Eindhoven University of Technology

MASTER

Thermomechanical behaviour of solder joints in electronic packages

van Houten, J.G.A.

Award date:1995

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Download date: 03. Jul. 2018

Thermomechanical behaviour of solder joints in electronic packages

J.G.A. van Houten

EUT, Faculty of Mechanical Engineering Report No. WFW 95.114

Master’s thesis

Professor: Prof.dr.ir H.E.H. Meijer

Coaches: Dr.ir. P.J.G. Schreurs (Eindhoven University of Technology) Ir. J.J.M. de Bever (Philips CFT)

Eindhoven, August 1995

Eindhoven University of Technology (EUT) Faculty of Mechanical Engineering Department of Fundamentals of Mechanical Engineering

Abstract

In many electronic packages solder joints are used to bond different materials together. Solder joints in electronic packages are subjected to thermally induced stresses due to a mismatch of thermal expansion coefficients of the materials bonded together. Because the thermomechanical behaviour of solder alloys significantly effects the thermally induced stresses in a package, constitutive models are investigated which describe the thermomechanical behaviour of solder alloys. It appeared that an elastic-plastic-creep constitutive model (extended Maxwell model) is the most widely used.

An extended Maxwell model is chosen to characterize the temperature- and time-dependent mechanical behaviour of the soft solder alloys PbSn5 and SnAg25SblO. The parameters in the extended Maxwell model are determined by uniaxial tensile and stress relaxation tests of solder wire at several temperatures. The extended Maxwell model is used for simulations with the finite element code MARC.

First, the reliability of the material parameters of both solder alloys is investigated by simulating the stress relaxation tests. The stress relaxation tests are simulated at a satisfactory level.

Second, the cooling down phase after a die (chip) is soldered onto a heatsink is simulated. Simulations are performed using an elastic-plastic constitutive model for both solder alloys by omitting the time- dependent creep behaviour of the extended Maxwell model. Furthermore, simulations are performed with the complete extended Maxwell model for SnAg25Sbl O. Simulations with the complete extended Maxwell model for PbSn5 failed due to numerical problems.

For SnAg25SblO the results of the simulations are compared with measurements of the radii of the die and heatsink 48 hours after die-attachment and with stress measurements in the top of the die. It can be concluded that the usage of an extended Maxwell model yields a better prediction of the radii of curvature than usage of an elastic-plastic constitutive model. Furthermore, the stresses do not match the numerical results. The latter may be caused by both measuring errors and modelling errors. In order to obtain a predictive constitutive model more reliable experiments have to be performed and compared with simulations. Moreover, the influence of the intermetallic layers should be investigated.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 BhyisiCa! prQb!em . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Thermomechanical modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Thermal loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3.1 Thermal loading during assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3.2 Thermal loading during thermal tests 3.3.3 Thermal loading during lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . 9 11

4 Mechanical behaviour of materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2 Constitutive models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.2.1 Linear elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2.2 Elasto-plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2.3 Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2.4 Elasto-visco-plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.3 Numerical procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3.2 Linear elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.3.3 Maxwell, extended Maxwell, and Bingham model . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction 13

22

5 Constitutive models for solder in literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Maxwell and extended Maxwell models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 27 28

5.2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 39 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2 Experiments for constitutive model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.2.2 Tensile tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.2.3 Stress relaxation tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.2.4 Analysis of stress relaxation data . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3 Experiments for verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 6.3.2 Radius of curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.3.3 Stresses in test chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

43 48

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7 Finite element simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7.2 Simulation of stress relaxation tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7.3 Simulation of cooling down phase of die-attachment process . . . . . . . . . . . . . . 54

8 Lifetime prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

9 Conclusions and recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

A Verification of the Maxwell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

B Experiments in literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 B.l Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 B.2 Specimen design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 B.3 Mechanical test conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Thermal. temperature controlled cycling . . . . . . . . . . . . . . . . . . . . . . B.3.2 Thermal. temperature and strain controlled cycling . . . . . . . . . . . . . . 78 B.3.1 77

C Measurement Focus System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

D Stress measinrememts with test chip TP7P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $1

E Elastic-plastic data in MARC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

F Finite element mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Chapter 1

Introduction In the electronic industry, most attention has always been on the electronic functionality of components. During the last decades, reliability of electronic packages has become an important issue in the electronic packaging industry. Packaging is a general term involving all levels of electronic assemblies. Reliability of electronic packages is critical because they are used to control operational and safety functions in aerospace, nuclear plants, telecommunications, consumer electronics and numerous other applications.

The research project described in this thesis is part of a long-term research program on the reliability of Discrete Element (DE) packages. The research program mainly concerns the thermomechanical reliability of these electronic packages, and is performed at Philips Semiconductors in cooperation with Philips Centre for Manufacturing Technology, both settled in the Netherlands. In many electronic packages, solder alloys are frequently used to bond different materials together. The thermomechanical behaviour of solder alloys appeared to have a significant effect on the thermally induced stresses in a package. Solder joints in electronic packages are subjected to thermally induced stresses due to a mismatch of thermal expansion coefficients of different materials bonded together. During assembly, testing, and lifetime, the electronic package continuously experiences temperature changes. A fast and large temperature change could induce a stress larger than the ultimate strength of the weakest component in the electronic package, resulting in crack initiation or rupture of this component. If the temperature change is small, no instantaneous crack initiation will occur. However, many temperature changes (thermal cycling) result in thermomechanical low-cycle fatigue of the solder joint.

As technology advanced, the size of solder joints became smaller and smaller, but reliability concerns increased exponentially. If the solder joint is both the mechanical and electrical interconnection, for example in surface mount technology (SMT), the failure of a single joint could put an entire electronic package out of operation. It is essential that the solder joints survive the package’s projected lifetime. Thus an accurate prediction of lifetime is essential. The most accurate method to determine solder joint reliability is to build a statistically significant number of packages and subject them to actual environments and determine when the joints fail. These experiments require fabrication of special test specimens and test equipment. Therefore, this method is geometry and material dependent, extremely time consuming and expensive. Thermomechanical modelling of solder joints offers the potential to predict the effects of different solder joint geometries and solder alloys on the reliability of solder joints. Prediction of reliability is important, because the costs of changing an assembly are very high.

The primary objective of the research project described in this thesis is to determine a constitutive model which is capable of describing the thermomechanical behaviour of solder alloys. The constitutive model can be used for finite element modelling of solder joints in DE packages, but also for solder joints in other electronic packages such as Integrated Circuit (IC) packages. The constitutive model has to be suitable for implementation in the finite element code MARC (MARC, 1994). The theory discussed in this thesis can be applied to many solder alloys.

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In chapter 2 we start with a discussion of the components and materials in an example of an electronic package, with emphasize on solder alloys and intermetallic layers. Chapter 3 deals with thermomechanical modelling of electronic packages. After a short general introduction, the geometry and the sources of thermal loading will be discussed. The mechanical behaviour of materials in general, and the implementation in the finite element code MARC are outlined in chapter 4. This chapter provides a background for the mechanical behaviour of solder alloys to be discussed in the next chapters. In chapter 5 constitutive models for soft solders in literature are discussed and summarized. In chapter 6 the (soft) solder alloys PbSn5 and SnAg25SblO (J-alloy) are modeled using one- dimensional experiments. The constitutive model obtained with these experiments serves as input for finite element simulations to be performed in chapter 7. Furthermore, supplementary experiments are described with a die (chip) soldered oiito a heatsink to verify the numerical results ~f the finite elenlent simulations. In chapter 7, finite element simulations are performed of the one-dimensional experiments to verify the constitutive model, and of the cooling down phase after a die (chip) is soldered onto a heatsink. Finally, chapter 8 deals with some theoretical aspects of lifetime prediction.

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Chapter 2

Physical problem

The mechanical response of solder cannot be isolated from the other components in an electronic package. Therefore the entire electronic package has to be considered. Electronic packages are designed in numerous different assemblies. For example, Discrete Element (DE), and Integrated Circuit (IC) packages. DE and IC packages are also designed in many different assemblies. A cross-section of a typical DE package is shown in figure 2.1. It consists of a die which is fixed onto a heatsink (die pad) by solder. For DE packages usually soft solders are used, such as PbSnS and SnAg25SblO. The solder together with the heatsink serves the purposes of heat dissipation and mechanical support. The bonding pads on the die are connected to the lead frames with thin bond wires. In figure 2.1 only one lead frame is shown. The lead frames serve first as holding fixture during the assembly process. Then, they provide the electrical connections between the die and Printed Circuit Board (PCB). The die and heatsink are encapsulated in plastic by transfer moulding. The plastic serves as protection of the die and bond wires against contamination and moisture penetration. The heatsink and the lead frames are soldered onto the metallizations of the PCB.

Bond wire L a d frame I Die /,f---- ___--

Plastic __

Solder \ \ -b i Solder

Figure 2.1: A cross-section of a typical DE package.

For electronic packages in general, the die-bonding layer may consist of solder, metal filled epoxy, or glass (Lau, 1993). Each of these materials has its application advantages and disadvantages. Die-attach material requirements include high adhesion, high thermal conductivity, fatigue resistance, inexpensive, easily to process, low processing temperature, low Young's modulus, and low yield stress. A low Young's modulus, low yield stress, and low processing temperature of the die-attach material reduce the thermally induced stresses in the die and heatsink. In table 2.1 some examples of material properties of electronic packaging materials are listed which could be used in the DE package in figure 2.1. The temperatures Tso,, and qiq, are the solidification and liquidus temperature, respectively. The values of the Young's modulus E and the yield stress oy are given at room temperature (20 "C). Silicon is a ceramic material and shows brittle behaviour. The stress at fracture in tension and compression is

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equal to about 169 and 600 MPa, respectively.

Material

Silicon (die) ~~

DLP Cu (heatsink)

KMC125 (plastic)

PbSn5 (solder)

SnAg25Sb 1 O (solder)

Table 2.1: Some examples of material properties of electronic package materials.

The mismatch in thermal expansion coefficients of the materials in table 2.1 results in large thermally induced stresses, especially if the electronic package is subjected to fast and large temperature changes and the width of the die is large. For high power applications of the electronic package, the die- bonding layer usually consists of soft solder which reduces the thermally induced stresses.

In this section, the discussion of electronic package materials is restricted to soft and hard solders. Furthermore, the existence of intermetallic layers at the solderíheatsink interface is illustrated. In the next chapter, thermomechanical modelling of electronic packages is desribed.

Soft solders

The most commonly used soft solders are tin-lead (Sn-Pb) alloys. They are generally inexpensive, have acceptable thermal conductivity, and their yield stresses are low, usually less than 50 MPa (Lau, 1993a). The low yield stress of the solder and low stress during plastic deformation prevents high stresses to be build up in the electronic package. However, the capability of plastic deformation makes soft solders subject to thermal fatigue, resulting in cracks in the bonding layer. These cracks will propagate during thermal cycling. The mechanical properties of soft solders will be discussed in chapter 5.

Hard solders

Hard solders are usually low-melting gold eutectics that have high yield strengths. The yield strengths at room temperature are above 185 MPa. The drawbacks of hard solders are their high cost (due to the gold content), and high stresses during thermal cycling. The high stresses can potentially exceed the ultimate strength of the silicon chip and cause cracking. The advantage of hard solders is that they all have high thermal conductivity and are free from thermal fatigue because the high yield strengths result in elastic deformation. Some material properties of hard solders alloys are listed in table 2.2.

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Table 2.2: Material properties of hard solder alloys (Lau, 1993a).

Intermetallic layers

Another concern is the possible formation of intermetallic compounds. Intermetallic compounds are formed by chemical reactions between the solder and the heatsink (Marshall et al., 1994). The most common interfacial intermetallics are Cu-Sn and Ni-Sn compounds that are formed by reaction between tin in the solder and copper or nickel in the heatsink. The most thoroughly studied intermetallics are the Cu-Sn compounds that form when Sn-Pb solders wet copper. There are two common compounds: Cu,Sn and Cu,Sn5. Cu,Sn forms preferentially when there is an excess of copper. For example, if lead- rich solders are used, such as PbSn5, there is a small amount of tin and a large amount of copper at the solder/heatsink interface. In contradiction, Cu,Sn, forms in excess of tin. For example, if tin-rich solders are used, such as eutectic Pb-Sn (SnPb37). The material properties of intermetallic phases contrast significantly with those of the bulk solder and the heatsink. For example, the intermetallic phases are generally less ductile, and less thermally and electrically conductive. Some material properties of the intermetallic phases and base metals are listed in table 2.3.

~~

Vicker's hardness [Kgímm']

Young's modulus [GPa]

Thermal expansion [ 10-6/oC]

Heat capacity [J/gm/deg]

Resistivity [pohm-cm]

Density [gm/cc]

Thermal conductivity [Watt/cm/"C]

Property 1 378155

85.5611.65

1 6.310.3

0.286+0.012

17.510.1

8.2810.02

0.34110.051

343?47 I 30 I 100

108.3k4.4 1 117 I 41

19.010.3 1 17.1 I 23 ~~

0.32610012 1 0.385 I 0.227

8.9310.02

8.910.02

0.70410.098 3.98 0.67

Table 2.3: Material properties of intermetallic phases and base metals at room temperature (Marshall et al., 1994).

The Vicker's hardness is greater for the intermetallic phases than for the base metals, indicating their greater stiffness, yield stress, and work hardening. No plastic deformation of the intermetallic layer should be expected under normal stress levels in solder joints. One might predict that failure would occur at the intermetallic layer due to the brittleness. However, thermomechanical low-cycle fatigue and ultimate failure of a soft solder joint occurs through the bulk of the solder because the deformation occurs in the bulk solder. The failure may occur in the solder in proximity to the interface due to stress concentrations at or near the interface between solder and the intermetallic.

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The intermetallic can grow and increase in thickness. The growth rate is limited by either the reaction rate at the growth site (reaction controlled), or bulk diffusion of elements to the reaction interface (diffusion controlled). The growth rate strongly depends on temperature. If there is a thick intermetallic layer, the failure mechanism evolves from bulk failure to intermetallic failure. The intermetallic layer may play a role as the solder joint continues to miniaturize. A thin die-bonding layer with a relative thick intermetallic layer reduces the height of the bulk solder, resulting in more plastic deformation (larger shear strain) of the remaining solder. If the thickness of the intermetallic layer is low compared to the thickness of the solder, it’s influence will be negligible. For this reason, it will initially be neglected in thermomechanical modelling of solder joints. Once the intermetallic layer appears to effect the mechanical properties significantly, it should be included in thermomechanical modelling.

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Chapter 3

Thermomechanical modelling

3.1 Introduction Thermomechanical modelling of electronic packages reduces the physical problem to a mathematical model which can be solved to quantify stresses, strains and other relevant parameters. Thermomechanical modelling requires: the constitutive equations, the geometry, the loading conditions, and the boundary conditions. Most of the mathematical models require numerical solutions for which the finite element method is frequently used. Over the past twenty years several finite element models have been used to model solder joints in electronic packages. These models range from a two-dimensional solder joint geometry with linear elastic constitutive equations to three-dimensional models that account for both plasticity and creep (Morgan, 1994). The finite element results can be used as input for a life prediction model, as shown in figure 3.1:

Figure 3.1: Life prediction of electronic packages.

The mathematical model provides a tool for investigating the effect of different solder joint geometries and different solder materials to optimize the life time of an electronic package. A reliable life prediction of electronic packages requires an accurate finite element analysis of the physical problem. Mathematical modelling of a physical problem inevitable involves the introduction of a number of idealizations, especially with concern to the constitutive behaviour of the materials. The accuracy of a finite element analysis depends on:

1. Accuracy of constitutive model and material parameters 2. Accuracy of geometry, loading and boundary conditions 3. Accuracy of numerical procedure

The geometry and (thermal) loading are discussed in section 3.2 and 3.3, respectively. Constitutive

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models are outlined in chapter 4 and 5.

3.2 Geometry In figure 2.1 an example of an electronic package is depicted. The geometry of the die, heatsink, plastic, and lead frame are defined by the designer. However, the solder joint geometry is a function of many variables including the geometry of the parts being soldered, mechanical properties of the solder and materials bonded together, surface roughness, cleanliness, and a variety of other process variables (Heinrich, 1994). These variables can significantly effect the reliability of the solder joint. Several methodologies are used to specify the solder joint geometry:

1. Assume the geometry 2. Measure actual cross-sections 3. Predict the geometry

The first two methodologies are the most widely used. Most of the existing models for predicting solder joint geometry are based upon surface tension theory (Heinrich, 1994).

3.3 Thermal loading During assembly, temperature cycle testing and lifetime of the electronic package, the components of the electronic package are subjected to thermal loading.

3.3.1 Thermal loading during assembly

The first thermal loading during assembly is caused by the die-attachment process. The die-attachment process is represented in diagram form in figure 3.2. At room temperature, To, the die and heatsink are unattached. If we assume the thermal expansion coefficients of the materials which are listed in table 2.1, then the heatsink expands more than the die during heating. At temperature T,, the solder is in the liquid phase and the die and heatsink are stress free. During cooling down from the solidification temperature of the solder, thermally induced stresses develop in the assembly due to the different thermal expansion coefficients of the die, heatsink, and solder. Because the coefficient of thermal expansion of the heatsink is larger than the coefficient of thermal expansion of the silicon die, the heatsink imposes a compressive force on the die, and the die imposes a tensile force on the heatsink. This results in bending of the assembly. The stress in the die is composed of a compressive stress and a bending stress. The stress in the heatsink is composed of a tensile stress and a bending stress. The resulting stresses at the dieholder interface and solder/heatsink interface impose a shear stress on the solder layer. This results in a nonuniform stress distribution throughout the die and heatsink thickness.

The second thermal loading during assembly is caused by the encapsulation of the die and heatsink in plastic by injection moulding. We assume that the plastic has a higher thermal expansion coefficient than the die and heatsink, and the heatsink has a higher thermal expansion coefficient than the die. Then, the die and heatsink will experience a shear force and a compressive force in the cooling down phase after injection moulding, due to the shrinkage of the plastic. The shear force acts parallel at the interface between the die/heatsink and plastic. The compressive force acts perpendicular to these interfaces. The shear force at the upper and lower interface mainly determines the bending. The resulting deformation is illustrated in figure 3.3, where Tg is the glass transition temperature of the plastic, and To is the environmental temperature.

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T O

Tension/ compression Bending

IJR - t o - . + O

Compressive T d e stress stress

I O

Figure 3.2: Deformation and stresses during die-attachment. It is assumed that the thermal expansion coefficient of the heatsink is larger than the thermal expansion coefficient of the die.

The figures 3.3a and 3.3b both represent the die or heatsink. The position in the plastic encapsulation determines wether the die or heatsink bends downwards (3.3a), or upwards (3.3b). Figure 3 . 3 ~ shows a die which is attached to a heatsink. Without plastic encapsulation this assembly will bend down. If the die and heatsink are encapsulated in plastic, the die and heatsink tend to bend downwards or upwards, depending on the stiffness of the die and heatsink, the differences in thermal expansion coefficients, and the position in the plastic encapsulation. Even when the die and heatsink are below the symmetry-axis of the plastic, the die and heatsink may bend down.

T g

T O

snesses:

Figure 3.3: Deformation and stresses of a die orland heatsink which are encapsulated in plastic.

3.3.2 Thermal loading during thermal tests

Because there is no standard temperature application range for electronic packages, the solder joints must be subjected to the worst case conditions to ensure that the joints will survive the devices projected lifetime. During temperature cycle testing, the electronic package has to sustain a large number of cycles, for example 1000. The cycle frequency is much higher than the frequency the package experiences during lifetime and is called accelerated testing. Tests are accelerated to complete the test in a reasonable length of time. The goal of an accelerated test is to detect the failure of an electronic module in an efficient way, without creating unnecessary failure modes which do not occur

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in the field (Pan, 1994). There are three means by which thermal loading is imposed:

1. Temperature cycle test (TCT) 2. Thermal shock test (TST) 3. Power cycling

The three test procedures are briefly discussed below, without the intention to be complete.

Temperature cycle test

In figure 3.4 an example of a temperature-time profile is depicted.

Figure 3.4: A typical temperature-time profile for temperature cycle testing

It is characterized by the initial temperature, maximum temperature, minimum temperature, ramp speed [“C/min], and hold time [min] at the minimum and maximum temperature. The wave shape and the temperature range may depend on the solder joint application. Other temperature-time profiles are sine wave shaped or sawtooth shaped (hold time in figure 3.4 is zero). The characteristics of the temperature-time profile determine the thermally induced stresses and strains in the solder joint. One should be aware that solder alloys with different melting points have different homologous temperatures at the temperature extremes. The homologous temperature is defined as Thorn = T/T,, where T is the temperature and T, is the melting point (or solidification temperature), both measured in degrees Kelvin.

Thermal shock test

During a thermal shock test the solder joint is cycled between two thermal baths (liquid) at opposite temperature extremes, possibly with a hold time at the temperature extremes to relax the stresses. The temperature ramp speed [Wmin] and temperature range during a thermal shock test are much higher than the temperature ramp speed and range during a temperature cycle test. Therefore, thermal shock tests are particularly useful to determine instantaneous rupture of the solder or the materials bonded together.

Power cycling

During power cycling the power components are turned on and off, resulting in energy dissipation. The temperature-time profile approaches the real use environment of the power components.

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3.3.3 Thermal loading during lifetime

During lifetime, the electronic package experiences temperature cycles due to energy dissipation of electronic devices and fluctuations in environmental temperature. Environmental temperature fluctuations are composed of daily fluctuations (day, night) and annual fluctuations (summer, winter). These fluctuations can be small or large. For example, computers in a climate-controlled environment experience small fluctuations, whereas electronic components used in avionics experience large fluctuations. The frequency of the temperature cycles due to energy dissipation depends on the use environment. For example, laptop and notebook applications are more frequently turned on and off than mainframe computers.

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- 112-

Chapter 4

Mechanical behaviour of materials

4.1 Introduction This chapter provides a theoretical background for the mechanical behaviour of solder alloys to be discussed in the next chapters. The physical nonlinear behaviour of solder alloys and the complex geometry of solder joints require numerical solutions for which the finite element code MARC will be used (MARC, 1994). Physical nonlinear behaviour results from the nonlinear relationship between stresses and strains. The models for these stress-strain relations are based on experimental data. The mechanical behaviour of materials is mathematically modeled with a constitutive model. In section 4.2 the most common constitutive models will be briefly discussed. In section 4.3 the numerical procedure in MARC to solve some of these constitutive models will be outlined.

4.2 Constitutive models The mechanical behaviour of materials can be represented by a combination of one or more springs, dampers, and plastic elements, where a spring represents a perfectly elastic material, and a damper represents a perfectly viscous material. A plastic element is inactive, i.e. no plastic strains occur, when the stress CJ is less than the yield stress CY,, of the material. In the finite element code MARC, visco- elastic behaviour can be represented by a Maxwell or a Kelvin-Voigt model. Elastic, plastic and viscous behaviour is combined in an extended Maxwell model or a Bingham model. Elasto-plasticity is represented by a spring and plastic element. The five models are shown in figure 4.1.

Figure 4.1: Examples of constitutive models.

It is customary to denote the elastic strain and plastic strain by ze' and E ~ ' , respectively. The strain in the damper is defined as creep strain, eer. In a Kelvin model the elastic strain ie' is equal to the creep strain eer, and both strains are equal to the total strain E. The constitutive equation of a Kelvin model in tensor notation is given by:

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where (r is the Cauchy stress tensor, 4C the fourth order constant elasticity tensor, and a dot denotes the time derivative. In the other models, the total strain is thought to be composed of an elastic and an inelastic strain component. The constitutive equation in tensor notation is given by:

where the inelastic strain rate depends on the type of model:

& Maxwell model

Bingham model Elastic - plastic model

Extended Maxwell model (4.3) cr + E P ~

&pl= &cr ~ & vp 1 & Pl

E hel =

The dot on the plastic strain does not represent a real time, but a virtual time. The question arises how a finite element code deals with the extended Maxwell model, which combines a real and a virtual time. The finite element code MARC seperates the elastic-plastic and creep calculation. This will be explained in more detail in section 4.3.

In a Maxwell model and an extended Maxwell model, the stress acting on each component (spring, damper, plastic element) is equal, whereas the strains are different. In order to determine the material parameters for a Maxwell model, it must be possible to split elastic and creep strains in the experimental test. For an extended Maxwell model it must be possible to split elastic, creep and plastic strains. In the Bingham model it is assumed that the plastic and creep strains are equal in magnitude, and these strains are defined as visco-plastic strain, E"~. The Maxwell model and the extended Maxwell model allow the material to creep at any stress level, whereas the Bingham model allows the material to creep only if a specific stress level is exceeded. The Bingham model can be transformed in a Maxwell model by setting the yield stress equal to zero. Degeneration into an elastic-plastic model is obtained by specification of a very high creep strain rate. The extended Maxwell model can be transformed into a Maxwell model by defining a very high yield stress.

In the subsequent sections the theory of linear elasticity, elasto-plasticity, creep (creep strain rate in Maxwell and extended Maxwell model), and elasto-visco-plasticity (Bingham model) will be briefly outlined. Although the Bingham model is also capable of describing creep behaviour, this model will be described in a seperate section. Because time-dependent material behaviour has special attention in this thesis, the implementation in MARC of the creep strain rate and the visco-plastic strain rate will be described.

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4.2.1 Linear elasticity

Linear elastic deformation is fully recoverable, time-independent deformation, with a linear relationship between stress and strain. As an example, the stress-strain relation for an isotropic linear elastic compressible material is given:

l7 Y - -

(1 +v)( 1 -2v)

l-v v v O O O

v 1-v v O O O

v v 1-v O O O O O O OS(1-2~) O O O 0 0 O 0.5(1-2~) O

O 0 0 O O OS(1-2~)

(4.4)

where E is the Young’s modulus and v is the Poisson’s ratio. The engineering shear strain yxy (yyz, yzx) should not be confused with the shear strain E,, (cy,, E~J:

Y, =2Eq (4.5)

4.2.2 Elas to- plasticity

Elasto-plasticity is defined as time-independent inelastic behaviour. During plastic deformation, permanent deformations are supposed to take place instantaneously. In reality, all deformations require a finite time. Plastic deformations simply refer to those whose characteristic times are orders of magnitude smaller than for example creep deformations. If the stress is below the yield stress oy of the material, the material behaves elastically and the stress will be proportional to the strain. If the stress is higher than the yield stress, plastic deformation takes place. Plasticity is characterized by the yield stress, and work hardening behaviour.

Elasto-plasticity in MARC

In MARC, the work hardening behaviour may be entered piecewise linear. MARC requires the input of the entire stress-strain curve at the lowest temperature during the simulation. The stress-strain curves at higher temperatures are defined by their yield stress and Young’s modulus with the accompanying temperature. The work hardening behaviour is not entered seperately at each temperature T, but as a ratio R of the work hardening slope Hp at the lowest temperature To:

HPU) =R(T).Hp(To) (4.6)

where the work hardening slope is defined by:

Because solder joints may be subjected to cyclic loading, it is worthwhile to describe some examples of work hardening behaviour under cyclic loading (tension, compression). In figure 4.2, the

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characteristics of isotropic work hardening (left) and kinematic work hardening (right) are illustrated (MARC, 1994). In this figure a constant work hardening slope H,, is assumed. The increasing numbers correspond with loading paths.

The isotropic work hardening rule states that the absolute value of the yield stress at reverse loading, o*,, is equal to the yield stress at loading, o,. The same holds for o,, and o,, respectively.

The kinematic work hardening rule states that the yield stress at reverse loading, o,, is equal to 02-20y, where oy is the initial yield stress (oy = o,). Loading from point 4 to point 5 shows that o, = 04+20y. Kinematic strain hardening implies that the difference in yield stress will always be equal to 2oy.

O

Y U

a -

U Y

C -

t

Figure 4.2: Isotropic work hardening (left) and kinematic work hardening (right).

4.2.3 Creep

Creep is defined as time-dependent inelastic strain under constant load and elevated temperature. For metals, elevated temperature begins at a homologous temperature (TIT,) of about 0.5. For many solder alloys, creep deformation becomes important even at room temperature due to the low melting points. For example, the melting point T, for PbSn5 is 578 K (305 OC). At room temperature (T=293 K), the homologous temperature for this solder alloy is 0.51. Creep behaviour depends on the combination of time, temperature, stress, strain, material properties, and chemical environment (Boresi, 1994). Creep behaviour includes the phenomenon of stress relaxation. Stress relaxation occurs in strain controlled loading cases. In a stress relaxation test a specimen is pulled to a predetermined strain, whereafter the displacement is fixed and the stress will relax. Creep curves are ordinarily obtained by uniaxial tension of bars at constant load, and presented as strain versus time curves. In figure 4.3, three characteristic regions can be distinguished:

I. Primary creep (continuously decreasing strain rate) 11. Secondary creep (constant strain rate) III. Tertiary creep (continuously increasing strain rate)

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t

o' * t Time

Figure 4.3: Creep behaviour at constant load and temperature.

The initial strain consists of either elastic strain, or partially elastic strain and partially plastic strain. During primary creep, the strain rate decreases because the effect of strain hardening is larger than the effect of increasing true stress. The true stress increases because the cross-sectional area decreases. The minimum creep rate occurs during the secondary creep stage, where these effects are in balance. If creep is maintained for a sufficiently long time, the strain rate increases rapidly and creep fracture occurs. The long term creep strength of solder is very low. Andrade (in Lau, 1993) considered primary and secondary creep as a superposition of transient creep (with a creep rate decreasing with time) and steady state creep (with a contant creep rate). Andrade's analysis is illustrated in figure 4.4:

, I _ , _ n I andsteadystate

steady state creep

Transient creep

4

Initial strain O t

Time

Figure 4.4: Andrade's analysis of primary and secondary creep

Because of the complexity of creep behaviour, mathematical modelling is often based on curve-fitting of experimental creep data. The equivalent creep strain rate is usually assumed to be dependent on the equivalent stress, equivalent creep strain, temperature, and time:

i "= S( O, E T,t) (4.8)

where O, E ", T , t are the equivalent stress, equivalent creep strain, temperature, and time,

respectively. TRe formula may be applied to multiaxial stress states through the use of the concept of

- 1 7 -

effective stress and effective strain. To model creep curves, it is customary to assume that the effects of stress o, creep strain ecr, temperature T, and time t are separable (MARC, 1994):

(4.9)

where A is a constant. Usually, one-dimensional experiments are performed allowing only one of the variables to change.

Creep strain rate in MARC

In this section the creep strain rate in MARC of the Maxwell model and extended Maxwell model will be described. Constitutive modelling of creep behaviour requires the introduction of a history parameter: the equivalent creep strain. The equivalent creep strain does not have a simply defined amplitude, but shows cumulative behaviour:

(4.10)

where the equivalent creep strain rate is obtained by transforming a multiaxial creep strain rate, expressed in a tensor, to a single scalar value:

-EC': ECr iicr =ii 3

The equivalent Von Mises stress is defined similarly by:

o=\/io":o" 2

where d is the deviatoric part of the Cauchy stress tensor, defined as:

od = (5 - ; tr((5)I

(4.11)

(4.12)

(4.13)

The equivalent creep strain and equivalent Von Mises stress can be expressed in the six independent components of the creep strain tensor and Cauchy stress tensor, respectively:

(4.14)

where the creep indexcr is omitted. Creep behaviour in MARC is based on a Von Mises creep potential

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with isotropic behaviour described by the creep strain rate:

(4.15)

where %/do Mises stress to each component of the Cauchy stress tensor gives:

is the outward normal to the current Von Mises stress surface. Differentiating the Von

In MARC, it is customary to separate the effects of stress, strain, temperature, and time:

There are four possible modes of input for the equivalent creep strain rate in MARC:

1. The functions f, g, h, and k in eq. (4.17) are each entered piecewise linear . 2. The functions f, g, h, and k are each entered in apower luw form:

(4.16)

(4.17)

(4.18)

3. The equivalent creep strain rate is directly defined with user subroutine CRPLAW. 4. The OAK Ridge National Laboratory Laws are used.

4.2.4 Elasto-visco-plasticity

Elasto-visco-plasticity is a mathematical model for rate-sensitive plastic materials (Lau, 1993). Elasto- visco-plastic models make no distinction between plastic and creep strains, but are capable of reproducing constant strain rate tests, creep tests, or relaxation tests (Frear et al., 1994). Furthermore, an elasto-visco-plastic material will not flow until the magnitude of an equivalent stress function F reaches an equivalent uniaxial stress state:

F = F ( o ) -Ouni (4.19)

Values of F < O indicate the elastic state at which no time-dependent deformation occurs. Usually, the

Von Mises function is used both for the equivalent stress function F , and the equivalent uniaxial stress state.

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Visco-plastic strain rate in MARC

The visco-plastic strain rate tensor in MARC is defined by:

where Q is a visco-plastic potential function, and cp is a flow function, with the properties:

(4.20)

(4.21)

4.3 Numerical procedure In this section the numerical procedure in MARC to solve the linear elasticity model, the Maxwell model, the extended Maxwell model, and the Bingham model will be outlined. We start with a general theory which provides a basis to solve these constitutive models.

4.3.1 General

Consider a domain i2 in R", where n denotes the dimension of the space. The domain R has a boundary

r, with unit outward normal Z , that is divided into r, and r, such that r = J?,ur, and rblnrp =

O. Along F, the essential boundary conditions (displacements) are prescribed, and along r, the natural boundary conditions (stresses, loads) are given. First the strong form of the quasi-static problem is

given: find the displacement field Ü such that (Baayens, 1991):

(4.22)

+ where d is the gradient operator, G is the Cauchy stress tensor, f is a body force, Üo is the

prescribed displacement field, and is the prescribed boundary load. The matrix representation of the

Cauchy stress tensor is given by:

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(4.23)

The Cauchy stress tensor is symmetric, i.e. ori = oyx, o,, = o,, oyz = oq. Due to symmetry the Cauchy

stress tensor has six independent components. Let denote the Cartesian vector base in I<, <, <I

IR3. Then, the position vector 2 and the displacement vector Ü=U(X) are defined as:

x=x<+ye'y+ z < u= u ëx + v q + w <

and the gradient operator is given by:

+ a + a - a - V=-ex+-e +-e ax ay y a2

(4.24)

(4.25)

In order to obtain the weak form of the elasticity problem, the first equation in (4.22) is premultiplied

with an arbitrary weighting function W and integrated over Q:

(4.26)

The weighting function w' has to satisfy specific continuity conditions. Furthermore,

satisfy the essential boundary conditions:

w' has to

+ (4.27) w'=o on ru

Integration by parts and substitution of the natural boundary condition gives the weak form of the

problem: find ü such that for all W :

(4.28)

Eq. (4.28) provides a basis for solving both elasticity and inelasticity problems. A relation between the stress and strain is required to solve this equation.

4.3.2 Linear elasticity

The constitutive model for linear compressible elastic materials is given by:

o =4c: E

where I is the infinitesimal strain tensor, defined as:

- 21 -

(4.29)

e=;{(au>.+(aq) (4.30)

where it again is assumed that the deformations are small, i.e. the geometrical linear theory is applicable. In the geometrical linear theory the relationship between strains and displacements is linear.

4.3.3 Maxwell, extended Maxwell, and Bingham model

Maxwell and Bingham mode!

Time-dependent inelastic material behaviour, described with a Maxwell and Bingham model, can be treated almost analogously to linear elastic material behaviour. The only difference is that a time stepping algorithm is required and the stress field, strain field, and inelastic strain field must be conserved. Both the creep strain I'' and visco-plastic strain eVp will be denoted by the inelastic strain

inel

Starting point is eq. (4.28) which holds for the geometrical linear theory. In a rate formulation eq. (4.28) becomes:

(4.31)

Due to the time dependence of the mechanical response, not only a spatial domain Q needs to be defined, but also a temporal domain S = [O,Te]. The temporal domain S is divided into N, time increments, such that:

(4.32)

Let the subscript IE signify that a quantity 5, is evaluated at time t = t,, then the time derivative of the quantity 5 can be approximated on Sn by:

(4.33)

Thus, it is assumed that the time derivative of the quantity at the start of the time increment remains constant during the increment, i.e. Euler integration is applied.The subscript I E in ~c~ will be omitted in the following formulas. Now we can write for the incremental formulation of eq. (4.31):

(4.34)

The incremental Cauchy stress is given by:

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AcT=4C: (A& -A&ine')

where the incremental total strain follows from eq. (4.30):

+ A& =;{(VA;). +(gfAÜ)}

and the incremental inelastic strain follows from eq. (4.33):

- * inel - & 'Atn

(4.35)

(4.36)

(4.37)

Substitution of eq. (4.35) in eq. (4.34) with use of eq. (4.36) yields:

h ( f w ' ) c : 4 C : ( f ~ Ü ) d Q = w ' * A T d Q + i w ' * ~ f l d r + S (?~?).:~c: Aeine'dQ w' (4.38) h P a

The integral

is called the pseudo-load integral due to the inelastic strain increment.

The second step is spatial discretization. The domain i2 is divided into nel elements a:

(4.39)

(4.40)

Within each element the variables u and w are approximated by polynomials P of order k (PJ and order I (PJ , respectively. These approximations are denoted by uh and wh. The polynomials are fully characterized by their order and by the value of uh and wh at a finite number of points (nodes). The original problem of finding the total field u for all x E is replaced by finding the nodal values that determine the approximate field uh. In the Galerkin method the interpolation polynomials of uh and wh are chosen equally (I = k), and the discretization is performed on the same mesh. Equation (4.38) can be rewritten as:

e=l e=l e=l

where

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(4.41)

K" = element stiffness matrix AR e = element incremental nodal force vector AQ - e = element pseudo load matrix

= element incremental displacement vector (AZ im')e = element incremental inelastic strain vector

and w e contains nodal values of the weighting function. The element stiffness matrix is given by:

(4.42)

where 0 is the matrix representation of 4C, and B results from the strain-displacement relation. The

incremental nodal force vector within element i2' can be written as:

AP' - =LaJJT~F - di2 +

(4.43)

where the column N consists of shape functions. The pseudo-load matrix within element Ge is given

by:

(4.44)

Equation (4.41) can be transformed into a global system of equations by an assembly process:

(4.45)

As eq. (4.45) must hold for all allowable - w , the displacement increments must satisfy the next set

of algebraic equations:

- 24 -

(4.46)

The inelastic strains are completely treated in the righthandside of the equations and hence the method is explicit. Implicit methods can also be used for elasto-visco-plastic analysis. The explicit method has the advantage that no assembly and solution of the stiffness matrix is required for each time step. A fully implicit formclation Is unconditionally stable for any choice of the time increment. Compared to the explicit procedure, the fully implicit procedure is more accurate, but the time increment may be more computationally expensive (MARC, 1994). Two important options in MARC will be explained: the option ‘AUTO CREEP’ and the option ‘AUTO THERM CREEP’.

The option AUTO CREEP is available to activate a time stepping algorithm for the Maxwell model and Bingham model. A period of time for the entire finite element simulation T,, and a suggested initial time increment at, has to be defined. The program automatically calculates the largest possible time increment for the subsequent time increments, that is consistent with the tolerance set on stress and strain increments.

The algorithm is as follows. For a given time increment at,, a solution of eq. (4.46) is obtained. The program then finds the largest values of stress change per stress, AO,,/O,, and inelastic strain per elastic strain ~&.Nfel/&,e‘. These values are compared to the tolerance values of the stress T, and strain TE. The value of p is calculated as the maximum of:

(4.47)

If p>l no convergence was obtained at and the time increment At, is reduced until convergence is obtained or the maximum number of recycles of the time increment is reached. The time increment is reduced as follows:

At,, = ( 0.8 /p) ‘At,, if p> l (4.48)

If p l l convergence was obtained at and the new time increment Atncl is calculated as:

= At, if 0 . 8 1 ~ 1 1 = 1 . 2 5 . ~ ~ if 0.65 Ip<O.8

At,,+l = 1.5 ’~t , , if p < 0.65 (4.49)

The option AUTO THERM CREEP is available to perform a thermally loaded elastic-creep stress analysis, based on a set of temperatures as a function of time. The program calculates its own set of temperature steps (increments). The times at all temperature steps are calculated for the creep analysis. At each temperature increment, an elastic analysis is carried out first to establish the stress level. An elastic-creep analysis is performed next for the time period between the current and previous temperature. Both the elastic analysis and the elastic-creep analysis is repeated until the total creep time is reached.

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In appendix A, the implementation of the Maxwell model in MARC is verified by running simple shear test cases. In these test cases, the option AUTO CREEP and the explicit procedure are used.

Extended Maxwell model

The extended Maxwell model is treated almost analogously to the Maxwell and Bingham model. The only difference is that an elastic-plastic analysis is carried out to establish the stress level, instead of an elastic analysis.

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Chapter 5

Constitutive models for solder in literature

5.1 Introduction One of the main problems in thermomechanical modelling of solder joints is the complex mechanical behaviour of solder alloys. It is generally recognized that the solder should be modeled with time-dependent nonlinear constitutive equations which account for plasticity, creep and temperature-dependent material parameters (Morgan, 1994). In the electronic packaging literature, several constitutive models for solder alloys are used, which range from linear elastic constitutive equations to elastic-plastic-creep and elasto-visco-plastic constitutive equations. All these constitutive models will show different mechanical responses to thermal cycling. What constitutes an adequate model depends on the competition between accuracy and efficiency (how accurately one can portray the mechanical behaviour of a solder with as few material parameters as possible). In this chapter a review will be given of the most frequently used constitutive equations for solder alloys, together with the values of the material parameters.

Under most thermomechanical loading conditions, the behaviour of a solder joint cannot be characterized as either pure (time-independent) plasticity or pure (time-dependent) creep. For a given loading condition, behaviour predicted with plasticity models is significantly different from behaviour predicted with secondary creep models (Ozmat, 1990). The creep model allows the stresses to relax when the macroscopic deformation is stopped, for example during the hold time of a thermal cycle, resulting in a redistribution of the stress- and strain-field. In contradiction, the stress- and strain-field calculated with an elastic-plastic constitutive model does not change. Generally, the thermomechanical loading determines the importance of the different deformation mechanisms in solder alloys, and therefore the constitutive model to be used. For example, if during thermal cycling the hold times at the maximum temperature are long, the deformation mechanisms which result in creep behaviour are dominant. Conversely, if the hold times are short, creep can be ignored. For this reason, some investigators subdivide the thermal cycle into parts in which different constitutive models are used. To simulate the mechanical behaviour of solder alloys during thermal cycling, the following constitutive models or combination of constitutive models could be used for finite element simulations:

1. 2. 3. 4. 5.

Elastic-plastic during ramp time and hold time Elastic-plastic during ramp time, elastic-creep during hold time Elastic-plastic-creep during ramp time, elastic-creep during hold time Elastic-creep during ramp time and hold time Elastic-visco-plastic during ramp time and hold time

An elastic-plastic finite element analysis does not incorporate time-dependent behaviour, such as stress relaxation at the hold times. An elastic-plastic analysis could be applied if the ramp times are relatively fast and the hold times short. Another application is to use it as a qualitative analysis. For example, to detect the maximum stress or strain in the solder joint. Elastic-plastic constitutive models will not

- 27 -

be discussed. Elasto-plasticity is standard in most finite element codes, and the user can suffice with the input of stress-strain curves obtained with uniaxial tensile tests. Furthermore, elastic-plastic behaviour of solder alloys is usually not extensively described in literature. In section 5.2 the Maxwell models (elastic-creep) and extended Maxwell models (elastic-plastic-creep) will be discussed. These constitutive models are the most widely used to model the mechanical behaviour of solder alloys. The use of elastic-visco-plastic models to model solder alloys is still in an early phase. Elasto-visco-plastic models are more complex than Maxwell models and extended Maxwell models, and they usually require more material parameters. For example, the model developed by Busso et al. (1992) needed ten material parameters. For these reasons, elastic-visco-plastic models will not be discussed. Once the Maxwell and extended Maxwell models have proved not to be capable of describing the mechanical behaviour of solder alloys at an adequate level, an option could be to investigate the suitability of elastic-visco-plastic models.

5.2 Maxwell and extended Maxwell models

5.2.1 General

In most constitutive models time-dependent elastic-plastic behaviour is modeled with a Maxwell model. In some constitutive models the elastic-creep behaviour is combined with plasticity in an extended Maxwell model. The plastic behaviour is usually not extensively discussed in literature. Most authors suffice with enumerating the yield stress and (constant) strain hardening slope at several temperatures. The constitutive models differ more explicitly from each other by the choice for the equivalent creep strain rate. The equivalent creep strain rate is described in literature by many empirical constitutive equations. A general equation is given by (see section 4.2.3):

The majority of creep equations for solder alloys concern the stress- and temperature-dependence of steady state creep, and temperature-dependent material parameters. No creep equations are found in literature in which the equivalent creep strain function g is included.

Stress-dependence of steady state creep

According to Darveaux and Banerji (1992), the steady state creep strain rate is powerlaw dependent on the stress at intermediate stresses, and an exponential function of stress at high stresses. This power law breakdown region could be described with a hyperbolic sine function. Most of the creep equations found in literature can be subdivided into equations with a power law function for the stress, and equations with a hyperbolic sine function for the stress:

In order to get an impression of the difference between the power law function and the hyperbolic sine function, both functions are plotted on a log-log scale in figure 5.1 with n = 1.5 and the value of a as parameter. The power law function has a constant slope for all stresses, whereas the hyperbolic sine function has a constant slope only for low stresses (equal to the slope of the power law function). The behaviour of the hyperbolic sine can be explained by writing:

- 28 -

[sinh(a6)]"=[0.5 fexp(a6) -exp(-aG))]"

For low values of the equivalent stress, eq. (5.3) becomes:

[sinh(aG)]" =(a@"

For high values of the equivalent stress, eq. (5.3) becomes:

[ sinh (a@]" =( 0.5 .exp( a6) >" (5.5)

The value of a prescribes the stress level at which eq. (5.3) transforms into eq. (5.4) or eq. (5.5).

Figure 5.1: Hyperbolic sine function (solid line) and power law function (-*-) with n = 1.5 and the value of a in the hyperbolic sine function as parameter.

Temperature-dependence of steady state creep

The temperature-dependence is usually associated with the Arrhenius law:

h( T ) =exp( -Q/RT)

where Q is the activation energy [J], R the universal gas constant (R = 8.3143 J.mol-'.K-'), and T the absolute temperature [KI. The value of Q can be determined by curve fitting of temperature-dependent stress-strain data. Instead of the universal gas constant R, the Boltzmann constant k is sometimes used (k = 1 .38062.10-23 J/K). Both physical constants are only used to make the dimensions consistent. The sensitivity of the function h to the activation energy Q is shown in figure 5.2. The value of the function h is always larger than O and smaller or equal to 1.

- 29 -

1 - Q=50 Q=100 Q=150 Q = 250

Q = 500

Q=1000

o.&!)O 250 300 350 400 450 500 55; Temperature [ K ]

Figure 5.2: The Arrhenius law as a function of temperature with the activation energy Q as parameter.

Time-dependence of creep

The time-dependent function k is mainly used to describe primary creep, but is usually chosen equal to 1, i.e. primary creep is neglected. Some expressions are (Boresi, 1993):

k ( t ) = { :-exp( -Pt> (5.7)

Temperature-dependent material parameters

Material parameters of solder alloys are usually temperature-dependent, for example the Young’s modulus E, the yield stress oy, or the coefficient of thermal expansion a. The equations used in literature to describe temperature-dependent parameters may be confusing, because the reference temperature and the unit (K, “C) may differ from article to article. In order to avoid confusion, all temperature dependent material parameters cp in the constitutive models to be discussed are rewritten to a Ph degree polynomial:

< P ( T ) = ( P , , + < P ~ * ( T - ~ ~ ~ ) + ( P ~ . ( T - ~ ~ ~ ) ~ + ... + ( ~ ~ * ( T - 2 7 3 ) ~ , T [KI (5.8)

Most material parameters are assumed to be linear dependent on temperature.

5.2.2 Examples

The examples of constitutive models found in literature are subdivided into models with a power law dependent stress in the creep strain rate (model’s P), and with a hyperbolic sine function for the stress (model’s H).

Model P1:

Paydar et al. (1993, 1994) used an extended Maxwell model, where the equivalent creep strain rate is described by Dom’s high temperature creep equation:

- 30 -

- E e r - -A- Gb [6][b)Doexp( - -Q/RT)

kT G d

Source

Grivas (1974)

Mohamed( 1976)

Lam (1979)

(5-9)

A [ - ] Q [kJ/mol] P [ - I n [ - I

3.2.10” 79.1 2.0 3 .O

1.5 -1 014 84.1 2.3 3 .O

40 44.0 1.6 2.4

where A is a dimensionless constant, G the temperature-dependent shear modulus, b the Burgers vector (characteristic length of crystal dislocation), k the Boltzmann’s constant, T the absolute temperature [KI, d the average grain size, n a constant stress exponent, p a constant grain size exponent, and Do a pre- exponential constant. The constants can be either obtained by imposing a constant strain rate, or by impcsing a c ~ n s t a ~ t true stress. Paydar et al. (2993) collected measurement data of eutectic PbSn solder (SnPb38) from three sources. In all cases, b = 3.2*10-7 [mm], d = 5.5~10-~ [mm], Bo = 100 [mm2/s], G = 2.2-1O4-16.1-(T-273) [MPa]. Furthermore, the Poisson’s ratio v was assumed to be constant, v = 0.4 [-l. The other four parameters given by several authors are listed in table 5.1.

Table 5.1: Material parameters in eq. (5.9).

Plasticity is characterized by a temperature-dependent yield stress oy, oY = 101.6-0.227-(T-273) [MPa], and temperature-dependent strain hardening slope E,, E, = O.OS.E(T). The Young’s modulus is related to the shear modulus by: E = 2G(l+v). In figure 5.3 the strain rate is plotted against the stress at room temperature (T = 298 K).

10- 10-1 ioo lo1 io’ 10’

Stress [MPa] io-*

Figure 5.3: Strain rate against stress for eutectic PbSn at room temperature.

From table 5.1 it can be concluded that there is considerable variation in the values of the material parameters for the same solder alloy. However, the author’s concluded that the total maximum strains in a solder joint calculated with the finite element method were rather insensitive to the variations in the material properties, while the stresses were. Furthermore, Paydar et al. (1994) observed that plastic

- 31 -

strains are dominant under faster ramp (ramp time e 2 min.), creep strains are dominant under slower ramp, and elastic strains are negligible. The equivalent total strain increases with decreasing ramp time. A higher Young's modulus resulted in a higher maximum Von Mises stress and equivalent plastic strain, but lower maximum equivalent creep strain.

Investigator

Pao (1993a)

Pao (1993b)

SQrensen (1994)

Model P2:

Solder alloy B [(sec.MPa")-'] Q [kJ/mol] ( 2- [KI ) E - 1 PbSnlO 100.6 7.42 4.25( T=3 13) 3.03( T=413)

SnCu3 9.1014.10'6 49.1 7.99(T=3 13) 6.30(T=413)

SnPb37 exp(25) 168 4305T' - 4.6

Pao et al. (1993a, 1993b) and Govila et al. (1994) used a Maxwell model with the following creep strain rate:

Y.. =B zn exp (-Q/R T ) (5.10)

where y " is the creep shear strain, B a material constant, z the shear stress, and IZ the stress exponent. In the constitutive model the grain size effect is included implicitly. To extract the material parameters, Pao et al. (1993a, 1993b) used stress relaxation data at temperature hold times during a thermal cycle test. The experimental procedure is discussed in appendix B. SQrensen (1994) performed uniaxial relaxation tests with dumbbell shaped test specimens (bulk solder), and used eq. (5.11) for uniaxial strains and stresses:

E C r =B on exp ( - Q / R T ) (5.11)

where ecr is the uniaxial creep strain, and o is the uniaxial stress. The material parameters in eq. (5.10) and eq. (5.11) are listed in table 5.3 for three solder alloys. Govila et al. (1994) didn't provide material parameters. All solder alloys showed a temperature-dependent stress exponent IZ.

In figure 5.4 the equivalent strain rate against the equivalent stress is shown at room temperature (T = 298 K).

- 32 -

IO' 10

lo-z lo-' loo Stress [MPal lo1 io2

Figure 5.4: Strain rate against stress for the three solder alloys.

Model H1:

Darveaux and Banerji (1992) used an extended Maxwell model:

(5.12) ? = y e 1 + y p l + j c r

The elastic strain is subtracted out of the data. The creep strain ycr is assumed to consist of transient and steady state creep (see Andrade's analysis in figure 4.4):

ycr=yf r ( 1 -exp(-Bt 3)) +-t dYS (5.13) d t d t

where y" is the transient creep strain at long time, B is the transient creep coefficient, and dys/dt is the steady state creep strain rate. According to Lau (1991), eq. (5.13) fails to give an accurate prediction of transient behaviour at high temperatures and low stresses. The steady state creep strain rate is given by :

(5.14) G a'I; T G

y" = C, - [ sinh (-]"exp( -QlR T )

where C,, a, and n are constants. G is the shear modulus and 'I; is the shear stress. The time- independent plastic strain ypl is described with a power law relation:

y"' = C, ('I;/ G )" (5.15)

where C, and m are constants. Tensile and shear loading is employed (constant displacement rate tests, constant load creep tests, and a limited number of stress relaxation tests) in the strain range between

- 33 -

and 10-1 sec-', and temperature range between 25 and 135 OC. The values of the parameters are listed in table 5.4 for the solder alloys SnPb40 and PbSn5.

Solder

SnPb40

PbSn5

Solder

SnPb40

PbSn5

Y" - 1 B [ - 1 C, [Wsec/MPa] a [ - ] n C - I

0.020 440 32.6 1300 3

0.030 482 1.75*109 1200 7

Q [kJ/mol] G, [MPa] GI [MPdK] C2C-3 mC-3

52.8 24.9 -1 O3 -8.96*103 2.3 4013 5.6

116 6.1 .lo3 -10 - -

Table 5.4: Material parameters in eq. (5.13-5.15) for the two solder alloys.

For both solders a Possion's ratio v of 0.35 is assumed. The transient creep parameters y" and B showed a considerable scatter in data, but for most of the test conditions primary creep could be neglected compared to secondary creep, i.e. secondary creep is achieved almost immediately. Below T/G = 0.001, the solder SnPb40 has essentially no time-independent plastic flow, because m = 5.6 in eq. (5.15). In figure 5.5 the steady state creep strain rate is plotted against the dimensionless shear stress.

lolo

. T = 2 9 3 K

Figure 5.5: Creep strain rate against dimensionless shear stress for the solder alloys SnPb40 and PbSn5.

Model H2:

Pan (1994) performed a finite element simulation of an eutectic PbSn solder joint subjected to thermal cycling. He modeled the slope of the temperature profile by a 'ladder' procedure with a combination of 'risers' and 'treads', as shown in figure 5.6.

The riser simulates the elastic-plastic portion of the deformation induced by an instantaneous temperature change. The tread simulates the creep portion of the deformation during a steady temperature. This modelling technique is similar to the auto-therm-creep option in the finite element

- 34 -

i

Solder A [sec-'pn-"] B [ma-'] n [ - I

SnPb38.1 2.9524-1 0' 0.125938 1.88882

* Time

m [ - I Q FJ/mol]

61.417 -3.01 1

Figure 5.6: Temperature-time profile.

code MARC (see section 4.3). Plasticity is not discussed in the article.

The secondary creep strain rate is given by:

(5.16) Ecr=A(sinhBo)"(d)"exp(-Q/RT)

Table 5.5: Material parameters in eq. (5.16 ) for eutectic solder.

Model H3:

The mechanical behaviour of SnPb37 solder is described with a Maxwell model and Bhatti et al. (1993):

E = ~ e l + ~ c r

The creep strain rate is composed of two power law functions for the stress:

where B, and B, are matenal constants [ ~ m - ~ ] , n, the stress exponent for low

by Wong et al. (1990)

(5.17)

(5.18)

stresses, n2 the stress exponent for high stresses, and Do a constant, Do = 1 [cm2/sec]. This expression does not include work hardening effects and it closely approximates the elastic-perfectly plastic behaviour of solder. The stress exponents n, and n2 in the creep equation (5.18) are found by determining the slope of the log(creep strain rate)-log(stress) relationship at low and high stresses, respectively. The values of the material

- 35 -

parameters are listed in table 5.6, where lower and upper bounds of B, and B, are given. Both investigators used the same experimental data.

n~

3

% BI (lower) BI (upper) B, (lower) B, (upper) E, [MPa] E,[MPa/K]

7 9.09.10’’ 8.11.10” 3.46*1d4 2.08-1026 3 2 ~ 1 0 ~ -88

Table 5.6: Material parameters in eq. (5.18).

413 K

293K 333K 373 K . .

C

In figure 5.7 the creep strain rate versus stress is shown for four temperatures on a log-log scale. The lower bounds of 23, and B, are used. Equation (5.18) approximates the behaviour of a hyperbolic sine function for the stress. A powerlaw breakdown is observed at OLE = 1 -10”. Finite element simulations of solder joints subjected to thermal cycling showed that the temperature profile strongly effected the cumulative creep strain (Bhatti et al., 1993).

;;;5:/E . u H : C .I

i

2 u

*

; I d o

10-l5 -

Figure 5.7: Creep strain rate against stress (left), and creep strain rate against dimensionless stress (right) at 4 temperatures.

5.3 Summary From the previous section it can be concluded that an extended Maxwell model covers most of the constitutive models found in literature:

For uniaxial loading, the elastic strain rate satisfies:

(5.19)

(5.20)

where E is the Young’s modulus, and (7 is the stress. The creep strain rate may be described by:

- 36 -

(5.21) &‘‘=A sZf(o) exp(-&IRT)

where A is a constant, sZ is a function of microstructural parameters (for example the grain size), AG) is a stress-dependent function, Q is the activation energy, R is the gas constant, and T is the absolute temperature. The stress-dependent function may be a hyperbolic sine function or a power law function. Microstructural parameters should only be included in eq. (5.21) if the effect of these parameters is significantly larger than the effect of measurement errors. For a new solder alloy to be characterized, it is more important to examine the stress and temperature dependence of the creep strain rate. The microstructural parameters are then implicitly included in the constant A . Once the stress and temperature dependence are h o w n and the measurement errors are relative small, the fornula for hle creep strain rate could be refined by including microstructural parameters.

In the next chapter the extended Maxwell model will be used to model the thermomechanical behaviour of the solder alloys PbSn5 and SnAg25SblO. Tensile tests and stress relaxation tests will be performed at several temperatures. The Young’s modulus E, the Poisson’s ratio v, the yield stress o,., the work hardening behaviour, and the formula for the creep strain rate (eq. 5.21) have to be dete&ned. The influence of microstructural parameters will be neglected.

- 37 -

Chapter 6

Experiments

6.1 Introduction The design of an experimental program for the determination of the material parameters depends on the choice for the constitutive model. The analysis of the experimental results will reveal wether the constitutive model is suitable for describing the mechanical behaviour or not. If the constitutive model is not accurate enough, it has to be modified. One of the main problems is the large amount of different solder materials used in the electronic packaging industry and the lack of useful mechanical test data. Furthermore, an extra complicating factor is the time-dependence of the mechanical behaviour (aging), due to a continuously developing microstructure, resulting in a considerable variation of mechanical properties of solder (Frear, 1994).

Experimental data of solder alloys provided by the manufacturer are usually restricted to stress-strain curves of uniaxial tensile tests at a few temperatures and strain rates. Tensile data are generally not sufficient to characterize the mechanical behaviour at a satisfactory level. Most of the constitutive models in chapter 5 incorporated time-dependent creep behaviour. To account for creep behaviour, a supplementary experimental program has to be designed. In appendix B measurements in literature are described.

In section 6.2 we will describe experiments with the soft solder alloys PbSn5 and SnAg25SblO (J- alloy). Both solder alloys are used for soldering a die onto a heatsink (see figure 2.1). These experiments are used to deveIop a constitutive model. The constitutive model will be used in chapter 7 as input for finite element simulations of the cooling down phase after die-attachment. During cooling down after die-attachment, stresses develop in the solder layer due to the different thermal expansion coefficients, and the die and heatsink will bend. The finite element simulations should be capable of predicting the radius of curvature of the die and heatsink, and the stresses in the die and heatsink, as a function of time.

In section 6.3, experiments with actual soldered components are described. The actual soldered components consist of a die soldered onto a heatsink. Two different experimental techniques are used to extract information out of the experiments. The first technique makes use of the changing geometry: bending of the die and heatsink. The radius of curvature of the heatsink and die are measured with a ‘Microfocus Measurement System’. The second technique makes use of the stresses on top of the die. These stresses are measured with a ‘test chip’. This test chip is capable of measuring stresses on it’s surface. The objective of these experiments is not to determine a constitutive model, but to compare the finite element output with the experimental results. This will give an impression of the reliability of the constitutive model.

- 39 -

6.2 Experiments for constitutive model

6.2.1 Introduction

In this section a constitutive model is developed for the solder alloys PbSn5 and SnAg25SblO (J-alloy). The extended Maxwell model is chosen to describe the mechanical behaviour of solder alloys:

This choice is based upon the constitutive models for solder found in literature (section 5.3) . At least two different types of experiments are necessary to determine the material parameters, because the time-independent elastic-plastic behaviour, and the time-dependent creep behaviour have to be separated. Furthermore, the experiments are performed at several temperatures to determine the temperature-dependent parameters.

Uniaxial tensile tests are performed to characterize the time-independent elastic-plastic behaviour. The elasticity-parameters to be determined are the Young’s modulus E, and the Poisson’s ratio v. Plasticity is characterized by the yield stress oy and the work hardening behaviour. Tensile tests are performed at a short time-scale to avoid creep strains, i.e. at a relative high strain rate, because the determination of the material parameters can be influenced by stress relaxation. Stress relaxation occurs at any stress level in a solder joint.

Stress relaxation tests are used to characterize the time-dependent creep behaviour. During a stress relaxation test, the specimen is firstly pulled to a predetermined strain at a constant strain rate. Then, the grip displacement of the tensile machine is fixed, and the stress will relax. During stress relaxation the stress decreases monotonically, and the yield stress will not be exceeded. Thus, the extended Maxwell model reduces to a Maxwell model:

where the following formula for the creep strain rate is chosen:

E “=Af(o)exp( -Q/R T ) (6.3)

The constant A, the activation energy Q, and the stress functionfio) have to be determined.

Finally, the elastic, plastic and creep part of the model are combined. It is assumed that the complete constitutive model is capable of predicting the short-term plastic strains, the long-term creep strains, and the interactions between them.

Specimen geometry and test equipment

Experiments are performed with solder wire from ‘Demetron’ , manufacturer of solder alloys. The diameter of PbSn5 and SnAg25SblO solder wire is approximately 1.15 mm and 0.95 mm, respectively. Solder wire is chosen as ‘test specimen’ for a number of reasons: no specimen fabrication is necessary (inexpensive and less time consuming), the solder is in a simple state of deformation, and there is no influence of intermetallic layers. A drawback is that no cyclic loading can be applied to examine the type of work hardening (isotropic, kinematic, etc. ). The tensile tests and relaxation tests are both

- 40 -

performed at Eindhoven University of Technology, Department of Mechanical Engineering, using a Frank tensile machine. A load cell of 500 N is used. The tensile machine is represented in diagram form in figure 6.1. Special grips are used to fix the solder wire.

1 MoWig part Of tensile machine

2 Pressureblock

3 Screw

4 Solderwire

Figure 6.1: Tensile machine in diagram form.

The elongation of the solder wire is measured by an extension-meter. The initial distance between the grips of the extension-meter and the tensile machine is chosen to be equal to 34 mm and 140 mm, respectively. The force, elongation, and time are recorded by a computer.

6.2.2 Tensile tests

Both solder alloys are tested at 20,50,75, 100, 125, and 150 OC. The strain rate is equal to 1 [Us] for all temperatures. At each combination of temperature and strain rate at least four experiments are performed to minimize measurement errors. The true stress-logarithmic strain curves of PbSn5 and SnAg25SblO are shown in figure 6.2. At each temperature a representive curve is depicted. The logarithmic strain is defined as E=ln(L/Lo), where Lo and L are the initial distance and the current distance between the grips of the extension-meter, respectively. The true stress is obtained by dividing the axial load F by the current cross-sectional area A, which can be expressed as:

where A, is the initial cross-sectional area. The Poisson’s ratio of both solder alloys is determined to be equal to 0.4. Only if the solder wire deforms homogeneously, then eq. (6.4) gives the right value of the true stress.

Both solder alloys show a strong temperature-dependence of the stress-strain behaviour. Furthermore, SnAg25SblO shows a more brittle behaviour and higher stresses, compared to PbSn5. In table 6.1 the values of the Young’s modulus and yield stress of PbSn5 and SnAg25SblO are listed. The measurements are averaged at each tempeature.

- 41 -

Uniaxial tension of PbSn5 35 1

150

I

3.1 5.7 9.2 29

2OC

50 c

75 c

loo c

125 c

15oc

Solder alloy Young's modulus [GPa]

PbSn5

SnAg25Sb 10

E = - 4 1 . 1 1 ~ + 2.02-io4

E = -104.9.T + 5.02*104

0.85 o:i o.i5 -2 p i 5 o:3 o.;, Logarithmic stram [-I

Standard deviation

1765

2829

Uniaxial tension of SnAgi5SblO

1 4 0 A

Y $ 8 ,

0.05 o. 1 0.15 Logarithmic strain [-I

2

Figure 6.2: True stress against logarithmic strain for PbSn5 (left) and SnAg25SblO (right) at six temperatures.

The Young's modulus as function of the temperature in degrees Kelvin is fitted in a least squares sense by a first order polynomial using all measurements at all temperatures. The results are listed in table 6.2.

Table 6.2: First order polynomial of the Young's modulus as function of the temperature [KI.

The elastic-plastic data as modeled in the finite element code MARC is listed in appendix E.

- 42 -

6.2.3 Stress relaxation tests

Stress relaxation tests are performed with both solder alloys at 20, 50, 75, 100, 125, and 150 'C. At each temperature, at least four stress relaxation tests are performed to minimize measurement errors. At all temperatures, PbSn5 solder wire is pulled to a final strain of 0.09 [-] at a strain rate of 0.01 [Us]. Due to the more brittle behaviour of SnAg25Sbl0, solder wire of this material is pulled to a final strain of 0.015 [-I, also at all temperatures and a strain rate of 0.01 [ U s ] . Stress relaxation is measured during approximately ten minutes. In order to minimize measurement errors, especially at low stresses, the stress-time curves are fitted in a least squares sense by MATLAB using the following formula:

o( t > = c e - 0 ~ t +c6e +c5 e -0.01' + c4 t -3 + c3 t -2 + cZ ( t - (to -0.5 ))-' + C; t + cG (6.5) I

where the tensile test starts at t = O, and stress relaxation starts at t = to. In figure 6.3 the stress-time curves of PbSn5 and SnAg25SblO are shown. Because of the high stresses of SnAg25SblO at the beginning of stress relaxation, the maximum of the y-axis is set to 60 MPa for a better visualization. From figure 6.3 it can be concluded that stress relaxation strongly depends on temperature and stress level.

Stress relaxation of PbSnS I

Stress relaxation of SnAg25SblO 601,

Figure 6.3: Stress-time curves of PbSn5 (left) and SnAg25SblO (right) at 20, 50, 75, 100, 125, and 150 OC.

6.2.4 Analysis of stress relaxation data

In this section the analysis of stress relaxation data is described for both solder alloys. During stress relaxation the total strain is constant, and eq. (6.2) becomes:

6 = 6 e l + k c r - --+Af(o)exp(-)=O t3 -& E RT

The constant A, the stress-dependent functionflo), and the activation energy Q have to be determined. The Young's modulus E is already determined by uniaxial tensile tests in section 6.2.2. Eq. (6.6) can be rewritten as:

- 43 -

-Q RT

-6 = E A f ( o)exg( -) (6.7)

and abbreviated to:

-0 = c ( T ) f( o)

where c(T) is given by:

c ( T ) =EAexp(-QlRT)

At each temperature, c(T) is constant. The time derivative of the stress as a function of time is obtained by differentiating eq. (6.5) with respect to time. A successful fit of the stress-time curve does not automatically imply a reliable calculation of the time derivative of the stress, especially if few measurement points are used at the beginning of stress relaxation. Therefore, the time derivative of the stress is also calculated with a central difference scheme. No differences for the time derivative of the stress were found.

Stress-dependent function

Several functions for the stress-dependence can be investigated, for example:

(6.10)

In figure 5.1 the behaviour of these stress-functions is shown after taking the logarithm. For low stresses, the hyperbolic sine function reduces to a power law function. For high stresses, the hyperbolic sine function reduces to an exponential function. Taking the natural logarithm of eq. (6.8) gives at a constant temperature:

in(-&) =in(c) +inCf(o))

Eq. (6.11) can be easily simplified in case of the power law function for the stress:

in(-&) =in(c) +nln(o)

and in case of the exponential function for the stress:

in(-&) =in(c) +no

Both eq. (6.12) as well as eq. (6.13) can be rewritten to a linear expression:

y =uo +u,x

where

- 44 -

(6.11)

(6.12)

(6.13)

(6.14)

y = in( -0) a,=ln(c) a, = n x=ln(o) , power law function X'CY , exponential function

(6.15)

and plotted with the test temperatures as parameter. If the power law function is correct, then eq. (6.14) with use of x = ln(o) should produce a straight line at each temperature with the stress exponent n as slope. Similarly, if the exponential function is correct, then eq. (6.14) with use of x = CY should produce a straight line at each temperacure with the siiess exponent n as slope. In figiire 5.4 m d 5.5, eq. (4.14) with use of x = in(@, respectively x = CY, is plotted for both PbSnS (left) and SnAg25SblO (right).

Stress relaxation of PbSn5 41

Stress relaxation of SnAg25SblO

O 1 2 3 4 5 -10' -2 -1 h(sigma) ln(sigma)

Figure 6.4: Eq. (6.14) with use of x = in(@ for PbSn.5 (left) and SnAg25SblO (right), with the temperature as parameter.

Stress relaxation of PbSn5 Stress relaxation of SnAg25SblO

150C 125C IOOC 7 5 C 5 0 C 2OC 6

4-

2 -

150 125 100 75 50 4

20 2-

i

'-4

-6

-8

ZO

.2-2

U

Ê v -

-4

-6

Figure 6.5: Eq. (6.14) with use of x = d for PbSn.5 (left) and SnAg25SblO (right), with the temperature as parameter.

From visco-elastic theory it follows that stress relaxation is influenced by the time required to impose the final strain (Govaert, 1995). For example, a short ramp time and a long ramp time will result in

- 45 -

different stress levels as a function of time. After some time both stress-time curves will converge. For this reason, stress relaxation data of PbSn5 is used after two times the loading time (after 20 seconds), and stress relaxation data of SnAg25SblO is used after 10 times the loading time (after 23 seconds). Stresses below approximately 1 MPa are inaccurate, because the force in the solder wire then approaches the sensitivity of the tensile machine's load cel. For this reason, stress relaxation measurements at 150 "C are omitted for the determination of the material parameters. Using these data reductions, it can be concluded from figure 6.4 and 6.5 that both PbSn5 and SnAg25SblO may be described by a power law function for the stress. The formula for the creep strain rate, eq. (6.3), becomes:

E"'=Ao" ex?( -QIRT) (6.16)

Figure 6.4 and 6.5 give only a visual impression which stress-function could be used. The parameter c and the stress exponent n in eq. (6.12) are determined by the MATLAB function 'fmins' in a least squares sense. This yields values of c and n for all stress relaxation tests at the temperatures 20, 50, 75, 100, and 125 "C. It appears that in(c) and n show approximately a linear dependence on T-' and T, respectively. The values of ln(c) are plotted against T-' EK-'] for PbSn5 (left) and SnAg25SblO (right) in figure 6.6. In figure 6.7 the values of the stress exponent n are plotted against the temperature T [KI. Each star represents a stress relaxation test. Some stars cover each other.

The linear relationship between h(c) and T [K-'1 is defined by:

ln (c )=k ,T- '+k ,

and the linear relationship between n and T [KI is defined by:

n =n,T+n,

(6.17)

(6.18)

The values of the constants k,, k,, nl, and no are listed in table 6.3, together with the standard deviation (sd).

PbSn5 -1.3 1 .lo4 27.3 0.67 -3.26 -1 0.' 14.8 0.33

29.4 0.59 -2.04.1 O-' 10.4 0.26

Table 6.3: Constants in eq. (6.17) and eq. (6.18) and the standard deviation (sd) for both solder alloys.

- 46 -

Stress relaxation of PbSn5 -4

-6-

-8

-10- h

2-12- - -14-

-16-

-

-18 t

5 -

c 4.5 .a

2 4 - pi

2 3.5 i G 3 -

2.5

2-

I -2!:4 2:6 2:s 3 3.2 3.4

Inverse of temperature [ 1/K]

-

-

-

Stress relaxation of SnAg25SblO

-7

c 4.5 B s 4- x : 3.5 i Gî 3 -

2.5

I

vi

2-

-6-

-8-

-10- h

2-12- - -14-

-16-

-18-

-

-

-

I 3 3.2 3.4

Inverse of temperature [UK]

Figure 6.6: The values of h(c) against T -' for PbSn5 (left) and SnAg25SblO (right).

Stress relaxation of PbSn5 6

5.5 x

'?$,O 300 320 340 360 380 400 4sO Temperature [KI

Stress relaxation of SnAg25SblO

6<

5 . 5 ~ 5

i

' 8 0 300 320 340 360 380 400 4:O Temperature [KI

Figure 6.7: Stress exponent n against temperature for PbSn5 (left) and SnAg25SblO (right).

Constant A and activation ener= Q

In order to determine the constant A and the activation energy Q, the temperature dependent parameter a, in eq. (6.14) is rewritten with use of eq. (6.9) to:

a,=ln(c) =ln(A)+ln(E)-QIRT (6.19)

We can produce a straight line if eq. (6.19) is expressed as:

y =bo +b1x (6.20)

where

- 47 -

y = ln(E)-In(c) bo= -ln(A) b, = QIR x z T - 1

Solder alloy

PbSn5

(6.21)

EO E1 Standard deviation

5.90 926 0.3 1

The constant A and the activation energy Q simply follow from bo and b,, respectively. If we use eq. (6.17) and determine a linear relationship between h(E) and T -’ [K -‘I:

ln(E) =E, T -’ +Eo (6.22)

SnAg25Sb 1 O

then the constants bo and b, follow from eq. (6.20) and eq. (6.21):

bo =Eo - ko b, =E, -k ,

6.26 1083 0.25

(6.23)

Solder alloy

PbSn5

SnAg25SblO

In table 6.4 the values of the constants E, and Eo in eq. (6.22) are listed.

A [ - I Q [JImoVK] n (T [KI)

1 .97.109 1.17 -1 0’ -3.26*10-2 T + 14.8

1. 12*1O1O 1.25*105 -2.04.10” T + 10.4

Table 6.4: Values of the constants in eq. (6.22), together with the standard deviation.

Finally, the values of A , Q, and the temperature-dependent stress exponent n in eq. (6.16) are summerized in table 6.5 for both solder alloys.

Table 6.5: Values of the creep parameters in eq. (6.16).

In section 7.2 finite element simulations of the stress relaxation tests are performed using the creep parameters in table 6.5.

6.3 Experiments for verification

6.3.1 Introduction

Testing actual soldered components has the advantage that the actual solder joint geometry is tested, and the effects of the bonded materials on the solder joint are included. In this section experiments are described with silicon dies (chips) which are soldered onto heatsinks (DLP Cu) by the soft solder alloys PbSnS and SnAg25SblO. In figure 6.8, the cross-section and dimensions [mm] of a die which is soldered onto a heatsink are shown.

- 48 -

0

r< 7 . .-

X ___)

Figure 6.8: Cross-section and dimensions [mm] of a die which is soldered onto a heatsink.

The dimensions of the die and heatsink perpendicular to the cross-section are approximately 4.5 mm and 8 mm, respectively. A chip is a silicon die with advanced electronic functions on it’s upper surface. The dies are soldered by Philips Semiconductors, Nijmegen. During cooling down, the die and heatsink will bend due to the different coefficients of thermal expansion. The bending of the die and heatsink depends on the time-dependent, thermomechanical behaviour of the solder layer. Measurements of the radius of curvature of the die and heatsink, and measurements of the stresses on top of the die (chip) yield indirectly information about the mechanical behaviour of the solder alloys. The chips used for stress measurements are not used for the radius of curvature measurements. The measurements are used to verify the numerical results obtained with finite element simulations of the die-attachment process in chapter 7.

4.3.2 Radius of curvature

With each solder alloy, 17 dies are soldered onto heatsinks. The radius of Curvature of both the die and heatsink is determined two days after the soldering process by the department of Measurement and Inspection at Philips CFT, Eindhoven. The y-position of the die and heatsink are measured as a function of the x-position by a Microfocus Measuring System of UBM Messtechnik GMBH. The working principle of this measurement system is described in appendix C. The radius of curvature of the die is determined along the entire top of the die, whereas the radius of curvature of the heatsink is determined between x = 4.0 and x = 7.0. The radius of curvature R of a beam is given by:

(6.24)

The measurement data y(x) is fitted by a second order polynomial in a least squares sense. Equation (6.24) with use of this second order polynomial yields a constant radius of curvature. The mean value and standard deviation are listed ir, table 6.6. Using PbSnS resillts in a larger radius of curvature (less bending), compared to SnAg25SblO. Furthermore, a large radius of curvature results in a relative large measurement error because the roughness of the surface influences the measurements.

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Solder alloy

PbSn5

SnAg25Sb 10

Table 6.6: Radius of curvature of the heatsink and die, two days after the soldering process.

~-

Radius of heatsink [mm] Radius of die [mm]

mean stand. deviation mean stand. deviation

2801 1336 3088 1744

86 1 125 974 314

6.3.3 Stresses in test chip

In this section stress measurements with a silicon test chip TP71 (Test Pattern No. 71) are described. These measurements are performed by Huybers (1995) at Philips Semiconductors, Nijmegen. The upper surface of the test chip is shown in diagram form in 6.9. The z-coordinate is perpendicular to cross- section in figure 6.8. The six fat lines represent the strain gauges. The working principle of the test chip is described in appendix D. In section 6.2 we have noticed the relative low stresses of PbSn5 compared to the stresses of SnAg25SblO. Furthermore, the radius of curvature of the die which is soldered with PbSn5 is relative large compared to SnAg25Sbl0, which also indicates lower stresses. Therefore, only measurements are performed with SnAg25SblO.

Figure 6.9: Layout of test chip

Forty test chips are soldered onto two leadframes. Both leadframes consist of 20 heatsinks. The leadframes serve as mechanical support for the heatsinks. Stress measurements are performed with both leadframes almost directly, and 24 hours after soldering. Furthermore, stress measurements are performed with the first leadframe after 360 hours (15 days), and the second leadframe after 120 hours (5 days). The measurement results in x-direction are listed in table 6.7, where xl = 0.75, x2 = 2.25, and x3 = 3.75 mm. The origin of the x-coordinate in figure 6.9 is different from the x-coordinate in figure 6.8. Actually, the values of the stress are average values along the strain gauges.

In the middle of the test chip compressive stresses are measured, whereas near the edges tensile stresses are measured. The stress components are composed of a bending stress and a compressive stress, which is illustrated in figure 3.2. The total stress is the compressive stress, summed with the bending stress.

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If the bending stress is larger than the compressive stress, then the bending of the die results in a tensile stress at the upper surface of the die. Furthermore, from table 6.7 it can be concluded that the stresses relax, as expected.

Leadframe 1 0, (xi) % (x2) 0, (x3)

‘almost directly’ 10.1 -19.1 6.7

24 hours 6.3 -6.4 3.6

360 hours 1.5 -4.8 -0.7

Leadframe 2 0, (xi> % (x2) 0, (x3)

‘almost directIy ’ 9.2 -18.6 7.8

24 hours 7.4 -4.4 7.6

120 hours 7.4 \- 1.4 8.2 L

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- 52 -

Chapter 7

Finite element simulations

7.1 Introduction In this chapter finite element simulations with the finite element code MARC are described. In section 7.2 finite element simulations are performed of the stress relaxation tests described in section 6.2. In section 7.3 the cooling down phase after a die is soldered onto a heatsink is simulated. The stresses on top of the die and the radius of curvature of the die and heatsink calculated with these simulations are compared with the experimental data in section 6.3.

In chapter 6 the parameters in the extended Maxwell model (eq. 6.1) are determined for the solder alloys PbSn5 and SnAg25SblO. The parameters in the formula for the creep strain rate (eq. 6.16) are listed in table 6.5. The formula for the creep strain rate is defined with user subroutine CRPLAW. The temperature-dependent elastic-plastic data of both solder alloys as modeled in MARC are described in appendix E. The elastic-plastic data used in the simulations of the stress relaxation tests are equal to the data which will be used in the simulations in section 7.3.

7.2 Simulation of stress relaxation tests A stress relaxation test with solder wire can be considered as a one-dimensional axisymmetrical problem. The finite element mesh and boundary conditions are shown in figure 7.1.

Figure 7.1: Finite element mesh of solder wire and boundary conditions.

The finite element mesh consists of 60 axisymmetrical elements (type lo), and is used for both solder alloys. First, the solder wire is pulled to a final strain using an elastic-plastic constitutive model. PbSn5 and SnAg25SblO are pulled to a final strain of 0.09 [-I and 0.015 [-I, respectively. Then, the displacement is fixed. Stress relaxation is calculated using an elastic-creep constitutive model (Maxwell model). The option AUTO CREEP and the explicit procedure are applied (see section 4.3.3).

Although the formula for the creep strain rate is determined with the stress relaxation tests at 25, 50, 75, 100, and 125 "C, the stress relaxation test at 150 "C is also simulated. In figure 7.2 the results of the finite element simulations are shown for PbSn5 (left) and SnAg25SblO (right). The elastic-plastic

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part of the simulations is not plotted. Because of the high stresses of SnAg25SblO at the beginning of stress relaxation, the y-axis is set to 60 MPa. From figure 7.2 we can conclude that the simulations of stress relaxation of PbSn5 approximate the measurements, except at the beginning of stress relaxation and at higher temperatures (125 and 150 "C). The simulations of stress relaxation of SnAg25SblO approximate the measurements at all temperatures.

Measurements (-), MARC (+)

2,h

O 100 200 300 400 500 600 700 Time [sec]

Measurements (-), MARC (+)

I 60

50

3 4 0 u 3

5 20

30 E o

10

'0 100 200 300 400 500 600 700 Time [sec]

Figure 7.2: Simulation of stress relaxation of PbSn5 (left) and SnAg25SblO (right) at 20, 50,75, 100, 125, and 150 "C. The solid lines represent the measurements, and the + denotes the numerical simulation by MARC.

7.3 Simulation of cooling down phase of die-attachment process

The assembly of die, heatsink and solder layer is considered as a 2-dimensional plane-strain problem. The cross-section and dimensions of the die, heatsink and solder layer as used in the experiments in section 6.3 are shown in figure 6.8. The same dimensions are used to create the fiiite element mesh. The finite element mesh consists of 996 plane strain elements (type l i ) , and is shown in appendix F. The die, heatsink, and solder layer consist of 240, 635, and 120 elements, respectively. Two elements are used for the thickness of the solder layer. It is implicitly assumed that the material interfaces are perfectly bonded and infinitely strong.

The temperature-time curve during the soldering process is measured by Philips Semiconductors, Nijmegen. The maximum temperature is approximately 360 "C for both solder alloys. The temperature- time curve during cooling down from the maximum temperature is also equal for both solder alloys. The solidification temperatures of PbSn5 and SnAg25SblO are equal to 305 and 228 "C, respectively. The temperature-time curve is measured to a temperature of 80 "C during the cooling down phase. The time required to cool down from 80 "C to room temperature (20 "C) is assumed to be equal to 60 seconds. The solidification temperatures are assumed to be the stress free temperatures of the assembly. The starting temperatuïe of a finite element simulation in MARC is equal :o the stress free temperature. Considering electronic packages in general, the starting temperature can be chosen to be equal to the solidification temperature of the solder, the moulding temperature of the plastic, or a temperature at which the stresses are fully relaxed. In MARC, the temperature-time curve is modeled piecewise linear. In table 7.1, the temperatures with the accompanying times are listed for both solder alloys. The temperature is assumed uniform throughout the model.

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Temperature VC] 305 228 140

Time [sec] (PbSn5) O - 18.3

Time [sec] (SnAg25SblO) - O 9.7

Table 7.1: The temperature as a function of time for both solder alloys as used in the finite element simulations.

80 20

34.5 94.5

25.9 85.9

Simulations are performed with both an elastic-plastic constitutive model as well as an extended Maxwell model for both soldeï alloys. The elastic-plastic data of the silicor, die and DLP Cu heatsink are listed in appendix E. The quantities of interest are the stress in x-direction along the top of the die, and the radius of curvature of the die and heatsink (see figure 6.8). Furthermore, the bending and compressive stress in the die are calculated for a better insight into the stress distribution in the die. The stress in x-direction along the top of the die is composed of a compressive and bending stress (see figure 3.2). The bending stress o, and the compressive stress o, in the die are given by:

1 *(‘xx, top -ox, botttom

Oc *( O x , top i- On, botttom )

where ox, top is the stress in x-direction along the top of the die, and ox, bottom is the stress along the bottom of the die.

Elastic-plastic constitutive model

In this section finite element results are described obtained with an elastic-plastic constitutive model for both solder alloys. The results of the simulations are given at 20 “C. Figure 7.3 (left) shows the stress in x-direction along the top of the die for PbSn5 (- -) and SnAg25SblO (solid line). The stars (*) denote the stress measurements with the test chip ‘almost directly’ after soldering (see table 6.7). The values of both leadframes are averaged. Because only stress measurements are performed with SnAg25SblO the stars have to be compared with the solid line. The stress measurements do not approach the finite element results. In the middle of the top of the die, the calculated stress is a tensile

Stress m x-direction along top of die Stress in x-direction aiong bottom of die

7nL I l i

-lo& -200 1 2 3 4 5

x position [mm]

Figure 7.3: Stress in x-direction along top of die (left) and bottom of die (right) for PbSn5 (- -) and SnAg25SblO (solid line). The stars (*) denote the stress measurements with the test chip.

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stress, whereas a compressive stress is measured. In figure 7.3 (right) the stress along the bottom of the die is shown. The resulting compressive and bending stress in the die, calculated with eq. (7.1), are shown in figure 7.4. Both the bending and compressive stress are lower for PbSn5.

Bending and compressive stress at 20 C - 150

I 1 - 1 W I I 2 50

B , 8

0 m

O 1 2 3 4 5 x-position [mm] -

8 - - _ - _ - - I -50 2 5 $-1oO ... . . . . . . , , . . . . . . . . . . . . . . . . . . . , . . . . . . .- g 6 -150

O 1 2 3 4 S x-position [mm]

Figure 7.4: Bending and compressive stress in die after cooling down to 20 "C for PbSn5 (- -) and SnAg25SblO (solid line).

In figure 7.5, the curvatures of the top of the die and bottom of heatsink are shown for PbSn5 (dotted) and SnAg25SblO (+). The y-position is set to zero at the lowest value of x where the measurements start, denoted by a star (*). The curvatures are fitted by a second order polynomial (solid line).

Curvature of top of die

Curvature of bottom of heatsink O - 4 -0.01

4 -0 02

8 ?-O 03

-0 04

.-

x

O 1 2 3 4 5 6 7 8 x-position [mm]

Figure 7.5: Curvature of top of die and bottom of heatsink for PbSn5 (dotted) and SnAg25SblO (+). The curvatures are fitted by a second order polynomial (solid line).

Applying eq. (6.24) to a second order polynomial yields a constant radius of curvature. The values of the radius of curvature are listed in table 7.2, together with the measurements of table 6.6.

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11 Radius of curvature I Finite element simulation I Measurement II

I 1) die (PbSn5) I 567 I 3088 II

die (SnAg25SblO) 288 974

~

heatsink (SnAg25Sb 1 O)

)I heatsink (PbSn5) I 48 1 I 2801 II 26 1 861

z

Table 7.2 : Radius of curvature of the die and heatsink calculated with an elastic-plastic constitutive model for both solder alloys, together with the measurements of tabie 6.6.

The radii of curvature calculated with an elastic-plastic constitutive model for both solder alloys do not approach the measurements. Including creep behaviour in the next section may reduce the bending, and thus increase the radii of curvature.

Extended Maxwell model

In this section only simulations with an extended Maxwell model for SnAg25SblO are described. Simulations with an extended Maxwell model for PbSn5 did not converge despite many efforts. The time increment in eq. (4.48) was set to such a low value that the finite element simulation was aborted.

During cooling down to 20 "C the AUTO THERM CREEP option is used (see section 4.3.3). At 20 "C, the AUTO CREEP option is applied to simulate the time-dependent creep behaviour for the next 48 hours. In figure 7.6 (left) the stress in x-direction along the top of the die is shown just after cooling down to 20 "C (-.-), and 24 hours after cooling down (dotted). The stress obtained with the elastic-plastic model in the previous section is also plotted (solid line). The stars (*) denote the stress measurements with the test chip after 24 hours (see section 6.3.3). The measurements of both leadframes are averaged. Thus, the stress measurements have to be compared with the dotted line. Again, the stress measurements do not approach the finite element results.

Stress in x-direction along top of die Stress in x-direction along bottom of die

Figure 7.6: Stress along top of die (left) and bottom of heatsink(right) just after cooling down (-.-), and 24 hours after cooling down (dotted). The stress obtained with the elastic-plastic constitutive model is also plotted (solid line).

- 57 -

The bending and compressive stress are shown in figure 7.7.

Bending and compressive stress at 20 C

. . . . . . . . . . . . . . . . . ..> ............ ... ........ ........ ....... .... / :

..... . . . . . . . . . . . . . ..... ......... O 1 2 3 4

x-position [mm]

o ...... Y

. . . . . . ...

- -.- ..

. . . . .

1 2 3 4 5 x-position [mm]

Figure 7.7: Bending and compressive stress just after cooling down (-.-) and 24 hours after cooling down (dotted), using an extended Maxwell model for SnAg25SblO. The solid line represents the results obtained with an elastic-plastic constitutive model.

Figure 7.8 shows the curvatures of the die and heatsink after 48 hours using an extended Maxwell model for SnAg25SblO (dotted). Furthermore, the results of using an elastic-plastic model are shown (+). The curvatures are fitted by a second order polynomial (solid line).

Curvature of top of die

Curvature of bottom of heatsink O

g o . 0 1 u

0 .s -0.02 .3

o

t - 0 . 0 3 ~ -0.04 O 1 2 3 4 5 6 7 8

x-position [mm]

Figure 7.8: Curvature of die and heatsink after 48 hours using an extended Maxwell model for SnAg25SblO (dotted). The curvatures using an elastic-plastic model are also shown (+).

The finite element predictions and measurements of the radii of curvature of the die and heatsink are listed in table 7.3. The elastic-plastic-creep constitutive model is more capable of predicting the radii of curvature than the elastic-plastic constitutive model. Considering the large standard deviation of the radius of curvature measurements, the elastic-plastic-creep model approximates the radii of curvature measurements.

- 58 -

I I I I I ' Constitutive model 1 for SnAg25SblO

Radius of curvature die

MARC Measurement

Radius of curvature heatsink

MARC Measurement

Elastic-plastic

Elastic-plastic-creep (85.9 sec.)

Elastic-plastic-creep (24 hours) I 576 1 - I 478 1 - II

288 974 26 1 861

317 28 1 -

Elastic-plastic-creep (48 hours)

So far, the temperature-dependent mechanical properties of the DLP Cu heatsink are neglected. A supplementary simulation is performed with the extended Maxwell model for SnAg25Sb 10 using temperature-dependent mechanical properties of DLP Cu. The temperature-dependent properties are listed in table E.2 in appendix E. In figure 7.9 the stress in x-direction along the top of the die is shown just after cooling down to 20 "C (-.-), and 24 hours after cooling down (dotted). Again, the stars (*) denote the stress measurements after 24 hours.

606 974 500 861

Stress in x-direction dong top of die 25 I

Temperature-inde pendent

Temperature-dependent

O 1 2 3 4 5 x-position

606 974 500 861

1084 974 915 861

Figure 7.9: Stress along top of die just after cooling down (-.-), and 24 hours after cooling down (dotted) using an extended Maxwell model for SnAg25SblO and temperature-dependent properties of DLP Cu.

In table 7.4 the radii of curvature after 48 hours are listed, showing the effect of temperature- dependent DLP Cu.

(1 Mechanical properties of I Radius of curvature die I Radius of curvature heatsink 11 (1 DLP Cu (heatsink): I MARC 1 Measurement 1 MARC 1 Measurement 11

Table 7.4: The effect of incorporating temperature-dependent properties of DLP Cu on the radii of curvature of the die and heatsink after 48 hours.

- 59 -

Incorporating temperature-dependent properties of DLP Cu strongly effects the stress level. The effect is almost as strong as the effect of incorporating time-dependent behaviour of the solder alloy. However, the stress measurements still don't match the stresses obtained with the finite element simulation. Considering the large standard deviation of the radii of curvature measurements, the radii of curvature of both the die and heatsink are predicted at a satisfactory level by MARC.

Discussion

The measurement results (especially the stress measurements) do not exactly match the finite element simulations because the measurements suffer from measurement errors, and the finite element model is a simplification of a compkx physical problem. Some sources of measurement errors are:

- The reliability of the test chip is uncertain. - The standard deviation of the radius of curvature measurements is very large. - The thickness of the solder layer may not be equal for all measurements. - The thickness of the solder layer may vary in x-direction, resulting in a not perfectly uniform stress

- The temperature-time history between 80°C during the cooling down phase, and the measurements state.

is not exactly known.

Some aspects of thermomechanical modelling which may introduce errors are:

- Uniaxial mechanical tests are performed to determine the constitutive model, whereas the solder layer

- A thin solder layer may behave different from bulk solder. - Stress relaxation is measured during ten minutes, whereas stress relaxation is simulated for 48 hours. - The elastic-plastic data of both solder alloys above 150 "C are assumed. - The parameters in the formula for the equivalent creep strain rate are based upon stress relaxation

tests in the temperature range of 20 to 150 "C. - Temperature-dependent mechanical properties of DLP Cu show a large variation in literature. - The influence of intermetallic layers is neglected. - The solder joint geometry is simplified. - The temperature is assumed to be uniform throughout the finite element model. - The numerical procedure in MARC may produce errors.

is mainly subjected to a shear loading.

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Chapter 8

rediction

A finite element analysis yields stresses and strains which can be used as input for a lifetime prediction model. If the stress exceeds the ultimate strength of a material, instantaneous rupture will occur and lifetime prediction is relative simple. Solder joints in electronic packages are subjected to many temperature changes during the lifetime of the electronic package. Thermal cycling results in thermomechanical low-cycle fatigue of the solder joint. Fatigue life prediction of solder joints is more complex. This chapter gives a short introduction about fatigue life prediction.

Several fatigue life prediction models have been proposed, for example strain-life relationship, and strain energy partitioning approaches. The strain-life relationship is generally based on a Coffin-Manson type relationship, which relates the number of cycles to failure to the total inelastic strain range:

where ~y~~~~ is the inelastic strain range, Nf is the number of cycles to failure, a is the Coffin-Manson exponent, and 6 is the fatigue ductility coefficient. Eq. (8.1) is plotted in figure 8.1 on a log-log scale. A smaller inelastic strain range increases the number of cycles to failure.

löll, ' " " " ' ^ ' ' " ' " . ' ' " ' ' " r ' ' " '

10' 10' io4 10' lo6 Number of cycles to failure

Figure 8.1: Coffin-Manson strain-life relationship.

The question arises what the definition of failure is. For that purpose a load drop parameter @(N) is introduced (Frear, 1994):

where AP, is the load range at the first cycle, and aP(N) the load range at the Nh cycle. A critical value

- 61 -

of the load drop parameter, for example 0.7, defines the number of cycles to failure. The latter is illustrated in figure 8.2 for uniaxial loading and shear loading. Uniaxial and shear loading show different fatigue behaviour. Most fatigue tests are uniaxial loading tests, whereas most solder joints are subjected to a shear loading. Fatigue life data of uniaxial loading experiments may not be applied to solder joints subjected to shear loading.

* f

Figure 8.2: Load drop as a function of the number of cycles for uniaxial loading (left) and shear loading (right).

At low thermal cycle frequencies the mechanical behaviour (stress relaxation) of solder will be significantly different from the mechanical behaviour at high frequencies. Therefore, a frequency modified version of the Coffin-Manson equation is introduced:

where v is the cycle frequency, and K defines the influence of the cycle frequency. Generally, K is between 1 and O. For K=l there is no frequency effect. Furthermore, most of the fatigue data is obtained by isothermal testing, ignoring the fact that in actual applications the temperature is varying.

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Chapter 9

Conclusions and recommendations

9.1 Conclusions Soft solder alloys show a strong temperature- and time-dependent mechanical behaviour

The extended Maxwell model is capable of reproducing the stress relaxation tests of both the solder alloys PbSn5 and SnAg25SblO at a satisfactory level. Wether the model is capable of predicting stresses in stress relaxation tests which are outside of the experimental range used to derive the parameters in the extended Maxwell model is of course not clear.

Using the solder alloy PbSnS results in lower thermally induced stresses in electronic packages compared to SnAg25SblO.

The radius of curvature measurements of the die and heatsink are better predicted by using an extended Maxwell model compared to an elastic-plastic constitutive model. Thus, incorporating time-dependent behaviour in a constitutive model for solder is necessary.

The stresses on top of the die measured by the test chips do not match the results of the finite element simulations.

9.2 Recommendations - The constitutive models obtained with stress relaxation tests are more reliable if they are capable

of predicting stresses and strains in experiments other than those used to derive the constitutive models. For example, the creep strain rate during a creep test could be predicted.

- Investigate the suitability of elasto-visco-plastic models (Bingham model) to describe the temperature- and time-dependent behaviour of solder alloys.

- Investigate the influence of the equivalent plastic strain and creep strain on the stress relaxation behaviour during a stress relaxation test. The creep strain rate may depend on these quantities.

- Investigate the influence of intermetallic layers at the solderíheatsink interface on the mechanical behaviour of a solder joint.

- In order to simulate thermal cycling, the type of work hardening should be investigated.

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- 64 -

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Pao, Y.H., Chen, K.L., Kuo, A.Y., ‘A nonlinear and time dependent finite element analysis of solder joints in surface mounted components under thermal cycling’, Materials Research Sociev Proceedings, Vol. 226, 1991, pp.23-28.

Pao, Y.H., Badgley, S., Jih, E., ‘Constitutive Behavior and Low Cycle Thermal Fatigue of 97Sn-3Cu Solder Joints’, Transactions of the ASME, Journal of Electronic Packaging, Vol. 115, June 1993, pp. 147-152.

Pan, T.Y., ‘Critical Accumulated Strain Energy (Case) Failure Criterion for Thermal Cycling Fatigue of Solder Joints’, Transactions of the ASME, Journal of Electronic Packaging, Vol. 116, September 1994, pp. 163-170.

Pao, Y.-H., Badgley, S., Govila, R., Baumgartner, L., Allor, R., Cooper, R.,‘Measurement of Mechanical Behavior of High Lead Lead-Tin Solder Joints Subjected to Thermal Cycling’, Transactions of the ASME, Journal of Electronic Packaging, Vol. 114, June 1992, pp.135-144.

Paydar, N., Tong, Y., Akay, H.U., ‘A Finite Element Study of Fatigue Life Prediction Methods for Thermally Loaded Solder Joints’, Advances in Electronic Packaging ASME, EEP-Vol. 4-2, 1993, pp. 1063- 1070.

Riemer, D.E., ‘Prediction of Temperature Cycling Life for SMT Solder Joints on TCE-Mismatched Substrates’, Proceedings of IEEE 40th Electronic Components and Technology Conference, May 1990, pp. 418-425.

Ross, R.G.Jr., Wen, L.C., Mon, G.R., ‘Solder Joint Creep and Stress Relaxation Dependence on Construction and Environmental Stress Parameters’, Transactions of the ASME, Journal of Electronic Packaging, Vol. 115, pp. 165-172, June 1993.

Sarihan, V., ‘Temperature Dependent Viscoplastic Simulation of Controlled Collapse Solder Joint

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Under Thermal Cycling’, Transactions of the ASME, Journal of Electronic Packaging, Vol. 115, March 1993.

Subrahmanyan, R., Wilcox, J.R., ‘A Damage Integral Approach to Thermal Fatigue of Solder Joints’, IEEE Transactions on Components, Hybrids, and Manufacturing Technology, Vol. 12, 1989, pp. 480-491.

Uegai, Y., Tani, S., Inoue, A., Yoshioka, S., Tamura, K.,‘A Method of Fatigue Life Prediction for Surface-Mount Solder Joints of Electronic Devices by Mechanical Fatigue Test’, Advances in Electronic Packaging ASME, EEP-Vol. 4-1, 1993, pp. 493-498.

Wong, B., Helling, D.E., ‘A Mechanistic Model for Solder Joint Failure Prediction Under Thermal Cycling’, Transactions of the ASME, Journal of Electronic Packaging, Vol. 112, June 1990, pp.104- 109.

Woychick, C.G., Senger, R.C., ‘Joining Materials and Processes in Electronic Packaging’, Principles of Electronic Packaging, Seraphim, D.P., Lasky, R.C., Li, C.-Y, eds., McGraw-Hill, New York, 1989, pp. 577-619.

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Appendix A

Verification of the Maxwell model

The implementation of the Maxwell model in the finite element code MARC is verified by running simple shear test cases. One 4 node square plane stress element (type 3) is used, sized 1 [-I by 1 [-I as shown in figure A.1.

Figure A.l: Simple shear of a square element.

The displacements of node 1 and 2 are prescribed to be zero in x- and y-direction. Node 3 and 4 are prescribed to have an equal displacement in x-direction and no displacement in y-direction, so E,= E,= O. The plane stress case is based on the assumption that ozz = oyz = o, = O and = E= = O, where x and y are the in-plane coordinates and the z-coordinate is perpendicular to the x-y plane. The stress- strain relation (4.4) for isotropic linear elastic material behaviour reduces to:

The normal strain E, is linearly dependent on the strains E, and

and for this reason it has not been included in matrix equation (A.1). From eqs. (4.2), (4.3) and (A.l) it follows that:

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In this particular case the Cauchy stress tensor is equal to the deviatoric part of the Cauchy stress tensor:

The Von Mises stress is given by:

The creep strain_ rate tensor is calculated with eq. (4.13):

The creep shear strain rate follows from eq. (A.6):

For simplicity, the equivalent creep strain rate is chosen to be power law dependent on the Von Mises stress:

Substitution of e¶. (AA) and eq. (A.5) in e¶. (A.7) yields:

Two load cases are considered. In the first load case a constant shear stress is applied to the upper plane of the element. Both the total shear strain and the creep shear strain will increase. In the second load case a constant total shear strain is prescribed. The shear stress will relax and the creep strain will increase with time. For both cases the explicit procedure is used (see section 4.3.3).

I Constant shear stress

When a constant shear stress is applied, it follows from eq. (A.3) that the shear strain rate is equal to the creep shear strain rate:

(A.lO)

The creep shear strain as a function of time is calculated with eq. (A.9):

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(A.11)

Analytic

The shear strain as a function of time is equal to the creep shear strain summed with the initial elastic shear strain:

MARC

(A.12)

I

In the numerical simulations the Young's modulus E = 1000, the Poisson's ratio v = 0.3, the constant

A yq (t = 100) yvcr (t = 100) y, (t = 100) (t = 100)

1E-4 3.26E- 1 3 .OE- 1 3.2600080E-1 3.00001OOE-1 7E-5 2.36E-1 2.1E-1 2.3 6OOO96E- 1 2.1OO009OE- 1 4E-5 1.46E- 1 1.2E- 1 1.460013 1E-1 1.2000 130E- 1 2E-5 I 8.6OE-2 I 6.OE-2 1 8.600048OE-2 1 6.0000482E-2

shear stress oq = 10, and the the constant A. In figure A.2 depicted.

0.35

0.31

- stress exponent rn = 1. The equivalent creep strain rate is varied with the shear strain against time and creep shear strain against time are

0 2 5 -

- I - 0 2 - m

O. 3

A=i .OE-4

A=7.OE-5 I 0.251

, I

O 05

LA 20

A=2 OE-5

20 40 so 80 1 O0 120 Tune 1 - 1 Time I - ]

I../------

A=l .OE-4

A=7.OE-5

A=4.OE-5

A=2.OE-5

Figure A.2: Shear strain y, against time (left) and creep shear strain y," against time (right) with the value of A as parameter. The x denotes the numerical solution (MARC), the solid line represents the analytical prediction.

The values of the numerical results and analytical predictions at t = 100 [-I are listed in table A.l

Table A.l: Analytical and numerical results of the simple shear test with a constant shear stress.

From the numerical simulations which are performed it can be concluded that the numerical results excellently agree with the analytical predictions.

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I1 Constant shear strain

From eq. (A.3) it follows that for a constant shear strain:

(A.13)

In order to obtain a linear differential equation for the shear stress the stress exponent m in the equivalent creep shear strain is chosen to be equal to 1. Now we can write for the creep shear strain rzte (ei. A.9):

= 3AoV

Substitution of equations (A.14) in equation (A.13) yields:

6,+CtOq=0

where

3EA C t = -

2( 1 +v)

The solution of this linear differential equation is given by:

o,( t ) =o,( t =O) e -at

where the shear stress at t = O follows from the linear elastic matrix eq. (A.1):

E oJt=O)= 2(1 +v)

(A.14)

(A.15)

(A.16)

(A.17)

(A.18)

The creep shear strain as a function of time is obtained by integrating eq. (A.14) with use of eq. (A.17):

(A.19)

In the numerical simulations the Young's modulus E = 1000, the Poisson's ratio v = 0.3, and the constant shear strain y.?Y = 0.2. The equivalent creep strain rate is varied with the constant A. In figure A.3 the creep shear strain against time and the shear stress against time are depicted.

The values of the numerical results and analytical predictions at t = 100 [-I are listed in table A.2. From the numerical simulations it can be concluded that the relative error in the shear stress increases with a higher creep shear strain rate; up to 23 % as A = 5.OE-5. The error is caused by the estimation

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of the creep strain increment during a time increment with eq. (4.37). The relative error in the creep shear strain can be neglected compared to the relative error in the shear stress.

Analytic MARC

0 , Z r ...............................

I I I I

80 i

I

O PO 40 60 80 100 120 Time [ - ]

1E-6 5E-6 1E-5 2E-5 5E-5

Figure A.3: Creep shear strain y," against time (left) and the shear stress ov against time (right) with the value of A as parameter. The x denotes the numerical solution (MARC), the solid line represents the analytical prediction.

6.85402660E+ 1 2.1795325OE-2 4.320 1 8296E+ 1 8.76752430E-2 2.4263 1750E+1 1,369 1 5745E- 1 7.653 12 15 8E+O 1.80 10 1 884E- 1 2.401 6601 9E- 1 1.99375568E-1

I I I I

A I O, (t = 100) I yncr(t = iOO) o, (t = 100)

6.8456001E+l 4.233 121 5E+1 2.3 1093 86E+ 1 6.82 17297E+0 1.8441780E- 1

yn=r (t = 100) /I 2.2001440E-2 8.99388OOE-2 1.399156OE-1 1 322635 1E- 1 1.995205OE-1

Table A.2: Values of the numerical results and analytical predictions at t = 100 [-l.

- 73 -

- 74 -

Appendix B

Experiments

Introduction

in literature

Much more articles found in literature deal with using constitutive models in finite element models, instead of determining these constitutive models. Even less articles are found which verify the numerical results of the finite element simulations. Because of the complex thermomechanical behaviour of solder alloys, the numerical results may become very doubtful. Therefore, it is worthwhile to discuss some experimental techniques.

In section B.2 we start with a discussion of specimen design. In section B.3 the mechanical test conditions to which these specimens can be subjected are categorized. Some categories are illustrated by interesting examples available in literature.

B.2 Specimen design Solder alloys can be tested in both actual soldered components as well as in simplified test specimens. The advantages and disadvantages of these specimens are briefly discussed below.

Actual soldered components

Testing actual soldered components, for example the electronic package in figure 2.1, has the advantage that the actual solder joint geometry is tested, and the effects of all other components on the solder joint are included. A disadvantage is the complicated stress- and strain-field in the solder, which are difficult to measure. Therefore, actual soldered components are not useful for determining a constitutive model. However, they can be used for determining the effect of different solder joint geometries and solder alloys on the fatigue life of a specific electronic package. The fatigue life is determined by imposing strain and/or temperature cycles. If the solder joint is both the mechanical and electrical interconnection, it is possible to monitor the electrical resistance as a function of the number of cycles. An increase in electrical resistance indicates crack initiation. The latter can also be applied to simplified test specimens (Frear, 1989).

Simplified test specimens

Simplified test specimens are designed to be in a simple state of deformation (simple shear, uniaxial loading). An advantage is that the constitutive model is more easily to determine, compared to actual soldered components. A drawback is the relative large volume of a test specimen (for example a dumbbell shaped specimen) compared to the volume of solder joints in actual components (for example the solder joint between the lead frame and the metallization in figure 2.1). Due to the small volume of solder joints in real components they more rapidly cool down to the environmental

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temperature than simplified test specimens. This may result in a different microstructure. The microstructure influences the mechanical behaviour of solder alloys.

Simple shear test specimens may consist of solder constraint between two metal plates. This allows the growth of intermetallic layers, dependent on the properties of the materials bonded together. Annealing (thermal aging) of the specimen will increase the intermetallic layer thickness (Frear, 1994), which offers the opportunity to investigate the influence of the intermetallic layer thickness on the stress-strain behaviour. Simple shear test specimens could introduce some experimental error, because the solder thickness may vary perpendicular to the metal plates, resulting in a not perfectly uniform stress and strain distribution (Darveaux,l992). Furthermore, the shear stress will not be perfectly uni fxn dce to ‘end-effects’ (Govaert, 1995). The she= stress Increases at the ends of the solder layer. An example of a double shear lab test specimens is shown in figure B.1. A double shear lab configuration is used to minimize bending.

, I/--so1aer , E 0.5 F

I 0.5 F

Figure B.l: Double shear lab configuration.

Uniaxial test specimens usually consist of bulk solder. For bulk solder, for example dumbbell shaped specimens or solder wire can be used. Dumbbell shaped specimens can be machined out of axisymmetric bars. Fabrication of those specimens is expensive and time consuming. Solder wire belongs to the standard assortment of solder alloy manufacturers. Therefore, tensile tests with solder wire are less expensive and less time consuming (no specimen fabrication). A drawback of solder wire is that no cyclic loading can be applied to examine the work hardening behaviour. Special attention has to be paid to the interaction between the specimen and the tensile machine. Compressive forces in the grips of the tensile machine introduce stress concentrations, resulting in fracture near those grips. In case of solder wire, this problem may be reduced by using special grips, as shown in figure 6.1. One of the concerns is the applicability of uniaxial test data of bulk solder to thin solder layers subjected to a shear loading.

B.3 Mechanical test conditions Mechanical test conditions of solder joints can be divided up into:

1. Thermal, temperature controlled cycling 2. Isothermal, strain controlled cycling 3. Thermal, temperature and strain controlled cycling 4. Isothermal creep tests (constant force) 5. Isothermal stress relaxation tests (constant displacement) 6. Isothermal tensile tests (constant displacement rate)

Dcrng thermal, temperatrire controlled cycling the deefomation solely results form the mismatch in the thermal coefficients of expansion of the materials soldered together; no external load is imposed. This test condition approaches the actual use environment of a solder joint. During isothermal, strain controlled cycling the specimen is cycled with a constant strain range at a constant temperature. An advantage is that the test procedure is much simpler than a thermal cycling test, and the stress-strain relation can be easily measured. One of the main concerns is the applicability of isothermal test data to thermal cycled solder joints. It is observed (Frear, 1989), that the development of the microstructure

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during isothermal cycling is significantly different from that found during thermal cycling.

All of the test conditions will show different mechanical responses of the solder alloy. Comparison among material parameters derived from different types of experiments gives an impression of the reliability of the constitutive model. A constitutive model becomes more reliable if it is capable of predicting experimental results under different test conditions than those used to derive the model. For example, a constitutive model derived from stress relaxation tests should be verified by predicting experimental results from creep tests. The cyclic experiments provide more information than the other categories. Especially with concern to the fatigue behaviour. Furthermore, the type of work hardening can be investigated. The test conditions can be applied to both actual soldered components as well as simplified specimens, although most actual soldered components are subjected to thermal, temperature controlled cycling.

In section B.3.1 an example of thermal, temperature controlled cycling is described. In section B.3.2 an example of thermal, temperature and strain controlled cycling is described.

In this section the experimental method designed by Pao et al. (1992) is shortly reviewed. The test specimen used for thermal cycling is shown in figure B.2.

I G-

. . . -F i E 1 Me 1?

.2: Specimen used for thermal cyling.

The specimen consists of a Alzo, beam (a = 6.1+10-6pC) and a Al 2024-T4 beam (OE= 20.7.10-6pC), bonded together at their ends with solder. Thermal cycling imposes a shear strain on the solder joint, resulting in a shear stress. Four high temperature strain gauges are mounted on the surface of the Al 2024-T4 beam to measure mechanical strains. These strains are related to the shear strain and shear stress in the solder joint by stress-strain relations. The stress-strain hysteresis loops are determined as a function of thermal cycles. At the hold times, the shear stress will relax due to creep of the solder. A Maxwell model is used to characterize he mechanical behaviour. The steady state creep parameters are related to the stress relaxation data at the hold times. A drawback is the cumbersome manner to obtain the stress-strain relation from simple beam and multilayer theory, which introduces errors. Ân advantage is that both the shear stress and shear strain change at the hold time, and the type of loading approaches the actual loading case (shear stress and bending moment).

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3.2 Ther temperature and strai controlled cycli The advantage of this method is that both the strain and temperature are controlled (imposed). In this section the experimental method of Frear (1989) is reviewed. Frear was capable of coupling temperature and strain controlled cycling. The strain ramps were actuated by signals from the temperature controller. Temperature cycling could be accurately performed from -60 "C to 150 "C with temperature ramps as fast as 1 "C per second. Temperature control was accurate to t l "C on heating and t 2 "C on cooling. The solder alloys SnPb40 and SnIn40Pb20 were tested in a double shear lap configuration. A temperature ramp rate of 1 "C per second imposed a shear strain rate of 0.11% per second. The strain rate is corrected for the thermal expansion of the specimen. The total strain range is 20% with a zero point at 25 "C. The temperature is cycled between -55 "C and 125 "C. In figure B.3 the test results for the solder alloy SnPb4O are depicted.

7 5

iI a 0

O m m 2 vi

- 7 5

-100 .& o too 12s 200

Time (minutes) Temperature ("C)

Figure B.3: Temperature versus time and shear stress versus time for the first few cycles on SnPb40 solder joints (left), and shear stress versus temperature (right).

Within the 3 minute hold time at the high temperature (125 "C), 65% of the stress has been relaxed. At the low temperature (-55 "C) stress relaxation was hardly observed. At the low temperature, cyclic hardening occurs during the first 20 cycles, followed by cyclic softening during the following cycles. At the high temperature, cyclic softening occurs from the first cycle.

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Appendix C

Microfocus Measuring System

In this appendix the working principle of the Microfocus Measuring System is described, as used in section 6.3.2. The working principle is illustrated in figure C.1. Infrared light from a laser (1) is focussed to a spot on the surface of the measurement object (13) by an objective lens (10). The light reflected by the surface is directed by a beam splitter (3) , through a prism with beam splitter (2) , and analysed by photodiodes (5). If the objective lens is precisely it’s focul distance from the surface, then the spot diameter is minimal. This minimum spot diameter equals 1 pm. The measurement system moves in a straight line over the surface of the measurement object. During this movement, the objective lens/surface distance changes due to the curvature of the measurement object. The change in spot diameter is observed by the photodiodes (5 ) and a focus error signal is generated. The error signal serves as input for a control circuit, which moves the objective lens up or down, until the spot diameter is minimized.

Figure C.l: Layout of the Microfocus Measurement system. The system consists of the following components: 1. Laser diode, 2. prism with beam splitter, 3. beam splitter, 4. window, 5. photodiodes, 6. leaf spring, 7. coil, 8. magnet, 9. collimator lens, 10. objective, 11. tube, 12. light barrier measurement system, 13. measurement object (die and heatsink), 14. microscope with illumination, 15. CCD camera.

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The vertical displacement of the objective lens is registered as a function of the horizontal displacement of the measurement system. A vertical measurement range of 150 pm is used. This range corresponds to a resolution of 10 nm.

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Appendix D

Stress measurements with test chip TP71

In this appendix stress measurements with the test chip TP71 (Test Pattern No. 71) are described. We start with a discussion of the physical principle of the stress measurements. The stress measurements are based on the effect of piezoresistance. This effect may be defied as the relative change in resistivity as a function of the applied stress or strain (Gestel, 1994). The piezoresistance effect is illustrated with a wire. The resistance R of a wire is given by:

R = - PL A

where p is the specific resistance, L the wire length, and A the cross-sectional area. A first order approximation of the change in resistance at constant temperature is given by:

where the subscript O denotes the initial state (zero stress), and v is the Poisson's ratio. For semiconductor materials, like silicon, the chance in specific resistance is the main term influencing the resistance. For conducting materials, like metals, the change in specific resistance is negligible and the change in resistance is mainly determined by the change in geometry. If we define a Gauge factor K:

then the Gauge factor for a metal is equal to (1+2v), and the Gauge factor for a semiconductor material could be almost 100 times the Gauge factor of a metal (Gestel, 1994). The change in specific resistance can be related to the change in stress by piezoresistance coefficients. If we neglect the influence of the change in geometry for silicon, and incorporate the temperature-dependence of the resistance, then eq. (D.2) becomes:

where R is the resistance [kOhm], X, are the piezoresistance coefficients [kOhm/MPa], and a is the temperature coefficient of resistivity [kOhdC] .

Silicon test chip TP71

The test chip "IT71 consists of isotropic silicon, type p (1 11). Borium tracks are applied at the surface

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of the silicon die by a diffusion process. These tracks serve as strain gauges. Since tracks with a large length-to-width relation are used, the current will always flow in the longitudinal direction. The test chip is only capable of measuring principle stresses at it’s surface. Therefore, the strain gauges are positioned at the symmetry-axis of the chip, which are free of shear stresses. The layout of the chip is shown in figure 6.9. Because there are only stress components in x- and y-direction, eq. (D.4) reduces to:

The ~ i e ~ ~ r e ~ i ~ t a f i c e coefficients are influenced by the dopiirg level of the resistor, and the oïkntbion of the crystallographic axes. Therefore, the values of the piezoresistance coefficients are determined with a bending test: nm= 0.0470, and n,,,,= -0.0122. The coefficient a is determined with a chip (not soldered onto a heatsink) subjected to a number of temperature changes: a = 0.2232 [kOhdC] .

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Appendix E

Material

Silicon

DLP Cu

Elastic-plastic data in MARC

oy [MPa] (T ( e p ' = 0.18) E [GPa] v [ - I a [10-6/K]

169 0.23 2.3

180 300 120 0.3 17.7

In this appendix the elastic-plastic data of silicon (die), DLP Cu (heatsink), and the solder alloys PbSn5 and SnAg25SblO are listed, as used in the finite element simulations in section 7.3. The thermal expansion coefficients are also given. The temperature-independent properties of silicon and DLP Cu are listed in table E.l.

Temperature ["C]

20

150

o y [mal E [GPa] (T (epz = 0.04) [ m a ]

279 95 322

279 53 -

In section 7.3, also finite element simulations are performed with temperature-dependent DLP Cu. The definition of temperature-dependent elastic-plastic data in MARC is described in section 4.2.2. The temperature-dependent elastic-plastic data of DLP Cu are listed in table E.2. The data in table E.2 differ from the data in table E.l because another source is used. The work hardening ratio R is equal to 1 for all temperatures.

300 248 46

Table E.2: Temperature-dependent elastic-plastic data of DLP Cu.

PbSnS and SnAg2SSbl O have temperature-dependent mechanical properties. The work hardening behaviour of PbSnS and SnAg25SblO at 20 "C, which is shown in figure 6.2, is modeled piecewise linear. The values of the stress with the accompanying plastic strain ep' are listed in table E.3. The values of the Young's modulus, yield stress, and work hardening ratio R are listed in table E.4 with the temperature as parameter. The linear fit for the Young's modulus is chosen because the parameters in the formula for the equivalent creep strain rate are calculated with eq. (6.22). The values above 150 "C are assumed. The Poisson's ratio was determined to be 0.4 for both solder alloys. The thermal expansion coefficients of PbSn5 and SnAg25SblO are equal to 29 and 19 [10-6/K], respectively.

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Table E.3: Values of the stress with the accompanying plastic strain at 20 "C.

Table E.4: Values of the Young's modulus, yield stress, and work hardening ratio R for PbSn5 and SnAg25SblO.

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inite de

The finite element mesh of the undeformed geometry of the die, heatsink, and solder layer, as used in section 7.3, is shown in figure F. 1. The deformed geometry of an arbitrary finite element simulation is shown in figure F.2 (magnification ten times).

Figure F.1: Finite element mesh of undeformed geometry of die, heatsink, and solder layer.

Figure F.2: Finite element mesh of deformed geometry (magnification ten times).

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