- Email: [email protected]
New possibilities of mechanical surface characterization with spherical indenters by comparison of experimental and theoretical results
Thin Solid Films 355±356 (1999) 284±289 www.elsevier.com/locate/tsf
New possibilities of mechanical surface characterization with spherical indenters by comparison of experimental and theoretical results T. Chudoba*, N. Schwarzer, F. Richter Technical University of Chemnitz, Institute of Physics, D-09107 Chemnitz, Germany
Abstract The topic of this work is a novel approach for the determination of mechanical properties of thin ®lms on a substrate based on the theoretical modeling of spherical indentation into a ®lm substrate system together with its adequate experimental realization. First, some results of a novel analytical solution were compared to results of ®nite element (FE) calculations and to ¯at punch models. Then, SiO2 layers on silicon were investigated as an example. The measured load-displacement curves for spherical indentations were compared to curves simulated by means of the theoretical model. When using appropriate elastic parameters of the ®lm and substrate a complete agreement can be achieved. Finally, the onset of plastic deformation within SiO2 was determined by a multiple partial unloading procedure with a 4 mm sphere. It was found that the load necessary to start plastic deformation in a 538 nm SiO2/Si system is only 60% of the value that was obtained for a 2007 nm SiO2 layer, although the plastic hardness is the same. Using the theoretical model it can be shown that the plastic deformation starts within the ®lm and that - despite the different critical loads measured - the critical von Mises stress is the same. q 1999 Elsevier Science S.A. All rights reserved. Keywords: Mechanical properties; Spherical indenter; Indentation measurements; Young'; s modulus
1. Introduction The topic of this work is a novel approach for the determination of mechanical properties of thin ®lms or systems of several ®lms on a substrate. The approach is based on the modeling of the indentation of a sphere into the ®lm/ substrate composite together with the corresponding experimental realization of that indentation. The theoretical model developed recently [1±4] represents the generalization of the indentation of a sphere into an in®nite homogeneous halfspace (Hertzian indentation [5]) to the indentation into an in®nite halfspace covered with one or more ®lms having different elastic properties. The model allows the analytical calculation of the complete elastic deformation and stress ®eld within the ®lm and the substrate. In particular it makes it possible to simulate the spherical indentation experiment, i.e. to calculate the loaddisplacement curve of the indenter for given elastic properties of the materials involved. This capability can be used to determine the elastic ®lm properties by a comparison between measured and calculated load-depth data. The application of a spherical indenter has the advantage * Corresponding author. Tel.: 149-371-531-3210; fax: 149-371-5313042. E-mail address: [email protected] (T. Chudoba)
that the indentation process can be split into an inelastic and a wholly elastic part that can be described with the analytical model. The model not only excludes accuracy problems at interfaces which might occur in FE calculations; not least it allows very time-ef®cient calculations. For instance, the calculation of a load-depth curve needs less than 5 s on a Pentium 300 MHz computer. The shape of interacting macroscopic bodies as well as the microscopic roughness of solid surfaces in mechanical contact may be approximated in many cases by interacting spheres. Therefore we believe that in the further course of this work it will be possible to use the model also for the description of more complex mechanical interactions in order to make progress towards the design of optimum ®lm systems for a particular mechanical interaction. In this paper, however, we will restrict ourselves to the presentation of some theoretical results for a single layer on a substrate. Further ®rst attempts are undertaken to con®rm the theoretical results by indentation measurements using a UMIS-2000 nanoindenter for different silicon oxide layers on silicon substrates. It is shown that a separation of the elastic ®lm properties from the measured compound data is possible and that the beginning of plastic deformation can be explained in dependence on the layer thickness.
0040-6090/99/$ - see front matter q 1999 Elsevier Science S.A. All rights reserved. PII: S00 40-6090(99)0044 5-9
T. Chudoba et al. / Thin Solid Films 355±356 (1999) 284±289
285
Table 1 Overview of the investigated layer and substrate materials Material
Thickness (nm)
Young's modulus (GPa)
Poisson's ratio
Fused silica Silicon (100) Sapphire (0001) Silicon oxide Silicon oxide Silicon oxide
Substrate Substrate Substrate 538 ^ 4 1012 ^ 2 2007 ^ 2
72 165 403 72 72 72
0.17 0.27 0.237 0.17 0.17 0.17
2. Experimental
3. Theoretical results
Indentations were made using a UMIS-2000 force-displacement measuring system. Details of this instrument have been published elsewhere [6]. The potential of the indentation system to allow highly accurate measurements is illustrated by the following parameters: displacement resolution 0.1 nm, internal noise uncertainty , 0.1 nm, force resolution 0.75 mN, and stage registration repeatability 0.2 mm. The samples were measured with different spherical diamond indenters of 4 and 49 mm radius. The exact radius was determined by means of reference materials utilizing a procedure described below. Load and depth were recorded simultaneously for a complete load-unload cycle. The contact force for the detection of the surface position was 0.02 mN. A software correction of the zero position was done to eliminate the small surface de¯ection at this force. The maximum force of 50 mN was held constant for only 1 s at maximum load in order to assure a short measurement time and thus reduce thermal drift problems. This was possible because the measurements were done in the elastic range. Between ®ve and ten indentations separated by a distance of 50 mm were made on each sample and averaged after zero point determination. The standard deviation for the depth was between 0.4 nm (at maximum depth of 117 nm) for 500 nm SiO2 on Si and 1.7 nm for fused silica (at maximum depth of 178 nm). This indicates that the in¯uence of the residual roughness could be minimized. Polished 4'' Si wafer were thermally oxidized in a water vapor atmosphere up to oxide thicknesses of about 500, 1000 and 2000 nm. The exact layer thickness was measured with an optical ®lm thickness probe FTP-500 (SENTECH Instruments). The results are given in Table 1.
At ®rst some theoretical results for certain ®lm-substrate combination will be presented in order to illustrate differences between coated and uncoated samples. Further, a comparison is made with ®nite element (FE) calculations to con®rm the reliability of the solution (formulas are given in Refs. [1±4]). The comparison is limited here to the radial and normal stress components. A detailed analysis will be published in a separate paper. For correspondence with the given FE results the calculation was done for the purely elastic case with the following parameters: indenter radius 200 mm, load 10 N, ®lm thickness 2.2 mm. The elastic properties of indenter, ®lm and substrate are given in Table 2. The contact radius at maximum load from the analytical solution was 20.7 mm. Fig. 1 shows the normal stress at the surface for the compound in relation to results for homogeneous samples with ®lm and substrate properties, respectively. The somewhat higher compliance of the layer in comparison to the substrate results in a smaller contact pressure in relation to an uncoated sample, while the contact radius is nearly unchanged. The agreement with the FE calculation is excellent. In Fig. 2 the radial stresses along the depth axis in the center of the impression are compared. Surprisingly the radial stress in the substrate for the coated sample is even
Table 2 Elastic parameter for the comparison with FE results Elastic parameter
Indenter
Film
Substrate
Young's modulus /GPa Poisson's ratio
1000 0.1
160 0.2
200 0.3
Fig. 1. Normal stress s z along the surface vs. distance r from the indentation center for a coated sample and homogeneous bodies with the same elastic properties as ®lm (160 GPa) and substrate (200 GPa). Additionally the result of a ®nite element calculation is shown.
286
T. Chudoba et al. / Thin Solid Films 355±356 (1999) 284±289
Fig. 2. Radial stress s r along the depth axis z in the center of a spherical impression for a coated sample and homogeneous bodies with the same elastic properties as ®lm (160 GPa) and substrate (200 GPa). For comparison the result of a ®nite element calculation is given.
higher than that of an uncoated sample although the layer has a lower modulus than the substrate. At the interface the stress is markedly reduced to a ®lm value that is lower than for either of the homogeneous samples. The reason for this effect is the mis®t of the elastic properties at the interface and the ideal adhesion between layer and substrate. The FE results agree with the analytical calculation. Because of the rough FE mesh in the substrate a small deviation exists at the interface. Some important consequences of the analytical solution for indentation experiments with layered samples are shown in the following ®gures. During an impression with a spherical indenter the load-depth function F(h) can be recorded and analyzed. The ®rst derivation of this function is the contact stiffness S that is often used for the calculation of the composite modulus Ec. Many models have been developed in the past to separate the Young's modulus of ®lm, Ef, and substrate, Es, [7±10] from the measured composite
Fig. 3. Normalized unloading stiffness vs. Young's modulus ratio of ®lm and substrate for spherical (our calculation) and ¯at punch geometry (taken from King [7]). Results for contact radius to ®lm thickness ratios of 0.5 and 2 are given.
modulus. However they are only empirical or approximation solutions or valid only for a ¯at punch geometry [7]. Fig. 3 shows the contact stiffness normalized by the stiffness of a homogeneous sample, S0, as a function of the modulus ratio of ®lm and substrate in comparison to the solutions of King [7] for a circular and a square ¯at punch. In contrast to us, King uses in his presentation simply the square root of the contact area A as the contact radius a. For better comparp ison we have therefore corrected his data by 1/ p in Fig. 3. Especially for a high Ef/Es ratio the stiffness for a spherical indenter is lower than for the punches because of the different stress distribution which has a stronger in¯uence of the layer. From the contact stiffness the composite modulus can be calculated by the formula p p S p 1 Ec 2 A This is done in Fig. 4 which shows Ec normalized by the reduced modulus Er of an uncoated sample as a function of the contact radius to ®lm thickness ratio a/t. The reduced modulus is de®ned by 1 1 2 n2i 1 2 n2s 1 Ei Es Er
2
and considers the indenter deformation with the values given in Table 2. Subscript i indicates the indenter and s the sample, n is the Poisson's ratio. The ®lm modulus has a strong in¯uence on the composite modulus and even for a very thin coating (high a/t ratio) the in¯uence of the ®lm cannot be neglected. Another effect of a layer is shown in Fig. 5. From the original Hertzian theory, i.e. indentation into a homogeneous body [5], follows a load-depth dependence F / h m with an exponent m 1:5. For a coated sample this exponent is no longer constant and changes in dependence on the Ef/Es and a/t ratios. In connection with that the contact depth to maximum depth ratio hc/h0 also varies. For a spherical indenter and a homogeneous body this ratio has a constant value of 0.5.
Fig. 4. Normalized composite modulus vs. contact radius to ®lm thickness ratio for different Young's modulus ratios of ®lm and substrate.
T. Chudoba et al. / Thin Solid Films 355±356 (1999) 284±289
Fig. 5. Exponent of the elastic load-depth function vs. contact radius to ®lm thickness ratio for a coated sample. Results for contact radius to ®lm thickness ratios of 0.25 and 4 are given.
4. Experimental results For the determination of the ®lm properties from the loaddepth data, the sphere radius and the instrument stiffness must be known as accurately as possible. No sphere is ideally shaped in practice and, especially in the small depth range investigated here, deviations from the nominal radius should be taken into consideration. Under the assumption that sphere radius R and instrument stiffness S are constant in the investigated depth and load range, both unknown values can be determined simultaneously by measuring two different materials with the reduced moduli Er1, Er2. Using equal force and the same indenter one gets the maximum indentation depths h1, h2 from which R and S can be calculated as follows !3 2 3 F2 1 1 2 2=3 3 R ÿ 2=3 4 Er2 h1 2 h2 3 Er1 S
h1 2
3 4
F 2=3 1=3 F l Er1 R
4
Fig. 6. Comparison of calculated to measured load-depth data for different materials.
287
This formulas are derived from the well known Hertzian [5] load-depth dependence under the assumption that the measured depth is the sum of indentation depth and the instrument deformation given by F/S. For high accuracy the elastic constants of the reference materials must be known very precise. Three materials were used: fused silica, Si(100) and sapphire (0001) single crystals. The advantage of the single crystalline materials is that the elastic constants are well known. However this advantage is accompanied by the disadvantage of elastic anisotropy. For an indentation with a sphere all crystallographic directions are involved in the deformation which reduces considerably the problem of anisotropy. In that case average values for Young's modulus and Poisson's ratio have to be used. They were calculated with elastic constants from the literature [11,12] according to the formulae of Hill [13]. Using the three substrate materials in pairs, three independent results for R and S can be obtained with equation (3) and (4). The arithmetic average of this results yield S 4900 ^ 200 mN/mm and R 49 ^ 1 mm. More details of the radius determination can be found in [14]. The result of this procedure is shown in Fig. 6 which compares the measured results to the loaddepth data calculated on the basis of the averaged R and S values. The mean deviation between calculated and measured depth during unloading is less than 1 nm for Si and sapphire and less than 2 nm for fused silica. Now the values for R and S can be used to con®rm results of the analytical theory for the load-depth data of coated samples. This is shown in Fig. 7 by using the same elastic properties for the SiO2 layer as for fused silica. The results in Fig. 7 show that the well-adhering oxide layer behaves completely elastic and the ®lm/substrate combination can be described with the model. At maximum depth the difference is less than 1.5 nm showing that, within the error of measurement, the SiO2 ®lm has the elastic properties of fused silica. A second experiment was carried out to determine the onset of plastic deformation in the SiO2 layers. Multiple partial unloading data with a 4 mm diamond sphere were
Fig. 7. Comparison of calculated to measured load-depth data for SiO2 on Si for different ®lm thickness values. The elastic parameter for the calculation are given in the ®gure.
288
T. Chudoba et al. / Thin Solid Films 355±356 (1999) 284±289
Fig. 9 the distribution of that stress is given along the axis of rotational symmetry for the three systems investigated. In each case the calculation was done for exactly that force at which plastic deformation started. In all three cases a maximum of s M occurred in the depth range between 390 and 440 nm. Despite the different forces the maximum value is one and the same within the error limits and represents obviously the critical s M for onset of plastic deformation. This shows that a thin surface layer with higher compliance (lower Young's modulus) may weaken the whole compound in relation to a homogeneous substrate. 5. Conclusions Fig. 8. Multiple partial unloading plot with the elastic-plastic transition for fused silica and a SiO2 layer on Si.
used. The penetration was measured at each of 40 load steps followed by a partial unloading sequence to 30% of the current maximum load. In the wholly elastic range, load and unload cycle agree and no depth difference for equal load can be observed. If plastic deformation starts the residual depth of an unload sequence differs more and more from the loading curve. This is shown in Fig. 8 for a fused silica sample and a 538 nm SiO2 layer on Si. Only the depth value for 70% unloading is shown in the unloading cycle. The point of ®rst deviation between loading and unloading cycle can be used for the determination of the force at which plastic deformation starts. The results are 22 mN for fused silica, 13.7 mN for the 538 nm SiO2 layer and 22.7 mN for the 2007 nm SiO2 layer. The remarkable and somewhat surprising result is that a much reduced force is required for onset of plastic deformation in case of the thinner SiO2 layer despite the fact that all three SiO2 materials (fused silica, 538 and 2007 nm SiO2 ®lms) had similar plastic hardness values within the (9.5 ^ 0.3) GPa range. To ®nd an explanation for that the von Mises comparison stress s M was calculated using our analytical model. In
A recently developed analytical solution for a Hertzian pressure distribution on a coated half space allows the complete calculation of stresses and deformations due to indentations with a sphere. Some results were compared with FE calculations and show a good agreement. The load-displacement behavior (contact stiffness, exponent of the load-depth function) and the position of the stress maximum for coated samples differ signi®cantly from homogeneous bodies. Experimental load-depth data from a UMIS2000 nanoindenter with a 49 mm radius spherical diamond indenter can be well described with the analytical model. Therefore elastic indentation experiments with a spherical indenter are a new possibility for the determination of Young's modulus of thin ®lms. Further it was shown, that the onset of plastic deformation can be determined by a combination of indentation experiments and stress calculations. The results for SiO2 layers on Si indicate that a thin layer may weaken the whole compound in comparison to a homogeneous substrate with ®lm properties. Acknowledgements The authors wish to thank A.C. Fisher-Cripps form the University of Technology, Sydney for making the FE results available. References
Fig. 9. Von Mises comparison stress along the depth-axis for fused silica and SiO2 layers on Si during indentation of a 4 mm radius diamond indenter and calculated for the respective critical load values.
[1] N. Schwarzer, M. Whittling, M. Swain, F. Richter, Thin Solid Films 270 (1995) 371. [2] N. Schwarzer, T. Chudoba, D. Billep, F. Richter, Investigation of coating substrate compounds using inclined spherical indentation, Surf. Coat. Technol (1999). [3] N. Schwarzer, F. Richter, G. Hecht, Elastic Field in a Coated Half Space under Hertzian Pressure Distribution, Surf. Coat. Technol. (1999). [4] N. Schwarzer, Arbitrary contact on a layered half space of transverse or complete isotropy, Int. J. Solids Structures (1999). [5] H. Hertz, Aus den Verhandlungen des Vereins zur FoÈrderung des Gewerbe¯eib es, Berlin, 1882, pp. 174±196. [6] T.J. Bell, A. Bendeli, J.S. Field, M.V. Swain, E.G. Thwaite, Metrologia 28 (1991) 463. [7] R.B. King, Int. J. Solids Structures 23 (1987) 1657. [8] H. Gao, C. -H, Chiu, , J. Jee, Int. J. Solids Structures 29 (1992) 2471.
T. Chudoba et al. / Thin Solid Films 355±356 (1999) 284±289 [9] M.F. Doerner, W.D. Nix, J. Mater. Sci. 1 (1986) 601. [10] J. Mencik, D. Munz, E. Quandt, E.R. Weppelmann, J. Mater. Res. 12 (9) (1997) 2475. [11] W.C. O'Mara, R.B. Herring, L.P. Hunt (Eds.), Handbook of Semiconductor Silicon Technology, Noyes Publications, Park Ridge, NJ. [12] G. Simmons, H. Wang, Single Crystal Elastic Constants and Calcu-
289
lated Aggregate Properties: A Handbook, 2, MIT Press, Cambridge, MA, 1991. [13] R. Hill, Proc. Phys. Soc. A 65 (1952) 349. [14] T. Chudoba, N. Schwarzer, F. Richter, Determination of elastic properties of thin ®lms by indentation measurements with a spherical indenter, Surf. Coat. Technol (1999).
New possibilities of mechanical surface characterization with spherical indenters by comparison of experimental and theoretical results T. Chudoba*, N. Schwarzer, F. Richter Technical University of Chemnitz, Institute of Physics, D-09107 Chemnitz, Germany
Abstract The topic of this work is a novel approach for the determination of mechanical properties of thin ®lms on a substrate based on the theoretical modeling of spherical indentation into a ®lm substrate system together with its adequate experimental realization. First, some results of a novel analytical solution were compared to results of ®nite element (FE) calculations and to ¯at punch models. Then, SiO2 layers on silicon were investigated as an example. The measured load-displacement curves for spherical indentations were compared to curves simulated by means of the theoretical model. When using appropriate elastic parameters of the ®lm and substrate a complete agreement can be achieved. Finally, the onset of plastic deformation within SiO2 was determined by a multiple partial unloading procedure with a 4 mm sphere. It was found that the load necessary to start plastic deformation in a 538 nm SiO2/Si system is only 60% of the value that was obtained for a 2007 nm SiO2 layer, although the plastic hardness is the same. Using the theoretical model it can be shown that the plastic deformation starts within the ®lm and that - despite the different critical loads measured - the critical von Mises stress is the same. q 1999 Elsevier Science S.A. All rights reserved. Keywords: Mechanical properties; Spherical indenter; Indentation measurements; Young'; s modulus
1. Introduction The topic of this work is a novel approach for the determination of mechanical properties of thin ®lms or systems of several ®lms on a substrate. The approach is based on the modeling of the indentation of a sphere into the ®lm/ substrate composite together with the corresponding experimental realization of that indentation. The theoretical model developed recently [1±4] represents the generalization of the indentation of a sphere into an in®nite homogeneous halfspace (Hertzian indentation [5]) to the indentation into an in®nite halfspace covered with one or more ®lms having different elastic properties. The model allows the analytical calculation of the complete elastic deformation and stress ®eld within the ®lm and the substrate. In particular it makes it possible to simulate the spherical indentation experiment, i.e. to calculate the loaddisplacement curve of the indenter for given elastic properties of the materials involved. This capability can be used to determine the elastic ®lm properties by a comparison between measured and calculated load-depth data. The application of a spherical indenter has the advantage * Corresponding author. Tel.: 149-371-531-3210; fax: 149-371-5313042. E-mail address: [email protected] (T. Chudoba)
that the indentation process can be split into an inelastic and a wholly elastic part that can be described with the analytical model. The model not only excludes accuracy problems at interfaces which might occur in FE calculations; not least it allows very time-ef®cient calculations. For instance, the calculation of a load-depth curve needs less than 5 s on a Pentium 300 MHz computer. The shape of interacting macroscopic bodies as well as the microscopic roughness of solid surfaces in mechanical contact may be approximated in many cases by interacting spheres. Therefore we believe that in the further course of this work it will be possible to use the model also for the description of more complex mechanical interactions in order to make progress towards the design of optimum ®lm systems for a particular mechanical interaction. In this paper, however, we will restrict ourselves to the presentation of some theoretical results for a single layer on a substrate. Further ®rst attempts are undertaken to con®rm the theoretical results by indentation measurements using a UMIS-2000 nanoindenter for different silicon oxide layers on silicon substrates. It is shown that a separation of the elastic ®lm properties from the measured compound data is possible and that the beginning of plastic deformation can be explained in dependence on the layer thickness.
0040-6090/99/$ - see front matter q 1999 Elsevier Science S.A. All rights reserved. PII: S00 40-6090(99)0044 5-9
T. Chudoba et al. / Thin Solid Films 355±356 (1999) 284±289
285
Table 1 Overview of the investigated layer and substrate materials Material
Thickness (nm)
Young's modulus (GPa)
Poisson's ratio
Fused silica Silicon (100) Sapphire (0001) Silicon oxide Silicon oxide Silicon oxide
Substrate Substrate Substrate 538 ^ 4 1012 ^ 2 2007 ^ 2
72 165 403 72 72 72
0.17 0.27 0.237 0.17 0.17 0.17
2. Experimental
3. Theoretical results
Indentations were made using a UMIS-2000 force-displacement measuring system. Details of this instrument have been published elsewhere [6]. The potential of the indentation system to allow highly accurate measurements is illustrated by the following parameters: displacement resolution 0.1 nm, internal noise uncertainty , 0.1 nm, force resolution 0.75 mN, and stage registration repeatability 0.2 mm. The samples were measured with different spherical diamond indenters of 4 and 49 mm radius. The exact radius was determined by means of reference materials utilizing a procedure described below. Load and depth were recorded simultaneously for a complete load-unload cycle. The contact force for the detection of the surface position was 0.02 mN. A software correction of the zero position was done to eliminate the small surface de¯ection at this force. The maximum force of 50 mN was held constant for only 1 s at maximum load in order to assure a short measurement time and thus reduce thermal drift problems. This was possible because the measurements were done in the elastic range. Between ®ve and ten indentations separated by a distance of 50 mm were made on each sample and averaged after zero point determination. The standard deviation for the depth was between 0.4 nm (at maximum depth of 117 nm) for 500 nm SiO2 on Si and 1.7 nm for fused silica (at maximum depth of 178 nm). This indicates that the in¯uence of the residual roughness could be minimized. Polished 4'' Si wafer were thermally oxidized in a water vapor atmosphere up to oxide thicknesses of about 500, 1000 and 2000 nm. The exact layer thickness was measured with an optical ®lm thickness probe FTP-500 (SENTECH Instruments). The results are given in Table 1.
At ®rst some theoretical results for certain ®lm-substrate combination will be presented in order to illustrate differences between coated and uncoated samples. Further, a comparison is made with ®nite element (FE) calculations to con®rm the reliability of the solution (formulas are given in Refs. [1±4]). The comparison is limited here to the radial and normal stress components. A detailed analysis will be published in a separate paper. For correspondence with the given FE results the calculation was done for the purely elastic case with the following parameters: indenter radius 200 mm, load 10 N, ®lm thickness 2.2 mm. The elastic properties of indenter, ®lm and substrate are given in Table 2. The contact radius at maximum load from the analytical solution was 20.7 mm. Fig. 1 shows the normal stress at the surface for the compound in relation to results for homogeneous samples with ®lm and substrate properties, respectively. The somewhat higher compliance of the layer in comparison to the substrate results in a smaller contact pressure in relation to an uncoated sample, while the contact radius is nearly unchanged. The agreement with the FE calculation is excellent. In Fig. 2 the radial stresses along the depth axis in the center of the impression are compared. Surprisingly the radial stress in the substrate for the coated sample is even
Table 2 Elastic parameter for the comparison with FE results Elastic parameter
Indenter
Film
Substrate
Young's modulus /GPa Poisson's ratio
1000 0.1
160 0.2
200 0.3
Fig. 1. Normal stress s z along the surface vs. distance r from the indentation center for a coated sample and homogeneous bodies with the same elastic properties as ®lm (160 GPa) and substrate (200 GPa). Additionally the result of a ®nite element calculation is shown.
286
T. Chudoba et al. / Thin Solid Films 355±356 (1999) 284±289
Fig. 2. Radial stress s r along the depth axis z in the center of a spherical impression for a coated sample and homogeneous bodies with the same elastic properties as ®lm (160 GPa) and substrate (200 GPa). For comparison the result of a ®nite element calculation is given.
higher than that of an uncoated sample although the layer has a lower modulus than the substrate. At the interface the stress is markedly reduced to a ®lm value that is lower than for either of the homogeneous samples. The reason for this effect is the mis®t of the elastic properties at the interface and the ideal adhesion between layer and substrate. The FE results agree with the analytical calculation. Because of the rough FE mesh in the substrate a small deviation exists at the interface. Some important consequences of the analytical solution for indentation experiments with layered samples are shown in the following ®gures. During an impression with a spherical indenter the load-depth function F(h) can be recorded and analyzed. The ®rst derivation of this function is the contact stiffness S that is often used for the calculation of the composite modulus Ec. Many models have been developed in the past to separate the Young's modulus of ®lm, Ef, and substrate, Es, [7±10] from the measured composite
Fig. 3. Normalized unloading stiffness vs. Young's modulus ratio of ®lm and substrate for spherical (our calculation) and ¯at punch geometry (taken from King [7]). Results for contact radius to ®lm thickness ratios of 0.5 and 2 are given.
modulus. However they are only empirical or approximation solutions or valid only for a ¯at punch geometry [7]. Fig. 3 shows the contact stiffness normalized by the stiffness of a homogeneous sample, S0, as a function of the modulus ratio of ®lm and substrate in comparison to the solutions of King [7] for a circular and a square ¯at punch. In contrast to us, King uses in his presentation simply the square root of the contact area A as the contact radius a. For better comparp ison we have therefore corrected his data by 1/ p in Fig. 3. Especially for a high Ef/Es ratio the stiffness for a spherical indenter is lower than for the punches because of the different stress distribution which has a stronger in¯uence of the layer. From the contact stiffness the composite modulus can be calculated by the formula p p S p 1 Ec 2 A This is done in Fig. 4 which shows Ec normalized by the reduced modulus Er of an uncoated sample as a function of the contact radius to ®lm thickness ratio a/t. The reduced modulus is de®ned by 1 1 2 n2i 1 2 n2s 1 Ei Es Er
2
and considers the indenter deformation with the values given in Table 2. Subscript i indicates the indenter and s the sample, n is the Poisson's ratio. The ®lm modulus has a strong in¯uence on the composite modulus and even for a very thin coating (high a/t ratio) the in¯uence of the ®lm cannot be neglected. Another effect of a layer is shown in Fig. 5. From the original Hertzian theory, i.e. indentation into a homogeneous body [5], follows a load-depth dependence F / h m with an exponent m 1:5. For a coated sample this exponent is no longer constant and changes in dependence on the Ef/Es and a/t ratios. In connection with that the contact depth to maximum depth ratio hc/h0 also varies. For a spherical indenter and a homogeneous body this ratio has a constant value of 0.5.
Fig. 4. Normalized composite modulus vs. contact radius to ®lm thickness ratio for different Young's modulus ratios of ®lm and substrate.
T. Chudoba et al. / Thin Solid Films 355±356 (1999) 284±289
Fig. 5. Exponent of the elastic load-depth function vs. contact radius to ®lm thickness ratio for a coated sample. Results for contact radius to ®lm thickness ratios of 0.25 and 4 are given.
4. Experimental results For the determination of the ®lm properties from the loaddepth data, the sphere radius and the instrument stiffness must be known as accurately as possible. No sphere is ideally shaped in practice and, especially in the small depth range investigated here, deviations from the nominal radius should be taken into consideration. Under the assumption that sphere radius R and instrument stiffness S are constant in the investigated depth and load range, both unknown values can be determined simultaneously by measuring two different materials with the reduced moduli Er1, Er2. Using equal force and the same indenter one gets the maximum indentation depths h1, h2 from which R and S can be calculated as follows !3 2 3 F2 1 1 2 2=3 3 R ÿ 2=3 4 Er2 h1 2 h2 3 Er1 S
h1 2
3 4
F 2=3 1=3 F l Er1 R
4
Fig. 6. Comparison of calculated to measured load-depth data for different materials.
287
This formulas are derived from the well known Hertzian [5] load-depth dependence under the assumption that the measured depth is the sum of indentation depth and the instrument deformation given by F/S. For high accuracy the elastic constants of the reference materials must be known very precise. Three materials were used: fused silica, Si(100) and sapphire (0001) single crystals. The advantage of the single crystalline materials is that the elastic constants are well known. However this advantage is accompanied by the disadvantage of elastic anisotropy. For an indentation with a sphere all crystallographic directions are involved in the deformation which reduces considerably the problem of anisotropy. In that case average values for Young's modulus and Poisson's ratio have to be used. They were calculated with elastic constants from the literature [11,12] according to the formulae of Hill [13]. Using the three substrate materials in pairs, three independent results for R and S can be obtained with equation (3) and (4). The arithmetic average of this results yield S 4900 ^ 200 mN/mm and R 49 ^ 1 mm. More details of the radius determination can be found in [14]. The result of this procedure is shown in Fig. 6 which compares the measured results to the loaddepth data calculated on the basis of the averaged R and S values. The mean deviation between calculated and measured depth during unloading is less than 1 nm for Si and sapphire and less than 2 nm for fused silica. Now the values for R and S can be used to con®rm results of the analytical theory for the load-depth data of coated samples. This is shown in Fig. 7 by using the same elastic properties for the SiO2 layer as for fused silica. The results in Fig. 7 show that the well-adhering oxide layer behaves completely elastic and the ®lm/substrate combination can be described with the model. At maximum depth the difference is less than 1.5 nm showing that, within the error of measurement, the SiO2 ®lm has the elastic properties of fused silica. A second experiment was carried out to determine the onset of plastic deformation in the SiO2 layers. Multiple partial unloading data with a 4 mm diamond sphere were
Fig. 7. Comparison of calculated to measured load-depth data for SiO2 on Si for different ®lm thickness values. The elastic parameter for the calculation are given in the ®gure.
288
T. Chudoba et al. / Thin Solid Films 355±356 (1999) 284±289
Fig. 9 the distribution of that stress is given along the axis of rotational symmetry for the three systems investigated. In each case the calculation was done for exactly that force at which plastic deformation started. In all three cases a maximum of s M occurred in the depth range between 390 and 440 nm. Despite the different forces the maximum value is one and the same within the error limits and represents obviously the critical s M for onset of plastic deformation. This shows that a thin surface layer with higher compliance (lower Young's modulus) may weaken the whole compound in relation to a homogeneous substrate. 5. Conclusions Fig. 8. Multiple partial unloading plot with the elastic-plastic transition for fused silica and a SiO2 layer on Si.
used. The penetration was measured at each of 40 load steps followed by a partial unloading sequence to 30% of the current maximum load. In the wholly elastic range, load and unload cycle agree and no depth difference for equal load can be observed. If plastic deformation starts the residual depth of an unload sequence differs more and more from the loading curve. This is shown in Fig. 8 for a fused silica sample and a 538 nm SiO2 layer on Si. Only the depth value for 70% unloading is shown in the unloading cycle. The point of ®rst deviation between loading and unloading cycle can be used for the determination of the force at which plastic deformation starts. The results are 22 mN for fused silica, 13.7 mN for the 538 nm SiO2 layer and 22.7 mN for the 2007 nm SiO2 layer. The remarkable and somewhat surprising result is that a much reduced force is required for onset of plastic deformation in case of the thinner SiO2 layer despite the fact that all three SiO2 materials (fused silica, 538 and 2007 nm SiO2 ®lms) had similar plastic hardness values within the (9.5 ^ 0.3) GPa range. To ®nd an explanation for that the von Mises comparison stress s M was calculated using our analytical model. In
A recently developed analytical solution for a Hertzian pressure distribution on a coated half space allows the complete calculation of stresses and deformations due to indentations with a sphere. Some results were compared with FE calculations and show a good agreement. The load-displacement behavior (contact stiffness, exponent of the load-depth function) and the position of the stress maximum for coated samples differ signi®cantly from homogeneous bodies. Experimental load-depth data from a UMIS2000 nanoindenter with a 49 mm radius spherical diamond indenter can be well described with the analytical model. Therefore elastic indentation experiments with a spherical indenter are a new possibility for the determination of Young's modulus of thin ®lms. Further it was shown, that the onset of plastic deformation can be determined by a combination of indentation experiments and stress calculations. The results for SiO2 layers on Si indicate that a thin layer may weaken the whole compound in comparison to a homogeneous substrate with ®lm properties. Acknowledgements The authors wish to thank A.C. Fisher-Cripps form the University of Technology, Sydney for making the FE results available. References
Fig. 9. Von Mises comparison stress along the depth-axis for fused silica and SiO2 layers on Si during indentation of a 4 mm radius diamond indenter and calculated for the respective critical load values.
[1] N. Schwarzer, M. Whittling, M. Swain, F. Richter, Thin Solid Films 270 (1995) 371. [2] N. Schwarzer, T. Chudoba, D. Billep, F. Richter, Investigation of coating substrate compounds using inclined spherical indentation, Surf. Coat. Technol (1999). [3] N. Schwarzer, F. Richter, G. Hecht, Elastic Field in a Coated Half Space under Hertzian Pressure Distribution, Surf. Coat. Technol. (1999). [4] N. Schwarzer, Arbitrary contact on a layered half space of transverse or complete isotropy, Int. J. Solids Structures (1999). [5] H. Hertz, Aus den Verhandlungen des Vereins zur FoÈrderung des Gewerbe¯eib es, Berlin, 1882, pp. 174±196. [6] T.J. Bell, A. Bendeli, J.S. Field, M.V. Swain, E.G. Thwaite, Metrologia 28 (1991) 463. [7] R.B. King, Int. J. Solids Structures 23 (1987) 1657. [8] H. Gao, C. -H, Chiu, , J. Jee, Int. J. Solids Structures 29 (1992) 2471.
T. Chudoba et al. / Thin Solid Films 355±356 (1999) 284±289 [9] M.F. Doerner, W.D. Nix, J. Mater. Sci. 1 (1986) 601. [10] J. Mencik, D. Munz, E. Quandt, E.R. Weppelmann, J. Mater. Res. 12 (9) (1997) 2475. [11] W.C. O'Mara, R.B. Herring, L.P. Hunt (Eds.), Handbook of Semiconductor Silicon Technology, Noyes Publications, Park Ridge, NJ. [12] G. Simmons, H. Wang, Single Crystal Elastic Constants and Calcu-
289
lated Aggregate Properties: A Handbook, 2, MIT Press, Cambridge, MA, 1991. [13] R. Hill, Proc. Phys. Soc. A 65 (1952) 349. [14] T. Chudoba, N. Schwarzer, F. Richter, Determination of elastic properties of thin ®lms by indentation measurements with a spherical indenter, Surf. Coat. Technol (1999).