Pad Scratching in Chemical-Mechanical Polishing: The Effects of Mechanical and Tribological Properties

If the asperity deformation is elastic, the average contact pressure, p_a, is given by the Hertz solution as²⁶

Asperity deformation reaches the elastic limit when the maximum contact pressure, p_o, is 1.5 times the yield strength of the asperity, σ_y,a.²⁷ Then the average contact pressure is given by

From Eqs. 1 and 2, the approach of distant points at the onset of asperity yielding, δ_y, can be written as

Beyond the elastic limit, analysis of the pressure distribution at an asperity contact is quite complex, since the asperity deformation comprises both elastic and plastic components. When the approach of distant points exceeds another critical limit, δ_f–p (= C_f–pδ_y), however, the deformation will be fully plastic everywhere in the asperity. The pressure then will be approximately uniform in the contact and is equal to the hardness of the asperity, which is three times the yield strength.^28–30 Therefore, the average contact pressure at a fully-plastically deformed asperity contact can be expressed as

Over the decades, several attempts have been made to characterize the transitional regime, elastic to fully plastic, using mathematical functions and by finite element analyzes.^31–34 Zhao et al. have suggested that the mean contact pressure can be represented by a logarithmic function based on the statistical results of spherical indentations by Francis.^25,35 Kogut and Etsion have analyzed the problem by the finite element method and showed that the logarithmic function better agrees with their results than the other models.³⁶ In the logarithmic function formulation, which is adopted here for simplicity, the average pressure at an elastic-plastic asperity contact can be expressed as²⁵

where C_f–p is defined as C_f–p = δ_f–p/δ_y.

Based on the experimental results using a spherical indenter, Johnson has proposed that the fully-plastic deformation state is reached when the asperity load is greater than 400 times the yield load.²⁷ From the elastic analysis, where the asperity load, P_a, is given as

The approach of distant points at the onset of fully-plastic flow is 54 times that at the onset of asperity yielding; i.e. C_f–p will be at least 54.²⁵

For frictionless or low frictional contact (0 ≤ μ < 0.3), the maximum shear stress, τ_max, for the Hertzian traction distribution is below the surface and the normalized maximum shear stress, τ_max/p_a, is given by²⁷

By the Tresca yield criterion, the surface layer will yield if the maximum shear stress reaches half the yield strength, σ_y,l. Thus, the average contact pressure at the onset of surface layer yielding, p_s, under elastically deformed asperity will be

As friction increases, however, the location of maximum shear stress will no longer be below the surface but rapidly rises to the surface. It has been shown by Hamilton and Goodman that if the coefficient of friction is greater than 0.3, the location of the maximum von Mises stress also moves to the surface.^37,38 Based on the closed-form, analytical solution, the normalized maximum von Mises stress, σ_M,max/p_a, in a elastic body under Hertizian traction distribution is given by³⁸

Although Eq. 9 depends on the Poisson's ratio of the surface layer, ν_l, the of the maximum von Mises from ν_l = 0.1 to 0.5, is not significantly different. Therefore, Eq. 9 can be simplified, for ν_l = 0.3, as

By the von Mises yield criterion, the surface layer will yield if the maximum equivalent stress reaches the yield strength. Then, the average asperity contact pressure at the onset of surface layer yielding will be

If the asperity deformation is fully plastic, the normal pressure distribution in the contact will be uniform. For frictionless or low-frictional contact (0 ≤ μ < 0.05), the maximum shear stress is located below the surface and the normalized maximum shear stress is given by^27,39

By the Tresca yield criterion, the average asperity contact pressure at the onset of surface layer yielding, under fully-plastically deformed asperity, will be

When the coefficient of friction is greater than 0.05, Eusner has shown by finite element analysis (FEA) that the location of the maximum von Mises stress for uniform contact pressure distribution moves to the surface.^23,40 Based on second-order polynomial regression of the FEA results, the magnitude of the normalized maximum von Mises stress for high friction is given by

Therefore, when the asperity deformation is fully plastic, the average contact pressure at the onset of surface layer yielding by the von Mises criterion will be

The normalized average contact pressures at the onset of surface layer yielding for both the Hertzian and the uniform traction distributions against the friction coefficient are plotted in Figure 3. In both cases the layer yields below the surface when friction coefficient is low and thus the effect of interfacial friction on layer yielding is neglegible. As the friction coefficient increases, however, the layer yields at the surface and the effect of friction then is significant.

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When the pad asperity deformation is elastic-plastic, the pressure distribution in the contact gradually changes from Hertzian to uniform as δ increases, Figure 2c. At such arbitrary condition, determination of the stress field in the surface layer is more complicated compared with the elastic and fully-plastic cases. Assuming that the magnitudes of maximum von Mises stress and maximum shear stress lie between the values of the two extremes cases, the normalized average contact pressure at the onset of yielding in the surface layer under elastic-plastically deformed asperity, can be approximated to be the pressure distribution of Hertzian or uniform. The difference between the Herzian and uniform pressure distributions is no more than 15% when μ ≥ 0.3.

Scratching of the surface layer initiates at the onset of plastic flow, i.e., yielding is a prerequisite for the scratching of ductile materials. Therefore, a sliding asperity will scratch only if the mean contact pressure, p_a, exceeds the mean contact pressure at the onset of surface layer yielding, p_s. From Eqs. 1, 8 and 11, an elastically deformed asperity (0 ≤ δ < δ_y) will scratch the layer surface if:

where δ_s,e is approach of distant points at the onset of surface layer yielding under an elastically deformed asperity in sliding contact, defined by

If the asperity deformation is elastic, but at the onset of yielding (δ = δ_y), from Eqs. 2, 8 and 11, the surface layer will be scratched if:

Thus whether an asperity can scratch the surface layer depends only on the yield strengths of the asperity and the surface layer, and the friction coefficient.

If the asperity deformation is elastic-plastic (δ_y < δ < δ_f–p), but the contact pressure distribution is uniform and given by Eq. 5, from Eqs. 9 and 12 the scratch criteria can be written as

where δ_s,p is approach of distant points at the onset of surface layer yielding under an elastic-plastically deformed asperity in sliding contact, defined by

Finally, in the extreme case of fully plastic asperity deformation (δ_f–p ≤ δ), from Eqs. 4, 9 and 12, the scratch criteria can be given as

In this extreme case, too, the condition whether an asperity can scratch the surface depends only on the yield strengths of the asperity and of the surface layer, and the friction coefficient.

It may be noted that the criteria for scratching in the cases of elastic and fully-plastic deformations depend only on the mechanical properties and the interfacial friction. That is, neither criterion is explicitly a function of the radius of curvature of the asperity, although the load required to induce plasticity in the asperity depends on the radius. Because the hardness is about three times the yield strength, the scratching criteria for elastically deformed asperities can be expressed from Eq. (18) as a function of asperity-to-layer hardness ratio and the coefficient of friction as

Thus, a scratch-regime map can be constructed with the ratio of pad hardness to layer hardness and with the coefficient of friction as coordinates, Figure 4a.^23,41 The map provides whether an elastically deformed asperity in sliding contact can scratch the surface layer. If the pad-to-layer hardness and the friction coefficient fall in the 'scratch regime,' an elastically deformed asperity, at the onset of yielding, will scratch the surface layer. An elastically deformed asperity may also scratch the surface if satisfy Eq. 16 is satisfied. However, an elastically deformed asperity, even at the onset of asperity yielding, cannot scratch the surface layer, if the conditions fall in the 'no-scratch regime.'

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Similarly, from Eq. (21) the scratching criteria for fully-plastically deformed asperities can be expressed as

and a scratch-regime map can be constructed as in Figure 4b. If pad-to-layer hardness and friction coefficient fall in the 'scratch regime,' a fully-plastically deformed asperity will scratch the surface layer. An elastic-plastically deformed asperity may also scratch the surface if Eq. 19 is satisfied. But neither an elastic-plastic nor a fully-plastically deformed asperity can scratch the surface layer, if the conditions fall in the 'no-scratch regime.'

The probability density function of exponentially distributed asperity heights, ϕ(z_a), can be written as

If n asperities per unit area are pressed against a smooth, flat surface layer, only asperities that are taller than the separation distance, d, will be in contact, therefore, the number of asperities in contact per unit area, n_c, may be written as

Figure 5 schematically shows the proportion of each asperity deformation mode depending on the approach of distant points, δ, equals to (z_a – d). Of the asperities in contact, the relatively short asperities, that have δ < δ_y, determined by Eq. 3, deform elastically, whereas the tall asperities, that have δ > δ_y, deform plastically—-elastic-plastic and fully plastic.

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The number of elastically and plastically deformed asperities per unit area, n_e and n_p, can be expressed, respectively, as

The proportions of elastically and plastically deformed asperities in the contact are

where ψ is the plasticity index, defined as

Thus, the relative proportions of elastically and plastically deformed pad asperities depend on the plasticity index, which depends on the ratio of asperity modulus to asperity hardness, E_a/H_a, and the ratio of standard deviation of asperity heights to asperity radius, σ_z/R_a. Based on Eq. (27), the probability of pad asperity deformation modes can be estimated if the plasticity index of the pad surface is known. As shown in Figure 6, ψ ≈ 1 is the transition region from elasticity-dominant contact to plasticity-dominant contact. If ψ ≪ 1, most asperities in contact deform only elastically. Then, Eq. (22) and Figure 4a should be considered as criteria for pad scratching. On the other hand, if ψ ≫ 1, a considerable number of asperities in contact will deform plastically, and therefore, Eq. (23) and Figure 4b should be the scratching criteria.

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While the scratch-regime maps provide the criteria for given pad-to-layer hardness ratio and the friction coefficient, they only indicate whether the pad may scratch the surface layer or not. For a quantitative analysis of pad scratching, therefore, the models must be further extended for estimating the relative proportion of asperities among those in contact that can scratch the layer.

When multiple asperities slide over the layer it is the approach of distant points, δ ( = z_a – d), which determines the mean pressure, p_a, at each asperity contact. The surface layer will yield only if p_a > p_s. That is, only the asperities that deform more than the approach of distant points at the onset of surface layer yielding, δ_s, which corresponds to p_s, can scratch the surface layer. Therefore, the number of scratches per unit area, n_s, can be estimated by

From Eqs. 25 and 29, the relative proportion of asperities that can scratch the layer among those in contact, n_s/n_c, is by

Determination of δ_s depends on where the hardness ratio and the interfacial friction fall in the scratch-regime maps. First, if the pad-to-layer hardness ratio and the friction coefficient fall in the 'scratch regime' of Figure 4a, i.e. if the conditions satisfy the Eq. (22), then δ_s = δ_s,e, given as Eq. 17. For multiple asperities in sliding contact the proportion of scratching asperities can be obtained from Eq. 30 as

Second, if the pad-to-layer hardness ratio and the friction coefficient fall in 'no-scratch regime' of Figure 4a, but fall in 'scratch regime' of Figure 4b, i.e., if the conditions do not satisfy the plastic criteria Eq. (22) but satisfy Eq. (23), then δ_s = δ_s,p, given as Eq. 20. Therefore, from Eq. 30,

Finally, if the pad-to-layer hardness ratio and the friction coefficient fall in the 'no- scratch regime' of Figure 4b, i.e., if the conditions do not satisfy Eq. (23), then the asperities in contact cannot scratch the surface and therefore,

To simplify Eqs. 31 through 33, scratching indexes for elastically and plastically deformed asperities, α_e and α_p respectively, are introduced as:

Then, the scratching criteria for the extreme cases of elastically and plastically deformed asperities, Eqs. (22) and Eqs. (23), respectively, can be expressed as

The proportion of asperities in sliding contact that can scratch a surface can be summarized, from Eqs. 31 to 36, as

Equation 37 suggests that the relative proportion of the asperities that can scratch the surface layer depends on the scratching index, α_e or α_p, and the plasticity index, ψ. The scratching indexes characterize the effects of the hardness ratio and the interfacial friction. If α_e ≥ 1, the larger the α_e value is the more are the elastically deformed asperities that scratch the surface layer. If α_e < 1 and α_p ≥ 0.33, none of the elastically deformed asperities can scratch the surface. However, plastically deformed asperities may scratch surface and the number of scratches may increases as the α_p increases. If α_p < 0.33, none of the asperities will scratch. Figure 7 shows the proportion of scratching asperities in contact according to the scratching index for different values of plasticity index. Typically, the hardness ratios of CMP pads and the surface layers vary from 0.01 to 0.5, and the friction coefficients are in the range 0.4 and 0.6. Therefore, the α_e value in a general CMP system is less than unity, and thus the elastically deformed asperities will not scratch the surface layers; the plastically deformed asperities are the primary source of pad scratching. Moreover, because typical pad surfaces have ψ > 1, most asperities in contact will deform plastically. It can be predicted that the softer pad asperities can scratch the relatively hard surfaces when α_p is greater than 0.33, and the number of scratches abruptly increases as α_p increases beyond 0.33 as shown in Figure 7.

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The surface profiles of a commercial pad, Pad A, and a standard CMP pad, IC1000, manufactured by Dow Chemical Co., were determined by a Tencor P16 profilometer, Figure 8. Sampling lengths of 5 mm were scanned at a data acquisition rate of 200 Hz. The tip radius of the stylus was 10 μm, the normal load was 20 μN and the scanning speed was 50 μm/s. Table I lists the statistics of the topographical parameters, heights, and radii of the asperities in the scanned length. In addition, the probability densities of the asperity heights are calculated and shown in Figure 9. It is apparent that the asperity-height distributions of both pads can be well expressed by exponential probability density functions, based on the standard deviation of asperity heights.

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The Young's modulus and the hardness, of the pad asperities and of the thin-film layers were determined by a Hysitron TriboIndenter, model TI900. Eight different monolithic layers, Al, Cu, SiO₂, Si₃N₄, TiN and three low-k dielectrics, all 1 μm thick and coated on Si wafers, were tested. Over 100 indentations on the pad and 49 indentations on each monolithic surface layer were made. In all cases, a Berkovich indenter was used and the maximum depth of indentation was 90 nm.

Table II lists the statistics of the hardness and modulus. Comparing the average values, not surprisingly the hardness values of the surface layers are much greater than those of the pad asperities. It may be noted, however, that the variation in pad properties is considerably large, probably due to the porosity. It was experimentally shown previously that the Young's modulus and hardness of CMP pads are log-normally distributed.²² Figure 10 shows the probability density of the logarithm of the normalized asperity hardness for both Pad A and IC1000. The data were normalized with the average value. Variation in the properties of Al and Cu is relatively large compared with those of the other layers. Hardness of the surface layers is normally distributed, as shown in Figure 11.

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Based on the measured topographical and mechanical properties of pad asperities, plasticity indexes of Pad A and IC1000 pad were calculated from Eq. 28 as 3.5 and 4.2, respectively. The probability of asperity deformation modes was estimated by Eq. (27). As listed in Table III, when the pads are pressed against the surface layers over 90% are expected to deform plastically.

Figure 12 shows the reciprocating sliding apparatus used for scratching experiments. Circular disks, 20 mm in diameter, of polishing pads were pressed at a normal load of 2 N, which corresponds to an average pressure of 7 kPa, and were slid over the wafers at a velocity of 7 mm/s. All sliding tests were conducted in deionized water, and the number of cycles in each run was 15. The friction between the pads and the surface layers during sliding was monitored by a strain-gage transducer. After each experiment, scratches on the surface layers of the wafer were characterized by optical and scanning electron microscopes.

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