Summary

In this study, two moiré techniques, namely reflection moiré and shadow moiré, were developed to measure the deformation of layered composite structures subjected to thermal loading. An artificial cross grating was also developed for reflection moiré in order to overcome the problems of measurement under temperature change, while at the same time preserving the improved resolution and the two orthogonal slopes.

The techniques developed to quantify fringe patterns and to improve the resolution for reflection moiré include phase stepping, phase unwrapping, and image processing, and these can also be applied to shadow moiré, as has been described previously in detail elsewhere [4]. It is notable for shadow moiré that, for general application, it uses a grating pitch finer than 100 um to reach the resolution of 10 um. However, reflection moiré can bear a relatively coarse grating pitch, such as 400 um, to equivalently obtain the resolution of 0.2 um for warpage, which also efficiently lowers the cost of grating fabrication.

Figure 2.1.1. An illustration of various mechanisms for phase shifting [1].

Figure 2.1.2. Schematic diagram of the relation between the wrapped and the unwrapped phase [2].

Figure 2.1.3. Schematic diagram of shadow moiré.

Figure 2.1.4. Shadow moiré fringes of TSOP with three-step shifting. (a) The first-step fringe pattern, (b) the second-step fringe pattern, (c) the third-step fringe pattern.

Figure 2.1.5. The wrapped phase map of TSOP.

Figure 2.1.6. The unwrapped phase map of TSOP.

Figure 2.1.7. The topography of TSOP reconstructed from the unwrapped phase.

Figure 2.2.1. Schematic diagram of the experimental setup for reflection moiré.

Figure 2.2.2. (a) The deformed grating recorded by the CCD. (b) The

wrapped phase map. (c) The profile of the dashed line in Figure 2(b). (d) The continous distrubution of the phase: unwrapped phase map.

Figure 2.2.3. (a) The topography measured from the copper-film surface with shadow moiré. (b) The wrapped phase map of the

copper-film/silicon sample with reflection moiré.

References

1. K. Creath, “Phase-measurement interferometry techniques,” Progress in Optics Vol. XXVI, pp. 351-393, 1998.

2. D. C. Ghiglia and M.D. Pritt, Two-dimensional Phase Unwrapping, 1998.

3. Image Processing Toolbox User’s Guide, The MathWorks, Inc., 1997.

4. I. Tsai, C. Z. Tsai, E. Wu, and C. A. Shao, “On Accurate Measurement of Warpage for Electronic Packages,” Proceedings of IMAPS Taiwan

Technique Symposium, pp. 290-297, 2001.

Chapter 3

Mechanical Properties of Sputtered Cu Film

Thin films have been widely used in various technological applications.

Though the principal function of thin film components is generally not structural, it is essential to acquire the knowledge of mechanical properties and responses of thin films, which usually dominate the stability and reliability of the devices.

For most methods, mechanical loading was applied, which didn’t deal with the CTE measurement in general. In this chapter, we present a method for obtaining mechanical properties of copper films on silicon substrates. The mechanical properties of silicon substrates also need to be known in practice, which are illustrated in Section 3.1 first. Finite element analyses with ANSYS were then performed, and the genetic algorithm (GA) was used to optimally obtain the mechanical parameters of copper films such as elastic modulus and CTE. The analysis method is presented in Section 3.2 to show how the

numerical algorithm works, and Section 3.3 describes the procedure of optical measurement in progress and the sample preparation. Reflection moiré developed in Chapter2 is adopted here for the mirror-like surface of silicon substrates. Finally, in Section 3.4, the results obtained are discussed with further examination.

3.1 Mechanical Properties of Si Substrates

To study the mechanical behavior of thin films, thin films on substrates are adopted most often. The most commonly used material for substrate is silicon, which has low CTE and is a prevalent semiconductor material. The theoretical values of elastic moduli of silicon were usually applied. However, the values might be different in practice. Before further study of thin-films, the actual mechanical properties of silicon substrates should be cautiously investigated first.

Silicon belongs to the cubic crystal class. In the original crystallographic orientation, there are three independent constants in elastic stiffness of silicon, which are C11=165.6 GPa, C12=63.9GPa, and C44=79.5 GPa [1]. For crystals with cubic symmetry, the axes of the coordinate system are generally aligned with the edges of the unit cell of the crystal, as shown in Figure 3.1.1. The elastic constants in other coordinate systems are obtained using tensor transformations:

mnpq,

where

s′

_ijkl and

c′

_ijkl are the compliance and stiffness tensors in the new

coordinate system, and

a

_ij is the direction cosine in terms of the Eulerian angles of the crystal coordinate system with respect to the new coordinate system. The transformation law (3-1b) is for

C

_ijkl, a fourth rank tensor. It requires tedious computation, and is not convenient for the 6x6 matrix

C

_αβ [2].

We introduce the simplified transformation for

C

_αβ. The 6x6 matrix

C

_αβ for cubic materials can be written as follows,



The compliance

s

_αβ can be expressed analogously. The special case of the transformation is the rotation about the z-axis through an angle θ , and the transformation laws are easily shown as:

T,

KCK

C

′=

^s

^′=

( ) ^K

^{−1 T}

^sK

^{−1, (3-3a)}



After transferring the matrixes C and s to the new coordination, the elastic modulus (E) in the new coordinate system is then defined by [2]:

( )( )

After the transformation conforming to the geometric orientation of (100) wafers with

4

θ

=

π

, the in-plane elastic moduli of (100) wafers calculated are

E<100> = 130.0 GPa, E<110> = E<-110> = 168.9 Gpa following the process

mentioned above. To measure the elastic moduli of silicon substrates, (100) and (110) silicon wafers with both side polished were adopted. The different orientations of specimens cut form (100) and (110) wafers are shown in Figure

3.1.2. Each specimen was cut into 6.0 cm x 1.5 cm. Three slices of (100) wafers with different thickness were used, which were 255 um, 350 um, and 525 um.

An ANSYS finite element model was used to examine whether the Euler beam equation is suitable for such specification with anisotropic material properties.

The silicon was simply supported and load was applied at the middle of the span as shown in Figure 3.1.3. Euler beam was chosen because it relate strains to the elastic modulus with a very simple equation as:

( ) z

where p is the loading force, b and h are the width and height of the beam, and E is the elastic modulus along the longitudinal direction. The strains calculated from 3D FEM using ANSYS were compared with the strain calculated from Euler beam equation in Figure 3.1.4. We can see that near the location where the load and boundaries applied, the strains from Euler beam equation deviated a little from that from FEM because of stress concentration. Therefore, we choose to locate the strain gages at the middle of the load force and each support where the strains from beam equation fit that from FEM well. The result shows that with proper strain gage locations, the in-plane elastic moduli could be successfully obtained with general Euler beam equation (3-5), which were

E<100>= 103.6 GPa, E<110>=140.7 GPa, E<-110>= 140.5 Gpa for the 255-um thick

wafer. The measurement agreed with the trend of theoretical values in each orientation. The results of <110> and <-110> directions only had 0.2%

difference. All the experimental data are listed in Table 3.1.1. The percentage of difference for each specimen cut from the same wafer was consistent to each other. The average difference increased from 7.2 % to 17.9 % with the thickness decreasing from 525 um to 255 um. The effect of defects became more obvious in thinner thickness of silicon wafer, and made the elastic moduli smaller than the theoretical values.

On the other hand, the theoretical elastics moduli of (110) wafers repeat every π rad., which is shown in Figure 3.1.5. The in-plane elastic moduli of (110) wafer were determined to be E<-111>= 170.1 GPa, E<1-12>=157.9 GPa with the same method, which were also smaller but close to the theoretical values with the consistent percentage. The theoretical and measured data of (110) wafers are also listed in Table 3.1.1. Use of the anisotropic (110) wafer aims to make characterization of material properties of the deposited thin film successful because it has different curvatures in each orthogonal directions which can offer more information to determine the thin film material properties.

The applications with thin silicon substrates are gradually increasing. For example, 3D system integration with thin silicon substrates can overcome the performance bottleneck for next IC generation. According to the results above, besides the thin thickness causes the specimens more flexible, the effect of defects in thinner substrates also causes the elastic moduli more deficient, which makes the substrate even more flexible. Therefore, use of thinner wafers always

comes with larger deformation, which might cause other integration problems and should be handled with more consideration.

3.2 Analysis Method

Finite element analyses adopted to calculate the deformation of the film-on-substrate composite structure under thermal loading were performed with ANSYS. A composite element, Shell99 in ANSYS, was introduced because the films on the substrates were of the micrometer order and the selected (100) and (110) silicon substrates were 355 um and 306 um thick respectively. Shell99 that takes the stiffness change along the thickness direction into consideration based on the composite theory is an ideal element for thin film analysis such as our cases here. The results with Shell99 used have been proved to be the same as those with Solid73, an 8-node solid element in ANSYS, as shown in Figure 3.2.1. The model is a 2um thick film on a 250 um thick substrate under temperature change. The slopes from models using these two elements fit well, except for a deviation near the edge, which is because of the unavoidable singularity near the boundary, even using finer mesh. Therefore, when we select data as the input for the searching algorithm, we should leave out the peripheral area, also, the reflection image of which is often affected by chipping effect. The number of the element needed with Shell99 was several times smaller than that with Solid73. In this way, the excessive aspect ratio issues, which were usually

encountered in the thin film analyses, could be avoided. Computation time could also be appreciably saved.

Each case was begun with randomly generating sets of mechanical parameters. ANSYS was employed to calculate the slope of warpage under temperature change. The stiffness matrix of the film-on-substrate composite structure was generated according to the guessed mechanical parameters. The following quadratic objective function was used to find the degree of deviation between the calculated and measured warpage slopes:

∑

= 

guessed mechanical parameters,

φ

_xi^exp and

φ

_yi^exp are the measured slope distributions, and N is the number of the measured data. The reason why the normalized form of the objective function is adopted is that we hope to make each measured point have the equal contribution to the fitness. Through the whole field optical method, we can obtain the points spreading all over the surface, which often have substantially different magnitude. We do not want the searching process to be dominated by some points with large magnitude only, so we choose to normalize the difference first. The ultimate goal was to reduce the deviation between the calculated and measured warpage slopes to its minimal value, and the minus sign was inserted in Equation (3-6) so that the objective

function can be fit to the algorithm set for searching maximum fitness. The genetic algorithm (GA) was adopted as the searching tool. Two mechanical constants, i.e., elastic modulus and coefficient of thermal expansion were then regenerated by GA and the loop repeated until a convergence criterion was satisfied. Use of the genetic algorithm assured convergence of the objective function to a globally maximum value, which was the critical part of this study as there were usually other local extreme values. A detailed description of the solution schemes for the inverse problem can be referred to [3, 4].

Verification of this searching algorithm was performed in imitation of the measurement for copper films on substrates. The specification used in the model is 2.5 cm squared for 350 um-thick (100) silicon substrate and 300 um-thick (110) silicon substrate with 3.7um-thick copper films on each top surface. Use of the anisotropic (110) wafer aims to offer more independent relations for determining the thin film material properties because its anisotropy resulted in different curvatures in each orthogonal directions. A noise of S/N ratio 10 was added to the slope of warpage under temperature change, which was calculated using ANSYS, to cover for the measured slope distributions,

φ

_xi^exp and

φ

_yi^exp, in Equation (3-6). Three mechanical parameters of thin films needed for the stiffness matrix of the finite element models are elastic modulus, CTE, and Poisson ratio, which are set as 96.6 Gpa, 38.8 ppm, and 0.19 respectively in this verification. The difficulty of GA in searching out the optimal solution is raised

with increasing number of variables. Since Poisson ratio is intrinsically less sensitive to the deformation than the other two parameters under such configuration, and has the clear and definite range in physics from –1 to 0.5, the value of Poisson ratio is man-made given with the increasing interval of 0.1 in each run with GA. With such certain noise, 9600 data points are needed to robustly obtain the result with Equation (3-6). Such a large amount of data can only be achieved easily with whole-field optical measurements as mentioned in Chapter 2. Best fitness was achieved with Poisson ratio from 0.19 to 0.2, and the optimally obtained elastic modulus is 95.1 GPa, and CTE is 39.1 ppm in average for thin film. We set the generation number as 250. If the best fitness happened at some generation, which is smaller than 20% away from the generation number we set, it is considered not stably converged yet. Then we will increase the generation number next time. With the condition in this case, the fitness converged at the 73rd generation as shown in Figure 3.2.2. The error is less than 2% for both elastic modulus and CTE, and the range of Poisson ratio also fits to the specified value that indicate the algorithm developed is stable and suitable for determining the material properties of thin film under such situation. The given and inversed parameters are all listed in Table 4.2.1.

3.3 Experimental Procedure

From the result of the analysis method mentioned in the previous section, up

to 9600 points of slope are needed to make the searching algorithm stabilize for the signal with S/N ratio 10. In addition, mirror-like surfaces of silicon substrates also make the whole-field reflection moiré the most expedient choice.

In the first step experiment, copper thin films were sputtered on 4-inch (100) and (110) silicon substrates with three different sets of thickness, which are 2.2 um, 3.7 um and 4.5 um. The sputtering parameters were 250 standard cm³/min Ar flow with 5kW power. The wafer thickness was measured using a micrometer before and after sputtering. All the samples were cut into a 2.5-centimeter square area. Cross section images of the samples were also recorded to measure the thickness of both films and substrates after all the deformation measurements were done.

The samples were then heated on the hotplate at 150°C for 10 minutes, and then taken out of the hotplate and placed carefully on the tripod made up of three thermal couples. The experimental setup has been illustrated in Figure 3.3.1. During the measurement, thermal loading was applied as the samples cooled down from an elevated temperature, and the images of deformed cross grating were recorded by the CCD every ten seconds. The cross grating was employed to measure the slope change of the two orthogonal directions. With numerically phase shifting and image processing techniques presented in Chapter 2, the history of slope change of the samples due to temperature change was then obtained. Figure 3.3.2 shows the slopes of the curves of slope change

vs. temperature change are not the same for the 1^st and 19^th day measurements.

That indicated the sample condition was not stable during the storage. The reason should be lack of an adhesive layer for copper and silicon substrate from the observation because most samples delaminated after long time storage.

Therefore, there is usually an adhesive and barrier layer for copper and silicon in practical applications. In this study, thin copper films were sputtered onto silicon substrates with titanium less than 0.1 um thick. The thickness percentage of titanium is less than 2.7 % of the thinnest copper film, and the mechanical properties are similar such as elastic modulus of titanium is 117 GPa, which is close to that of copper when they are bulk. The effect of titanium on structural deformation can be reasonably neglected. Figure 3.3.3 shows the curves of slope change vs. temperature change remained the same even after two-month storage, which illustrates titanium as adhesive layer provides excellent adhesion for copper film to silicon substrates and stabilizes the specimen condition.

Since we can obtain stable and repeatable measurement data, we are finally relieved to use them as the measured slopes,

φ

_xi^exp and

φ

_yi^exp, in the object function for the searching algorithm demonstrated in Section 3.2. The algorithm was respectively operated for copper films with 2.2 um, 3.7 um and 4.5 um thickness. Best fitness was achieved for all three thickness with Poisson ratio from 0.35 to 0.4 using 0.1 increasing interval. For 2.2 um copper film, the optimally obtained elastic modulus is 105.8 GPa, and CTE is 32.5 ppm. For 3.7

um, the elastic modulus is 97.5 GPa, and CTE is 30.0 ppm. As for 4.5 um, the elastic modulus is 92.5 GPa, and CTE is 29.9 ppm. According to the verification test in previous section, the results obtained with the algorithm developed can be regarded reliable in these applications.

3.4 Discussions

In the pervious section, the material properties we obtained for sputtered copper film didn’t conform to that of copper in bulk, which is 110~126 GPa for elastic modulus and 16.6~17.6 ppm for CTE. Table 3.4.1 lists of material properties of copper as thin film and in bulk. For all these three copper film with different thickness, elastic moduli are less than that in bulk, but increase monotonically with decreasing thickness. However, CTE are larger than that in bulk, and decrease monotonically with decreasing thickness.

The related results of copper thin films in literatures are summarized in Table 3.4.2. We can see that the records of CTE are extremely limited so that the systematic comparison of CTE is not available. Most literatures focused on the measurement of elastic modulus, which, according to records listed in Table 3.4.2, varies with deposition methods and thickness. Most elastic moduli are close to 100Gpa except for Read’s 7.32 um electrodeposited film [5] and Koh’s 4 um nanocrystalline film [6]. Read imputed the deficient elastic modulus of electrodeposited copper film to defects in the microstructure such as porosity,

while nanocrystalline (nc) metals have been recognized as possessing some appealing mechanical properties with progress in the processing of materials, such as ultra-high yield and fracture strengths, decreased elongation and toughness. Koh et al. [6] employed nanoindentation techniques to characterize elastic modulus normal to the interface, which has high-modulus textures oriented, and increase the elastic modulus up to 138.7 Gpa. For the samples using sputtering deposition of relatively similar order of thickness, such as 2.2 um in this study, 0.99 um of Read, and 1.6 um of Ho et al., the corresponding

while nanocrystalline (nc) metals have been recognized as possessing some appealing mechanical properties with progress in the processing of materials, such as ultra-high yield and fracture strengths, decreased elongation and toughness. Koh et al. [6] employed nanoindentation techniques to characterize elastic modulus normal to the interface, which has high-modulus textures oriented, and increase the elastic modulus up to 138.7 Gpa. For the samples using sputtering deposition of relatively similar order of thickness, such as 2.2 um in this study, 0.99 um of Read, and 1.6 um of Ho et al., the corresponding

在文檔中 薄膜機械特性及殘留應力之反算偵測(3/3) (頁 22-0)