- Email: [email protected]
Study of nano-mechanical properties for thin porous films through instrumented indentation: SiO2 low dielectric constant films as an example
Microelectronic Engineering 85 (2008) 2172–2174
Contents lists available at ScienceDirect
Microelectronic Engineering journal homepage: www.elsevier.com/locate/mee
Study of nano-mechanical properties for thin porous films through instrumented indentation: SiO2 low dielectric constant films as an example M. Herrmann a,*, F. Richter a, S.E. Schulz b a b
Institute of Physics, Solid State Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany Center for Microtechnologies, Chemnitz University of Technology, 09107 Chemnitz, Germany
a r t i c l e
i n f o
Article history: Received 12 March 2008 Accepted 13 March 2008 Available online 18 March 2008 Keywords: Nanoindentation Porous low-k dielectrics Young’s modulus Yield stress
a b s t r a c t Much research has been focused on the mechanical properties of porous materials such as films of silica xerogels because of their potential for application to microelectronic interconnects. To accurately probe the film properties, one has to challenge with the porosity as well as the large differences between film and substrate properties. In this paper, a study is presented for the investigation of Young’s modulus and yield stress of these porous films by instrumented indentation under complete consideration of the substrate influence by using the approach of the ‘effectively shaped indenter concept’. This concept provides the basis of a more appropriate analysis for thin films in case of elastic–plastic contact situations as given for porous low-k films. It was found that the ratio of yield stress to Young’s modulus, which equals the yield strain of the stress–strain curve, is not constant and changes with porosity. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction A continuous reduction of the metal line width in microelectronic interconnects has driven the development of materials with a low dielectric constant. One promising class of these materials are the porous low-k dielectrics, which allow to reduce the k-value below 2.2 [1,2–4,5–9], whereas the introduction of pores leads to a dramatic reduction of the mechanical stability. Hence, successful integration of this new material into technology requires not only fulfilment of electrical parameters but also full compatibility to chemical mechanical polishing and other process steps which can lead to material failure such as crack formation or delamination [2,5,8,10–13,14]. Hence, a great interest exists to accurately probe the mechanical properties as well as to understand the material behaviour of porous materials under these loading conditions. To access the mechanical response, instrumented indentation experiments were carried out in order to investigate film properties like Young’s modulus (E) and yield stress (Y) which are more suitable to correlate mechanical stability and interconnect qualification than hardness. 2. Concept of the effectively shaped indenter The development of the effectively shaped indenter concept has contributed to a deeper understanding of the elastic–plastic contact during indentation and goes back to Pharr and co-workers * Corresponding author. Tel.: +49 371 531 33570. E-mail address: [email protected] (M. Herrmann). 0167-9317/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.mee.2008.03.006
[15,16]. An utilisation of the effective indenter became possible with the introduction of Schwarzer’s extended Hertzian approach (EHA) [17] in combination with the so-called image load method [18] into this concept. Therewith, the concept of the effectively shaped indenter can be applied to nanoindentation data to determine Young’s modulus and yield stress, when certain assumptions are fulfilled. In the following, a short introduction is given on how one has to use it in order to obtain the above-mentioned mechanical properties. To begin, the physical situation during indentation is as follows: if an indenter is pressed into a sample, the material undergoes elastic–plastic deformation, while it is assumed that during unloading only elastic recovery of the material takes place. The concept says that during elastic unloading the real indenter can be substituted by an effectively shaped indenter which has the same distance u(r) with respect to a flat elastic surface as the real indenter has regarding the plastically deformed surface (cf. Fig. 1). The concept is realised in the following steps: (i) a Young’s modulus is determined from the initial slope of the unloading curve following the method of Oliver–Pharr [19]. In case of a bulk sample, this procedure delivers us the sample modulus while for a film on a substrate this modulus is just an effective modules containing information of both parts. In step, (ii) we take the modulus obtained in (i) and do a second, more sophisticated fit using the unloading curve as a whole rather than only the initial slope. This provides us with the shape of the effective indenter as expressed in terms of a set of geometrical coefficients di (i = 0, 2, 4, 6) which are defined in the EHA. In step (iii) (for layered samples only) the second fit is repeated considering a film on a substrate instead of a homogeneous halfspace. If substrate modulus and film thickness
2173
M. Herrmann et al. / Microelectronic Engineering 85 (2008) 2172–2174
Fig. 1. Illustration of the G. Pharr’s ‘effectively shaped indenter’ concept.
are known, this gives us the Young’s modulus of the film. After all relevant Young’s moduli have been determined, with the known coefficients di of the effective indenter the surface pressure distribution under the indenter is obtained and from that, the complete stress and deformation field can be calculated for homogeneous [17] or layered [18,20,21] materials. This calculation can be done for any point of the elastic unloading curve including the state of maximum load which was still characterised also by plastic deformation. To analyse this situation, the von Mises yield criterion has been used which is written in the following form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 rM ¼ pffiffiffi ðr1 r2 Þ2 þ ðr2 r3 Þ2 þ ðr3 r1 Þ2 ¼ Y ðat yieldÞ; ð1Þ 2 where r1, r2, r3 are the principal stresses of the elastic field and Y is the material specific yield stress measured in an uniaxial tensile test. In principle, yielding occurs when the von Mises stress rM exceeds the critical value Y. If the residual stresses due to the plastic deformation are small, the stress state within the sample is mainly attributed to the elastic stresses. Therefore, it is assumed that the von Mises stress distribution corresponding to the elastic field of the effectively shaped indenter at maximum load adequately describes the real physical situation with respect to yield and the spatial maximum value of rM represents the yield strength. It was shown that this approach is valid for a variety of films as well as bulk materials [20,21]. To summarize, with this concept, experimental unloading curves of rather complex contact situations can be analysed as it is the case for low-k dielectric films on much stiffer substrates. 3. Experimental details A series of mesoporous SiO2 dielectrics films (xerogel films) were prepared by a conventional sol–gel method and spun onto silicon wafers [4]. Different film porosities were obtained by slightly varying the pre- and post-gelling time of the sol as well as the gelling ambient. To determine the film porosity, open and full, measurements have been performed by using spectroscopic ellipsometry (WVASE 32, Woollam Co) combined with an effective medium approach (EMA), ellipsometric porosimetry (EP) as well as X-ray reflectometry (XRR) for comparison purposes. The k-value was measured by a standard C–V measurement technique with a mercury probe. A detailed description of the investigations is given in a later publication of the authors.
Indentation experiments were carried out using a Berkovich indenter and spherical indenter of radius 48 lm with a UMIS 2000 (CSIRO, Australia) having a depth resolution of about 0.1 nm and a force resolution of about 0.75 lN. All indenters were made of diamond. The real indenter shapes in form of an area function (Berkovich, sphere) or radius function (sphere) as well as instrument stiffness have been determined by means of reference measurements in fused silica and sapphire. Measurements with varying maximum loads were carried out on every sample, where each of them was repeated 10–20 times. In a first step, standard indentations with a Berkovich indenter were carried out for a series of varying maximum loads with 0.1, 0.2, 0.3, 0.4, 0.5, 0.6 mN. Then, a modified Oliver–Pharr method was used where a modification was implemented with respect to radial displacement correction as a function of the ratio H/E and variable e factor [23]. The value of Poisson’s ratio was assumed to be 0.2 in all calculations. To determine the elastic modulus of the films, the effectively shaped indenter concept has been applied to the data obtained by the Berkovich data as discussed in Section 2. Therefore, only the upper 50% of the unloading curve were fitted to ensure that the recovery was dominated by elastic stresses. The complex fit procedures are implemented in a special software package which was used for the analysis [24]. To determine the yield stress of the porous films, a maximum load series consisting of 0.8, 1.0, 2.0, 4.0, 6.0, 8.0 and 10.0 mN have been used in combination with the 48 lm indenter. Again, the upper part of the unloading curve was fitted and therewith delivered the effective indenter shape as already mentioned above. The inspection of these load-depth curves showed nearly full elastic recovery for the lowest maximum loads. Hence, to ensure the dominance of elastic stresses during unloading, only the loaddepth curves at the lowest maximum loads have been used for the determination of the yield stress. 3. Results and discussion The approach of the effectively shaped indenter concept was used for the analysis of Young’s modulus and yield stress of the films, where the results are given in Table 1. The values for Young’s modulus lie between 2.7 GPa and 0.8 GPa and decrease with increasing porosity, whilst the yield stress values decrease from about 137 MPa to 73 MPa. The values of both properties are reduced with increasing porosity as it was expected. A deeper insight into the mechanical behaviour is provided by the ratio of yield stress to Young’s modulus, which is equal to the yield strain, eY. The resulting yield strain (eY = Y/E) is increased from 5% to 9% with porosity. This means that the maximum allowable strain during loading depends on the porosity, which leads to the necessity to consider the loading conditions more in detail. This issue is illustrated in Fig. 2 where the stress–strain curves have been plotted for the linear elastic range up to the yield point. It can be seen, that samples having higher porosities (i.e. sample 5) are able to carry higher strains than samples having lower porosities as long as the stresses are low. On the other hand, samples
Table 1 Overview of film properties Sample
Film thickness (nm)
k-value
Open porosity by EP (%)
Full porosity by EMA (%)
Effective film density by XRR (g/cm3)
Young’s modulus (GPa)
Yield stress (GPa)
X1 X2 X3 X4 X5
609 ± 6 714 ± 6 596 ± 5 678 ± 4 658 ± 6
2.63 2.56 2.42 2.38 2.08
38.9 39.7 44.7 44.5 51.4
43.04 ± 0.14 45.27 ± 0.13 47.65 ± 0.12 52.11 ± 0.13 57.37 ± 0.12
1.071 ± 0.013 1.024 ± 0.005 0.902 ± 0.005 0.827 ± 0.008 0.725 ± 0.003
2.67 2.35 1.91 1.42 0.83
0.137 0.122 0.118 0.096 0.073
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M. Herrmann et al. / Microelectronic Engineering 85 (2008) 2172–2174
0.20
X1 X2 X3 X4 X5
Stress (GPa)
0.15
0.10
0.05
0.00 0.00
0.02
0.04
0.06
0.08
0.10
Strain Fig. 2. Linear elastic range of the stress–strain curve for xerogel samples having different porosities. The dots mark the yield point.
von Mises stress (MPa)
4. Conclusion The concept of the effectively shaped indenter has been used for the determination of the film properties like E and Y which provide the basis for a deeper understanding of stress and strain states within the material during loading. Furthermore, it seems that shear stresses come into play due to the layered structure and combined lateral boundaries as well as thermal stresses.
300
200
Acknowledgements The authors highly acknowledge financial support by the Deutsche Forschungsgemeinschaft and the methodical support from N. Schwarzer (SIO, Eilenburg, Germany), and thank the Saxonian Institute of Surface Mechanics (SIO) for the support in using the Film Doctor software.
100
0 0.00
responding to the given structure have also to be taken into account to understand the occurrence of failures. To do so, the influence of a stiff barrier layer on the load-carrying capacity of the low-k-film-substrate stack in case of the contact with a spherical particle of a radius of 10 lm has been investigated (Fig. 3). The von Mises stress distribution was plotted along the indentation axis, where the onset of plastic flow would most likely occur. Nearly the same stress level is reached when the layer-substrate compounds are loaded with 1 mN. In both cases the plastic flow starts within the low-k film. The cap layer provides no improvement of the load-carrying capacity of the compound. But, the von Mises stress distribution shows a neutral fibre effect within the cap layer, which is attributed to the bending of the stiff cap layer into the low-k film as in case of a plate-like bending. Hence, other failure types have now to be considered due to tensile stresses within the top layer possibly leading to fracture or delamination due to high shear stresses at the interface.
0.25
0.50
0.75
1.00
References Fig. 3. Distribution of the von Mises stress along the indentation axis for (i) low-k film on silicon substrate, and (ii) cap layer / low-k film / silicon substrate. Nearly the same stress level is reached when the layer-substrate compounds are loaded with 1 mN. The cap layer provides no improvement of the load-carrying capacity of the compound. In both cases the plastic flow starts within the low-k film.
having low porosities can withstand higher stresses than samples with high porosities provided the strain is low. In conclusion, we observe a reverse dependence with respect to stress or strain controlled loading conditions which controls the mechanical deformation behaviour of the porous samples under load and has to be considered for a proper discussion of mechanical stability issues. The ratio of hardness to elastic modulus, H/E, is often used to characterise the resistance of samples against the onset of plastic deformation because its value is proportional to the yield strain given as eY = Y/E / H/E with H = C Y. However, this is only valid as long as the ratio of hardness to yield strength, C = H/Y, is constant. However, the direct determination of yield strain above has shown that its value varied with porosity and thus the H/E ratio cannot be constant in this case. Hence, the H/E-ratios should not directly be related to the yield strain for these materials. The loading conditions for xerogels embedded in integrated structures are in reality much more complex and thus real-acting stress distributions cor-
[1] [2] [3] [4]
[5] [6]
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [23] [24]
K. Maex, M.R. Baklanov, D. Shamiryan, et al., J. Appl. Phys. 93 (11) (2003) 8793. C. Jin, S. Lin, J.T. Wetzel, J. Electron. Mater. 30 (4) (2001) 284. A. Jain, S. Rogojevic, S. Ponoth, et al., Thin Solid Films 398–399 (2001) 513. S. Fruehauf, I. Streiter, S.E. Schulz, et al., Conf. Proc. ULSI XVII – Advanced Metallization Conference 2001 (AMC 2001), Materials Research Society, Warrendale, 2002, p. 287. A.K. Sikder, I.M. Irfan, A. Kumar, et al., J. Electron. Mater. 30 (12) (2001) 1527. S.E. Schulz, T. Gessner, in: S.P. Murarka, M. Eizenberg, A.K. Sinha (Eds.), Interlayer Dielectrics for Semiconductor Technologies, Elsevier Academic Press, 2003, p. 234. J.B. Vella, I.S. Adhihetty, K. Junker, et al., Int. J. Fract. 119/120 (2003) 487. S.Y. Chang, H.L. Chang, Y.C. Lu, et al., Thin Solid Films 460 (2004) 67. S.Y. Chang, T.J. Chou, Y.C. Lu, et al., J. Electrochem. Soc. 151 (2004) 146. A.A. Volinsky, J.B. Vella, W.W. Gerberich, Thin Solid Films 429 (2003) 201. S. Balakumar, X.T. Chen, Y.W. Chen, et al., Thin Solid Films 462–463 (2004) 161. Y.H. Wang, M.R. Moitreyee, R. Kumar, et al., Thin Solid Films 462–463 (2004) 227. L. Shen, K. Zeng, Microelectron. Eng. 71 (2004) 221. F. Iacopi, S.H. Brongersma, B. Vandevelde, et al., Microelectron. Eng. 75 (2004) 54. G.M. Pharr, A. Bolshakov, J. Mater. Res. 17 (2002) 2660. A. Bolshakov, W.C. Oliver, G.M. Pharr, MRS Symp. Proc. 356 (1995) 675. N. Schwarzer, J. Phys. D: Appl. Phys. 37 (2004) 2761. N. Schwarzer, ASME J. Tribol. 122 (2000) 672. W.C. Oliver, G.M. Pharr, J. Mater. Res. 7 (1992) 1564. N. Schwarzer, G.M. Pharr, Thin Solid Films 469–470 (2004) 194. N. Schwarzer, Thin Solid Films 494 (2006) 168. T. Chudoba, M. Griepentrog, Z. Metallkd. 96 (2005) 1242. FilmDoctor: software for the evaluation of the elastic field, available at: (contact: [email protected]).
Contents lists available at ScienceDirect
Microelectronic Engineering journal homepage: www.elsevier.com/locate/mee
Study of nano-mechanical properties for thin porous films through instrumented indentation: SiO2 low dielectric constant films as an example M. Herrmann a,*, F. Richter a, S.E. Schulz b a b
Institute of Physics, Solid State Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany Center for Microtechnologies, Chemnitz University of Technology, 09107 Chemnitz, Germany
a r t i c l e
i n f o
Article history: Received 12 March 2008 Accepted 13 March 2008 Available online 18 March 2008 Keywords: Nanoindentation Porous low-k dielectrics Young’s modulus Yield stress
a b s t r a c t Much research has been focused on the mechanical properties of porous materials such as films of silica xerogels because of their potential for application to microelectronic interconnects. To accurately probe the film properties, one has to challenge with the porosity as well as the large differences between film and substrate properties. In this paper, a study is presented for the investigation of Young’s modulus and yield stress of these porous films by instrumented indentation under complete consideration of the substrate influence by using the approach of the ‘effectively shaped indenter concept’. This concept provides the basis of a more appropriate analysis for thin films in case of elastic–plastic contact situations as given for porous low-k films. It was found that the ratio of yield stress to Young’s modulus, which equals the yield strain of the stress–strain curve, is not constant and changes with porosity. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction A continuous reduction of the metal line width in microelectronic interconnects has driven the development of materials with a low dielectric constant. One promising class of these materials are the porous low-k dielectrics, which allow to reduce the k-value below 2.2 [1,2–4,5–9], whereas the introduction of pores leads to a dramatic reduction of the mechanical stability. Hence, successful integration of this new material into technology requires not only fulfilment of electrical parameters but also full compatibility to chemical mechanical polishing and other process steps which can lead to material failure such as crack formation or delamination [2,5,8,10–13,14]. Hence, a great interest exists to accurately probe the mechanical properties as well as to understand the material behaviour of porous materials under these loading conditions. To access the mechanical response, instrumented indentation experiments were carried out in order to investigate film properties like Young’s modulus (E) and yield stress (Y) which are more suitable to correlate mechanical stability and interconnect qualification than hardness. 2. Concept of the effectively shaped indenter The development of the effectively shaped indenter concept has contributed to a deeper understanding of the elastic–plastic contact during indentation and goes back to Pharr and co-workers * Corresponding author. Tel.: +49 371 531 33570. E-mail address: [email protected] (M. Herrmann). 0167-9317/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.mee.2008.03.006
[15,16]. An utilisation of the effective indenter became possible with the introduction of Schwarzer’s extended Hertzian approach (EHA) [17] in combination with the so-called image load method [18] into this concept. Therewith, the concept of the effectively shaped indenter can be applied to nanoindentation data to determine Young’s modulus and yield stress, when certain assumptions are fulfilled. In the following, a short introduction is given on how one has to use it in order to obtain the above-mentioned mechanical properties. To begin, the physical situation during indentation is as follows: if an indenter is pressed into a sample, the material undergoes elastic–plastic deformation, while it is assumed that during unloading only elastic recovery of the material takes place. The concept says that during elastic unloading the real indenter can be substituted by an effectively shaped indenter which has the same distance u(r) with respect to a flat elastic surface as the real indenter has regarding the plastically deformed surface (cf. Fig. 1). The concept is realised in the following steps: (i) a Young’s modulus is determined from the initial slope of the unloading curve following the method of Oliver–Pharr [19]. In case of a bulk sample, this procedure delivers us the sample modulus while for a film on a substrate this modulus is just an effective modules containing information of both parts. In step, (ii) we take the modulus obtained in (i) and do a second, more sophisticated fit using the unloading curve as a whole rather than only the initial slope. This provides us with the shape of the effective indenter as expressed in terms of a set of geometrical coefficients di (i = 0, 2, 4, 6) which are defined in the EHA. In step (iii) (for layered samples only) the second fit is repeated considering a film on a substrate instead of a homogeneous halfspace. If substrate modulus and film thickness
2173
M. Herrmann et al. / Microelectronic Engineering 85 (2008) 2172–2174
Fig. 1. Illustration of the G. Pharr’s ‘effectively shaped indenter’ concept.
are known, this gives us the Young’s modulus of the film. After all relevant Young’s moduli have been determined, with the known coefficients di of the effective indenter the surface pressure distribution under the indenter is obtained and from that, the complete stress and deformation field can be calculated for homogeneous [17] or layered [18,20,21] materials. This calculation can be done for any point of the elastic unloading curve including the state of maximum load which was still characterised also by plastic deformation. To analyse this situation, the von Mises yield criterion has been used which is written in the following form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 rM ¼ pffiffiffi ðr1 r2 Þ2 þ ðr2 r3 Þ2 þ ðr3 r1 Þ2 ¼ Y ðat yieldÞ; ð1Þ 2 where r1, r2, r3 are the principal stresses of the elastic field and Y is the material specific yield stress measured in an uniaxial tensile test. In principle, yielding occurs when the von Mises stress rM exceeds the critical value Y. If the residual stresses due to the plastic deformation are small, the stress state within the sample is mainly attributed to the elastic stresses. Therefore, it is assumed that the von Mises stress distribution corresponding to the elastic field of the effectively shaped indenter at maximum load adequately describes the real physical situation with respect to yield and the spatial maximum value of rM represents the yield strength. It was shown that this approach is valid for a variety of films as well as bulk materials [20,21]. To summarize, with this concept, experimental unloading curves of rather complex contact situations can be analysed as it is the case for low-k dielectric films on much stiffer substrates. 3. Experimental details A series of mesoporous SiO2 dielectrics films (xerogel films) were prepared by a conventional sol–gel method and spun onto silicon wafers [4]. Different film porosities were obtained by slightly varying the pre- and post-gelling time of the sol as well as the gelling ambient. To determine the film porosity, open and full, measurements have been performed by using spectroscopic ellipsometry (WVASE 32, Woollam Co) combined with an effective medium approach (EMA), ellipsometric porosimetry (EP) as well as X-ray reflectometry (XRR) for comparison purposes. The k-value was measured by a standard C–V measurement technique with a mercury probe. A detailed description of the investigations is given in a later publication of the authors.
Indentation experiments were carried out using a Berkovich indenter and spherical indenter of radius 48 lm with a UMIS 2000 (CSIRO, Australia) having a depth resolution of about 0.1 nm and a force resolution of about 0.75 lN. All indenters were made of diamond. The real indenter shapes in form of an area function (Berkovich, sphere) or radius function (sphere) as well as instrument stiffness have been determined by means of reference measurements in fused silica and sapphire. Measurements with varying maximum loads were carried out on every sample, where each of them was repeated 10–20 times. In a first step, standard indentations with a Berkovich indenter were carried out for a series of varying maximum loads with 0.1, 0.2, 0.3, 0.4, 0.5, 0.6 mN. Then, a modified Oliver–Pharr method was used where a modification was implemented with respect to radial displacement correction as a function of the ratio H/E and variable e factor [23]. The value of Poisson’s ratio was assumed to be 0.2 in all calculations. To determine the elastic modulus of the films, the effectively shaped indenter concept has been applied to the data obtained by the Berkovich data as discussed in Section 2. Therefore, only the upper 50% of the unloading curve were fitted to ensure that the recovery was dominated by elastic stresses. The complex fit procedures are implemented in a special software package which was used for the analysis [24]. To determine the yield stress of the porous films, a maximum load series consisting of 0.8, 1.0, 2.0, 4.0, 6.0, 8.0 and 10.0 mN have been used in combination with the 48 lm indenter. Again, the upper part of the unloading curve was fitted and therewith delivered the effective indenter shape as already mentioned above. The inspection of these load-depth curves showed nearly full elastic recovery for the lowest maximum loads. Hence, to ensure the dominance of elastic stresses during unloading, only the loaddepth curves at the lowest maximum loads have been used for the determination of the yield stress. 3. Results and discussion The approach of the effectively shaped indenter concept was used for the analysis of Young’s modulus and yield stress of the films, where the results are given in Table 1. The values for Young’s modulus lie between 2.7 GPa and 0.8 GPa and decrease with increasing porosity, whilst the yield stress values decrease from about 137 MPa to 73 MPa. The values of both properties are reduced with increasing porosity as it was expected. A deeper insight into the mechanical behaviour is provided by the ratio of yield stress to Young’s modulus, which is equal to the yield strain, eY. The resulting yield strain (eY = Y/E) is increased from 5% to 9% with porosity. This means that the maximum allowable strain during loading depends on the porosity, which leads to the necessity to consider the loading conditions more in detail. This issue is illustrated in Fig. 2 where the stress–strain curves have been plotted for the linear elastic range up to the yield point. It can be seen, that samples having higher porosities (i.e. sample 5) are able to carry higher strains than samples having lower porosities as long as the stresses are low. On the other hand, samples
Table 1 Overview of film properties Sample
Film thickness (nm)
k-value
Open porosity by EP (%)
Full porosity by EMA (%)
Effective film density by XRR (g/cm3)
Young’s modulus (GPa)
Yield stress (GPa)
X1 X2 X3 X4 X5
609 ± 6 714 ± 6 596 ± 5 678 ± 4 658 ± 6
2.63 2.56 2.42 2.38 2.08
38.9 39.7 44.7 44.5 51.4
43.04 ± 0.14 45.27 ± 0.13 47.65 ± 0.12 52.11 ± 0.13 57.37 ± 0.12
1.071 ± 0.013 1.024 ± 0.005 0.902 ± 0.005 0.827 ± 0.008 0.725 ± 0.003
2.67 2.35 1.91 1.42 0.83
0.137 0.122 0.118 0.096 0.073
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M. Herrmann et al. / Microelectronic Engineering 85 (2008) 2172–2174
0.20
X1 X2 X3 X4 X5
Stress (GPa)
0.15
0.10
0.05
0.00 0.00
0.02
0.04
0.06
0.08
0.10
Strain Fig. 2. Linear elastic range of the stress–strain curve for xerogel samples having different porosities. The dots mark the yield point.
von Mises stress (MPa)
4. Conclusion The concept of the effectively shaped indenter has been used for the determination of the film properties like E and Y which provide the basis for a deeper understanding of stress and strain states within the material during loading. Furthermore, it seems that shear stresses come into play due to the layered structure and combined lateral boundaries as well as thermal stresses.
300
200
Acknowledgements The authors highly acknowledge financial support by the Deutsche Forschungsgemeinschaft and the methodical support from N. Schwarzer (SIO, Eilenburg, Germany), and thank the Saxonian Institute of Surface Mechanics (SIO) for the support in using the Film Doctor software.
100
0 0.00
responding to the given structure have also to be taken into account to understand the occurrence of failures. To do so, the influence of a stiff barrier layer on the load-carrying capacity of the low-k-film-substrate stack in case of the contact with a spherical particle of a radius of 10 lm has been investigated (Fig. 3). The von Mises stress distribution was plotted along the indentation axis, where the onset of plastic flow would most likely occur. Nearly the same stress level is reached when the layer-substrate compounds are loaded with 1 mN. In both cases the plastic flow starts within the low-k film. The cap layer provides no improvement of the load-carrying capacity of the compound. But, the von Mises stress distribution shows a neutral fibre effect within the cap layer, which is attributed to the bending of the stiff cap layer into the low-k film as in case of a plate-like bending. Hence, other failure types have now to be considered due to tensile stresses within the top layer possibly leading to fracture or delamination due to high shear stresses at the interface.
0.25
0.50
0.75
1.00
References Fig. 3. Distribution of the von Mises stress along the indentation axis for (i) low-k film on silicon substrate, and (ii) cap layer / low-k film / silicon substrate. Nearly the same stress level is reached when the layer-substrate compounds are loaded with 1 mN. The cap layer provides no improvement of the load-carrying capacity of the compound. In both cases the plastic flow starts within the low-k film.
having low porosities can withstand higher stresses than samples with high porosities provided the strain is low. In conclusion, we observe a reverse dependence with respect to stress or strain controlled loading conditions which controls the mechanical deformation behaviour of the porous samples under load and has to be considered for a proper discussion of mechanical stability issues. The ratio of hardness to elastic modulus, H/E, is often used to characterise the resistance of samples against the onset of plastic deformation because its value is proportional to the yield strain given as eY = Y/E / H/E with H = C Y. However, this is only valid as long as the ratio of hardness to yield strength, C = H/Y, is constant. However, the direct determination of yield strain above has shown that its value varied with porosity and thus the H/E ratio cannot be constant in this case. Hence, the H/E-ratios should not directly be related to the yield strain for these materials. The loading conditions for xerogels embedded in integrated structures are in reality much more complex and thus real-acting stress distributions cor-
[1] [2] [3] [4]
[5] [6]
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [23] [24]
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