Effects of an elastic substrate on the interfacial adhesion of thin films

Surface & Coatings Technology 200 (2006) 5003 – 5008 www.elsevier.com/locate/surfcoat

Effects of an elastic substrate on the interfacial adhesion of thin films Xian-Fang Li* Institute of Mechanics and Sensor Technology, School of Civil Engineering and Architecture, Central South University, Changsha, Hunan 410083, China Received 8 April 2005; accepted in revised form 9 May 2005 Available online 24 June 2005

Abstract Nanoindentation is a powerful technique for measuring the mechanical properties of materials such as elastic modulus, hardness, fracture toughness and interfacial adhesion, especially for some structures in micron thickness. Conventional nanoindentation techniques produce large plastic deformation directly affecting the elastic properties to be measured. As compared to conventional nanoindentation, the crosssectional nanoindentation lowers the influence of plastic deformation on measured elastic properties, and measured data are more reliable. For the cross-sectional nanoindentation, a modified theoretical model based on the elastic plate theory with elastically restrained edges is presented to evaluate the energy release rate of an interfacial delamination between a thin film and an elastic substrate. A closed-form solution is given and the predicted interfacial adhesion is in satisfactory agreement with existing experimental data. D 2005 Elsevier B.V. All rights reserved. Keywords: Cross-sectional nanoindentation; Elastic plate theory; Closed-form solution; Energy release rate; Interfacial adhesion

1. Introduction For a thin film of thickness of micron order coated on an elastic structure, nanoindentation provides a simple, easy, and powerful technique for measuring the mechanical properties of materials such as elastic modulus, hardness, interfacial adhesion and fracture toughness [1]. A lot of different experimental techniques and theoretical models have been proposed for these purposes [2,3]. For example, indentation may be performed at the surface of a thin film coating [4,5]. In general, plastic deformation inevitably takes place in the vicinity of the impressed region for sharp indenters such as Berkovich indenters or Vickers indenters since the stress in the region close to the sharp apex rises immediately a value high enough to produce plastic deformation, and even for a spherical indenter without sharp apex when applied load exceeds the first pop-in load [6]. As compared to indentation impressed on the surface of the coating, another nanoindentation technique for measur-

* Tel.: +86 731 887 7750. E-mail address: [email protected]. 0257-8972/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.surfcoat.2005.05.009

ing the interfacial adhesion and fracture toughness is performed in a cross-section termed as cross-sectional nanoindentation (CSN), either into the substrate near the interface [7] or directly at the interface between the thin film and the substrate [8]. For the former CSN technique, there is evident advantage that the plastic zone induced by a nanoindentation is far away from the interface. So far several theoretical models have been proposed to analyze interfacial adhesion. Because of the very thin coating film, Sanchez et al. [7] put forward a one-dimensional model of an assembly of tapered beams. However, the accuracy of this analytic model is not too good, which exceeds 15% for a certain range. In place of a purely elastic interface of two brittle materials, Scherban et al. [9] further studied the behavior of adhesion of a metal – dielectric interface where plastic deformation must be taken into account by means of finite element method. On the other hand, by considering the influence of transverse shear on elastic bending energy of the thin plate, a modified analytical approximation has been suggested by Zheng et al. [10], who employed the bending theory of an anisotropic cylindrical plate instead of that of an isotropic thin plate. In the above-mentioned works, the elastic plate is assumed to be clamped at the boundary of the interface debonding. Obviously, this

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X.-F. Li / Surface & Coatings Technology 200 (2006) 5003 – 5008

assumption completely neglects the presence of the elastic substrate, and the single isolated plate with clamped edges is only analyzed. In order to capture the characteristic that the interfacial adhesion is dependent on not only the material properties of the thin film but also the material properties of the substrate, a novel theoretical model of the CSN technique is presented in this paper. That is, the interface debonding between a thin film and an elastic substrate is modeled as the elastic bending of an elastic thin plate with elastically restrained edges, and the influence of the elastic substrate is described by two elastic constants. The stored elastic energy of the bending plate is evaluated and the energy release rate for interface debonding is then obtained. In particular, when elastically supported edges reduce to clamped edges, a simple and explicit closed-form solution of the energy release rate is given. The influence of the material properties of the elastic substrate on the energy release rate is analyzed. The theoretical predictions for the interfacial adhesion based on the present model are compared with previous results.

a) Applied load Wedge 2a 2b w0 Substrate Thin film

Oxide-layer

b)

2a 2b

w0

2. Theoretical model According to the CSN technique for measuring the interface adhesion of a thin film coated on the surface of substrate, a Berkovich indenter normal to the wafer crosssection and close to the interface of interest at a distance d is impressed into the substrate (Fig. 1), where one side of the triangular pyramid is required to be parallel to the interface. When applied load is raised to exceed a certain critical value, the propagation of a pre-cracking interface crack can be observed. Since the indenter position may be away from the interface at an arbitrary distance d, it is possible to quite weaken the influence of plastic deformation around the sharp apex on the interfacial adhesion to be measured. The analytical model is sketched in Fig. 1c, where b and a denote the radii of the inner semi-circle and the precracking interface debonding semi-circle, respectively. In the inner contact region, the deflection of the elastic plate of thickness h is supposed to be prescribed constant w 0. For convenience, the effects of the diameter boundary of the semi-circle is disregarded, and the results for a semi-circle can then be treated as one half of those for the entire circle. Consequently, the boundary condition at the inner edge can be stated as

c)

2b

w0 K1

K2

2a

Fig. 1. Sketch of (a) CSN test configuration, (b) the effect of the elastic substrate on the deflection of the thin film (top view), (c) the model of an elastic plate with elastically restrained edges.

neglected; so at the outer edge we adopt the following elastically supported boundary conditions [11,12]. D

d dr 

D



   d 2 w 1 dw   ; þ ¼ K w 1   2 dr r dr r¼a r¼a

  d 2 w m dw  dw  þ ¼  K ; 2 dr2 r dr r¼a dr r¼a

ð3Þ

ð4Þ

where Eh3 12ð1  m2 Þ

ð5Þ

wðbÞ ¼ w0 ;

ð1Þ

D¼

dw  ¼ 0:  dr r¼b

ð2Þ

is the bending stiffness of the plate, E and m being Young’s modulus and Poisson’s ratio, K 1 and K 2 are the translational and rotational flexibility parameters, which can be used to describe the influence of the substrate, the dimensions of which are N/m2 and N, respectively. In particular, if letting

On the other hand, owing to the perfect bonding between the thin film and the substrate, the influence of the elastic substrate on the interfacial adhesion cannot be completely

X.-F. Li / Surface & Coatings Technology 200 (2006) 5003 – 5008

K 1 and K 2 tend to infinity, the above boundary conditions reduce to

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A0 ð2klnk þ 1Þ þ 2B0 k þ C0 ¼ 0;

ð14Þ

4K1 A0  B0  ðK1 þ 1ÞC0  D0 ¼ 0;

ð15Þ

wðaÞ ¼ 0;

ð6Þ

dw  ¼ 0;  dr r¼a

ð7Þ

ð16Þ

corresponding to the case of clamped edges. Even for the latter case, up to now an explicit closed-form solution is not available. It is worth pointing out that for the case of clamped edges, Sanchez et al. [7] employed a numerical technique to derive its numerical results. Moreover, they gave an explicit approximate solution for a simplified model. That is, the plate is modeled as an assembly of tapered beams, in which only radial stresses r rr is considered. Consequently, for a one-dimensional bending beam, from the basic governing equation

If denoting the coefficient matrix of the resulting linear algebraic equations as D(k), one easily gets the desired solution as follows

w00 ðrÞ ¼

M ðr Þ ; EI ðrÞ

½K2 ð3 þ mÞþ 1A0 þ 2½K2 ð1 þ mÞ þ 1B0 þ ðK2 m þ 1ÞC0 ¼ 0:

A0 ¼

2 ; 1  k2 þ 2klnk

ð18Þ

B0 ¼ 

1  k  2klnk

; ð1  kÞ 1  k2 þ 2klnk

ð19Þ

C0 ¼ 

4klnk

; ð1  kÞ 1  k2 þ 2klnk

ð20Þ

D0 ¼ ð9Þ

ð17Þ

where the superscript  1 and T stands for the inverse of a matrix, and transposition of a matrix, respectively. Especially, when K 1, K 2 Y V, we have

ð8Þ

where M(r) =  M 0 + F 0(r  b), M 0 and F 0 being an unknown bending moment and an unknown point load applied at r = b, respectively, which can be determined through given boundary conditions, and I(r) is the moment of inertia of the cross-section of the beam, depending on the radius as follows prh3 I ðr Þ ¼ : 12

ðA0 ; B0 ; C0 ; D0 ÞT ¼ DðkÞ1 ðw0 ; 0; 0; 0ÞT ;

1  k þ 2klnk

: ð1  kÞ 1  k2 þ 2klnk

ð21Þ

Then, by employing the boundary conditions Eq. (1), Eq. (2) at r = b and conditions Eq. (6), Eq. (7) at r = a, a closedform deflection can be obtained as   2w0 lnk  r 2 r  r ln  1 w¼ þ a a ð1 þ kÞlnk þ 2ð1  kÞ 2ð1  kÞ a  lnk r lnk  þ þ1 ; ð10Þ 1  k a 2ð 1  kÞ

According to the viewpoint of energy balance, the driving force of the interface debonding results from the released energy when interface crack extends radially per length. In other words, once the stored elastic energy U of the semi-circle is found, the energy release rate for the interface crack propagation can be determined through the negative gradient or ratio of the stored elastic energy with respect to the area of the debonding region, i.e.

where k = b / a. In the following, according to the bending theory of elastic plate [13], the transverse deflection of the elastic plate, w, obeys the following differential equation 2 2 l w ¼ 0; ð11Þ

G¼ 

where l2 is the two-dimensional Laplace operator. By solving the above differential equation, a general solution is readily found to be  r 2 r  r 2 r w ¼ A0 ln þ B0 þ C0 þ D0 ; ð12Þ a a a a where A 0, . . ., D 0 are unknown and to be determined. To this end, substituting the above general solution into the given four boundary conditions leads to A0 k2 lnk þ B0 k2 þ C0 k þ D0 ¼ w0 ;

ð13Þ

BU 1 dU ¼  ; BA pa da

ð22Þ

where A is the area of the debonding region. Clearly, in order to determine the energy release rate of the interface debonding, it is sufficient to give the stored energy due to bending of the plate, which can be expressed in terms of the transverse deflection w for the present analysis as follows  Z

2 i 1 pD a h 00 2 U¼ ðw Þ þ 2mr1 w00 w V þ r1 w V rdr 2 2 b pa pa K1 wðaÞ2 þ K2 w VðaÞ2 ; þ ð23Þ 2 2

g

where the prime stands for differentiation with respect to r, d / dr. It is noted that the last two terms are attributed to the contribution of the elastic substrate.

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X.-F. Li / Surface & Coatings Technology 200 (2006) 5003 – 5008

Now we substitute the result of w in Eq. (17) into Eq. (23). After some algebra one can get   4pDw20 K 1 a3 K2 a F U¼ F F ð k Þ þ ð k Þ þ ð k Þ ; ð24Þ 0 1 2 D a2 D

tapered beams with varying inertia moment and found the corresponding deflection given by Eq. (10), from which one can evaluate the energy and energy release rate in closed form expressed, respectively, by

where the expressions for Fj(k)( j = 0, 1, 2) have been omitted here for saving space. In particular, for a rigid clamped boundary conditions the above results reduces to

U¼

4pDw20 Ucl ¼ Fcl ðkÞ; a2

ð25Þ

4pDw20 F ð kÞ a2

ð31Þ

  8Dw20 k FV ð k Þ F ð k Þ þ 2 a4

ð32Þ

and G¼ with

with 2

1  k2 þ 4klnk  Fcl ðkÞ ¼



3

2k ln k 2

ð 1  kÞ :

2 1  k þ 2klnk 2

ð26Þ

With the elastic energy U at hand, one can further evaluate the energy release rate through Eq. (22) and find  8Dw20 k b kb G¼ F0 ðkÞ þ FV0 ðkÞ  1 F1 ðkÞ þ 1 FV1 ðkÞ 2 a4 2 2  b kb þ 2 F2 ðkÞ þ 2 FV2 ðkÞ ; ð27Þ 2 2 where we have denoted two dimensionless parameters b 1 and b 2 as, respectively, b1 ¼

K1 a3 ; D

ð28Þ

b2 ¼

K2 a : D

ð29Þ

In particular, for the case of an elastic plate with clamped edges, we further obtain   8Dw20 k Gcl ¼ Fcl ðkÞ þ FVcl ðkÞ ; ð30Þ 2 a4 where F cl(k) is given by Eq. (26). Obviously, if setting k Y 0, i.e. b Y 0, it is readily found that the energy release rate G cl for a point load at the center can be recovered from the above result, which indicates that this result is more accurate than that derived through an assembly of tapered beams since the latter case does not provide a suitable approximation for small values of k [7].

1þk 2 ln k 2lnk þ 1  m2 1k F ð kÞ ¼ : 8 ½ð1 þ kÞlnk þ 2ð1  kÞ2

ð33Þ

Just as pointed out, the results from Eq. (33) are not applicable for small values of k, and the errors can reach 10% when k > 0.4. Fig. 2 gives a comparison of two results, showing the variations of the normalized energy release rate G / G 0, G 0 = 8Dw 02 / (a  b)4. It is interesting to note that the present analytic results are identical to the numerical results for the semi-circular plate in Sanchez et al. [7], who did not give an explicit expression for this case. Furthermore, from Fig. 2 it is seen that the present results from Eq. (26) tend to 1 as k = b / a approaches 0, equivalent to that the energy reduces to that of the circle plate subjected to a point load at the center. Nevertheless, the result expressed by Eq. (33) when k Y 0 gives (1  m 2) / 8, dependent on the Poisson’s ratio. To obtain the interfacial adhesion, one may calculate the critical energy release rate G cr by using the prescribed data at the onset of fracture. In what follows we evaluate the critical energy release rate G cr based on several different analytic models. Here we employ the experimental data

A B

3. Results and discussions First, for the case of clamped edges, from the previous section the energy and energy release rate have been given by explicit expression Eqs. (25) and (30), respectively. However, they are only determined numerically in Ref. [7]. Instead of an exact solution, Sanchez et al. [7] oversimplified an essential two-dimensional problem to solve analytically a one-dimensional bending of an assembly of

Fig. 2. Comparison of the normalized energy release rate by using two different models (solid line: the bending of a semi-circle plate, and dashed lines: the bending of an assembly of tapered beams, m = 0.25 (A) and m = 0.3 (B)).

X.-F. Li / Surface & Coatings Technology 200 (2006) 5003 – 5008

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Table 1 Interfacial adhesion of SiN thin film evaluated by the different models Test

1

2

3

4

5

6

7

8

9

10

11

12

13

14

d(Am) k G cr [7] G cr [10] G cr

0.74 0.21 3.0 3.9 3.8

0.81 0.03 1.8 2.7 3.5

1.09 0.14 2.2 3.1 3.1

1.47 0.18 1.6 2.1 2.1

1.52 0.39 2.9 3.4 3.3

1.76 0.28 2.1 2.6 2.5

1.86 0.31 2.6 3.1 3.1

2.21 0.32 2.6 3.1 3.1

2.78 0.25 1.3 1.6 1.6

3.00 0.29 1.4 1.6 1.7

3.50 0.30 1.4 1.6 1.7

4.07 0.26 1.4 1.7 1.7

4.50 0.31 2.2 2.6 2.6

5.21 0.28 1.3 1.4 1.6

given in Ref. [7], who applied the CSN technique to measure the interfacial adhesion of the silicon nitride – silicon oxide thin film of thickness 1 Am. In our computations, the Young’s modulus of the material is taken as 171 GPa by the method proposed by Oliver and Pharr [1], and the Poisson’s ratio is taken as 0 . 3. Besides the data including the radius of the interface crack a, and the maximum deflection of the thin film w 0, it is desirable for us to know the values of k for each test. From the given data together with the derived formula (i.e. the above Eq. (32)) in Ref. [7], one can find a set of the corresponding k, which ranges from 0.14 to 0.39 except for k = 0 . 033 for the second test. Using these data, we evaluate the corresponding critical energy release rates G cr for each test, which are tabulated in Table 1. For comparison, the results evaluated by other analytic models [7,10] are also listed in Table 1. It is easy to find that our results are very close to those given in Ref. [10]. The results given in Ref. [7] are generally underestimated as compared to those for the circle plate, in agreement with the conclusion in Ref. [7]. Due to plastic deformation in the vicinity of the sharp indenter apex, it is believed that when distance d is small enough, the effects of plastic deformation are negligible. In Table 1, the energy release rate G cr is seen to reach a maximum value for d = 1 Am, implying an evident influence of plastic deformation when the indenter apex is close to the interface. On the other hand, taking into account plastic deformation, the distribution of stress in the entire

space is difficult [14]. However, compared to stress distribution, the maximum deflection w 0 is easily measured in experiment. Moreover, w 0 can be taken as a constant value for a certain range of k. The effects of the elastic substrate on the energy release rate of interface debonding are examined in Figs. 3 and 4. Figs. 3 and 4 illustrate the energy release rate G as a function of the ratio b / a with the radius of the semi-circle a = 25 Am and the central deflection w 0 = 1 Am for various values of 1 / b 2 when 1 / b 1 = 0, 0.002, respectively. Clearly, from Fig. 3 it is observed that with increasing b / a, the energy release rate rises, as expected. In addition, for positive values of 1 / b 2, the curve of G is seen to lie below the curve of G cl. Moreover, the larger 1 / b 2 is, the lower the corresponding curve is. For the case of 1 / b 1 = 0.002, similar trends can be found in Fig. 4. It implies that the evaluated energy release rate G for an elastic substrate is always less than that for a rigid clamped substrate. The reason is that deformation of the elastic substrate consumes a partial energy. This effect may be neglected for small values of k, and becomes more and more pronounced when k is raised.

Fig. 3. Energy release rate G as a function of the ratio b = a with the radius of the semi-circle a = 25 Am, the central deflection w 0 = 1 Am and 1 / b 1 = 0 for various values 1 / b 2.

Fig. 4. Energy release rate G as a function of the ratio b / a with the radius of the semi-circle a = 25 Am, the central deflection w 0 = 1 Am and 1 / b 1 = 0.002 for various values 1 / b 2.

4. Conclusions The interfacial debonding between a thin film and an elastic substrate is analyzed by using the elastic plate theory

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X.-F. Li / Surface & Coatings Technology 200 (2006) 5003 – 5008

with elastically restrained edges, which describes the effects of the elastic substrate on the interfacial adhesion. This method is employed to the cross-sectional nanoindentation, finding that for the rigid clamped edges, the energy release rate of the interface crack propagation is determined in explicit analytic form. In particular, for the case of elastically restrained edges, the evaluated energy release rate decreases since the elastic substrate absorbs a partial energy.

Acknowledgements The author is grateful to Prof. L. Roy Xu (Vanderbilt University, USA) for some fruitful discussions on this topic.

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