Analysis of whisker growth on a surface of revolution

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Analysis of whisker growth on a surface of revolution Fuqian Yang a,∗ , Yu Shi b a b

Department of Chemical and Materials Engineering, University of Kentucky, Lexington, KY 40506, United States State Key Laboratory of Advanced Processing and Recycling of Nonferrous Metals, Lanzhou University of Technology, Lanzhou 730050, China

a r t i c l e

i n f o

Article history: Received 9 May 2017 Received in revised form 21 June 2017 Accepted 28 June 2017 Available online xxxx Communicated by M. Wu Keywords: Whisker growth Grain boundary diffusion Growth rate Stress relaxation

a b s t r a c t A general mass transport equation for diffusion-controlled whisker growth on a surface of revolution is formulated. Two limiting cases for the whisker growth of a circular cylinder-like whisker are analyzed; one surface is planner, and the other one is spherical. The growth rate is proportional to the concentration difference for the growth of the whisker on both the planar surface and the spherical surface. Using the relation between mechanical work and chemical energy, the growth rate is found to be proportional to the pressure difference. Assuming the whisker growth as the mechanism for stress relaxation of a thin film, the whisker length and the stress relaxation associated with the whisker growth are found to be exponential functions of time. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Tin (Sn) whiskers, grown on the surface of Sn-plated films and Sn-based alloys, have imposed a great challenge to the reliability of micro- and nanoelectronics due to the possibility of short circuit induced by Sn whiskers. The random-like formation of Sn whiskers has complicated the understanding of the fundamental mechanisms controlling the nucleation and growth of Sn whiskers and the efforts to possibly mitigate the formation and growth of Sn whiskers. In general, the nucleation and growth of Sn whiskers can be considered as a local mechanism/process for stress relaxation, which competes with other mechanisms/processes, such as dislocation motion and grain boundary sliding, for stress relaxation. It is the energy gradient (stress gradient or concentration gradient) which determines the growth of Sn-whiskers. Considering the confinement effect of a surface oxide layer on the stress relaxation in a Sn film, Tu [1] proposed that Sn whiskers initiate at sites with the fracture of the surface oxide layer and atomic migration is driven by a long-range stress gradient. Galyon and Palmer [2] pointed out the importance of the mass transport through grain boundary network in regulating the growth of Sn whiskers. Boettinger et al. [3] suggested that localized creep of columnar grain structures to relieve the compressive stress in thin films contributes to the growth of Sn whiskers. Yang [4] studied diffusion-limited growth of whiskers from the framework of lat-

*

Corresponding author. E-mail address: [email protected] (F. Yang).

http://dx.doi.org/10.1016/j.physleta.2017.06.050 0375-9601/© 2017 Elsevier B.V. All rights reserved.

tice diffusion. Based on the concept of grain boundary fluid flow [5,6], Tu and Li [7] derived the growth rate of a whisker as a function of viscosity and the pressure gradient, which reduces to the one controlled by grain boundary diffusion [1] when the Einstein relation is used. Buchovecky et al. [8] suggested that Sn whiskers initiate from soft Sn-grains in a Sn surface coating and considered the contribution of both time-independent plastic deformation and grain boundary diffusion in numerical simulation of the growth of Sn-whiskers. Chason and co-workers [9] discussed the role of intermetallic growth, stress evolution, and plastic deformation in the whisker formation. Considering the mass transport through grain boundary network, Li and co-workers [10–12] analyzed the growth of whiskers and hillocks controlled by interfacial flow. Pei et al. [13] assumed that the change rate of average volume is proportional to the stress above a critical value and analyzed the stress relaxation during the growth of Sn whiskers. Wang et al. [14] developed a whisker pinch-off model to analyze the morphological changes of Sn whiskers and the decrease in the whisker density during thermal cycling. Recently, Yang [15] proposed a nonlinear viscous model for the growth of Sn whiskers and obtained the evolution of the whisker length as a function of time controlled by the stress relaxation during the whisker growth. All of the analyses have been based on the condition that the surface of Sn films is planar. No analysis has been focused on curved surfaces and from the theory of diffusion. In the study of indentation-induced growth of Sn whiskers, Williams et al. [16] sketched a possible path for stress-assisted diffusion of atoms to the root of a whisker during the whisker growth, in which the atomic diffusion likely starts from the surface

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n = ( J x+x r x+x − J x r x ) y ϑ + ( J y + y − J y )lx t = ( J x r ) y ϑ + r  J y ϑx

(2)

The change of the amount of atoms in the volumetric element can be calculated from the concentration, c, as r c x y ϑ . Using the concentration, c, and letting t → 0, one obtains a differential equation from Eq. (2) as

∂ c 1 ∂( J x r ) ∂ J y = + ∂t r ∂s ∂y Fig. 1. Schematic diagram of the growth of a whisker on a surface of revolution and a volumetric element ( J x and J y are the components of atomic flux in the x- and y-direction, respectively).

of a hemisphere instead of a planar surface. Considering the observation of the growth of Sn whiskers from pure Sn-plated relay terminal as shown in NASA website (https://nepp.nasa.gov/whisker/) and possible stress-assisted diffusion of atoms from the surface of a hemisphere for the indentation-induced whisker growth, we analyze the whisker growth on a surface of revolution from the framework of grain boundary/interface diffusion in contrast to the use of a long-range stress gradient as the driving force by Tu [1] and the use of grain boundary fluid flow by Tu and Li [7] and Yang [15]. The site for the growth of a whisker is assumed to be at the intersection between axisymmetric axis and the surface of revolution. A general equation for the diffusion on the surface of revolution is derived. Two limiting cases are analyzed for the whisker growth; one surface is spherical, and the other is planar.

Yang and Li [17] pointed out that the stress-driven diffusion of atoms (vacancies) from high pressure to low pressure may not be complete without knowing the mechanism of diffusion since there still exists the diffusion of atoms (vacancies) driven by the gradient of concentration even under the condition of uniform pressure. The appropriate approach is to analyze the diffusion of atoms (vacancies) driven by the gradient of concentration and assume triple junctions, surfaces, and grain boundaries as the sinks and sources of vacancies. Here, we consider the diffusion in a grain boundary, which is driven by the gradient of concentration, for the analysis of the whisker growth. Fig. 1 shows schematically the growth of a whisker on a surface of revolution with the axis of revolution being z axis and a volumetric element for the diffusion of atoms in the film abutted to an impermeable substrate. The surface of revolution is defined by a function r (x), representing the radius of the median of the film, and an intrinsic curvilinear, orthogonal coordinate system (x, ϑ, y) is also depicted in Fig. 1. Due to axisymmetric characteristic of the problem, the non-zero components of atomic flux are J x and J y in the x- and y-direction, respectively. The change of the resultant amount of atoms in the volumetric element per unit time can be calculated as

n = ( J x+x l x+x − J x l x ) y + ( J y + y − J y )lx t

for the diffusion of atoms in a thin film on a surface of revolution with axisymmetric characteristic. The relationships between the flux components and the concentration of atoms in the curvilinear coordinate system are

J x = −D

(1)

where n is the change of the resultant amount of atoms in the volumetric element in the time increment of t, l x and l x+x are the chord lengths of the element at x and x + x, respectively, l is the average chord length of the element, x is the increment of x (which is equal to the incremental arc length measured along the median of the film from the axis of revolution, i.e. s), and  y is the increment of y. For the volumetric element with both l x and l x+x subtending the same angle, Eq. (1) can be simplified to the first order of approximation as

∂c ∂s

and

J y = −D

∂c ∂y

(4)

with D being the diffusion coefficient of atoms in the film. For the diffusion of atoms in a thin film with impermeable substrate and a surface confined by an ultrathin oxide layer, the boundary conditions for the flux component of J y are





J y  y =0 = J y  y =h = 0

(5)

with h being the film thickness. Integrating Eq. (3) with respect to y from 0 to h yields

∂ ∂t

2. Problem formulation

(3)

h cdy =

1 ∂

h J x rdy

r ∂s

0

(6)

0

For quasi-steady state diffusion, Eq. (6) gives

1 ∂

h

h J x rdy = 0,

r ∂s 0

J x rdy = const.

i.e.

(7)

0

Define the average concentration of atoms, < c >, as

=

1

h cdy

h

(8)

0

Substituting Eqs. (4) and (8) in Eq. (7), one obtains

r

∂< c > = const. ∂s

(9)

which is the differential equation describing the diffusion of atoms in a thin film abutted to a surface of revolution under the condition that the diffusion is quasi-static and axisymmetric about the axisymmetric axis. Using Eq. (9) and boundary conditions, one can calculate the concentration distribution of atoms in the thin film and the flux of atoms into the root of a whisker. From the mass balance, the growth rate of the whisker then can be determined. In the following, only two limiting cases are discussed; one surface is spherical, and the other is planar. 3. Limiting cases Here, we assume that the whisker growth is controlled by grain boundary diffusion driven by the gradient of concentration. There are h = δ (the thickness of grain boundary) and D = D gb (the diffusivity of grain boundary).

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Case I. Planar surface For the surface of a Sn film being planar, one has r = s. Equation (9) becomes

r

∂< c > = const. ∂r

(10)

whose solution is

(11)

Here, a and b are two constants to be determined. The boundary conditions are

< c >|r =r1 = < c 1 > and

< c >|r =r2 = < c 2 >

(12)

where r1 is the radius of the region contributing to the whisker growth associated with the triple junction from the intersection of the grain boundary and other grain boundaries, and r2 is the radius of the whisker associated with the triple junction from the intersection of the whisker and the grain boundary. Substituting the boundary conditions of (12) into Eq. (11) yields

< c2 > − < c1 > r ln ln(r2 /r1 ) r1

(13)

From Eq. (13), the flux of atoms toward the root of the whisker is

J = − D gb

 d< c >   dr 

r =r2

= D gb

< c1 > − < c2 > 1 ln(r2 /r1 ) r2

(14)

According to the mass balance, the growth rate of the whisker, ˙ is L,

L˙ =

2π r2 δ(− J )

π r22

= 2 D gb

kT

c max  ln  c

0

δ ln

kT

r =r1

c max  c

0



r =r2

(15)

dy = p 1 δ

and

< c 1 > − c max c max

dy = p 2 δ

(16)

= p 1  and kT

< c 2 > − c max c max

= p2 

(17)

Substituting Eq. (17) in Eq. (15) yields

L˙ =

22 δ c max D gb p 1 − p 2 r22 ln(r1 /r2 )

sin θ

s = Rθ

and

(19)

∂< c > = const. ∂θ

kT

=

p1 − p2

r22 ln(r1 /r2 )

kT

a

1 + cos θ

2

1 − cos θ

< c > = − ln

+b

(21)

The boundary conditions are

< c >|θ =θ1 = < c 1 > and

< c >|θ =θ2 = < c 2 >

(22)

where θ1 is the polar angle for the edge of the region contributing to the whisker growth, and θ2 is the polar angle for the edge of the whisker root. Substituting the boundary conditions of (22) into Eq. (21) yields

< c > = < c 1 > + (< c 1 > − < c 2 >) f (θ1 , θ2 )   1 + cos θ 1 + cos θ1 × ln − ln 1 − cos θ 1 − cos θ1

(23)

with

  f (θ1 , θ2 ) = ln

(1 + cos θ1 ) (1 − cos θ2 ) (1 − cos θ1 ) (1 + cos θ2 )

−1 (24)

    −1  −1  R θ2 r2 θ2 −1 (1 − cos θ2 ) ≈ 2 ln ln = 2 ln = 2 ln (1 − cos θ1 ) θ1 R θ1 r1   (1 + cos θ2 ) ≈1 and (25) (1 + cos θ1 ) 

Equation (23) becomes

< c > = < c1 > +

(18)

which is the same as the result given by Tu and Li [7]. The growth rate is proportional to the pressure difference.

< c2 > − < c1 > r ln ln(r2 /r1 ) r1

(26)

which is the same as Eq. (13). From Eq. (23), the flux of atoms toward the root of the whisker is

 < c1 > − < c2 >  = 2D gb f (θ1 , θ2 ) Rdθ θ =θ2 R sin θ2

d< c > 

(27)

According to the mass balance, the growth rate of the whisker, ˙ is L,

L˙ =

2π r2 δ(− J )

π

r22

=−

4δ D gb f (θ1 , θ2 ) < c 1 > − < c 2 > R2

sin2 θ2

(28)

The growth rate of the whisker is proportional to the concentration difference, and inversely proportional to the square of the radius of the spherical surface. Define dimensionless growth rate of λ as

λ=

2δ D gb

(20)

The solution of Eq. (20) is

J = − D gb

with k being the Boltzmann constant, p 1 and p 2 being the pressures at r = r1 and r2 , respectively, and  being atomic volume. The c max is the maximum number of atoms per unit volume, and c max  = 1. To the first order of approximation, Eq. (16) gives

kT

r = R sin θ

For 0 < θ2 < θ < θ1 << 1, there is

< c1 > − < c2 > δ ln(r1 /r2 ) r22

The growth rate of the whisker is proportional to the concentration difference. It is known that one of the mechanisms controlling stress relaxation is the growth of whiskers, i.e. the whisker growth is a function of the local pressures at r = r1 and r2 , which are acted onto the atoms. According to the theory of thermodynamics, there must exist the conversion of mechanical work to chemical energy for the diffusion of atoms at triple junctions, i.e. [18]

δ

Case II. Spherical surface For the surface of a Sn film being spherical, the polar angle of θ is used to describe a point in the grain boundary/interface. There are

and Eq. (9) becomes

< c > = a ln r + b

< c > = < c1 > +

3

R 2 L˙ 4δ D gb (< c 1 >− < c 2 >)

=−

f (θ1 , θ2 ) sin2 θ2

(29)

Fig. 2 shows variation of the dimensionless growth rate with θ1 for several values of θ2 . It is evident that the smaller θ2 the larger is the dimensionless growth rate for the same value of θ1 . One expects that a longer whisker will be formed for a smaller θ2 for

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hydrostatic pressure and biaxial stress was used in Eq. (33). The energy rate needed for the diffusion of atoms is the change rate of the difference of the chemical energies associated with the diffusion of atoms (vacancies). From Eq. (16), the change rate of the difference of the chemical energies can be calculated from the difference of the work rates done by the pressures of p 1 and p 2 as

˙ = ( p 1 − p 2 )π r22 L˙ = U˙ c = ( p 1 − p 2 ) M

2π δ D gb ( p 1 − p 2 )2 ln(r1 /r2 )

kT (34)

Substituting Eqs. (33) and (34) in Eq. (32) yields

d( p 1 − p 2 ) dt Fig. 2. Variation of dimensionless growth rate of λ with θ1 for several values of θ2 .

the same volume of whiskers. Such a result is in accord with the general experimental observation that the smaller the diameter of a whisker, the longer is the whisker. Following the same procedure for the derivation of Eq. (17), one can have the same relationships between the concentration of atoms and the local pressure as Eq. (17). Using the relationships of Eq. (17), Eq. (28) can be written as

L˙ = −

4δ D gb f (θ1 , θ2 ) p 1 − p 2 R 2 sin2 θ2

(30)

kT

The growth rate of the whisker is proportional to the pressure difference. For 0 < θ2 < θ < θ1 << 1, Eq. (30) reduces to

L˙ =

2δ D gb

p1 − p2

r22 ln(r1 /r2 )

kT

with the characteristic time of

τd−1 =

L˙ =

(31)

= −U˙ c

(32)

where U E is the strain energy of the thin film, and U˙ c is the energy rate needed for the diffusion of atoms through the grain boundary/interface. The strain energy stored in a thin film of thickness h and radius r1 on a stiff substrate can be calculated as

Mf

4 M f δ D gb

(36)

τd as

1

(37)

9 r12 h ln(r1 /r2 ) kT

2δ D gb r22 ln(r1 /r2 )

p 1 (0) − p 2 (0) −t /τd e kT

(38)

which gives

As discussed above, the gradient of concentration is the driving force for the growth of whiskers. In analyzing the whisker growth controlled by the flow of grain boundary fluid, Yang [15] pointed out that the growth of whiskers in a thin film leads to stress relaxation in the thin film. Following the approach given by Yang [15], one can assume that the contribution of the surface energy and kinetic energy of the growing whiskers to the decrease of the strain energy in the thin film is negligible. The energy dissipation due to the grain boundary/interface diffusion of atoms for the whisker growth mainly determines the release rate of the strain energy stored in the thin film if there are no mechanisms, such as the formation of intermetallic compounds, for the generation of stresses (strain energy). There is

π r12 h

(35)

kT

The characteristic time for the growth of the whisker controlled by the grain boundary/interface diffusion is inversely proportional to the diffusivity and biaxial modulus (M f ). The pressure difference decreases with the increase of the relaxation time, as expected. Substituting Eq. (36) in Eq. (18), one obtains the time dependence of the growth rate of the whisker as

L=

U E = π r12 hM f ε 2 =

9 r12 h ln(r1 /r2 )

p 1 − p 2 = [ p 1 (0) − p 2 (0)]e −t /τd

4. Kinetic analysis of the whisker growth on a planar surface

dt

4 M f δ D gb ( p 1 − p 2 )

The solution of Eq. (35) is

which is the same as Eq. (18).

dU E

=−

(σ1 − σ2 )2 =

9π r12 h 4M f

( p 1 − p 2 )2

(33)

where M f is the biaxial modulus of the film and ε is the extensional mismatch strain, isotropic in the plane of the interface between the film and the substrate, σ1 and σ2 are the biaxial stresses at r = r1 and r2 , respectively. The relation of p = −2σ /3 between

9 r12 h p 1 (0) − p 2 (0) 2 r22



Mf

1−



p1 − p2

(39)

p 1 (0) − p 2 (0)

The whisker length is a linear function of the pressure difference. For t /τd << 1, the temporal evolution of the whisker length to the first order approximation of t /τd is

L=

δ D gb

8

p 1 (0) − p 2 (0)

9 r22 ln(r1 /r2 )

kT

t

(40)

which is proportional to the diffusivity. For t → ∞, Eq. (39) gives

L∞ =

9 r12 h p 1 (0) − p 2 (0) 2 r22

(41)

Mf

which is the same as the result given by Yang [15]. Note that a factor of 4/9 in Eq. (37) and associated equations in Yang’s work [15] is missed. Using the characteristic time of τ v for the whisker growth controlled by grain boundary fluid given by Yang [15] and Eq. (37), one obtains

 3

3M f τv δ = τd 2ηr12 h ln(r1 /r2 ) 2

kT r12 h ln(r1 /r2 ) M f δ D gb

=

3kT

 2 δ

4η D gb

2 (42)

Assume that the Einstein equation for grain boundary fluid is valid, i.e. [7]

D gb η kT

=

1 2π 1/3

(43)

with η being the viscosity of grain boundary fluid. Substituting Eq. (43) in Eq. (42) yields

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τv 3π δ 2 = τd 82/3

(44)

which is similar to the ratio of the growth rate of the whisker controlled by the grain boundary fluid to the growth rate of the whisker controlled by the grain boundary diffusion, as given by Tu and Li [7]. As pointed out by Tu and Li [7], the right side of Eq. (44) is not far from unity for Sn. This result reveals the equivalence of the grain boundary fluid and the grain boundary diffusion in analyzing the growth of Sn whiskers. Using Eq. (39), one obtains the temporal evolution of the volume of a whisker as

V =

9π r12 h p 1 (0) − p 2 (0)  2

Mf





1 − e −t /τd = V ∞ 1 − e −t /τd



(45)

where V is the volume of the whisker at time t, V ∞ is the volume of the whisker for t → ∞. Cheng et al. [11] statistically studied the growth of tin whiskers, and observed that the overall growth rate of tin whiskers decreased with time with a tendency for saturation. They used the Avrami equation to analyze the temporal evolution of the volume of tin whiskers, and obtained

V = V ∞ (1 − e

−0.386t 1.394

)

(46)

which is qualitatively in accord with Eq. (45). The difference between the power indices (1 and 1.394) is likely due to statistical result in determining the growth of individual whiskers, which represents the growth of multiple whiskers. Generally, it is impossible to experimentally determine the growth behavior of a single whisker due to the random characteristics of the nucleation of whiskers. 5. Summary The use of Pb-free solders in microelectronics has renewed the interest of mitigating the formation and growth of Sn whiskers, which requires better understanding of the mechanisms controlling the growth of Sn whiskers. Based on the principle of atomic diffusion, a general mass transport equation has been formulated for the diffusion of atoms in a thin film on a surface of revolution. Assuming grain boundary/interface diffusion as the dominant

5

mechanism for the whisker growth, the growth rate is found to be proportional to the concentration difference for the whisker growth on both a planar surface and a spherical surface. Using the relation between mechanical work and chemical energy, the growth rate is found to be proportional to the pressure difference, the same as the result given by Tu [1]. Considering the whisker growth as the mechanism for stress relaxation, kinetic analysis of the whisker growth has been performed. Both the whisker length and the stress relaxation associated with the whisker growth are exponential functions of time, and the characteristic time controlling the temporal evolution of the whisker length is inversely proportional to the diffusivity and biaxial modulus of the film. Acknowledgements YS is grateful for the support from National Natural Science Foundation of China (Grant No. 51675256). References [1] K.N. Tu, Phys. Rev. B 49 (1994) 2030–2034. [2] G.T. Galyon, L. Palmer, IEEE Trans. Electron. Packag. Manuf. 28 (2005) 17–30. [3] W.J. Boettinger, C.E. Johnson, L.A. Bendersky, K.W. Moon, M.E. Williams, G.R. Stafford, Acta Mater. 53 (2005) 5033–5050. [4] F.Q. Yang, J. Phys. D, Appl. Phys. 40 (2007) 4034–4038. [5] F.Q. Yang, J.C.M. Li, Mater. Sci. Eng. A, Struct. Mater.: Prop. Microstruct. Process. 210 (1996) 135. [6] F.Q. Yang, J. Phys. D, Appl. Phys. 30 (1997) 286–288. [7] K.N. Tu, J.C.M. Li, Mater. Sci. Eng. A, Struct. Mater.: Prop. Microstruct. Process. 409 (2005) 131–139. [8] E.J. Buchovecky, N.N. Du, A.F. Bower, Appl. Phys. Lett. 94 (2009) 191904. [9] E. Chason, N. Jadhav, W.L. Chan, L. Reinbold, K.S. Kumar, Appl. Phys. Lett. 92 (2008) 171901. [10] J. Cheng, P.T. Vianco, J.C.M. Li, Appl. Phys. Lett. 96 (2010) 184102. [11] J. Cheng, P.T. Vianco, B. Zhang, J.C.M. Li, Appl. Phys. Lett. 98 (2011) 241910. [12] J. Cheng, F.Q. Yang, P.T. Vianco, B. Zhang, J.C.M. Li, J. Electron. Mater. 40 (2011) 2069–2075. [13] F. Pei, A.F. Bower, E. Chason, J. Electron. Mater. 45 (2016) 21–29. [14] Y. Wang, J.E. Blendell, C.A. Handwerker, J. Mater. Sci. 49 (2014) 1099–1113. [15] F.Q. Yang, Metall. Mater. Trans. A, Phys. Metall. Mater. Sci. 47 (2016) 5882–5889. [16] J. Williams, N. Chapman, N. Chawla, J. Electron. Mater. 42 (2013) 224–229. [17] F.Q. Yang, J.C.M. Li, Scr. Metall. Mater. 32 (1994) 139–144. [18] F.Q. Yang, J.C.M. Li, J. Appl. Phys. 74 (1993) 4382–4389.