# Inelastic Strains in Solder Joint Interconnections

*“Everyone knows that we live in the era of engineering, however, he rarely realizes that literally all our engineering is based on mathematics and physics."*

—Bartel Leendert van der Waerden, Dutch mathematician

## Background/Motivation

Solder materials are widely used in microelectronics and optoelectronics. Tin- and tin-lead-based solders have been used in radio engineering to provide electrical connection since the early 1920s. Since the mid-1980s, tin-lead-eutectic-based solder joint interconnections (SJI) were and still are used also as a mechanical support in flip-chip designs (first level interconnections). Ball-grid array (BGA) and column- grid array (CGA) interconnections are also used to attach a package to its substrate (second level of interconnections). SJIs are the bottleneck of electronic and photonic packaging reliability. This is mostly because they are typically subjected to low- cycle fatigue conditions, such as experiencing inelastic deformations in operation conditions, not to mention accelerated testing, such as temperature cycling or drop tests. Inelastic strains in solder joints have been addressed by numerous investigators (see, for example, [1-10]).

Interfacial thermal stresses, whether elastic or inelastic, concentrate at the peripheral portions of solderly bonded assemblies. Figure 4.1 explains the physics of this phenomenon. Consider a bimaterial bonded assembly manufactured at an elevated temperature and subsequently cooled dowrn to a low (operation or testing) temperature. The adherend (bonded) element in the midportion of an assembly component is subjected either to tension [the component with a higher coefficient of thermal expansion (CTE)] or compression (the component with a lower GTE), and the induced forces acting on this element are more or less the same on both sides of the element. The situation is different for an element located at the assembly end. There are no external forces acting on it from the outside. Because of that, the force acting on the inner side of the element has to be equilibrated by the other assembly component through the assembly interface. This circumstance leads to the interfacial shearing stress at the assembly end. But that is not all. The force acting in the component’s body and the interfacial force act in different planes and form a bending moment and, since there is no external bending moment acting at the end of the component, this bending moment has to be equilibrated by the other assembly component through the interface. This circumstance causes the stress acting in the through-thickness direction of the assembly—the pealing stress. The interfacial shearing stress is antisymmetric: it is zero in the mid-cross section of the assembly

FIGURE 4.1 Thermally induced interfacial stresses concentrate at the bonded assembly ends.

and acts in the opposite directions at the assembly “halves.” The peeling stress, however, has to be self-equilibrated, because there are no external forces acting on the assembly in its through-thickness direction. This means that the areas of the peeling stress diagram above and below the interface should be equal (although they are, as a rule, configured differently: the portion of the peeling stress diagram at the assembly end is “sharper” than its inner portion). The peeling stress is symmetric with respect to the mean cross section of the assembly. Since the interfacial stresses, both shearing and peeling, are the largest at the assembly ends, it is the peripheral portions of a soldered assembly that are most likely to experience plastic deformations, and this happens if the maximum thermally induced strain exceeds the yield stress of the bonding material—the solder. The low-cycle fatigue conditions, when a soldered assembly is subjected to temperature cycling during testing or in actual operation, make such a bonding material vulnerable and thereby responsible for the fatigue strength of the assembly.

As to the effect of the assembly size (Figure 4.2), it has been shown [11-14] that

this effect depends not only on this size per se, but also on the parameter *к =*

the interfacial shearing stress. This parameter, as this formula indicates, increases with an increase in the axial compliance *X* of the adherends and decreases with an increase in its interfacial compliance к. That is why a reliable adhesively bonded or

FIGURE 4.2 Effect of the assembly size on the interfacial shearing stress.

soldered assembly is the one that is characterized by stiff adherends and a compliant bond. When the product *kl* is above 3.5-4.0, the further increase in the assembly size does not affect the level and the distribution of the stresses: the stress fields at the end portions of long and/or stiff enough assemblies do not interact. Mathematically (see Figure 4.2) this circumstance manifests itself for the interfacial shearing stress through the factor of tanh *kl,* that becomes equal to one for *kl* values exceeding 3.5-4.0.

In the subsequent analysis, we develop an approximate analytical model for the assessment of interfacial stresses in a bimaterial soldered assembly with a low-yield stress of the bonding material. The analysis has been carried out initially for a photonic assembly, in which a tin-silver solder was used to attach a vulnerable GaAs photonic chip to a copper submount (Figure 4.3) [7] and was extended later to BGA and other SJI designs [8-11].

The analysis is carried out under the major assumption that the bonding material is linearly elastic at the strain level below its yield strain and is ideally plastic at the levels exceeding this strain. It is clear that the previously obtained elastic solution (see, for example, [12-14]), on one hand, and the present ideally-elastic/ideally- plastic solution, on the other, addresses the two extreme cases in the behavior of the bonding material. The more general and a more realistic situation, when the bonding material experiences elasto-plastic deformations above the yield point, is beyond the scope of the present analysis.

FIGURE 4.3 GaAs photonic chip mounted on a copper submount (substrate) using tin-silver solder, characterized by a low-yield strain. (From E. Suhir, “Interfacial Thermal Stresses in a Bi-Material Assembly with a Low-Yield-Stress Bonding Layer,” Journal of Applied Physics-D, Modeling and Simulation in Materials Science and Engineering, vol. 14, 2006.)

## Assumptions

The following major assumptions are used in this analysis:

- • Only the longitudinal cross section of the package-substrate assembly can be considered.
- • The bonded components (the chip and the submount/substrate) can be treated, from the standpoint of stress/structural analysis, as elongated rectangular plates that experience linear elastic deformations.
- • Approximate methods of structural analysis (strength-of-materials) and materials physics, rather than rigorous methods of elasticity and plasticity, can be used to evaluate stresses and displacements.
- • At least one of the assembly components (substrate/submount) is thick and stiff enough, so that this component and the assembly as a whole do not experience bending deformations. The thinner component, the chip, might experience, however, some bending with respect to the thicker component.
- • The bonding material (solder) behaves in a linearly elastic fashion, when the induced shearing strain is below its yield point, and is ideally plastic, when this strain exceeds the materials yield strain.

• The yield stress in shear, *x _{Y},* if unknown, can be assessed from the measured yield stress in tension, g

*by the von Mises formula*

_{y},

- • The interfacial shearing stresses can be evaluated without considering the effect of peeling; the peeling stress can be determined, if necessary, from the evaluated interfacial shearing stress.
- • The peeling stress is proportional to the deflections of the thinner component of the assembly—the chip, with respect to the thicker component—the submount.

## Shearing Stress

### Basic Equation

Let an elongated soldered bimaterial assembly (Figure 4.3) be manufactured at an elevated temperature and subsequently cooled down to a low (room or testing) temperature. In an approximate analysis, the longitudinal interfacial displacements, к, (.V) and *u _{2}(x),* of the adherends (the assembly components) can be sought, within the assembly’s elastic midportion,

*-x,* (±x* are the abscissas of the boundaries of the elastic midportion of the assembly), in the form [12]:

where cq and a_{2} are the coefficients of thermal expansion (contraction) of the materials, *At* is the change in temperature,

are the longitudinal axial compliances of the assembly components, and *h _{2}* are the component thicknesses (in accordance with one of our assumptions, the thickness,

*h*of the thicker component—the submount—is significantly greater than the thickness, /г

_{2},_{ь}of the thinner component), £| and

*E*are the Young’s moduli of the component materials, v, and v

_{2}_{2}are their Poisson’s ratios,

are the interfacial compliances of the assembly components, G| and *G _{2}* are the shear moduli of the component materials,

*x(x)*is the interfacial shearing stress,

are the thermally induced forces acting in the cross sections of the assembly components, *x _{Y}* is the yield stress of the bonding material, and /«is the length of the plastic zone at the ends of the assembly. The length, /*, can be defined as /* =

*l-x*,*where / is half the assembly length. The origin, 0, of the coordinate, л:, is in the mid-cross section of the assembly.

The first terms in the right parts of the expressions (4.2) are unrestricted (stress- free) displacements. The second terms determine the displacements due to the thermally induced forces, *T(x*), that arise in the cross sections of the assembly components, because of the thermal contraction mismatch of the dissimilar materials of the soldered components. These terms are defined based on Hooke’s law assuming that all the points of the given cross section have the same longitudinal displacements, that is, the assembly’s cross sections remain flat (undistorted) despite the change in the states of stress and strain. The third terms in the right part of equations (4.2) account for the inaccuracy of such an assumption and consider the fact that the interfacial displacements are somewhat larger than the displacements of the inner points of the cross sections. The structure of these additional terms reflects an assumption that the displacements, which are responsible for the distortion in the planarity of the component’s cross section, are proportional to the interfacial shearing stress acting in this cross section (Figure 4.4). It is also assumed that these additional displacements are not affected by the stresses and strains in the adjacent cross sections and can be assessed as the product of the interfacial compliance of the

FIGURE 4.4 The longitudinal displacements of the given cross section of the adherend are

somewhat larger than the displacements of the inner points of the cross section.

assembly component (which is known in advance) and the sought induced shearing stress acting in this cross section.

While the structural analysis approach is used in this book for the evaluation of stresses and displacements, the coefficients of proportionality (interfacial compliances) between the interfacial displacements and the interfacial shearing stresses are evaluated on the basis of the theory of elasticity solution (see, for example [38]). This solution was obtained using Ribiere treatment of the problems for long-and-narrow strips subjected to the distributed shearing loads applied to one or to both of their long edges.

The condition of the compatibility of the interfacial displacements, *щ(х)* and *u _{2}(x),* can be written, considering the compliance

of the bonding layer, as

Here, *E _{f)}* and v

_{0}are the elastic constants of the bonding material (the solder below the yield stress), and /г

_{0}is the thickness of the bonding layer. Introducing formulas (4.2) into the compatibility condition (4.7), we obtain the following basic integral equation for the shearing stress function, x(x), in the elastic midportion of the assembly:

Here, Да = а_{2} -ai is the thermal expansion (contraction) mismatch of the materials of the soldered components (the adherends), X = X| + X_{2} is the total longitudinal axial compliance of the assembly, and к = к_{0} + к, + к_{2} is its total longitudinal interfacial compliance. It is noteworthy that, in the case of a thin and/or low modulus bonding layer, only the two soldered components (the adherends) determine the axial compliance of the assembly. As to the interfacial compliance, both the soldered components (the adherends) and the bonding layer (“adhesive”) contribute to the interfacial compliance: the role of a thin and low modulus bonding layer is typically comparable with the role of thick and high modulus bonded components (“adherends”), as one could see from the numerical example at the end of this analysis (see also Chapter 2).

### Boundary Conditions

In the case when plastic strains occur in the bonding material, the following conditions must be fulfilled at the boundary, *x = x*,* between the “inner” (linearly elastic) and the “outer” (ideally plastic) zones:

The first condition in (4.9) indicates that the shearing stress at the boundary between the elastic and the plastic zones must be equal to the yield stress of the bonding material. The second condition follows from formula (4.5): the shearing stress, t(x), is self-equilibrated, and therefore the integral in (4.5) is zero forx = *x*.* Physically, this condition is due to the fact that, since the interfacial shearing stress at the peripheral portions of the assembly is constant (is equal to the yield stress, *x _{Y}),* the thermally induced force

*T(x)*changes linearly at these portions, from

*x*at the boundary of the elastic and the plastic zones, to zero at the assembly ends. The sign “minus” in front of the second boundary condition in (4.9) indicates that the force at the boundary should be compressive (negative) for the compressed component of the assembly. In the case of a purely elastic state of strain (/* = 0), the following boundary condition should be fulfilled:

_{Y}U,

This condition reflects the fact that there are no external longitudinal forces acting at the end cross sections of the assembly components.

### Elasto-Plastic Solution

From (4.8), we find, by differentiation with respect to the coordinate, *x:*

The next differentiation, considering the relationship (4.5), yields where

is the parameter of the interfacial shearing stress. Equation (4.11) has the following solution in the elastic midportion of the assembly:

It is clear that this solution satisfies the first condition in (4.9). Introducing sought solution (4.14) into formula (4.5), we conclude that the second condition in (4.9) is also satisfied. Introducing solution (4.14) into basic equation (4.8), we find that the

*и*

relative length *—* of the plastic zone could be determined from the following transcendental equation:

where

is the maximum elastic interfacial shearing stress at the end of an infinitely long assembly [11-14]. As evident from equation (4.15), no plastic zones could possibly

T°°

occur (/* < 0), if the ratio of the maximum elastic shearing stress in an infinitely

*Ту*

long assembly to the yield stress of the bonding material is equal or smaller than coth *kl:*

Indeed, for long (large / values) and/or stiff (large *к* values) assemblies, when coth *kl* could be considered equal to one, condition (4.17) is equivalent to the requirement that the yield stress is simply larger than the maximum elastic interfacial shearing stress. In such a situation, no plastic stresses could possibly occur. If the *kl* value is small, then condition (4.17) yields

Thus, no plastic deformations could possibly occur, if, in a short and/or compliant assembly, condition (4.18) is fulfilled, that is, if the yield stress *T _{y}* is high, the assembly compliance к in the denominator in the right part of condition (4.18) is significant, the thermal strain ДаД? in the numerator is low, and the size, /, of the assembly is small.

Equation (4.15), if solved for the lengths ratio in the parentheses, yields

T°°

If the yield stress *T _{y}* is low' and, for this reason, the stress ratio —^ is significantly

larger than one, then, as evident from (4.19), *l* =* /, such as the entire interfacial zone is occupied by the plastic strains (stresses), regardless of whether the *kl* value is large or small. If the above stress ratio is significantly smaller than one, then equation (4.19) yields

As evident from this equation, plastic deformations might still take place if the *к *value is large, despite the low / value.

### Possible Numerical Procedure for Solving the Elasto-Plastic Equations

The transcendental equations (4.15) and (4.19) can be solved numerically. Let, for

T°°

example, the stress ratio be —^ = 2, and the parameter *kl* is also *kl =* 2. Then, equa- tion (4.15) yields

*U*

This equation has the following solution: *—* = 0.4002. This result could be obtained

by simply assuming different length ratios, plotting the function in the right part of equation (4.21) and accepting, as a solution, the length ratio, which is equal to the computed value of this function in the left part of the equation. The same result can be obtained from equation (4.19). Another approach is to employ a rapidly converging iterative numerical procedure based on the well-known Newton’s formula

for solving a transcendental equation/(x) = 0. The formula of the (4.22) type can be obtained from equation (4.15) as follows:

where the notation

is used. For = 2 and *kl* = 2, assuming, in the zero approximation, ^y j = 0.5, we obtain:

No iterations are necessary for sufficiently long and stiff assemblies, characterized by large *kl* values, since the hyperbolic cotangent wall be equal to “one” for any (but large enough) value of the argument. Indeed, let *kl* be equal to *kl =* 10. Then, assuming that the у ratio is significantly smaller than one, we have z* ~ 1 and equation (4.23) yields

### Predicted Lengths of the Plastic Zones Based on an Elastic Solution

We proceed from equation (4.12) and seek its elastic solution in the form similar to (4.14):

Introducing (4.25) into equation (4.12), we conclude that the following relationships must be fulfilled:

The first formula in (4.26) is the same as formula (4.13). This is because formula (4.13) defines the parameter *к* of the elastic interfacial shearing stress. With formulas (4.26), solution (4.25) yields

where t",_{ax} is the maximum shearing stress at the end of a long and/or stiff assembly (£/—»<*>). This stress is expressed by formula (4.16). Putting *x _{Y}* = t(x<) in formula

/* *X**

(4.27), we obtain the following formula for the relative length — = 1- — of the plastic zone:

where

For = 2 and *kl =* 2, formula (4.28) yields — = 0.3054. Comparing this value

*Ту l*

*L*

with the value, *—* = 0.4002 obtained using the elasto-plastic solution, we conclude

that in the case in question the prediction based on an elastic solution underestimates considerably (by about 24%) the length of the plastic zones. The underestimation is even greater (about 80%) in the numerical example carried out in the last section of this analysis.