Sensitivity analysis on the fatigue life of solid state drive solder joints by the finite element method and Monte Carlo simulation

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1 ( ().,-volV)( ().,-volV) TECHNICAL PAPER Sensitivity analysis on the fatigue life of solid state drive solder joints by the finite element method and Monte Carlo simulation Yeungjung Cho 1 Jinwoo Jang 1 Gunhee Jang 1 Received: 20 October 2017 / Accepted: 25 February 2018 Ó Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract We proposed a method to determine the probabilistic fatigue-life distribution of solid state drives (SSDs) due under thermal cycling through finite element method and Monte Carlo simulation and the effect of their design and operating variables on failure through sensitivity analysis. In the developed finite element model, we utilized the Anand model to represent the viscoplastic behavior of solder balls and utilized the Prony series to represent the viscoelastic behavior of a polymer material in the underfill. In order to verify the developed finite element model, we compared the simulated strain with that measured in a thermal chamber. Using the developed finite element model, thermal cycling analysis was performed to determine the fatigue life of the SSD using the simulated results and Morrow s energy-based fatigue model. The finite element analysis results showed that the outermost solder ball at the corner of the dynamic random access memory (DRAM), was the most vulnerable component under thermal cycling. Monte Carlo simulation was performed to analyze the fatigue life distribution according to the manufacturing tolerances of the design and operating variables of the SSD. Sensitivity analysis was also performed to determine the effects of variables on number of cycles to failure. The analysis results showed that the minimum temperature of the thermal cycling profile condition and the diameter of the DRAM solder balls dominantly affected the fatigue life of the SSD. 1 Introduction Solid state drives (SSDs) have recently and quickly replaced hard disk drives due to their advantages of low noise, small size, light weight, and high data processing speed. However, whenever SSDs are energized, the electrical energy needed for operation is converted to heat, during which the SSDs are exposed to thermal excitation. Evaluating the reliability and fatigue life of SSDs under thermal cycling has become an important topic even in the initial design stage. Steinberg (1988) reported that 55% of electronic products are destroyed due to thermal factors. SSDs are electronic products in which packages are connected by printed circuit board (PCB) through the surface mounting of solder joints, which is directly related to the lifespan of electronic devices. The temperature of an SSD changes continuously during operation, and the solder balls & Gunhee Jang ghjang@hanyang.ac.kr 1 PREM, Department of Mechanical Convergence Engineering, Hanyang University, Wangsimni-ro 222, Seongdong-Gu, Seoul 04763, Korea are subjected to repeated stresses due to differences in coefficients of thermal expansion (CTE) between the package and PCB. If the generated stress is higher than the fatigue limit, cracks will occur and propagate, finally leading to fatigue failure. With regard to the fatigue lifespan of SSDs caused by continuous changes in temperature, the Joint Electron Device Engineering Council (JEDEC) (1989), an international standardization organization, has stipulated an evaluation method to assess the fatigue failure of electronic products due to thermal cycling. Many researchers have developed methods to predict fatigue life due to thermal cycling using the finite element method not only because thermal cycling experiments requiring sophisticated equipment are costly and timeconsuming, but also because it is difficult to reveal differences in the fatigue life characteristics of SSDs due to variations in design variables. To predict the fatigue life of solder balls, Pang et al. (2004) derived material constants with respect to temperature and frequency for two fatigue failure models. Using the derived material constants, Schubert et al. (2003) predicted the difference in fatigue life between SnPb solder and lead-free solder (SnAgCu) under thermal cycling condition for an FCOB (flip chip on

2 board). They showed that lead-free solder lasted longer than SnPb at higher strain, whereas SnPb lasted longer than lead-free solder at lower strain. Zhang et al. (2014) performed finite element analysis to compare the fatigue life of CSP (chip scale on package) under thermal cycling condition according to solder material. SnAgCu solder joints exhibited higher fatigue life than SnPb solder joints. Chen et al. (2001) calculated the fatigue life of solder balls in FCOB through finite element methods under thermal cycling condition. However, only a single value was provided with regard to the fatigue life of their models. Although various studies have been performed to predict the fatigue life of solder joints in a single value, fatigue life depend on the manufacturing tolerances of SSDs. Probabilistic distributions were also introduced to consider manufacturing tolerances. Evans et al. (2000) obtained a distribution of the fatigue life using a simple Engelmaier model and Monte Carlo simulation. However, their study not only used a specific equation, but also considered the diagonal distance from one corner joint to another corner joint in the failing row of solder balls, the height of the solder joint, and the CTE of the ball grid array. Yang et al. (2004) calculated stress values according to the height of the solder ball, pitch between solder balls, and PCB thickness by considering the viscoplastic behavior of FCOB solder joints and the viscoelastic behavior of the underfill. Sensitivity analysis was performed through finite element analysis and response surface methods to calculate the stress distribution in consideration of the manufacturing tolerances of solder joints. However, they did not further investigate the fatigue life of the FCOB. We proposed a method to determine the probabilistic fatigue-life distribution of SSDs under thermal cycling through finite element method and Monte Carlo simulation and the effect of their design and operating variables on failure through sensitivity analysis. In the developed finite element model, we utilized the Anand model to represent the viscoplastic behavior of solder balls and the Prony series to represent the viscoelastic behavior of a polymer material in the underfill. The developed finite element model was verified by comparing the simulated strain of the SSD with the measured strain in a thermal chamber. We determined the fatigue life of the SSD using Morrow s energy-based fatigue model. Finally, we determined the fatigue life distribution according to the manufacturing tolerances of the design and operating variables of the SSD through Monte Carlo simulation. We also analyzed the effects of the design and operating variables on failure through sensitivity analysis. 2 Method of analysis 2.1 Development and verification of the finite element model We developed a finite element model of a SSD, which can be seen in Fig. 1. NAND, dynamic random access memory (DRAM), and controller packages were mounted to the PCB through solder balls, and underfill was packed in the empty space between the PCB and packages. The developed finite element model consisted of 44,120 elements with an 8-node cubic element. Package, solder ball, and underfill were connected to each other through node sharing. A contact element was used between the PCB and solder ball, underfill with the always bonded option. Figure 2 shows the boundary conditions applied to the developed finite element model. In order to construct the operating environment, fixed boundary conditions were applied to both the front and rear parts. We developed an ANSYS APDL script that automatically created the finite element model when a user entered the design variables, material properties, and thermal cycling profile. The solder balls were assumed to consist of viscoplastic material. Underfill was assumed to consist of viscoelastic material. The PCB and packages were assumed to consist of an elastic material. We utilized the Anand model to represent viscoplastic behavior that has plasto-elasticity of the solids and fluidic viscous properties of the solder balls. The Anand model determined the strain rate by introducing the deformation resistance as follows. h _e p ¼ Ae Q RT sinh n r i1 m ð1þ s n _s ¼ h 0 1 s asign s o 1 _e s s p ð2þ s ¼ ^s _e n p A e Q RT ; ð3þ where _e p, s, Q, R, A, n, m, h 0, ^s, n, a, T, and s represent the inelastic strain rate, deformation resistance, activation energy, universal gas constant, pre-exponential factor, multiplier of stress, strain rate sensitivity of stress, hardening constant, coefficient for deformation resistance saturation value, strain rate sensitivity of saturation value of deformation resistance, strain rate sensitivity of hardening, absolute temperature, and saturation value of deformation resistance, respectively. Table 1 shows the Anand model parameters of the SAC305 (Chang et al. 2006). The viscoelastic behavior of the underfill was represented by a combination of solid elasticity and fluid viscosity. The underfill was composed of a polymer material and was implemented using a Maxwell element in ANSYS, the master curve of the stress relaxation behavior with respect

3 Fig. 1 SSD and finite element model Fig. 2 Finite element model and boundary condition of the SSD to time and temperature was represented by the following Prony series. Gðt; TÞ ¼G 1 þ Xn i¼1 G i e t As i ; ð4þ where G 1, G i, n, s i, and A are the fully relaxed shear modulus, shear modulus, the number of the Maxwell element, the relaxation time, and shift factor, respectively. The underfill values of G i and s i were obtained by referring Kim et al. (2011, 2012). The underfill was assumed to be a resin composite (A-type) with the same glass transition temperature in the reference. Additionally, the value of G 1 was provided by manufacturer. The Young s modulus of the underfill changed dramatically as it passed the glass transition temperature. The Prony series represented the stress relaxation behavior over time, the behavior of temperature and total time could be expressed using the shift function. For the polymer material such as underfill, the shift function used the following Williams-Landau-Ferry (WLF) shift function. log 10 ðaþ ¼ C 1ðT T R Þ ; ð5þ C 2 þ T T R where T R, C 1, and C 2 are the reference temperature and empirical constants with value of 300 K, 10.62, and - 180, respectively. The T R was the room temperature where we performed the experiment. Since we already knew the shape of master curve and temperature-shear modulus values, the C1 and C2 values were calculated by shifting the master curve. Figure 3 shows the master curve of the underfill. The master curve shows that the shear modulus Table 1 Anand parameters of a solder joint Parameter Description Value of SAC305 s Initial value of deformation resistance [MPa] 45.9 Q R Activation energy, Universal gas constant [K] 7460 A Pre-exponential factor [1/s] 5.87E6 n Multiplier of stress [ ] 2.00 m Strain rate sensitivity of stress [ ] h 0 Hardening constant [MPa] 9350 ^s Coefficient for deformation resistance saturation [MPa] 58.3 n Strain rate sensitivity of saturation value [ ] a Strain rate sensitivity of hardening [ ] 1.5

4 Fig. 4 Strain and temperature measurement setup of the SSD Fig. 3 Glass transition temperature and master curve of the underfill was about 3 GPa at room temperature and changed rapidly at temperatures above the glass transition temperature prior to converging to 30 MPa. Table 2 shows the material properties of all components including the PCB and packages. In this paper, we assumed that these materials exhibited elastic behavior under yield stress. In order to verify the developed finite element model, the strain and temperature of the SSD were simultaneously measured in a thermal chamber. Figure 4 shows the experimental equipment used to measure strain and temperature. As shown in Fig. 4, there was no device to constrain the SSD in the experimental environment. To establish the same environment as the experiment, it was assumed to be in free state without boundary conditions. A strain gauge and a thermocouple were attached to the PCB and the distance off-set between the thermocouple and the strain gauge was minimized as small as possible asshowninfig.4. Then, the SSD was placed into the thermal chamber. A glass cover was placed on top of the experimental equipment to create a thermal chamber environment. Figure 5 shows the measured temperature and strain results, showing that the strain tended to increase with an increase in temperature. The maximum strain was measured to be at 358 K. In order to verify the experimental results, strain on the PCB was simulated with the developed finite element model under the same conditions as the experiment. The strain of the SSD was calculated to be The difference between the experiment and simulation was less than 5.5%, showing that the developed finite element model well represented SSD deformation due to changes in temperature. 2.2 Thermal cycling analysis Thermal cycling condition was applied to the developed SSD finite element model according to the JEDEC standard. Figure 6 shows the thermal cycling profile of the JEDEC standard N condition (1989). The minimum and maximum temperatures were 233 and 358 K for 10 min, respectively. The ramp rate was 12.5 K/min and 5 cycles were iterated. The reference temperature was assumed to be 300 K, which is room temperature. The thermal cycling profile was applied to all nodes of the finite element model. The number of cycles to failure was calculated using the thermal cycling analysis results and the Morrow energybased fatigue law (1964). Nf m DW ¼ K; ð6þ where N f, m, DW, and K are the number of cycles to failure, fatigue exponent, strain energy density, and material ductility coefficients. m and K were obtained from Mustafa s research results (2014), where m and K were and 4.504, respectively. The number of cycles to failure was calculated using DW. Table 2 Material properties of a SSD Component Density [kg/m 3 ] Young s modulus [MPa] Poisson s ratio CTE [ppm/k] PCB , NAND , DRAM Controller ,000 13,000 Solder ball , Underfill

5 Fig. 5 Experimental results using thermal chamber a Measured temperature of the SSD. b Measured strain of the SSD distribution. We assumed a 1% standard deviation of the nominal value for all variables. Second, a finite element model was created using a randomly generated number and the developed ANSYS APDL script. Third, thermal cycling analysis was performed, and the applied thermal cycling profile was generated as a random number. Finally, the process was repeated 2000 times for a total of 2000 results. Through a total of 2000 iterations, the SSD probability models were analyzed in consideration of manufacturing tolerances. 3 Results and discussion Fig. 6 Thermal cycling profile according to the JEDEC standard 2.3 Monte Carlo simulation process The design variables of the SSDs produced by actual manufacturing processes are not determined with unique values but are somewhat distributed depending on the manufacturing process. In such a probabilistic model, the Monte Carlo method is one of the methods used to obtain numerical results. The Monte Carlo simulation is a computational algorithm that relies on repeated random sampling to obtain numerical results through iterative processes. Monte Carlo simulation is widely used in a variety of research fields, including those related to manufacturing tolerances such as Lee s research (2015). We developed an analysis method that combined Monte Carlo simulation with the finite element method. The fatigue life under thermal cycling condition was analyzed to consider the effects of manufacturing tolerances, variations in material properties, and temperature profiles. Figure 7 illustrates the developed analysis process. First, we generated random numbers for all variables using the Latin hypercube sampling method. Design variables and material properties of the PCB, solder balls, and packages, as well as temperature variations were assumed to be normal 3.1 Thermal cycling results Thermal cycling analysis was performed as explained in Sect. 3.1 through the ANSYS. Figure 8 shows the distribution of plastic strain energy density of the solder balls calculated from the finite element analysis, showing that the maximum plastic strain energy density occurred in the outermost solder ball at the lower right end of the DRAM. Therefore, solder ball at the lower right end of the DRAM was determined to be the most vulnerable part of the SSD under thermal cycling condition. The number of cycles to failure of SSD was calculated based on failure of the solder ball at the most vulnerable part of the SSD. Figure 9 shows the plastic strain and elastic strain of the lower right end of the DRAM solder ball. Plastic strain was more dominant than elastic strain. Additionally, the plastic strain was shown to gradually increase with time. Through this result, creep effect of solder ball could be confirmed. The hysteresis loop of the lower right end of the DRAM solder ball is shown in Fig. 10. The loop moved to the right due to creep effects as the cycle progressed. The loop was also confirmed to gradually converge as it repeated. In the last cycle, which converged sufficiently, the number of cycles to failure of the SSD was calculated by applying the plastic strain energy density in Eq. (6), which was calculated to be 453 cycles. Through the thermal cycling analysis, the

6 Fig. 7 Monte Carlo simulation process with the finite element model Fig. 8 Plastic strain energy density distribution of solder balls vulnerable part of the SSD was determined in the thermal cycling condition, and the fatigue life of SSD was derived under the thermal cycling condition. 3.2 Monte Carlo simulation results Monte Carlo simulation was performed as explained in Sect. 3.2 through the ANSYS. Figure 11 shows the distribution of number of cycles to failure including manufacturing tolerances as explained in Sect The average and standard deviation of the cycles to failure for SSD were and cycles, respectively. The fatigue life prediction process of the SSD with the manufacturing tolerance was constructed. Sensitivity analysis was performed with the Pearson correlation coefficient to analyze the correlations between variables and number of cycles to failure. Figure 12 shows the effects of the variables on the number of cycles to failure excluding factors less than 1%. The effect of minimum temperature of the thermal cycling profile had the greatest influence on number of cycles to failure of the SSD. The minimum temperature of the thermal cycling profile had a more dominant effect than the maximum temperature because the difference between the reference temperature and minimum temperature (67 K) was bigger than the difference between the reference temperature and the maximum temperature (58 K). The

7 Fig. 9 Plastic and elastic exz strain under thermal cycling of vulnerable solder balls Fig. 10 Hysteresis loop under thermal cycling of solder balls Fig. 12 Pie chart of the design variables diameter of the DRAM solder ball was the second major factor because the maximum strain energy density occurred in the DRAM solder ball. The influence of solder ball diameter was determined to be more dominant than that of height. Also, the x-axis pitch of the DRAM solder balls was the third major factor. The width, thickness, and height of the PCB had a significant effect on the number of cycles to failure because PCB was the largest volume of SSD. Moreover, dimensions of PCB were an easy factor to control during the manufacturing process, which is expected to increase the fatigue life of the SSD. Overall, the material properties had a weaker effect on the cycles to failure than did the geometry. However, the CTE of the PCB and underfill, which occupied a large volume of the SSD, had some influence on the cycles to failure. Although the minimum temperature of the thermal cycling profile was identified to have the greatest influence on the number of cycles to failure, it was not a manufacturing tolerance determined at the production stage. In terms of manufacturing tolerances that could be considered during the design and production stage, the diameter of DRAM solder balls was the most important factor that affected the number of cycles to failure. In addition, the second most important factor was the x-axis pitch of the DRAM solder balls. Therefore, the robust design of the SSD could be developed by increasing the diameter of the DRAM solder balls and decreasing the X-axis pitch of the DRAM solder balls. 4 Conclusions Fig. 11 Histogram of cycles to failure We proposed a method to estimate the distribution of fatigue life of SSDs due to thermal cycling using the finite element method and Monte Carlo simulation. In the developed finite element model, we utilized the Anand

8 model to represent the viscoplastic behavior of solder balls and the Prony series to represent the viscoelastic behavior of the polymer material in underfill. In order to verify the developed finite element model, we compared the simulated strain with that measured in a thermal chamber. Using the developed finite element model, thermal cycling analysis was performed to determine fatigue life of the SSD using the simulated results and Morrow s energy-based fatigue model. We confirmed that the outermost solder ball at the corner of the DRAM of the SSD was the most vulnerable component under thermal cycling and determined the fatigue life of the SSD under thermal cycling. We showed that the major operating and design variables affecting the fatigue life of SSDs were the temperature profile and diameter of solder balls in the SSD. We expect that a more accurate fatigue life distribution could be obtained using the actual probability distribution of major SSD variables. The proposed method could contribute to the development of SSDs robust to thermal cycling condition. Acknowledgements This research was supported by a Semiconductor Industry Collaborative Project between Hanyang University and Samsung Electronics Co. Ltd. References Association EI, others (1989) JEDEC STANDARD: Temperature Cycling JESD22-A104-D Chang J, Wang L, Dirk J et al (2006) Finite element modeling predicts the effects of voids on thermal shock reliability and thermal resistance of power device. Weld J 85:63 70 Chen L, Zhang Q, Wang G et al (2001) The effects of underfill and its material models on thermomechanical behaviors of a flip chip package. IEEE Trans Adv Packag 24:17 24 Evans JW, Evans JY, Ghaffarian R et al (2000) Simulation of fatigue distributions for ball grid arrays by the Monte Carlo method. Microelectron Reliab 40: Kim M-K, Joo J-W (2012) Thermo-mechanical Behavior of WB- PBGA Packages Considering Viscoelastic Material Properties. J Microelectron Packag Soc 19:17 28 Kim YK, Gang JH, Lee B-Y (2011) Material property effects on solder failure analyses. Microelectron Reliab 51: Lee JH, Lee MH, Kim KB et al (2015) Monte Carlo simulation of the manufacturing tolerance in FDBs to identify the sensitive design variables affecting the performance of a disk-spindle system. Microsyst Technol 21: Morrow JD (1964) ASTM STP 378 Mustafa M, Roberts JC, Suhling JC, Lall P (2014) The effects of aging on the fatigue life of lead free solders. In: 2014 IEEE 64th Electronic Components and Technology Conference (ECTC). pp Pang JH, Xiong BS, Low TH (2004) Creep and fatigue characterization of lead free 95.5 Sn-3.8 Ag-0.7 Cu solder. In: Electronic Components and Technology Conference, Proceedings. 54th. IEEE, pp Schubert A, Dudek R, Auerswald E, et al (2003) Fatigue life models for SnAgCu and SnPb solder joints evaluated by experiments and simulation. In: Electronic Components and Technology Conference, Proceedings. 53rd. pp Steinberg Dave S (1988) Vibration analysis for electronic equipment. Wiley, United States of America Yang D-G, Liang JS, Li Q-Y et al (2004) Parametric study on flip chip package with lead-free solder joints by using the probabilistic designing approach. Microelectron Reliab 44: Zhang L, Sun L, Guo Y, He C (2014) Reliability of lead-free solder joints in CSP device under thermal cycling. J Mater Sci: Mater Electron 25: