Effects of the substrate on the determination of thin film mechanical properties by nanoindentation

Effects of the substrate on the determination of thin film mechanical properties by nanoindentation

Acta Materialia 50 (2002) 23–38 www.elsevier.com/locate/actamat Effects of the substrate on the determination of thin film mechanical properties by n...

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Acta Materialia 50 (2002) 23–38 www.elsevier.com/locate/actamat

Effects of the substrate on the determination of thin film mechanical properties by nanoindentation Ranjana Saha, William D. Nix* Stanford University, Department of Materials Science and Engineering, Stanford, CA 94305-2205, USA Received 3 July 2001; received in revised form 5 September 2001; accepted 5 September 2001

Abstract We examine the effects of the substrate on the determination of mechanical properties of thin films by nanoindentation. The properties of aluminum and tungsten films on the following substrates have been studied: aluminum, glass, silicon and sapphire. By studying both soft films on hard substrates and hard films on soft substrates we are able to assess the effects of elastic and plastic inhomogeneity, as well as material pile-up, on the nanoindentation response. The data set includes Al/glass and W/sapphire, with the film and substrate having nearly the same elastic properties. These systems permit the true contact area and true hardness of the film to be determined from the measured contact stiffness, irrespective of the effects of pile-up or sink-in. Knowledge of the true hardness of the film permits a study of the effects of the elastic modulus mismatch on the nanoindentation properties, using the measured contact stiffness as a function of depth of indentation.  2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Nanoindentation; Thin films; Substrate effects; Aluminum; Tungsten

1. Introduction Determination of the mechanical properties of thin films on substrates by indentation has always been difficult because of the influence of the substrate on the measured properties [1–12]. The indentation response of a thin film on a substrate is a complex function of the elastic and plastic properties of both the film and substrate. The standard methods that are used for extracting properties from the measured load–displacement data were

* Corresponding author. Tel.: +1-650-725-2605; fax: +1650-725-4034. E-mail address: [email protected] (W. D. Nix).

developed primarily for monolithic materials [13]. These same methods are often applied to film/substrate systems for determining film properties without explicit consideration of how the substrate influences the measurements. In order to measure ‘film-only’ properties, a commonly used rule of thumb is to limit the indentation depth to less than 10% of the film thickness [13]. While using this rule is experimentally feasible for films that are greater than about a micrometer in thickness, this approach cannot be used for very thin films. Other methods must be developed for dealing with substrate effects if nanoindentation techniques are to be useful in the mechanical characterization of very thin films. Numerous investigators have used both experimental [14–22]

1359-6454/02/$22.00  2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 1 ) 0 0 3 2 8 - 7

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and theoretical [23–29] methods to study the problem of extracting ‘true’ film properties from nanoindentation of film/substrate composites. We have chosen to study a wide range of film/substrate systems, including both soft films on hard substrates and hard films on soft substrates. Specifically, we have studied the indentation properties of aluminum and tungsten films on the following substrates: aluminum, glass, silicon and sapphire. This set of film/substrate systems includes two, Al/glass and W/sapphire, which are nearly elastically homogeneous. For these film/substrate pairs, a measurement of the contact stiffness during indentation, together with knowledge of the elastic modulus of the system, permits a determination of the true contact area and true hardness, irrespective of the effects of pile-up or sink-in around the indenter. Knowledge of the true hardness of the film permits a study of the effects of the elastic modulus mismatch on the nanoindentation properties, using the measured contact stiffness as a function of depth of indentation relative to the film thickness for films on different substrates. The experimental results compare favorably with a modified form of King’s model for the effect of elastic modulus mismatch on the nanoindentation properties of thin films on substrates. By conducting experiments on all of these film/substrate pairs we are able to investigate some of the effects that the substrate may have on the determination of mechanical properties of thin films by nanoindentation.

2. Experiment 2.1. Thin film deposition Both Al and W were deposited onto the different substrates by sputtering. In the case of Al, the base pressure in the chamber prior to sputtering was 6– 10×10⫺9 Torr and the Ar gas pressure during sputtering was 1.5 mTorr. The sputtering conditions ˚ /s. For the W resulted in a deposition rate of 1.6 A deposition, the base pressure in the system prior to sputtering was 1×10⫺6 Torr or better and the Ar gas pressure during sputtering was 12 mTorr. The

sputtering conditions were 176 W of power with the current set at 0.5 A. Film thicknesses were measured using a Dektak IIa profilometer. The Al films had nominal thicknesses of 0.5, 1.0, 1.5, and 2.0 µm while the W films had nominal thicknesses of 0.64 and 2.16 µm. 2.2. Structural characterization The texture of these films was determined by xray diffraction. The Al films had a (111) texture, which is expected for an fcc metal sputter deposited at room temperature. All four films displayed the Al (111) peak and the second order Al (222) peak. The W films were primarily (110), which is also the expected film texture for a bcc metal sputter deposited at room temperature. Along with the W (110) and the second order W (220) peaks, the W (211) peaks were also present. 2.3. Nanoindentation The mechanical properties of the substrates and films were characterized using a Nano II and a Nano XP (MTS Nano Innovation Center, Oak Ridge, TN) with a Berkovich indenter tip. Both instruments were operated in the continuous stiffness mode (CSM) and the indentations were made using a constant nominal strain rate (P˙/P) of 0.05 s⫺1. Five or ten indentations were made in each sample and the results presented are an average of these indentations. Hardness and Young’s modulus were first determined using the Oliver and Pharr analysis [13]. In the discussion and figures the hardness and elastic modulus determined using the Oliver–Pharr analysis are referred to as H (O&P) and E (O&P), respectively. It should be noted that Oliver and Pharr [13] developed their method of analysis for monolithic materials. Nevertheless, the O&P method has become the standard method of analysis for nanoindentation and is frequently used for thin film studies. The present work is an attempt to determine the limitations of the O&P method of analysis for thin films on substrates. Another way of characterizing the mechanical properties of thin films is by analyzing the parameter P/S2 as a function of the depth of indentation relative to the film thickness. Here P is the load

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and S is the contact stiffness. Since the parameter involves only load and stiffness, which are directly measured during an indentation test, it is independent of indenter tip shape calibration and assumptions about material behavior. Joslin and Oliver [17] were the first to suggest that the parameter P/S2 is a material characteristic. Starting with the classical definition of hardness, H = P/A, we may write P ⫽ H∗A,

(1)

where A is the projected contact area. Using relations developed by Sneddon [30], the contact area in turn may be related to the measured contact stiffness by S⫽b

2 E √A, √p r

(2)

where b is a constant that depends on the geometry of the indenter and Er is the reduced modulus, expressed in terms of the elastic properties of the indenter (i) and film (f) as (1⫺n2i ) (1⫺n2f ) 1 ⫽ ⫹ . Er Ei Ef

(3)

This accounts for the effect of a non-rigid indenter on the load–displacement behavior. Eliminating contact area, A, from Eqs. (1) and (2), we get 1 p H P . 2 ⫽ 2 S b 4 E2r

Fig. 1. Plot of P/S2 of fused silica as a function of indentation depth. Since H and E of a bulk material like fused silica is independent of depth of indentation, P/S2 is constant with indentation depth.

(4)

We see that P/S2 is directly proportional to hardness, H, and inversely proportional to the square of the reduced modulus, Er. For homogeneous materials, H and E are usually constant with indentation depth. As a result P/S2 is also constant with indentation depth. This can be seen in Fig. 1, which is a plot of P/S2 as a function of indentation depth for fused silica. For a film/substrate, on the other hand, P/S2 will vary with depth depending on the hardness and elastic modulus of the film and substrate. Analyzing P/S2 provides a more complete picture of the nanoindentation properties of a film/substrate system than can be obtained using the Oliver–Pharr method alone.

2.4. Substrate characterization The substrates used in this study were: aluminum, glass, silicon and sapphire. The Al substrate was a piece of the Al–Mg alloy hard disk that is used in the data storage industry. A piece of a microscope glass slide was used as a glass substrate. The Si substrate used was a piece of a single crystal Si wafer with the (100) orientation. The sapphire substrate was also a single crystal with the basal plane (0001) orientation. The hardness and modulus of the bare substrates were calculated from nanoindentation data using the Oliver–Pharr method [13] and are plotted in Fig. 2. Mesarovic and Fleck [31] have shown that Poisson’s ratio has a minor effect on the indentation results. Hence, a Poisson’s ratio of 0.25 was used to calculate the Young’s modulus of the substrates. The results are an average of ten indentations made to a depth of 500 nm. Sapphire has the highest hardness of 30 GPa and the highest modulus of 440 GPa. Silicon is next with a hardness of 12.75 GPa and modulus of 172 GPa. Glass has a hardness of 6.8 GPa and its modulus is 73 GPa. Aluminum is the softest of the substrates with a hardness of 0.95 GPa and modulus of 75 GPa. Glass and Al have very similar moduli; we will make use of this in our analysis later.

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Fig. 3. Load vs. indentation depth for the 0.5 µm Al films on sapphire, silicon, glass, and aluminum substrates.

Fig. 2. (a) Hardness and (b) elastic modulus of the four substrates calculated using the Oliver–Pharr method.

3. Results 3.1. Aluminum thin films A series of indentations were made in the Al films with depths ranging from 250 nm to 2500 nm depending on the thickness of the film. The load and stiffness recorded as a function of indentation depth during the experiment were used to calculate the hardness and modulus of the films. Fig. 3 is a plot of the indentation load as a function of indenter displacement to a depth of 1000 nm for the 0.5 µm Al thin films on the four different substrates. The observed load–displacement relations are consistent with intuitive expectations. Fig. 4 is a plot of the contact stiffness as a function of the indenter displacement to a depth of 1000

Fig. 4. Plot of contact stiffness as a function of indentation depth for the 0.5 µm Al films on different substrates.

nm for the 0.5 µm Al thin films on different substrates. As shown in (Eq. (2)), stiffness is directly proportional to the reduced modulus and square root of the contact area. Hence for a homogeneous material, stiffness is expected to increase linearly with indentation depth since elastic modulus does not vary with depth. However, for a thin film on a substrate, if the film and substrate have different moduli, then the measured contact stiffness will deviate from linearity as the indentation depth increases. This is shown in Fig. 4 as a positive deviation from linearity for the stiffer substrates.

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Fig. 5 compares the hardness and modulus of the 0.5 µm Al film on the four different substrates. The results have been plotted as a function of the indentation depth normalized by the film thickness. Fig. 5(a) shows the hardnesses of these films calculated using the method of Oliver and Pharr [13]. The hardness is observed to decrease with increasing depth of indentation at extremely small depths (h/t⬍0.05). This indentation size effect is expected for soft metal films and has been related to strain gradient plasticity [32]. The hardness is constant and the same for all the substrates at depths between 0.05⬍h/t⬍0.2. This value of 0.6 GPa is close to that expected for an Al film. For 0.2⬍h/t⬍1, the hardness begins to increase with increasing depth of indentation. The observed

Fig. 5. (a) Hardness and (b) elastic modulus of 0.5 µm Al films on different substrates plotted as a function of the indentation depth normalized by the film thickness.

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increase in hardness might be explained by the influence of the substrate on the measured hardness, since the observed rise follows the relative hardnesses of the substrates. However, as explained below, the observed changes in hardness are an artifact of inaccuracies in the determination of the contact area during indentation. A more pronounced increase in hardness with increasing depth is observed for h/t⬎1. This is to be expected as the diamond starts to penetrate the hard substrates at such depths. Since the film is extremely soft compared to the substrate, it yields at a much smaller load compared to the substrate and all the plastic deformation occurs in the film when the indentation depth is less than the film thickness. The load (stress) required for the substrate to yield is reached only when the indenter is at the film/substrate interface. Fig. 5(b) is a plot of the Young’s moduli of the 0.5 µm Al films on different substrates determined using the method of Oliver and Pharr. The modulus of the film is around 65 GPa at extremely small depths (h/t⬍0.05) and then it increases with increasing depth of indentation. Again the rate of increase scales with the modulus of the substrates, i.e. the largest increase in modulus is observed in the case of the Al film on sapphire since it has the largest mismatch in modulus while the modulus of Al/Al is relatively constant. Compared to the hardness, the measured elastic moduli are more strongly affected by the substrate. Also the substrate influence is observed at a smaller indentation depth. This is to be expected because the elastic field under the indenter is not confined to the film itself; rather it is a long-range field that extends into the substrate especially when the film thickness is small. As a result, the substrate stiffness is observed to influence the measured contact stiffness at small depths of indentation, in this case ⬍50 nm. Fig. 6 compares the hardness and modulus of the 0.5, 1.0, 1.5, and 2.0 µm thick Al films on glass. The results obtained are similar to those shown in Fig. 5. The Young’s moduli for these films range from 50 to 90 GPa. Because Al and glass have very similar elastic moduli, between 70 and 73 GPa, the measured modulus is expected to be constant with indentation depth. The discrep-

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Fig. 7. P/S2 vs. indentation depth of the 0.5 µm Al films on the four different substrates. The P/S2 parameter scales inversely with the modulus of the substrate.

Fig. 6. (a) Hardness and (b) elastic modulus of different thickness Al films on glass plotted as a function of indentation depth normalized by the film thickness.

ancy between the measurements and the expectation is due to limitations in the Oliver–Pharr method for determining contact area. Since the Al films are extremely soft compared to the substrate, the material tends to pile-up along the sides of the indenter. The Oliver–Pharr method of calculating contact area does not account for this ‘extra’ area due to pile-up and hence underestimates the contact area and overestimates the hardness and elastic modulus. The limitations in the Oliver–Pharr method are fully explained below. A more accurate picture of the deformation of the Al films can be obtained by analyzing the P/S2 data. Fig. 7 is a plot of P/S2 as a function of indentation depth for the 0.5 µm Al films on different substrates. We observe that after the initial drop,

which can be attributed to an indentation size effect, P/S2 scales inversely with the modulus of the substrates i.e. Al/sapphire has the lowest P/S2 value and Al/Al and Al/glass have the highest value with Al/Si in between. This is expected because P/S2 is inversely proportional to E2r . P/S2 for Al/Al is constant with indentation depth. This is expected because Al/Al can be considered to be a homogeneous system, and as seen in the case of fused silica, P/S2 is constant with indentation depth for a homogeneous material. Al/glass is observed to have the same P/S2 value as that of Al/Al until the indentation depth is close to the film thickness and then it increases for indentation depths greater than the film thickness. The composites Al/Al and Al/glass are elastically homogeneous because Al and glass have very similar elastic moduli. But the similarity in P/S2 for indentation depths less than the film thickness indicates that the two films also have the same hardness for that displacement range. This implies that for Al/glass, the hard glass substrate is not influencing the deformation behavior of the composite and the soft Al film is accommodating all the plastic deformation until the indenter is close to the film/substrate interface. The parameter, P/S2, increases only when the indentation depth is equal to and greater than the film thickness because the indenter is now indenting the hard glass substrate. For the case of Al/Si and Al/sapphire, P/S2 is observed to decrease

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steadily until the indentation depth is close to the film thickness. This is because Er in these two cases is continuously increasing with indentation depth. As the indenter approaches the film/substrate interface, there is an increasing influence of the stiffer substrate on the measured contact stiffness. The rise in P/S2 with indentation depth greater than the film thickness can be attributed to the indentation of the harder substrate by the indenter. The hardness of the Al film on glass was observed to be constant with indentation depth until the indenter was close to the film/substrate interface, i.e. the hardness of the glass substrate did not influence the measured hardness of the Al film. This will also hold true for Al/Si and Al/sapphire since Si and sapphire are harder than glass. Hence from the analysis of the P/S2 data we expect the hardness of the Al films on the different substrates to be the same until the indentation depth is close to the film/substrate interface. This is in contrast to the Oliver–Pharr prediction of hardness, which was observed to gradually increase with indentation depth (Fig. 5(a)).

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A series of indentations were made in the 640 nm and 2160 nm W thin films. Fig. 8 compares the load–displacement response of the 640 nm W films on the four different substrates indented to a load of 150 mN. The observed load–displacement

relations are consistent with intuitive expectations. A close look at the load–displacement data of the W film on Al shows a significant decrease in the slope of the load–displacement curve at an indentation depth of 苲150 nm. This indicates yielding of the Al substrate. Fig. 9 compares the contact stiffness of the 640 nm W films on different substrates plotted as a function on indentation depth. The stiffness of the W film on sapphire is approximately linear with depth suggesting that elastic modulus of the W film is similar to that of the sapphire substrate. The negative deviations from linearity of the stiffness of the W films on silicon, glass and aluminum indicate that the film has a higher modulus than these substrates. Fig. 10 compares the hardness and elastic modulus of the 640 nm W film on the four different substrates. The results are plotted as a function of the indentation depth. Fig. 10(a) is a plot of the hardness calculated using the method of Oliver and Pharr [13]. We see that the hardness at small depths is between 13 and 14 GPa except for W/glass, which has a lower hardness. As the depth of indentation increases, the hardness decreases for the case of W on Al and glass, stays almost constant for the W film on Si, and increases for the case of W on sapphire. The observed behavior is expected based on the influence of the substrate hardness on the measured hardness. Fig. 10(b) is a plot of the Young’s moduli of

Fig. 8. Load vs. indentation depth of 640 nm W films on sapphire, silicon, glass, and aluminum substrates.

Fig. 9. Plot comparing the measured contact stiffness of the 640 nm W films on different substrates.

3.2. Tungsten thin films

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Fig. 10. (a) Hardness and (b) elastic modulus of 640 nm W films on different substrates.

the 640 nm W films on different substrates, again determined using the method of Oliver and Pharr. The modulus of the W films on different substrates do not ‘match-up’ at small indentation depths, as observed for the case of Al films on these substrates (Fig. 5(b)). The effect of the substrate is observed over the entire range of indentation depths. The probable reason for this is that the W film is significantly stiffer (bulk W has a modulus of 410 GPa) and harder than the Al film. As a result, at a given displacement, larger stresses develop in the W film than in the Al films. Also, Al, glass, and Si are compliant compared to W. Hence, the substrate effect is ‘seen’ sooner in the W film compared to the Al film. Fig. 11 compares the hardness and modulus calculated for the 640 nm and 2160 nm W films on sapphire substrates. The results have been plotted

Fig. 11. (a) Hardness and (b) elastic modulus of 640 nm and 2160 nm W films on sapphire substrates plotted as a function of indentation depth normalized by the film thickness.

as a function of the indentation depth normalized by the film thickness. Fig. 11(a) is a plot of the hardnesses for these two films. We see that the hardness at shallow depths is 14–15 GPa and increases with increasing depth of indentation. The W films are soft compared to the sapphire substrate, which has a hardness of 30 GPa. An increase in hardness is therefore expected, but only at larger displacements where the diamond indenter penetrates the sapphire substrate. The difference in hardness between the thick and thin films, and the increasing hardness with indentation depth is due to an incorrect contact area determination by the Oliver and Pharr analysis, as explained earlier and discussed below. The Young’s moduli of these films are plotted

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in Fig. 11(b) and lie between 400 and 440 GPa at shallow depths. The values then level off between 480 and 520 GPa for normalized indentation depth ratios greater than 0.3. The sapphire substrate has a modulus of 440 GPa and the literature value for the elastic modulus of bulk W is 410 GPa [13]. Again the discrepancy in the modulus measurement is due to the incorrect contact area determination. Fig. 12 is a plot of P/S2 as a function of the depth of indentation for the 640 nm W films on Al, glass, Si, and sapphire substrates. We observe that P/S2 for the case of W/sapphire is approximately constant for indentation depths less than the film thickness. This is again expected because W and sapphire have very similar moduli, resulting in a constant Er for the case of W/sapphire. Since W is softer than sapphire (HW=14 GPa and Hsapphire =30 GPa), the effect of the harder substrate on the measured hardness and hence on P/S2 would be observed only when the indenter penetrates the sapphire substrate. W and Si have very similar hardness. Hence P/S2 is influenced solely by Er for the case of W/Si. Si is more compliant (E=170 GPa) than W (E苲400 GPa). As a result, Er decreases as indentation depth increases, consistent with the observed increase in P/S2 with indentation depth for this case. Since Al and glass are also more compliant than W, one would expect P/S2 for the W film on these two substrates to also increase with indentation depth. This is observed in the case

Fig. 12. P/S2 of 640 nm W films on different substrates plotted as a function of indentation depth.

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of W/glass and in the case of W/Al the parameter P/S2 increases with indentation depth for h⬍100 nm. P/S2 for W/Al and W/glass are the same at indentation depths ⬍100 nm. This is to be expected because both Al and glass have the same elastic modulus and the measured hardness is the same for both these films at indentation depths ⬍100 nm. At indentation depths ⬎100 nm, P/S2 for W/Al starts to decrease. The likely explanation for this behavior is that the Al substrate begins to yield once the indentation depth is greater than 100 nm. Al has a hardness of only 1 GPa while the W film has a hardness of 13–14 GPa. Since P/S2 is directly proportional to H, when the substrate begins to yield, the effective hardness of the composite starts to decrease and hence P/S2 also decreases. This effect is also evident in the plot of hardness as a function indentation depth in Fig. 10(a). The hardness reaches a maximum for the W/Al composite at 苲100 nm followed by a sharp drop in hardness. A similar decrease in P/S2 (Fig. 12) should be observed for the case of W/glass. The absence of such an effect may be due to the less significant difference in hardness between the W film and glass substrate, relative to the case of W and Al. 4. Intrinsic film hardness calculated from P/S2 using the constant modulus assumption The problem with determining the intrinsic hardness of thin films is that the measurement is influenced by the properties of the substrates especially when the film is extremely thin. The hardness measurement of a soft film is enhanced by the hard substrate while the reverse is true for a hard film on a soft substrate. Another problem involves the calculation of the true contact area. Soft films on hard substrates tend to pile-up when indented, while hard films on soft substrates tend to sink-in. Hence the true contact depth is underestimated in the case of a soft film on a hard substrate and overestimated in the case of a hard film on a soft substrate, when compared to the contact depth calculated using the Oliver–Pharr method. Since contact area is a function of contact depth, the calculated hardness is overestimated for a soft film on a hard

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substrate and underestimated for a hard film on a soft substrate. In their analysis of indentation data, Oliver and Pharr assumed that there is always some downward elastic deflection (sink-in) of the material surface. However, this is not observed for soft films on hard substrates. Since the films are soft compared to the substrates, in some cases an order of magnitude softer than the substrate, material tends to pile-up around the indentation to a much greater degree than in the case of monolithic materials. This is due to the severe constraint imposed on plasticity in the film by the relatively undeformable substrate. Pile-up is clearly evident in Fig. 13(a) which is an AFM image of a 750 nm deep indentation into the 1.0 µm Al film on glass substrate and in Fig. 13(b) which is the plot of a section through the indentation. Significant pile-up is observed along the sides of the triangular indentation and the height of this pile-up is 377 nm, which is a significant fraction of the indentation depth. The actual contact area includes the extra area associated with the piled-up material. This area, which can contribute significantly to the load bearing capacity of the contact, is not accounted for in the Oliver–Pharr method of calculating contact area and leads to errors in the calculation of the true contact pressure or hardness. The true contact pressure or hardness of the film can be determined by accounting for the pile-up effects on the contact area. This can be accomplished by adopting the method of Joslin and Oliver [17] i.e. using the P/S2 parameter. This parameter can be used only when the material is elastically homogeneous and the indentation modulus is

known. For the problem of a film on a substrate, it is necessary to choose a film/substrate combination that is elastically homogeneous. Al/glass and W/sapphire satisfy this requirement. In both of these cases, the film and substrate have very similar elastic properties and these systems can be regarded as elastically homogeneous. Re-writing Eq. (4), we get

冉冊

P 4 H(E) ⫽ b2E2r Ł 2 ł. p S

(5)

Thus the true contact pressure or hardness may be determined by measuring the load and contact stiffness, even when pile-up occurs. Joslin and Oliver [17] were the first to use the P/S2 parameter in their study of the mechanical properties of ion implanted Ti alloys. They pointed out that since it is proportional to H, it is a direct measure of the true hardness. Page et al. [20] have also used this term to characterize coated systems. 4.1. Aluminum films The constant modulus assumption for determining film hardness can be applied to the Al films on glass. Using the known modulus of the Al/glass system, EAl⬇Eglass⬇73 GPa, the true hardness for this system is determined from the load, P, and contact stiffness, S, through the parameter P/S2. The true hardnesses for the four different film thicknesses calculated in this way are shown in Fig. 14, plotted as a function of the depth of indentation normalized by the film thickness. We see that at small depths of indentation the hardness falls with increasing depth, as expected on the

Fig. 13. (a) AFM image of a 750 nm indentation made in the 1.0 µm Al film on glass substrate. The light area around the edges of the triangle is the piled-up material. (b) Cross-section of the indentation showing the height of the pile-up.

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4.2. Tungsten films

Fig. 14. Hardness of the Al/glass films calculated using the constant modulus assumption, H(E), as a function of indentation depth normalized by the film thickness.

basis of indentation size effects in bulk materials. At deeper indentations the hardnesses reach constant values of about 0.6 GPa for the 0.5 and 1.0 µm film and 0.45 GPa for the 1.5 and 2.0 µm film. The hardnesses are approximately constant until the indentation depth is about 0.75 times the film thickness. Then the hardness starts to increase with increasing depth of indentation. This plateau in hardness and subsequent increase with increasing depth of indentation is not observed in bulk materials and may be caused by hardening associated with the strong gradients of plastic strain in the film between the indenter and the substrate [33– 35]. The continued increase in hardness with increasing depth of indentation after the indenter reaches the film/substrate interface is caused mainly by the penetration of the harder substrate by the diamond indenter. The observation of indentation size effects at small depths and substrateinduced strain gradient effects at large depths complicates the identification of the true plastic properties of the film. The plateaus we have identified in the hardness vs. depth curves may be influenced by the competition between these two strain gradient effects. However, we also note that the difference in ‘plateau’ hardness for the different film thicknesses (1 µm vs. 2 µm) are consistent with other measures of the yield strengths of thin Al films on substrates [33]. Thus we take these plateaus to be good estimates of the true properties of the film.

In the case of W films, W/sapphire satisfies the condition of elastic homogeneity. Using a modulus of EW⬇Esapphire⬇440 GPa, the true hardness for this composite is determined through the parameter P/S2. The true hardnesses for the 640 nm and 2160 nm W films are shown in Fig. 15 as a function of the indentation depth normalized by the film thickness. The behavior is similar to that observed for the case of Al/glass. Here too we see that at small depths of indentation, h/t⬍0.05, the hardness falls with increasing depth due to indentation size effects. At deeper indentations, the hardness reaches a constant value of 14–15 GPa for both the films. The hardnesses are constant until the indenter displacement is approximately 0.6 times the film thickness. Then the hardness starts to increase with increasing depth of indentation and is attributed to the hardening caused by strong gradients of plastic strain in the film between the indenter and the substrate [33–35]. An even greater increase in hardness is expected for indentation depths greater than the film thickness because of the penetration of the harder substrate by the diamond indenter.

Fig. 15. Hardness of the 640 nm and 2160 nm W/sapphire films calculated using the constant modulus assumption, H(E), as a function of indentation depth normalized by the film thickness.

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5. Modeling the elastic properties of thin films on substrates The Oliver–Pharr method [13] was developed for monolithic materials. Re-writing Eq. (3), the elastic contact modulus or reduced modulus, Er, is defined as (1⫺n2i ) (1⫺n2) 1 ⫽ ⫹ , Er Ei E

(6)

where ni and Ei are the Poisson’s ratio and elastic modulus, respectively, of the diamond indenter and v and E are the Poisson’s ratio and elastic modulus of the material being indented. This equation is applicable to a thin film on a substrate only if the film and substrate have the same elastic properties, i.e. Ef=Es and vf⬇vs. A difference in the Poisson’s ratio of the film and substrate has a small effect on the reduced modulus [31]. However, if the film and substrate are elastically inhomogeneous, then Eq. (6) returns the wrong film modulus, especially if the film is extremely thin and/or has a modulus very different from that of the substrate. This is because the substrate modulus also contributes to the measured elastic contact modulus and its influence on the contact modulus increases as the depth of indentation increases. Hence the Oliver and Pharr method cannot predict the correct film modulus unless the film and substrate have the same elastic properties.

layered isotropic elastic half-space. He modified the Doerner and Nix equation for the reduced modulus and defined it as at 1⫺n2i 1⫺n2f 1 ⫽ ⫹ (1⫺e⫺ a ) Er Ei Ef 1⫺n2s ⫺at ⫹ (e a ) Es

(7)

where a is the square root of the projected contact area, t is the thickness of the film below the punch and a is a numerically determined scaling parameter that is a function of a/t, the normalized punch size, and is different for different indenter geometries. Fig. 16 shows the variation of a with a/t for a triangular punch. The hollow circles are the numerically determined values by King. King’s values were fitted using the equation shown in the figure and extrapolated to larger ratios of a/t. Eq. (7) also varies between the limits of ‘film-only’ properties when a/t→0 and ‘substrate-only’ properties when a/t→a. The validity of Eq. (7) in determining the true film modulus was tested by analyzing the nanoindentation data obtained for the Al and W films on different substrates. 7. Aluminum films The Al films were deposited on substrates that either had similar moduli as that of the film (glass

6. King model Doerner and Nix [3] developed a model that attempted to account for the influence of the substrate compliance by including a term due to the substrate in the reduced modulus equation. However, the constants that scaled the relative contributions of the film and substrate were determined empirically and were valid only for the particular film on substrate case that they studied. King [26] modified the solution presented by Doerner and Nix and made it applicable to all film/substrate systems. He used numerical methods to study the problem of flat-ended cylindrical, quadrilateral, and triangular punches indenting a

Fig. 16. Plot of a as a function of normalized punch size, a/t. The result of the curve fit to King’s analysis is shown in the box and the a values calculated using this equation are plotted as a line.

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and Al) or were stiffer (Si and sapphire). Also, the hardnesses of the Al films were less than those of the substrates. Hence we can assume that the film accommodates all the plastic deformation and the substrate begins to yield only when the indenter is close to the film/substrate interface, as shown earlier. As a result we will also assume that: the hardness calculated for the Al/glass films using the constant modulus assumption is also the hardness of the Al films on sapphire, Si, and Al substrates. Once the hardness is known, we can calculate the variation in reduced modulus with indenter displacement for the Al films on sapphire, silicon, and aluminum substrates from the P/S2 data using the following equation. Er ⫽



1 p S2 H(E). b2 4 P

(8)

The true contact area can be calculated either from load, P, and hardness, H(E) or from the measured contact stiffness using A⫽

冉冊

1 p S 2 , b2 4 Er

(9)

once the correct reduced modulus is known. The reduced modulus, Er calculated using Eq. (8) and the true contact area calculated using Eq. (9) will be used to evaluate the solutions presented by King to calculate film modulus. We need to modify the analysis presented by King in order to use Eq. (7) to calculate the modulus of the Al film from the reduced modulus data. King assumed a flat triangular punch in his analysis, as shown in Fig. 17(a). However, we have an

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indenter with the Berkovich geometry (three-face pyramid). In order to extend King’s analysis to our case, we imagine that we too have a flat punch that is situated at the tip of the Berkovich indenter as shown in Fig. 17(b). Hence in our case the effective film thickness, ‘t’, is equal to (t⫺h) where h is the total indenter displacement. The modified solution is now a(t⫺h) 1⫺n2i 1⫺n2f 1 ⫽ ⫹ (1⫺e⫺ a ) Er Ei Ef 1⫺n2s ⫺a(t⫺h) (e a ). ⫹ Es

(10)

As before, the indenter properties ni and Ei are 0.07 and 1140 GPa, respectively, the Al film modulus, Ef, is 73 GPa, and the substrate modulus, Es, is 440 GPa for sapphire, 170 GPa for Si, and 73 GPa for Al. Poisson’s ratio of the film and substrate are assumed to be equal to 0.25. Fig. 18 is a plot of the reduced modulus, Er (King), predicted by Eq. (10) for the 0.5 µm and 2.0 µm Al films. Reduced moduli calculated from the experimental data using Eq. (8), Er (Expt.), is also shown for comparison. We see that Er is the same at very small displacements and is equal to the reduced modulus for just film and indenter. This indicates that at extremely shallow displacements, we measure the ‘film-only’ modulus. As the indenter displacement increases, the reduced modulus increases for the case of Al on sapphire and Si and remains constant for the case of Al on Al which is expected. We see that Er (King) matches Er (Expt.) reasonably well for indentation depths less than 50% of the film thickness. The match is better in case of the 2.0 µm film compared

Fig. 17. (a) King’s analysis assumes a flat punch indenting a film of thickness t. (b) In order to extend King’s analysis to our case, we assume the flat punch to be located at the tip of the Berkovich indenter. Hence the effective film thickness below the punch is (t⫺h).

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Fig. 18. Plot of reduced moduli of the (a) 0.5 µm and (b) 2.0 µm Al films on different substrates calculated using the modified King solution (Er (King)). Reduced moduli calculated from the experimental data, Er (Expt.) have also been plotted.

to the 0.5 µm film. We observe that the predicted reduced modulus is greater than the reduced modulus calculated from experimental data at large indentation depths. This implies that the modified King model is predicting a larger contribution of the substrate to the reduced modulus. As a result, if we used the modified King solution to calculate film modulus from Er (Expt.), the film modulus would be underestimated. This can be seen in Fig. 19, which is a plot of the predicted film modulus as a function of indentation depth. For indentation depths greater than 50% of the film thickness, the predicted modulus decreases as the depth of indentation increases. One of the reasons why the modified King solution does not work for the entire displacement range is that King’s model is based on

Fig. 19. Plot of film modulus predicted by the modified King solution for the (a) 0.5 µm and (b) 2.0 µm Al films on different substrates. The prediction matches the expected film modulus of 73 GPa reasonably well for indentation depths less than 50% of the film thickness.

a flat triangular punch whereas we have an indenter tip with the Berkovich geometry. If we consider the case when the indenter displacement is equal to the film thickness, then (t⫺h) is equal to zero, and we should measure only the substrate stiffness. However, because of the pyramidal shape of the indenter, only the tip of the indenter is at the film/substrate interface while the sides of the indenter are still in contact with the film. Hence the measured stiffness will be less than that of the substrate. Thus by assuming that the flat punch is at the tip of the Berkovich indenter, the present model overestimates the contribution of the substrate stiffness to the measured contact stiffness, thus underestimating the film modulus.

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8. Tungsten films The W films were deposited on substrates that were either more compliant than the film (Al, glass, and Si) or had a similar modulus (sapphire). Also the hardnesses of the W films were greater than the Al and glass substrates, equal to the Si substrate and less than the sapphire substrate. Hence for the case of W on Al and glass, the substrate will begin to yield well before the indenter tip reaches the film/substrate interface, as shown earlier. Because of this, the assumption made in the case of the Al films, that the film hardness, as a function of indentation depth, is constant until the indenter tip is close to the film/substrate interface, cannot be applied to the case of W on Al and glass. As a result the reduced moduli and true contact areas required for evaluating the elastic modulus models could not be calculated for the case of W on Al and glass. The modified King model was evaluated using the W/Si data. The hardness values required for calculating the correct reduced modulus using Eq. (8) could be obtained from two different sources: The W film and Si substrate have very similar hardnesses. Hence either of the two hardness could be used in Eq. (8). Or we could use the hardness calculated from the W/sapphire case using the constant modulus assumption. The film hardness was assumed to be 14 GPa and the reduced modulus and true contact areas were calculated for the 640 nm W film on Si. The film modulus calculated using the modified King solution, Eq. (10), is plotted in Fig. 20. The model does a reasonably good job of predicting the film modulus, especially at small indentation depths.

9. Summary The objective of this research was to study the effect of substrate properties on the nanoindentation measurement of film properties and to develop methods for extracting the intrinsic film properties from nanoindentation experiments. We tested and analyzed eight different film/substrate systems: Al and W films on aluminum, glass, silicon, and sapphire substrates. By depositing the same film on different substrates we were able to

Fig. 20. Plot of film modulus predicted by the modified King solution for the 640 nm W film on Si. The prediction matches the expected film modulus of 410 GPa reasonably well.

show how the substrate properties affect the measured film properties. We used this data to develop a model for extracting film hardness and elastic modulus from nanoindentation data. The effect of the substrate hardness on the film hardness was negligible in the case of soft films on hard substrates because the plastic deformation was contained within the film and the substrate yielded plastically only when the indenter penetrated the substrate. This was observed to be true in the cases of Al on glass, silicon, and sapphire and W on sapphire when we were able to measure the contact areas accurately by accounting for pileup. Hardness was observed to be constant for indentation depths less than the film thickness, t. However, when contact areas were determined by the Oliver–Pharr method, which did not account for pile-up, the hardnesses were observed to exhibit a ‘substrate-effect’, i.e. hardness was observed to increase with indentation depth. These film/substrate systems also exhibited strain gradient effects. An increase in hardness due to strain gradients was observed for indentation depths less than 30% of the film thickness and greater than 75% of the film thickness. Substrate hardness was observed to affect the measured film hardness for the case of a hard film on a soft substrate. This is because the substrate yields at indentation depths less than the film thickness. In such cases the true hardness of the film

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could be determined from the indentation data only if the indentation depth was less than 10% of the film thickness, as observed in the cases of W films on Al and glass. One of the simplest ways to estimate the intrinsic or true hardness of a film is to deposit it on a substrate that has the same elastic modulus as that of the film. Then by using the constant modulus assumption, one can easily determine the hardness from a measure of both the load, P, and contact stiffness, S. Compared to hardness, the nanoindentation measurement of the elastic modulus of thin films is more strongly affected by the substrate. This is because the elastic field under the indenter is not confined to the film itself; rather it is a long-range field that extends into the substrate, especially when the film thickness is small. As a result, the substrate stiffness is observed to influence the measured contact stiffness at small indentation depths relative to the film thickness, i.e. at h/t⬍0.1. Hence it is difficult to estimate the film modulus just from nanoindentation data, especially when there is a large elastic mismatch between the film and substrate. In such cases, the modified King’s analysis can be used to estimate the film modulus. Acknowledgements Support for this work by the Division of Materials Science, office of Basic Energy Sciences of the United States Department of Energy through grant DE-FG03-89ER45387 is gratefully acknowledged. Vidya Ramaswamy is also acknowledged for her help in depositing the Al films. References [1] Burnett PJ, Rickerby DS. Surf Engng 1987;3:69. [2] Chechenin NG, Bottiger J, Krog JP. Thin Solid Films 1995;261:219.

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