COMPARISON OF SOLDER JOINT FATIGUE PREDICTIONS BASED ...

COMPARISON OF SOLDER JOINT FATIGUE PREDICTIONS BASED ON THE DARVEAUX’S MODEL AND THE COHESIVE ZONE MODEL L. Benabou1), Z. Sun1), P.R. Dahoo2) 1)

2)

LISV, Université de Versailles Saint Quentin-en-Yvelines, 78140 Vélizy, France LATMOS, Université de Versailles Saint Quentin-en-Yvelines, 78280 Guyancourt, France

ABSTRACT The fatigue cracking process of a lead-free solder joint under power cycling is modeled through two approaches. The first approach consists in using a cohesive zone model describing the decohesion at the die-solder interface and accounting for the mixed-mode crack development. The second approach gives the damage evolution in the solder based on the inelastic strain energy per stabilized cycle. In both approaches, the solder material is characterized by a viscoplastic behaviour, with temperature and strain rate having strong effects. The models are implemented in the finite element code Abaqus in order to simulate the fatigue of the connecting solder used in a simplified power module under a given mission profile.

KEYWORDS Creep behaviour, solder joint, cyclic loading, finite element analysis, cohesive zone model, inelastic hysteresis strain energy, lifetime prediction

INTRODUCTION Thermo-mechanical reliability of power electronics is a major issue for reasons related to safety and manufacturing costs. In service, electronic components are subjected to thermal cycles due to environmental temperature changes as well as heat dissipation of siliconbased chips under power cycling. Mismatches of the coefficient of thermal expansion between the different materials of a power module give rise to significant thermal stresses. In the case of IGBT (Insulated-Gate Bipolar Transistor) modules, solder joint delamination is one of the main failure modes along with wire bond cracking. These modules are complex systems (Fig. 1), and thus only the components relevant to the fatigue in the solder are considered in this study. For the sake of simplicity, a three-layer arrangement (composed of the chip, the solder and the substrate) is modelled with proper specification of the thermal boundary conditions and the electric power loading. The structure of this paper is as follows: the constitutive law of the SnAgCu lead-free solder is first detailed. The Anand’s model is used to account for the temperature and strain rate dependence of the solder material. Then, a first approach, based on the cohesive zone model, is presented so that crack initiation and evolution can be described at the solder-die interface during power cycles. A second approach, using the Darveaux’s model and giving the degradation evolution in the bulk solder, is also considered. The module lifetime is finally computed according to both approaches, allowing for their comparison.

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Fig. 1: Schematic S r representat ion of the power modu ule

CONST TITUTIVE BEHAVIOUR B R OF THE SOLDER S The An nand’s visccoplastic model m is commonly c d th he time-de ependent used to describe constitu utive behaviour of the solder s in ele ectronic pac ckages. It was impleme ented in the Abaqus finite ele ement code e with the user u subrou utine CREE EP. There are a two specific feature es in the Anand model m [1,2]. Firstly, the e model nee eds no explicit yield co ondition, no or loading/un nloading criterion n. Secondlyy, it uses a single sccalar s as an interna al variable for express sing the average ed isotropicc deformation resistan nce to plas stic flow. The T set of Anand con nstitutive relationss accountss for various physical mechanism ms such ass temperatu ure and strrain rate dependency, their history effe ects, strain hardening, h etc. The eq quivalent ine elastic strain rate is given byy:

    v  Asinh    s  

1m

 Q  exp   R   RT

(1)

and the evolution relation r for the t deforma ation resista ance s is asssumed to b be of the form: n a   v s    Q   s  h0 1  s sgn  ˆ n1    v with s  s  exp  R  s  s   RT  A

(2)

T is the abso e s stress, olute tempe erature, m is the strrain rate where  is the equivalent sensitivvity of stresss, h0 is the hardening constant, a is the stra ain rate sensitivity of ha ardening, he saturation value of s , and n iss the strain rate sensittivity of saturation valu ue. Note s  is th that the e deformattion resistance has th he initial value v T valuess of all the e model s 0 . The parame eters, found in [3], are specified s forr the lead-frree solder alloy a in Tablle 1.

 A m s0 Q R -1 ) (s (MPa) (°K) 3.2992 9883 15.773 1.0673 0.3686 Table 1: Anand con nstants for SnAgCu so older

h0 (MPa) 1076.9

sˆ (MPa) 3.1505

a

n

1.6833

0.0352

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COHEZIVE ZONE APPROACH In this approach, a user interface element is implemented in the user subroutine UEL to bridge classical bulk elements. Depending on the cohesive law adopted for this interface element, fatigue crack initiation and propagation can be predicted. The common feature of the various traction-separation laws is that the cohesive traction reaches a peak value and, then, drops towards zero with gradual increase of the opening displacement. Thus, the key parameters of the cohesive law are the cohesive strength (peak value of the curve) and the cohesive energy or fracture energy (total area under the curve). Under mixed-mode crack development, the thermodynamic potential, expressed with respect to the local displacement jumps at the interface, is as follows [4]:

 u, d  



1 1  D  k n u n 2

2 



 k t u t2 

1  k n u n 2

2 

(3)

where D  0,1 is a scalar damage variable, and the constants k n , k t , k n are the interface stiffnesses in traction, shear and compression respectively. The thermodynamic forces in the interface are obtained by differentiating the potential  with respect to the state variables:





    T  u  1  D  k n u n  n  k t u t t  k n u n - n  Y     Y  Y I II  m D

(4)

In order to account for the coupling between the different modes, an equivalent relative displacement [5] and an equivalent cohesive traction are introduced:

m 

u n

2 

  2 u t2 ;

Tm 

1  2 k n  m with   k t k n 2

(5)

Description of damage initiation and growth is done with the use of a failure criterion, f Ym , d   0 , accounting for the irreversibility of damage [4]: t f Ym , d    1  D  Ym d  Gm 0  Gmc  Gm 0 D  0

(6)

where Gm0 is the equivalent energy density for damage initiation, and G mc represents the mixed-mode fracture energy. These two equivalent parameters can be computed based on single-mode parameters. The consistency condition, f  0 , allows for the derivation of the damage rate:

1  D k n  dD  m dt Gmc  Gm 0

d m dt

(7)

DARVEAUX’S APPROACH

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In this approach, progressive damage is not restricted to a cohesive layer between the die and the solder, but it can affect the bulk solder material wherever inelastic strain accumulates. The stress tensor in the material is updated over the cycles as:

  1  D

(8)

where  is the undamaged stress tensor that would exist in the material in the absence of damage computed in the current increment. Damage initiation and evolution are controlled by the inelastic hysteresis strain energy as proposed by Darveaux [6]. The cycle number in which damage is initiated at a material point is given by:

N 0  c1w c2

(9)

where c1 and c 2 are material constants, and w is the inelastic hysteresis strain energy at the stabilized cycle. Damage evolution is also described based on w with the damage rate per cycle calculated as: dD c 3  w c 4 dN L

(10)

where c 3 and c 4 are two more material constants, and L is the characteristic length associated with an integration point so that mesh dependency may be alleviated. Contrary to the cohesive zone model implemented in the UEL subroutine, the Darveaux model can be associated with the direct cyclic procedure in Abaqus. Indeed, the cohesive layer method is computationally expensive because it requires to simulate the entire loading history, cycleby-cycle, whereas the progressive damage description of Darveaux, which can be coupled with the direct cyclic technique, makes it possible to obtain directly the stabilized response of the material under cyclic loading. Thus, this technique results in considerable computational effectiveness for fatigue predictions of the solder joint submitted to intermittent electric power load.

NUMERICAL RESULTS The module is modelled as a three-layer arrangement consisting of the die, the solder and the substrate. The effect of water-cooling on the system is simulated with boundary convection on the bottom surface of the substrate by defining a reference sink temperature and a film coefficient. All three bulk materials are meshed with four-node plane strain elements. In the cohesive zone approach, the electrical loading prescribed to the module is reproduced by applying an equivalent thermal loading which is determined by assuming that electrical energy is completely converted into thermal energy through Joule heating. Also, a very thin layer, located at the die-solder interface where solder failure is expected to occur, is meshed with cohesive elements. As for the Darveaux’s approach, a two-step procedure is used: first a coupled thermal-electric analysis is carried out to find the temperature solution, and then a stress analysis is done based on this temperature solution. In Fig. 2, the temperature evolution is plotted during some cycles at three locations in the system, in the die, solder and substrate respectively. The active power cycling of the system (total current delivered of 75 A with period of 0.5s on & 0.5s off) leads to maximum temperature of approximately 80C and temperature swing of less than 10 C . At such level of temperatures, creep processes are expected to dominate the deformation kinetics. For

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both approaches, based on the computed thermal stresses and strains, the continuous update of the damage variable throughout the cycles will allow for the prediction of the number of cycles to rupture. In the cohesive zone approach, the path failure is known a priori, and thus crack propagation is restrained to the interface layer made of cohesive elements. In the Darveaux’s approach, damage can initiate and develop everywhere in the solder elements. However, for the present case study, the damage zone predicted by the Darveaux’s approach is restricted to a narrow band with a thickness of one finite element. This is due to important thermal stresses at the die-solder junction, resulting from CTE mismatch between the die and the solder. This damage zone propagates and remains just below the interface, which is consistent with the interface delamination mechanism modelled by the cohesive zone model.

Fig. 2: Temperatures cycles obtained from the electrical-thermal resolution in the Darveaux’s approach

Fig. 3: Evolution of the damage zone size with respect to the number of power cycles according to both approaches The lifetime of the system can be obtained by considering that the component fails when the crack extension covers all the solder width. In reality, the solder joint may be considered faulty long before the crack crosses the solder from end to end. In semi-empirical studies of

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solder reliability, such as in the Coffin-Manson law, the number of cycles to failure is defined with the 50% load drop criterion. Indeed, during fatigue experiment of solders, it is observed that there is an exponential decrease in load with increasing number of cycles. Hence, the number of cycles at half of the initial load is taken practically as the number of cycles to failure. The system lifetime, thus defined, seems more relevant in the context of industrial specifications related to the reliability of electronic components. Therefore, according to this criterion, the computed lifetime is obtained when the damage zone extends to nearly half of the solder width (load carrying capacity decreased by half), which corresponds roughly to 6000 cycles for both approaches (Fig. 3).

CONCLUSION In this work, a simplified IGBT module has been modeled for prediction of solder reliability under power cycling through two different approaches. The first approach gives the damage evolution in interface elements based on their traction-separation law. The second approach uses the stabilized inelastic strain energy per cycle to update the damage level in bulk elements of the solder. For both cases, the solder viscoplastic behavior is described by the Anand constitutive law. Damage initiation and propagation near the die-solder interface are consistently reproduced by both models whose lifetime predictions are in good agreement.

REFERENCES [1]

[2]

[3]

[4]

[5]

[6]

Anand, L.: Constitutive equations for hot working of metals International Journal of Plasticity 1 (1985), pp. 213-231 Brown, S.B.; Kim, K.H.; Anand, L.: An internal variable constitutive model for hot working of metals International Journal of Plasticity 5 (1989), pp. 95-130 Bhate, D.; Chan, D.; Subbarayan, G.; Chiu, T.C.; Gupta, V.; Edwards, D.R.: Constitutive Behavior of Sn3.8Ag0.7Cu and Sn1.0Ag0.5Cu alloys at creep and low strain rate regimes IEEE Transactions on Components and Packaging Technologies 31 (2008), pp. 622633 Benabou, L.; Sun, Z.; Dahoo, P.R.: A thermo-mechanical cohesive zone model for solder joint lifetime prediction International Journal of Fatigue 49 (2013), pp. 18-30 Camanho, P.P.; Davila, C.G.: Mixed-mode decohesion finite elements for the simulation of delamination in composite materials NASA/TM-2002-211737 (2002), pp. 1-37 Darveaux, R.: Effect of simulation methodology on solder joint crack growth correlation and fatigue life prediction Journal of Electronic Packaging 124 (2002), pp. 147-154

Corresponding author: [email protected]

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