Computational analysis of the interfacial effect on electromigration in flip chip solder joints

Microelectronic Engineering 86 (2009) 2132–2137

Contents lists available at ScienceDirect

Microelectronic Engineering journal homepage: www.elsevier.com/locate/mee

Computational analysis of the interfacial effect on electromigration in flip chip solder joints Dongchoul Kim * Department of Mechanical Engineering, Sogang University, 1 Shinsoo-dong, Mapo-gu, Seoul 121-742, Republic of Korea

a r t i c l e

i n f o

Article history: Received 26 September 2008 Received in revised form 27 February 2009 Accepted 4 March 2009 Available online 13 March 2009 Keywords: Electromigration Solder joint The interface energy Diffuse interface model

a b s t r a c t Recent experiments have shown failure in flip chip micro solder joints induced by electromigration. This paper proposes a three-dimensional computational model to simulate the evolution of micro and sub-micro scale solder joints due to electromigration induced diffusion. The evolving morphology of the solder joint related with multiple mechanisms causes a computational challenge. This is addressed by employing a diffuse interface model with multiple concentrations. To efficiently resolve complications, a semi-implicit Fourier spectral scheme and a biconjugate-gradient method are adopted. Results have demonstrated rich dynamics and solder breakage at the interface on the cathode side. Furthermore, the effect of the designed interface region on the reliability of solder joints has been investigated with the developed model. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction Flip chip is one of the rapidly growing microelectronics packaging technologies due to its advantages such as high input/output counts, better electrical performance, high throughput and low profile. In flip chip, solder joints are the interconnection between a die and the package carrier. Along with the trend of miniaturization, the diameter of a solder joint has reduced from 100 to 50 lm and smaller. The average current density easily reaches 104 A/cm2 and higher [1]. Under such high current density, electromigration in solder joints becomes a major concern for the reliability of flip chip. The transport of momentum from electrons to atoms causes a flux of metal atoms diffuse in the direction of the electron flow. At the sites of flux divergence, atoms will either deplete or accumulate, leading to the development of voids or hillocks which may cause failure of solder joints. Experimental techniques have been developed to quantitatively investigate the electromigration process in a solder joint. Characteristics such as activation energy, critical length, threshold current density, effective charge numbers and electromigration rate have been measured [2,3]. From experiments performed during the operation of a real flip chip, it has been also observed that void formation starts near the current crowding region of the solder contact, and spreads quickly across the contact area after nucleation [4]. Voids developed at the contact area reduce the contact crosssection. The associated increase of local current density further promotes electromigration and eventually leads to failure such as * Tel.: +82 2 705 8643. E-mail address: [email protected] 0167-9317/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.mee.2009.03.044

breakage. Prediction of the solder reliability demands advanced computational tools which can describe the entire evolution process of the solder joint. Several studies have presented the twodimensional simulations with a void assumed to have a pancake shape [5,6]. In this paper, we present a three-dimensional model for the evolution of flip chip solder joints. This model incorporates the interface energy in addition to the electromigration process, which would play a significant role in the evolution of voids at the contact cross-section. The recent miniaturization of solder joints leads to the considerable effect of the interface energy on the evolution of the microstructures. Especially, for novel microscale systems, such as flip chip, whose reliability is substantially influenced by the contact area between two different materials, the interface energy becomes inevitable for modeling the system. However, there has been no attempt to incorporate the interface energy in computational modeling of flip chip solder joints. Furthermore, previous two-dimensional modeling studies assumed a preexisting void which evolves following the guided direction and shape within a steady solder bump. Our three-dimensional dynamic model allows us to investigate the dynamic deformation of the contact region between the evolving solder bump and flip chip. Moreover, with the model that incorporates the interface energy, we investigate the effect of the designed geometry in the contact region on the reliability of solder joints and propose effective ways to improve the reliability. An evolving solder joint and multiple mechanisms in the system cause a computational challenge. This is addressed by employing a diffuse interface model with multiple concentrations. Three concentrations generate three different regions, flip chip, solder, and air. Each region with different material properties

D. Kim / Microelectronic Engineering 86 (2009) 2132–2137

evolves simultaneously. To efficiently resolve these complications, a robust numerical scheme is adopted. 2. Model Fig. 1 shows a schematic drawing of a flip chip solder joint sandwiched between two copper plates. The solder may change its morphology by diffusion via electron wind. The arrows marked by e indicate the direction of electron flow. In order to efficiently present the morphological change of this system that incorporates multiple kinetics and energetics, a diffuse interface model is employed. We have demonstrated its efficiency in simulating dynamic evolutions of various nano/microstructures involving multiple mechanisms [7–10]. Here, three concentration fields, C1, C2, and C3, are introduced to represent the lead-free solder, the copper plates, and air, respectively. Define a concentration C1 by the volume fraction of the lead-free solder and a concentration C2 by the volume fraction of the copper plates. Air in the domain is represented by C3 which is 1  C1  C2. The concentrations are regarded as spatially continuous and time-dependent functions, C1(x1, x2, x3, t) and C2(x1, x2, x3, t). For simplicity, we assume isotropic surface energy and diffusivity. Following Cahn-Hilliard, the free energy of the system with multiple concentrations can be defined as

G¼

Z n o g c ðC 1 ; C 2 Þ þ h11 ðrC 1 Þ2 þ h22 ðrC 2 Þ2 þ h12 ðrC 1 rC 2 Þ dV: ð1Þ

The first term represents the chemical energy in the system. Other terms account for the interface energy between three components, where h11, h22, and h12 are material constants. Atoms diffuse in the solder bump from high chemical potential regions to low chemical potential regions, which causes the solder bump to change its shape. The corresponding driving forces for diffusion in the solder bump and the plates are attributed to the chemical potential by ¼ rl01 and Fchem ¼ rl02 , respectively. The chemical potenFchem 1 2 tial relates to the free energy by l01 ¼ dG=dC 1 and l02 ¼ dG=dC 2 . In an electric field, one driving force is attributed to the electron-wind force. Thus, the total driving force can be expressed by elec þ Felec and F2 ¼ Fchem þ Felec and Felec are the F1 ¼ Fchem 1 1 2 2 , where F1 2 driving force related with the electric field in the solder bump and the plates, respectively. The total driving force leads to the atomic flux of J1 ¼ M1 F1 and J2 ¼ M2 F2 . By combining with the mass conservation relation, @C 1 =@t þ r  J1 ¼ 0 and @C 2 =@t þ r  J2 ¼ 0, Cahn-Hilliard equations are obtained as followings:

     @C 1 dg c  2h11 r2 C 1  h12 r2 C 2 þ Felec ¼ r M1 r ; 1 @t dC 1      @C 2 dg c :  2h22 r2 C 2  h12 r2 C 1 þ Felec ¼ r M2 r 2 @t dC 2

ð2Þ ð3Þ

The electron-wind force refers to the effect of the momentum exchange between the moving electrons and the ionic atoms and is described as

e-

Cu (C1=0, C2=1)

e-

Lead-Free Solder

(C1=1, C2=0)

x3

Cu (C1=0, C2=1)

e-

x1 Fig. 1. A schematic drawing of a flip chip solder joint. The two concentration fields, C1 and C2, represent a lead-free solder and plates, respectively.

Felec ¼ f1e r/; 1

2133

Felec ¼ f2e r/; 2

ð4Þ

where f1e ¼ jejZ 1 =X1 and f2e ¼ jejZ 2 =X2 . X1 and X2 are the atomic volume of tin and copper, respectively. Z 1 and Z 2 are the phenomenological effective valence of the atom of tin and copper, respectively. e is the charge of an electron. The negative sign in equations means that the force is in the direction of the electron flow. Due to the conservation of electric charges and the Ohm’s law, the electric potential obeys the Laplace equation,

r  fnðC 1 ; C 2 Þr/g ¼ 0;

ð5Þ

where n is the conductivity of the media, which is linearly interpolated by those of the solder bump and the copper plates, namely, nðCÞ ¼ n1 C 1 þ n2 C 2 . This partial differential equation, together with boundary conditions, determines the electric potential. The chemical energy with three components have been derived from the studies of two-phase solidification [11],

g c ¼ f0

hn o C 21 ð1  C 1 Þ2 þ C 22 ð1  C 2 Þ2 þ C 23 ð1  C 3 Þ2

  o C 21 n 15ð1  C 1 Þ 1 þ C 1  ðC 3  C 2 Þ2 þ C 1 9C 21  5 4  o  C 22 n 15ð1  C 2 Þ 1 þ C 2  ðC 3  C 1 Þ2 þ C 2 9C 22  5 þ a2 4  o  C 23 n þ a3 15ð1  C 3 Þ 1 þ C 3  ðC 1  C 2 Þ2 þ C 3 9C 23  5 : 4 ð6Þ

þ a1

By taking a1 = a2 = a3 = 1 for simplicity and C3 = 1  C1  C2, the chemical potentials are expressed by

n  o @g c ¼ f0 2ð2C 1 þ C 2  1Þ 2C 21  2C 1 þ 2C 1 C 2  C 2 þ 2C 22 ; @C 1 n o @g c ¼ f0 2ðC 1 þ 2C 2  1Þð2C 21  C 1 þ 2C 1 C 2  2C 2 þ 2C 22 Þ : @C 2

ð7Þ ð8Þ

By normalization with the characteristic length, Lc, time, tc,

   @C 1 1 2 2 ¼ r M1 r l01  Ch11 r2 C 1  Ch12 r2 C 2  E/ ; 2 @t    @C 2 1 2 2 2 RM ¼ r M2 r l02  Ch22 r C 2  Ch12 r2 C 1  E/ ; 2 @t

ð9Þ ð10Þ

 where the chemical potentials are l01 ¼ ð2C 1 þ C 2  1Þ 2C 21  2C 1 þ  2C 1 C 2  C 2 þ 2C 22 and l02 ¼ ðC 1 þ 2C 2  1Þ 2C 21  C 1 þ 2C 1 C 2  2C 2 þ 2C 22 . The characteristic variables are tc ¼ L2c =2M1c f0 , Ch11 ¼ h11 =L2c f0 , Ch12 ¼ h12 =L2c f0 , and Ch22 ¼ h22 =L2c f0 . RM is the ratio of the characteristic mobilities, RM ¼ M1c =M2c and E ¼ /c =2f 0 . Ch11 , Ch22 , and Ch12 are the Cahn numbers which represent the relative significance of the interface energy between materials. 3. Numerical implementation The numerical approach needs to have a high spatial resolution to resolve the high order derivatives in the diffusion equation as well as the large gradients at the interface region. The approach should also be efficient and stable for the time integration, which is especially important for three-dimensional simulations. We propose an efficient semi-implicit Fourier spectral method for both high spatial resolution and fast computation. The central idea of the semi-implicit method is to treat the linear term implicitly and the nonlinear term explicitly to allow for larger time steps without losing numerical stability [12,13]. In contrast, a fully implicit treatment yields expensive schemes while explicit discretization quickly leads to numerical instability or needs impractical time-step constraint,

2134

D. Kim / Microelectronic Engineering 86 (2009) 2132–2137

@C 1 1 2 2 ¼ 2s1 r2 C 1 þ s1 r2 C 2  Ch11 r4 C 1  Ch12 r4 C 2 2 @t    1 2 2 þ r  M1 r l01  Ch11 r2 C 1  Ch12 r2 C 2  E/ 2  1 2 2 2 2 4 2s1 r C 1  s1 r C 2 þ Ch11 r C 1 þ Ch12 r4 C 2 ; 2 RM

ð11Þ

ð12Þ Note that the stability is achieved by the extrapolated Gear (SBDF) scheme for time integration. Among the second order multi-step methods, SBDF has the strongest high modal decay [14]. This provides the required damping for the very high frequencies in the diffusion equation without a severe time-step constraint. Applying the scheme to the normalized diffusion equation, we obtain the following discretizated form:

1 2 2 ¼ Dt 2s1 r2 C nþ1 þ s1 r2 C nþ1  Ch11 r4 C 1nþ1  Ch12 r4 C 2nþ1 1 2 2

ð16Þ

By applying the Fourier transform to Eqs. (13) and (14), we effectively perform calculations with

@C 2 1 2 2 ¼ 2s2 r2 C 2 þ s2 r2 C 1  Ch22 r4 C 2  Ch12 r4 C 1 2 @t    1 2 2 2 2 0 þ r  M 2 r l2  Ch22 r C 2  Ch12 r C 1  E/ 2  1 2 2 2 2 4 2s2 r C 2  s2 r C 1 þ Ch22 r C 2 þ Ch12 r4 C 1 ; 2

3 nþ1 1  2C n þ C n1 C 2 1  1 2 1

n  2 2 2 n 2 n Pn1 ¼ Dt r  M1 r l0n 1  Ch11 r C 1  Ch12 r C 2 =2  E/ o 2 2 2s1 r2 C n1  s1 r2 C n2 þ Ch11 r4 C n1 þ Ch12 r2 C n2 =2 ;



þ 2P n1  P1n1 ; ð13Þ 3 nþ1 1  2C n2 þ C n1 C 2 2 2 2   Dt 1 2 4 nþ1 2 2 nþ1 4 nþ1 ¼ 2s2 r C 2 þ s2 r2 C nþ1  Ch r C  r C Ch 22 1 2 1 RM 2 12

   ^ n  1 C^ n1 þ 2P^ n  P ^ nþ1 ¼ 1 L4 2C ^ n1 C 1 1 1 1 Det 2 1   ^n  1 C ^ n1 þ 2P^n  P ^ n1 L2 2C ; 2 2 2 2 2

ð17Þ

   ^ n1 þ 2P ^ nþ1 ¼ 1 L3 2C^ n  1 C ^n  P ^ n1 C 2 1 1 1 Det 2 1   ^n  1 C ^ n1 þ 2P^n  P ^ n1 þL1 2C ; 2 2 2 2 2

ð18Þ

2

2

4

2

2

4

where L1 ¼ 3=2 þ 2s1 Dtk þ Ch11 Dtk , L2 ¼ s1 Dtk þ Ch12 Dtk =2, L3 ¼ s2 Dtk2 =RM þ Ch212 Dtk4 =ð2RM Þ, L4 ¼ 3=2 þ 2s2 Dtk2 =RM þ Ch222 Dtk4 =RM , Det ¼ L1 L4  L2 L3 . The caret ‘^’ stands for the Fourier transform, 2 2 2 2 and k ¼ k1 þ k2 þ k3 . The electric field is determined by r fnðC 1 ; C 2 Þr/g ¼ 0, which can also be written as nr2 / þ rn r/ ¼ 0. We apply the second order discretization and compute the matrix with the preconditioned biconjugate-gradient method [15]. The Jacobi preconditioner is adopted. The approach allows efficient computation of large 3D domains. In the following, we outline the procedure to compute C nþ1 and 1 from C n1 and C n2 . First, we compute the electric potential field C nþ1 2 /n that corresponds to the concentration distributions C n1 and C n2 . ^n, Then, using the solution of /n , compute ðlc1 Þn and ðlc2 Þn . Using C 1 ^ n , and / ^ n , we compute the convective Cahn-Hilliard equation in C 2 ^ nþ1 . The new concentrations ^ nþ1 and C Fourier space to obtain C 1 2 nþ1 ^ nþ1 and C ^ nþ1 by the inverse and C are obtained from C C nþ1 1 2 1 2 Fourier transform. The procedure is repeated repeats until a prescribed time.

þ 2Pn2  P2n1 : ð14Þ

4. Results and discussions

where

n  2 2 2 n 2 n P n1 ¼ Dt r  M 1 r l0n 1  Ch11 r C 1  Ch12 r C 2 =2  E/ o 2 2 2s1 r2 C n1  s1 r2 C n2 þ Ch11 r4 C n1 þ Ch12 r2 C n2 =2 ;

4.1. The interface energy of the solder joint

ð15Þ

The proposed three-dimensional model and numerical approach allow a comprehensive computational study of the

Fig. 2. The simulation results with different combinations of Cahn numbers: (a) Ch11 = 1.0, Ch22 = 1.0, Ch12 = 1.0, and (b) Ch11 = 1.0, Ch22 = 1.0, Ch12 = 0.1.

2135

D. Kim / Microelectronic Engineering 86 (2009) 2132–2137

dynamic evolution of solder joints. Representative results are presented in this section. The calculation domain size is 70  70  50. The diameter of the solder bump is 50 that corresponds a physical diameter of 50 lm since the characteristic length is taken to be 1 lm. The plates at the top and the bottom of the solder bump have a thickness of 5 lm. Visualized three-dimensional graphs present the solder surface and the plates by tracking where C1 and C2 have 0.5. Fig. 2 shows an evolution sequence from t = 0 to t = 200 when the electric field is not applied with different combination of the surface energies. Since the diffusivity of Cu at 150 °C is around 1012 cm2/s and that of Sn is 1.3  1010 cm2/s, the plates diffuse slowly and almost remain flat while the solder bump undergoes the morphological evolution during simulation. From the diffusivity of Cu at 150 °C, the characteristic time becomes around 700 s. Hence the normalized time t = 200 corresponds to around 40 h. Fig. 2a shows an evolution when the interface energies at the contact region between the solder bump and the plates have the same value of that of the solder bump and the plates; Ch11 = 1.0, Ch22 = 1.0, and Ch12 = 1.0. If a separated solder bump is considered, the solder bump should evolve into a spherical shape to minimize its surface energy. However, there is the interface energy at the contact area between the solder bump and the plates, which plays an important role during the evolution process. In Fig. 2a, the whole system can be regarded as a single structure because the interface energy of the contact region is the same with that of the solder bump and the plates. Thus, the solder bump evolves into a cylindrical shape and the contact area increases to minimize the total interface energy of the system. As the interface energy of the contact area becomes smaller than that of the solder and the plates as shown in Fig. 2b, the contact area does not increase and the solder bump tends to remain a spherical shape to reduce its surface energy which is larger than that of the contact area. The reliability of solder joints correlates with the contact area which is considerably dependent on the relation between the interface energies of each material. Hence, chemical or geometric manipulations to increase the interface energy between the solder and the plates will improve the reliability of the solder joint. As shown in Fig. 2c, when the Cahn numbers are assigned as Ch11 = 1.0, Ch22 = 1.0, and Ch12 = 0.1, which the surface energy between the solder bump and the plates is 10 times smaller than that of the solder bump and the plates, the simulation without an electric field shows a reasonably

a

1400 1200 1000

Ac

E = 0.5

800

E = 1.0

600

E = 1.5

400 200 0 0

E = 2.0 10

20

30

40

50

60

70

t

b

100 80 60

tf 40 20 0 0

0.5

1

1.5

2

2.5

A Fig. 4. (a) The contact region between the solder bump and the top plate (Ac) is plotted as a function of time with different strengths of the applied electric fields (E). (b) The failure time (tf) is exponentially decreased as the strength of the applied electric field increases. The assigned Cahn numbers are Ch11 = 1.0, Ch22 = 1.0, and Ch12 = 0.1.

maintained sphere shape of the solder bump similar to empirical observations. Thus, the Cahn numbers of Ch11 = 1.0, Ch22 = 1.0, and Ch12 = 0.1 are assigned in the following simulations. 4.2. Electromigration in the solder joint Fig. 3 presents the evolution of a solder joint under an electric field. Electron wind flows from the left of the top plate to the right

Fig. 3. (a) The simulated evolution of the solder joint is presented. (b) The solder bump starts to break apart from the top plate at the left side of the solder bump. The assigned Cahn numbers are Ch11 = 1.0, Ch22 = 1.0, and Ch12 = 0.1.

2136

a

D. Kim / Microelectronic Engineering 86 (2009) 2132–2137

1500

Ch12 = 0.1 Ch12 = 0.5 Ch12 = 0.7 Ch12 = 0.9

1000

Ac 500 0

0

100

200

300

400

500

t

4.3. The effect of the designed contact region

b 500 400

tf

The evolution sequence shows that the electron wind causes solder material to diffuse from top to bottom, i.e. from cathode to anode. In Fig. 4, we calculate the failure time by the contact area between the solder bump and the top plate applying various strengths of the electric field. Fig. 4a shows the decreasing contact area as time goes on and the stronger electric field induces the quickly decreasing contact area, which leads to the accelerated failure time. As shown in Fig. 4b, the failure time is almost exponentially decreased as the strength of the applied electric field increases.

300 200 100 0 0

0.2

0.4

0.6

0.8

1

Ch12 Fig. 5. The enhanced interface energy at the contact region induces the improved reliability of the solder joint. (a) The contact area (Ac) is plotted as a function of time with the different Cahn numbers (Ch12). (b) The failure time (tf) is exponentially increased as the interface energy of the contact region increases.

of the bottom plate as shown in Fig. 1. The solder bump is subjected to a normalized electric potential of 1 that corresponds to the current density of 104 A/cm2, which is typical in experiments. The current crowding occurs at the contact region between the top plate and the solder bump. As shown by experimental study, the formation of a void is observed at this contact area. In our simulations, the solder bump starts to break apart from the top plate at the left side of the solder bump as shown in Fig. 3b and eventually the solder bump and the top plate break apart as shown in Fig. 3a. This simulation result is consistent with experimental observations under which a void tends to initiate where current crowding occurs. When a void forms, it blocks the original electrical path and changes the current crowding region to travel around the void. The growth of the void reduces the effective contact area at the interface region and increases the current density in the remaining contact area. Then, this void will make the current density higher and eventually causes the circuit failure. In simulations, the contact area between the solder bump and the top plate decreases.

Here, we have investigated the effect of the chemically and geometrically designed contact region on the reliability of the solder joint. The reliability of the solder joint considerably depends on the condition of the contact region where the failure happens. Thus, the chemical and geometrical manipulation to increase the interface energy will improve the reliability of the solder joint. Fig. 5 shows the simulation results with the various interface energies of the contact region, which can be employed by the chemical manipulation. As shown in Fig. 5, the enhanced interface energy by the chemical manipulation improves the reliability of the solder joint by increasing the failure time. To investigate the effect of the geometrically designed contact region, we have performed simulations with patterned plates. Fig. 6 shows the designed surfaces and the corresponding initial solders. The top plate is designed with stripes or bumps. Patterns have a width of 5 lm and a depth of 8 lm. Stripes are patterned in the x1 and x2 directions to investigate the effect of the patterned direction. The electrons flow from the left of the top plate to the right of the bottom plate along the x1 direction as shown in Fig. 1. Thus, the stripe pattern in Fig. 6a has the direction normal to the direction of the electron flow. Our three-dimensional model makes it possible to study the effect of various patterns. Fig. 7 shows the simulation results with the designed plates. In Fig. 7a, the area of the contact region between the solder bump and the top plate (Ac) is plotted as a function of time until the initial contact area decreases to a significantly small contact are, 100, which can be regarded as failure. The patterns have the same width and depth, which are 5 and 8, respectively. The contact area with the flat top surface as shown in Fig. 4 is 1153 and this area of the contact region is increased by the designed patterns in Fig. 6a and b, which becomes 4728. As shown in Fig. 6c, the contact area increases a little more with the bump pattern, which is 4947, than

Fig. 6. The designed surfaces of the top plates. (a) The top plate is patterned with stripes having a width of 5 and a depth of 8 in the x1 direction and (b) in the x2 direction. (c) The top plate is patterned with bumps having a width of 5 and a depth of 8.

D. Kim / Microelectronic Engineering 86 (2009) 2132–2137

a

Flat Stripes (x1) Stripes (x2) Bumps

4000

Ac

transport driven by the electron flow. This simulation result suggests that the evolution of the solder joint can not be determined only by energetics but also it is limited by the kinetic process. From the same reason, the reliability of the solder joint with the bump pattern is not better than that with the stripe pattern in the x1 direction, even it has the larger contact area as shown in Fig. 7b. The available kinetic route for mass transport at the contact area significantly influences the reliability of solder joints. These simulations suggest that the reliability of solder joints can be significantly improved by the designed plates.

6000 5000

3000 2000 1000 0 0

b

250

100

150

200

250

t

5. Conclusions

3000

This paper presents a three-dimensional model to simulate the evolution of solder joints under electric current. Previous twodimensional modeling studies assumed preexisting voids which evolve guided evolving directions and shapes within steady solders. Our three-dimensional dynamic model allows us to investigate the dynamic deformation of the solder joint. The solder bump starts to break apart from the current crowding region and the failure time is exponentially decreased as the strength of the applied electric field increases. The simulations present that the chemically enhanced interface energy improves the reliability of the solder joint. Moreover, the simulations with the geometrically designed pattern on the top plate demonstrate that the reliability of solder joints is not limited by energy minimization, but by an interplay between the energetics and kinetics. The increased interface energy and the lengthened kinetic path improve the reliability of solder joints. This study provides an efficient model which presents the dynamic evolution process of the solder joint and can be a solid base for further analysis incorporating additional mechanisms.

Stripes (x1) Stripes (x2) Bumps

200

tf

50

150 100

flat

50 0 0

1000

2000

2137

4000

5000

6000

Ac (initial) Fig. 7. (a) The area of the contact region between the solder bump and the top plate (Ac) is plotted as a function of time with different patterns of the top plate. The patterns have a width of 5 and a depth of 8. (b) The time (tf) when the area of the contact region decreases to 100 is affected by the shape and the size of patterns. The designed patterns have various widths of 5, 7, and 10, which are denoted by ‘N’, ‘s’, and ‘’, respectively.

with the stripe pattern. As shown in Fig. 7a, the solder joint with the designed patterns shows improved reliability than that without a pattern. When t = 36, the contact area without a pattern decreases to 100 while that with the designed pattern becomes 100 at over t = 200. The solder joint with the designed patterns takes approximately 6 times more time than that without a pattern to reduce its contact area to a very small area that can cause the failure. The designed patterns on the top plate increase the contact area between the solder bump and the top plate where the failure happens. The increased contact area increases the interface energy of the contact area, which leads to increased reliability of the solder joints. Fig. 7b shows the time when the contact area of a solder joint becomes 100 with different sizes of patterns. Simulations with the patterns having width of 5, 7, and 10 are performed. The increased number of stripes or bumps by reducing the width of a pattern induces the increased contact area between the solder bump and the top plate. Thus, the interface energy increases and the reliability of the solder joint improves. The simulation results also present the effect of the pattern geometry. Comparing the results with the stripe pattern having a width of 5, the solder with the strip pattern in the x1 direction that is normal to the direction of the electron flow shows better reliability than that with the strip pattern in the x2 direction even though they have the same contact area. The stripe pattern in the x1 direction lengthens the path of the electron flow at the contact area so that it takes more time for mass

Acknowledgement This work was supported by the Sogang University Research Grant of 2007118. References [1] K.N. Tu, J. Appl. Phys. 94 (2003) 5451–5473. [2] Y.T. Yeh, C.K. Chou, Y.C. Hsu, C. Chen, K.N. Tu, Appl. Phys. Lett. 86 (2005) 203504. [3] C. Basaran, H. Ye, D.C. Hopkins, D. Frear, J.K. Lin, J. Elect. Pack. 127 (2005) 157– 163. [4] E.C.C. Yeh, W.J. Choi, K.N. Tu, P. Elenius, H. Balkan, Appl. Phys. Lett. 80 (2002) 580–582. [5] L. Zhang, S. Ou, J. Huang, K.N. Tu, S. Gee, L. Nguyen, Appl. Phys. Lett. 88 (2006) 012106. [6] S.W. Liang, Y.W. Chang, T.L. Shao, C. Chen, K.N. Tu, Appl. Phys. Lett. 89 (2006) 022117. [7] D. Kim, W. Lu, Phys. Rev. B 73 (2006) 035206. [8] D. Kim, W. Lu, J. Mech, Phys. Solids 54 (2006) 2554–2568. [9] W. Lu, D. Kim, Acta. Mater 53 (2005) 3689–3694. [10] W. Lu, D. Kim, Nano Lett. 4 (2004) 313–316. [11] R. Folch, M. Plapp, Phys. Rev. E 72 (2005) 011602. [12] W. Lu, Z. Suo, J. Mech, Phys. Solids 49 (2001) 1937–1950. [13] J. Zhu, L.-Q. Chen, J. Shen, Phys. Rev. E 60 (1999) 3564–3572. [14] U.M. Ascher, S.J. Ruuth, B.T.R. Wetton, SIAM J. Numer. Annal. 30 (1995) 797– 823. [15] G.H. Golub, C.F.V. Loan, Matrix Computation, Johns Hopkins University Press, Baltimore, 1989.