Determination of Strain Gradient Plasticity Length Scale for Microelectronics Solder Alloys

Juan Gomez Universidad EAFIT, Medellin, Colombia Cemal Basaran Electronic Packaging Laboratory, University at Buffalo, State University of New York, 212 Ketter Hall, Buffalo, NY 14260 e-mail: cjb@buffalo.edu Determination of Strain Gradient Plasticity Length Scale for Microelectronics Solder Alloys Strain gradient plasticity theories that have emerged during recent years to provide an explanation for size dependent behavior exhibited by some materials have also created a need for additional material parameters. In this study on Pb/ Sn solder alloys material length scale, which is needed for use in strain gradient plasticity type constitutive models, is determined. The value of length scale is in agreement with values available in literature for different materials like copper, nickel, and aluminum. DOI: 10.1115/1.2721082 Keywords: Length scale, nanoindentation, strain gradient plasticity, size effects Introduction It has been observed that response of many metals and its alloys at small scales is substantially different from that in bulk specimens. For instance, Fleck et al. 1 have shown that torsional strength of thin copper wires increases as wire diameter decreases from 170 m to12 m. During nanoindentation experiments on copper specimens McElhaney et al. 2 identified a dependence of measured hardness on the indentation depth. Stolken and Evans 3 found a dependence of bending moment on the thickness as beam thickness went from 50 m to 12.5 m on thin nickel microbeams. Xue et al. 4 identified a dependence on the resistance of a particle reinforced metal matrix on the reinforcing particle diameter for a constant particle volume fraction. A computational mechanics description of all of the above experiments requires the consideration of a length scale into the constitutive models representing the material behavior. Classical plasticity theories have no internal length scale and therefore are unable to predict any size dependent behavior. Recently, several theories have emerged in order to explain these size dependent phenomena. The proposed theories start from the assumption that in fact there is an intrinsic material length scale parameter and that size effects are triggered when characteristic structural dimensions approach this intrinsic length scale. The extra hardening at small scales has been explained in terms of additional dislocation introduced by the presence of nonuniform plastic strain fields 5,1,6 9. It is well known that material deformation in metals promotes dislocation formation, motion, and storage. Material hardening is due to dislocation storage. The dislocations stored by trapping each other in a random way are generally referred to like statistically stored dislocations 10. On the other hand, in the presence of nonuniform plastic deformation fields there are strain gradients that have to be accommodated by additional dislocations. These types of dislocations are generally referred to like geometrically necessary dislocations 11,12. The storage of geometrically necessary dislocations causes extra hardening by becoming obstacles to the evolving glide dislocations. The available models that attempt to explain the size dependent phenomena introduce the coupling between statistically and geometrically necessary dislocations in different forms. For instance, in the so called phenomenological theories of Fleck and Hutchisnon 10,13 and theories derived from those like the mechanism based strain gradient MSG model of Gao et al. 14, hardening is introduced following the Contributed by the Electrical and Electronic Packaging Division of ASME for publication in the JOURNAL OF ELECTRONIC PACKAGING. Manuscript received June 16, 2005; final manuscript received October 4, 2006. Review conducted by Stephen McKeown. Taylor model from dislocation mechanics and after enhancing the shear flow stress by the density of geometrically necessary dislocations according to = Gb S + l G 1 where is the shear flow stress; is a material coefficient; G is the shear modulus; b is the Burger s vector magnitude; S and G are the densities of statistically stored and geometrically necessary dislocations, respectively; and l is the material length scale. In this type of model the dislocation densities are described in terms of the plastic strain and gradients of the plastic strain. In contrast, the strain gradient plasticity theory of Aifantis 15,16 introduces the length scale like an enhancing parameter that factors the Laplacian of the equivalent plastic strain in the flow stress term. Although the mathematical coupling between the statistically stored dislocations and the geometrically necessary dislocations has been given different forms by different researches the common point is the presence of the length scale parameter. The predictive capabilities for structures at small scales are gaining considerable interest due to the strong development of the microelectronics industry during the recent years. The success of the computational treatment of the problem is however tied to the availability of material properties at such small scales and in the particular case of strain gradient plasticity theories to the determination of the length scale material parameter. The length scale parameter has been determined for a series of materials mainly by curve fitting of computational models to experimental data. Fleck et al. 1 determined an approximate value of the length scale for copper of 3.7 m by fitting their strain gradient plasticity theory to experiments on microtorsion of thin wires. Stolken and Evans 3 performed microbending on thin nickel foils and found a value of the length scale of 5.0 m by curve fitting the Fleckand Hutchinson 13 gradient theory to their experimental data. Begley and Hutchinson 17 curve fitted the nanoindentation results from Ma and Clarke 6 and Stelemashenko et al. 5 on silver single crystals and tungsten single crystals and inferred values of the length scale on the order 0.2 0.6 m. Nix and Gao 18 curve fitted their MSG theory to the nanoindentation experimental data of McElhaney et al. 2 on cold worked polycrystalline copper and single crystal copper and found values of the length scale of 12 m and 5.84 m, respectively. Recently Abu Al-Rub and Voyiadjis 19 and motivated by the Nix and Gao s 18 theory, proposed a method to determine the length scale based solely on nanoindentation experimental results without the need to fit any specific strain gradient theory. Both, the Nix and Gao and Al-Rub and Voyiadjis models are based on the Taylor model from dislocation mechanics. However, they differ in the form of coupling between the densities of statistically stored and geometrically necessary 120 / Vol. 129, JUNE 2007 Copyright 2007 by ASME Transactions of the ASME

dislocations. In this work we follow this approach to determine the length scale for Pb/Sn eutectic solder alloy but assuming the coupling from Nix and Gao 18. Other forms of coupling were explored but we found that the linear coupling of Nix and Gao 18 better fits our experimental data. The current study is motivated by the trend of miniaturization of microelectronics solder joints due to the continuing decreasing size of microelectronics packages. Here we want to explore whether or not solder alloy exhibits size effects and how important they are in predicting behavior at small scales. Potential applications are thermo-mechanical fatigue problems at small scales which are induced by the coefficient of thermal expansion mismatch between the soldered parts. Although nanoindentation studies have been previously performed on this type of material, none of them have been aimed at finding the strain gradient plasticity length scale. That is the primary objective of this study. The work is organized as follows: The first part describes the meaning of the length scale and its interpretation within the gradient plasticity theories. The second part makes a description of the nanoindentation technique and discusses the method used and sample preparation as well as some characterization of the possible sources of error within our experiments. The final part shows the most relevant results and some conclusions and further work. Length Scale Interpretation Consideration of size dependent response in gradient type models has been done through the introduction of a more general definition of the equivalent plastic strain E which in general form can be written like E = p + l 1/ 2 where,, are interaction coefficients that describe the coupling between the equivalent plastic strain p measure of statistically stored dislocations and the gradient term measure of the geometrically necessary dislocations ; and l is the length scale that enhances the gradient terms. For instance, Fleck and Hutchinson 10,13 and his co-workers have considered solids where = = =2 and l 2 = l 2 sg 1 l ijk 1 ijk + l 2 cs ij ij l where l sg and l cs are phenomelogical length scales making explicit the contribution between rotation gradients curvatures and gradients of stretch. On the other side of the spectrum the lower-order theories of Aifantis 15,16 and co-workers consider the case = = =1 and l =l p. Looking at these two theories it is clear that whereas the length scale plastic parameter l can be understood based on micromechanical reasoning, a direct physical explanation of the origin of l sg and l cs is not yet available and we can think of them as phenomelogical factors. In the type of theories promulgated by Fleck and Hutchinson 10,13 or theories derived from those, the gradients of strain are introduced into the models with their corresponding higher-order conjugate variables in the so called Cosserat type of solids. This type of formulation introduces additional field variables and boundary conditions, whereas the Aifantis type of theories keeps the original structure from the classical plasticity theory. Moreover, a direct physical explanation for the source of the higher-order stress terms has not been fully clarified and the controversy still remains open about their validity. Furthermore, experimental work has been attempted toward the measurement of l in Eq. 2. However, to the best of our knowledge there are no experiments available to determine l sg and l cs. Nanoindentation Experiments The nanoindenter has the capability of high-resolution depth sensing and ultralow-load sensitivity. The technique of nanoindentation is a relatively new form of mechanical testing of elastic and Batch_Id Table 1 Length scales from nanoindentation results H 0 GPa Ave. h nm Ave. length scale nm 1_1_1 0.18 318.3 341.5 1_1_2 0.18 297.78 319.5 1_1_3 0.18 372.6 399.8 1_2_2 0.18 195.02 209.25 1_2_500 0.18 320.17 343.5 1_2_3 0.18 224.42 240.8 1_3_1 0.18 268.32 287.9 1_4_1 0.18 230.35 247.16 1_7_1 0.18 300.2 322.1 plastic material properties for structures with small size. When the indenter tip is driven into the surface of the material, both elastic and plastic deformation processes occur, producing a loading and unloading cycle. After unloading, elastic deformation is recovered, so one can separate the elastic properties. In each load unload cycle, the data of the load versus displacement indentation depth are recorded. In the case of the Berkovich indenter the area of the impression is computed out of the following area shape function for a perfect tip A = 24.5h 2 3 and the hardness is calculated like H= P/A. In this work we used the continuous stiffness measurement CSM technique. When the CSM feature is used during testing, the stiffness is measured continuously by superimposing a small high-frequency oscillation on the primary loading signal and analyzing the response of the system by means of a frequency specific amplifier. This method returns hardness and modulus as a continuous function of penetration into the test surface. The CSM oscillation frequency and amplitude are set to a harmonic frequency target of 45 Hz and a harmonic displacement target of 2 nm, respectively. The phase shift between the excitation oscillation and the displacement oscillation is zeroed. The indenter tip begins approaching the surface from a distance above the surface of approximately 1000 nm. The approach velocity is determined by surface approach velocity when the indenter determines that it has contacted the test surface, according to the criteria surface approach sensitivity, the indenter penetrates the surface at a rate of 0.05 s 1. When the surface penetration reaches the depth limit, the load on the indenter is held constant for 10 s. The indenter is then withdrawn from the sample but not completely at a rate equal to the maximum loading rate. When the load on the sample reaches 10% of the maximum load on sample, the load on the sample is held constant for 100 s. The indenter is then withdrawn from the sample completely and the sample is moved into the position for the next test see Table 1. Length Scale Determination. In this work and in order to determine the strain gradient plasticity length scale l in Eq. 2 we will follow the approach from Abu Al-Rub and Voyiadjis 20. Their method is based on a dislocation mechanics model that yields the density of geometrically necessary dislocations underneath a spherical indenter. They used a model originally proposed by De Guzman et al. 21 and that has also been used by Nix and Gao 18 where the density of geometrically necessary dislocations is expressed as G = 3h V 2ba 2 4 where is the total length of injected dislocation loops in a plastic hemispherical volume V located underneath the indenter; h and a are the contact depth and contact radius, respectively; and b is the magnitude of Burger s vector. Furthermore, Abu Al-Rub and Voyiadjis 20 assumed that the total dislocation density is given by Eq. 5. Journal of Electronic Packaging JUNE 2007, Vol. 129 / 121

T = S /2 + F G /2 2/ 5 where F and are factors expressing different couplings between dislocation densities. Equation 5 reduces to the Nix and Gao 18 linear coupling for F=1 and =2 and to the Fleck and Hutchinson 10 harmonic coupling for F=1 and =4. Abu Al-Rub and Voyiadjis 19 related the density of statistically stored dislocations to the indentation test parameters for conical and pyramidal indenters and have proposed the following model that directly yields the length scale from the nanoindentation experiment H H 0 =1+ h* h p /2 6 where h * = l with =3/2cr tan and being the angle of indented surface that remains constant; H is the size dependent hardness; and H 0 is the hardness at large indentation depths. In Eq. 6 r is the Nye s factor introduced by Arselinis and Parks 7 to reflect the scalar measure of geometrically necessary dislocations density resultant from macroscopic plastic strain gradients and c is a material constant. When the data corresponding to the indentation test are plotted like H/H 0 versus h /2 the length scale parameter l can be identified from the slope h * /2. The relation between the hardness and penetration depth given by Eq. 6 follows from assuming that the flow stress in tension and the shear strength as given by Eq. 1 are related according to the Taylor law and such that = Z = Z Gb T where Z is a coefficient that relates yield stress in tension to yield strength in pure shear; is a material coefficient; G is shear modulus; b is Burger s vector; and T is the dislocation density. Additionally it is also assumed that the nanoindentation hardness H can be related to the tensile flow stress through Tabor s factor as H =3 7 8 Fig. 1 Scanning electron microscope picture showing the location of 100 200 nm depth nanoindentation batch to determine reliability of the results and the end polished surface of the specimen Sample Preparation and Experimental Results. The indentation experiments were performed using a Nano Indenter XP equipped with the CSM technique. The indenter was located in an isolated low noise environment at room temperature. The temperature was not controlled during the test. Sample cross sections were prepared via sectioning using a low-speed diamond saw followed by epoxy mounting and manual polishing using 240,600 and 1200 silicon carbide paper. Final polishing was done with 0.3 m and 0.05 m colloidal alumina suspension in a vibratory polisher until a scratch free surface was achieved. The specimen was later characterized using scanning electron microscopy SEM. A completely perfect surface is impossible to achieve since in the particular case of eutectic solders, the material creeps at room temperature, and this creates limitations in the polishing process. A typical scanning electron microscope picture showing the polished surface is shown in Fig. 1. Care was taken during testing to locate the indentation points away from visible scratches. In order to determine the minimum depth of indentation at which we can obtain reliable results and free of errors due to surface contamination or uncertainties in the contact surface, we performed an initial batch of 200 nm depth indentations as shown in Fig. 1. Selected hardness results corresponding to the encircled locations are shown in Fig. 2. It can be seen that approximately In this work we have used a pyramidal Berkovich indenter. For proving properties such as hardness and elastic modulus at the smallest possible scales, the Berkovich triangular pyramidal indenter is preferred over the four sided Vickers or Knoop indenter because a three sided pyramid is more easily ground to a sharp point 22. For the Berkovich indenter =24.7 deg. The additional parameters for the length scale determination are Nye s factor r =1.85, 7 and c=0.4. Furthermore, a value of =2 as originally proposed by Nix and Gao 18 has been used. Using the above set of parameters and once we know h * from our experimental results we can find the length scale from l = 2h* cr 3 tan 9 Fig. 2 Nanoindentation at 200 nm depth to determine reliability of results 122 / Vol. 129, JUNE 2007 Transactions of the ASME

Fig. 3 Hardness versus depth of indentation and normalized hardness square versus inverse of indentation depth for Specimen 1-Solder 1 batch test Number 2. The data below 150 nm there is considerable scatter in the results. Moreover, the large hardness values below 50 nm are totally unrealistic and are just due to errors in locating the contact surface. Other sources of error are due to the soft character of the material. During the polishing process small particles from the abrasive paper may become embedded within the surface as revealed on some of the SEM pictures. In the spirit of isolating the size effects we have considered results at depths larger than 0.3 m only. Below this value of indentation depth it is likely that uncertainties in the contact area would arise. In order to measure the hardness and identify the presence of size effects we performed a series of approximately 100 indentations in different solders at maximum depths close to 4.0 m. All the solders were obtained from the same actual package provided by Intel. At this depth the hardness Journal of Electronic Packaging JUNE 2007, Vol. 129 / 123

Fig. 4 Hardness versus depth of indentation and normalized hardness square versus inverse of indentation depth for Specimen 1-Solder 3 batch test Number 2. The data has already stabilized to an average value of 0.18 GPa. In order to determine the strain gradient plasticity length scale we have plotted the data corresponding to H/H 0 2 versus 1/h and performed gross curve fitting to determine the parameter h * in Eq. 5. Selected results are shown in Figs. 3 7 together with the corresponding fitting parameter. When plotted in this form the data follow the linear trend from Nix and Gao 18 and corresponding to =2 in Abu-Al Rub and Voyiadjis 20. Considering all our experimental results we have calculated an average length scale l=301.3 nm 0.30 m. This value of the length scale can have implications on applications when typical structural dimensions approach the micron scale. In particular, solder joints can experience low-cycle fatigue failure due to the coefficient of thermal expansion mismatch between the soldered parts even at room temperature. The 124 / Vol. 129, JUNE 2007 Transactions of the ASME

Fig. 5 Hardness versus depth of indentation and normalized hardness square versus inverse of indentation depth for Specimen 1-Solder 7 batch test Number 1. The data additional strength when present at small scales could be translated into longer fatigue life of the components. This incidence will be reported elsewhere. Finally, the scatter in the hardness data is expected due to voids and preexisting defects in the solders; however, the experimental results are very consistent and we are confident that the value of the length scale found is a good approximation of the material parameter. Conclusions Size dependent response of microelectronics Pb/ Sn solder joints has been studied using nanoindentation. A normalized hardness on the order of 1.55 has been measured which may be compared to the values of 3.0 and 2.0 corresponding to single crystal and polycrystalline copper, respectively. Using a method recently proposed by Abu Al-Rub and Voyiadjis 20 and based on the Journal of Electronic Packaging JUNE 2007, Vol. 129 / 125

Fig. 6 Hardness versus depth of indentation and normalized hardness square versus inverse of indentation depth for Specimen 1-Solder 1 batch test Number 1. The data work of Nix and Gao 18 we found an average value of the plastic length scale of the material to be l=301.3 nm which is of the same order of magnitude as values found for other metallic materials. We also verified that the material studied closely follows the linear trend corresponding to =2, which describes the coupling between the equivalent plastic strain p a measure of statistically stored dislocations and the gradient term a measure of the geometrically necessary dislocations in the work of Nix and Gao 18. Based on experimental data presented in this paper and computer simulations presented in Gomez and Basaran 8,9 it is safe to assert that length scale should be considered in modeling solder alloy if the solder joint is to be used in below room temperatures. Above room temperature most solder alloys are in 0.4 T m or at 126 / Vol. 129, JUNE 2007 Transactions of the ASME

Fig. 7 Hardness versus depth of indentation and normalized hardness square versus inverse of indentation depth for Specimen 1-Solder 4 batch test Number 1. The data higher and geometrically necessary dislocations are easily annihilated by the recovery and recrystalization process. The methodology used in this paper for determining length scale is strictly limited to metals and alloys. Results presented in this paper are for the strain rate used and room temperature. Length scale is known to be strongly temperature and strain rate dependent, hence these results would not be applicable to higher temperatures or different strain rates. Acknowledgment This project was partially funded by a grant from the National Science Foundation Civil and Mechanical Systems Division, Material Design, and Surface Engineering Program and partially by a grant from the US Navy ONR Advanced Electrical Power Systems. The authors gratefully acknowledge help received from Dr. Hua Ye during nanoindentation testing. Journal of Electronic Packaging JUNE 2007, Vol. 129 / 127

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