- Email: [email protected]
Accurate extraction of the mechanical properties of thin films by nanoindentation for the design of reliable MEMS
Microelectronics Reliability 50 (2010) 1621–1625
Contents lists available at ScienceDirect
Microelectronics Reliability journal homepage: www.elsevier.com/locate/microrel
Accurate extraction of the mechanical properties of thin films by nanoindentation for the design of reliable MEMS Yuji Sasaki a, Mauro Ciappa b,*, Takayuki Masunaga a, Wolfgang Fichtner b a b
Toshiba Corporation, Corporate Manufacturing Engineering Center, 33, Shin-Isogo-Cho, Isogo-Ku, Yokohama 235-0017, Japan Swiss Federal Institute of Technology (ETH), Integrated Systems Laboratory, ETH-Zentrum, CH-8092 Zurich, Switzerland
a r t i c l e
i n f o
Article history: Received 5 July 2010 Accepted 19 July 2010 Available online 21 August 2010
a b s t r a c t Traditional procedures for the extraction of mechanical properties of thin films by nanoindentation measurements have shown problems in terms of accuracy and in the ability to support sophisticated constitutive models. In this paper, an inverse modeling procedure based on finite element analysis is presented to solve these limitations. Finite element simulation is used to predict the relationships between the indentation load and depth. The developed approach is applied to extract the viscoplastic properties of aluminum single grain, the viscoelastic properties of acrylic resin films, and the residual strain in stainless steel. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction The demand of new techniques for accurate extraction of the mechanical properties related to the microscopic structure of thin films used for semiconductor devices and Micro Electro Mechanical Systems (MEMS) is increasing. This is mainly driven by the need to improve the structural designs and manufacturing processes by numerical analysis. Generally, bulk material properties can be extracted by tensile and bend tests. However, these are no longer possible in thin films. Usually, the mechanical properties at microscopic level are measured by nanoindentation, i.e. by forcing a diamond indenter into the film surface in order to acquire the relationship between indentation load and indentation depth (L–D curve). Forcing the indenter into the surface results not only into elastic but also in plastic and viscous deformation. The only parameter directly obtained from nanoindentation measurements is the Young’s modulus, which is extracted from the slope of the unloading L–D curve. Nevertheless, inelastic material properties are often indispensable to improve structural designs. In this respect, techniques to determine inelastic material properties from measured L–D curves are under development. At present, research covers plastic [1,2] and viscous property [3,4]. The stress and strain distribution beneath an indenter tip is quite complicated. Therefore, numerical inverse analysis of the experimental curves is used to extract the mechanical properties that are needed to calibrate accurately the constitutive equations. This paper, the procedure is presented to extract the viscoplastic properties of aluminum single grain, the viscoelastic properties * Corresponding author. Tel.: +41 446322436; fax: +41 446321194. E-mail address: [email protected] (M. Ciappa). 0026-2714/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.microrel.2010.07.119
of acrylic resin films, and the residual strain in stainless steel (SUS304).
2. Extraction of the viscoplastic parameters The structural design by numerical analysis requires knowing the viscoplastic properties of the materials to predict the stress and displacement in the device under operating conditions. Generally, viscoplastic properties are obtained from tensile tests. However, in the case of thin films, this is not feasible because of the large dimensional errors involved, as well as due to the impossibility to manufacture proper supports for the samples.
2.1. Experimental Nanoindentation measurements have been performed by the Hysitron TI-950 nanoindenter at a room temperature and with a diamond conical tip. The shape of the indenter has been chosen for axisymmetric FEM modeling. Both load and depth are continuously measured during the measurement cycle. Fig. 1 shows the two different load conditions with a loading rate of 1 mN/s and 0.05 mN/s, respectively. In both cases, the maximum load is 5 mN. In Table 1 the related loading, hold, and unloading time are listed in detail. The L–D curve measured in an aluminum single grain is shown in Fig. 2. The indentation load increases with increasing indentation depth. During the constant force phase, the probe slightly deepens into the material, by indicating a viscous behavior. While unloading, the indentation load rapidly decreases and only elastic recovery load is measured. The gradient of the L–D curve in the
1622
Y. Sasaki et al. / Microelectronics Reliability 50 (2010) 1621–1625
Fig. 1. Load conditions for the extraction of the viscoplastic properties. Fig. 3. FEM model. Table 1 Loading cycle parameters.
Cycl. 1 Cycl. 2
Loading time (s)
Hold time (s)
Unloading time (s)
5 100
2 20
5 2.5
Fig. 2. Measured L–D curve in an aluminum single grain.
loading phase is different than the gradient in the unloading phase. This is an indication that plastic deformation occurs. 2.2. Simulation The FEM simulation has been carried out by the non-linear structural analysis tool of the MSC MARC. 2.2.1. FEM modeling Although the most common indenter probes are of the Berkovich type, a trigonal–pyramidal shape is not suited for numerical simulation, since it produces numerical instabilities. Therefore, a conical shape with a 90° angle and 1 lm curvature radius has been assumed leading to the model shown in Fig. 3. For the diamond indenter a Young modulus of 1000 GPa has been assumed, resulting into a maximum strain of 2.5 10–4. Thus, since the load with a deformable indenter is the same as for rigid indenter, in the model the indenter is treated as a rigid body. The strain behavior of the surface strongly depends on the friction coefficient. In case of no friction (coefficient 0.0), the surface shows material pile-up. On the contrary, if the friction coefficient
exceeds 0.3, the surface shows sink-in, because the sample material is forced downwards by the indenter. Since experimental data exhibit pile-up of the sample material, no friction has been assumed for the simulation. The problem to be simulated is multiscale in nature. In fact, the model has to cover at the same time the indenter tip with a size of few micrometers, and the sample with the size of several millimeters. This imposes to use an adaptive meshing strategy to obtain the best simulation accuracy with the minimum computing time and memory usage. Since the nanoindentation mainly produces local effects around the tip, the optimum set of model width and height (W and H, resp.) in combination with the indentation depth D has been investigated. This parametric analysis has shown that the optimum ratios W/L and H/D are above ten and three, respectively. In this case, the simulation becomes independent on the boundary conditions. Generally, the analysis accuracy is reduced if the strain of meshes is very large. Therefore, an adaptive mesh technique, which divides large-strained mesh elements into small-strained mesh elements, is adopted. In particular, this happens if the inner mesh angle becomes less than 5° and over 175°. The optimum size of the refined area has been found to be twice the indentation depth. 2.2.2. Constitutive model For viscoplasticity, a power-law has been assumed as constitutive equation, which accounts for work hardening and viscosity
ry ¼ Aðe0 þ ep Þn þ Be_ mp e0 ¼ ðE=AÞ1=n1
ð1Þ ð2Þ
where ry is the yield stress, e0 the initial equivalent strain, ep the plastic strain, A the work hardening coefficient, n the work hardening exponent, e_ p the strain rate, B the strain rate coefficient, and m the strain rate exponent. The values of the Young modulus E and of the Poisson ratio have been assumed 67.7 GPa and 0.3, respectively. The coefficients A, n, B, and m are unknown parameters to be determined from the experimental data by FEM inverse analysis. Numerical analysis of the simulation data requires to introduce a characteristic value to represent the complex strain rate field around the tip. This is done by plotting the change of the indentation depth h during the hold period as a function of the hold time, yielding [5].
e_ ¼
1 dh h dt
ð3Þ
Furthermore, the applied stress r is assumed to be the applied load F divided by the projected tip area crosssection Ap.
Y. Sasaki et al. / Microelectronics Reliability 50 (2010) 1621–1625
r¼
F Ap
1623
ð4Þ
Ap for a conical tip is simply calculated from the contact radius a and the contact depth hc. 2
ð5Þ
2
ð6Þ
Ap ¼ pa2 þ pðhc r 2 Þ a ¼ pð2hq r hq Þ
Here, the contact depth hc is assumed to be equal to the total measured depth h. In Eqs. (5) and (6), r is the curvature radius of tip and hq the depth of spherical part. The linear relationship between the strain rate and applied stress is showed in Fig. 4 as a log–log plot. A strain rate exponent m is obtained from the gradient of stress– strain rate relationship. Fig. 6. Experimental data vs. simulation.
2.2.3. Inverse modeling Usually, material parameters are extracted from experimental load–indentations plots (L–D, t–D) by model based curve analysis techniques. The experience has shown that this traditional approach is too inaccurate and does not provide sufficient flexibility for the calibration of the sophisticated models, which are required for instance for polymer thin films. In order to face these limitations, an inverse modeling procedure assisted by FEM simulation
Fig. 7. Load condition for the extraction of viscosity parameters.
Fig. 4. Extraction of the strain rate exponent.
FEM analysis taking of L-D curve change parameters and recalculation
compared to experimental result by least square method
convergence
No
Yes decision of the parameters for constitutive equation Fig. 5. Scheme for FEM-assisted inverse modeling.
has been developed here to process the data obtained by nanoindentation. Fig. 5 shows the generic procedure used to extract the parameters of interest from the experimental data. According to the scheme in Fig. 6, L–D curves have been calculated as a function of the load and of the indentation depth, and compared with experimental result by the least square method. New L–D curves are then recalculated by refining the parameters A and n until the desired level of accuracy has been reached. The final result of the iteration is shown in Fig. 6. Here, the value extracted for A and n is 175 MPa and 0.30, respectively. The fact that the simulated data are in excellent agreement with the experimental measurement for (bulk) aluminum validates the accuracy of the proposed extraction technique. 3. Extraction of viscosity parameters The characterization of viscosity parameters is of paramount when investigating long-term loading effects at high temperatures. Acrylic resin films are known to exhibit a significant viscous behavior, therefore also in this case inverse modeling of nanoindentation data is needed to extract to calibrate the constitutive equations. 3.1. Experimental As in Section 2, nanoindentation measurements have been carried out by the Hysitron TI-950 system. Nanoindentation curves
1624
Y. Sasaki et al. / Microelectronics Reliability 50 (2010) 1621–1625
have been acquired at 40 °C and 50 °C by diamond spherical indenter with a tip curvature radius of 20 lm. Fig. 7 shows the loading conditions for the constant load method that has been used. A step load and unload of 10 mN is applied for 10 s, while the hold time at maximum load is 300 s. Fig. 10 shows the measured L–t curve. During the constant load, the indentation depth respectively increases up to 1–1.5 lm evidencing the creeping behavior.
E1* η1
E2*
3.2. Simulation procedure
η2
In this Section the constitutive equations are first calibrated according to the traditional techniques for comparison. Then the same parameters are extracted by the inverse modeling procedure presented in Section 2.2.3. Fig. 9. Maxwell–Voigt model.
3.2.1. Power law equation The constitutive equation for the temperature dependence of the viscoelasticity in polymer thin films is expressed by (R is the gas constant) [6].
e_ ¼ C rn exp
Q RT
ð7Þ
Here the parameters to be calibrated by the nanoindentation measurements at different temperatures T are the coefficient C, the exponent n, and the activation energy Q. Additional material parameters measured by bend test are listed in Table 2. 3.2.2. Traditional extraction procedure At first, the stress exponent n is determined by the same procedure as used in Section 2.2.2 for single aluminum grains. The projected area of the spherical tip is
Ap ¼ pa2
ð8Þ
where the contact radius a is obtained by Eq. (6). The linear relationship between the strain rate and applied stress at both temper-
Table 2 Material parameters of the acrylic resin. Temperature (°C)
Young modulus (GPa)
Poisson ratio
40 50
2.166 1.470
0.4 0.4
Fig. 10. Experimental data and power-law vs. Maxwell–Voigt model.
atures is shown in Fig. 8 (top) in a log–log representation, which is used to extract the parameter n. The strain rate during the hold time is calculated from the change in time of the indentation depth and plotted in the Arrhenius representation in Fig. 8 (bottom) for the extraction of the activation energy Q. The values extracted for n and Q are 4.46 and 132.6 kJ/mol, respectively. In Fig. 10, the simulation results using the parameters extracted above is compared with the experimental data. Here, the temperature-dependent parameter C has been assumed, being 1.7 10– 30 and 3.0 10–32 for 40 °C and 50 °C, respectively. 3.2.3. Maxwell–Voigt four-element model The accuracy of the inverse modeling procedure presented in Section 3.2.2 can be considerably improved by assuming a different set of constitutive equations. The model used in the following is the combination of elastic and viscous elements shown in Fig. 9, which results from the series connection of the Maxwell and Voigt models. The related constitutive equation solved for the indentation depth h is [7] 3=2
h
Fig. 8. Plots for the calibration of the constitutive equation.
ðtÞ ¼
E 2 3 P0 1 1 1 pffiffiffi þ 1 etg2 þ t 4 r E1 E2 g1
ð9Þ
where P0 is the steady applied load, r is the curvature radius of the tip. E* is the combination of elastic modulus and Poisson ratio. The elastic and viscous material properties are given in terms of E* and g.
Y. Sasaki et al. / Microelectronics Reliability 50 (2010) 1621–1625
E1 (GPa)
E2 (GPa)
g1 (GPa s)
g2 (GPa s)
based product design. Even in this case, the main parameters are extracted by FEM-assisted inverse modeling of the experimental data.
1.8 0.75
1.45 0.33
230 800
25 31
4.1. Experimental
Table 3 Parameters of the four-element model.
40 °C 50 °C
1625
The nanoindentation measurements have been carried out by the Hysitron TI-950 system at room temperature and with a diamond conical indenter, whose angle and curvature radius are 90° and 1 lm, respectively. The sample material is SUS304 stainless steel, which has been treated by a shaving process (removal of a thin metal layer from the surface of the blanked part). Fig. 11 shows the load condition, which consists of a loading phase with a loading rate of 0.02 mN/s of a hold phase with 2 s, duration and finally of an unload phase. The corresponding L–D curve is represented in Fig. 12. 4.2. Simulation The constitutive equation for plasticity used here for FEM analysis is the same power-law in Eqs. (1) and (2). The (bulk) value assumed for the Young modulus E is 197 GPa, for the Poisson ratio 0.3, for the work hardening coefficient A 1300 MPa, and for the work hardening exponent n 0.45. Strain rate effects are neglected. Inverse modeling by FEM simulation and least square method is carried out according to Section 2.2.3 to replicate the experimental data. The equivalent plastic strain is considered as a residual strain to be used as initial condition in the FEM simulation. Fig. 12 shows that if an initial equivalent plastic strain of 1.2 is assumed, the simulation curve is in excellent agreement with the experimental data. On the contrary, if no initial stress is assumed, the simulation strongly overestimates the indentation depth at a given load.
Fig. 11. Load condition for the extraction of the residual stress.
5. Summary and conclusions
ð10Þ
Nanoindentation in conjunction with a FEM-based inverse modeling scheme has been shown to be an applicable technique for the extraction of mechanical properties of microelectronic materials, in particular of thin films used for semiconductor devices and MEMS. A procedure has been proposed, to improve the reliability of MEMS based on the inverse modeling of experimental L–D curves. This scheme has been applied to extract accurately the viscoplastic properties of aluminum single grains, the viscoelastic properties of acrylic resin films, and the residual strain of SUS304. The proposed procedure has been validated successfully by comparison of experimental results with nanoindentation curves obtained from the constitutive equations.
ð11Þ
References
Fig. 12. L–D curve for the calibration of residual strain.
FEM simulation requires deriving the stress–strain relation from Eq. (9), i.e.
eðtÞ ¼ es þ ec ¼ r es ¼
r
E 1 1 1 t 2 1 e g2 þ t þ E1 E2 g1
E1
where es is elastic strain and ec the creep strain. According to the procedure presented in Section 2.2.3, the FEM inverse analysis has been applied to fit the experimental D–t curves in Fig. 10, delivering the parameters listed in Table 3. As shown in Fig. 10, the calibration based on the more realistic Maxwell–Voigt model leads to much more accurate calibration than the simpler constitutive equation in form of a power-law. 4. Extraction of residual strain Residual strain occurring during the processing of the devices is a very relevant parameter to be considered in the FEM simulation
[1] Beghini M, Bertini L, Fontanari V. Evaluation of the stress–strain curve of metallic materials by spherical indentation. Int J Solids Struct 2006;43: 2441–59. [2] Bouzakis K-D, Michailidis N. An accurate and fast approach for determining materials stress–strain curves by nanoindentation and its FEM-based simulation. Mater Charact 2006;56:147–57. [3] Beake B. Modelling indentation creep of polymers: a phenomenological approach. J Phys D: Appl Phys 2006;39:4478–85. [4] Li James CM. Impression creep and other localized tests. Mater Sci Eng A 2002;322:23–42. [5] Mayo MJ, NIX WD. A micro-indentation study of superplasticity in Pb, Sn, and Sn-38 wt% Pb. Acta Metall 1998;36:2183–92. [6] Sargent PM, Ashby MF. Indentation creep. J Mater Sci Technol 1992;8:594–601. [7] Fischer-C AC. A simple phenomenological approach to nanoindentation creep. Mater Sci Eng A 2004;385:74–82.
Contents lists available at ScienceDirect
Microelectronics Reliability journal homepage: www.elsevier.com/locate/microrel
Accurate extraction of the mechanical properties of thin films by nanoindentation for the design of reliable MEMS Yuji Sasaki a, Mauro Ciappa b,*, Takayuki Masunaga a, Wolfgang Fichtner b a b
Toshiba Corporation, Corporate Manufacturing Engineering Center, 33, Shin-Isogo-Cho, Isogo-Ku, Yokohama 235-0017, Japan Swiss Federal Institute of Technology (ETH), Integrated Systems Laboratory, ETH-Zentrum, CH-8092 Zurich, Switzerland
a r t i c l e
i n f o
Article history: Received 5 July 2010 Accepted 19 July 2010 Available online 21 August 2010
a b s t r a c t Traditional procedures for the extraction of mechanical properties of thin films by nanoindentation measurements have shown problems in terms of accuracy and in the ability to support sophisticated constitutive models. In this paper, an inverse modeling procedure based on finite element analysis is presented to solve these limitations. Finite element simulation is used to predict the relationships between the indentation load and depth. The developed approach is applied to extract the viscoplastic properties of aluminum single grain, the viscoelastic properties of acrylic resin films, and the residual strain in stainless steel. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction The demand of new techniques for accurate extraction of the mechanical properties related to the microscopic structure of thin films used for semiconductor devices and Micro Electro Mechanical Systems (MEMS) is increasing. This is mainly driven by the need to improve the structural designs and manufacturing processes by numerical analysis. Generally, bulk material properties can be extracted by tensile and bend tests. However, these are no longer possible in thin films. Usually, the mechanical properties at microscopic level are measured by nanoindentation, i.e. by forcing a diamond indenter into the film surface in order to acquire the relationship between indentation load and indentation depth (L–D curve). Forcing the indenter into the surface results not only into elastic but also in plastic and viscous deformation. The only parameter directly obtained from nanoindentation measurements is the Young’s modulus, which is extracted from the slope of the unloading L–D curve. Nevertheless, inelastic material properties are often indispensable to improve structural designs. In this respect, techniques to determine inelastic material properties from measured L–D curves are under development. At present, research covers plastic [1,2] and viscous property [3,4]. The stress and strain distribution beneath an indenter tip is quite complicated. Therefore, numerical inverse analysis of the experimental curves is used to extract the mechanical properties that are needed to calibrate accurately the constitutive equations. This paper, the procedure is presented to extract the viscoplastic properties of aluminum single grain, the viscoelastic properties * Corresponding author. Tel.: +41 446322436; fax: +41 446321194. E-mail address: [email protected] (M. Ciappa). 0026-2714/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.microrel.2010.07.119
of acrylic resin films, and the residual strain in stainless steel (SUS304).
2. Extraction of the viscoplastic parameters The structural design by numerical analysis requires knowing the viscoplastic properties of the materials to predict the stress and displacement in the device under operating conditions. Generally, viscoplastic properties are obtained from tensile tests. However, in the case of thin films, this is not feasible because of the large dimensional errors involved, as well as due to the impossibility to manufacture proper supports for the samples.
2.1. Experimental Nanoindentation measurements have been performed by the Hysitron TI-950 nanoindenter at a room temperature and with a diamond conical tip. The shape of the indenter has been chosen for axisymmetric FEM modeling. Both load and depth are continuously measured during the measurement cycle. Fig. 1 shows the two different load conditions with a loading rate of 1 mN/s and 0.05 mN/s, respectively. In both cases, the maximum load is 5 mN. In Table 1 the related loading, hold, and unloading time are listed in detail. The L–D curve measured in an aluminum single grain is shown in Fig. 2. The indentation load increases with increasing indentation depth. During the constant force phase, the probe slightly deepens into the material, by indicating a viscous behavior. While unloading, the indentation load rapidly decreases and only elastic recovery load is measured. The gradient of the L–D curve in the
1622
Y. Sasaki et al. / Microelectronics Reliability 50 (2010) 1621–1625
Fig. 1. Load conditions for the extraction of the viscoplastic properties. Fig. 3. FEM model. Table 1 Loading cycle parameters.
Cycl. 1 Cycl. 2
Loading time (s)
Hold time (s)
Unloading time (s)
5 100
2 20
5 2.5
Fig. 2. Measured L–D curve in an aluminum single grain.
loading phase is different than the gradient in the unloading phase. This is an indication that plastic deformation occurs. 2.2. Simulation The FEM simulation has been carried out by the non-linear structural analysis tool of the MSC MARC. 2.2.1. FEM modeling Although the most common indenter probes are of the Berkovich type, a trigonal–pyramidal shape is not suited for numerical simulation, since it produces numerical instabilities. Therefore, a conical shape with a 90° angle and 1 lm curvature radius has been assumed leading to the model shown in Fig. 3. For the diamond indenter a Young modulus of 1000 GPa has been assumed, resulting into a maximum strain of 2.5 10–4. Thus, since the load with a deformable indenter is the same as for rigid indenter, in the model the indenter is treated as a rigid body. The strain behavior of the surface strongly depends on the friction coefficient. In case of no friction (coefficient 0.0), the surface shows material pile-up. On the contrary, if the friction coefficient
exceeds 0.3, the surface shows sink-in, because the sample material is forced downwards by the indenter. Since experimental data exhibit pile-up of the sample material, no friction has been assumed for the simulation. The problem to be simulated is multiscale in nature. In fact, the model has to cover at the same time the indenter tip with a size of few micrometers, and the sample with the size of several millimeters. This imposes to use an adaptive meshing strategy to obtain the best simulation accuracy with the minimum computing time and memory usage. Since the nanoindentation mainly produces local effects around the tip, the optimum set of model width and height (W and H, resp.) in combination with the indentation depth D has been investigated. This parametric analysis has shown that the optimum ratios W/L and H/D are above ten and three, respectively. In this case, the simulation becomes independent on the boundary conditions. Generally, the analysis accuracy is reduced if the strain of meshes is very large. Therefore, an adaptive mesh technique, which divides large-strained mesh elements into small-strained mesh elements, is adopted. In particular, this happens if the inner mesh angle becomes less than 5° and over 175°. The optimum size of the refined area has been found to be twice the indentation depth. 2.2.2. Constitutive model For viscoplasticity, a power-law has been assumed as constitutive equation, which accounts for work hardening and viscosity
ry ¼ Aðe0 þ ep Þn þ Be_ mp e0 ¼ ðE=AÞ1=n1
ð1Þ ð2Þ
where ry is the yield stress, e0 the initial equivalent strain, ep the plastic strain, A the work hardening coefficient, n the work hardening exponent, e_ p the strain rate, B the strain rate coefficient, and m the strain rate exponent. The values of the Young modulus E and of the Poisson ratio have been assumed 67.7 GPa and 0.3, respectively. The coefficients A, n, B, and m are unknown parameters to be determined from the experimental data by FEM inverse analysis. Numerical analysis of the simulation data requires to introduce a characteristic value to represent the complex strain rate field around the tip. This is done by plotting the change of the indentation depth h during the hold period as a function of the hold time, yielding [5].
e_ ¼
1 dh h dt
ð3Þ
Furthermore, the applied stress r is assumed to be the applied load F divided by the projected tip area crosssection Ap.
Y. Sasaki et al. / Microelectronics Reliability 50 (2010) 1621–1625
r¼
F Ap
1623
ð4Þ
Ap for a conical tip is simply calculated from the contact radius a and the contact depth hc. 2
ð5Þ
2
ð6Þ
Ap ¼ pa2 þ pðhc r 2 Þ a ¼ pð2hq r hq Þ
Here, the contact depth hc is assumed to be equal to the total measured depth h. In Eqs. (5) and (6), r is the curvature radius of tip and hq the depth of spherical part. The linear relationship between the strain rate and applied stress is showed in Fig. 4 as a log–log plot. A strain rate exponent m is obtained from the gradient of stress– strain rate relationship. Fig. 6. Experimental data vs. simulation.
2.2.3. Inverse modeling Usually, material parameters are extracted from experimental load–indentations plots (L–D, t–D) by model based curve analysis techniques. The experience has shown that this traditional approach is too inaccurate and does not provide sufficient flexibility for the calibration of the sophisticated models, which are required for instance for polymer thin films. In order to face these limitations, an inverse modeling procedure assisted by FEM simulation
Fig. 7. Load condition for the extraction of viscosity parameters.
Fig. 4. Extraction of the strain rate exponent.
FEM analysis taking of L-D curve change parameters and recalculation
compared to experimental result by least square method
convergence
No
Yes decision of the parameters for constitutive equation Fig. 5. Scheme for FEM-assisted inverse modeling.
has been developed here to process the data obtained by nanoindentation. Fig. 5 shows the generic procedure used to extract the parameters of interest from the experimental data. According to the scheme in Fig. 6, L–D curves have been calculated as a function of the load and of the indentation depth, and compared with experimental result by the least square method. New L–D curves are then recalculated by refining the parameters A and n until the desired level of accuracy has been reached. The final result of the iteration is shown in Fig. 6. Here, the value extracted for A and n is 175 MPa and 0.30, respectively. The fact that the simulated data are in excellent agreement with the experimental measurement for (bulk) aluminum validates the accuracy of the proposed extraction technique. 3. Extraction of viscosity parameters The characterization of viscosity parameters is of paramount when investigating long-term loading effects at high temperatures. Acrylic resin films are known to exhibit a significant viscous behavior, therefore also in this case inverse modeling of nanoindentation data is needed to extract to calibrate the constitutive equations. 3.1. Experimental As in Section 2, nanoindentation measurements have been carried out by the Hysitron TI-950 system. Nanoindentation curves
1624
Y. Sasaki et al. / Microelectronics Reliability 50 (2010) 1621–1625
have been acquired at 40 °C and 50 °C by diamond spherical indenter with a tip curvature radius of 20 lm. Fig. 7 shows the loading conditions for the constant load method that has been used. A step load and unload of 10 mN is applied for 10 s, while the hold time at maximum load is 300 s. Fig. 10 shows the measured L–t curve. During the constant load, the indentation depth respectively increases up to 1–1.5 lm evidencing the creeping behavior.
E1* η1
E2*
3.2. Simulation procedure
η2
In this Section the constitutive equations are first calibrated according to the traditional techniques for comparison. Then the same parameters are extracted by the inverse modeling procedure presented in Section 2.2.3. Fig. 9. Maxwell–Voigt model.
3.2.1. Power law equation The constitutive equation for the temperature dependence of the viscoelasticity in polymer thin films is expressed by (R is the gas constant) [6].
e_ ¼ C rn exp
Q RT
ð7Þ
Here the parameters to be calibrated by the nanoindentation measurements at different temperatures T are the coefficient C, the exponent n, and the activation energy Q. Additional material parameters measured by bend test are listed in Table 2. 3.2.2. Traditional extraction procedure At first, the stress exponent n is determined by the same procedure as used in Section 2.2.2 for single aluminum grains. The projected area of the spherical tip is
Ap ¼ pa2
ð8Þ
where the contact radius a is obtained by Eq. (6). The linear relationship between the strain rate and applied stress at both temper-
Table 2 Material parameters of the acrylic resin. Temperature (°C)
Young modulus (GPa)
Poisson ratio
40 50
2.166 1.470
0.4 0.4
Fig. 10. Experimental data and power-law vs. Maxwell–Voigt model.
atures is shown in Fig. 8 (top) in a log–log representation, which is used to extract the parameter n. The strain rate during the hold time is calculated from the change in time of the indentation depth and plotted in the Arrhenius representation in Fig. 8 (bottom) for the extraction of the activation energy Q. The values extracted for n and Q are 4.46 and 132.6 kJ/mol, respectively. In Fig. 10, the simulation results using the parameters extracted above is compared with the experimental data. Here, the temperature-dependent parameter C has been assumed, being 1.7 10– 30 and 3.0 10–32 for 40 °C and 50 °C, respectively. 3.2.3. Maxwell–Voigt four-element model The accuracy of the inverse modeling procedure presented in Section 3.2.2 can be considerably improved by assuming a different set of constitutive equations. The model used in the following is the combination of elastic and viscous elements shown in Fig. 9, which results from the series connection of the Maxwell and Voigt models. The related constitutive equation solved for the indentation depth h is [7] 3=2
h
Fig. 8. Plots for the calibration of the constitutive equation.
ðtÞ ¼
E 2 3 P0 1 1 1 pffiffiffi þ 1 etg2 þ t 4 r E1 E2 g1
ð9Þ
where P0 is the steady applied load, r is the curvature radius of the tip. E* is the combination of elastic modulus and Poisson ratio. The elastic and viscous material properties are given in terms of E* and g.
Y. Sasaki et al. / Microelectronics Reliability 50 (2010) 1621–1625
E1 (GPa)
E2 (GPa)
g1 (GPa s)
g2 (GPa s)
based product design. Even in this case, the main parameters are extracted by FEM-assisted inverse modeling of the experimental data.
1.8 0.75
1.45 0.33
230 800
25 31
4.1. Experimental
Table 3 Parameters of the four-element model.
40 °C 50 °C
1625
The nanoindentation measurements have been carried out by the Hysitron TI-950 system at room temperature and with a diamond conical indenter, whose angle and curvature radius are 90° and 1 lm, respectively. The sample material is SUS304 stainless steel, which has been treated by a shaving process (removal of a thin metal layer from the surface of the blanked part). Fig. 11 shows the load condition, which consists of a loading phase with a loading rate of 0.02 mN/s of a hold phase with 2 s, duration and finally of an unload phase. The corresponding L–D curve is represented in Fig. 12. 4.2. Simulation The constitutive equation for plasticity used here for FEM analysis is the same power-law in Eqs. (1) and (2). The (bulk) value assumed for the Young modulus E is 197 GPa, for the Poisson ratio 0.3, for the work hardening coefficient A 1300 MPa, and for the work hardening exponent n 0.45. Strain rate effects are neglected. Inverse modeling by FEM simulation and least square method is carried out according to Section 2.2.3 to replicate the experimental data. The equivalent plastic strain is considered as a residual strain to be used as initial condition in the FEM simulation. Fig. 12 shows that if an initial equivalent plastic strain of 1.2 is assumed, the simulation curve is in excellent agreement with the experimental data. On the contrary, if no initial stress is assumed, the simulation strongly overestimates the indentation depth at a given load.
Fig. 11. Load condition for the extraction of the residual stress.
5. Summary and conclusions
ð10Þ
Nanoindentation in conjunction with a FEM-based inverse modeling scheme has been shown to be an applicable technique for the extraction of mechanical properties of microelectronic materials, in particular of thin films used for semiconductor devices and MEMS. A procedure has been proposed, to improve the reliability of MEMS based on the inverse modeling of experimental L–D curves. This scheme has been applied to extract accurately the viscoplastic properties of aluminum single grains, the viscoelastic properties of acrylic resin films, and the residual strain of SUS304. The proposed procedure has been validated successfully by comparison of experimental results with nanoindentation curves obtained from the constitutive equations.
ð11Þ
References
Fig. 12. L–D curve for the calibration of residual strain.
FEM simulation requires deriving the stress–strain relation from Eq. (9), i.e.
eðtÞ ¼ es þ ec ¼ r es ¼
r
E 1 1 1 t 2 1 e g2 þ t þ E1 E2 g1
E1
where es is elastic strain and ec the creep strain. According to the procedure presented in Section 2.2.3, the FEM inverse analysis has been applied to fit the experimental D–t curves in Fig. 10, delivering the parameters listed in Table 3. As shown in Fig. 10, the calibration based on the more realistic Maxwell–Voigt model leads to much more accurate calibration than the simpler constitutive equation in form of a power-law. 4. Extraction of residual strain Residual strain occurring during the processing of the devices is a very relevant parameter to be considered in the FEM simulation
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