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Evaluation of intrinsic film stress distributions from induced substrate deformation
Microelectronic Engineering 78–79 (2005) 404–409 www.elsevier.com/locate/mee
Evaluation of intrinsic film stress distributions from induced substrate deformation R.L. Engelstad, Z. Feng *, E.G. Lovell, A.R. Mikkelson, J. Sohn Computational Mechanics Center, University of Wisconsin-Madison, 1513 University Avenue, Madison, WI 53706, USA Available online 21 January 2005
Abstract Measuring and controlling intrinsic stress in films deposited on substrates is a basic and essential challenge in microand nano-engineering. Indirect measuring methods for determining film stress are commonly used. Generally, commercial tools measure the induced mask shape, and relate it to film stress via the two-dimensional form of StoneyÕs equation, applied in a point-wise manner. This equation gives incorrect results if the stress distribution is not uniform. As an alternative, a numerical method based upon finite element (FE) analysis has been developed. It contrasts traditional structural FE methods since the out-of-plane displacements, curvatures or cross-section rotations measured on the substrate surface are used as input data. Numerical tests and applications to lithographic masks demonstrated that the FE technique provided accurate film stress even if magnitudes varied from point to point and principal directions changed over the surface. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Thin films; Lithography; Nanofabrication; Stress measurement; Finite element methods
1. Introduction In a film–substrate system, intrinsic stress deforms the substrate, affects the film strain, strength, material properties, surface fatigue, crack generation and product characteristics. In lithographic masks the stress can cause significant image distortions, which compromise the accuracy of image resolution and placement. Consequently, the ability to accurately measure the intensity and *
Corresponding author. E-mail address: [email protected] (Z. Feng).
spatial distribution of film stress is especially important for fabrication process optimization. Equipment for directly measuring localized thinfilm stress is expensive and not well-suited for production environments. In recent years, indirect measuring methods have been increasingly adopted for determining film stress. Laser scanning or interferometric methods have been employed to determine the out-of-plane displacement (OPD), cross-section rotation or the curvature of the film– substrate system. The film stress is then calculated from the measured data by the local application of StoneyÕs equation. Although most measuring tools
0167-9317/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.mee.2005.01.011
R.L. Engelstad et al. / Microelectronic Engineering 78–79 (2005) 404–409
provide accurate information for the induced substrate shape, the calculated stress magnitudes and distributions are not reliable, since the local application of StoneyÕs equation is not valid. The limitations of the equation will be discussed in the next section. A new technique based on finite element (FE) analysis has been developed at the University of Wisconsin-Computational Mechanics Center (UW-CMC). The technique overcomes the deficiencies of StoneyÕs equation. 2. StoneyÕs equation The original StoneyÕs equation was derived for a beam flexed by a uniformly stressed film [1] rf ¼ Es t2s K=6tf ;
ð1Þ
where rf is the film normal stress, Es is the elastic modulus of the substrate, ts is the substrate thickness, tf is the film thickness, and K is the curvature caused by intrinsic stress. If the film stress is uniform, Eq. (1) can be extended to evaluate the film stress in a two-dimensional film–substrate system by replacing Es with the biaxial modulus rf ¼ Es t2s K=6tf ð1 ms Þ; where ms is PoissonÕs ratio of the substrate.
ð2Þ
405
The stress rf in Eq. (2) is assumed to be uniform; thus the curvature K should be constant, and the deformed substrate surface is spherical. Eq. (2) has also been applied locally to calculate film stress point-by-point, when the curvature is not constant. In these cases, the constant curvature K in Eq. (2) is replaced by local curvature, which depends on the positions of data points and measured directions. Although the local application of StoneyÕs equation is widely used, its validity cannot be established. Membrane resonant frequency test results demonstrated that the local application of StoneyÕs equation provided incorrect film stress [2]. Numerical tests showed it can cause substantial errors, as illustrated in Section 4. The applicability of StoneyÕs equation is significantly limited, since the deformed shapes of most film-coated wafers and lithographic masks are not spherical. Thirty deformed substrates, consisting of stressed wafers and reticles, were tested at the UW-CMC. None were spherical. Fig. 1 shows curvature contours of a typical stressed wafer. Its material and geometry properties are cited in Table 1. The curvatures varied in magnitude and direction over the surface. To obtained reliable film stress from such aspherical shapes, new methods are needed.
Fig. 1. Curvature contours for (a) Kx and (b) Ky of a wafer substrate, flexed by film deposition.
406
R.L. Engelstad et al. / Microelectronic Engineering 78–79 (2005) 404–409
Table 1 Material and geometric properties of a test wafer Parameter
Elastic modulus E (GPa)
PoissonÕs ratio m
Thickness t (lm)
Outer radius r0 (mm)
Substrate (Si) Film (Cr)
160 248
0.12 0.30
625 0.07
50 50
3. UW-CMC technique As an alternative to StoneyÕs equation, a new technique has been developed at the UW-CMC. It is based upon finite element analysis, but contrasts ordinary FE techniques. Traditional FE methods are widely used to analyze structural components with known external loads and were also applied to compute the film stress in film–substrate systems when the thermal and lattice mismatch strains were determined or estimated [3–6]. In the cases discussed here, the external loading, intrinsic stress, and mismatch strain are unknown. Only the measured out-of-plane displacements, cross-section rotations, or curvatures can be used as input data for the FE analysis. Mathematically, the film stress cannot be determined from the measured raw data uniquely, since displacements and curvatures are only collected at a finite number of discrete points on the mask surface. To eliminate the indeterminacy and evaluate the input data at specific finite element nodes, which may not coincide with measured data points, an interpolation technique is introduced to preprocess the raw data. Ordinary polynomials (and some special polynomials) can be used as relatively simple interpolation functions. The polynomial order, which directly affects the accuracy in film stress, can be chosen by the following procedure. The measured substrate OPD is represented by an nth-order polynomial Wn(x, y), and it is evaluated at FE nodes (x1, y1), (x2, y2), . . ., (xi, yi), . . ., (xm, ym). The integer m is the total number of nodes. The order n is a positive integer and varies from zero to infinity. The Wn(x1, y1), . . ., Wn(xi, yi) . . . and Wn(xm, ym) form an m-dimensional vector W n W n ¼ fW n ðx1 ; y 1 Þ; . . . ; W n ðxi ; y i Þ; . . . ; W n ðxm ; y m Þg ð3Þ and W 0 ; W 1 ; W 2 ; . . . ; W n ; . . . and W 1 form a metric space. The distance (difference) between
W k and W l [7] is normalized and the result is denoted as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m P 2 ½W k ðxi ; y i Þ W l ðxi ; y i Þ DðW k ; W l Þ ¼
i¼1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m P 2 W l ðxi ; y i Þ
:
ð4Þ
i¼1
When a tolerance of the normalized distance is specified (denoted as g) a polynomial order N0 is selected as an initial guess, and the normalized distance between W N 0 1 and W N 0 is calculated by Eq. (4) to determine whether the inequality (5) is satisfied DðW N 0 1 ; W N 0 Þ < g:
ð5Þ
If it is not satisfied, N0 has to be increased and the inequality of Eq. (5) should be checked again. These steps are repeatedly performed until the inequality is satisfied and the necessary order is identified. Algorithms for the determination of the necessary polynomial order, interpolation and FE analysis have been developed, coded and tested. The assumption of uniform film stress is a fundamental necessity for correctly applying StoneyÕs equation, but is unnecessary for performing the FE analysis. Thus, the present technique can be used to analyze a film–substrate system in which the film stress is a function of location and direction.
4. Examination of the StoneyÕs equation and the FE technique Numerical tests can be utilized to assess the accuracy of techniques linking film stress to substrate deformation. The test case that was chosen is a special axisymmetric example, in which the film radial stress and circumferential stress vary linearly from a maximum value at the center to a minimum at the outer edge (Fig. 2). An exact solution for this
R.L. Engelstad et al. / Microelectronic Engineering 78–79 (2005) 404–409
407
z, w
σθ σr r
Film Substrate
r0 Fig. 2. A special axisymmetric case with linearly varying film stress. Its analytic solution relating film stress to substrate deformation was used as a benchmark.
250
case has been developed. The filmÕs radial stress rr and circumferential stress rh are given by rr ¼ rc ð1 r=ro Þ;
200
rh ¼ rc ½1 ð1 þ 2mf Þr=ð2 þ mf Þro ;
Esr
ð6Þ
ð7Þ This analytical solution is based on elasticity theory and satisfies equilibrium, compatibility and boundary conditions; thus it can be used as a benchmark for StoneyÕs equation and the UWCMC FE code. A numerical test case, based upon the analytical solution was performed. The maximum stress at the center was 500 MPa. The material and geometric parameters are listed in Table 1. Fig. 3 illustrates the errors resulting from the utilization of both the localized StoneyÕs equation and the UW-CMC technique. Using localized StoneyÕs equation with the radial coordinate varied from 0 to 97.5% of the outer radius, the error in radial stress increased from 7.8% to 211%, and in circumferential stress
Error (%)
where rc is the maximum radial and circumferential stress at the center of the film, r is the radial coordinate, ro is the outer radius and mf is PoissonÕs ratio of the film. The corresponding OPD of the substrate is given by " 2 3 # 3rc r2o tf ð1 m2s Þ ð2 þ ms Þ r 2 r w¼ : Es t2s ð2 þ mf Þ ð1 þ ms Þ ro 3 ro
150 Es
100 Efr and Ef
50 0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
r/r0 Fig. 3. Errors from the localized StoneyÕs equation and the UW-CMC code: Esr and Esh, errors in rr and rh, from the localized StoneyÕs equation; Efr and Efh, errors in rr and rh, from the FE code.
from 7.8% to 109%. The error from the UWCMC technique was negligible; for radial stress it ranged from 0.01% to 0.53%, and for circumferential stress from 0.08% to 0.51%. The FE technique was clearly superior to the 2-D Stoney equation.
5. Applying the UW-CMC code to lithographic masks The UW-CMC code was used to analyze commercial wafers and reticles. The deformation data
408
R.L. Engelstad et al. / Microelectronic Engineering 78–79 (2005) 404–409
were measured by various tools. As an example, test results for a stressed wafer are reported here. (The material and geometric properties are cited in Table 1.) Fig. 4 shows the substrate OPD, from which film stress was evaluated. The deformed surface was aspherical (Fig. 4(a)); thus StoneyÕs equa-
tion would not produce meaningful results and the FE technique was used. Fig. 5 shows the contours of principal stresses in the film plane. The magnitudes of the principal stresses, r1 and r2, were not equal to each other at most points, and their distribution was not uniform. Based on the princi-
Fig. 4. Measured flatness of a wafer: (a) 3-D OPD shape, (b) OPD contours.
Fig. 5. Contours of principal stresses in the film plane.
Fig. 6. Contours of Von Mises stress rmises and average stress ra in film; ra = (r1 + r2)/2.
R.L. Engelstad et al. / Microelectronic Engineering 78–79 (2005) 404–409
pal stresses, various parameters for stress control were calculated. Average stress is convenient and commonly used in the semiconductor industry. The Von Mises stress is important for special purposes, such as the prediction of the dislocation generation [3]. Contours of the both stresses are illustrated in Fig. 6.
409
Acknowledgements This research has been funded by DARPA/ ARL. Additional computer support was provided by the Intel Corporation and Microsoft.
References 6. Conclusions The localized use of StoneyÕs equation cannot provide accurate film stress for most commercial film–substrate systems, since the film stress is not uniform. The present FE technique has very small computational errors and can be readily implemented with existing tools. It is very promising for the determination of film stress with arbitrary gradients, and thus for the optimization of fabrication processes.
[1] G.G. Stoney, Proc. Roy. Soc. A 82 (1909) 172. [2] M.P. Schlax, R.L. Engelstad, E.G. Lovell, C. Brooks, C. Magg, Proc. SPIE, Emerg. Litho. Tech. IV 3997 (2000) 539. [3] Z. Feng, E.G. Lovell, R.L. Engelstad, T.F. Kuech, S.E. Babcock, Appl. Phys. Lett. 80 (2002) 1547. [4] Z. Feng, E.G. Lovell, R.L. Engelstad, T.F. Kuech, Mat. Res. Soc. Symp. Proc. 695 (2002) 65. [5] Q.K.K. Liu, A. Hoffmann, H. Siegle, A. Kaschner, C. Thomsen, J. Christen, F. Bertram, Appl. Phys. Lett. 74 (1999) 3122. [6] Y.L. Shen, S. Suresh, I.A. Blech, J. Appl. Phys. 80 (1996) 1388. [7] B.Z. Vulikh, in: I.N. Sneddon (Ed.). Introduction to Functional Analysis for Scientists and Technologists (English Translation), Pergamon Press, Oxford, 1963.
Evaluation of intrinsic film stress distributions from induced substrate deformation R.L. Engelstad, Z. Feng *, E.G. Lovell, A.R. Mikkelson, J. Sohn Computational Mechanics Center, University of Wisconsin-Madison, 1513 University Avenue, Madison, WI 53706, USA Available online 21 January 2005
Abstract Measuring and controlling intrinsic stress in films deposited on substrates is a basic and essential challenge in microand nano-engineering. Indirect measuring methods for determining film stress are commonly used. Generally, commercial tools measure the induced mask shape, and relate it to film stress via the two-dimensional form of StoneyÕs equation, applied in a point-wise manner. This equation gives incorrect results if the stress distribution is not uniform. As an alternative, a numerical method based upon finite element (FE) analysis has been developed. It contrasts traditional structural FE methods since the out-of-plane displacements, curvatures or cross-section rotations measured on the substrate surface are used as input data. Numerical tests and applications to lithographic masks demonstrated that the FE technique provided accurate film stress even if magnitudes varied from point to point and principal directions changed over the surface. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Thin films; Lithography; Nanofabrication; Stress measurement; Finite element methods
1. Introduction In a film–substrate system, intrinsic stress deforms the substrate, affects the film strain, strength, material properties, surface fatigue, crack generation and product characteristics. In lithographic masks the stress can cause significant image distortions, which compromise the accuracy of image resolution and placement. Consequently, the ability to accurately measure the intensity and *
Corresponding author. E-mail address: [email protected] (Z. Feng).
spatial distribution of film stress is especially important for fabrication process optimization. Equipment for directly measuring localized thinfilm stress is expensive and not well-suited for production environments. In recent years, indirect measuring methods have been increasingly adopted for determining film stress. Laser scanning or interferometric methods have been employed to determine the out-of-plane displacement (OPD), cross-section rotation or the curvature of the film– substrate system. The film stress is then calculated from the measured data by the local application of StoneyÕs equation. Although most measuring tools
0167-9317/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.mee.2005.01.011
R.L. Engelstad et al. / Microelectronic Engineering 78–79 (2005) 404–409
provide accurate information for the induced substrate shape, the calculated stress magnitudes and distributions are not reliable, since the local application of StoneyÕs equation is not valid. The limitations of the equation will be discussed in the next section. A new technique based on finite element (FE) analysis has been developed at the University of Wisconsin-Computational Mechanics Center (UW-CMC). The technique overcomes the deficiencies of StoneyÕs equation. 2. StoneyÕs equation The original StoneyÕs equation was derived for a beam flexed by a uniformly stressed film [1] rf ¼ Es t2s K=6tf ;
ð1Þ
where rf is the film normal stress, Es is the elastic modulus of the substrate, ts is the substrate thickness, tf is the film thickness, and K is the curvature caused by intrinsic stress. If the film stress is uniform, Eq. (1) can be extended to evaluate the film stress in a two-dimensional film–substrate system by replacing Es with the biaxial modulus rf ¼ Es t2s K=6tf ð1 ms Þ; where ms is PoissonÕs ratio of the substrate.
ð2Þ
405
The stress rf in Eq. (2) is assumed to be uniform; thus the curvature K should be constant, and the deformed substrate surface is spherical. Eq. (2) has also been applied locally to calculate film stress point-by-point, when the curvature is not constant. In these cases, the constant curvature K in Eq. (2) is replaced by local curvature, which depends on the positions of data points and measured directions. Although the local application of StoneyÕs equation is widely used, its validity cannot be established. Membrane resonant frequency test results demonstrated that the local application of StoneyÕs equation provided incorrect film stress [2]. Numerical tests showed it can cause substantial errors, as illustrated in Section 4. The applicability of StoneyÕs equation is significantly limited, since the deformed shapes of most film-coated wafers and lithographic masks are not spherical. Thirty deformed substrates, consisting of stressed wafers and reticles, were tested at the UW-CMC. None were spherical. Fig. 1 shows curvature contours of a typical stressed wafer. Its material and geometry properties are cited in Table 1. The curvatures varied in magnitude and direction over the surface. To obtained reliable film stress from such aspherical shapes, new methods are needed.
Fig. 1. Curvature contours for (a) Kx and (b) Ky of a wafer substrate, flexed by film deposition.
406
R.L. Engelstad et al. / Microelectronic Engineering 78–79 (2005) 404–409
Table 1 Material and geometric properties of a test wafer Parameter
Elastic modulus E (GPa)
PoissonÕs ratio m
Thickness t (lm)
Outer radius r0 (mm)
Substrate (Si) Film (Cr)
160 248
0.12 0.30
625 0.07
50 50
3. UW-CMC technique As an alternative to StoneyÕs equation, a new technique has been developed at the UW-CMC. It is based upon finite element analysis, but contrasts ordinary FE techniques. Traditional FE methods are widely used to analyze structural components with known external loads and were also applied to compute the film stress in film–substrate systems when the thermal and lattice mismatch strains were determined or estimated [3–6]. In the cases discussed here, the external loading, intrinsic stress, and mismatch strain are unknown. Only the measured out-of-plane displacements, cross-section rotations, or curvatures can be used as input data for the FE analysis. Mathematically, the film stress cannot be determined from the measured raw data uniquely, since displacements and curvatures are only collected at a finite number of discrete points on the mask surface. To eliminate the indeterminacy and evaluate the input data at specific finite element nodes, which may not coincide with measured data points, an interpolation technique is introduced to preprocess the raw data. Ordinary polynomials (and some special polynomials) can be used as relatively simple interpolation functions. The polynomial order, which directly affects the accuracy in film stress, can be chosen by the following procedure. The measured substrate OPD is represented by an nth-order polynomial Wn(x, y), and it is evaluated at FE nodes (x1, y1), (x2, y2), . . ., (xi, yi), . . ., (xm, ym). The integer m is the total number of nodes. The order n is a positive integer and varies from zero to infinity. The Wn(x1, y1), . . ., Wn(xi, yi) . . . and Wn(xm, ym) form an m-dimensional vector W n W n ¼ fW n ðx1 ; y 1 Þ; . . . ; W n ðxi ; y i Þ; . . . ; W n ðxm ; y m Þg ð3Þ and W 0 ; W 1 ; W 2 ; . . . ; W n ; . . . and W 1 form a metric space. The distance (difference) between
W k and W l [7] is normalized and the result is denoted as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m P 2 ½W k ðxi ; y i Þ W l ðxi ; y i Þ DðW k ; W l Þ ¼
i¼1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m P 2 W l ðxi ; y i Þ
:
ð4Þ
i¼1
When a tolerance of the normalized distance is specified (denoted as g) a polynomial order N0 is selected as an initial guess, and the normalized distance between W N 0 1 and W N 0 is calculated by Eq. (4) to determine whether the inequality (5) is satisfied DðW N 0 1 ; W N 0 Þ < g:
ð5Þ
If it is not satisfied, N0 has to be increased and the inequality of Eq. (5) should be checked again. These steps are repeatedly performed until the inequality is satisfied and the necessary order is identified. Algorithms for the determination of the necessary polynomial order, interpolation and FE analysis have been developed, coded and tested. The assumption of uniform film stress is a fundamental necessity for correctly applying StoneyÕs equation, but is unnecessary for performing the FE analysis. Thus, the present technique can be used to analyze a film–substrate system in which the film stress is a function of location and direction.
4. Examination of the StoneyÕs equation and the FE technique Numerical tests can be utilized to assess the accuracy of techniques linking film stress to substrate deformation. The test case that was chosen is a special axisymmetric example, in which the film radial stress and circumferential stress vary linearly from a maximum value at the center to a minimum at the outer edge (Fig. 2). An exact solution for this
R.L. Engelstad et al. / Microelectronic Engineering 78–79 (2005) 404–409
407
z, w
σθ σr r
Film Substrate
r0 Fig. 2. A special axisymmetric case with linearly varying film stress. Its analytic solution relating film stress to substrate deformation was used as a benchmark.
250
case has been developed. The filmÕs radial stress rr and circumferential stress rh are given by rr ¼ rc ð1 r=ro Þ;
200
rh ¼ rc ½1 ð1 þ 2mf Þr=ð2 þ mf Þro ;
Esr
ð6Þ
ð7Þ This analytical solution is based on elasticity theory and satisfies equilibrium, compatibility and boundary conditions; thus it can be used as a benchmark for StoneyÕs equation and the UWCMC FE code. A numerical test case, based upon the analytical solution was performed. The maximum stress at the center was 500 MPa. The material and geometric parameters are listed in Table 1. Fig. 3 illustrates the errors resulting from the utilization of both the localized StoneyÕs equation and the UW-CMC technique. Using localized StoneyÕs equation with the radial coordinate varied from 0 to 97.5% of the outer radius, the error in radial stress increased from 7.8% to 211%, and in circumferential stress
Error (%)
where rc is the maximum radial and circumferential stress at the center of the film, r is the radial coordinate, ro is the outer radius and mf is PoissonÕs ratio of the film. The corresponding OPD of the substrate is given by " 2 3 # 3rc r2o tf ð1 m2s Þ ð2 þ ms Þ r 2 r w¼ : Es t2s ð2 þ mf Þ ð1 þ ms Þ ro 3 ro
150 Es
100 Efr and Ef
50 0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
r/r0 Fig. 3. Errors from the localized StoneyÕs equation and the UW-CMC code: Esr and Esh, errors in rr and rh, from the localized StoneyÕs equation; Efr and Efh, errors in rr and rh, from the FE code.
from 7.8% to 109%. The error from the UWCMC technique was negligible; for radial stress it ranged from 0.01% to 0.53%, and for circumferential stress from 0.08% to 0.51%. The FE technique was clearly superior to the 2-D Stoney equation.
5. Applying the UW-CMC code to lithographic masks The UW-CMC code was used to analyze commercial wafers and reticles. The deformation data
408
R.L. Engelstad et al. / Microelectronic Engineering 78–79 (2005) 404–409
were measured by various tools. As an example, test results for a stressed wafer are reported here. (The material and geometric properties are cited in Table 1.) Fig. 4 shows the substrate OPD, from which film stress was evaluated. The deformed surface was aspherical (Fig. 4(a)); thus StoneyÕs equa-
tion would not produce meaningful results and the FE technique was used. Fig. 5 shows the contours of principal stresses in the film plane. The magnitudes of the principal stresses, r1 and r2, were not equal to each other at most points, and their distribution was not uniform. Based on the princi-
Fig. 4. Measured flatness of a wafer: (a) 3-D OPD shape, (b) OPD contours.
Fig. 5. Contours of principal stresses in the film plane.
Fig. 6. Contours of Von Mises stress rmises and average stress ra in film; ra = (r1 + r2)/2.
R.L. Engelstad et al. / Microelectronic Engineering 78–79 (2005) 404–409
pal stresses, various parameters for stress control were calculated. Average stress is convenient and commonly used in the semiconductor industry. The Von Mises stress is important for special purposes, such as the prediction of the dislocation generation [3]. Contours of the both stresses are illustrated in Fig. 6.
409
Acknowledgements This research has been funded by DARPA/ ARL. Additional computer support was provided by the Intel Corporation and Microsoft.
References 6. Conclusions The localized use of StoneyÕs equation cannot provide accurate film stress for most commercial film–substrate systems, since the film stress is not uniform. The present FE technique has very small computational errors and can be readily implemented with existing tools. It is very promising for the determination of film stress with arbitrary gradients, and thus for the optimization of fabrication processes.
[1] G.G. Stoney, Proc. Roy. Soc. A 82 (1909) 172. [2] M.P. Schlax, R.L. Engelstad, E.G. Lovell, C. Brooks, C. Magg, Proc. SPIE, Emerg. Litho. Tech. IV 3997 (2000) 539. [3] Z. Feng, E.G. Lovell, R.L. Engelstad, T.F. Kuech, S.E. Babcock, Appl. Phys. Lett. 80 (2002) 1547. [4] Z. Feng, E.G. Lovell, R.L. Engelstad, T.F. Kuech, Mat. Res. Soc. Symp. Proc. 695 (2002) 65. [5] Q.K.K. Liu, A. Hoffmann, H. Siegle, A. Kaschner, C. Thomsen, J. Christen, F. Bertram, Appl. Phys. Lett. 74 (1999) 3122. [6] Y.L. Shen, S. Suresh, I.A. Blech, J. Appl. Phys. 80 (1996) 1388. [7] B.Z. Vulikh, in: I.N. Sneddon (Ed.). Introduction to Functional Analysis for Scientists and Technologists (English Translation), Pergamon Press, Oxford, 1963.