the y-direction component of the current density vector j are j _x and j _y , respectively, and the angle between the crystal grain boundary I and the x axis is θ, the current at the ends of the crystal grain boundaries I, II, and III As shown in FIG. 2, the density vector component and the temperature are expressed as follows.

By substituting the above-described components of the current density vector and the temperature into the equations (2) and (3), the atomic flux along the crystal grain boundary at each crystal grain boundary end can be obtained. Here, the direction going out of the element is defined as positive. Multiplying the atomic flux at the ends of the grain boundaries I, II and III by the width δ and unit thickness of the grain boundaries to obtain the number of atom movements per unit time at each grain boundary edge, Add. And ignoring the small term, further simplifies the equation by using the current conservation law, be divided by a volume √3b ^2/4 elements. Thus, the unit time in the grain boundary diffusion when the angle between the grain boundary I and the x axis is θ, the number of atoms decreased per unit volume, that is, the divergence AFD _{gbθ of the} atomic flux is as follows: Given.

Here, C = ND ₀ Z ^* e / k, ρ is the local resistivity depending on temperature, and Q _gb is the activation energy at the grain boundary. The first term in the brackets on the right side of Equation (5) is a term related to the divergence of the atomic flux at the grain boundary triple point, and the second and third terms are the atomic flow at the grain boundary itself. It is a term related to the divergence of the bundle. Moreover, when AFD _gb ( _theta ) takes a positive value, formation of a void (gap part) is meant, and when negative, formation of a hillock (metal atom lump) is meant.

When actual wiring is considered, θ takes an arbitrary value. Therefore, it is necessary to obtain the flux divergence in consideration of all the possible range of θ (0 <θ <2π). Here, focusing only on void formation, an expected value of only a positive value of AFD _gbθ when θ changes from 0 to 2π is obtained. Here, since the negative value of AFD _gbθ does not contribute to void formation, it is regarded as 0. I'm dividing by 2 takes the sum of the absolute value of _AFD gbθ and _AFD gbθ, can be extracted only contribution to the formation of voids _AFD gbθ. Thus, the divergence AFD _gb of the atomic flux related to the void formation at the crystal grain boundary of the polycrystalline wiring is derived as follows.

Next, crystal grain boundaries in bamboo wiring are handled. In the bamboo wiring, there is almost no crystal grain boundary, and even if it exists, it is perpendicular to the wiring length direction. Therefore, the divergence of the atomic flux at the crystal grain boundary in the bamboo wiring can be ignored. Therefore, the following equation is obtained.
[Equation 9]
AFD _gb = 0 (bamboo wiring) (7)

ここにＱ_latは格子拡張の支配的な粒内拡散の活性化エネルギである。さらに結晶粒界における原子流束の発散ＡＦＤ_gbの導出と同様に、ボイド形成のみに着目すると、多結晶配線およびバンブー配線の結晶粒内におけるボイド形成に関する原子流束の発散は次式にように導出される。

(Atomic flux divergence in crystal grains)
Consider lattice diffusion by EM in a crystal grain. Lattice diffusion can be handled in the same way in both polycrystalline wiring and bamboo wiring. In the crystal grains, vector analysis can be performed with respect to the atomic flux vector J. Based on the equations (2), (3), and (4b), the number AFD ′ _lat of atomic reduction per unit time and unit volume is obtained by the following equation.

Here, Q _lat is the activation energy of intragranular diffusion, which is dominant in lattice expansion. Further, as in the derivation of atomic flux divergence AFD _gb at grain boundaries, focusing only on void formation, the divergence of atomic flux related to void formation in crystal grains of polycrystalline wiring and bamboo wiring is as follows: Derived.

式（１０）および式（５）より、多結晶配線における原子流束の発散には、電流密度、電流密度勾配、温度および温度勾配が影響を及ぼしていることが分かる。 (EM damage control parameters for polycrystalline wiring and bamboo wiring)
Based on the total AFD _gen of the atomic flux divergence expressed by the sum of AFD _gb and AFD _lat in equation (1), divergence of the atomic flux in the polycrystalline wiring and the bamboo wiring is considered. Now, at a general use temperature, the divergence of the atomic flux within the crystal grains is negligibly small compared with that at the grain boundaries. Therefore, the divergence of the atomic flux in the polycrystalline wiring is sufficient considering only the divergence of the atomic flux at the crystal grain boundary. Therefore, the dominant parameter of EM damage in the polycrystalline wiring is expressed by the following equation.

From the equations (10) and (5), it can be seen that the current density, the current density gradient, the temperature, and the temperature gradient influence the divergence of the atomic flux in the polycrystalline wiring.

  Next, let us consider the divergence of atomic flux in bamboo wiring. Since the divergence of the atomic flux in the bamboo wiring is expressed by the sum of the equations (7) and (9), only the divergence within the crystal grains needs to be considered. Therefore, the dominant parameter of EM damage in bamboo wiring is expressed by the following equation.

式に与える電流密度および温度の分布は数値解析により求める。解析の基礎式は、次のように表される。電界のポテンシャルφ_eを支配する式は

で表される。オームの法則は、

と表される。定常熱伝導方程式は、

ここで、電流問題における抵抗率は一定と仮定しており、Ｈは配線から基板への熱の流れに関する定数で、∇²＝∂²／∂ｘ²＋∂²／∂ｙ²である。

The current density and temperature distribution given to the equation are obtained by numerical analysis. The basic formula of the analysis is expressed as follows. The equation governing the electric field potential φ _e is

It is represented by Ohm's law is

It is expressed. The steady state heat conduction equation is

Here, it is assumed that the resistivity in the current problem is constant, and H is a constant related to the heat flow from the wiring to the substrate, and ∇ ² = ∂ ² / ∂x ² + ∂ ² / ∂y ² .

The physical property constant of the wiring material is determined by an acceleration test using linear wiring. The constants ρ ₀ and α can be obtained by measuring the electrical resistance of the linear metal wiring. The constant H is determined so that the electric resistance value of the metal wiring calculated based on the temperature distribution obtained from the numerical analysis is equal to the measured value. The activation energy Q _gb for the polycrystalline wiring is given from the slope of the plot of lnVT / (j _in ρ) with respect to 1 / T. Here, j _in is the input current density, and V is the volume of the void in the central region of the wiring after energization for a predetermined time in the case of three acceleration conditions. The void volume is estimated by multiplying the total area of the void measured by a scanning electron microscope (SEM) by the thickness of the thin film. On the other hand, the activation energy Q _lat in the crystal grains in the bamboo wiring is obtained from the slope of a straight line in which lnVT ² / ρ is plotted against 1 / T. Here, V is obtained by inputting a current to the linear bamboo wiring at three different substrate concentrations for a certain period of time and measuring the vicinity of the wiring cathode end with an atomic force microscope (Atomic Force Microscope AFM) after each energization. Void volume. The constant Δφ is determined so that the calculated value of the ratio of the void volume in the central region of the wiring and the void volume in the vicinity of the cathode side end is equal to the ratio of the measured values. The constant C _gb is obtained from the relationship between the void volume measured in the experiment and the calculated value. In this way, all unknown constants can be determined only from experimental results using straight wiring.

(Numerical simulation using AFD _gen )
The lifetime and disconnection location in the metal wiring are predicted by performing numerical simulation of processes from the initial generation and growth of voids to disconnection failure using AFD _gen of polycrystalline wiring and bamboo wiring. Considered here, a change in the distribution of the current density and temperature due to the void growth in the calculation of AFD _gen.

In the numerical simulation, the metal wiring is divided into elements. Using a smaller element size can give a more realistic result. The thickness of the element is changed by the procedure shown in the flowchart of FIG. First, the current density and temperature distribution are obtained by a numerical analysis method such as two-dimensional finite element analysis (FEM analysis) (S304). The divergence AFD _gen of the atomic flux of each element is calculated by using these distributions and physical property constants (S306) of the wiring material determined in advance by an acceleration test (S308). The volume reduction per calculation step in the simulation (S312) is given by multiplying the volume of each element, the time corresponding to one calculation step, and the atomic volume by the calculated divergence of the atomic flux (S310). Here, an actual time is assigned to one calculation time. The volume reduction amount is converted into a thickness reduction amount in each element (S314). An element with reduced thickness indicates that a void has been formed, and the depth of the void corresponds to the amount by which the thickness of the element is reduced. Numerical analysis of the current density and temperature distribution in the metal wiring is performed again in consideration of the thickness of each element (S304). In this way, the calculation shown in FIG. 3 is repeated.

(Numerical simulation in polycrystalline wiring)
The numerical simulation for the polycrystalline wiring by this procedure can sufficiently predict the distribution of voids and the failure location after a certain period of time. In order to further predict the life, the following must be considered in the simulation. This is a void growth morphology in a polycrystalline wiring, in which voids that grow selectively along the crystal grain boundary and grow in a slit shape extend in the direction of the wiring width while being coupled to each other. The parameter AFD _gen is derived based on the assumption that the void formation is performed at the crystal grain boundary, but is finally expanded to the expected value of the void formation at an arbitrary point of the metal wiring. Here, the form of void formation is once again converted to the formation of slit-like voids along the crystal grain boundaries. In the element division of the wiring, dedicated elements for forming slit-like voids are arranged as shown in FIG. Here, FIG. 4A shows an example of a finite element model of wiring used later in FIG. 4B, 4C, and 4D are enlarged views of the respective portions.

The thickness of a dedicated element for a slit-like void is reduced based on the calculation of AFD _gen in that element and adjacent elements. The pitch of the slits is determined by the average crystal grain size. The width of the slit is one of the physical constants of the wiring material, and can be obtained according to the procedure shown in FIG.

In FIG. 5, while performing SEM observation, an acceleration test is performed until a disconnection failure occurs (S504). From the SEM image of the metal wiring obtained just before the failure occurs, a region where slit-like voids are concentrated, that is, a place where a disconnection failure is about to occur is extracted (S502). The length of the dense area along the longitudinal axis of the wiring (S506) is divided by the slit pitch to obtain the number of slits included in the dense area (S510). On the other hand, the total area of the slit-like voids in the dense region is measured (S508). If the total area of the void is divided by the number of slits and the wiring width, the effective width of the slit in the simulation is obtained (S512). The experiment for determining the slit width is performed with the sample used to determine the constants ρ _o , α, H, Δφ, and C _gb .

  In consideration of changes in the thickness of elements for forming slit-like voids, the numerical simulation calculation process for life prediction is based on the condition that the element that penetrates the thickness occupies the wiring width or the element or temperature that penetrates the thickness is the material. The process is repeatedly performed until the failure of the metal wiring defined as the state in which the element exceeding the melting point occupies the wiring width. As described above, the numerical simulation can predict the failure location together with the life of the metal wiring in the operating condition.

(Verification of prediction method for polycrystalline wiring)
Two aluminum polycrystalline wires shown in FIG. 6 were used for prediction of lifetime and failure location. The bent metal wiring has a two-dimensional distribution of current density and temperature. The constants necessary for prediction are given by a simple acceleration test using straight wiring. The two wirings are called sample 1 and sample 2. As shown in FIG. 6A, they differ not only in shape but also in test conditions.

High current density and temperature were selected as test conditions compared to common operating conditions. The reason for this is to shorten the time required for verification by experiment. The physical constants for calculating AFD _gen are obtained as shown in the table of FIG. The average grain size is measured using a focused ion beam (FIB) apparatus. By executing the numerical simulation, a disconnection failure due to electromigration is predicted for each of the sample 1 and the sample 2. The changes with time in the AFD _gen distribution and the void distribution for the sample 1 are shown in FIGS. 7 and 8, respectively. The changes with time in the AFD _gen distribution and the void distribution for Sample 2 are shown in FIGS. 9 and 10, respectively. The change in the void distribution with respect to time is indicated by the contour line of the film thickness. Due to changes in current density and temperature distribution due to void growth, the AFD _gen distribution changes with time. In the case of Sample 1, the metal wiring failure occurs with a lifetime of 7700 s, and the failure location is predicted to be the cathode end of the wiring. On the other hand, the failure with respect to the sample 2 occurs at a lifetime of 3400 s, and the failure portion is predicted to be on the cathode side of the corner.

  In order to verify this prediction result, an experiment was conducted with the same wiring dimensions and conditions as in the simulation. 11 specimens were used as sample 1 and 12 specimens were used as sample 2. An aluminum thin film is formed on a silicon substrate covered with a silicon oxide film by vacuum deposition. The specimen is patterned by etching after annealing. The test piece is tested using the apparatus shown in FIG. 11 until the metal wiring is opened. Thereafter, the specimen is observed with SEM.

  12 and 13 show the experimental results of the frequency distribution of disconnection faults and the average time until failure. As shown in FIG. 12A, in the case of sample 1, the average time to failure obtained from all 11 test pieces is 6731 s. The most frequent failure location is the cathode end of the wiring. The average failure time of the predicted failure location, that is, the six test pieces opened at the cathode end is 6820 s, which is close to the average time to failure obtained from the 11 test pieces. On the other hand, in Sample 2, the average time to failure obtained from all 12 test pieces is 3655 s, and the cathode side of the corner is one of the most frequently failed locations (see FIG. 13A).

(Numerical simulation in bamboo wiring)
A numerical simulation of the process from formation and growth of voids by EM to disconnection is performed using the dominant parameter AFD _gen of the bamboo wiring EM damage. As a result, it is possible to predict the wiring life and the disconnection location in consideration of the temporal change of the current density distribution and the temperature distribution accompanying the formation and growth of voids.

In the simulation, the assumed wiring is divided into elements as shown in FIG. 15, and the thickness of each element is changed by the method shown in the flowchart of FIG. The current density and temperature distribution in the wiring is obtained by a numerical analysis method such as two-dimensional finite element analysis (S304), and the AFD _{gen of} each element is obtained using the result and the physical property constant (S306) of the wiring obtained in advance by experiment. Is calculated (S308). By multiplying this by the volume of each element, the time of one time step on the simulation, and the atomic volume (S310), the volume that decreases during one time step is calculated for each element (S312). Here, the time of one time step corresponds to the actual time. The volume of each element is reduced, and the thickness of each element is changed accordingly (S314). In the element whose thickness is reduced, it is considered that a void having a thickness corresponding to the reduced amount is formed. Next, the electric resistance change and the heat conduction change corresponding to the thickness change are taken into consideration for each element, and then numerical analysis of the current density and temperature is performed again, and the subsequent calculation is repeated. When the element whose thickness is considered to be sufficiently zero compared to the initial thickness penetrates in the wiring width direction, or the element whose thickness is zero or the element whose temperature exceeds the melting point of the material occupied the wiring width The time is defined as a disconnection in the simulation, and the calculation ends.

(Verification of prediction method in bamboo wiring)
Three types of Al bamboo wiring shown in FIG. 16 were used as the disconnection prediction target. In the wiring bent here, the current density distribution and the temperature distribution exhibit a two-dimensional distribution. The length from the wiring corner to the anode end is A, and the length from the wiring corner to the cathode end is B. A = 14.0 μm, B = 8.0 μm wiring is called ASYM (+), A = 11.2 μm, B = 10.9 μm wiring is called SYM, A = 8.0 μm, B = 13.9 μm wiring is called ASYM (−). To. In each shape, the test conditions of the input current density and the substrate temperature were the same. The wiring width is not constant as shown in FIG. 16, and the wiring width on the anode side is slightly narrower than the cathode side than the corner. In order to shorten the time required for the verification experiment, a high density current of about 15 MA / cm ² and a temperature of 393 K, which are higher than general practical conditions, were selected as test conditions. The amount of current passed is 72.0 mA. The physical property values of the thin film necessary for the AFD _gen calculation were obtained as shown in the table of FIG. In this prediction, the constants H and λ are the wiring resistance value calculated based on the simulation of the temperature distribution by two-dimensional finite element analysis and the wiring resistance measured in the experiment for each shape, for each of the linear wiring and the bent wiring. The values were determined to match. In general, it is said that the thermal conductivity of the thin film is lower than that of the bulk, but the obtained λ is 1.55 × 10 ⁻⁴ W / (μm · K), which is lower than the bulk value.
Numerical simulation was performed using the physical properties as described above, and disconnection due to EM was predicted for each of the three types of wirings ASYM (+), SYM, and ASYM (−).

FIGS. 18 and 19 show the changes over time in the distribution of AFD _{gen and} the distribution of voids in the case of ASYM (+), respectively. Similarly, the case of SYM is shown in FIGS. 20 and 21, and the case of ASYM (−) is shown in FIGS. 22 and 23, respectively. Here, the void distribution is indicated by an isoline of the wiring thickness. The distribution of AFD _gen changes over time after the start of energization due to changes in current density and temperature distribution accompanying the growth of voids. For ASYM (+), disconnection after 7100 s after the start of energization was predicted on the anode side of the wiring corner. For SYM, a disconnection after 7000 s was predicted on the corner anode side. On the other hand, for ASYM (−), a disconnection after 5200 s was predicted near the cathode end of the wiring.

  In order to verify the effectiveness of this disconnection prediction method, a disconnection test was performed under the same test conditions using three types of wiring assumed in the disconnection prediction. In this disconnection test, energization was performed until disconnection using the experimental apparatus shown in FIG. After the disconnection, observation was performed with a field emission electron microscope (FE-SEM). In the experiment, 9 specimens with ASYM (+), 10 specimens with SYM, and 11 specimens with ASYM (-) were used. About the above disconnection experiment, the experimental result of the test piece of ASYM (+), SYM, and ASYM (-) is shown to FIGS. 24-26, respectively. The figure shows the frequency distribution of wire breakage and the average value of wire life. In the case of ASYM (+), the average disconnection time of the nine test pieces was 9160 s, and the portion where the wiring broke most was the wiring corner portion anode side. The average disconnection time of the four test pieces disconnected on the corner anode side, which is the disconnection point predicted by the numerical simulation, was 7965 s, which was close to the average disconnection time of all nine test pieces. On the other hand, in the case of SYM, the average disconnection time of the 10 test pieces was 7836 s, and the portion where the wiring was broken most was on the wiring corner portion anode side. The average disconnection time of the five disconnected wires was 7344 s, which was close to the average disconnection time of all ten test pieces as in ASYM (+). Further, in the case of ASYM (−), the average disconnection time of the 11 test pieces was 6996 s, and the most broken part was in the vicinity of the cathode end of the wiring. The average disconnection time of the six test pieces disconnected at the wiring cathode end, which was the disconnection point predicted by the numerical simulation, was 6160 s, which was close to the average disconnection time of all 11 test pieces.

(Conclusion)
The prediction and the experimental result showed good agreement in both the wiring life and the disconnection location. Although there is some variation in the disconnection location of the experiment, this prediction method was able to predict the location where the wiring was broken most. From this, given the physical properties and practical conditions of the thin film that constitutes the wiring, by performing a numerical simulation using the AFD _gen that is the dominant parameter for EM damage, It was shown that it was possible to predict the life and breakage location under practical conditions, and the effectiveness of this prediction method could be verified.

Void formation induced by electromigration is not only the current density, temperature and gradient, but also electrical resistivity, average grain size, activation energy, relative angle between grain boundaries, atomic density, diffusion coefficient, net charge It depends on the material properties such as the effective width of the grain boundaries. The parameter AFD _gen determined as a function of these factors governs void formation. The disconnection failure of metal wiring occurs as a result of void formation and its growth. The failure location changes depending on the combination of these factors determined by the operating conditions such as the wiring shape, substrate temperature, and input current density. That is, in some cases, a failure occurs at the corner of the bent metal wiring, and in other cases, it occurs at the cathode end of the bent wiring. By the simulation of the present invention based on AFD _gen , the life of the metal wiring and the location of the disconnection failure are accurately predicted.

  The present invention may be applied not only to a stand-alone computer system, but also to, for example, a client / server system composed of a plurality of systems. The configuration of the present invention can be realized by reading and executing the program from the storage medium storing the program for performing prediction related to the present invention. Examples of the recording medium include a floppy disk, a CD-ROM, a magnetic tape, and a ROM cassette.

It is a figure which shows the atomic flux and current density in a crystal grain boundary. It is a figure which shows the model of the crystal grain boundary structure of the square element of unit thickness which contains only one triple point comprised by three crystal grain boundaries inside. It is a flowchart which shows the process of numerical simulation. It is a figure which shows the example of the element division | segmentation used for the numerical simulation of a polycrystalline wiring. It is a flowchart which shows the procedure for calculating | requiring the effective width | variety of the slit which is a physical property constant of wiring material. It is the figure shown about two samples which perform the numerical simulation of a polycrystalline wiring. It is a figure which shows the numerical simulation result of _AFDgen distribution in a polycrystalline wiring (sample 1). It is a figure which shows the numerical simulation result of the void distribution in a polycrystalline wiring (sample 1). It is a figure which shows the numerical simulation result of AFD _gen distribution in a polycrystalline wiring (sample 2). It is a figure which shows the numerical simulation result of the void distribution in a polycrystalline wiring (sample 2). It is a block diagram which shows the apparatus for verification experiment. It is a figure which shows the experimental result in a polycrystalline wiring (sample 1). It is a figure which shows the experimental result in a polycrystalline wiring (sample 2). It is a constant of material properties used for numerical simulation of polycrystalline wiring. It is a figure which shows the example of the element division | segmentation used for the numerical simulation of bamboo wiring. It is the figure shown about three samples which perform the numerical simulation of bamboo wiring. This is a constant of material properties used for numerical simulation of bamboo wiring (ASYM (+)). It is a figure which shows the numerical simulation result of AFD _gen distribution in a bamboo wiring (ASYM (+)). It is a figure which shows the numerical simulation result of the void distribution in a bamboo wiring (ASYM (+)). It is a figure which shows the numerical simulation result of AFD _gen distribution in a bamboo wiring (SYM). It is a figure which shows the numerical simulation result of the void distribution in a bamboo wiring (SYM). It is a figure which shows the numerical simulation result of AFD _gen distribution in a bamboo wiring (ASYM (-)). It is a figure which shows the numerical simulation result of the void distribution in a bamboo wiring (ASYM (-)). It is a figure which shows the experimental result in a bamboo wiring (ASYM (+)). It is a figure which shows the experimental result in a bamboo wiring (SYM). It is a figure which shows the experimental result in a bamboo wiring (ASYM (-)).

Claims (3)

A means to divide the bamboo wiring into elements and obtain the current density and temperature distribution by numerical analysis method;
The atomic flux divergence formula AFD _lat in the crystal grain is obtained, and the atomic flux divergence (AFD _gen ) in the bamboo wiring is derived in advance by the following formula:

Means for calculating the atomic flux divergence of each of the divided elements by using the derived equation of atomic flux divergence (AFD _gen ) and the current density and temperature distribution and the physical constant of the wiring material;
A means for determining the volume reduction of each element from the calculated atomic flux divergence;
Means for determining a change in thickness of each element,
By repeating the operation of each means, processing is performed until the element that penetrates the thickness occupies the wiring width, or the element that penetrates the thickness or the element whose temperature exceeds the melting point of the material occupies the wiring width. An apparatus for evaluating the reliability of metal wiring, characterized by predicting wiring life and disconnection locations. Dividing the bamboo wiring into elements and obtaining the current density and temperature distribution by numerical analysis,
The atomic flux divergence formula AFD _lat in the crystal grain is obtained, and the atomic flux divergence (AFD _gen ) in the bamboo wiring is derived in advance by the following formula:

Calculating the atomic flux divergence of each of the divided elements using the derived equation of atomic flux divergence (AFD _gen ) and the current density and temperature distribution and the physical constants of the wiring material;
Calculating the volume reduction of each element from the calculated atomic flux divergence;
Determining the change in thickness of each element,
By repeating the operation of each step, processing is performed until the element that penetrates the thickness occupies the wiring width, or the element that penetrates the thickness or the element whose temperature exceeds the melting point of the material occupies the wiring width. A method for evaluating the reliability of metal wiring, characterized by predicting wiring life and disconnection location.   A recording medium storing a program for causing a computer to execute each step of the reliability evaluation method according to claim 2.

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