## Abstract

This paper reviews the previous research on the methodologies for evaluating structural integrity of wire bonds and die-attachments in power modules. Under power module operation, these parts are subjected to repeated temperature variations which induce repeated thermal stress due to the mismatch in coefficients of thermal expansion (CTE) of the constituent materials. Thus, thermal fatigue phenomena are critical issues for the structural integrity of power modules. In the present paper, we also deal with the evaluation methodologies for thermal fatigue in the temperatures over 200$°C$, which are expected operational temperatures for wide bandgap semiconductor power modules. The failure models based on the temperature range $ΔT$ widely used in the power electronics community are critically reviewed from a mechanical engineering viewpoint. Detailed discussion is given concerning the superiority of failure models based on the physical quantities such as the inelastic strain range $Δεin$, the inelastic strain energy density range $ΔWin$, and the nonlinear fracture mechanics parameter range $ΔT$* over the conventional $ΔT$-based failure models. It is also pointed out that the distributed state concept (DSC) approaches based on the unified constitutive modeling and the unified mechanics theory are promising for evaluating the structural integrity of power modules. Two kinds of test methods, a power cycling test (PCT) and a thermal cycling test (TCT), are discussed in the relation to evaluating the lifetimes of wire-liftoff and die attach cracking.

## 1 Introduction

Among various kinds of energy, the ratio of electric power consumption to total energy consumption increases year by year, because electric power is safe, clean, and easy to use. Power modules are utilized for electric power control and play a key role in efficient energy conversion. They are embedded in various systems such as electric power transmission systems, railway systems, automobiles, factory automation systems, home appliances, and so on. Thus, research on highly reliable and long-life power modules contributes to realizing energy-saving and sustainable society.

Let us consider an insulated gate bipolar transistor (IGBT) shown in Fig. 1. Key locations for the structural integrity are the connection between a chip and a bonding wire, a wire-bonding location, and the connection between a chip and a copper substrate by means of a die attachment material. Under power module operation, these connections are subjected to repeated temperature cycles which induce repeated variations of thermal stress due to the mismatch of coefficients of thermal expansion (CTE) of materials consisting of a power module. Thus, thermal fatigue phenomena are critical issues for the structural integrity of power modules. They are wire-liftoff, in which a bonding wire is delaminated from a chip surface, heel cracking, in which the heel part of a bonding wire is fractured, and die attach cracking, in which a crack occurs and propagates in a die attach layer. In the present review paper, we deal with these thermal fatigue phenomena. Power modules using wide bandgap semiconductors such as SiC and GaN have been developed in recent years. Their operating temperatures are expected to be over 200 °C. In this paper, we also deal with the thermal fatigue phenomena at elevated temperature over 200 °C.

Fig. 1
Fig. 1
Close modal

Historically, the academic disciplines concerning structural integrity evaluation have advanced in the safety design and operation of large-scaled structures used as civil engineering structures, hull structures, aircraft structures, pressure vessels and piping, etc. Especially, fracture mechanics and fatigue strength have greatly advanced. Electronic packaging can be regarded as a small-scaled or miniature structure which has sources of stress concentration such as interfaces between dissimilar materials. Thus, methodologies for the structural integrity evaluation of large-scaled structures have been necessarily applied to electronic packaging [1]. In 1986, a Japanese automobile company decided to recall faulty automobiles causing sudden start. The reason for such an incident was the fatigue failure of solder joints in integrating circuits (ICs) of an autodrive computer. This trouble attracted attention to the structural integrity of electronic packaging. Research on the structural integrity of IC packaging has been actively performed since 1980s. On the other hand, such research for power devices has been performed since 1990s [2,3], and several review papers have been published [48]. According to the authors' opinion, main concern of the structural integrity of power modules is condition monitoring and remaining life estimation of in-service power devices, as shown in Refs. [5], [7], and [8].

Fewer power modules are utilized in electric power transmission systems and railway vehicles than in the automobiles and home electric appliances, and their structural integrity can be confirmed by in-service condition monitoring and periodic inspection. On the other hand, an enormous number of power modules are utilized in the automobiles and home electric appliances without proper inspection. We cannot deny that a malfunction of electronic devices including power modules in automobiles may cause a serious accident as described before, and recall of an enormous number of automobiles may be possibly caused. It is necessary to confirm the structural integrity of electronic devices in the design phase to avoid their malfunction during in-service period. The CAE-based stress analysis, fracture mechanics, and strength of materials have been applied to securing not only the structural integrity of large-scaled structures but also that of IC packaging structures.

In the present paper, we review the previous research on the methodologies for the thermal fatigue evaluation of power modules. The review is made based on a mechanical engineering viewpoint such as fracture mechanics and strength of materials. In addition, we will propose the research policy for securing the structural integrity of power modules in the design phase.

## 2 Failure of Bonding Wire

Here, we deal with heel cracking and wire-liftoff as the failure modes of a bonding wire.

### 2.1 Heel Cracking.

When a wire is connected to different components with different CTEs, it experiences relative motion during temperature variation. Under the repeated temperature cycles, high local stress is repeatedly generated at the heel part of a bonding wire, and heel cracking is caused as a fatigue phenomenon. In this case, the number of cycles to fatigue failure (fatigue life or lifetime) $Nf$ is given by the Coffin–Manson law [9,10] as follows [1113]:
$Nf=C(Δεp)p$
(1)
where $Δεp$ is the plastic strain range per cycle, and C and p are material constants. At elevated temperature, creep strain is induced in addition to plastic strain, the Coffin–Manson law, Eq. (1), is modified by changing $Δεp$ into the inelastic strain range per cycle $Δεin$ given by
$Δεin=Δεp+Δεc$
(2)

where $Δεc$ is the creep strain range per cycle. $Δεp$ and $Δεc$ can be obtained by the inelastic finite element analysis considering both plastic and creep strains.

Another methodology for predicting the lifetime of heel cracking is an energy approach [14]. Celnikier et al. propose the following equation for the lifetime estimation of an aluminum ribbon [14]:
$Nf(I)=wpcr/Δwp(I)$
(3)
where $Δwp(I)$ is the plastic energy density per cycle under current I, and $wpcr$ is the limit of the plastic energy density for heel cracking determined from $Nf(50A)=250,000$ cycles. In Ref. [14], the lifetimes of heel cracking obtained from Eq. (3) are compared with those obtained from the Coffin–Manson law, Eq. (1). Although the authors of Ref. [14] conclude that the results obtained from both approaches are in good agreement, the agreement is not good from a quantitative viewpoint. Such discrepancy results from the method for determining $wpcr$. If Eq. (3) is valid for an arbitrary value of I, the following failure model should be true for an arbitrary value of I:
$Nf(I)=C(Δwp(I) )−1, (C: constant)$
(4)
As shown in Eq. (1), the failure model based on $Δwp(I)$ should be generally the following form:
$Nf(I)=C(Δwp(I) )p$
(5)

There is no logical necessity of p = –1 for an arbitrary value of I.

References [4] and [5] state that the heel cracking of a bonding wire rarely occurs in advanced IGBT multichip modules, although it can be observed mainly after long endurance tests in the case where the ultrasonic bonding process is not optimized, and conclude that the main failure mode of a bonding wire is wire-liftoff. According to the recent experimental research [15], wire-liftoff always occurs in the IGBTs molded by silicone-gel, while, in the IGBTs molded by resin, wire-liftoff occurs for large temperature range ($ΔT=80 °C, 100 °C$) and both wire-liftoff and heel cracking is possible to occur for small temperature range ($ΔT=60 °C$). Resin constrains the movement of a bonding wire more strongly than silicone-gel, so that the stress and strain in a bonding wire are greatly different between the silicone-gel molded IGBTs and the resin molded IGBTs. This is the reason why the failure mode of a bonding wire is different between these two types of IGBTs. Heel cracking has been recognized again when a molding material is changed from silicone-gel to resin. For the resin-molded IGBTs, similar heel cracking is observed by Choi et al. [16] and Zeng et al. [17].

The residual stresses induced in a bonding wire during the wire-bonding process has an impact on the lifetime of heel cracking. The scientific knowledge of the impact of residual stresses on fatigue crack propagation rate was gained by investigating the midair explosion accident of Comet I, the first commercial jet airliner, over the Mediterranean Sea, which occurred in 1954 [18]. The following scientific knowledge was gained from the accident investigation. The fatigue crack propagation rate accelerates or decelerates according as the residual stress component perpendicular to the crack propagation direction is tensile or compressive. It is therefore expected that crack initiation lifetime becomes longer under residual compressive stress and shorter under residual tensile stress. According to Ref. [14], the residual stress induced in a bonding wire during the wire-bonding process has an impact on reducing heel cracking lifetime. This implies that the region around heel cracking is dominated by residual tensile stress. The wire-bonding process is composed of a prepress step, in which a wire is attached and pressed on the surface of a chip/substrate and an ultrasonic step, in which ultrasonic vibration is added to the bonding part of a wire, keeping a prepress load constant. We need information on the residual stress state around the location where heel cracking occurs to discuss the impact of residual stress on the lifetime of heel cracking. It seems that the authors of Ref. [14] did not carry out the stress analysis of the wire-bonding process to obtain a residual stress state around the location where heel cracking occurs. Conclusively sufficient information is not provided in Ref. [14] to discuss the relation between the residual stress induced during the wire-bonding process and the lifetime of heel cracking. Examples of the stress analysis and the stress measurement during the wire-bonding process are given in Refs. [19] and [20].

### 2.2 Wire-Liftoff

#### 2.2.1 Conventional Failure Models.

As described in the Sec. 2.1, wire-liftoff is a main failure mode for aluminum wires of IGBTs, and a lot of papers have been published on this failure mode. In these papers, thermal fatigue life has been related to a temperature ranges. The fatigue life $Nf$ in a wire-liftoff phenomenon has been represented by two parameters, the temperature range per cycle $ΔT$ and the mean temperature Tm [3]. In a mathematical expression, it is given by
$Nf=C1(ΔT)p1$
(6)

where $p1$ is a material constant, and C1 is a function of Tm. Moreover, it is expected that the temperature swing duration $tΔT$ has an impact on the lifetime prediction [21], and C1 is a function of both Tm and $tΔT$.

From a mechanical engineering viewpoint, the fatigue life $Nf$ is given by the Coffin–Manson law as
$Nf=C2(Δεp)p2$
(7)
If the elastic strain range per cycle $Δεe$ is very small compared with the plastic strain range per cycle $Δεp$, Eq. (6) is equivalent to the Coffin–Manson law under a pure thermal fatigue condition without mechanical loading [3]. As pointed out by Zhang [22], Eq. (7) gives erroneous results when $Δεe$ cannot be neglected in comparison with $Δεp$, that is, in the case of high cycle fatigue. At elevated temperature, creep strain is induced in addition to plastic strain. In such a case, the Coffin–Manson law, Eq. (7), is modified using the inelastic strain range per cycle $Δεin$(= $Δεp+Δεc)$ as follows [23]:
$Nf=C3(Δεin)p3$
(8)

where $p3$ and C3 are material constants. In the thermal fatigue of a bonding wire subjected to repeated temperature variations, Eq. (8) should be used if the effect of creep strain cannot be neglected. Unlike $ΔT$ in Eq. (6), $Δεin$ in Eq. (8) is a physical quantity difficult to measure experimentally, but its accurate value can be obtained from the finite element analysis considering both plastic and creep deformations.

The $ΔT$-based failure models have been developed and used exclusively to estimate the wire-liftoff lifetime for power modules by European researchers. That is not a consensus for researchers engaged in the reliability studies of IC electronic packaging, in which Eq. (8) is exclusively utilized. Why does there such difference exist? In the present authors' opinion, main concern is directed to monitoring power modules under operation in the power electronics community, while the lifetime estimations of IC packages in the design phase are of crucial importance in the IC community. An enormous number of IC packages are supplied to the market after a short R&D period and their services periods are relatively short compared with power modules'. From a viewpoint of economic risk management, the lifetime estimation of IC packages in the design phase is more important than their in-service monitoring. $ΔT$ is easy to measure for monitoring power modules under operation, and various failure models are proposed based on Eq. (6), as shown in Ref. [22]:

• Coffin–Manson model [24]
$Nf=AΔT −n$
(9)
• General Coffin–Manson model [25]
$Nf=A(ΔT−ΔT0)−n$
(10)
• Modified Coffin-Manson model [3]
$Nf=AΔT −nexp(Ea/kbTm)$
(11)
• General modified Coffin–Manson model [22]
$Nf=A(ΔT−ΔT0)−n exp (Ea/kbTm)$
(12)
• Bayerer's model [26]
$Nf=AΔT −nexp(β2/Tmin)tonβ3Iβ4Vβ5Dβ6$
(13)
• General Bayerer's model [22]:
$Nf=A(ΔT−ΔT0)−nexp (β2/Tmin)tonβ3Iβ4Vβ5Dβ6$
(14)

Here, kb is the Boltzmann constant. In these failure models shown in Eqs. (10)(14), additional parameters are included to accommodate Eq. (9) to various operating conditions. Such additional parameters are the temperature range $ΔT0$ corresponding to the elastic strain range, the mean absolute temperature Tm, the minimum absolute temperature $Tmin$, the power-on-time ton, the current per wire bond I, the blocking voltage of the chip V, and the diameter of bonding wire D. The additional parameters are needed, because the effects of plastic and creep strains on the lifetime of power modules are not represented only by the temperature range $ΔT$. The constants A, n, Ea,$β2$, $β3$, $β4, β5$, and $β6$ are those determined from experimental data. As the number of additional parameters increases, we need more experimental data to determine the constants relevant to the additional parameters.

In the failure model based on $Δεin (=Δεp+Δεc)$, Eq. (8), the effect of the temperature parameter $ΔT0$ is taken into account, because the elastic strain range is excluded in $Δεin$. Other temperature parameters Tm and Tmin are parameters relevant to both plastic and creep strains. The time parameter ton is a parameter relevant to creep strain. In several studies on die attach cracking [2729] described in Sec. 3 of this paper, the thermal cycling frequency f is introduced into the failure model based on $ΔT$ as a time parameter to take account of the effect of creep strain on the lifetime of power modules. These time parameters have an impact on creep strain. If we perform thermoplastic-creep analysis using a temperature distribution obtained from electrothermal analysis, the effects of these time factors are included in $Δεin.$ The effects of other parameters I, V, and D are also considered in the same way as the time factors. Consequently, the effects of all parameters included in the failure models based on $ΔT$ are represented by $Δεin$, so that no additional parameter is required, when the failure model based on $Δεin$, Eq. (8), is used for estimating the lifetime of power modules. On the other hand, it must be considered what type of the failure model shown in Ref. [22] is selected for lifetime estimation, when the failure model based on $ΔT$, Eq. (6), is used. It is concluded that the failure model based on $Δεin$ is superior to those based on $ΔT$, because the former being simpler than the latter, the number of constants to be determined by experimental data is less in the former model than in the latter ones. As a result, the former require a smaller number of experimental data than the latter to formulate failure models. Furthermore, as pointed out in Ref. [8], the values of the constants relevant to the parameters in the failure models based on $ΔT$, Eqs. (9)(14), are determined for a specific power module, and they would need to be recalibrated for another power module type. On the other hand, such recalibration is not required for the failure model based on $Δεin$, Eq. (8), regardless of power module dimensions if power modules are composed of the same materials.

Wire-liftoff is regarded as a crack propagation phenomenon due to thermal fatigue, and the strain intensity factor is applied to a wire-liftoff phenomenon [30]. The crack propagation rate per cycle is given by the following Paris law:
$dadN=C(ΔKεeq)n$
(15)

where a, N, and $ΔKεeq$ denote the crack length, the number of cycle, and the equivalent strain intensity factor range considering both the mode I (opening mode) and the mode II (in-plane shearing mode), respectively, and C and n are material constants. The remaining length of a jointed section is calculated based on Eq. (15) after constant temperature cycles, and compared with experimental data. Both results are relatively in good agreement. Although this methodology is reasonable from strength of materials, this cannot be applied to the failure at elevated temperature, in which plastic and creep deformations are significant.

#### 2.2.2 Failure Models at Elevated Temperatures.

Power devices using wide bandgap semiconductors such as SiC and GaN are expected to operate over 200 $°C$. Matsunaga and Uegai [31] report that aluminum wire-liftoff lifetime becomes saturated in the wire–bond junction temperatures more than $200 °C$ as shown in Fig. 2. The similar phenomenon is observed by Yamada et al. [32] and Agyakwa et al. [33]. In the lifetime estimation based on Eq. (6), much more additional temperature parameters would be necessary to represent the lifetime saturation phenomenon. The inelastic strain range $Δεin$ increases monotonically with increase in temperature because of increase in creep strain and increase in plastic strain due to the reduction of the yield stress of an aluminum wire. According to the lifetime estimation based on Eq. (8), the lifetime of aluminum wire-liftoff decreases monotonically with increase in temperature, and Eq. (8) does not explain the saturation phenomenon of the aluminum wire-liftoff lifetime shown in Fig. 2. Yang et al. [34] perform a study to explain this phenomenon. They propose a damage model of the interface between an aluminum wire and a silicon die which includes the effect of temperature and time-dependent material properties. The proposed model takes account of the damage removal due to recrystallization at high temperature as well as the damage build-up. This model has five parameters. The values of the constants relevant to the parameters are determined so as to represent experimental results of a particular system [33]. The time variations of interface damage and crack length can be obtained by solving simultaneous ordinary differential equations. It is not certain that the values of the constants determined so as to represent a particular system has generality, that is, they are applicable to a system with the same material as a particular system but different shape and dimensions.

Fig. 2
Fig. 2
Close modal

Hereafter, much attention is paid to the failure models of wire-liftoff applicable to the wire-bond junction temperature over $200 °C$, in which the aluminum wire-liftoff lifetime becomes saturated. Any failure models described above are not suitable for representing such a saturation phenomenon. Above $200 °C$ large stress relaxation occurs in an aluminum wire due to the large creep deformation and the large plastic deformation resulted from drastic reduction of the yield stress. The impact of stress relaxation on the wire-liftoff lifetime should be taken into account in the failure models.

##### 2.2.2.1 Inelastic strain energy density range $ΔWin$⁠.
The inelastic strain energy density range per cycle $ΔWin$ is the area of a cyclic stress–inelastic strain hysteresis loop, and denoted by the following integral:
$ΔWin=∮Cσijdεijin$
(16)
where $σij$ and $εijin$ are the stress tensor and the inelastic strain tensor, respectively, and C denotes an integral path corresponding to a hysteresis loop of a stress–inelastic strain curve. The failure model is given by
$Nf=C4( ΔWin)p4$
(17)

where $p4$ and $C4$ are material constants.

Equation (17) is proposed by Morrow [35] as an energy model for low cycle fatigue. This model contains not only inelastic strain but also stress, so that $ΔWin$ is a physical quantity taking account of stress relaxation. Kanda et al. [36] show that Eq. (17) is equivalent to Eq. (8) in the case where a steady-state stress–inelastic stain loop is obtained under a cyclic loading condition. The energy model, Eq. (17), is applied to the lifetime prediction of solder joints in electronics packaging [29,3739], and also to a wire-liftoff lifetime [40].

##### 2.2.2.2 Nonlinear fracture mechanics parameter T* range $ΔT$*.
Other possible physical quantities useful to the lifetime prediction of thermal fatigue at elevated temperatures are fracture mechanics parameters. They fall into two categories. One is a linear fracture mechanics parameter such as the stress intensity factor K. The other is a nonlinear fracture mechanics parameter such as the J-integral. At elevated temperature, large inelastic strain occurs due to the reduction of yield stress and creep deformation, so that nonlinear fracture mechanics parameters should be utilized to discuss the fracture at elevated temperatures. The J-integral proposed by Rice [41] is given by
$J=∫Γ0[Wn1−tiui,1]dΓ$
(18)
where W, ni, and ti denote the stain energy density, the outward normal vector and the traction vector, respectively, and W and ti are defined by
$W=∫0εijσijdεij$
(19)
$ti=σijnj$
(20)

$Γ0$ is an arbitrary integral path enclosing the crack tip, as shown in Fig. 3. The physical meaning of the J-integral is the energy flux flowing into a fracture process zone in the vicinity of a crack tip. Under the condition of the deformation theory of plasticity, the J-integral has path-independence which enables calculating the accurate value of the J-integral by using far-field values of $σij$ and $εij$ around a crack tip. This feature is important because a crack tip is a singular point where $σij$ and $εij$ diverge.

Fig. 3
Fig. 3
Close modal
Afterward the J-integral is extended to the J-integral range $ΔJ$ for the application of fatigue crack growth under cyclic loading [4245]. $ΔJ$ is defined as follows:
$ΔJ=∫Γ0[ΔWn1−ΔtiΔui,1]dΓ$
(21)
where
$ΔW=∫0ΔεijΔσijdΔεij$
(22)
In Eq. (22), $Δσij$ and $Δεij$ are the stress range and the strain range, respectively, and defined as the difference between a current state and a reference state on the loading process in cyclic loading
$Δσij=σij−σij(0)$
(23)
$Δεij=εij−εij(0)$
(24)
The superscript (0) denotes a reference state. Similarly, the ranges of traction $Δti$ and displacement $Δui$ are defined by
$Δti=ti−ti(0)$
(25)
$Δui=ui−ui(0)$
(26)

According to Lamba [42], Wüthrich [44], and Tanaka [45], the J-integral range $ΔJ$ has the same path-independence as the J-integral, if a crack is open and the strain energy density range $ΔW$ is given by a single valued function of $Δεij$. Kubo et al. [46] has a detailed discussion on the path-independence of $ΔJ$ when crack closure occurs during a load cycle.

The J-integral defined by Eq. (18) loses path-independence for thermal stress problems, creep deformation problems, flow theory of plasticity problems, and dynamic problems. For such problems, Atluri et al. [47,48] propose a path-independent nonlinear fracture mechanics parameter T*-integral defined by
$T* =∫Γε[Wn1−tiui,1]dΓ=∫Γ0[Wn1−tiui,1]dΓ−∫A0−Aε[W,1−(σijui,1),j]dA$
(27)

where the integral paths $Γε$ and $Γ0$, and the domains $A0$ and $Aε$ are shown in Fig. 3. Here, $Γ0$ is an arbitrary integral path enclosing a crack tip, and $Γε$ is an integral path enclosing a fracture process zone existing near a crack tip. For a problem subjected to the deformation theory of plasticity, in which no unloading occurs, the T*-integral is identical to the J-integral, because the second term of the right-hand side in Eq. (27) becomes equal to zero. Thus, the T*-integral is generalization of the J-integral for various loading cases.

For thermal fatigue problems under repeated temperature cycles, in which both thermoelastic–plastic strain and creep strain occur, the authors of Ref. [40] newly propose the T*-integral range $Δ$T* defined in the same way as $Δ$J
$ΔT*=∫Γε[ΔWn1−ΔtiΔui,1]dΓ=∫Γ0[ΔWn1−ΔtiΔui,1]dΓ−∫A0−Aε[ΔW,1−(ΔσijΔui,1),j]dA$
(28)
This expression is valid in the case where a crack is open. When a crack is closed, $Δ$T* in Eq. (28) is modified by adding the correction term expressing a crack closure effect. For more details refer to Ref. [40]. The failure model based on $Δ$T* are given by
$Nf=C5(ΔT*)p5$
(29)
##### 2.2.2.3 Application of failure models at elevated temperatures.

The results of application of the failure models at elevated temperature, Eqs. (17) and (29), are shown in Ref. [40], where the wire-liftoff lifetimes predicted by these models are compared with the experimental data of Matsunaga and Uegai [31]. Figure 4 shows a finite element model of the wire bonding structure. The part of wire bonding is subjected to temperature cycles with the initial temperature of 25 $°C$, the period of 2 s and the fixed temperature range $ΔT$ of $80 °C$. Finite element thermoelastic–plastic-creep analyses were performed for five cases of the maximum temperature Tmax, that is, Tmax = 105, 150, 200, 250, and 300 $°C$.

Fig. 4
Fig. 4
Close modal
In the practical application of $ΔWin$ given by Eq. (17) to the finite element analysis, we employ the area averaged inelastic strain energy density range $ΔWin¯$ defined by
$ΔWin¯=∑k=1N{(ΔWin)k×Ak}/∑k=1NAk$
(30)

where N and $Ak$ indicate the number of finite elements contained in the selected domain and the area of element k, respectively. In the lifetime calculation, $ΔWin$ in Eq. (17) should be replaced by $ΔWin¯$.

The exact calculation of $Δ$T* can be done by using Eq. (28). There is no function of calculating the exact $Δ$T* values in a commercial finite element program such as marc, ansys, etc. Thus, the following approximate calculation method for evaluating $Δ$T* is employed to apply the commercial finite element program MSC.MARC2015 to the study on a wire-liftoff problem:

1. Smelser and Gurtin [49] show that the J-integral defined by Eq. (18) is valid for a bimaterial body if a crack is located along a bond line. Thus, the J-integral values can be calculated for several integral paths using a standard function of the general purpose finite element program MSC MARC2015.

2. The near-field J-integral value at the integral path $Γε$ in the vicinity of a crack tip is obtained by using the extrapolation of the far-field J-integral values calculated in the process (1). As shown in Eq. (27), the near-field J-integral value is equivalent to the T*-integral value.

3. The near-field $ΔJ$-value is calculated from
$ΔJ=Jmax−Jmin$
(31)
where $Jmax$ and $Jmin$ are the maximum and the minimum of near-field J-integral values during per loading cycle.
4. According to Eq. (28), the near-field $ΔJ$-value is regarded as $Δ$T*-value.

Theoretically the J-integral range $ΔJ$ defined by Eq. (31) is incorrect [50] but such a J-integral range is used in a lot of papers [5156], because fatigue crack propagation rate is well correlated by it. Moreover, Ref. [57] shows that the $ΔJ$-value calculated from Eq. (31) agrees reasonably well with that calculated from the exact definition of $ΔJ$, Eq. (21), for a thermal fatigue problem.

The area average of the inelastic strain energy density range $ΔWin¯$ defined by Eq. (30) are calculated from the finite elements located on the extended line of a crack from the crack tip X =0 to Xave. Figure 5 shows the variation of $ΔWin¯$ with distance from crack tip. It is clearly found that $ΔWin¯$ depends on the domain selected for the calculation of $ΔWin¯$. Thus the domain selection cannot be uniquely determined. The similar situation occurs when the inelastic strain range $Δεin$ is utilized in the lifetime estimation.

Fig. 5
Fig. 5
Close modal

The variations of the normalized number of cycle to failure with the maximum temperature $Tmax$ are shown in Fig. 6, where the results obtained from failure models based on $Δεin$, $ΔWin¯$, and $Δ$T* are compared with the experimental data Matsunaga and Uegai [31]. In comparison with the experimental data, the failure model based on $Δ$T* provides the best result.

Fig. 6
Fig. 6
Close modal

## 3 Die Attach Cracking

A power device chip is joined to a substrate by die attach. A die attach layer is subjected to repeated thermal stress due to the CTE mismatch between a semiconductor chip and a substrate, when temperature varies repeatedly. Such repeated thermal stress induces the thermal fatigue of a die attach layer, which results in die attach cracking. Although several review papers were published on the methodologies for estimating the thermal fatigue lifetime of solder joints in electronics packaging and the constitutive equations of solder materials [5861], we pay attention to the papers on the lifetime estimation of die attach cracking in power modules, and review them critically.

Almost all the previous papers on die attach cracking deal with crack propagating in the horizontal direction of a die attach layer, and research on such die attach cracking is reviewed in Secs. 3.1 and 3.2. The die attach cracking propagating in the thickness or vertical direction of a die attach layer has been recently observed in power modules operated under elevated temperatures. Such a type of die attach cracking is also treated in Sec. 3.3.

Total lifetime of die attach cracking is generally the sum of crack initiation lifetime and crack propagation lifetime. Crack propagation lifetime can be neglected compared with total lifetime in the cases of wire-liftoff and the fatigue failure of solder bumps in flip-chip packaging, because the dimension of a wire jointed on a chip surface and that of solder bumps in flip-chip packaging are at most a few millimeters. On the other hand, the dimension of a jointed part is a few centimeters in the case where the surface of a power device chip fully adheres to a substrate. Thus, the ratio of crack propagation lifetime to total lifetime becomes large. Here, we describe the lifetime estimation methodologies of die attach cracking without and with consideration of crack propagation.

### 3.1 Lifetime Estimation Methodologies Without Consideration of Crack Growth

#### 3.1.1 Lifetime Estimation Formulae Using Temperature Range.

Inspired by the paper of Norris and Landzberg [27], the following estimation formulae using the temperature range $ΔT$ are proposed:

• Refs. [27] and [62]:
$Nf=AfmΔT−nexp(Ea/kbTmax)$
(32)
• Refs. [62] and [63]:
$Nf=AfmΔT−nexp(Ea/kbTmax)[corr(ΔT)]−1/c$
(33)
• Ref. [64]:
$Nf=A(1/thot)mΔT−nexp(Ea/kbTmax)$
(34)
where A, m, n, and $Ea$ are the material constants determined from experimental data, $kb$ denotes the Boltzmann constant, and $Tmax$ is the maximum absolute temperature. Equation (32) is proposed for Pb–Sn eutectic solder alloys. Equation (32) is extended to the application of Sn–Ag–Cu lead-free solder alloys, and Eqs. (33) and (34) are such modified versions of Eq. (32). In Eqs. (32) and (33), f denotes the frequency of temperature cycle, and $[Corr(ΔT)]$ is a correction term given by
$corr(ΔT)=aln(ΔT)+b$
(35)

where a and b are package and solder material specific constants provided in Ref. [65], and c denotes a constant depending on temperature profile [65]. The parameter $thot$ in Eq. (34) is the time per cycle, during which the component is hot. As described in Sec. 2.2, additional parameters are added to the lifetime estimation formulae based on $ΔT$ to express complex phenomena.

We can define the acceleration factor AF as follows:
$AF=(Nf)2(Nf)1$
(36)
where the subscripts 1 and 2 indicate separate states. Based on this equation, the lifetime in the service condition of a power module $(Nf)2$ can be estimated from that of an acceleration test $(Nf)1$. The expression of AF for Eq. (32) is given by
$AF=(Nf)2(Nf)1=(f2f1)m(ΔT1ΔT2)nexp [(Ea/kb){1/(Tmax)2−1/(Tmax)1}]$
(37)

In deriving the above equation, we utilize the fact that the values of A, m, n, and $Ea$ are identical both in the state 1 and in the state 2. The formulae similar to Eq. (37) can be obtained for Eqs. (33) and (34).

#### 3.1.2 Lifetime Estimation Formula Using Inelastic Strain Range.

The lifetime predictions can be done based on the Coffin–Manson law, Eq. (8), that is,
$Nf=C3(Δεin)p3$
(38)

Being different from the temperature range $ΔT$, the inelastic strain range $Δεin$ is difficult to measure, but its accurate value can be obtained from inelastic finite element analysis.

The lifetime estimation formulae based on $ΔT$, Eqs. (32)(34), include the maximum temperature $Tmax$, the temperature cycle frequency f, and the hot dwell time per temperature cycle $thot$ in addition to $ΔT$ to accommodate these formulae to complex operating conditions. The constants relevant to these parameters, A, m, n, and $Ea$, are determined so as to fit experimental data. All the additional parameters $Tmax$, f, and $thot$ are related to plastic and creep strains, so that the inelastic strain range $Δεin$ obtained from inelastic finite element analysis automatically includes the effects of $Tmax$, f, and $thot$. Consequently, no additional parameter is needed in the failure model based on the inelastic strain range $Δεin$ even under complex operating conditions.

The lifetime estimation formula based on $Δεin$, Eq. (38), has been widely used for solder joints in IC packaging since 1990s [23,6670], but the applications of the failure model based on $Δεin$, Eq. (38), to the lifetime of die attach cracking in power modules [7174] are fewer than those of the failure models based on $ΔT$, Eqs. (32)(34).

#### 3.1.3 Lifetime Estimation Formula Using Inelastic Strain Energy Density Range.

Although the estimation formula using the inelastic strain energy density range is shown in the wire-liftoff lifetime, this formula can be also applied to predicting the die attach cracking lifetime. The formula is given again as follows:
$Nf=C4( ΔWin)p4$
(39)

The above formula has been applied to the lifetime predictions of small bumps in IC packages [37,68,70,7578], in which crack propagation lifetime can be neglected in comparison with total lifetime. In Refs. [29], [39], [79], and [80], the formula, Eq. (39), is applied to the lifetime prediction of die attach cracking in power modules. The size of a die attach layer is a few centimeters and much larger than the wire bonding size, so that the estimation formula for die attach cracking lifetime, Eq. (39), is usually used together with the formula for crack propagation lifetime as described later.

In the practical application of Eq. (39), $ΔWin$ is replaced by the area/volume-averaged inelastic strain energy density range $ΔWin¯$ given by Eq. (30). In such a case, we must consider what domain is selected as the inelastic strain energy density range concerned with thermal fatigue failure. Fatigue failure generally occurs at a stress concentration part. Thus, we should select a domain including a stress concentration part, but we cannot determine such a domain uniquely. Nakajima et al. [81] deal with a crack in a homogeneous material, assuming that stress and strain around a crack tip are governed by the HRR-fields [82,83]. They consider a square domain with the side length of Larea, the center of which is located at the crack tip. Let $ΔWin$ be the inelastic strain energy density range averaged in the square domain with Larea. The following relationship holds, because $ΔWin$ and Larea are inversely proportional to each other
$ΔWin−c=ΔWin·Larea$
(40)

where $ΔWin−c$ is a proportional constant independent of $Larea$. It is therefore possible to discuss die attach cracking lifetime using the new parameter $ΔWin−c$ independent of the domain selected for the calculation of $ΔWin$. Equation (40) is true in the cases of plastic deformation under the deformation theory of plasticity and steady-state creep deformation. Otherwise $ΔWin−c$ is domain-dependent, for example, in the cases of plastic deformation with unloading and transient creep deformation where the stress and strain around a crack tip is not governed by the HRR-fields. Also the stress singularity at a notch and a corner is not represented by HRR-fields, so that $ΔWin−c$ is domain-dependent.

### 3.2 Lifetime Estimation Methodologies With Consideration of Crack Propagation.

Fatigue crack propagation rate, that is, crack propagation per cycle da/dN is written by the Paris law [84]
$dadN=C1(ΔK)n1$
(41)
where $C1$ and $n1$ are material constants, and $ΔK$ denotes the stress intensity factor range per cycle. Equation (41) is valid under the condition that small-scale yielding is satisfied around a crack tip. A crack in a ductile material does not satisfy this condition. In such a case, the original Paris law Eq. (41) should be modified using the inelastic strain range $Δεin$ defined by Eq. (2), the inelastic strain energy density range $ΔWin$ defined by Eq. (16), and the J-integral range $Δ$J defined by Eq. (21). They are given by
$dadN=C2(Δεin)n2$
(42)
$dadN=C3(ΔWin)n3$
(43)
$dadN=C4(ΔJ)n4$
(44)
As $Δ$J loses path-independence in the cases of plastic deformation with unloading and creep deformation, the T*-integral range $Δ$T* should be used instead of $Δ$J
$dadN=C5(ΔT*)n5$
(45)
The lifetime estimation formulae shown in Sec. 3.1 basically provide fatigue crack initiation lifetime and good predictions of the fatigue lifetime for small-sized solder joints such as microbumps. The die attach size of power modules is so large that crack propagation lifetime cannot be neglected. Let $N0$ and $N1$ be fatigue crack initiation lifetime and crack propagation lifetime, respectively, then total lifetime N is given by
$N=N0+N1=N0+Lc(da/dN)$
(46)

where $Lc$ denotes the critical crack length, at which a rapid increase of temperature or a rapid variation of electric quantities occurs.

#### 3.2.1 Lifetime Estimation Methodology Using Inelastic Strain Range.

In this methodology the formulae Eqs. (38) and (42) are, respectively, used for crack initiation lifetime and for crack propagation lifetime.
$Formula for crack initiation lifetime: N0=α1(Δεin)β1$
(47)
$Formula for crack propagation rate:dadN=γ1(Δεin)δ1$
(48)

The point at issue for the application of the above formulae is as follows. As shown in Sec. 2.2.2.3, a fatigue crack arises at a stress concentration region. There is arbitrariness in selecting the evaluation point of $Δεin$, because a crack tip is a stress and strain singular point after crack initiation. In the original Paris law shown in Eq. (41), the stress intensity factor range $ΔK$ is utilized to avoid such arbitrariness. The stress intensity factor $K$ is one of the indicator representing the intensity of stress singularity. The $J$-integral and T*-integral, respectively, given in Eqs. (44) and (45) are also the similar indicators to the stress intensity factor K, and determine the intensity of stress singularity uniquely, if they have path-independent nature.

Déplanque et al. [85] applied this methodology to the fatigue lifetime estimation of the die attach cracking for a Sn–Pb eutectic solder alloy and a Sn–Ag–Cu lead free solder alloy in a power-MOS transistor. They used the accumulated inelastic (creep) strain averaged from a crack tip (x =0) to a certain point (x = u) along the extended line of a crack. Although the definition of accumulated inelastic (creep) strain is not clear in this paper, it is equivalent to the inelastic (creep) strain range $Δεin$ used in Eqs. (47) and (48) if it represents the accumulated inelastic (creep) strain per temperature cycle. Other researchers also use Eq. (48) to obtain the crack propagation lifetime of a power module [8688]. They divide a crack propagation path into multiple subsections, each of which is a side of the finite element allocated along a crack path, and the lifetime of crack propagation that makes advance of each subsection is obtained from Eq. (48). It is not clear whether crack initiation lifetime is considered in this methodology. If it is not considered, the present methodology provides shorter lifetime.

Japanese researchers propose a methodology for simulating not only fatigue crack initiation but also a crack propagation path [89,90]. In this methodology, a damage parameter D is obtained both from the Coffin–Manson law and the linear cumulative damage rule (Miner's rule) based on the finite element stress analysis taking account of both plastic and creep strains, and the elements satisfying $D=1$ are removed by reducing the Young's modulus at a small value compared with an undamaged material. This methodology is utilized to predict the crack propagation path of QFP solder joints [89] and BGA solder joints [90] under thermal fatigue conditions, and also applied to the crack propagation in a die attach layers of a IGBT power module [91,92]. This methodology can deal with the propagation of a two-dimensional crack in a three-dimensional structure, although the methodology employed in Refs. [8588] deals with only that of a one-dimensional crack in a two-dimensional structure.

#### 3.2.2 Lifetime Estimation Methodology Using Inelastic Strain Energy Density Range.

In this methodology, the formulae Eqs. (39) and (43) are, respectively, used for crack initiation lifetime and for crack propagation lifetime
$Formula of crack initiation lifetime: N0=α2(ΔWin)β2$
(49)
$Formula of crack propagation rate:dadN=γ2(ΔWin)δ2$
(50)

This methodology is proposed by Darveaux for evaluating the lifetime of die attach cracking in IC packages [93,94], and is used by other authors not only for the estimation of die attach cracking lifetime of IC packages [95,96] but also for that of power modules [9799]. As described in Sec. 2.2.2.3 and 3.1.3, in the practical application of $ΔWin$ to Eqs. (49) and (50), Darveaux proposed the use of volume averaged $ΔWin$, $(ΔWin)ave$, which is basically equivalent to $ΔWin¯$ given by Eq. (30) described in 2.2.2.3. Then what region is chosen to calculate $(ΔWin)ave$? Any answers have never been provided for this question by the authors using the inelastic strain energy density range, except that the region where the highest stress is expected should be always included.

Kariya et al. [100] obtain the constants $γ2$ and $δ2$ in a formula of crack growth rate Eq. (50) for sintered Ag nanoparticles which is expected as a die attachment material of next-generation power modules. In Ref. [100], they use the volume-averaged $ΔWin$, the value of which depends on the domain selected for the calculation. So they propose the use of $ΔWin−c$ defined by Eq. (40) instead of volume-averaged $ΔWin$, because $ΔWin−c$ is independent of the domain selected for the calculation of $ΔWin$ and uniquely determined [101]. In this case, fatigue crack propagation rate is given by
$dadN=γ2(ΔWin−c)δ2$
(51)

They perform fatigue crack propagation tests under a displacement-controlled pulsating tensile–tensile mode using single-edge notched specimens made of sintered Ag nanoparticles and determine the constants $γ2$ and $δ2$ in Eq. (51) from experimental data. As described in Sec. 3.1.3, the use of the formula Eq. (51) is limited to plastic deformation without unloading obeying the deformation theory of plasticity and steady-state creep deformation.

#### 3.2.3 Lifetime Estimation Methodology Using Fracture Mechanics Parameters.

The formulae of crack propagation rate using fracture mechanics parameters are Eqs. (41), (44), and (45). The inelastic strain range $Δεin$ and the inelastic strain energy density range $ΔWin$ depend on the point or domain selected for their estimations and they are not determined uniquely. On the other hand, the stress and strain around a crack tip are uniquely determined by fracture mechanics parameters, and $Δ$J and $Δ$T* defined by path integral have an advantage of numerical analysis because the values of $Δ$J and $Δ$T* can be evaluated from the far-field mechanical quantities if they have path-independence. The parameter $ΔWin−c$ proposed by Kariya et al. can be also regarded as one of the fracture mechanics parameters.

Thermal fatigue lifetime analysis of a power module is performed using the original Paris law, Eq. (41), as a formula of crack propagation lifetime together with the Coffin–Manson law, Eq. (38), as a formula of crack initiation lifetime [102]. According to the results of the analysis, almost all the total lifetime is occupied by the crack initiation lifetime, and the crack propagation lifetime is only 3.6% of the total lifetime. The die attachment material considered in the analysis is a 95Pb–5Sn solder alloy. The use of the original Paris law written by the stress intensity factor range $Δ$K is not valid because a 95Pb–5Sn solder alloy is ductile and the small-scale yielding condition is not satisfied, and provide shorter crack propagation life time.

Pao performs pioneering research on the thermal fatigue life prediction of solder joints in electronic packaging [103]. He uses the $Δ$J-based equation, Eq. (44), as a formula of crack propagation rate due to thermal fatigue and the C*-based equation [104] as a formula of crack propagation rate due to creep, that is
$Formula of crack propagation rate due to thermal fatigue:dadN=γ3(ΔJ)δ3$
(52)
$Formula of crack propagation rate due to creep:dadt=γ3′(C*)δ3′$
(53)

where t denotes time, and the total lifetime $Nf$ is determined from the number of cycles, at which a crack length reaches the critical one $Lc$. Although crack initiation life time is not considered in the analysis, the lifetime obtained from the analysis agrees well with experimental data. The reason for good agreement would be due to the fact that the critical crack length $Lc$ of one half of solder joint width, a large crack length, is assumed in the analysis.

Kariya and coworkers [105] carry out fatigue crack propagation tests of single-edge notch specimens made of sintered Ag nanoparticles under displacement-controlled tensile–tensile loading condition and the isothermal temperature condition of 140$°C$. They employ the $Δ$J-based relationship as a formula of fatigue crack propagation rate:
$dadN=γ3(ΔJ)δ3$
(54)

and the constants $γ3$ and $δ3$ are determined from experimental data. In applying Eq. (54), $ΔJ$ value is calculated from a simplified evaluation formula proposed by Asada et al. [106]. If thermal and creep deformations are taken into account, $Δ$J loses path-independence and cannot be uniquely determined like $Δεin$ and $ΔWin$.

The fracture mechanics parameters T* and $Δ$T* described in detail in Sec. 2.2.2.2 are path-independent and determined uniquely under thermal and creep deformations; thus valid fracture mechanics parameters for such deformations. The present authors of this paper recommend that the following formula be used to represent fatigue crack propagation rate under elevated temperature condition, in which thermal and creep deformations are significant in addition to plastic deformation:
$dadN=γ4(ΔT*)δ4$
(55)

A crack must be assumed in the methodologies for the lifetime estimation using fracture mechanics parameters such as $ΔT*$. Thus, an important problem in applying a fracture mechanics parameter to lifetime estimation is how to estimate fatigue crack initiation lifetime. The following are solutions of this problem:

1. Fatigue crack initiation lifetime is estimated by the methodologies without assuming a crack, that is, Eqs. (47) and (49).

2. Fatigue crack initiation lifetime $N0$ is obtained from the following formula:
$N0=α4(ΔT*)β4$
(56)

by assuming a short-length crack. In this case, a problem what length of a crack should be regarded as a short-length crack is left unresolved. One of the method to resolve this problem is to carry out sensitivity analysis of $N0$ by varying a crack length. We regard a crack with the length, below which the sensitivity of crack length to $N0$ becomes low, as a short-length crack.

#### 3.2.4 Other Lifetime Estimation Methodologies

##### 3.2.4.1 Continuum damage mechanics.

The continuum damage mechanics deals with the failure process from the generation and coalescence of microdefects such as voids to macrocrack propagation within the framework of the continuum mechanics by introducing a damage variable D ranging from zero to one. The values of D equal to zero and one represent no damage and full damage, respectively. D is usually defined as the ratio of the damaged area to the initial area. The damage variable D has a direct influence on stress and an indirect influence on constitutive equations of materials via stress. A detailed description of the continuum damage mechanics is given in the monograph published by Murakami [107]. In this approach, it is possible to deal with a failure problem of a structure without assuming a crack explicitly.

The applications of the continuum damage mechanics to the failure of solder joints are presented in Refs. [108] and [109]. In these papers, the damage levels of solder joints are obtained for simple test systems and loading conditions. The damage mechanics is applied to the die attach cracking of power modules [110,111]. These studies use the approach similar to that employed by Yang et al. in the estimation of wire-liftoff lifetime [34]. In accordance with Yang et al.'s approach, the factors impacting on the variation of D are expressed by the functions of the temperature T, the inelastic strain $εin$, the damage variable D, and the distance from a crack tip x, and the constants included in each function are determined from the experimental data on the relationship between the crack length and the number of thermal cycles shown in Ref. [112]. The time variations of damage variable and crack length can be obtained by solving simultaneous ordinary differential equations. In the continuum damage mechanics, the damage variable D is incorporated into the framework of the continuum mechanics and included in the governing equations of the continuum mechanics. On the other hand, in the approach used in Refs. [110] and [111], the selections of the functions for representing the damage variable D are empirical rather than theoretical. So it would not be suitable to classify this approach as the continuum damage mechanics.

Desai et al. [113116] apply another type of the continuum damage mechanics approach. Their approach is named the distributed state concept (DSC) approach based on the unified constitutive modeling and takes account of elastic, plastic and creep strains, microcracking, damage and degradation or softening, and postpeak stiffening under thermomechanical loading for materials. The basic assumption of the DSC approach is as follows: the deforming material element is the mixture of material parts in two reference state, relative intact (RI) and fully adjusted (FA) states. Under this assumption, the observed stress tensor, $σija$, is given by
$σija=(1−D)σiji+Dσijc$
(57)

where the superscripts a, i, and c denote the observed, RI and FA states, respectively, and D is a scalar disturbance function or the ratio of the change in disorder parameter to the original reference state, which is regarded as degradation/damage metric. In Refs. [113116], a degradation function D is empirically obtained by curve fitting to test data. This approach is applied to the thermal fatigue problems of Sn–Pb solder joints of electronic packaging. A good review on the DSC approach based on the unified constitutive modeling is given by Desai [117]. Furthermore Basaran and Yan [118] and Basaran and Tang [119] propose the unified mechanics theory, that is, the unification of Newton's universal laws of motion and laws of thermodynamics, and derive the expression of D from the entropy, a measure of disorder in the system. We can obtain the expression of D theoretically without test data in the unified mechanics theory. The DSC approach based on the unified mechanics theory is applied to low cycle fatigue [118], thermal fatigue [119], dynamic loading with harmonic vibration [120], and concurrent thermal cycle and dynamic loading [121]. Sophisticated derivations of both analytical and computational three-dimensional models based on the unified mechanics theory have been recently provided for predicting damage and fatigue life by a research team led by Basaran and coworkers [122].

We can easily incorporate the DSC approach into nonlinear finite element formulation [115,120], and obtain the variation of D in the process of fatigue using a nonlinear finite element analysis. Although the DSC approaches based on the unified constitutive modeling and the unified mechanics theory have not yet been utilized to estimate the lifetime of die attach cracking in power modules, they are the promising approaches. We are expecting that these approaches are utilized to structural integrity evaluation of power modules.

##### 3.2.4.2 Cohesive zone model.

The cohesive zone model (CZM) is a model for dealing with a crack propagation phenomenon in ductile materials. In this model, cohesive force acts on crack surfaces even after the separation of crack surfaces occurs, and a crack gradually propagates under the resistance of the cohesive force. The region of crack surfaces on which cohesive force acts is called a cohesive zone. It is assumed that the crack surfaces completely separate when cohesive force becomes zero. Detailed description of the CZM is given in Ref. [123]. The model can deal with the failure of a structure without assuming a crack with a crack tip having stress and strain singularity. The CZM is applied not only to the thermal fatigue of microsolder bumps [124] but also that of a die attach layer in a power module [125].

### 3.3 Lifetime Estimation Methodologies for Die Attach Failure in the Thickness Direction.

Demands of the high temperature operation of power modules have been increasing in recent years. Die attach failure in the thickness direction of a die attach layer (hereafter abbreviated as “vertical failure”) has been observed in such an environment [126128]. In Ref. [126], the vertical failure of a Sn–Cu die-attach solder joint is observed, when power cycling tests (PCTs) are performed under the temperature conditions of ($Tjmin=50 °C, Tjmax=150 °C)$ and ($Tjmin=50 °C, Tjmax=175 °C$), where $Tjmin$ and $Tjmax$ denote the minimum junction temperature and the maximum junction temperature, respectively. In Ref. [127], three kinds of die-attach solder alloys, Pb–5Sn–1.5Ag, Sn–7Cu, and Sn–3Cu–10Sb, are tested under the power cycling condition of $Tjmax=175 °C$ and $ΔT=$150 $°C$. As a result, the dominant die attach failure mode of the Pb-based solder alloy, Pb–5Sn–1.5Ag, is the failure of the horizontal direction in a die attach layer (hereafter abbreviated as “horizontal failure”), while that of the Sn-based solders alloys, Sn–7Cu and Sn–3Cu–10Sb, is vertical failure. In Ref. [128], the thermal cycling tests (TCTs) ranging from 50 $°C$ to 175 $°C$ are performed for four kinds of test specimens consisting of a Si-die, a Sn–Ag–Cu–Sb die attach layer and four kinds of substrates with different CTEs. The test results are summarized as follows:

1. The horizontal failure is mainly observed in the test specimens consisting of the substrate with large CTE, that is, in the test specimens with the large CTE mismatch between a Si-die and a substrate.

2. The vertical failure is mainly observed in the test specimens consisting of the substrate with small CTE, that is, in the test specimens with the small CTE mismatch between a Si-die and a substrate.

A study on the vertical failure mechanism of a Sn-based solder layer is also performed based on the microstructure images of solder layers [129]. According to this study, fatigue cracks are originated from the intermetallic compounds such as Cu6Sn5 around $β$-Sn dendrite boundaries. They propagate along the dendrite boundaries/the grain boundaries, and then, the vertical direction in a solder layer under the normal biaxial-stress condition caused by cyclic temperature variation. The failure mechanism mentioned above is confirmed by a finite element simulation.

Now, let us show the lifetime estimation methodologies for vertical failure. Harubeppu et al. [130] propose a lifetime estimation formula, assuming that voids are generated and grown on the grain boundaries by creep deformation at high temperature, and further that the grain boundary voids are connected and form intergranular cracks by the tensile stress at low temperature. They define the creep damage per cycle $Dc$ as accumulated creep strain per cycle and the fatigue damage per cycle $Df$ as a power of temperature range $ΔT$. The creep damage is caused only by creep deformation, so that $Dc$ is independent of $Df$. On the other hand, the fatigue damage promotes the coalescence of grain boundary voids generated by creep damage, so that the contribution of fatigue damage to the total damage is small when $Dc$ is small and becomes large when $Dc$ increases. Considering these facts, they propose the total damage $D$ per cycle as $D=Dc(1+Df).$ Then, the lifetime $Nf$ is given by $Nf∝1/D$. The proposed lifetime estimation formula contains three material properties characterizing creep strain rate, three material properties characterizing thermal fatigue, and one coefficient dependent on the structure of a power module, which is the so-called fitting parameter, determined from the lifetime experiments of a specific power module. Conclusively, the lifetime estimation formula proposed by Harubeppu et al. [130] cannot be used as the lifetime estimation in the design phase of a power module, because the lifetime of a power module is not estimated only by material properties.

Studies on the creep damage caused by repeated thermal stress due to temperature cycle have been performed in the structural reliability studies of the equipment of chemical plants and nuclear plants. The linear cumulative damage rule [131] and the strain range-partitioning method [132,133] are representative methodologies for dealing with fatigue-creep interaction problems that both fatigue damage and creep damage have to be simultaneously taken into consideration. In the former methodology, the summation of time fractions is used as a measure of creep damage proposed by Robinson [134], and that of cycle fractions is used as a measure of fatigue damage called Miner's rule. Total damage is defined as the summation of both damages. The continuum damage mechanics provides more reasonable creep damage than that proposed by Robinson [135]. In the latter methodology, we consider two strain modes, that is, plastic flow and creep, and consider four combinations of plasticity and creep depending on tensile and compression as follows:

• $Δεpp$: tensile plastic strain reversed by compressive plastic strain.

• $Δεcp$: tensile creep strain reversed by compressive plastic strain.

• $Δεpc$: tensile plastic strain reversed by compressive creep strain.

• $Δεcc$: tensile plastic creep reversed by compressive creep strain.

Then, we divide total inelastic strain range into the above four stain range, and obtain the fraction of each strain range $Fij$($ij=pp, cp,pc cc)$. Based on the partitioned strain range–lifetime relationships $Δεij−Nf,ij (ij=pp, cp,pc, cc)$ obtained from the experiments, we can estimate the lifetime $Nf$ as
$FppNf,pp+ FcpNf,cp+FpcNf,pc+FccNf,cc+1Nf$
(58)

The lifetime of the vertical failure in the die attach of power modules can be estimated using the above methodologies, but such a study is not yet performed.

## 4 Power Cycling Test and Thermal Cycling Test

Both a power cycling test (PCT) or a thermal cycling test (TCT) have been utilized to the structural integrity evaluation of power modules. A PCT is done by setting test conditions such as voltage, current, cooling method, and ambient temperature of a test specimen. Thus, a test specimen is subjected to temperature cycling due to the joule heating of a power module. In a TCT, a test specimen is subjected to temperature cycling in a temperature cycle chamber. A temperature distribution is heterogeneous for a PCT and homogeneous for a TCT. As shown in Refs. [3] and [26], wire-liftoff lifetimes are usually obtained from PCTs. In such cases, the junction temperature is evaluated through the collector–emitter voltage VCE as described in IEC 60747 standard [136]. The junction temperature obtained from VCE represents the average chip temperature and does not represent the temperature at the bonding part of a wire. A wire-liftoff phenomenon is a local failure of a bonding wire caused by the CTE mismatch of a bonding wire and a semiconductor die. The local temperature at the bonding part contributes to the wire-liftoff. In some cases, there exists much difference between such a local temperature and the average chip temperature obtained from VCE, as shown in Ref. [137]. The temperature of the bonding part in a wire measured by infrared cameras or thermocouples, or the temperatures calculated using electrothermal simulation tools such as ansys [136] should be used instead of the temperatures obtained from VCE to formulate the wire-liftoff failure models shown in Eqs. (9)(14). When Eq. (8) is used for evaluating the wire-liftoff lifetime, $Δεin$ is an indicator representing a local inelastic strain of the bonding part in a wire which directly influences a local failure mode called wire-liftoff. Other physical quantities such as the inelastic strain energy density range $ΔWin$, the nonlinear fracture mechanics parameter J-integral range $Δ$J and the nonlinear fracture mechanics parameter T*-integral range $Δ$T* are similar to $Δεin$. From a scientific view point, the temperature range $ΔT$ obtained from VCE in a PCT is not a valid parameter for describing wire-liftoff lifetime. If we insist on using the $ΔT$-based wire-liftoff failure models, we should perform TCTs instead of PCTs or should use the local temperature of a bonding part measured experimentally or calculated by CAE tools. Better failure models for predicting wire-liftoff lifetime are those based on $Δεin$, $ΔWin$, $Δ$J, and $Δ$T*.

Now, let us consider the horizontal failure of a die attach layer. All the lifetime estimation formulae given in Eqs. (32)(34) are obtained from TCTs. A temperature distribution is homogeneous in a TCT, so that the temperature of a substrate is the same as that of a semiconductor die. The driving force of the horizontal failure of a die attach layer in a TCT is the shear stress caused by a global mismatch in the CTEs of a semiconductor die and a substrate, $Δ$CTE, and given by
$Driving force∝(αD−αS)(Tmax−Tmin)=ΔCTE×ΔT$
(59)
where $αD$ and $αS$ denote the CTEs of a semiconductor die and a substrate, respectively, and $Tmax$ and $Tmin$ denote the maximum temperature and the minimum temperature in a temperature cycle, respectively. In a PCT, there is a temperature gap between a semiconductor die and a substrate. Let $Δmax$ and $Δmin$ be the temperature gaps at the temperatures of $Tmax$ and $Tmin$, respectively. The driving force of the horizontal failure of a die attach layer in a PCT is given by
$Driving force∝(αD−αS)( Tmax−Tmin)+αS(Δmax −Δmin)= ΔCTE ×ΔT+αS(Δmax −Δmin)$
(60)

The temperature gaps $Δmax$ and $Δmin$ are expected to be very small, because a die attach layer is very thin compared with a semiconductor die and a substrate. Thus, the second term can be neglected in Eq. (60), and the driving force of the horizontal failure of a die attach layer is almost the same both in a TCT and a PCT. It is concluded that the failure models obtained from TCTs, Eqs. (32)(34), can be applied to the lifetime estimation of power modules under operational conditions.

## 5 Summaries of Previous Research

In the present paper, we review the methodologies for estimating the thermal fatigue lifetimes of bonding wires and die attach layers in power modules. The following are summaries of the present review:

1. The purely empirical formulae based on the temperature range $ΔT$ have been proposed to estimate a wire-liftoff lifetime and a die attach cracking lifetime mainly by European and American researchers. They are Eqs. (9)(14) for wire-liftoff lifetime and Eqs. (32)(34) for die attach cracking lifetime. In these formulae, the lifetime $Nf$ is expressed by the product of physical quantities influencing $Nf$, and the constants relevant to the physical quantities are determined so as to fit experimental data, and a lot of experimental data are required to determine these constants for complex operating conditions of power modules. Furthermore, the use of these formula is limited to a specific power module, from which experimental data are obtained to formulate the equations for lifetime estimation.

2. The lifetime estimation formulae for wire-liftoff and die attach cracking based on the inelastic strain range $Δεin$, the inelastic strain energy density range $ΔWin$, the nonlinear fracture mechanics parameter J-integral range $Δ$J and the nonlinear fracture mechanics parameter T*-integral range $Δ$T* are simpler than those based on the temperature range $ΔT$, because the above-mentioned quantities are physical quantities directly influencing thermal fatigue. This is because the lifetime estimation formula based on $Δεin$, for example, contains all the temperature factors and the time factors of that based on the temperature range $ΔT$. Other physical quantities except for the temperature range $ΔT$ are similar to $Δεin$.

3. Large stress relaxation occurs due to large plastic deformation resulted from reduction of yield stress and creep deformation. Thus, the lifetime estimation formula based on $Δεin$ cannot represent the phenomena peculiar to the high temperature operation of power modules. The impact of stress relaxation on the thermal fatigue lifetime should be taken into account in the formulae. The formulae based on $ΔWin$, $Δ$J, and $Δ$T* are effective in the case of the high temperature operation of power modules. Among them, $ΔWin$ depends on the domain selected for their estimation and is not determined uniquely. As for the fracture mechanics parameters such as $Δ$J and $Δ$T*, their values are uniquely determined if they have path-independence. Under the condition of thermal and creep deformation, $Δ$J is path-dependent, while $Δ$T* is path-independent. Thus, the use of the formula based on $Δ$T* is recommended to estimate the thermal fatigue lifetime at elevated temperature.

4. In wire-liftoff lifetime, crack propagation lifetime can be neglected, because a wire-bonding length is very short. On the other hand, the die attach size of power modules is so large compared with a wire-bonding length that crack propagation lifetime should be considered in the lifetime estimation of die attach cracking by using the fatigue crack propagation rate given by the Paris law.

5. The lifetime estimation methodologies based on the continuum damage mechanics and the cohesive zone model are able to deal with both crack initiation and crack propagation without assuming a crack with a crack tip having stress and strain singularity. Among the methodologies based on the continuum damage mechanics, Desai and Basaran et al. proposed the DSC approaches based on the unified constitutive modeling and the unified mechanics theory, and successfully applied these approaches to lifetime estimations of IC packaging. We are expecting that these approaches would be powerful methodologies for structural integrity evaluation of power modules, especially for lifetime estimation of die attach crack.

Commercial finite element codes such as marc, ansys, etc., do not have the standard functions for the continuum damage mechanics and the cohesive zone model. So great effort is required to add these functions to the commercial finite element codes or to develop new special-purpose finite element programs. The rainflow method [138140] has been frequently utilized as a cycle count algorithm to predict the lifetime for complex load versus time histories. The lifetime estimation methodologies based on the continuum damage mechanics and the cohesive zone model are ill-matched to the rainflow method, because they are different from the Coffin–Manson type failure models. Thus, these methodologies need to calculate a damage variable D step by step. An enormous amount of computational time would be required for such a calculation.

6. New mechanical reliability problems of power modules are becoming apparent due to the change of manufacturing technology and operating environment for power modules. For example, heel cracking is becoming apparent as a failure mode of a bonding wire, when a molding material of power modules changes from silicone-gel to hard resin. Die attach failure in the thickness direction is becoming obvious, when a substrate with a low CTE is used to cope with high operating temperature. It becomes important to understand why such a failure mode is becoming apparent, and to develop a methodology for the lifetime estimation of such a failure mode.

7. The use of the failure models based on the temperature range $ΔT$ evaluated through the collector-emitter voltage VCE in a power cycling test, which is described in IEC 60747 standard, is unsuitable for evaluating the lifetime of a local failure such as wire-liftoff. This is because the temperature obtained from VCE is the average temperature of a Si/semiconductor die and not the local temperature at a failure location. The local temperature at the bonding part contributes to wire-liftoff. In summary, although the formulae for the wire-liftoff lifetimes obtained from power cycling tests based on IEC 60747 standard are useful for estimating the residual lifetimes of in-service power modules and for predicting the lifetimes of existing power modules, it is inappropriate to use them for estimating the wire-liftoff lifetimes in the design phase, because the temperatures determined based on IEC 60747 standard do not represent the local temperatures at the wire-bonding parts. The wire-liftoff lifetime data obtained from thermal cycling tests should be used for predicting the lifetimes in the design phase.

## 6 Proposals for Future Research

We should make a paradigm shift from estimating residual lifetimes/monitoring of in-service power modules using the purely empirical failure models such as $ΔT$-based failure models to the prior assessment of lifetime in the design phase of power modules using the physically/mechanically based failure models such as $Δεin$-, $ΔWin$- and $Δ$T*-based failure models or the DSC approaches based on the unified constitutive modeling and the unified mechanics theory. As described in Sec. 1, this is because enormous numbers of power modules will be utilized from now on in various fields, and, in such a circumstance, the malfunction and recall of power modules would be enormously unfavorable impact on our daily lives. So power device manufactures should provide reliable power modules to the market. From such a viewpoint, the present authors emphasize the need of the paradigm shift mentioned above, and propose the research policy for securing the structural integrity of power modules in the design phase, as follows:

1. Material testing is performed to obtain the material properties of a bond wire and a die attachment material. The following material characteristics are necessary:

• elastoplastic characteristics,

• creep characteristics,

• fatigue characteristics of the Coffin–Manson law type, and

• fatigue crack propagation characteristics of the Paris law type.

2. The physical quantity representing (c) and (d) should be a fracture mechanics parameter that can be uniquely determined. Among fracture mechanics parameters, a nonlinear fracture mechanics parameter should be selected, because ductile materials are utilized as a bonding wire/ribbon and die attach. We recommend the nonlinear fracture mechanics parameter T* range $Δ$T*, because it has path-independence for various loading conditions including creep deformation as well as thermo-elastoplastic deformation.

3. We propose that mechanical loading tests are used to obtain the fatigue characteristics of the Coffin–Manson law type, and the fatigue crack propagation characteristics of the Paris law type as shown in Ref. [105] instead of time-consuming thermal cycle tests and power cycle tests. These test data are useful for the lifetime estimation of crack initiation and crack propagation in a die attach layer of a power module. The mechanical loading tests are also proposed for the estimation of wire-liftoff lifetime [141,142].

4. The lifetime estimation of power modules in design phase can be done by making full use of CAE tools such as finite element software together with the inelastic material properties of the materials consisting of a power module obtained from material testing and using the following procedure:

• Step 1: Based on circuit design, the temperature distribution in a power module under the operating condition is obtained from electrothermal analysis.

• Step 2: Based on the temperature distribution in a power module obtained in step 1 and the design dimensions of a power module, the physical quantities such as displacements, strains, and stresses are obtained from the thermo-elastoplastic creep analysis using the elastoplastic and creep characteristics of a bond wire and a die attachment material.

• Step 3: The physical quantities relevant to fatigue characteristics and fatigue crack propagation characteristics such as $Δεin$, $ΔWin$, $Δ$J, $Δ$T*, etc., are obtained from those obtained in step 2.

• Step 4: Based on the result of step 3, wire-liftoff lifetime and die attach cracking lifetime are, respectively, obtained from the fatigue characteristics of the Coffin–Manson law type and the fatigue crack propagation characteristics of the Paris law type.

The lifetime estimation in the design phase of a power module is performed according to the procedure mentioned above, and the lifetime obtained is checked whether or not it satisfies the design goal. If it does not satisfy the design goal, design change is necessary to secure the structural integrity of a power module.

## Funding Data

• JSPS Grant-in-Aid for Scientific Research (Grant No. JP18K03863; Funder ID: 10.13039/501100001691).

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