## Abstract

Miniaturization as well as manufacturing processes that electronics devices are subjected to often results in to increase in operational parameters such as current density, temperature, mechanical load, and with potential to induce stresses that may be detrimental to device reliability. Past studies have identified some failure mechanisms common to these devices. Examples of these failure mechanisms include fatigue, electromigration, stress induced voiding, corrosion, conduction filament formation, and time-dependent dielectric breakdown. While some review activities related to reliability model development based on these failure mechanisms can be easily found in literature, to the best of our knowledge, a single review paper, which captures the reliability model progresses made over the past four decades across these failure mechanisms in comparison with Standards such as Joint Electron Device Engineering Council (JEDEC) and Institute for Printed Circuits (IPC) is to the best of our knowledge lacking. To fill this gap, a detailed review of failure mechanism driven reliability models, with emphasis on physics of failure (PoF) for power electronics was carried out in this paper. Although, other failure mechanisms exist, our review is only limited to fatigue, electromigration, stress induced voiding, corrosion, conduction filament formation, and time-dependent dielectric breakdown. It was found that most reliability research modeling efforts are yet to be fully integrated into Standards.

## 1 Introduction

Power modules play a key role in delivering flexible and efficient energy conversion. Tougher environmental protection laws and the ever growing need to cut down greenhouse gas emissions have led to the rising demand for high performance power electronics, especially for aerospace and automotive applications [1]. This is due to expansion of power solution from a predominately secondary system toward a significantly higher energy requirement to power not only vehicle entertainment systems, but also environment control devices, electric motors, safety systems, and sensors in vehicles. Figure 1 depicts schematics of a hypothetical power module.

Fig. 1
Fig. 1
Close modal

Diode and insulator gate bipolar transistor (IGBT) with nickel/gold metallization both on front and back side allows for soldering process, to be implemented. This facilitated the development of wire-less power module. By replacing the Al wire on the traditional with copper spacer and copper clip, the electric, thermal performance, and reliability can be improved [2]. Several factors such as thermomechanical stress caused by coefficient of thermal expansion (CTE) mismatch, environmental induced degradation, fatigue, electromigration as well as their combine effects complicate failure mechanisms and thereby, pose reliability or lifetime prediction difficulties [1,3].

Reliability models based on traditional methods contained in standard handbooks such as MIL-HDBK 217F, 217-Plus, PRISM, Telcordia, FIDES, and CNET, are strongly discouraged as a reference material for reliability prediction purposes [3]. Mainly, because these handbooks assume components to have constant failure rates that can be modified using separate modifiers to account for a various operating, quality, and environmental conditions. Many studies have enumerated concerns associated with this type of reliability modeling approaches [36]. Generally, reliability prediction models derived from these handbooks are inaccurate and provide highly misleading predictions, with potential to result in inferior designs as well as product decisions.

IEEE 1413 standards and its associated guidebook, IEEE 1413.1 are regarded as improvement to the previously named handbooks, because they provided reliability prediction assessment methods based on field data as well as information on the benefits of reliability prediction using physics of failure (PoF). Reliability models based on PoF [7,8] are generally preferred, because it puts emphasis on failure analysis, root cause of failure, as well as failure mechanisms [610]. This paper, review's reliability modeling activities related to power electronics components based on failure mechanisms. In Ref. [2] electronics failure mechanisms were classified into six. That is fatigue, electromigration (EM), corrosion, conductive filament formation (CFF), stress-driven diffusion voiding, and time-dependent dielectric breakdown (TDDB). Numerous research efforts have been carried out on each of these failure mechanisms. For instance, Zhao et al. [9], studied the effects of lead-free solder joints aging on their reliability using various pitch sizes and ball arrangement. The study shows that the degradation rate slows with aging time and the reliability degrades up to 70% after two years of aging at elevated temperature.

Xu et al. [10], carried out an accelerated EM test on four types of interconnect structure comprising of conventional ball grid array (BGA); solder ball with a copper via on top; an individual copper via in the substrate; and an individual copper plated-through-hole (PTH). The result shows higher probability of EM on the copper via in the substrate when placed in the vicinity of the solder ball. The effect of the humidity and temperature cycling on corrosion-based reliability was assessed by quantifying the leakage current (LC) on interdigitated test comb patterns, precontaminated with sodium chloride [11]. The results showed that the temperature cycling led to significant variation in the water–vapor concentration. In the work of Sood and Pecht [12], it was shown that smaller conductor spacing lowers CFF time to failure.

GaN power device TDDB was studied with various substrates [13]. The results show a decreasing shape parameter and an increase in scale parameter under positive substrate biases. In the study of stress induced voiding [14], combined electromigration as well stress migration to develop a unified model. Although some past studies have conducted review on specific power electronics failure mechanisms such as solder fatigue [15] and electromigration [16], a comprehensive review of reliability modeling efforts that captures the six failure mechanisms, with the view to highlight future expectation is to the best of our knowledge lacking. This paper, therefore, seeks to fill this gap. The remaining of this paper is organized as follows. In Sec. 2, reliability modeling based on fatigue failure mechanism is reviewed. In Secs. 3 and 4 reliability models based on electromigration and corrosion are discussed, respectively. In Secs. 5 and 6 reliability models based on CFF and SIV failure mechanisms are discussed, respectively. In Sec. 7, TDDB failure mechanism is discussed, and Sec. 8 concludes the study.

## 2 Fatigue

Power electronics devices are usually exposed to mechanical vibration and thermal cyclic loads. The effects of these loads include fatigue crack initiation, propagation, and finally fracture. In Ref. [3] die attach, wire bond/tab, solder leads, bond pads, traces, via/PTHs, and interfaces were identified as the major fatigue associated failure site in electronic packaging. For over four decades, numerous PoF-based fatigue life prediction models have been developed for these fatigue prone failure sites [1722]. In Ref. [15], it was reported that the properties of the solder material play a prominent role in the life of the solder joint. A detailed classification of the common materials used for solder joint can be found in Ref. [23]. Generally, based on health and environmental concern, there is strong shift from the well-known lead-tin (PbSn) solder to the lead-free solders such as the tin-silver-copper (SAC) family as well as its doped variants such as innolot. Rare earth elements have equally been shown to be a promising dopant [24,25]. Due to cost, melting temperature, and mechanical strength of SAC solder, the next generation of lead-free solder alloy will focus on doped solder alloy [24].

Successful prediction of fatigue failure is often based on the ability to accurately model the solder joint. Understanding the crack initiation and propagation mechanism in solder joint is critical to efficient fatigue-based life modeling. Apparently, it is quiet challenging to track and monitor the initiation as well as propagation of fatigue cracks in electronics packaging. Hence, a lot of fatigue failures models often adopt failure criteria appropriate to the purpose of their study. Lee et al. [26] carried out a review of solder joint life prediction model. Subsequently, Su et al. [15], provided an updated review on solder joint life models, capturing some fatigue models not covered in Ref. [26]. In this work, some new models with emphasis on recent incorporation of microstructural effects are covered with the view of highlighting prevailing challenges as well as direction of future research on electronic packaging fatigue life modeling. Section 2.1 discusses solder joint failure models.

### 2.1 Solder Joint Failure Models.

Solder joint fatigue life prediction models are generally categorized into four, comprising plastic-strain, creep-strain, energy, and damage accumulation models. These categories are based on the mechanism used to induce failure. In the strain-based models, thermal induced strain is applied, leading to stresses within the system [15]. The strain could be plastic, shear, or a combination of plastic and shear. Generally, the strain effect is either time-dependent plastic or creep-based [26]. For energy-based models, the overall stress–strain hysteresis energy of the solder joint is used as a basis of the model development. Fracture mechanics or creep and fatigue mechanisms often used to compute damage accumulation caused by fatigue crack propagation is the basis of damage accumulation models. Past reviews of these models can be found in Refs. [15] and [26]. Figure 2 and Table 1 represent the classification of these fatigue modeling methods. The parameters associated with each model are defined where they appear. In this section emphasizes is place on recent direction of research, associated with each of the solder joint failure model categories.

Fig. 2
Fig. 2
Close modal
Table 1

Fatigue models and classification

Failure model categoryModelDamage parameterPackage conditionMeritDemeritPackage ApplicationReferences
Plastic strainCoffin–MasonPlastic strainLCFSimple to implementIt ignored elastic strain, creep strain, and stress effectAll[27]
Kilinski et al.Strain rangeHCF and LCFElastic component was consideredIt ignored creep strain and stress effectSurface mount technology (SMT)[28]
Shi et al.Frequency-based plastic strainLCFReveals fatigue exponent and ductility coefficient dependency on temperature and frequencyIt ignored creep strain and stress effectN/A[29]
SolomonPlastic shear strainLCFShows the influence of plain strain on a cyclic frequency60/40 solder[30]
EugelmaierPlastic shear strain rangePower and Thermal cyclingSimple input parametersPBGA[33]
CreepKnecht and FoxMatrix creep shear strain rangeThermal cyclingCaptures creep behaviorIt ignored plastic strain and stress effect. It ignored grain boundary creepSolder joints in SMD[37]
SyedGrain boundary and matrix creepPower cyclingConsidered both grain boundary and matrix creepIt ignored plastic strain and can't be used for high homologous temperature solderPBGA[38]
Ohguchi et al.Total strainStepped ramp wave loadCombined creep, elastic and inelastic creepIt ignored stress effect and microstructural effectN/A[41,42]
Zhiwen et al. Zhang et al.Creep strainANN + creep model Improve prediction accuracyComputationally expensiveWLCSP[45,46]
Zhao et al.Creep strainThermal cyclingBGA/Memory module[46,47]
EnergyMorrowCyclic plastic strain energyCyclic loadStress and strains were consideredIt ignored creep and microstructural effectN/A[48]
Dasgupta et al.Total strain energyThermal cyclingElastic, plastic and creep strain energy were consideredIt ignored creep fatigue interactionsLCC/TSOP[49]
Akay et al.Total strain energyThermal cycling + thermal profileConsidered creep fatigue interactionsNot well tested68-pin LCC, 20-pin LCC[42]
Emeka et al. Deng et al.Strain energy per unit volumeThermal cyclingIgnored complete in-service parametersFlip chip assembly[52] [53]
PanStrain energy densityThermal cyclingIntegrated in service parametersThe use of energy weighting factor is requiredEutectic Sn-Pb solder[56]
DarveaukEnergy densityThermal cyclingHigh prediction accuracyComputational expensive Variation alloying element composition was ignoredFilm capacitor assembly[57,58]
Zhang et al.Total strain energy +Ag %Temperature cyclingAlloying element composition was factored.Only one alloying element was considered Not well testedPower semiconductor SAC 305[61]
Damage othersMinerDamage per cycle to total damageFatigue cyclingValid for stress independent materialNot valid for stress dependent material Overestimate fatigue lifeN/A[62]
Zhu et al.Plastic strain + creep deformation rateFatigue test Creep test Creep +fatigue testSimple and accurate for high strain rate conditionDoes not account for fatigue under low strain rate.SAC 305[64]
HamashaDamage per cycle + stress effectLCF and HCFRealistic service conditions were usedBased on miner rule is known to overestimate lifeLead free solders (SAC 105, SAC-Ni, SAC 305)[63]
Abdul-Baqi et al.Average effective damageMechanical load cycleCan model materials that fail in a quasi-brittle mannerFailed to capture crack propagationLead free package interphase[67]
Forrest and IbrahimDamage area percentMechanical load cycle and Thermal cyclingImprove accuracyGeometry specific and depend on the amount of FEA modeling skillsWafer level chip scale package, BGA with SAC 305 Array[68]
Failure model categoryModelDamage parameterPackage conditionMeritDemeritPackage ApplicationReferences
Plastic strainCoffin–MasonPlastic strainLCFSimple to implementIt ignored elastic strain, creep strain, and stress effectAll[27]
Kilinski et al.Strain rangeHCF and LCFElastic component was consideredIt ignored creep strain and stress effectSurface mount technology (SMT)[28]
Shi et al.Frequency-based plastic strainLCFReveals fatigue exponent and ductility coefficient dependency on temperature and frequencyIt ignored creep strain and stress effectN/A[29]
SolomonPlastic shear strainLCFShows the influence of plain strain on a cyclic frequency60/40 solder[30]
EugelmaierPlastic shear strain rangePower and Thermal cyclingSimple input parametersPBGA[33]
CreepKnecht and FoxMatrix creep shear strain rangeThermal cyclingCaptures creep behaviorIt ignored plastic strain and stress effect. It ignored grain boundary creepSolder joints in SMD[37]
SyedGrain boundary and matrix creepPower cyclingConsidered both grain boundary and matrix creepIt ignored plastic strain and can't be used for high homologous temperature solderPBGA[38]
Ohguchi et al.Total strainStepped ramp wave loadCombined creep, elastic and inelastic creepIt ignored stress effect and microstructural effectN/A[41,42]
Zhiwen et al. Zhang et al.Creep strainANN + creep model Improve prediction accuracyComputationally expensiveWLCSP[45,46]
Zhao et al.Creep strainThermal cyclingBGA/Memory module[46,47]
EnergyMorrowCyclic plastic strain energyCyclic loadStress and strains were consideredIt ignored creep and microstructural effectN/A[48]
Dasgupta et al.Total strain energyThermal cyclingElastic, plastic and creep strain energy were consideredIt ignored creep fatigue interactionsLCC/TSOP[49]
Akay et al.Total strain energyThermal cycling + thermal profileConsidered creep fatigue interactionsNot well tested68-pin LCC, 20-pin LCC[42]
Emeka et al. Deng et al.Strain energy per unit volumeThermal cyclingIgnored complete in-service parametersFlip chip assembly[52] [53]
PanStrain energy densityThermal cyclingIntegrated in service parametersThe use of energy weighting factor is requiredEutectic Sn-Pb solder[56]
DarveaukEnergy densityThermal cyclingHigh prediction accuracyComputational expensive Variation alloying element composition was ignoredFilm capacitor assembly[57,58]
Zhang et al.Total strain energy +Ag %Temperature cyclingAlloying element composition was factored.Only one alloying element was considered Not well testedPower semiconductor SAC 305[61]
Damage othersMinerDamage per cycle to total damageFatigue cyclingValid for stress independent materialNot valid for stress dependent material Overestimate fatigue lifeN/A[62]
Zhu et al.Plastic strain + creep deformation rateFatigue test Creep test Creep +fatigue testSimple and accurate for high strain rate conditionDoes not account for fatigue under low strain rate.SAC 305[64]
HamashaDamage per cycle + stress effectLCF and HCFRealistic service conditions were usedBased on miner rule is known to overestimate lifeLead free solders (SAC 105, SAC-Ni, SAC 305)[63]
Abdul-Baqi et al.Average effective damageMechanical load cycleCan model materials that fail in a quasi-brittle mannerFailed to capture crack propagationLead free package interphase[67]
Forrest and IbrahimDamage area percentMechanical load cycle and Thermal cyclingImprove accuracyGeometry specific and depend on the amount of FEA modeling skillsWafer level chip scale package, BGA with SAC 305 Array[68]

#### 2.1.1 Plastic Strain-Based Models.

Coffin–Mason (C–M) model [27] given in Eq. (1) is among the pioneer models in this category. Although has been widely used, it ignored the effect of elastic strain. To factor elastic strain, Basquin equation is often combined with C–M model to produce life prediction model usually referred to as total strain equation [28] given in Eq. (2).
$Nf=.5[Δεpa2εf]1/c$
(1)
$Nf= .5 [eln[EΔε2σf]−lnεf(b+c)]$
(2)
where $Nf, Δεpa$, $σf, Δε, εf$, $b$, and$c$ represents number of failure cycle, plastic strain amplitude, fatigue strength coefficient, strain range, elastic modulus, fatigue ductility coefficient, and fatigue ductility exponent, respectively. Another modification to the C–M model is the Shi model given in Eq. (3), which considered the effect of cyclic frequency [29]
$Nf=.5[Δεp2εf]1/cvk−1$
(3)
The frequency exponent $k,$ depends on the frequency value $v$. $k$ approximately take value of 0.91 for 10−3 to 1 Hz range and 0.42 for 10−3 to 10−4 Hz range. $c=−0.442 x 6x104Ts+1.74x102 ln(1+v)$ and $Ts$ represents mean cyclic solder joint temperature. Instead of fatigue life modeling based on plain plastic strain, Solomon [30] develop shear strain-based model given in Eq. (4). where $α and Ø$ are constants, whose values depend on solder material and thermal cycling test profile. $Δγps$represents the plastic shear strain range. A major limitation of Solomon's model is the lack of creep consideration, hence has limited use. To improve on both Solomon and Coffin-Mason's models, Engelmaier proposed a fatigue life model depicted in Eq. (5).
$Nf=[ØΔγps]1/α$
(4)
$Nf=.5[Δγps2εf]1/μ$
(5)
where $μ$ is the fatigue toughness exponent [31]. All the plastic strain models require some geometry related information to compute the fatigue life. Experimental data or finite element analysis (FEA) is usually used to provide such information. Commercial FEA software such as ansys or abaqus use constitutive model which combines the time-dependent plastic and visco-plastic phenomenon to extract relevant data such as plastic strain range. Stress–strain relationships in solders under varying temperature and strain rates are generally modeled using the Anand model shown in the following equation:
$ε˙p=Aexp(−QRT)[sinh((ϵσ¯s))]1/m$
(6)
where $ε˙p, A, m, Q, RT$, and$ϵ$ represents the inelastic strain rate, pre-exponential factor, strain rate sensitivity, activation energy, universal gas constant, temperature, and material constant, respectively. Furthermore, s represents the internal state variable. The internal variable evolution equation $s˙$ and the saturated value $s*$ are expressed as in the following equations, respectively,
$s˙={ho|1−ss*|asign(1−ss*)ε˙p}$
(7)
$s*= s̃[ε˙pAexp(QRT)]n$
(8)

where $ho$ and $a$ represents the material strain hardening parameters. $s̃$ denotes a coefficient of saturation, and $n$ represents the saturation value of deformation resistance strain rate sensitivity. The nine material parameters, such as $A, Q, m, n, j, s̃, a, ho,$ and $s$ can be easily determined from extensive experimental tests and are often available from manufacturer data sheet. In Zhang et al. [32], Anand and Engelmaier models were combined to predict the fatigue life of Sn-Ag-Cu-Zn lead-free solder joints in a chip-scale packaging device. It was found that the use of Zn as a dopant improves the solder fatigue life. In Refs. [22] and [33], Anand model assigned from abaqus was combined with Engelmaier model to investigate the fatigue life of PBGA under temperature cycling. It was shown that solder height and temperature ramp rate significantly affected fatigue life. Also, the joint corner was assumed to determine the life of the joint. In Ref. [34] C–M model and machine learning were used to assess the failure life prediction of wafer level package. In a recent study, FEA and strain-based models (C–M, Eugelmaier, Solomon, and Syed) were used to assess the fatigue life of ball grid array (BGA) joints comprising of various alloy compositions [35,36]. The results show lack of consistency in the predicted values among the strain-based models compared. The study attributed the lack of consistency to the critical role played by damage parameters as well as the diverse failure modes associated with fatigue damage.

#### 2.1.2 Creep-Based Fatigue Life Models.

Elevated temperature and constant load often trigger creep behavior in solder joint. Creep in solder materials is characterized by three phases. That is primary creep phase, secondary creep phase, and tertiary creep phase. Before the onset of creep phases, the initial deformation that occurs upon loading of the test specimen includes both elastic and plastic strains. During primary creep phase, the material strain hardens, which results in a decrease in the strain rate. In the secondary creep phase, specimen deformations proceed at a steady strain rate, while during the tertiary creep phase, it experiences rapid rise in strain rates due to the formation and growth of voids until the test specimen ruptures. Material stress level and creep rates determine the prevalent creep mechanism. Generally, grain boundary creep occurs at low stress and creep rates, while matrix creep occurs at high stresses and creep rates. Based on these mechanisms in Knecht and Fox [37], matrix creep fatigue model (Eq. (9)) was proposed. The model relates, the microstructure and the matrix creep shear strain range to the solder fatigue life
$Nf=KΔγmc$
(9)
where $K$is the solder microstructure and failure criteria dependent constant; and $Δγmc$ is strain-range associated with the matrix creep. Syed [3840] on the other hand, modeled creep fatigue life as a function of grain boundary sliding ($Dgbs$) and the matrix creep accumulated equivalent creep strain ($Dmc$) per cycle respectively as shown in the following equation:
$Nf=10.022Dgbs+0.063Dmc$
(10)
A major drawback associated with Syed's fatigue model is the fact that plastic-strain effects, was ignored. Ohguchi et al. [41,42], factored inelastic and elastic strains to the creep strain to predict fatigue life (Eq. (11)) based on a model considered to be a modified C–M model, where $Δεcr$ is the creep strain amplitude, $Δεcr=Δεe+Δεp+Δεc$, and $Δεe, Δεp, and Δεc$ denotes the elastic strain, plastic strain and creep strain, respectively,
$Nf=(11.6Δεcr)1/0.557$
(11)
Similarly, Hsieh [43] used modified C–M fatigue model and implemented the modified model for the fatigue life prediction of a wafer-level chip scale package (WLCSP). In Yoshiharu et al. [44], the complete inelastic strain range was partitioned into four components as given in Eq. (12). $Npp$ represents the number of cycles to failure associated with time-independent plasticity reversed by time-independent plasticity; $Npc$ represents number of cycles to failure associated with time-independent plasticity reversed by creep; $Ncp$ represents number of cycles to failure associated with creep reversed by time-independent plasticity; $Ncc$ represents number of cycles to failure associated with creep reversed by creep. Subsequently, miner rule was used to predict the fatigue life as given in the following equation:
$1Nf=1Npp+1Ncc+1Ncp+1Npc$
(12)
A good prediction of fatigue failure depends on the accessibility of well-developed constitutive models, which describes the solder-creep behavior under a variety of stress conditions. This is because, constitution models facilitate derivation of key stress, strain, and strain energy-related information. Most constitutive model mainly considers the secondary creep rate. These models typically relate stress and temperature in hyperbolic sine function Eq. (13), single or double power law Eq. (14), and Anand model Eq. (6)
$ε˙p=A1sinh(ασ)ne(−Q1kT)$
(13)
$ε˙p=A1e(−Q1kT)(σσ1)n1+A2e(−Q1kT)(σσ1)n2$
(14)

Although, many FEA software such as ansys and abaqus depend on steady-state constitutive model to model creep behavior, the prediction accuracy is believed to be less than optimal. To increase the prediction accuracy, FEA has recently been combined with heuristic algorithms. To illustrate, Chen et al., [44] combined ansys and artificial intelligent (AI) method for solder joint fatigue life prediction in WLCSP. Similarly, Zhang et al. [44], evaluated the reliability of WLCSP using a combination of FEA with ANN. In both studies modified C–M model was used for the fatigue life prediction. Zhao et al. [46], reported that analytical life prediction without incorporating microstructural effects due to aging or impurities may be erroneous. In Morooka and Yoshiharu, [20] this concept was considered in a fatigue life prediction of a BGA solder joints. Validation of the microstructural effect requires more studies. Therefore, constitute interesting area to concentrate on creep-based fatigue life prediction models [47,48]. Also, constitutive models that integrate microstructural effects due to recrystallization during aging or variation in test sample quality will equally be an interest area to consider in the future.

#### 2.1.3 Energy-Based Fatigue Life Models.

These models basically use hysteresis energy concept or some form of volume-weighted average stress–strain history, to predict fatigue failure. Many studies have argued that these models are better than creep-based or strain-based models, because it relates the stress and strain that the solder joint experience during testing. Energy-based lifetime models use strain range, namely, total cyclic strain energy [49] and partition cyclic strain energy [50]; strain energy per volume range; and stress range to characterize the thermal loading [51,52]. Morrow's model was among the earliest energy-based fatigue life prediction models. Based on Morrow's model, fatigue life characterized by the number of cycles to failure is related to the cyclic strain energy (hysteresis energy) through a power law formulation as shown in the following equation:
$Nf=(ΔwC)nc$
(15)
where $C, Δw, and nc$ represents the solder material fatigue constant, cyclic strain energy, and solder material fatigue exponent, respectively. The model in Eq. (15), ignores creep effects. Lack of creep consideration has tendency to affect the accuracy of the fatigue modeling effort. To enhance accuracy, Dasgupta et al. [50], modeled solder joint life using the partitioning strain energy method. In their study, the strain energy was partitioned into three portions. The energy associated with the three strain portions are represented in the following equations:
$we=weoNe elastic strain energy$
(16)
$wp=wpoNp plastic strain energy$
(17)
$wcr=wcrNcr creep strain energy$
(18)
where $weo, wpo, wcr$ are the elastic, plastic, and creep strain energy density, respectively, $weo and e; wpo and p; and wcr and cr$ are fatigue constant relating to elastic, plastic, and creep damage, respectively. $N$ is the cycle to failure and total damage ($D$) per cycle through the solder joint is therefore given as in Eq. (19). where $De, Dp, and Dcr$ are the elastic, plastic, and creep damage per cycle, respectively,
$D= De+ Dp+Dcr$
(19)
The damage caused by each strain component is given by $Di=1Ni$. In Refs. [53] and [54], the strain energy per unit volume range (Eq. (20)) was used to model fatigue life. $W$ and $wacc$represents creep energy density and accumulated creep energy density per cycle
$Nf=(Wwacc)−1$
(20)
Akay et al. [55], implemented volume-weighed averaging method for the computation of fatigue life in solder alloys. In their work, Engelmaier's model given in Eq. (21), as well as energy-partitioning approach, were used to evaluate creep–fatigue interaction
$Nf=.5(Δγεf)1/k$
(21)
where $Δγ$ is volume-weighed average effective creep shear strain range. Pan [56] used the critical accumulated strain energy (CASE), to model fatigue life in solder joint. Their model is as given in the following equation:
$Nf= CPazEp˙+ bzEc˙$
(22)
where $CP$ denotes the critical strain-energy density. $Ep˙$ and $Ec˙$ represents the time-dependent plastic and creep energy, respectively. FEA was used to estimate the values of $Ep˙$ and $Ec˙$. $az and bz$ represent energy-based weighting factors. Darveaux [57], proposed a model (Eq. (23)), that relies on a combination of crack initiation and propagation components for fatigue life prediction
$Nf= Nos+ac−(Nos−Nop)dapdNdasdN+dapdN$
(23)
$dapdN$ and $dasdN$ are the primary and secondary crack propagation term, respectively. $Nos−Nop$and $Nos$ the primary and secondary crack initiation energy-based terms, respectively. $ac$ is the total crack length. $Nop$ and $dadN$ are determined as a function of the energy density term ($Δw$) calculated from the stress–strain hysteresis curve as given in the following equations:
$Nop=k1Δwk2$
(24)
$dadN=k3Δwk4$
(25)
The prediction accuracy of Darveaux model was compared to Coffin–Manson model and the result show that Darveaux model had a lower prediction error [58]. Similarly, Lee et al. [59], used Darveaux model to predict the fatigue life of film capacitor assemblies under thermal cycling. Liu et al. [60] used energy conservation law, entropy conservation law, and deformation mechanics, to develop a new fatigue life prediction model expressed in the following equation:
$Nf=6.903 x10−6(ΔεpΔεt)8.075ΔWt$
(26)
where $ΔWt, Δεp, and Δεt$ represents the total strain energy density, plastic strain, and elastic strain, respectively. The model was used to prediction fatigue life of a BGA solder joints. Validation of its accuracy requires more experimental data. Studies have shown that the accuracy of fatigue life prediction model depends on the incorporation of material variation [54,61]. To address that, Chen et al. [54], used two strain energy-based life models called total energy life model as expressed in Eq. (27) and partitioned energy fatigue life model given in Eq. (28) to predict the fatigue life of a silver-based solder materials
$Nf=(Δw21,500*Ag+34,950)−0.588$
(27)
$Nf=(Δw32,930*Ag+50,010)−0.563$
(28)

The Ag in the models represents %Ag content by weight. Geometry plays a key role in the energy-based fatigue models [38]. The complexity of FEA implemented in an energy-based fatigue analysis is dependent on the size and shape of the solder joint. The accuracy of recently developed energy-based fatigue model will require further study. Most importantly, extension of the single element variation is currently accounted for in Eqs. (27) and (28) to more than one element will be expected, to account for multiple doped solders.

#### 2.1.4 Damage-Based Models.

These models are derived based on the principle of fracture, creep, and fatigue mechanisms. They are generally classified into two categories, namely, linear damage, and nonlinear damage cumulative model. The driving force for the use of linear damage models is the close approximation to reality and simplicity. Miner rule given in Eq. (29) is commonly used to predict the fatigue life of a material exposed to cyclic loading conditions with varying stress amplitudes [62]
$∑i=1kniNi=1$
(29)
where $ni$ represents accumulated failure cycles at stress amplitude $i$ and $Ni$ denotes the number of failure cycles at the same stress amplitude. k indicates the number of stress levels. In Ref. [63] Miner's rule was reported as, unsuitable for fatigue life prediction in lead-free solder joints. This is due to damage interactive tendency and stress dependency of lead-free solder material. However, in Zhu et al. [64], a linear damage rule for high strain rate conditions using creep and fatigue mechanism for lead free solder was presented. The creep damage portion modeled based on Monkman–Grant equation accounted for the holding stage and the fatigue damage, which captures the ramp stage was evaluated by implementing Coffin–Manson model. This model is depicted in the following equation:
$Nf= 1(.76Δγp)−2.38−Δtε˙s0.78.5$
(30)
where $Δγp$is the plastic strain range and $ε˙s$ is the stable creep deformation rate. Although, this method cannot reflect the fatigue life under low strain rate, its accuracy and simplicity for high strain rate condition will support its high applicability. Integration of the bilinear interaction diagram method proposed in Hamasha et al. [65], with Zhu's model could improve the accuracy of the model. Nonlinear accumulative damage model proposed by Corten and Dolan expressed as Eq. (31) could account for the effects of load interaction during fatigue cycling [66]
$∑i=1kniN1(σiσ1)d=1$
(31)
where $N1$ is the fatigue life at the highest stress amplitude $σ1$, $ni$ is the number of cycles at stress amplitude $σi$, d is the material constant. Geometry and microstructure play a key role in the accuracy of the damage-based model. While pioneering damage models, Refs. [62] and [66] didn't factor in microstructural effects. More recent fatigue life studies, consider the effect of microstructure in solder joint modeling using cohesive zone approach [67]. Although crack propagation was considered in their implementation of two-dimensional model with an effective damage parameter, it failed to capture explicitly the propagating nature of fatigue cracks. Recently, Baber and Guven, [68] used peri-dynamic approach to model the fatigue life of solder joint. In their method, the damage area percent was estimated using the following equation:
$DAP= damage area ≥0.4total possible damage area$
(32)
Then, using Eq. (33) similar to Paris law, the damage growth rate can be linked to the DAP through empirically determined material constant $k1$ and $k2$. Upon determination of the rate of crack growth, the number of cycles to failure can be determined using Eqs. (34) and (35):
$dAdN=k1(dØdd(DAP))k2$
(33)
$N=A(dAdN)$
(34)
$Ntotal = ∑i=1kNi=Ai(dAdN)i$
(35)

#### 2.1.5 Other Models.

Lau and Pao [69] carried out an estimation of reliability function based on Norris and Landzberg fatigue model as given in Eq. (36), where the acceleration factors AF, which is defined as the ratio of the life at operating condition (opt-subscript) to the life at accelerated condition (acc-subscript) are given in Eqs. (37)(39). $φoptφacc$ is the maximum temperature fatigue life ratio under isothermal condition. $β and η$ are the shape and scale parameters
$Ro(t)=e−(tAF.η)β$
(36)
$AF=(ΔTaccΔTopt)2(ΔfoptΔfacc)1/3(φoptφacc)$
(37)
$AF=(ΔTaccΔTopt)2(ΔfoptΔfacc)1/3e1414(1ΔTopt−1ΔTacc)$
(38)
$AF=(ΔγaccΔγopt)2(ΔfoptΔfacc)1/3e1414(1ΔTopt−1ΔTacc)$
(39)
Luan et al. [70], implemented a fatigue life prediction model expressed as Eq. (40) based on drop test. Where $Nfd$ is the fatigue life, which corresponds to the number of drops without failure and $θ$ is a constant
$Nfd=e− θβ$
(40)
$β$ and $θ$ can be obtained using the expression given in Eq. (41). where $N$ is the impact of life at corresponding failure rate
$F= θ+ βln(N)$
(41)

In Ref. [71] fatigue life model for solder joints under corresponding failure rate ($F$) based on Weibull equation was proposed from thermal cycling tests using Eq. (41). After careful review fatigue related publications, we observed that there is lack of clear definition of solder joint fatigue. Some model definition for solder joint is based on JESD22-A122A [72] for power cycling, which stated that failure criteria shall include, but not be limited to, hermeticity for hermetic devices, parametric limits, functional limits, and mechanical damage resulting in failure of the test point of interest. In the case of IPC JSTD change is resistance suffix [73]. Some others use mechanical behaviors like amplitude stress. Clearly, lack of global fatigue failure standard for all kinds of fatigue test, could result in misleading fatigue life comparison among various testing condition. Although microstructural effect has been recent capture in some fatigue models, a lot of research activities, is required to enhance their prediction accuracy.

## 3 Electromigration

Electromigration (EM) is a mass movement of atoms, caused by the momentum transfer between conducting electrons and metal atoms. Migration of atom along a line, often results in flux divergences. At sites of flux divergences, material accumulation (hillock) or depletion (void) takes place [74]. Typically, damage caused by voids is much more probable than damage caused by extrusions. Within a power device, electromigration predominantly occurs along interconnect lines. For wire bonds, it occurs either at the attachment spot or at the middle of the wire between two attachment points, while for solder balls it occurs along the interphase between the solder and intermetallic compound. Figure 3 depicts electromigration induced void in an interconnect line, wire bond, and solder ball.

Fig. 3
Fig. 3
Close modal

Typically, the formation and growth of voids cause a significant increase in the resistance in a line and, finally, leads to open circuit. Material, temperature, grain structure, stress gradients current density are the common sources of flux divergence [78]. For over 50 years EM reported by the Fik's and Huntington has been an area of active research [79,80]. This trend has been driven by integrated circuit (IC) miniaturization to nanoscale, which often results in to increase in current density. Despite numerous research articles already published on EM, persistent miniaturization of electronic devices constantly presents the need for EM induced failure evaluation due to complex stresses and stress interaction that exist at nanoscale [80]. With the growing shift from traditional EM-based models to physics-based models as well as the availability of finite element analysis tools like ansys and comsol, clarity and prediction accuracy of EM failure mechanism is growing. Many review articles on the EM have been published [16,78]. It is not the intention of this section to reproduce the content already available in those articles. Rather this work will highlight notable progress made in electromigration reliability model development up to the time of writing this paper. We note that not citing a paper does not diminish its significance. During the EM process, atomic flux diffusion (AFD) varies with diffusion paths and time. Generally, reduction in the AFD, increases interconnects life. Based on the concept of AFD, we categorized EM reliability models into two; the diffusion path model and the driving force model.

### 3.1 Diffusion Path Model.

Diffusion path reliability model assumes electron wind force to be the cause of EM mass transport. The atom flux associated with wind force is expressed as the following equation:
$j= NkTZeiρDfe−EgbkTj$
(42)
where $j, ρ, N, k, T, Df$ E, $ei, and Z$ is the current density, electrical resistivity, atomic concentration, Boltzmann's constant, temperature, atomic diffusivity, grain-boundary self-diffusion activation energy, electric charge, and effective atomic charge number, respectively. Generally, inhomogeneity exists in interconnect microstructure. This homogeneity results in diverse diffusivities along various diffusion path. The common EM diffusion path includes surface, interface, bulks, pipes, grain boundaries, and lattice [81]. The fastest diffusion path drives EM failure. Drift velocity ($vd$) given in Eq. (43), is often used to represent the rate of diffusion
$vd=DekT(Zeeiρj−Ωδσδx)$
(43)
Based on Eq. (43), $DeZe$ can be expressed as Eq. (44) if the back-stress gradient $δσδx$ is ignored
$DeZe=DsZsfs+DbZbfb+DiZifi+DpZpfp+DgbZgbfgb$
(44)

where the subscript s, b, i, p, and gb denote surface, bulk, interface, pipe, and grain boundary, while $fq(q=s,b,i,p, and gb)$ which depends on the interconnect geometry and structure is the proportion of atoms diffusing through a given path. $Ω$ is the atomic volume. At temperature less than 400 °C, bulk and pipe diffusion are basically negligible. Hence, surface, interphase, and grain boundary diffusion dominate in most line interconnect. Concise description of diffusion path reported in literature with respect to commonly used interconnect line materials is provided. Due to many triple points peculiar to aluminum microstructure, grain boundary activation energy (AE) is lower, compared to the AE at its high protective surface or interphase. Hence, grain boundary diffusion is the predominant diffusion path in aluminum line [81]. On the contrary, copper with less protective surface, exhibits lower activation energy at its interface compared to order paths. Other electromigration studies, affirm that surface migration is the most rapid diffusive pathway, even faster than both grain boundary [82,83] and interface diffusion [84] in copper and its alloys-based interconnect. Due to the high rise in copper line resistance associated with miniaturization, strong research efforts are currently channeled toward materials such as cobalt, ruthenium as well as topological metals [85,86]. Cobalt interconnects have already been implemented in advanced nodes [85]. The attractive feature of these new materials is their ability to maintain or experience only very low resistance rise as the line thickness is reduced to nanoscale. While, grain boundary diffusion has been reported as the predominant diffusion path in cobalt and ruthenium [87], in topologically metals the fastest diffusion path is bulk [85]. Table 2, depicts interconnect material and the predominant diffusion path.

Table 2

Common interconnect material diffusion path

MetalsResistivity ($nΩm$)Predominant diffusionReferences
Aluminum and its alloys0.265Grain boundary[16]
Copper and its alloys0.168Grain boundary surface/interphase[81,82]
Cobalt0.56Grain boundary[83]
Ruthenium71 degGrain boundary[84]
Topological materials (NbAs, CoSi, etc.)7–80Bulk[84]
MetalsResistivity ($nΩm$)Predominant diffusionReferences
Aluminum and its alloys0.265Grain boundary[16]
Copper and its alloys0.168Grain boundary surface/interphase[81,82]
Cobalt0.56Grain boundary[83]
Ruthenium71 degGrain boundary[84]
Topological materials (NbAs, CoSi, etc.)7–80Bulk[84]
Reliability modeling from the perceptive of diffusion approach found in literature is predominantly based on Black's model. The weak mode AE [87] is often used to determine the dominant diffusion mechanism as expressed in the following equation:
$MTF50=Aej−2e(EddkT)$
(45)
where $Ae$ is a constant and $Edd$ denotes activation energy. Since EM is a diffusion-controlled process, the $Edd$ is dependent on the diffusion path. For a system containing network of interconnect, weak-link approximation approach is commonly used as a means of computing the system reliability. This assumes that a serial circuit of N interconnects function only if the weakest of the interconnect does not fail [88]. Thus, given that $τ (=1,…,n)$ are likely random failure time from a population of $n$ interconnects. Failure of the system is defined by the occurrence of any component failure in the series. Under such conditions, the reliability of the system within the time interval (0, t] is expressed as the following equation:
$RN(t)=1−[1−[1−F1(t)]n]$
(46)

Generally, reliability models based on diffusion approach, is known to be insufficient, as other factors such as stress gradient, thermal gradient as well as impurities, and resistivity scaling are known to facilitate EM. Developing a reliability model which captures other EM contributory factors, facilitated a lot of research attention to driving force-based reliability models.

### 3.2 Driving Force Models.

Atomic flux diffusion has been described as the primary cause of EM. Several factors, such as electron wind force, stress gradient, thermal gradient, and atomic concentration gradient have been reported to drive EM, which has led to numerous reliability models [8994]. In this study, we classify these models into traditional and physics-based models. Table 3 summarizes the major driving force inspired by EM reliability models found in literature. It is important to note that Table 3 does not capture all related references mentioned in-text.

Table 3

Classification of electromigration modeling activities

ClassificationModelDriving forceAnalysis method/principleMeritDemeritApplicationReferences
Black'sEWFEmpiricalSimple well testedOver estimation of reliability.Single line interconnect SAC 305[89] [90,91]
Blech'sEWFEmpiricalIdentified critical line lengthSingle line interconnect[92,93]
It ignored hydrostatic stress component
Shatzkes and LioydEWF + vacancy concentrationNumerical (Laplace transformation)Established current density exponent as 2Single line interconnect[94]
Not applicable to multiline interconnect
Physics-basedKorhonen et al.EWF/thermal gradientNumerical/ Hydrostatic stress-based FEM/FDMBetter prediction accuracy, when compared to traditional modelsFailed to account for multisegment interconnectSingle line aluminum interconnect Single/multiline interconnect[96] [97,98]
Time to failure is only based nucleation
Three-phase compact modelEWF/thermal gradientNumericalTime to failure is based on nucleation, incubation and, growthIt ignored recovery effectVLSI Full chip[99]
Extended to multiline interconnect
Voltage-base EM modelEWF/thermal gradient/stress gradientFEM FEMApplicable to single and multiline interconnectIgnored joule heatingMultisegment interconnect[101106]
It implemented separate computation of joule heating, hence reduction in accuracy.
Incorporated Joule heating
Saturated volume-based modelEWF/thermal gradientFEMEnhanced prediction accuracyComputationally expensiveMultisegment interconnect[107110]
Void-based modelEWF/thermal gradientStatistical correlationConsidered life dependency on variables such as voids, geometry, and experimental factors.Very slow prediction approachSingle line copper interconnect[111]
Mass divergence modelEWF/thermal gradient/stress gradientNumerical simulationCaptured other migration mechanismComputationally expensiveCSP package with PCB Cu-via structure[112] [113]
Not clear how well it fits experimental observation
Other modelsEWFStatistical/FEM Eigenfunction SOV/ASOV Stochastic Time dependentAccount for residual stressIt required discretization of partial differential equationsVSLI full chip Multisegment interconnect signal line/power grid[98] [121] [114,115] [122124,127] [120,121]
Fast computation
It ignored recovery effect
Very efficient for very large scale lineIt ignored recovery effect
It ignored stress interaction computationally expensive
Simple
Simple to implement
Captured recovery effect and improve accuracy
ClassificationModelDriving forceAnalysis method/principleMeritDemeritApplicationReferences
Black'sEWFEmpiricalSimple well testedOver estimation of reliability.Single line interconnect SAC 305[89] [90,91]
Blech'sEWFEmpiricalIdentified critical line lengthSingle line interconnect[92,93]
It ignored hydrostatic stress component
Shatzkes and LioydEWF + vacancy concentrationNumerical (Laplace transformation)Established current density exponent as 2Single line interconnect[94]
Not applicable to multiline interconnect
Physics-basedKorhonen et al.EWF/thermal gradientNumerical/ Hydrostatic stress-based FEM/FDMBetter prediction accuracy, when compared to traditional modelsFailed to account for multisegment interconnectSingle line aluminum interconnect Single/multiline interconnect[96] [97,98]
Time to failure is only based nucleation
Three-phase compact modelEWF/thermal gradientNumericalTime to failure is based on nucleation, incubation and, growthIt ignored recovery effectVLSI Full chip[99]
Extended to multiline interconnect
Voltage-base EM modelEWF/thermal gradient/stress gradientFEM FEMApplicable to single and multiline interconnectIgnored joule heatingMultisegment interconnect[101106]
It implemented separate computation of joule heating, hence reduction in accuracy.
Incorporated Joule heating
Saturated volume-based modelEWF/thermal gradientFEMEnhanced prediction accuracyComputationally expensiveMultisegment interconnect[107110]
Void-based modelEWF/thermal gradientStatistical correlationConsidered life dependency on variables such as voids, geometry, and experimental factors.Very slow prediction approachSingle line copper interconnect[111]
Mass divergence modelEWF/thermal gradient/stress gradientNumerical simulationCaptured other migration mechanismComputationally expensiveCSP package with PCB Cu-via structure[112] [113]
Not clear how well it fits experimental observation
Other modelsEWFStatistical/FEM Eigenfunction SOV/ASOV Stochastic Time dependentAccount for residual stressIt required discretization of partial differential equationsVSLI full chip Multisegment interconnect signal line/power grid[98] [121] [114,115] [122124,127] [120,121]
Fast computation
It ignored recovery effect
Very efficient for very large scale lineIt ignored recovery effect
It ignored stress interaction computationally expensive
Simple
Simple to implement
Captured recovery effect and improve accuracy
Traditional EM models: EM modeling effort reported in Ref. [89] pioneered activities associated with traditional EM models. This model is today popularly known as the Black's model. In his study, a semi-empirical equation was developed and was used to compute the median time to failure $(t50)$ of a group of Al-based interconnects under same test condition
$t50=Aj−2e(EaekT)$
(47)
The difference between Eqs. (45) and (47) that in Eq. (45) the $Edd$ represents the activation energy of the dominant diffusion path, while in Eq. (47), $Eae$ represents activation energy associated with void formation. Black model is still broadly applied today as a typical means of $t50$ estimation. In Ref. [90], current crowding-induced electromigration on SAC 305 was studied and Black's model was used to compute the time to failure. Recently, in Tu and Gusak [91], the rate of entropy production was used to validate Black's model. The authors consequently used the entropy production method and black's model to analytically derive the time to failure for thermo-migration as well as stress-migration. Blech in his EM study found the “Blech effect” concept and proposed a model based on the “back flow” effect of atoms in the opposite direction of electron flow due to back stress in the interconnect when voids and hillocks are formed [92,93]. Consequently, the steady-state solution arises as soon as the driving force due to generated stress is equivalent to that due to EM. Hence, failure will occur only if Eq. (48) is satisfied
$jl>(σc−σo)ΩZeiρ$
(48)
The reliability implication of Blech effect (47) is that short lines may be immortal to EM. Shatzkes and Lloyd [94], proposed a model given in Eq. (49) where the driving force comprises of the EM term and an opposing concentration gradient term acting on the blocking boundary with a vanishing atomic flux. They recommended a modification of Black equation, based on the assumption that nucleation dominates EM failure and growth, and failure occurs after nucleation
$MTF=BT2j−2e(EaekT)$
(49)

Lloyd and Kitchin [95] implemented numerical method and found that the current density exponent associated with Black's model is also two. Many of the traditional models have faced overwhelming criticism due to conservativeness and their applicability to single wire segment. In addition, stress contributed by other connected wires, or temperature variations is often handled by overestimating the EM effect. To address these problems with the view to enhance prediction accuracy, for multisegment interconnect facilitated the development of physics-based EM models.

Physics EM models: EM models based on PoF offers the benefit of better reliability prediction in terms of accuracy. In this paper, physics-based models found in literature are classified into five. That is, Korhonen's model, three phase compact model, voltage-based model, saturated volume-based, and void based. All other models which do not fall under the above-mentioned five are classified as other model.

Korhonen's model: Pioneering physics-based EM modeling could be traced to a model developed by Korhonen et al. [96], which is currently referred to as Korhonen's model. To illustrate the basis of their model development, consider the dual damascene interconnect depicted in Fig. 4(a). Electrons flow from the (M1) interconnect line to M2 interconnect line. In this structure, M2 is separated from M1 using Ta barrier layer. After some time, rise in hydrostatic stress and stress gradient acting counterforce to atomic migration occurs due to unidirectional electrical load. Generally, the longer the line, the higher the tendency for the stress to reach critical level, thereby resulting in a void nucleation at the cathode or hillock at the anode.

Fig. 4
Fig. 4
Close modal
In Ref. [93] transient evolution of hydrostatic stress caused by EM in a confined metal $σ(s,t)$ was described by a partial differential equation expressed in the following equation:
$∂σ(s,t)∂t=∂∂t[k(∂σ(s,t)∂t+G)], 0≤s≤L, t>0BC: ∂σ(0,t)∂t=−G, t>0BC: ∂σ(L,t)∂t=−G, t>0IC: σ(s,0)= σT$
(50)
where, $κ = DaBΩ/kT$, and $Da$ = $Doe(− Ed−Ω σTkT)$ is the effective atomic diffusivity. $Ed$ represents atomic diffusion activation energy, $T$ is the absolute temperature, $k$ is the Boltzmann constant, $B$ is the effective bulk elasticity modulus and $G= eiZρ jΩ$. $σT$ is thermal stress developed in the metal line during cooling from the zero-stress temperature down to temperature of use condition. Based on Korhonen's model the stress development over time in a metal is depicted in Fig. 4(b). Void is created if stress at the cathode node exceeds the critical stress. The time taken to reach the critical stress is called nucleation time and is expressed as the following equation:
$tnc=πk[σ2G]2$
(51)

The expression of $σ$ for finite line with and without void under constant diffusion coefficient as well as case of stress-dependent diffusion coefficient can be found in Ref. [95]. Based on Korhonen's model many hydrostatic stress-based kinetics models, which are generally more accurate than traditional model have been developed [96103]. Although Korhonen's model is fundamental to the development of EM physics of failure, it was only applied to a single wire segment. In addition, recent studies show that void nucleation does not cause increase in resistance or failure, except it grows to a critical size often corresponding to the diameter of via or cross-sectional area of the interconnect [97,98].

To capture void incubation and growth period facilitated the development of three phase compact EM models. This model which has been experimentally validated is based on the principle that a nucleated void must grow to a critical size, often the cross-sectional area of the interconnect before current will flow over the interconnect layer, which results in high current density as well as joules heating. Experimental plot verifying these three regions can be found in Ref. [99]. Figure 5 depicts these phases, and the resultant resistance rises with time. During the nucleation phase, a void is not formed until time $tnc$. Therefore, the resistance remains constant. During the incubation phase, defined by the time from $tnc$ to $ti$, the void has nucleated, but its size is not significant. Similarly, the resistance remains basically unchanged. The incubation time ($ti$ − $tnc$) for single and multisegment interconnect is expressed as the following equations, respectively.

Fig. 5
Fig. 5
Close modal
$ti − tnc=ΔLkTDaeZρj$
(52)
$ti − tnc=ΔLkTWDaeZρ∑jjjWj$
(53)
Growth phase is characterized by the time from $ti$ to $t50$. During the growth phase, the void attains its critical size and force the current to flow through the liner or barrier layers. Due to higher line resistivity, the resistance will rise due to joule heating [100]. A critical aspect of EM validation in the design flow is immortality check. Bleck limit, which mainly focused on a single wire segment states that, a wire is immortal for EM if $jL<(jL)crit$. While efficient for single wire segments evaluation, it is not efficient for a multisegments interconnect evaluation. To address EM immortality check for multisegment interconnect, inspired the development of voltage-based models. In some studies [101103], voltage-based EM immortality check for nucleation phase in multiline interconnect was proposed as expressed in the following equation:
$Vcrit>Ve$
(54)
where $Ve=12A∑d≠gNadVd$ and $Vcrit=ΩeZ(σcrit−σo)$and $ad$ is the total area of the branches connected to the node been analyzed, $σcrit$ and $σo$ are the critical and initial stress. Multisegment interconnect is said to be immortal if Eq. (54) is satisfied. A major drawback of this model is its failure to factor joules heating. To address that, in Refs. [104106] EM reliability check model, which considers joules heating was proposed and expressed as the following equation:
$Vcrit>VeT−ViT$
(55)
where $ViT=Vi−QZeln(Ti)$ and $VeT=1/A∑iϵo(Vi−QZeln(Ti))Ai$. $Q$ denotes the specific heat of transport, $Ai$ is the area of the $i$th line, A represents the total area of all lines, $Vi$ and $Ti$ are the voltage and temperature at node $i$, respectively. $VeT$ represents the EM voltage for the whole tree, and $ViT$ is effective voltage considering temperature at node $i$. $o$ denotes all branches in the interconnect tree. In Ref. [107], EMspice tool, was used to evaluate a novel model, which combines the EM mortality check at the nucleation and incubation phase and implemented the model for full-chip power grid analysis. Although joules heating was considered, it was computed separately, which resulted to reduction in prediction accuracy. Nonetheless, studies have shown that interconnect lines can still be immortal even after void nucleation due to void not attaining a critical size [108110]. Two kinds of models exist in this regard. The first is based on void volume, while the second is based on void area. In Ref. [107] a fast saturation volume estimation model for multiline interconnect in incubation phase was proposed and expressed as the following equation:
$νst=h×∑i(−2σc,i+jiLieiZΩ)×LiWi2Bm$
(56)
where $νst$ represents the total saturation volume of a multiline wire, $ji, Li, Wi$ denotes the current density, length, and width of $ith$ segment, respectively. $h$ represents the thickness of the wire. $Bm$ is the bulk elastic modulus, while $σc,i$ is the steady-state stress of segment $i$ cathode, that becomes zero upon void nucleation. Void that does not attain the saturated volume is assumed insignificantly. EM reliability void area-based model have also been reported. For instance, in Ref. [111] the lifetime of 0.18 um wide copper interconnect was estimated using the model given in the following equation:
$t=CAvoidTACuDie[−EkT]$
(57)
where $Avoid$ and $ACu$ denotes the area of the void and copper line, respectively. Equation (57) also indicates that the lifetime depends not only on the void area but also on the temperature, interconnect line cross-sectional area, and line diffusivity. Mass divergence model have equally been reported [112,113]. In these studies, the lifetime of the interconnects was estimated using the equation given as the following equation:
$t=−lnNiNdiv(Jtotal)$
(58)
where $div(Jtotal)$ which represent the total divergence due to electromigration, thermal and stress migration is represented using the following equation:
$div(Jtotal)=(EakT2−1T+αρoρo(1+α(T−To))).NDkTZ*eρj.∇T+ (EakT2−3T+αρoρo(1+α(T−To))).NDkTQ.∇2TT+NDQ*3k3T3j2ρ2ei2.e(−EakT)+(EakT2−1T).NDkTΩ.∇σ∇T−2ENΩDoα13(1−v)kTe(−EakT).(1T+αρoρo(1+α(T−To))).∇2T−2ENΩDoα13(1−v)kTe(−EakT). j2ρ2ei23k2T$
(59)

where $α$ denotes the line CTE, $∇σ, and ∇T$ are the stress and thermal gradient, respectively. $v$ denotes the Poisson ratio. A key merit of the mass divergence model is that it considers the effect of other migration mechanisms such as stress migration and thermo-migration nearly impossible to isolate from EM.

Other EM models, have also been reported. Most of these models are based on some of the above-described variants, with modification in the solution solving technique. For instance, in Ref. [114] semi-analytical technique, based on separation of variables (SOV) method was used to model EM reliability for multiline interconnect. Similarly, in Ref. [115] semi-analytic solution based on accelerated separation of variable (ASOV) method was implemented to solve Korhonen's equation for multiline interconnects. In Refs. [116] and [117], analytical solutions, which can be applied to multiline interconnect were used to solve EM reliability problem for a variety of interconnect such as single, T, and cross-shaped lines. EM dynamic models with capability to model recovery effects under time varying current density and temperature changes were proposed in other studies [118120]. Some other studies reported that the recovery effects consideration at system level could facilitate improvement in EM lifetime [121,122]. In Refs. [123] and [125] the stochastic behavior and EM impact of power grid and signal line interconnect carrying AC currents was investigated. In Ref. [126] a detailed AC EM analysis for signal interconnect based on Monte Carlo method was presented. In Ref. [127], EM-susceptible wires were tracked using a hierarchical EM mortality check algorithm. They compared their result with Korhonen's model for a wire with a finite and semifinite length and reported that Korhonen's model may not be accurate for very long interconnect line. In Ref. [128], power grids stochastic EM analysis, was proposed using the Hermite polynomial chaos-based stochastic analysis. In Ref. [129] thermo-mechanical stress and EM stress on the array of vias for copper wires was proposed and showed that a via in a via array has a different lifetime due to layout dependency. Recently, in Ref. [130] two methods were implemented to trigger interconnect failure. In the first method, an active wire segment is converted to a passive sink, while in the second method, a passive sink is converted to an active sink. In Ref. [131], the two methods described in Ref. [130] were modified by a direct conversion of an active wire segment into an active sink. This approach facilitated easy mortality monitoring of the interconnect structure during operation by simply changing the direction of current flow in the sink segment. Finite different method (FDM) and finite element method (FEM) are well known to be very accurate, but inefficient for very large interconnect wires analysis. To address these FDM and FEM deficiencies, an analytic approach, which basically used eigenfunction to solve Korhonen's equation was proposed in Ref. [121]. Their method has potential to give the exact solution of stress evolution for multiline interconnect for both nucleation and growth phases at a specified time.

Notwithstanding over three thousand articles already published on EM, most experimental data collection effort has mainly concentrated on only current density and temperature as the acceleration stress factors, even though a lot of research works have recognized mechanical stress as a significant EM accelerating factor. To address that, in Ref. [132], a test vehicle designed to apply mechanical stress, current density, and temperature was built. In the study, reliability data were collected based on change in resistance. However, physics-based analysis was not carried out to augment experimental effort. We believe that future research effort will standardize EM three stress factors test plan as well as develop more physics-based models that correctly represent the type of data collected. Also, despite numerous publications affirming the inaccuracy of Black's model for EM time to failure prediction, JEDEC standards [133136], for testing and time to failure estimation are still based on Black's model. In addition, little effort was found to be channeled toward the assessment of the effect of stress factor interaction on the reliability metric for EM. Not considering stress interaction for multistress prevailing would likely reduce the accuracy of any estimation of interest.

## 4 Corrosion

Moisture present in electronics devices is well known to greatly affect its reliability [137]. The deterioration often takes place through a variety of mechanisms, which we broadly classified into two: hydromechanical intrusion and electrochemical corrosion. The hydromechanical mechanism occurs due to hygroscopic stress developed, which could cause delamination in substrate and crack in the package. Soft-moulds used for encapsulated devices are highly susceptible to moisture absorption. Metal is therefore attacked by ions from the solution which diffuses through the encapsulant [138]. Electrochemical corrosion of metals on the other hand also results in electronic devices deterioration under high humid conditions [139]. Factors such as film thickness, geometrical heterogeneities, inclusion/alloying elements, mechanical damage, electrolyte composite, and intermetallic compounds affect the rate of corrosion [140]. Although the present of moisture leads to the deterioration of both metallic and organic electronic device materials, only electrochemical attack on metal (corrosion) is discussed in this section, and emphasis is based on corrosion-based reliability models. For corrosion to occur, oxidation and reduction reaction must occur. Although the oxidation of metals such as Ni, Al, Zn, Si used in packaging is limited, based on the characteristics of the electrolytic solution and that of the metal conductor, metal ions can corrode due to the collapse of the oxide film, which further exposes the metal and facilitates reaction of the metal to form metal complexes and/or salts [141]. While readers can consult [142144] for details on corrosion, a brief description of corrosion process applicable to interconnect is provided. In a neutral solution, metals with a thick film like Aluminum corrosion start with the electrolysis of the water adsorbed to the insulator surface. The oxidation process produces H+ ions and O2 gas in accordance with the reaction given below:
$2H2O→O2(g)+4H++4e−$
Similarly, the reduction process forms OH ions and H gas in a neutral solution or neutral solution containing dissolved oxygen at the cathode as expressed in the reaction below:
$2H2O+ 2e−→H2(g)+4(OH)−2H2O+O2(g)+ 4e−→H2(g)+4(OH)−$

Solution pH greatly affects metal corrosion. The pH-potential diagrams depicted in Fig. 6, are a plot of equilibrium potential (E) against pH for Aluminum. Even though Fig. 6, provides limited information on the corrosion process kinetics, it clearly shows the oxide layer thermodynamic stability boundaries.

Fig. 6
Fig. 6
Close modal

The pH–potential diagram for aluminum comprises of three regions: immunity (I), passivation (III), and corrosion (II and IV). Pure aluminum in the present water develops aluminum hydroxide $(Al(OH)3)$ coating on its surface, which facilitates outstanding oxidation resistance [145,146]. From, the pH diagram, $Al(OH)3$ thermodynamic stability is only between pH 4–9. The $Al(OH)3$, can react with an acid or a base [147]. It oxidizes to form Al3+-ions in an acid solution. Al3+-ions are unstable as well as possess low affinity to migrate [141]. At the cathode, it forms aluminates when it reacts with hydroxide ions.

Since aluminates are stable, they usually migrate through solution and gets precipitated at the anode, except they react to form complex ions during the cause of migration [145,146]. The schematic of this corrosion process is shown in Fig. 7(a). Other metals, such Cu and Ag, easily forms thin metal oxide layer, based on the reaction given below:
$M→Mn++ne−$
Fig. 7
Fig. 7
Close modal
In the present of applied electric field, the metal ions migrate toward the cathode, through the electrolyte [139], where they recombine to form neutral metal atoms as given in the reaction below:
$Mn++ne−→M$

Figure 7(b) depicts the electrochemical migration failure mechanism. The potential difference of metals plays a major role on the rate of corrosion. For galvanically connected metal, the higher the difference in potential, the higher the magnitude of the driving force. Table 4 shows the potential for most metals used in electronic devices.

Table 4

Electrochemical potentials of metals in electronic devices [145]

10% flux solution10% sweat solution
MaterialElectrochemical potential (mV)MaterialElectrochemical potential (mV)
BulkAu+186Au+156
Ag+147Ag+58
316 L SS+146316 L SS+46
Cu+50Cu–26
Ni–110Ni–171
SAC solder–391Tin–480
Sn–438SAC solder–502
Al–532Al–668
Coated on PCBAAg- rolled bonded+167Ag-rolled bonded+156
Au-immersion coated+115Au-immersion coated+58
Mobile dome steel ASI 202+102Mobile dome steel ASI 202+18
Cu-electroplated+39Cu-electroplated–31
Ni electroless coating–241Ni electroless coating–287
Tin surface finished–329Tin surface finshed–462
SAC solder HASL–418SAC solder HASL–478
Al-sputter coated–502Al-sputter coated–583
10% flux solution10% sweat solution
MaterialElectrochemical potential (mV)MaterialElectrochemical potential (mV)
BulkAu+186Au+156
Ag+147Ag+58
316 L SS+146316 L SS+46
Cu+50Cu–26
Ni–110Ni–171
SAC solder–391Tin–480
Sn–438SAC solder–502
Al–532Al–668
Coated on PCBAAg- rolled bonded+167Ag-rolled bonded+156
Au-immersion coated+115Au-immersion coated+58
Mobile dome steel ASI 202+102Mobile dome steel ASI 202+18
Cu-electroplated+39Cu-electroplated–31
Ni electroless coating–241Ni electroless coating–287
Tin surface finished–329Tin surface finshed–462
SAC solder HASL–418SAC solder HASL–478
Al-sputter coated–502Al-sputter coated–583

In general, corrosion process basically involves three steps: oxidation, metal ion migration, and reduction [148]. It is well known to be affected by the humidity, temperature, pH, electric field, process, and service-related contaminants, as well as intermetallic compound [149,150]. Based on the corrosion predominant stress factors, many reliability models have been developed. We classify these models into two: classical models and physics-based models. Table 5 summarizes some notable corrosion-based reliability models found in literature. In Ref. [140] other corrosion classical models were reported. We observed that they are basically a repetition of one of the models given in Eqs. (60)(64), with slight variation in model parameters. Hence, were ignored because the intension of this review was not to repeat an already published review, but rather to capture newly models developed since Pecht et al. [140] was published.

Table 5

Notable corrosion-based reliability models

ClassificationAuthorsTest conditionFailure distributionMeritDemeritFailure criteriaApplicationReferences
ClassicalReich and Hakim80 °C/80%RH 94 °C/92%RH 121 °C/100%RHAssumes exponential, but not verifiedSimple to implementEffect of environment stress and process defect not considered$I cb$> 1 mA, $hcb=400 Vcb=1.75V$PNP devices[151]
Lawson70 °C, 85 °C, 108 °C with 45%-97%RHAssumes log-normal, but not verifiedEstablished the quadratic dependency of relative humidity to MTTF$I cb<$ 2 nA,Small signal NPN[152] [153] [154]
Lycoudes80 °C/80%RH 121 °C/100%RHAssumes log-normal, but not verifiedSimple to implementLack of functionalityEight pin epoxy W/O a die coat[155]
Gunn85 °C/81%RH 115 °C/81%RH 130 °C/81%RH 150 °C/81%RHAssumes log-normal, but not verifiedSimple to implementLack of functionalityBipolar IC and NMOS DRAMs[156]
Peck85 °C/85%RH 121 °C/100%RH 135 °C/94%RH 140 °C/94%RH 140 °C/100%RH 150 °C/100%RHAssumes log-normal, but not verifiedAccount for ion concentrationIgnores ionic contaminationLack of functionalityEpoxy encapsulated devices[157] [158]
Failed to address critical ionic concentration or conditions below which corrosion cannot occur
HornungVoltage/temperatureAssumes log-normal, but not verifiedShowed voltage dependency on ionicmigrationIt does not account for humidity and ionic concentrationN/ASilver in borosilicate glass[159]
Osenbach85 °C/85%RH/5V 135 °C/85%RH/5VAssumes log-normalThe combined effect of temperature, relative humidity, and voltage on lifetime was captured.Doesn't normalize voltage used, by substituting electric field with electric field strengthI > 20 nAInP planar PIN photodiode[160]
Physics-basedLall et al.130 °C/100%RHAssumes log-normalAccounted for ion concentration, Intermetallics and pHComputationally expensiveNot discussedCu-Al wirebond[150]
Improve prediction accuracy through multiphysics coupling in comsol
Jeffrey et al.Various combination of temperature/relative humidity/concentration of ionsVerified exponentialAddressed nondeterministic modeling parameterRelies on extensive experiment to generate model parameters, which varies from location to location.$ΔR/Ro=10%$Au/Al wirebond[162]
Identifies and mathematical linkage between degradation modes and electric circuit models High prediction accuracy
ClassificationAuthorsTest conditionFailure distributionMeritDemeritFailure criteriaApplicationReferences
ClassicalReich and Hakim80 °C/80%RH 94 °C/92%RH 121 °C/100%RHAssumes exponential, but not verifiedSimple to implementEffect of environment stress and process defect not considered$I cb$> 1 mA, $hcb=400 Vcb=1.75V$PNP devices[151]
Lawson70 °C, 85 °C, 108 °C with 45%-97%RHAssumes log-normal, but not verifiedEstablished the quadratic dependency of relative humidity to MTTF$I cb<$ 2 nA,Small signal NPN[152] [153] [154]
Lycoudes80 °C/80%RH 121 °C/100%RHAssumes log-normal, but not verifiedSimple to implementLack of functionalityEight pin epoxy W/O a die coat[155]
Gunn85 °C/81%RH 115 °C/81%RH 130 °C/81%RH 150 °C/81%RHAssumes log-normal, but not verifiedSimple to implementLack of functionalityBipolar IC and NMOS DRAMs[156]
Peck85 °C/85%RH 121 °C/100%RH 135 °C/94%RH 140 °C/94%RH 140 °C/100%RH 150 °C/100%RHAssumes log-normal, but not verifiedAccount for ion concentrationIgnores ionic contaminationLack of functionalityEpoxy encapsulated devices[157] [158]
Failed to address critical ionic concentration or conditions below which corrosion cannot occur
HornungVoltage/temperatureAssumes log-normal, but not verifiedShowed voltage dependency on ionicmigrationIt does not account for humidity and ionic concentrationN/ASilver in borosilicate glass[159]
Osenbach85 °C/85%RH/5V 135 °C/85%RH/5VAssumes log-normalThe combined effect of temperature, relative humidity, and voltage on lifetime was captured.Doesn't normalize voltage used, by substituting electric field with electric field strengthI > 20 nAInP planar PIN photodiode[160]
Physics-basedLall et al.130 °C/100%RHAssumes log-normalAccounted for ion concentration, Intermetallics and pHComputationally expensiveNot discussedCu-Al wirebond[150]
Improve prediction accuracy through multiphysics coupling in comsol
Jeffrey et al.Various combination of temperature/relative humidity/concentration of ionsVerified exponentialAddressed nondeterministic modeling parameterRelies on extensive experiment to generate model parameters, which varies from location to location.$ΔR/Ro=10%$Au/Al wirebond[162]
Identifies and mathematical linkage between degradation modes and electric circuit models High prediction accuracy
Some classical-based corrosion models are briefly described. In Reich and Hakin [151], the failure data from various PNP devices as well as integrated circuits (ICs) operating in Panama was studied. From the study, the time to failure model given in Eq. (60) was derived. Where $RH$ is the relative humidity, $Az and Bz$ are constant
$tf=e[Az+Bz(T+RH)]$
(60)
The study found that encapsulant materials have a substantial impact on microcircuit devices reliability. In Ref. [152], corrosion of small signal NPN devices were studied, and observed the quadratic dependency of time to failure to RH as expressed in Eq. (61). In addition, encapsulant resin type and purity considerably found to influence device corrosion
$tf=e[EaKT+Bz.RH2]$
(61)
Other studies [153,154], used Eq. (61) to study the corrosion of CMOS 4011 encapsulated in epoxy and SRAM, respectively. In Ref. [155], electronic devices coated with thermoset material were studied and fitted the result in the expression given in the following equation:
$tf=e[EaKT+BzRH]$
(62)
In Ref. [156] bipolar ICs were studied at different temperature and humidity condition. In the study, a model given in the following equation was derived
$tf=e[EaKT]$
(63)
Peck [157], studied various devices encapsulated with epoxy, under different test condition. From the study, a regression-based model expressed in Eq. (64) and popularly known as Peck's model was derived. Other studies have implemented Peck's model. For instance, Shirley and Hong [158], used Peck's model to study the corrosion SRAM with 84 and 100 pins
$tf=RHcce[EaKT]$
(64)
In Ref. [159], the corrosion of silver in borosilicate glass was studied and a model which indicated the time to failure dependency on voltage and temperature given in Eq. (65) was derived, which is also known as Hornung model. Leveraging on Peck's and Hornung's models, Osenbach [160] developed a corrosion reliability model, which shows the lifetime dependency of electronic parts to voltage, relative humidity, and temperature as expressed in Eq. (66). Zorn and Kaminski [141], used Eq. (66), to assess the corrosion of an insulated gate bipolar transistor (IGBT) module. where V is the voltage, $cc$ and $gg$ are relative humidity and voltage exponents, respectively,
$t50= ψdGVe[EaKT]$
(65)
$t50= VggRHcce[EaKT]$
(66)
A major demerit of these classical reliability models is that they either basically assume an exponential or lognormal failure distribution without verification or occasionally correlated time to failure as a function of one or more environmental conditions. In addition, they are mainly activation energy-based model. Sadly, activation energy-based model does not provide user with the root cause of a failure. To address these concerns, physics of failure corrosion reliability-based methods are preferable [161]. In Ref. [149] a multiphysics modeling of corrosion of Cu–Al interconnect in high humidity environments was conducted. In the study, Tafel parameters inclusive of open circuit potential and polarization curve slope measured for both Cu and Al at varying ions concentrations as well as pH values were integrated into comsol model, to facilitate the corrosion current prediction at the Cu–Al bond pad. Although it is possible to use the corrosion current to estimate corrosion rate and time to failure, the study didn't establish failure time for the interconnect studied. To address this reliability prediction model based on physics of failure deficiency. In Ref. [150] a physics of failure inspired time to failure model given in the following equation was developed:
$t50=Ao.RH−Ne[A1T0].CoA2.IMCoA3 |pH−7|oA3$
(67)
where $A0$, $A1$, $A2$, $A3$ are constant, $IMCo$, $Co$ are the amount intermetallic and concentration at used condition. Generally, Eq. (67) appears to be an extension of Peck's model with the addition of concentration, intermetallic, and pH adjustment factor. While the model additional parameters were justified by physics, its performance or validity for other interconnect material like Si, Ni, Ag still requires further study. Similarly, in Ref. [162] physics of failure approach was used to study the corrosion effect on the reliability of wire bonds. Based on the experimentally observed effect of humidity, temperature, and chloride ion concentration on the resistance change of the wire bond, a deterministic expression of rate of resistance given in the following equation was established:
$δ(ΔR/Ro)dt= koPcl2[1−e(−(RHζ)β)].e(−EakT)$
(68)

where $ΔR/Ro$ is the change in wirebond resistance, $ko$ is the surface reaction rate constant, and $Pcl2$ is the concentration of gaseous chlorine. Compared to fatigue and electromigration failure mechanisms, physics-based corrosion models have not received a lot of attention. We feel that insufficient understanding of mathematical linkage between critical aspects dealing with a wide range of potentially relevant corrosion degradation factors, as well as difficulty in addressing the existence of nondeterministic (uncertain) modeling parameters, are among factors affecting the low attention given to corrosion mechanism-based reliability models. In addition, JESD22-A107B [163] as well as other standard on corrosion mainly focus on test protocol. Lack of reliability-based prediction model based on corrosion failure mechanism, incorporated into standards may cast doubt on the validity of prediction models found in literature. Nonetheless, more research activities are required in this area to facilitate development of more robust physics-based models.

## 5 Conductive Filament Formation

In Sec. 4 reliability models associated with corrosion failure mechanism in electronics devices were described. Most of the models in Sec. 4, didn't consider the effect of electric field. In this section, a failure mechanism driven by the combined presence of an applied electric field and liquid medium known as CFF is reviewed. This failure mechanism was first observed in 1955, by researchers at Bell Laboratories [164] on silver. In their investigation, silver in contact with insulating materials was found to be removed from the anode and deposited in a different location, which resulted in breakdown of the insulating material. In Refs. [165167], Au, Cu, Ni, Sn, and Pb were found to migrate, which led to microcircuits insulation breakdown. The failure mode for this mechanism includes formation of shorts that grows through the substrate and between conductors of same size as a substrate [168]. Generally, CFF is known to occur in two steps: organic material delamination and moisture absorption at the interface and metal migration leading to loss of insulation resistance. Several reliability models for time to failure due to CFF have been reported and are subsequently discussed.

In some studies, Refs. [169] and [170] two-step time to failure models were developed. One is dependent on humidity, while the other is dependent on voltage. It was further shown that the two-steps failure was serial in nature, thus the combined reliability model commonly called Welsher equation [171] is given as the following equation:
$tf=aRHxxe[EART]+dL2V$
(69)
Ready and Turbini [172], studied the effects of voltage and spacing on CFF failures for a H–H test pattern. Based on experimental data, they derived a modified Welsher's equation given as the following equation:
$tf=ce[EakT]+dL4V2$
(70)
In Ref. [171] the two-step filament formation was confirmed. Although the degradation was found to be dependent on relative humidity and temperature, voltage dependence was not established. Based on that, the time to failure due to filament formation is expressed as Eq. (71). Equation (71) is only valid for temperatures between 50 °C and 100 °C and $RH$ between 60–95%
$tf=α(1+μLnV)RHxxe[EART]$
(71)
Rudra et al. [169], derived the threshold humidity ($Mt$) model for CFF. This helped to show the inefficiency of linear acceleration factor models for reliability extrapolation. The $RHth$ is expressed as the following equation:
$Mt=e[0.4383+0.1828ln(c)+1917T−0.278ln(V)]$
(72)
Using Eq. (72) the filament formation time to failure for $M>Mt$ is given as the following equation:
$tf=af[1000Le]nVm.(M−Mt), M>Mt$
(73)
where $a$ is the acceleration factor for filament formation, $f$ is a multiplayer correction factor, $n$ is a geometry acceleration factor, $V$ is the applied voltage, $m$ is a voltage acceleration factor, $M$ represents the moisture content percentage, and $Mt$ denotes threshold moisture content percentage. $Le=kL$. This model has been used for CFF time to failure prediction [173,174]. Di-Giacomo investigated [175] the migration characteristics of silver (Ag) in encapsulated packages and developed a physics-based model derived from Butler–Volmer equation expressed as the following equation:
$tf=QcβJt$
(74)

where $Qc$ represents the quantity of metal ions that should migrate to enable dendritic growth across a gap, $β$ denotes metal to metal degree of oxidation, $Jt$ represents the current density at the dendrite tip. The industrial relevance of Di-Giacomo's model diminishes due to current density at the dendritic tip requirement. Although, many research effort has been put into development of reliability models, which can be used so long as users are aware of their limitation, many of the developed models are rather not physics based. Furthermore, little effort is channeled toward models that capture interaction between stress factors. As stated in Sec. 4, there is a strong variation in CFF test protocol by standards. To illustrate, JEDEC-A101-B specifies CAF test condition as 85 °C/85%RH for 1000 h [176]. On the other hand, IPC-9691A specifies, test conditions of 85 °C and 87% RH for 500 or 1000 h [177]. There is therefore a need to develop a unified model for electrochemical migration time to failure evaluation. This unified model in our view will required advance surface layer measurement of water (ellipsometry). In addition, rigorous characterization of plating material, board surface, flux chemistry, and their interactions are extremely essential if a robust unified model is desired.

## 6 Stress Induced Voiding Formation

Stress induced voiding (SIV) is a key reliability issue common to line interconnect. The induced stress occurs due to processing activities associated with both single and multilayered interconnect system. Common sources of these stresses on metal line include the deposition of the interlevel dielectric and the subsequent cooling, CTE mismatch associated with the dielectric material and the metal following thermal treatment, as well as other thermal treatments that may have an influence on the stress state of the metal. The main driving force for SIV is stress gradient. Void formation and growth effect in an interconnect depends on the location. Owing to the sudden open failure related to voids below the via, these failure modes are referred to as SIV, while the partial resistance rise at the via is often called stress migration. This characteristic is experimentally correlated with metal geometry [178,179]. Over the years, many physics-based models have been developed to facilitate reliability prediction for SIV mechanism. We classified these models into two: those based on vacancy movement and those based on stress relaxation.

McPherson and Dunn [180] proposed SIV time to failure model (see Eq. (71)) derived from Fick's law. The study assumes that vacancy movement caused by thermally induced stress was the basis of SIV formation. $Esiv$ represents SIV formation activation energy, $Bsiv$ is a constant independent of stress and activation energy, $μz$ is a constant and $σ$ is the stress.
$tf=Bsivσ−μze(EsivkT)$
(75)
Ogawa et al. [181] improved on Eq. (71), through the concept of active diffusion volume. Thus, only those vacancies within the active diffusion volume, took part in the SIV process. Based on that, the time to failure given in the following equation was developed:
$tf=(2w23δ) (f.(D0e(EsivkT)kT).ϖ.ΔσΔx)$
(76)
where $w$ is the via diameter, $δ$ is the width of the diffusion path, $ϖ$ is the atomic volume, $f$ is fraction of vacancy and $ΔσΔx$ is the stress gradient. In the models described above, lattice relaxation due to strain energy was not considered. In Refs. [182] and [183], it was shown that the effect of SIV will be underrated if the impacts from lattice relaxation are not considered. To address this gap, Fischer and Zitzelsberger [184], proposed SIV time to failure model given as Eq. (73), which considered relaxed stress
$tf=CT(To−T)eEsivkT$
(77)
where $C$ represents a specific constant. $To$ is the process temperature, while $T$ represents the test temperature. Tan and Hou, [185] proposed SIV stress relaxation lifetime model from the energy perspective for line Eq. (74), surface Eq. (75) and cubic Eq. (76) region of an interconnect as well as the critical temperature given in Eq. (77) that will facilitate its occurrence
$tlf=C1γ2TBsiv3D1(To−T)4eEsivkT$
(78)
$tsf=C2γTBsiv2D1(To−T)2eEsivkT$
(79)
$tcf=C3γ2/3TBsiv5/3 (3D1DG12+DG32) (To−T)4/3eEsivkT$
(80)
By rearranging the temperature independent terms as a constant C from Eqs. (74)(76), the unified expression for the SIV time to failure expressed in Eq. (73) was derived
$Tc=2EsivTo(Esiv−kT)2+4EsivTokN+ Esiv+EATo$
(81)

$tlf$, $tsf$, and $tcf$ are the time to failure developed for line, surface, and cubic interconnect, respectively. $C1−C3$ are constants, which depend on the void surface area at failure; cap layer/Cu interconnect thickness and width at the interfacial layer; the interconnection width and thickness. $B and T$ represents the effective bulk modulus and temperature, respectively.

In Ref. [186], physics associated with void growth, lead to the classification of void growth into three sequential stages, referred to as SIV plate, long line, and short line. The time to failure model associated with the three sequential stages is given as Eqs. (79)(81), respectively. Several factors such as stress, effective modulus, diffusivity, and temperature are accounted for in the time to failure models given in the following equations:
$SIV plate tp= tpkt2πσoDΩ if Ψ≪1$
(82)
$SIV long line tl= Vr2πBkT4σo2w2DΩ if 1<Ψ≪L/w$
(83)
$SIV short line ts= (4kTL24σo2π2DBΩ)ln([1−tfBσoLw]π28) if Ψ∼L/w$
(84)
The time to failure was simplified as the following equation:
$tf=max[tp,tl,ts]$
(85)
where $tp,tl, and ts$ represents the time to failure for the plate, long line as well as short line, respectively. Gavin and Allman [187] developed a SIV model, based on the assumption that TDDB, is controlled by a set of weak links. The TDDB based on weak links was developed using Gibbs-Maxwell quasi-steady-state model as given in the following equation:
$(tf)weak=dkTh2B′2πσoDΩ.ro3EσoVdf(1+3Φ2roλn).eEsivkT$
(86)
where $Φ= 12π(nλ)3/2+2r2π(nλ)3/2+2ro2(nλ)1/2$, $Vdf$ is the diffusion volume, $h$ is the trench depth, B represent Gibbs' prefactor, $d$is the average grain size, $n$is the number of FRUs, $E$ is the effective elastic modulus, $ro$ is the via radius, $D$ is the diffusivity. In Ref. [188] the weak link concept was evaluated using extreme value statistical theory. This model has been used in stress migration and SIV in Fischer and Zitzelsberger [184], to derive an AFT model in Al-based metallization. In addition, it has been used by Refs. [189] and [190] to describe stress-relaxation in encapsulated Cu films. Gavin and Allman [191], extend their previously published model in Ref. [190] for SIV given as Eq. (84), where $g(t)$ is given as Eq. (85) for a case where diffusion proceeds according to a power-law as given in Eq. (86)
$R=1−e(sμnvπλ(gv(t)−γ)μ)$
(87)
$g(t)=1−e(−α1(t−to), M=11−[1+ξM(t−to]−1M−1, M>1, γ=ros, 0
(88)
$tf=A(ro3Vint)(ΔαefΔT)−me(EsivkT)$
(89)

where $nv$ is the via chain length, $λ$ is the probability per unit area of a void nucleus, the interconnect volume $Vint. The power−law parameter M ∈ (1, ∼ 20)$. While the inclusion of $Vint$ and $ro$ in Eq. (86) represent a notable improvement, we believe that a more robust SIV time to failure model could be achieved if microstructural variations, adhesion of the capping layer as well as other variables that may affect void nucleation are integrated into the model. Although, SIV models contained JESD214.01 [192], were derived based on physics of failure, specifically in agreement with the models of Refs. [178182], effect of microstructure, as well as adhesion of capping barrier, were ignored and constitute an interesting research area to explore in future.

## 7 Time Dependent Dielectric Breakdown

Time dependent dielectric breakdown is a failure mechanism whereby breakdown takes place, often below the breakdown strength of the material (dielectric) with time when stored at a nonvarying electric-field (e-field). In the present of electric field, several conduction mechanisms have been reported to facilitate TDDB [193,194]. Ionic conduction, direct tunneling, Schottky emission, space-charge-limited conduction, Poole–Frenkel (PF) emission, ohmic conduction, and Fowler–Nordheim (FN) tunneling, were reported as the common conduction mechanisms. Detailed description of these conduction mechanisms, although can be found in Ref. [194] is not the interest of this paper. Rather, a summary of the common TDDB reliability models found in literature with emphasis on their merits and demerits, especially for low-k dielectric is the primary concern of this section. Generally, TDDB reliability models found in literature are developed based on either current, field, or field-current induced degradation collected data. We classify these models into two; intrinsic and extrinsic TDDB models. Table 6 provides the summary of these intrinsic and extrinsic models.

Table 6

Intrinsic and extrinsic TDDB models

TDDB model categoriesModelsConduction mechanismMetal ion migrationDielectric applicabilityFailure distributionMeritDemeritReferences
IntrinsicLloyd $(1E+E)$Poole-FrenkelNoUltralow -$k$ interlevel dielectricLog-normalVery simpleNot applicable to thin gate dielectric with thickness less than the mean free path of electrons[197]
It doesn't depend on the mechanism causing the damage.
Thermochemical (E)Fowler–Nordheim (FN)NoThin gate oxidesWeibullAcceleration parameters of the electric field, as well as activation energy, can be estimated quantitativelyLacks explanation for polarity dependence deficiency[198,200]
Low-k dielectric
1/EFowler–Nordheim tunnelingNoThin gate oxidesWeibullCommonly used reliability modelLow prediction accuracy for film thickness < 5 nm[201,202]
Low-k dielectric
Very low efficiency of injected hole induced defects
Lack quantitative explanation for the high temperature dependence associated with TDDB testing.
Haase modelPoole-FrenkelNoLow-k dielectricLog-normalFailure criterion is based on time to minimum currentIt validity of 100 TTMC as a TTF estimation criterion for a different type of dielectric is unclear[204]
Power law voltageFowler–Nordheim tunnelingNoHyper thin oxide filmWeibullCaptured hydrogen induced breakdownNot suitable for thick oxide films No explanation for the high temperature dependence observed during TDDB testing[205]
It suffers from dilution problems
ExtrinsicCu drift E modelPoole–FrenkelYesLow-k dielectricWeibullA set of parameters (activation energy and periodicity of the potential) can ensure obtaining a good fit over a variety of temperature and e-fieldLacks explanation for polarity dependence deficiency[206,207]
$E1/2$For high quality Si02 dielectric (FN) Other dielectric (Poole-Frenkel or Schottky)YesMetal-SiN-metal capacitorWeibullEfficient for the prediction of impurity induced defect.Lack quantitative explanation for the high temperature dependence associated with TDDB testing.[208]
Low-k dielectric
$E2$Poole–FrenkelYesLow-k dielectricWeibullA set of model parameters is required to fit all experimental data satisfactoryLacks explanation for polarity dependence deficiency[212]
TDDB model categoriesModelsConduction mechanismMetal ion migrationDielectric applicabilityFailure distributionMeritDemeritReferences
IntrinsicLloyd $(1E+E)$Poole-FrenkelNoUltralow -$k$ interlevel dielectricLog-normalVery simpleNot applicable to thin gate dielectric with thickness less than the mean free path of electrons[197]
It doesn't depend on the mechanism causing the damage.
Thermochemical (E)Fowler–Nordheim (FN)NoThin gate oxidesWeibullAcceleration parameters of the electric field, as well as activation energy, can be estimated quantitativelyLacks explanation for polarity dependence deficiency[198,200]
Low-k dielectric
1/EFowler–Nordheim tunnelingNoThin gate oxidesWeibullCommonly used reliability modelLow prediction accuracy for film thickness < 5 nm[201,202]
Low-k dielectric
Very low efficiency of injected hole induced defects
Lack quantitative explanation for the high temperature dependence associated with TDDB testing.
Haase modelPoole-FrenkelNoLow-k dielectricLog-normalFailure criterion is based on time to minimum currentIt validity of 100 TTMC as a TTF estimation criterion for a different type of dielectric is unclear[204]
Power law voltageFowler–Nordheim tunnelingNoHyper thin oxide filmWeibullCaptured hydrogen induced breakdownNot suitable for thick oxide films No explanation for the high temperature dependence observed during TDDB testing[205]
It suffers from dilution problems
ExtrinsicCu drift E modelPoole–FrenkelYesLow-k dielectricWeibullA set of parameters (activation energy and periodicity of the potential) can ensure obtaining a good fit over a variety of temperature and e-fieldLacks explanation for polarity dependence deficiency[206,207]
$E1/2$For high quality Si02 dielectric (FN) Other dielectric (Poole-Frenkel or Schottky)YesMetal-SiN-metal capacitorWeibullEfficient for the prediction of impurity induced defect.Lack quantitative explanation for the high temperature dependence associated with TDDB testing.[208]
Low-k dielectric
$E2$Poole–FrenkelYesLow-k dielectricWeibullA set of model parameters is required to fit all experimental data satisfactoryLacks explanation for polarity dependence deficiency[212]

Intrinsic TDDB models. These models were developed on the assumption that dielectric breakdown is only driven by either applied field or current or both. Contamination from metallic ions and its associated contributory role to dielectric breakdown is completely ignored. Intrinsic TDDB constituted a key concern in integrated circuit (IC) applications before 2000s [195]. Aluminum gates were the primary fabrication technique for IC during that period. SiO2 dielectric material reacts with Al, to form Al2O3. The aluminum oxide compound builds a conductive path in SiO2 because of SiO2 breakdown. Numerous intrinsic TDDB related studies have been published [194199]. Following a careful survey of these publications, we classified intrinsic reliability models into five, comprising of Lloyd, thermochemical (E), 1/E, Haase, and power law, and are consequently described.

Lloyd model was developed based on the assumption that excited electrons caused failure in low-k dielectric. Upon passing current to the dielectric, electrons speed is enhanced by the current induced e-field, which results in the electron gaining energy, required to travel some distance within the dielectric. After a certain distance, it undergoes scattering and subsequently dissipation of its acquired energy. If the electron energy is less or equal to a threshold energy value, a fresh defect, also called trap is created. The buildup of defects in the dielectric ultimately results in TDDB. The speed of defect generation is a function of the product of the current and the probability that when scattering incident occurs, the electron has sufficient energy to generate defect. Based on this assumption, Poole–Frenkel mechanism equation [197] and Poisson distribution were used to derive the time to failure given as the following equation:
$tf=Nfl−NoAEe(−γE+EtμqE)$
(90)

$No$ is the number of pre-existing defects, $Nfl$ is the number of defects needed to promote breakdown. $A$ and $γ$ are the Poole–Frenkel parameter constants. $Et$ represents the threshold energy, $μ$ is the mean free path of the electron. The $E$ dependence in Eq. (87) is because of the Poole–Frenkel conduction mechanism, while the 1/E dependence is due to the exponential probability distribution function.

Thermochemical (E) model: in this model, defect is generated due to field-driven process [198]. E-model is based on the theory that a conductive path forms between an anode and a cathode under stress conditions [199]. Separated Si–O bonds form because of the applied stress, which results in the formation of the conduction path. Derivation of E model-based dielectric time to failure is described in McPherson and Mogul [200]. The dielectric lifetime is dependent on the speed of bond-breakage. The E model time to failure is given as the following equation:
$tf=Ae(ΔEtdkbT −aE)$
(91)

where $ΔEtd$ stands for the activation energy associated with bond breakage, $T$ is the temperature and $a$ represent the dipole polarization related acceleration parameter.

The$1/E$model was developed on the concept, that during voltage stressing, electron travel into the dielectric material from the cathode by FN tunneling. Upon arrival of the electron at the anode, they thermalize, and holes are generated from their energy. Chen et al. [201] proposed 1/E model, to explain TDDB in thin gate oxides. Their model was later modified by Schuegraf and Hu [202]. In accord with Schuegraf and Hu, and Suehle [203], $1/E$ model time to failure, is expressed as the following equation:
$tf=τo(T)e(ψE)$
(92)

where $ψ=Ge+H$, is the field acceleration parameter. $Ge$ and $He$ are associated with electron and hole tunneling , respectively. $τo(T)$ is the temperature-dependent prefactor.

Haase model is slightly different from the TDDB models briefly discussed above. Instead of concentrating on developing time to failure model for dielectric, it tries to simulate the leakage current as a function of time. The dielectric failure criterion is based on the time to minimum current [204]. The rationale for this approach is based on the argument that some of the conduction mechanisms associated with the previously described dielectric models lack empirical explanation.

Power-law voltage model also called the anode hydrogen release (AHR) model is developed on the concept that excitation (coherent or incoherent) processes, facilitate hydrogen released into the Si–O bonds. The released hydrogen interaction with the weak silica bonds is assumed the cause of bond defect generation [205 ]. This model is of the form expressed as given in the following equation:
$tf=AoT V−N$
(93)

A major drawback of the intrinsic models is the lack of consideration of contaminant especially from metallic ions. This necessitated the development of extrinsic TDDB models.

Extrinsic model as previously mentioned, breakdown can sometimes be due to metal ion contamination. TDDB models that factor this metal contaminant are referred to as extrinsic models. Some extrinsic TDDB models include Cu drift E model, E1/2 model, E2 model. A brief description of these models is presented.

Cu drift E model: Muraka et al. [206] evaluated drift of Cu+ inside a dielectric (SiO2). They found that Cu in Si forms defects in the band gap of the Si. That is harmful to metal oxide semiconductor device operation. Wu et al. [207] developed a TDDB reliability model based on the diffusion of copper ions in a periodic potential with an external bias. From their work, the derived $tf$ is expressed as the following equation:
$tf=Be[Eaka]e[qγEkaT] −e[− qγEkaT]$
(94)

where $B$ is a constant that depends on $γ$. For large fields, this $tf$ relationship reduces to 1/E model with an acceleration parameter given by q$γkBT.$

E1/2 model: Aller [208] pioneered E1/2dielectric reliability model. In his study, SiN capping layer was deposited on a Cu interconnect and PF conduction was assumed to be predominant. Upon application of voltage stress, copper at the anode top get transferred to the interphase between the low-k dielectric and capping layer. With passage of time, the copper arriving at the cathode will accrue as a sheet of positive charge. When the concentration exceeds a critical value, TDDB occurs. Based on Poole–Frenkel conduction mechanism, the time to failure was derived as expressed in the following equation:
$tf=1Ee[−qkbT qπεok E]$
(95)
Conversely, in Ref. [209], Schottky emission conduction mechanism was used to derive the TDDB time to failure model given in the following equation:
$tf=Ccrit2 lo2DoAoT4e[1kbT (ED+2φs−2βsE)]$
(96)

where $lo$is the wire total length; $φs$ is the barrier height and $βs=(q34πkεo)1/2$. Other studies [210,211] validated experimentally the E1/2 model low-k TDDB involving SiOCH dielectric material.

E2 model: Zhao et al. [212] proposed the E2 model. Like the $E1/2$models, copper at the anode is transferred to the cathode in the present of electric field. With passage of time, the copper arriving at the cathode will accrue, which will subsequently lead to failure. The time to failure model is given as the following equation:
$tf=Ae[ED−γEapp2kbT ](f(Ce, T,Eapp) )$
(97)

where $f(Ce, T,Eapp)$is the time required for the electric field at the cathode to attain the breakdown field $ED$. This time is dependent on Cu2+ solubility in the dielectric, $Ce$, while T and $Eapp$ are the absolute temperature and applied field. To validate the TDDB reliability models available in literature, Allers [208] evaluated Cu/SiO2/Si structure for electric field range of 3.5–10 MV/cm. The study results show that the 1/E model outperformed the other models considered in terms of time to failure for TDDB. Rodriguez-Fernandez et al. [213,214] compared three intrinsic models (E model, AHR model, and 1/E model), to identify the acceleration law, which drives the breakdown of $HfO2$ dielectric in ReRAM devices. Their result reveals, that the E-models exhibited the lowest dispersion in the considered acceleration factor. Yeap et al. [215], used power law and $E1/2$ models, to assess the effect of chip population line spacing on its lifetime prediction. It was found that for the chip population with line spacing smaller than $4 nm$ the power law and $E1/2$ model show no significant difference in lifetime prediction. While for line spacing larger than 4 nm the power law results in overestimation. Rodriguez-Fernandez et al. [213], tested $E1/2$ model on $HfO2$, while $E2$ model was tested on $SiO2$ [216]. Most of the other models have been verified only on $SiOCH$. So many other dielectric models mainly focus on predicting the time to breakdown at a specific electric field and temperature, based on data collection from accelerated testing. While many models have been developed on the assumption of uniform electric field. Nonuniform field may occur due to nonuniform lithography process, in addition to porosity and pattern line edge roughness. Also, clarity of operation definition of breakdown especially for future dielectric material such as porou k- material. Though many physics-based models have been developed for time to breakdown estimation, the JESD 92 [216] and JEP159A [217] standards for ultrathin gate dielectrics and low-k/metal inter/intralevel dielectric respectively agree with some of the physics-based TDDB model determined with Weibull distribution predominantly used.

## 8 Integrated Reliability Modeling for Power Electronics Systems

Weibull distribution, which is suitable for the various failure mechanisms, from fatigue to TDDB is notably one of the well-recognized mathematical models used in reliability engineering [218]. Its probability density function is given in Eq. (94). $γ$ is the location parameter
$f(t)= βη.(t−γη)β−1.e−(t−γη)β$
(98)
If failure can occur at any time within the life of the system the location parameter $γ$=0. Generally, the shape parameter can be obtained from probability plot and varies as a function of stress level. For solder joint fatigue and Si IGBT modules, LESIT [219] and CIPS2008 [220] models given in the following equations can be used to estimate the scale parameter, respectively,
$Nf= Axc.(ΔTj)α.e−Ek(Tmn+273)$
(99)
$Nf= Kxc.(ΔTj)β1.eβ2(Tj+273).tnβ3.Iβ4.Vβ5.Dβ6$
(100)
where $Axc$, $Kxc$, α, $β1$$β6$ are the material constants. In addition, $ΔTj$, $Tmn$ and $Tj$are amplitude of temperature swing per cycle, mean junction temperature, and absolute maximum junction temperature, respectively. For the CIPS2008 model $tn$, I, V, D represents time, current per wire bond, chip blocking voltage, and diameter of bonding wire, respectively. In Ref. [221], a modified CIPS2008 model was compared to LESIT model for a multiparameter reliability model of a SiC power MOSFET subjected to repetitive thermomechanical load. It was found that CIP2008 show lower estimation error. Many power electronics reliability modeling efforts are often based on one of the failure mechanisms described above [50,96,150,171,180,213]. Realistically, failure of power electronics devices is seldom caused by only one failure mechanism. Rather, a combination of failure mechanisms due to integration of many parts or components. For instance, electromigration along interconnects line can occur simultaneously with fatigue related failure at the attachment points. In addition, MOSFETs, insulated gate bipolar transistors (IGBT), capacitors within a power module can fail along with the degradation driven mechanism already described above. Hence, there is a need for an integrated reliability models that consider all potential failure mechanisms or modes associated with all components that make up a power electronics system. In this section, integrated reliability models are concisely described with emphasis on power electronics systems. We classified integrated lifetime prediction models into three; weakest link, configuration-based, and data driven approaches. In the weakest link approach, the failure criteria of each module component are defined and their lifetime estimated based on one of the mechanisms described above. The lifetime of the component with the least life is considered the life of the modules. Some studies [222,223] used the weakest link approach to predict the life of (IGBT) comprising of busbar, wire bond, chip solder, and substrate solder, under thermal cycling loading of 40 °C and 125 °C with 15 min ramps and dwells. In their study, the busbar was found to have the lowest lifetime of 946 cylces. Thus, the IGBT lifetime was assumed to be 946 cycles. A major drawback of the weakest link approach is it inability to account accurately for the reliability of multiple devices within a module based on configuration or functional layout. For instance, a power module with two MOSFETs functionally connected in parallel. The failure of one of the MOSFET does not imply the failure of the power module. Configuration-based reliability modeling approach accounts for the variation in device configuration and its effect on the overall module reliability. Generally, series, parallel, k-out-n, and mixed configuration are commonly used in power electronics. Detail description of reliability models based on configuration was can be found in Refs. [224226]. We briefly highlight some power module configuration-based modeling studies. In Qiu et al. [227] reliability model comprising of D and M number of diodes and MOSFET functionally connected in series was presented as depicted in the following equation:
$R(t)= ∏j−1DRdiode(t).∏j−1MRMOSFET(t)$
(101)
where $R diode(t)$ and $RMOSFET(t)$ is the reliability of the diodes and MOSFET at time $t$, respectively. In addition, the study also modeled the reliability of a power module comprising of k-out-n wire bond connect in series to a chip as given in Eq. (98). where $Rcp(t)$ and $Rwb(t)$ is the reliability of the diodes and MOSFET at time $t$, respectively,
$R(t)=Rcp(t) .∑j−rnCnjRwb(t)j[1−Rwb(t)]n−j$
(102)

Tu et al. [228] implemented configuration approach in reliability- and cost-based redundancy design for modular multilevel converter. Uwe and Kai [229], used Boolean algebra to solve an integrated data center infrastructure comprising of a power grid and cooling systems. Peyghami et al. [230], used configuration-based approach for reliability modeling of power electronics converter. A major drawback of the configuration-based approach is its real-time reliability prediction implementation difficulty. Thus, recent effort is channeled toward data-driven approaches. Generally, field data collected via experiment or degradation monitoring using sensors are used to train dataset based on physic of failure for prediction purposes. In Ref. [223] reduced order models were proposed to predict the stresses in the power electronic module package when subjected to operating and environmental loads. Ahsan et al. [231], used machine learning-based data-driven prognostic models to predict degradation behavior of IGBT as well as determine remaining useful life based on degradation raw data collected from accelerated aging tests under thermal overstress condition. Improvement in power electronics module reliability depends on appropriate material; geometry and size optimization as well as operating and effective stress control strategies. Reliability assessment which combines finite element modeling, experimental validation, and data-driven prognostics approach has potential to improve system reliability prediction and understanding. Considerable research effort is still required to develop data-driven reliability tools for prognostic purposes.

## 9 Conclusion

In this paper, reliability models associated with six failure mechanisms comprising of fatigue, EM, corrosion, CFF, SIV, and TDDB commonly experienced by electronic devices found in literature were studied in addition to integrated modeling approaches of power electronics models, with the view to highlight progress made and areas requiring improvement. Based on approximately 58, 73, 27, 12, 17, and 24 papers reviewed on fatigue, EM, corrosion, CFF, SIV, and TDDB, respectively, the following condition is reached.

There is lack of clear definition of solder joint fatigue within Standards, which could lead to inaccurate basis of data comparison.

• Although, five fatigue model categories were identified, creep and energy-based fatigue models received most recent research attention, with recent modeling effort geared to factoring microstructural effects into models. While many constitutive equations are readily available, careful examination is required before application to a specific solder joint.

• Despite, extensive electromigration physics of failure articles published, which considers the hydrostatics stresses due to electron wind force, indicating the presence of three stress factors (current density, temperature, and mechanical stress), EM accelerated test are mainly based on two stress factors (current density, temperature). In addition, very few electromigration models consider the interaction between stress factors. To enhance better prediction accuracy models, effect of stress interaction needs to be considered in future studies.

• Voided interconnects are analyzed after electromigration data collection has been concluded. There is a big doubt, as to the relevancy of this practice, from the perspective of nucleation site analysis reported in many past studies. Hence, future research on EM will most likely focus on systematic void nucleation and migration monitoring in real-time.

• JEDEC standard is still based on Black's even though it has been over emphasized to be very inaccurate. It is therefore essential to update JEDEC standards to reflect some well tested physics-based models.

• Physics of failure-based corrosion and conductive filament formation models have received extremely low attention lately, with most of the models based on Arrhenius model. This observation could be due to lack of comprehension of mathematical linkage of critical degradation factors. In addition, available standards were found to be inconsistent in test procedure.

• TDDB PoF reliability models are mainly developed based on the assumption of uniform electric field. Nonuniform electric could exist. Thus, the application uniform field-based model to nonuniform field distribution could be misleading.

• In general, most of the recent successful research modeling efforts is not integrating quickly into standards.

• A critical observation from this review is that most modeling efforts only focused on a single failure mechanism. In practice, components may experience combined failure mechanism. For instance, EM and fatigue. Models based on such combine failure mechanism may be more accurate from a reliability prediction perspective.

• Integrated reliability models based on data-driven approach, though still developing show great potential for real time power electronics real-time reliability prediction.

## Funding Data

• Office of Naval Research (Contract No. FA9550-21-1-0205; Funder ID: 10.13039/100000006).

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