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Determining mechanical properties of thin films from the loading curve of nanoindentation testing
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Thin Solid Films 516 (2008) 7571 – 7580
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Determining mechanical properties of thin films from the loading curve of nanoindentation testing Manhong Zhao a , Yong Xiang b , Jessica Xu c , Nagahisa Ogasawara d , Norimasa Chiba d , Xi Chen a,⁎ a b c
Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY 10027-6699, USA Intel Corporation, Materials Technology Department, 3065 Bowers Avenue, MS SC2-24, Santa Clara, CA 95054, USA Intel Corporation, Logical Technology Development, 2501 NW 229th Avenue, MS RA3-402, Hillsboro, OR 97124, USA d Department of Mechanical Engineering, National Defense Academy, Hashirimizu, Yokosuka 239-8686, Japan Received 14 August 2007; received in revised form 11 March 2008; accepted 12 March 2008 Available online 21 March 2008
Abstract Nanoindentation has been widely used to evaluate material properties. In this study, we propose a method that utilizes only the loading curves of an indentation test to extract the elastoplastic properties of an elastic-perfectly plastic thin film as well as the plastic properties of a work hardening thin film. The use of loading curve circumvents some common difficulties encountered during the post-processing of experimental unloading curves. Measurements are taken at two different indentation depths, which have different levels of substrate effects and lead to the establishment of independent equations that correlate the material properties with indentation responses. Effective reverse analysis algorithms are proposed by following which the desired film properties can be determined from a sharp indentation test. The extracted material properties agree well with that measured from a bulge test. © 2008 Elsevier B.V. All rights reserved. Keywords: Nanoindentation; Thin film; Elastic property; Plastic property
1. Introduction 1.1. Background of indentation on thin films One of the most important material systems in microelectronics consists of one or more layers of thin films deposited onto a substrate of another material; the films can be metallic, ceramic, or dielectric, and the substrate is typically Si or other hard and stiff material in many cases. While the film Young's modulus often remains close to its bulk counterpart, due to the presence of the film–substrate interface and small grain size, thin metal films often support stresses that would be relaxed through plastic flow in bulk materials, leading to a higher yield stress and/or work hardening exponent (i.e. different plastic
⁎ Corresponding author. Tel.: +1 212 854 3787; fax: +1 212 854 6267. E-mail address: [email protected] (X. Chen). 0040-6090/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2008.03.018
properties than bulk) [1–6]. Measuring the elastic and plastic properties of thin films is always the first critical step to analyze their mechanical integrity. This is often considered difficult since the conventional techniques, such as the micro-tensile or bulge test on freestanding thin films [5, 7–11], or the microbeam cantilever deflection technique [12–14], require extensive micromachining efforts to remove the substrate while trying not to alter the film properties. Among alternative techniques, nanoindentation is arguably the simplest approach for measuring the mechanical properties of small material structures including thin films [15–20]. In an indentation test, a diamond indenter is pressed into materials (Fig. 1(a)) and the indentation load (P) and indentation depth (δ) are continuously measured with high accuracy (Fig. 1(b)). One of the most widely used indenter geometry is a pyramidal Berkovich tip; it has been established that for two sharp indenters with different shapes (e.g. pyramidal Berkovich and conical indenters), as long as their ratios of cross-sectional area to depth are the same, these two indenters yield almost identical
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[15,16,22,24] (for elastic-perfectly plastic and stress-free bulk material) to relate the measured hardness (H) and contact stiffness (S) with the film yield stress (σy) and Young's modulus (E), respectively. In practice, however, such conventional approach has several disadvantages:
Fig. 1. (a) Schematic of indentation on a film/substrate material system, and (b) schematic of the indentation load–depth curve: data will be taken at two different indentation depths (loads).
P–δ curves with difference in general less than 1% for bulk materials [20,21] and we have verified that for thin films. To simplify the analysis, the Berkovich indenter is usually modeled as a rigid cone with half apex angle α = 70.3° (Fig. 1a) [20,22]. Note that due to the complicated stress and strain fields resulting from the finite deformation, the P–δ curve is only implicitly related with the material elastoplastic properties as well as the material/system structure (e.g. the presence of substrate in a film/substrate system) — functional relationships need to be established such that through a reverse analysis, the most important and intrinsic elastic and plastic parameters of the thin film may be derived from the experimental data [20]. 1.2. Usefulness of substrate effect The most attractive aspect of nanoindentation is that the test can be performed quickly and does not require removal of the film from its substrate, and it has been widely used to measure the mechanical properties of bulk materials and thin films (including both metals and ceramics) [16,20,23,24]. Conventionally, experiments with contact depths of less than 10–20% of the film thickness (h) are required to obtain the intrinsic properties of the film [20], and it is assumed that under such circumstance, the substrate influence on the measurement is negligible. Under this assumption, one could use the well established indentation theory
(1) As the indenter penetrates the specimen, the material either produces a plastic pile-up or elastic sink-in at the crater rim depending on the yield strain, σy / E [25], which would make it difficult to measure the projected contact area A and hardness accurately (especially at the maximum penetration when it needs to be measured), becoming one of the primary error sources. Therefore, it is more desirable to use the reliable P–δ curve measured from an indentation test [20,26–28]. (2) Although the elastic-perfectly plastic model is widely used for ceramics and high-strength metals upon indentation [29–33], for many other metal films, the effects of work hardening on indentation behaviors need to be accounted for. Thus, an effective framework should be applicable to both elastic-perfectly plastic material and materials with work hardening. (3) Although δ is intentionally to be kept much smaller than h in the conventional approach, in some cases the substrate effect is still not negligible at very shallow indentation and thus the bulk indentation theory may not work well [24]. Even if indentation depths as small as several nm can be made by a high-end commercial nanoindenter, there are still many experimental issues interfering with measurements at this scale [34,35], such as indenter tipfilm surface adhesion, specimen surface roughness, indenter tip bluntness, and strain gradient effects [36,37] — these effects make it difficult to accurately measure P–δ curves and/or obtain intrinsic film properties. An alternative approach to circumvent these problems is to keep δ moderately deep and to explore the film properties based on an understanding of the substrate effect using continuum mechanics. In our previous work on thin films deposited on a rigid substrate [38], the concept of using the substrate effect to measure the thin film elastoplastic parameters has been validated; however, the rigid substrate assumption requires that the substrate be at least 10 times stiffer than film, which limits the applicability of the previous study [38]. In this study, the rigid substrate assumption is alleviated and the substrate elastic deformation is also accounted for, which is applicable to many technologically important films including those deposited on Si substrate (the Si deform elastically during indentation as long as δ is not too deep). We develop a theoretical framework that can be applied to general elastic substrates, and use that to guide indentation experiments of thin films on a Si substrate. 1.3. Usefulness of loading curve Another important difference between this paper and our previous work [38] is that the present paper only uses the data
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from the loading section of P–δ curves, based on the following considerations: (1) In a nanoindentation test, very often the stress relaxation is observed at the maximum load when unloading just occurs, which is because the strain beneath indenter is quite large and the strain rate is also very large at this point (since the loading rate is usually finite during the experiment), and the sudden withdraw of indenter causes oscillations in measurement whereas holding at maximum load may lead to uncertainties in the measured quantities [16]. In addition, the initial portion of unloading is more prone to thermal drift than the loading curve. Many experimental schemes have been proposed to minimize these effects, nevertheless, the possible existence of these aspects requires that the experiment must be carried out very carefully, otherwise the first unloading may not be entirely elastic, the unloading curve may not be very reproducible, and the unloading curve must be used with caution (see below for an example). (2) In many theoretical and numerical indentation studies, a rigid indenter is used [20,24,32,38,39] whereas in practice, a diamond indenter tip is employed. It is well known that the unloading behavior is strongly affected by the compliance of indenter tip [16]. Therefore, in order to apply the developed indentation method based on rigid indenter, a real P–δ curve measured from experiment must be converted to that by a rigid indenter — for example, our previous work [38] utilizes the unloading work in indentation analysis, which requires the full information the unloading P–δ curve with a rigid indenter, and if the diamond tip contribution is not properly removed from experimental data, plus any of the abovementioned error due to the nonelastic effects in the unloading P–δ curve, large error may occur based on our error sensitivity analysis [38]. For example, we have shown [40] that in a recent study by Guelorget et al. [41], because their sharp and spherical indentation experiments were not carefully conducted, in particular the scatter of unloading data was not properly treated, and the contribution of elastic deformation of the diamond tip was not removed from the measurement of contact stiffness, the material properties extracted from reverse analyses had very large error. It comes to our attention that quite a few users (e.g. [41]) may not fully aware of how to carefully handle the unloading data obtained from an experiment; therefore, perhaps a way to temporarily circumvent such issue is to avoid the use of unloading data, whose accurate measurement and post-processing may require extra care. Another benefit of avoiding the use of unloading curve is that, as remarked above, the experimental P–δ curve measured with a diamond indenter (regardless of what experimental scheme one uses) must be converted to that of a rigid indenter, so as to apply many well established theories based on rigid indenters. The conversion of the entire P–δ curve is often very difficult due to the unknown exact indenter tip geometry (e.g.
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effective “height” of the tip), which will in particular impact the unloading curve (since the variation of δ is small, which may be on the same order of diamond tip deformation). Since the diamond tip compliance has very small influence on the loading P–δ curve (especially at moderate indentation depth adopted in this paper, which is much larger than the diamond tip deformation), we therefore propose to circumvent such difficulty by using only the perhaps more reliable loading P– δ curve measured from an experiment. During loading as the indenter approaches the interface, the varying degree to which the substrate affects the indentation measurement provides rich information for extracting the film elastoplastic properties. The data are taken when the indentation depth δ is 1/3 and 2/3 of film thickness h, respectively — these two depths are chosen such that 1) they are not too small to interfere with the aforementioned small scale effects; 2) they are also not too large so as to prevent cracking in practice; 3) they are separated sufficiently apart, where the different substrate effects would induce different loading curvatures (see below) and provide two independent equations that can be used to solve two independent elastoplastic parameters — the principle is similar to [38] but an important difference is that the substrate elastic deformation is accounted for in this paper (and the substrate property is not fixed during the establishment of indentation method, although later in this paper, Si substrate is chosen in parallel experiment), and that we only use information obtained from loading so as to avoid uncertainties at unloading. The indentation load at the two depths are denoted as P1 and P2, respectively (Fig. 1b). 1.4. Organization of paper An effective reverse analysis algorithm is established in this paper such that, once the substrate modulus is known, useful elastic and/or plastic properties of the film material can be readily obtained from the indentation P–δ loading curve via one indentation test. Theories are developed for two categories of thin films: (1) Deriving Young's modulus and yield stress of elasticperfectly plastic films. As mentioned earlier, the classic indentation theory [15,16,20,22] is focused on elasticperfectly plastic bulk materials and such theory has been applied to ceramics and high-strength metals [16], to extract their effective elastic and plastic properties upon indentation. Here we extend the technique to thin films, by taking into account the substrate effect under the assumption that the substrate deforms elastically; in addition, the approach does not require the determination of projected contact area. The technique is applicable to ceramic or high-strength metal thin films whose indentation behaviors can be effective described by the phenomenological elastic-perfectly plastic material model [29–33] as long as significant cracking does not occur, and it will be applied to study a SiN film on Si substrate. (2) Solving the plastic properties of thin films with work hardening. In many cases the lattice structure of the film
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remains close to that of its bulk counterpart and thus Young's modulus can be assumed to be known. It is then critical to derive the film's plastic properties including the yield stress and work hardening exponent (for materials conform to the power-law hardening model). The proposed theory, applicable to metallic film, will be demonstrated via a Cu film on Si substrate and compared with the measurements obtained from a bulge test on freestanding film. For either category of thin films, two material properties can be solved from the two independent equations established from the loading curves. In this study, SiN and Cu films have been chosen to demonstrate the proposed techniques because of their technological importance in microelectronics applications. Cu is widely used as interconnects in integrated circuits, while SiN is often used as passivating layers. The proposed techniques may be readily extended to other widely used thin films and provide an effective method to measure their elastoplastic properties.
During the forward analysis, E / σy is varied from 10 to 4000, and E / Es is varied between 1/8 and 8. As shown in Fig. 2, the normalized indentation loads are plotted as two three-dimensional surfaces that are functions of material properties. The normalized indentation loads increase as E / σy increases or E / Es decreases. The presence of substrate has a remarkable effect on the normalized indentation load when E / σy is small (i.e. more elastic materials), whereas the substrate effect is smaller for materials with larger E / σy (more plastic materials). While the plastic properties are more localized due to plastic flow, the influence zone of elastic responses is much larger and thus the normalized indentation load of the more elastic materials are more sensitive to the variation of E / Es, i.e. the substrate effect. In other words, indentation on the more elastic film may better utilize the more prominent substrate effect with the proposed technique. The functions in Fig. 2 can be fitted in the following form with less than 1% maximum error over the material space considered: P1 ln ry d21
2. Finite element analysis The relationships between the indentation response and material parameters are established via the finite element method (FEM) [42], where the material properties are varied over a wide range in the numerical experiments (forward analysis). By taking advantage of the axisymmetry of conical indenter, the film is meshed into over 2000 4-node axisymmetric elements (with reduced integration), and the substrate is made up of about 9000 elements. The substrate dimensions (both width and height) are taken to be at least 100 times larger than the film thickness, so as to simulate the semi-infinite substrate. Coulomb's friction law is used between contact surfaces with a friction coefficient of 0.15 [43]. As long as this value is relatively small, friction is a minor factor in indentation [20,44].
ln
P2 ry d22
!
!
¼ p1 þ p2 n þ p3 n2 þ p4 g þ p5 g2 þ p6 g3 = 1 þ p7 n þ p8 g þ p9 g2
ð3Þ ¼ p1 þ p2 n þ p3 n2 þ p4 g þ p5 g2 þ p6 g3 = 1 þ p7 n þ p8 n2 þ p9 g þ p10 g2
ð4Þ Here, ξ ≡ ln(E / Es) and η ≡ ln(E / σy). The coefficients in Eqs. (3) and (4) are tabulated in Table 1. These two functions relate the indentation response with the film elastoplastic properties as well as film/substrate elastic mismatch; once calibrated via FEM, they can be employed in the reverse analysis for determining the film properties, discussed next. 3.1.2. General reverse analysis algorithm If one could first flip over the film/substrate system and carry out an indentation test on the very thick substrate, then Young's modulus of substrate, Es, can be measured in a straightforward manner according to the classic bulk indentation theory [16].
3. Determining Young's modulus and yield stress of elastic-perfectly plastic films 3.1. Indentation theory establishment 3.1.1. Forward analysis For an elastic-perfectly plastic thin film, its Young's modulus is E and its yield stress is σy, and the substrate modulus is Es. From the principle of dimensional analysis illustrated in Ref. [20], the following functional forms can be established to relate the normalized indentation load with film/substrate properties: P1 E E ¼ F ; ð1Þ 1 Es ry ry d21 P2 E E ¼ F2 ; Es ry ry d22
ð2Þ
where δ1 = h / 3 and δ2 = 2h / 3. Both functions can be verified and determined from extensive finite element analyses by varying the material properties in a large range, so as to be applicable to more systems.
Fig. 2. The normalized indentation load plotted as functions of material properties for elastic-perfectly plastic thin films.
M. Zhao et al. / Thin Solid Films 516 (2008) 7571–7580 Table 1 Fitting coefficients in Eqs. (3) and (4) Coefficients in Eq. (3) p1 p2 p3 p4 p5 p6 p7 p8 p9
1.26920 − 0.51126 − 0.00229 0.50848 − 0.11675 0.00861 − 0.10571 − 0.18639 0.01522
Coefficients in Eq. (4) p1 p2 p3 p4 p5 p6 p7 p8 p9 p10
1.461204 − 0.83322 0.1417 0.24664 − 0.09268 0.01047 − 0.17243 0.03102 − 0.25201 0.02526
For the silicon substrate used in the experimental study, its Young's modulus is taken to be 120 GPa, which is also consistent with the average value found in literature [45,46]. From an indentation experiment on a film with known thickness, the loading P–δ curve is measured with the maximum penetration equals to (or exceeds) 2/3 of film thickness. The most important data to be taken are the loading curvatures, P1 / δ12 = 9P1 / h2 and P2 / δ22 = 9P2 / 4h2 . During the reverse analysis, the postulated film properties (E, σy) are varied over a large range and substituted into Eqs. (3) and (4) along with the measured indentation data — the combination of (E, σy) that minimizes the total error in Eqs. (3) and (4) and represents the identified elastoplastic properties of the film. A practical issue is that multiple local minima may occur during the numerical reverse analysis procedure. However, according to our recent work [39] the solution of film indentation with moderately deep penetration depth is unique. Therefore, if one indeed encounters multiple local minima, the corresponding candidate material properties are substituted back into Eqs. (3) and (4) to examine if the theoretical loading curvatures agree with those measured — finally, the correct solution of material properties should be able to reproduce the loading P–δ curve measured from experiment. Next, we apply the established technique (theory) to nanoindentation on a thin film sample and to extract its mechanical properties.
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loading rate of 5 nm/s and a maximum indentation depth of approximately 300 nm (i.e., 2/3 of the film thickness h = 450 nm), followed by a complete unloading. Note that during the experiment, we do not have to indent the specimen exactly to 300 nm and then unload. In fact, we could make the maximum penetration above 300 nm but only use the data of the measured load and depth during loading, at 150 nm and 300 nm, respectively. Indeed, in Fig. 3(a) the maximum depth in experiment is slightly above 300 nm. This is because the unloading curve is not used in this study, which is another practical advantage of the technique proposed in this paper. Note that some commercial nanoindenters cannot perform displacement-controlled experiment, yet with the framework proposed in this paper, we do not have to control the maximum indentation depth exactly. Macroscopic cracking was not observed, which enables the employment of the phenomenological elastic-perfectly plastic model (see below for discussions). The measured indentation load–depth curves for the loading segment are plotted in Fig. 3(a) as lines. The data measured from different indents are fairly close, with a maximum difference (scatter) of about 2%.
3.2. Application: determining Young's modulus and yield stress of a SiN thin film 3.2.1. Experimental procedure Since the proposed technique emphasizes the substrate effect, a fairly elastic material, SiN, is chosen to be the film — we expect that its Young's modulus and effective yield strength (during compression) may be effectively identified based on the discussion in Section 3.1.1. In addition, the strain gradient effect that is usually associated with metal films [37,38] should be absent in SiN, thus a relatively thin sample film can be employed. 450 nm SiN films were deposited onto 730 μm (100) Si substrate using the plasma enhanced chemical vapor deposition (PECVD) technique with a proprietary process. Eight indents were carried out on the SiN film/Si substrate material system using a MTS nanoindenter (Eden Prairie, MN, USA) with a Berkovich tip. The instrument was operated in continuous stiffness mode (CSM) with a nominally constant displacement
Fig. 3. (a) The eight different nanoindentation load–depth curves measured on a SiN film/Si substrate (lines); (b) The determined uniaxial stress–strain curves of the SiN film, correspond to the eight experiments; such constitutive behavior (assuming perfectly plastic material) corresponds to that in compression. Based on the averaged determined phenomenological properties, a numerical indentation test is carried out on such material and the resulting load–depth curve is given as symbol in (a).
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3.2.2. Determination of SiN thin film properties from indentation tests Based on the experimental curves, the reverse analysis is applied to extract both Young's modulus E and yield stress σy of SiN film. For each individual test, reverse analysis based on Section 3.1.2 is carried out to extract (E, σy) of the film, and the resulting uniaxial stress–strain curve of the identified film properties is given in Fig. 3(b); a total of 8 results are shown which correspond to the 8 indentation experiments. The averaged measured Young's modulus is 196.2 GPa and the averaged identified effective yield stress is 9.88 GPa — the identified Young's modulus is very close to that measured by the continuous stiffness measurement (CSM) at shallow indentation depth, which is about 200 GPa, as well as the reported value of a plasma-CVD SiN film, 210 GPa, measured by a deflection test [47]. It is difficult to independently verify the identified effective compressive yield strength of SiN, since during a parallel bulge test of the freestanding SiN film, the film fractured at tensile stress (which was absent during the indentation experiment); moreover, to our knowledge it is impossible to carry out a uniaxial compression test on a freestanding SiN film with a large strain. Therefore, we will evaluate the plastic property determination of the present indentation technique in the next section, through bulge test on a ductile Cu film specimen. We remark that for the SiN film, we adopt the widely-used model for ceramics under indentation and assume that its behavior upon indentation is elastic-perfectly plastic [19,29–33]. First, elastic-perfectly plastic is the simplest elastoplastic behavior and thus theoretically it is important, moreover if one could verify that a thin film specimen indeed nearly conforms to the elastic-perfectly plastic behavior, then the formulation established in this section is applicable. It is noted again that it is impossible to carry out an uniaxial compression test on the SiN film sample we used, nevertheless in the literature, the compression test on a silicon nitride cylinder shows a behavior that is not far away from the elastic-perfectly model [48], and similar behaviors can be found for many other ceramic-like materials, such as GaAs, Si, etc. [49–52]. Second, the elasticperfectly plastic model is a phenomenological model in the sense that even if there are other mechanisms (e.g. microcracks in a brittle material), the overall compressive stress–strain behavior upon indentation (especially that during loading) may still be roughly described by such an ideal model. Note that microcracking in a brittle material may lead to a different unloading behavior which may invalidate the elastic-perfectly plastic model, however, only the loading curve from indentation is used in this paper and the unloading behavior is not concerned. In other words, when such an ideal model is applied to the SiN film in our experiment, and with its modulus and effective yield stress derived above, the indentation behavior of such model is consistent with that of the real specimen regardless of the specimen's microstructural details; that is to say, if we carry out a numerical indentation test on a virtual material that obeys the ideal elastic-perfectly plastic relationship and with the derived material parameters given in Fig. 3(b), the resulting P–δ curve (open circular symbol in Fig. 3(a)) agrees well with the measured P–δ curves (lines in Fig. 3(a)). Hence, the elastic-perfectly
plastic model could be effectively applied to describe (and to reproduce) the indentation loading behavior of SiN in this study, at least in a phenomenological way, and such model may be applied to other similar ceramic materials [19,29–33]. 3.2.3. Error sensitivity of the proposed technique During nanoindentation experiments, the noise (error) of measurements of the indentation loads P1 and P2 is inevitable, and thus the error sensitivity of the film properties determined from the reverse analysis needs to be investigated. By differentiating Eqs. (3) and (4), the dependences of the variations of the identified Young's modulus and yield stress with respect to the errors of P1 and P2 can be explicitly derived. In the case of indentation on SiN film/Si substrate, with Es = 120 GPa, E = 196.2 GPa, and σy = 9.88 GPa, the error sensitivity analysis leads to dE dP1 dP2 ¼ 7:34 9:94 E P1 P2
ð5Þ
dry dP1 dP2 ¼ 4:83 þ 10:66 ry P1 P2
ð6Þ
where the perturbations of the identified material properties (left hand side of the equations) are related with the errors of measured indentation loads (right hand side of equations). Note that such errors are that generated from possible perturbation of experimental data, and independent of the error of the fitting functions, Eqs. (3) and (4). We illustrate the application of the error sensitivity equations, Eqs. (5) and (6), for SiN/Si film/substrate system: the errors of measured P1 and P2 will be magnified by 7.34 and − 9.94 times, respectively, and becoming the error of the determined Young's modulus; the errors of measured P1 and P2 will be magnified by − 4.83 and 10.66 times, respectively, and becoming the error of the determined yield strength. Note that in both Eqs. (5) and (6), the signs of two parameters related with dP1 and dP2 are opposite. Therefore, any systematical error of the measured indentation load (i.e. when the errors of P1 and P2 are of the same order of magnitude and sign, such as those arisen from instrument calibration error or diamond compliance contribution) will induce a partial error cancellation effect on the identified material properties (Young's modulus and yield stress), such that the reverse analysis results are relatively stable and insensitive to the systematic errors — which can also be demonstrated through the small variations of the identified film properties for 8 different indentation tests shown in Fig. 3(b). Of course, such argument does not hold for the statistical measurement error if the signs of the errors of P1 and P2 are different. 4. Measuring plastic properties of elastoplastic film with work hardening 4.1. Indentation theory establishment 4.1.1. Forward analysis For many metal films (except those high-strength films whose strain hardening may be ignored), work hardening needs to be incorporated at finite plastic deformation. The widely used
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power-law hardening model is a good approximation for most metals and alloys including thin films [20], whose uniaxial stress–strain relationship is: r ¼ Ee
for
eVry =E
and
r ¼ Ren
for
ezry =E; ð7Þ
where n is the work hardening exponent, and R ≡ σy(E / σy)n is the work hardening rate. We note that the power-law hardening model is the simplest and most widely used phenomenological continuum model used in indentation mechanics to describe the elastoplastic behaviors of a specimen [20,39], and the adoption of an effective continuum model is required for extracting material properties from any indentation analysis. We neglect the inhomogeneity induced by film microstructure and strain gradient effect in the theoretical study, by assuming that the indentation depth is sufficiently deep to overcome such effects. Other forms of material hardening model may be explored in future. Within the context of power-law hardening elastoplastic thin film/elastic substrate system, the dimensionless functions relating the normalized loading curvatures and material properties become P1 E E ¼ G1 ; ;n ð8Þ Es ry ry d21 P2 E E ¼ G ; ; n 2 Es ry ry d22
ð9Þ
During the forward analysis of calibrating these two equations, both E /σy and E / Es are varied over a wide range (E /σy from 10 to 4000, and E / Es from 1/8 to 8) so as to ensure broader potential applications; moreover, n is varied between 0 and 0.6 (a typical range for most metals and alloys [39]). Both functions can be determined from extensive finite element analyses: P1 ln ry d21
!
¼ p1 þ p2 n þ p3 n2 þ p4 g þ p5 g2 þ p6 g 3 ð1 þ p7 n þ p8 g þ p9 g2 Þ þ p10 þ p11 n þ p12 n2 þ p13 n3 þ p14 g þ p15 g2 1 þ p16 n þ p17 n2 þ p18 g þ p19 g2 n þ p20 þ p21 n þ p22 n2 þ p23 n3 þ p24 g þ p25 g2 1 þ p26 n þ p27 n2 þ p28 n3 þ p29 g þ p30 g2 n2 ð10Þ
P2 ln ry d22
!
¼ p1 þ p2 n þ p3 n2 þ p4 g þ p5 g2 þ p6 g 3 1 þ p7 n þ p8 n2 þ p9 g þ p10 g2 þ p11 þ p12 n þ p13 n2 þ p14 g þ p15 g2 þ p16 g3 1 þ p17 n þ p18 n2 þ p19 n3 þ p20 g þ p21 g2 n þ p22 þ p23 n þ p24 n2 þ p25 n3 þ p26 g þ p27 g2 1 þ p28 n þ p29 n2 þ p30 g þ p31 g2 n2 ð11Þ
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The maximum error of fitting is less than 1%, with ξ ≡ ln(E /Es) and η ≡ ln(E /σy). The coefficients in Eqs. (10) and (11) are tabulated in Table 2. When n = 0, Eqs. (10) and (11) reduce to Eqs. (3) and (4), respectively. 4.1.2. General reverse analysis algorithm In the reverse analysis of the current problem, we assume that both the film and substrate modulus are known (i.e. both E and E /Es are known) where the film modulus can be either taken from that of its bulk counterpart, or measured by other developed methods. Therefore, the unknown plastic parameters in this problem are σy and n, which may be solved from the two equations above. For a film with known thickness, from the loading P–δ curve, the important shape factors, P1 /δ12 = 9P1 /h2 and P2 /δ22 = 9P2 / 4h2, are measured. Reverse analysis can be carried out by minimizing the total error of Eqs. (10) and (11) with respect to the candidate material properties and experimental data. Similar to the procedure described in Section 3.1.2, numerical iteration is used to filter out pseudo-local minima and obtain the best candidate material whose indentation behavior could reproduce that of experimental data — such iteration can be carried out based on Eqs. (10) and (11), and without additional finite element simulations. Next, we apply the established method to nanoindentation on an elastoplastic thin film sample and to extract its plastic properties. 4.2. Application: determining plastic properties of a Cu thin film 4.2.1. Experimental procedure Nanoindentation test is performed on a copper film/Si substrate system with film thickness of 900 nm. Copper has close tensile and compressive behaviors [53] — the compressive constitutive behavior can be measured by the proposed indentation technique, and the tensile behavior of the same material system is characterized using the bulge test technique. The details of material preparation, the bulge test procedure, and the results have been published elsewhere [4,54]. Briefly, Cu films of 0.9 μm thickness were sputtered directly onto long, rectangular, freestanding Si3N4/TaN membranes that were prepared using micromachining. Immediately after deposition, Table 2 Fitting coefficients in Eqs. (10) and (11) Coefficients in equation p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15 p16
1.26920 − 0.51126 − 0.00229 0.50848 − 0.11675 0.00861 − 0.10571 − 0.18639 0.01522 − 0.57366 − 0.11405 − 0.03021 0.00159 0.28659 − 0.01072 − 0.02451
p17 p18 p19 p20 p21 p22 p23 p24 p25 p26 p27 p28 p29 p30
Coefficients in equation − 0.00687 − 0.18099 0.01019 0.24978 0.01800 0.06218 − 0.00979 − 0.12963 0.00655 − 0.77508 0.26357 − 0.03224 − 0.03436 0.00144
p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15 p16
1.46120 − 0.83322 0.14174 0.24664 − 0.09268 0.01047 − 0.17243 0.03102 − 0.25201 0.02526 − 2.13940 − 0.09858 − 0.00672 1.59180 − 0.36257 0.02827
p17 p18 p19 p20 p21 p22 p23 p24 p25 p26 p27 p28 p29 p30 p31
−0.01184 7.4909E-4 3.1286E-5 −0.31600 0.02833 0.65379 0.16761 0.02865 −0.00153 −0.33428 0.01972 −0.22611 0.03513 −0.16255 0.01077
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the Cu film was annealed in-situ at 200 °C for 10 min, resulting in an average grain size of 0.5 μm and a mixed crystallographic texture that is dominated by (111) and (100) fiber components. The Si3N4/TaN was removed with a reactive ion etch and the resulting freestanding Cu film was subsequently deformed in plane strain by applying pressure to the rectangular membrane. The applied pressure and the corresponding membrane deflection
were measured and converted into film stress and strain. The resulting plane strain stress–strain curve (consisting of multiple loading and unloading segments) of the sputter-deposited Cu film is plotted as the solid line (with open circle symbols) in Fig. 4(a). In the nanoindentation experiment, four indents were carried out on the Cu/Si system using the MTS nanoindenter with a nominally constant displacement loading rate of 5 nm/s and a maximum indentation depth of approximately 600 nm (i.e., 2/3 of the film thickness h = 900 nm). The indentation load–depth curves for the loading segment are plotted in Fig. 4(b). 4.2.2. Determination of plastic properties of Cu thin film from indentation tests In the reverse indentation analysis, Young's modulus of Cu is taken to be that is measured from the bulge test, 125 GPa, which is also close to that of bulk Cu [55,56], and Young's modulus of Si substrate is taken to be 120 GPa [45,46]. From the load–depth curves in Fig. 4(b), the indentation loads at indentation depths of 300 nm and 600 nm (corresponding to 1/3 and 2/3 of film thickness) are measured and then substituted into Eqs. (10) and (11). With the postulated film properties varying over a large range, the film plastic properties are determined by following the procedures described in Section 4.1.2. With respect to the scatter of indentation data in Fig. 4(b), the plastic properties identified from the reverse analysis have a relatively small fluctuation, demonstrated in Fig. 4(c) in terms of the identified uniaxial stress–strain curves (with uniaxial strain up to 15%). The averaged yield stress and work hardening exponent of the copper film are determined as σy = 201 MPa and n = 0.205, respectively. In order to compare the material properties determined from indentation test and bulge test, the plane strain stress–strain curves based on the film parameters identified from the 4 nanoindentation tests (when the strain is varied in a comparable range with that in the bulge test) are also given in Fig. 4(a). We note that the plane strain stress–strain curve of the material determined from the nanoindentation test is higher than that obtained from bulge test. Such discrepancy may be attributed to: (1) the presence of the film/substrate interface that restricts the mobility of gliding dislocations during the indentation; (2) the strain gradient effect due to relatively shallow penetration depth in the indentation test, since the thickness of the Cu film under investigation is relatively small; (3) the tip bluntness effect (the indenter tip used in experiment has a radius of about 70 nm). 4.2.3. Error sensitivity Following an error sensitivity analysis similar to that in Section 3.2.3, the perturbations of the identified material properties (with respect to the averaged Cu film properties reported above) are expressed as functions of the errors of the measured indentation loads:
Fig. 4. (a) The plane strain stress–strain curves of a Cu film (on Si substrate) identified from the reverse analysis of nanoindentation tests on a Cu film/Si substrate system, and the results determined from the nanoindentation test also agree well with that measured from a bulge test. (b) Four nanoindentation tests on the Cu film on Si substrate. (c) The uniaxial stress–strain curves of the Cu film identified from the reverse analysis of four nanoindentation tests, as the strain is varied in a larger range.
dry dP1 dP2 ¼ 6:91 5:61 ry P1 P2
ð12Þ
dn dP1 dP2 ¼ 5:97 þ 6:14 n P1 P2
ð13Þ
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The parameters on the right hand sides of Eqs. (12) and (13) represent the magnifying factors of the errors in reverse analysis. The error of the identified yield stress is 6.91 times of error of P1 and − 5.61 times of the error of P2, and the error of the identified work hardening exponent is − 5.97 times of perturbation of P1 and 6.14 times of the perturbation of P2. Again, the signs of the magnifying factors in either Eq. (12) or Eq. (13) are opposite, and therefore the perturbations of identified material properties from systematic perturbations of P1 and P2 will be reduced by the partial cancellation effect. In addition, since the magnifying factors of σy and n are of opposite sign, for any source of error, the resulting effect on the overall plastic property may be partially cancelled — that is, if the yield strength is overestimated by an error in measurement, then the work hardening exponent is underestimated — the overall effect leads to a more consistent trend of the identified plastic property (c.f. Fig. 4(c)). 5. Conclusion In this paper, an indentation technique is developed to extract the mechanical properties of thin films by utilizing the substrate effect. We start with studying the substrate effect, “amplifying” and then utilizing such effect, and finally subtracting off such effect to obtain the thin film mechanical properties. The proposed technique expands our previous work [38] and incorporates the important substrate elastic deformation; moreover, only the loading segment of an indentation test is needed, which circumvents the common sources of measurement errors encountered during unloading and increases the potential applicability. From the numerical forward analysis, the relationships between the material properties and the indentation parameters are established based on extensive finite element simulations. Based on these relationships, the film material properties can be determined from the reverse analysis by performing an indentation test on the sample material, and measuring data at two different indentation depths (i.e. using two different substrate effects). Two versions of this technique are presented: (a) to extract the material properties of an elastic-perfectly plastic film (applicable to ceramic and high-strength metal/alloy films upon compression) and (b) with the film elastic modulus known (or taken to be that of the bulk), to extract the yield stress and work hardening exponent of a work hardened metal/alloy film. These two approaches are applied to a SiN film and a Cu film, respectively. In both cases plausible results and good agreements with literature or parallel bulge test are reported. In addition, error sensitivity analyses of both applications are carried out, which reveals that the present film indentation technique is reasonably stable. Acknowledgments The work of MZ and XC is supported by NSF CMS0407743 and CMMI-0643726. The bulge test was done at Harvard University and the authors have benefited from discussion with J. Vlassak (Harvard).
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References [1] W.D. Nix, Metall. Mater. Trans. A 20 (1989) 2217. [2] R. Venkatraman, J.C. Bravman, W.D. Nix, P.W. Davies, P.A. Flinn, D.B. Fraser, J. Electron. Mater. 19 (1990) 1231. [3] R.P. Vinci, J.J. Vlassak, Annu. Rev. Mater. Sci. 26 (1996) 431. [4] L. Nicola, Y. Xiang, J.J. Vlassak, E.V.d. Giessen, A. Needleman, J. Mech. Phys. Solids 54 (2006) 2089. [5] H.D. Espinosa, B.C. Prorok, J. Mater. Sci. 38 (2003) 4125. [6] Y. Xiang, T.Y. Tsui, J.J. Vlassak, J. Mater. Res. 21 (2006) 1607. [7] Y. Xiang, X. Chen, J.J. Vlassak, J. Mater. Res. 20 (2005) 2360. [8] J.J. Vlassak, W.D. Nix, J. Mater. Res. 7 (1992) 3242. [9] Y. Xiang, X. Chen, J.J. Vlassak, Mater. Res. Soc. Sym. Proc. 695 (2001) 491. [10] H. Huang, F. Spaepen, Acta Mater. 48 (2000) 3261. [11] M.A. Haque, T. Saif, Thin Solid Films 484 (2005) 364. [12] S.P. Baker, W.D. Nix, J. Mater. Res. 9 (1994) 3145. [13] S.P. Baker, W.D. Nix, J. Mater. Res. 9 (1994) 3131. [14] T.P. Weihs, S. Hong, J.C. Bravman, W.D. Nix, J. Mater. Res. 13 (1998) 931. [15] F. Doerner, W.D. Nix, J. Mater. Res. 1 (1986) 601. [16] W.C. Oliver, G.M. Pharr, J. Mater. Res. 7 (1992) 1564. [17] T.Y. Tsui, J.J. Vlassak, W.D. Nix, J. Mater. Res. 14 (1999) 2196. [18] T.Y. Tsui, J.J. Vlassak, W.D. Nix, J. Mater. Res. 14 (1999) 2204. [19] X. Chen, Y. Xiang, J.J. Vlassak, J. Mater. Res. 21 (2006) 715. [20] Y.T. Cheng, C.M. Cheng, Mater. Sci. Eng. R44 (2004) 91. [21] M. Li, W.M. Chen, N.G. Liang, L.D. Wang, J. Mater. Res. 18 (2004) 73. [22] G.M. Pharr, Mater. Sci. Eng. A 253 (1998) 151. [23] R. Saha, W.D. Nix, Acta Mater. 50 (2002) 23. [24] X. Chen, J.J. Vlassak, J. Mater. Res. 16 (2001) 2974. [25] M. Zhao, X. Chen, Y. Xiang, J.J. Vlassak, D. Lee, N. Ogasawara, N. Chiba, Y.X. Gan, Acta Mater. 55 (2007) 6260. [26] N. Ogasawara, N. Chiba, X. Chen, J. Mater. Res. 20 (2005) 2225. [27] N. Ogasawara, N. Chiba, X. Chen, J. Mater. Res. 21 (2006) 947. [28] N. Ogasawara, N. Chiba, X. Chen, Scripta Mater. 54 (2006) 65. [29] X. Chen, J.W. Hutchinson, J. Mech. Phys. Solids 50 (2002) 2669. [30] X. Chen, J.W. Hutchinson, A.G. Evans, Acta Mater. 52 (2004) 565. [31] X. Chen, J.W. Hutchinson, A.G. Evans, J. Am. Ceram. Soc. 88 (2005) 1233. [32] Y. Sun, A. Bloyce, T. Bell, Thin Solid Films 271 (1995) 122. [33] G. Care, A.C. Fischer-Cripps, J. Mater. Sci. 32 (1997) 5653. [34] R. Saha, Z. Xue, Y. Huang, W.D. Nix, J. Mech. Phys. Solids 49 (2001) 1997. [35] J.G. Swadener, E.P. George, G.M. Pharr, J. Mech. Phys. Solids 50 (2002) 681. [36] N.A. Fleck, J.W. Hutchinson, Adv. Appl. Mech. 33 (1997) 295. [37] H. Gao, Y. Huang, W.D. Nix, J.W. Hutchinson, J. Mech. Phys. Solids 47 (1999) 1239. [38] M. Zhao, X. Chen, N. Ogasawara, A.C. Razvan, N. Chiba, D. Lee, Y.X. Gan, J. Mater. Res. 21 (2006) 3134. [39] X. Chen, N. Ogasawara, M. Zhao, N. Chiba, J. Mech. Phys. Solids 55 (2007) 1618. [40] N. Ogasawara, M. Zhao, N. Chiba, X. Chen, J. Mater. Res. 23 (2008) 598. [41] B. Guelorget, M. Francois, C. Liu, J. Lu, J. Mater. Res. 22 (2007) 1512. [42] ABAQUS, ABAQUS 6.4 User's Manual, ABAQUS Inc., Pawtucket, Rhode Island, 2004. [43] F.P. Bowden, D. Tabor, The Friction and Lubrications of Solids, Oxford University Press, Oxford, 1950. [44] J.L. Bucaille, S. Stauss, E. Felder, J. Michler, Acta Mater. 51 (2003) 1663. [45] D.R. Lide, CRC Handbook of Chemistry and Physics, CRC Press, Boca Raton, FL, 1998. [46] B. Bhushan, X. Li, J. Mater. Res. 12 (1997) 54. [47] O. Tabata, K. Kawahata, S. Sugiyama, I. Igarashi, Sens. Actuators, A 20 (1989) 135. [48] M. Gei, D. Bigoni, S. Guicciardi, Mech. Mater. 36 (2004) 335. [49] F.P. Francois, A. Lefebvre, G. Vanderschaeve, Phys. Status Solidi, A 109 (1988) 187.
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[50] W. Jiang, K. Qiu, F. Liu, H. Choo, P.K. Liaw, Adv. Eng. Mater. 9 (2006) 147. [51] K.F. Yao, F. Ruan, Y.Q. Yang, N. Chen, Appl. Phys. Lett. 88 (2006) 122106. [52] J. Rabier, J.L. Demenet, Phys. Status Solidi, B 222 (2000) 63. [53] M.S. Paterson, J. Sci. Instrum. 32 (1955) 356.
[54] Y. Xiang, J.J. Vlassak, Acta Mater. 54 (2006) 5449. [55] M.F. Ashby, D.R.H. Jones, Engineering Materials I, Butterworth Heinemann, 1996. [56] J. Dolbow, M. Gosz, Mech. Mater. 23 (1996) 311.
Thin Solid Films 516 (2008) 7571 – 7580
www.elsevier.com/locate/tsf
Determining mechanical properties of thin films from the loading curve of nanoindentation testing Manhong Zhao a , Yong Xiang b , Jessica Xu c , Nagahisa Ogasawara d , Norimasa Chiba d , Xi Chen a,⁎ a b c
Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY 10027-6699, USA Intel Corporation, Materials Technology Department, 3065 Bowers Avenue, MS SC2-24, Santa Clara, CA 95054, USA Intel Corporation, Logical Technology Development, 2501 NW 229th Avenue, MS RA3-402, Hillsboro, OR 97124, USA d Department of Mechanical Engineering, National Defense Academy, Hashirimizu, Yokosuka 239-8686, Japan Received 14 August 2007; received in revised form 11 March 2008; accepted 12 March 2008 Available online 21 March 2008
Abstract Nanoindentation has been widely used to evaluate material properties. In this study, we propose a method that utilizes only the loading curves of an indentation test to extract the elastoplastic properties of an elastic-perfectly plastic thin film as well as the plastic properties of a work hardening thin film. The use of loading curve circumvents some common difficulties encountered during the post-processing of experimental unloading curves. Measurements are taken at two different indentation depths, which have different levels of substrate effects and lead to the establishment of independent equations that correlate the material properties with indentation responses. Effective reverse analysis algorithms are proposed by following which the desired film properties can be determined from a sharp indentation test. The extracted material properties agree well with that measured from a bulge test. © 2008 Elsevier B.V. All rights reserved. Keywords: Nanoindentation; Thin film; Elastic property; Plastic property
1. Introduction 1.1. Background of indentation on thin films One of the most important material systems in microelectronics consists of one or more layers of thin films deposited onto a substrate of another material; the films can be metallic, ceramic, or dielectric, and the substrate is typically Si or other hard and stiff material in many cases. While the film Young's modulus often remains close to its bulk counterpart, due to the presence of the film–substrate interface and small grain size, thin metal films often support stresses that would be relaxed through plastic flow in bulk materials, leading to a higher yield stress and/or work hardening exponent (i.e. different plastic
⁎ Corresponding author. Tel.: +1 212 854 3787; fax: +1 212 854 6267. E-mail address: [email protected] (X. Chen). 0040-6090/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2008.03.018
properties than bulk) [1–6]. Measuring the elastic and plastic properties of thin films is always the first critical step to analyze their mechanical integrity. This is often considered difficult since the conventional techniques, such as the micro-tensile or bulge test on freestanding thin films [5, 7–11], or the microbeam cantilever deflection technique [12–14], require extensive micromachining efforts to remove the substrate while trying not to alter the film properties. Among alternative techniques, nanoindentation is arguably the simplest approach for measuring the mechanical properties of small material structures including thin films [15–20]. In an indentation test, a diamond indenter is pressed into materials (Fig. 1(a)) and the indentation load (P) and indentation depth (δ) are continuously measured with high accuracy (Fig. 1(b)). One of the most widely used indenter geometry is a pyramidal Berkovich tip; it has been established that for two sharp indenters with different shapes (e.g. pyramidal Berkovich and conical indenters), as long as their ratios of cross-sectional area to depth are the same, these two indenters yield almost identical
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[15,16,22,24] (for elastic-perfectly plastic and stress-free bulk material) to relate the measured hardness (H) and contact stiffness (S) with the film yield stress (σy) and Young's modulus (E), respectively. In practice, however, such conventional approach has several disadvantages:
Fig. 1. (a) Schematic of indentation on a film/substrate material system, and (b) schematic of the indentation load–depth curve: data will be taken at two different indentation depths (loads).
P–δ curves with difference in general less than 1% for bulk materials [20,21] and we have verified that for thin films. To simplify the analysis, the Berkovich indenter is usually modeled as a rigid cone with half apex angle α = 70.3° (Fig. 1a) [20,22]. Note that due to the complicated stress and strain fields resulting from the finite deformation, the P–δ curve is only implicitly related with the material elastoplastic properties as well as the material/system structure (e.g. the presence of substrate in a film/substrate system) — functional relationships need to be established such that through a reverse analysis, the most important and intrinsic elastic and plastic parameters of the thin film may be derived from the experimental data [20]. 1.2. Usefulness of substrate effect The most attractive aspect of nanoindentation is that the test can be performed quickly and does not require removal of the film from its substrate, and it has been widely used to measure the mechanical properties of bulk materials and thin films (including both metals and ceramics) [16,20,23,24]. Conventionally, experiments with contact depths of less than 10–20% of the film thickness (h) are required to obtain the intrinsic properties of the film [20], and it is assumed that under such circumstance, the substrate influence on the measurement is negligible. Under this assumption, one could use the well established indentation theory
(1) As the indenter penetrates the specimen, the material either produces a plastic pile-up or elastic sink-in at the crater rim depending on the yield strain, σy / E [25], which would make it difficult to measure the projected contact area A and hardness accurately (especially at the maximum penetration when it needs to be measured), becoming one of the primary error sources. Therefore, it is more desirable to use the reliable P–δ curve measured from an indentation test [20,26–28]. (2) Although the elastic-perfectly plastic model is widely used for ceramics and high-strength metals upon indentation [29–33], for many other metal films, the effects of work hardening on indentation behaviors need to be accounted for. Thus, an effective framework should be applicable to both elastic-perfectly plastic material and materials with work hardening. (3) Although δ is intentionally to be kept much smaller than h in the conventional approach, in some cases the substrate effect is still not negligible at very shallow indentation and thus the bulk indentation theory may not work well [24]. Even if indentation depths as small as several nm can be made by a high-end commercial nanoindenter, there are still many experimental issues interfering with measurements at this scale [34,35], such as indenter tipfilm surface adhesion, specimen surface roughness, indenter tip bluntness, and strain gradient effects [36,37] — these effects make it difficult to accurately measure P–δ curves and/or obtain intrinsic film properties. An alternative approach to circumvent these problems is to keep δ moderately deep and to explore the film properties based on an understanding of the substrate effect using continuum mechanics. In our previous work on thin films deposited on a rigid substrate [38], the concept of using the substrate effect to measure the thin film elastoplastic parameters has been validated; however, the rigid substrate assumption requires that the substrate be at least 10 times stiffer than film, which limits the applicability of the previous study [38]. In this study, the rigid substrate assumption is alleviated and the substrate elastic deformation is also accounted for, which is applicable to many technologically important films including those deposited on Si substrate (the Si deform elastically during indentation as long as δ is not too deep). We develop a theoretical framework that can be applied to general elastic substrates, and use that to guide indentation experiments of thin films on a Si substrate. 1.3. Usefulness of loading curve Another important difference between this paper and our previous work [38] is that the present paper only uses the data
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from the loading section of P–δ curves, based on the following considerations: (1) In a nanoindentation test, very often the stress relaxation is observed at the maximum load when unloading just occurs, which is because the strain beneath indenter is quite large and the strain rate is also very large at this point (since the loading rate is usually finite during the experiment), and the sudden withdraw of indenter causes oscillations in measurement whereas holding at maximum load may lead to uncertainties in the measured quantities [16]. In addition, the initial portion of unloading is more prone to thermal drift than the loading curve. Many experimental schemes have been proposed to minimize these effects, nevertheless, the possible existence of these aspects requires that the experiment must be carried out very carefully, otherwise the first unloading may not be entirely elastic, the unloading curve may not be very reproducible, and the unloading curve must be used with caution (see below for an example). (2) In many theoretical and numerical indentation studies, a rigid indenter is used [20,24,32,38,39] whereas in practice, a diamond indenter tip is employed. It is well known that the unloading behavior is strongly affected by the compliance of indenter tip [16]. Therefore, in order to apply the developed indentation method based on rigid indenter, a real P–δ curve measured from experiment must be converted to that by a rigid indenter — for example, our previous work [38] utilizes the unloading work in indentation analysis, which requires the full information the unloading P–δ curve with a rigid indenter, and if the diamond tip contribution is not properly removed from experimental data, plus any of the abovementioned error due to the nonelastic effects in the unloading P–δ curve, large error may occur based on our error sensitivity analysis [38]. For example, we have shown [40] that in a recent study by Guelorget et al. [41], because their sharp and spherical indentation experiments were not carefully conducted, in particular the scatter of unloading data was not properly treated, and the contribution of elastic deformation of the diamond tip was not removed from the measurement of contact stiffness, the material properties extracted from reverse analyses had very large error. It comes to our attention that quite a few users (e.g. [41]) may not fully aware of how to carefully handle the unloading data obtained from an experiment; therefore, perhaps a way to temporarily circumvent such issue is to avoid the use of unloading data, whose accurate measurement and post-processing may require extra care. Another benefit of avoiding the use of unloading curve is that, as remarked above, the experimental P–δ curve measured with a diamond indenter (regardless of what experimental scheme one uses) must be converted to that of a rigid indenter, so as to apply many well established theories based on rigid indenters. The conversion of the entire P–δ curve is often very difficult due to the unknown exact indenter tip geometry (e.g.
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effective “height” of the tip), which will in particular impact the unloading curve (since the variation of δ is small, which may be on the same order of diamond tip deformation). Since the diamond tip compliance has very small influence on the loading P–δ curve (especially at moderate indentation depth adopted in this paper, which is much larger than the diamond tip deformation), we therefore propose to circumvent such difficulty by using only the perhaps more reliable loading P– δ curve measured from an experiment. During loading as the indenter approaches the interface, the varying degree to which the substrate affects the indentation measurement provides rich information for extracting the film elastoplastic properties. The data are taken when the indentation depth δ is 1/3 and 2/3 of film thickness h, respectively — these two depths are chosen such that 1) they are not too small to interfere with the aforementioned small scale effects; 2) they are also not too large so as to prevent cracking in practice; 3) they are separated sufficiently apart, where the different substrate effects would induce different loading curvatures (see below) and provide two independent equations that can be used to solve two independent elastoplastic parameters — the principle is similar to [38] but an important difference is that the substrate elastic deformation is accounted for in this paper (and the substrate property is not fixed during the establishment of indentation method, although later in this paper, Si substrate is chosen in parallel experiment), and that we only use information obtained from loading so as to avoid uncertainties at unloading. The indentation load at the two depths are denoted as P1 and P2, respectively (Fig. 1b). 1.4. Organization of paper An effective reverse analysis algorithm is established in this paper such that, once the substrate modulus is known, useful elastic and/or plastic properties of the film material can be readily obtained from the indentation P–δ loading curve via one indentation test. Theories are developed for two categories of thin films: (1) Deriving Young's modulus and yield stress of elasticperfectly plastic films. As mentioned earlier, the classic indentation theory [15,16,20,22] is focused on elasticperfectly plastic bulk materials and such theory has been applied to ceramics and high-strength metals [16], to extract their effective elastic and plastic properties upon indentation. Here we extend the technique to thin films, by taking into account the substrate effect under the assumption that the substrate deforms elastically; in addition, the approach does not require the determination of projected contact area. The technique is applicable to ceramic or high-strength metal thin films whose indentation behaviors can be effective described by the phenomenological elastic-perfectly plastic material model [29–33] as long as significant cracking does not occur, and it will be applied to study a SiN film on Si substrate. (2) Solving the plastic properties of thin films with work hardening. In many cases the lattice structure of the film
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remains close to that of its bulk counterpart and thus Young's modulus can be assumed to be known. It is then critical to derive the film's plastic properties including the yield stress and work hardening exponent (for materials conform to the power-law hardening model). The proposed theory, applicable to metallic film, will be demonstrated via a Cu film on Si substrate and compared with the measurements obtained from a bulge test on freestanding film. For either category of thin films, two material properties can be solved from the two independent equations established from the loading curves. In this study, SiN and Cu films have been chosen to demonstrate the proposed techniques because of their technological importance in microelectronics applications. Cu is widely used as interconnects in integrated circuits, while SiN is often used as passivating layers. The proposed techniques may be readily extended to other widely used thin films and provide an effective method to measure their elastoplastic properties.
During the forward analysis, E / σy is varied from 10 to 4000, and E / Es is varied between 1/8 and 8. As shown in Fig. 2, the normalized indentation loads are plotted as two three-dimensional surfaces that are functions of material properties. The normalized indentation loads increase as E / σy increases or E / Es decreases. The presence of substrate has a remarkable effect on the normalized indentation load when E / σy is small (i.e. more elastic materials), whereas the substrate effect is smaller for materials with larger E / σy (more plastic materials). While the plastic properties are more localized due to plastic flow, the influence zone of elastic responses is much larger and thus the normalized indentation load of the more elastic materials are more sensitive to the variation of E / Es, i.e. the substrate effect. In other words, indentation on the more elastic film may better utilize the more prominent substrate effect with the proposed technique. The functions in Fig. 2 can be fitted in the following form with less than 1% maximum error over the material space considered: P1 ln ry d21
2. Finite element analysis The relationships between the indentation response and material parameters are established via the finite element method (FEM) [42], where the material properties are varied over a wide range in the numerical experiments (forward analysis). By taking advantage of the axisymmetry of conical indenter, the film is meshed into over 2000 4-node axisymmetric elements (with reduced integration), and the substrate is made up of about 9000 elements. The substrate dimensions (both width and height) are taken to be at least 100 times larger than the film thickness, so as to simulate the semi-infinite substrate. Coulomb's friction law is used between contact surfaces with a friction coefficient of 0.15 [43]. As long as this value is relatively small, friction is a minor factor in indentation [20,44].
ln
P2 ry d22
!
!
¼ p1 þ p2 n þ p3 n2 þ p4 g þ p5 g2 þ p6 g3 = 1 þ p7 n þ p8 g þ p9 g2
ð3Þ ¼ p1 þ p2 n þ p3 n2 þ p4 g þ p5 g2 þ p6 g3 = 1 þ p7 n þ p8 n2 þ p9 g þ p10 g2
ð4Þ Here, ξ ≡ ln(E / Es) and η ≡ ln(E / σy). The coefficients in Eqs. (3) and (4) are tabulated in Table 1. These two functions relate the indentation response with the film elastoplastic properties as well as film/substrate elastic mismatch; once calibrated via FEM, they can be employed in the reverse analysis for determining the film properties, discussed next. 3.1.2. General reverse analysis algorithm If one could first flip over the film/substrate system and carry out an indentation test on the very thick substrate, then Young's modulus of substrate, Es, can be measured in a straightforward manner according to the classic bulk indentation theory [16].
3. Determining Young's modulus and yield stress of elastic-perfectly plastic films 3.1. Indentation theory establishment 3.1.1. Forward analysis For an elastic-perfectly plastic thin film, its Young's modulus is E and its yield stress is σy, and the substrate modulus is Es. From the principle of dimensional analysis illustrated in Ref. [20], the following functional forms can be established to relate the normalized indentation load with film/substrate properties: P1 E E ¼ F ; ð1Þ 1 Es ry ry d21 P2 E E ¼ F2 ; Es ry ry d22
ð2Þ
where δ1 = h / 3 and δ2 = 2h / 3. Both functions can be verified and determined from extensive finite element analyses by varying the material properties in a large range, so as to be applicable to more systems.
Fig. 2. The normalized indentation load plotted as functions of material properties for elastic-perfectly plastic thin films.
M. Zhao et al. / Thin Solid Films 516 (2008) 7571–7580 Table 1 Fitting coefficients in Eqs. (3) and (4) Coefficients in Eq. (3) p1 p2 p3 p4 p5 p6 p7 p8 p9
1.26920 − 0.51126 − 0.00229 0.50848 − 0.11675 0.00861 − 0.10571 − 0.18639 0.01522
Coefficients in Eq. (4) p1 p2 p3 p4 p5 p6 p7 p8 p9 p10
1.461204 − 0.83322 0.1417 0.24664 − 0.09268 0.01047 − 0.17243 0.03102 − 0.25201 0.02526
For the silicon substrate used in the experimental study, its Young's modulus is taken to be 120 GPa, which is also consistent with the average value found in literature [45,46]. From an indentation experiment on a film with known thickness, the loading P–δ curve is measured with the maximum penetration equals to (or exceeds) 2/3 of film thickness. The most important data to be taken are the loading curvatures, P1 / δ12 = 9P1 / h2 and P2 / δ22 = 9P2 / 4h2 . During the reverse analysis, the postulated film properties (E, σy) are varied over a large range and substituted into Eqs. (3) and (4) along with the measured indentation data — the combination of (E, σy) that minimizes the total error in Eqs. (3) and (4) and represents the identified elastoplastic properties of the film. A practical issue is that multiple local minima may occur during the numerical reverse analysis procedure. However, according to our recent work [39] the solution of film indentation with moderately deep penetration depth is unique. Therefore, if one indeed encounters multiple local minima, the corresponding candidate material properties are substituted back into Eqs. (3) and (4) to examine if the theoretical loading curvatures agree with those measured — finally, the correct solution of material properties should be able to reproduce the loading P–δ curve measured from experiment. Next, we apply the established technique (theory) to nanoindentation on a thin film sample and to extract its mechanical properties.
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loading rate of 5 nm/s and a maximum indentation depth of approximately 300 nm (i.e., 2/3 of the film thickness h = 450 nm), followed by a complete unloading. Note that during the experiment, we do not have to indent the specimen exactly to 300 nm and then unload. In fact, we could make the maximum penetration above 300 nm but only use the data of the measured load and depth during loading, at 150 nm and 300 nm, respectively. Indeed, in Fig. 3(a) the maximum depth in experiment is slightly above 300 nm. This is because the unloading curve is not used in this study, which is another practical advantage of the technique proposed in this paper. Note that some commercial nanoindenters cannot perform displacement-controlled experiment, yet with the framework proposed in this paper, we do not have to control the maximum indentation depth exactly. Macroscopic cracking was not observed, which enables the employment of the phenomenological elastic-perfectly plastic model (see below for discussions). The measured indentation load–depth curves for the loading segment are plotted in Fig. 3(a) as lines. The data measured from different indents are fairly close, with a maximum difference (scatter) of about 2%.
3.2. Application: determining Young's modulus and yield stress of a SiN thin film 3.2.1. Experimental procedure Since the proposed technique emphasizes the substrate effect, a fairly elastic material, SiN, is chosen to be the film — we expect that its Young's modulus and effective yield strength (during compression) may be effectively identified based on the discussion in Section 3.1.1. In addition, the strain gradient effect that is usually associated with metal films [37,38] should be absent in SiN, thus a relatively thin sample film can be employed. 450 nm SiN films were deposited onto 730 μm (100) Si substrate using the plasma enhanced chemical vapor deposition (PECVD) technique with a proprietary process. Eight indents were carried out on the SiN film/Si substrate material system using a MTS nanoindenter (Eden Prairie, MN, USA) with a Berkovich tip. The instrument was operated in continuous stiffness mode (CSM) with a nominally constant displacement
Fig. 3. (a) The eight different nanoindentation load–depth curves measured on a SiN film/Si substrate (lines); (b) The determined uniaxial stress–strain curves of the SiN film, correspond to the eight experiments; such constitutive behavior (assuming perfectly plastic material) corresponds to that in compression. Based on the averaged determined phenomenological properties, a numerical indentation test is carried out on such material and the resulting load–depth curve is given as symbol in (a).
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3.2.2. Determination of SiN thin film properties from indentation tests Based on the experimental curves, the reverse analysis is applied to extract both Young's modulus E and yield stress σy of SiN film. For each individual test, reverse analysis based on Section 3.1.2 is carried out to extract (E, σy) of the film, and the resulting uniaxial stress–strain curve of the identified film properties is given in Fig. 3(b); a total of 8 results are shown which correspond to the 8 indentation experiments. The averaged measured Young's modulus is 196.2 GPa and the averaged identified effective yield stress is 9.88 GPa — the identified Young's modulus is very close to that measured by the continuous stiffness measurement (CSM) at shallow indentation depth, which is about 200 GPa, as well as the reported value of a plasma-CVD SiN film, 210 GPa, measured by a deflection test [47]. It is difficult to independently verify the identified effective compressive yield strength of SiN, since during a parallel bulge test of the freestanding SiN film, the film fractured at tensile stress (which was absent during the indentation experiment); moreover, to our knowledge it is impossible to carry out a uniaxial compression test on a freestanding SiN film with a large strain. Therefore, we will evaluate the plastic property determination of the present indentation technique in the next section, through bulge test on a ductile Cu film specimen. We remark that for the SiN film, we adopt the widely-used model for ceramics under indentation and assume that its behavior upon indentation is elastic-perfectly plastic [19,29–33]. First, elastic-perfectly plastic is the simplest elastoplastic behavior and thus theoretically it is important, moreover if one could verify that a thin film specimen indeed nearly conforms to the elastic-perfectly plastic behavior, then the formulation established in this section is applicable. It is noted again that it is impossible to carry out an uniaxial compression test on the SiN film sample we used, nevertheless in the literature, the compression test on a silicon nitride cylinder shows a behavior that is not far away from the elastic-perfectly model [48], and similar behaviors can be found for many other ceramic-like materials, such as GaAs, Si, etc. [49–52]. Second, the elasticperfectly plastic model is a phenomenological model in the sense that even if there are other mechanisms (e.g. microcracks in a brittle material), the overall compressive stress–strain behavior upon indentation (especially that during loading) may still be roughly described by such an ideal model. Note that microcracking in a brittle material may lead to a different unloading behavior which may invalidate the elastic-perfectly plastic model, however, only the loading curve from indentation is used in this paper and the unloading behavior is not concerned. In other words, when such an ideal model is applied to the SiN film in our experiment, and with its modulus and effective yield stress derived above, the indentation behavior of such model is consistent with that of the real specimen regardless of the specimen's microstructural details; that is to say, if we carry out a numerical indentation test on a virtual material that obeys the ideal elastic-perfectly plastic relationship and with the derived material parameters given in Fig. 3(b), the resulting P–δ curve (open circular symbol in Fig. 3(a)) agrees well with the measured P–δ curves (lines in Fig. 3(a)). Hence, the elastic-perfectly
plastic model could be effectively applied to describe (and to reproduce) the indentation loading behavior of SiN in this study, at least in a phenomenological way, and such model may be applied to other similar ceramic materials [19,29–33]. 3.2.3. Error sensitivity of the proposed technique During nanoindentation experiments, the noise (error) of measurements of the indentation loads P1 and P2 is inevitable, and thus the error sensitivity of the film properties determined from the reverse analysis needs to be investigated. By differentiating Eqs. (3) and (4), the dependences of the variations of the identified Young's modulus and yield stress with respect to the errors of P1 and P2 can be explicitly derived. In the case of indentation on SiN film/Si substrate, with Es = 120 GPa, E = 196.2 GPa, and σy = 9.88 GPa, the error sensitivity analysis leads to dE dP1 dP2 ¼ 7:34 9:94 E P1 P2
ð5Þ
dry dP1 dP2 ¼ 4:83 þ 10:66 ry P1 P2
ð6Þ
where the perturbations of the identified material properties (left hand side of the equations) are related with the errors of measured indentation loads (right hand side of equations). Note that such errors are that generated from possible perturbation of experimental data, and independent of the error of the fitting functions, Eqs. (3) and (4). We illustrate the application of the error sensitivity equations, Eqs. (5) and (6), for SiN/Si film/substrate system: the errors of measured P1 and P2 will be magnified by 7.34 and − 9.94 times, respectively, and becoming the error of the determined Young's modulus; the errors of measured P1 and P2 will be magnified by − 4.83 and 10.66 times, respectively, and becoming the error of the determined yield strength. Note that in both Eqs. (5) and (6), the signs of two parameters related with dP1 and dP2 are opposite. Therefore, any systematical error of the measured indentation load (i.e. when the errors of P1 and P2 are of the same order of magnitude and sign, such as those arisen from instrument calibration error or diamond compliance contribution) will induce a partial error cancellation effect on the identified material properties (Young's modulus and yield stress), such that the reverse analysis results are relatively stable and insensitive to the systematic errors — which can also be demonstrated through the small variations of the identified film properties for 8 different indentation tests shown in Fig. 3(b). Of course, such argument does not hold for the statistical measurement error if the signs of the errors of P1 and P2 are different. 4. Measuring plastic properties of elastoplastic film with work hardening 4.1. Indentation theory establishment 4.1.1. Forward analysis For many metal films (except those high-strength films whose strain hardening may be ignored), work hardening needs to be incorporated at finite plastic deformation. The widely used
M. Zhao et al. / Thin Solid Films 516 (2008) 7571–7580
power-law hardening model is a good approximation for most metals and alloys including thin films [20], whose uniaxial stress–strain relationship is: r ¼ Ee
for
eVry =E
and
r ¼ Ren
for
ezry =E; ð7Þ
where n is the work hardening exponent, and R ≡ σy(E / σy)n is the work hardening rate. We note that the power-law hardening model is the simplest and most widely used phenomenological continuum model used in indentation mechanics to describe the elastoplastic behaviors of a specimen [20,39], and the adoption of an effective continuum model is required for extracting material properties from any indentation analysis. We neglect the inhomogeneity induced by film microstructure and strain gradient effect in the theoretical study, by assuming that the indentation depth is sufficiently deep to overcome such effects. Other forms of material hardening model may be explored in future. Within the context of power-law hardening elastoplastic thin film/elastic substrate system, the dimensionless functions relating the normalized loading curvatures and material properties become P1 E E ¼ G1 ; ;n ð8Þ Es ry ry d21 P2 E E ¼ G ; ; n 2 Es ry ry d22
ð9Þ
During the forward analysis of calibrating these two equations, both E /σy and E / Es are varied over a wide range (E /σy from 10 to 4000, and E / Es from 1/8 to 8) so as to ensure broader potential applications; moreover, n is varied between 0 and 0.6 (a typical range for most metals and alloys [39]). Both functions can be determined from extensive finite element analyses: P1 ln ry d21
!
¼ p1 þ p2 n þ p3 n2 þ p4 g þ p5 g2 þ p6 g 3 ð1 þ p7 n þ p8 g þ p9 g2 Þ þ p10 þ p11 n þ p12 n2 þ p13 n3 þ p14 g þ p15 g2 1 þ p16 n þ p17 n2 þ p18 g þ p19 g2 n þ p20 þ p21 n þ p22 n2 þ p23 n3 þ p24 g þ p25 g2 1 þ p26 n þ p27 n2 þ p28 n3 þ p29 g þ p30 g2 n2 ð10Þ
P2 ln ry d22
!
¼ p1 þ p2 n þ p3 n2 þ p4 g þ p5 g2 þ p6 g 3 1 þ p7 n þ p8 n2 þ p9 g þ p10 g2 þ p11 þ p12 n þ p13 n2 þ p14 g þ p15 g2 þ p16 g3 1 þ p17 n þ p18 n2 þ p19 n3 þ p20 g þ p21 g2 n þ p22 þ p23 n þ p24 n2 þ p25 n3 þ p26 g þ p27 g2 1 þ p28 n þ p29 n2 þ p30 g þ p31 g2 n2 ð11Þ
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The maximum error of fitting is less than 1%, with ξ ≡ ln(E /Es) and η ≡ ln(E /σy). The coefficients in Eqs. (10) and (11) are tabulated in Table 2. When n = 0, Eqs. (10) and (11) reduce to Eqs. (3) and (4), respectively. 4.1.2. General reverse analysis algorithm In the reverse analysis of the current problem, we assume that both the film and substrate modulus are known (i.e. both E and E /Es are known) where the film modulus can be either taken from that of its bulk counterpart, or measured by other developed methods. Therefore, the unknown plastic parameters in this problem are σy and n, which may be solved from the two equations above. For a film with known thickness, from the loading P–δ curve, the important shape factors, P1 /δ12 = 9P1 /h2 and P2 /δ22 = 9P2 / 4h2, are measured. Reverse analysis can be carried out by minimizing the total error of Eqs. (10) and (11) with respect to the candidate material properties and experimental data. Similar to the procedure described in Section 3.1.2, numerical iteration is used to filter out pseudo-local minima and obtain the best candidate material whose indentation behavior could reproduce that of experimental data — such iteration can be carried out based on Eqs. (10) and (11), and without additional finite element simulations. Next, we apply the established method to nanoindentation on an elastoplastic thin film sample and to extract its plastic properties. 4.2. Application: determining plastic properties of a Cu thin film 4.2.1. Experimental procedure Nanoindentation test is performed on a copper film/Si substrate system with film thickness of 900 nm. Copper has close tensile and compressive behaviors [53] — the compressive constitutive behavior can be measured by the proposed indentation technique, and the tensile behavior of the same material system is characterized using the bulge test technique. The details of material preparation, the bulge test procedure, and the results have been published elsewhere [4,54]. Briefly, Cu films of 0.9 μm thickness were sputtered directly onto long, rectangular, freestanding Si3N4/TaN membranes that were prepared using micromachining. Immediately after deposition, Table 2 Fitting coefficients in Eqs. (10) and (11) Coefficients in equation p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15 p16
1.26920 − 0.51126 − 0.00229 0.50848 − 0.11675 0.00861 − 0.10571 − 0.18639 0.01522 − 0.57366 − 0.11405 − 0.03021 0.00159 0.28659 − 0.01072 − 0.02451
p17 p18 p19 p20 p21 p22 p23 p24 p25 p26 p27 p28 p29 p30
Coefficients in equation − 0.00687 − 0.18099 0.01019 0.24978 0.01800 0.06218 − 0.00979 − 0.12963 0.00655 − 0.77508 0.26357 − 0.03224 − 0.03436 0.00144
p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15 p16
1.46120 − 0.83322 0.14174 0.24664 − 0.09268 0.01047 − 0.17243 0.03102 − 0.25201 0.02526 − 2.13940 − 0.09858 − 0.00672 1.59180 − 0.36257 0.02827
p17 p18 p19 p20 p21 p22 p23 p24 p25 p26 p27 p28 p29 p30 p31
−0.01184 7.4909E-4 3.1286E-5 −0.31600 0.02833 0.65379 0.16761 0.02865 −0.00153 −0.33428 0.01972 −0.22611 0.03513 −0.16255 0.01077
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the Cu film was annealed in-situ at 200 °C for 10 min, resulting in an average grain size of 0.5 μm and a mixed crystallographic texture that is dominated by (111) and (100) fiber components. The Si3N4/TaN was removed with a reactive ion etch and the resulting freestanding Cu film was subsequently deformed in plane strain by applying pressure to the rectangular membrane. The applied pressure and the corresponding membrane deflection
were measured and converted into film stress and strain. The resulting plane strain stress–strain curve (consisting of multiple loading and unloading segments) of the sputter-deposited Cu film is plotted as the solid line (with open circle symbols) in Fig. 4(a). In the nanoindentation experiment, four indents were carried out on the Cu/Si system using the MTS nanoindenter with a nominally constant displacement loading rate of 5 nm/s and a maximum indentation depth of approximately 600 nm (i.e., 2/3 of the film thickness h = 900 nm). The indentation load–depth curves for the loading segment are plotted in Fig. 4(b). 4.2.2. Determination of plastic properties of Cu thin film from indentation tests In the reverse indentation analysis, Young's modulus of Cu is taken to be that is measured from the bulge test, 125 GPa, which is also close to that of bulk Cu [55,56], and Young's modulus of Si substrate is taken to be 120 GPa [45,46]. From the load–depth curves in Fig. 4(b), the indentation loads at indentation depths of 300 nm and 600 nm (corresponding to 1/3 and 2/3 of film thickness) are measured and then substituted into Eqs. (10) and (11). With the postulated film properties varying over a large range, the film plastic properties are determined by following the procedures described in Section 4.1.2. With respect to the scatter of indentation data in Fig. 4(b), the plastic properties identified from the reverse analysis have a relatively small fluctuation, demonstrated in Fig. 4(c) in terms of the identified uniaxial stress–strain curves (with uniaxial strain up to 15%). The averaged yield stress and work hardening exponent of the copper film are determined as σy = 201 MPa and n = 0.205, respectively. In order to compare the material properties determined from indentation test and bulge test, the plane strain stress–strain curves based on the film parameters identified from the 4 nanoindentation tests (when the strain is varied in a comparable range with that in the bulge test) are also given in Fig. 4(a). We note that the plane strain stress–strain curve of the material determined from the nanoindentation test is higher than that obtained from bulge test. Such discrepancy may be attributed to: (1) the presence of the film/substrate interface that restricts the mobility of gliding dislocations during the indentation; (2) the strain gradient effect due to relatively shallow penetration depth in the indentation test, since the thickness of the Cu film under investigation is relatively small; (3) the tip bluntness effect (the indenter tip used in experiment has a radius of about 70 nm). 4.2.3. Error sensitivity Following an error sensitivity analysis similar to that in Section 3.2.3, the perturbations of the identified material properties (with respect to the averaged Cu film properties reported above) are expressed as functions of the errors of the measured indentation loads:
Fig. 4. (a) The plane strain stress–strain curves of a Cu film (on Si substrate) identified from the reverse analysis of nanoindentation tests on a Cu film/Si substrate system, and the results determined from the nanoindentation test also agree well with that measured from a bulge test. (b) Four nanoindentation tests on the Cu film on Si substrate. (c) The uniaxial stress–strain curves of the Cu film identified from the reverse analysis of four nanoindentation tests, as the strain is varied in a larger range.
dry dP1 dP2 ¼ 6:91 5:61 ry P1 P2
ð12Þ
dn dP1 dP2 ¼ 5:97 þ 6:14 n P1 P2
ð13Þ
M. Zhao et al. / Thin Solid Films 516 (2008) 7571–7580
The parameters on the right hand sides of Eqs. (12) and (13) represent the magnifying factors of the errors in reverse analysis. The error of the identified yield stress is 6.91 times of error of P1 and − 5.61 times of the error of P2, and the error of the identified work hardening exponent is − 5.97 times of perturbation of P1 and 6.14 times of the perturbation of P2. Again, the signs of the magnifying factors in either Eq. (12) or Eq. (13) are opposite, and therefore the perturbations of identified material properties from systematic perturbations of P1 and P2 will be reduced by the partial cancellation effect. In addition, since the magnifying factors of σy and n are of opposite sign, for any source of error, the resulting effect on the overall plastic property may be partially cancelled — that is, if the yield strength is overestimated by an error in measurement, then the work hardening exponent is underestimated — the overall effect leads to a more consistent trend of the identified plastic property (c.f. Fig. 4(c)). 5. Conclusion In this paper, an indentation technique is developed to extract the mechanical properties of thin films by utilizing the substrate effect. We start with studying the substrate effect, “amplifying” and then utilizing such effect, and finally subtracting off such effect to obtain the thin film mechanical properties. The proposed technique expands our previous work [38] and incorporates the important substrate elastic deformation; moreover, only the loading segment of an indentation test is needed, which circumvents the common sources of measurement errors encountered during unloading and increases the potential applicability. From the numerical forward analysis, the relationships between the material properties and the indentation parameters are established based on extensive finite element simulations. Based on these relationships, the film material properties can be determined from the reverse analysis by performing an indentation test on the sample material, and measuring data at two different indentation depths (i.e. using two different substrate effects). Two versions of this technique are presented: (a) to extract the material properties of an elastic-perfectly plastic film (applicable to ceramic and high-strength metal/alloy films upon compression) and (b) with the film elastic modulus known (or taken to be that of the bulk), to extract the yield stress and work hardening exponent of a work hardened metal/alloy film. These two approaches are applied to a SiN film and a Cu film, respectively. In both cases plausible results and good agreements with literature or parallel bulge test are reported. In addition, error sensitivity analyses of both applications are carried out, which reveals that the present film indentation technique is reasonably stable. Acknowledgments The work of MZ and XC is supported by NSF CMS0407743 and CMMI-0643726. The bulge test was done at Harvard University and the authors have benefited from discussion with J. Vlassak (Harvard).
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References [1] W.D. Nix, Metall. Mater. Trans. A 20 (1989) 2217. [2] R. Venkatraman, J.C. Bravman, W.D. Nix, P.W. Davies, P.A. Flinn, D.B. Fraser, J. Electron. Mater. 19 (1990) 1231. [3] R.P. Vinci, J.J. Vlassak, Annu. Rev. Mater. Sci. 26 (1996) 431. [4] L. Nicola, Y. Xiang, J.J. Vlassak, E.V.d. Giessen, A. Needleman, J. Mech. Phys. Solids 54 (2006) 2089. [5] H.D. Espinosa, B.C. Prorok, J. Mater. Sci. 38 (2003) 4125. [6] Y. Xiang, T.Y. Tsui, J.J. Vlassak, J. Mater. Res. 21 (2006) 1607. [7] Y. Xiang, X. Chen, J.J. Vlassak, J. Mater. Res. 20 (2005) 2360. [8] J.J. Vlassak, W.D. Nix, J. Mater. Res. 7 (1992) 3242. [9] Y. Xiang, X. Chen, J.J. Vlassak, Mater. Res. Soc. Sym. Proc. 695 (2001) 491. [10] H. Huang, F. Spaepen, Acta Mater. 48 (2000) 3261. [11] M.A. Haque, T. Saif, Thin Solid Films 484 (2005) 364. [12] S.P. Baker, W.D. Nix, J. Mater. Res. 9 (1994) 3145. [13] S.P. Baker, W.D. Nix, J. Mater. Res. 9 (1994) 3131. [14] T.P. Weihs, S. Hong, J.C. Bravman, W.D. Nix, J. Mater. Res. 13 (1998) 931. [15] F. Doerner, W.D. Nix, J. Mater. Res. 1 (1986) 601. [16] W.C. Oliver, G.M. Pharr, J. Mater. Res. 7 (1992) 1564. [17] T.Y. Tsui, J.J. Vlassak, W.D. Nix, J. Mater. Res. 14 (1999) 2196. [18] T.Y. Tsui, J.J. Vlassak, W.D. Nix, J. Mater. Res. 14 (1999) 2204. [19] X. Chen, Y. Xiang, J.J. Vlassak, J. Mater. Res. 21 (2006) 715. [20] Y.T. Cheng, C.M. Cheng, Mater. Sci. Eng. R44 (2004) 91. [21] M. Li, W.M. Chen, N.G. Liang, L.D. Wang, J. Mater. Res. 18 (2004) 73. [22] G.M. Pharr, Mater. Sci. Eng. A 253 (1998) 151. [23] R. Saha, W.D. Nix, Acta Mater. 50 (2002) 23. [24] X. Chen, J.J. Vlassak, J. Mater. Res. 16 (2001) 2974. [25] M. Zhao, X. Chen, Y. Xiang, J.J. Vlassak, D. Lee, N. Ogasawara, N. Chiba, Y.X. Gan, Acta Mater. 55 (2007) 6260. [26] N. Ogasawara, N. Chiba, X. Chen, J. Mater. Res. 20 (2005) 2225. [27] N. Ogasawara, N. Chiba, X. Chen, J. Mater. Res. 21 (2006) 947. [28] N. Ogasawara, N. Chiba, X. Chen, Scripta Mater. 54 (2006) 65. [29] X. Chen, J.W. Hutchinson, J. Mech. Phys. Solids 50 (2002) 2669. [30] X. Chen, J.W. Hutchinson, A.G. Evans, Acta Mater. 52 (2004) 565. [31] X. Chen, J.W. Hutchinson, A.G. Evans, J. Am. Ceram. Soc. 88 (2005) 1233. [32] Y. Sun, A. Bloyce, T. Bell, Thin Solid Films 271 (1995) 122. [33] G. Care, A.C. Fischer-Cripps, J. Mater. Sci. 32 (1997) 5653. [34] R. Saha, Z. Xue, Y. Huang, W.D. Nix, J. Mech. Phys. Solids 49 (2001) 1997. [35] J.G. Swadener, E.P. George, G.M. Pharr, J. Mech. Phys. Solids 50 (2002) 681. [36] N.A. Fleck, J.W. Hutchinson, Adv. Appl. Mech. 33 (1997) 295. [37] H. Gao, Y. Huang, W.D. Nix, J.W. Hutchinson, J. Mech. Phys. Solids 47 (1999) 1239. [38] M. Zhao, X. Chen, N. Ogasawara, A.C. Razvan, N. Chiba, D. Lee, Y.X. Gan, J. Mater. Res. 21 (2006) 3134. [39] X. Chen, N. Ogasawara, M. Zhao, N. Chiba, J. Mech. Phys. Solids 55 (2007) 1618. [40] N. Ogasawara, M. Zhao, N. Chiba, X. Chen, J. Mater. Res. 23 (2008) 598. [41] B. Guelorget, M. Francois, C. Liu, J. Lu, J. Mater. Res. 22 (2007) 1512. [42] ABAQUS, ABAQUS 6.4 User's Manual, ABAQUS Inc., Pawtucket, Rhode Island, 2004. [43] F.P. Bowden, D. Tabor, The Friction and Lubrications of Solids, Oxford University Press, Oxford, 1950. [44] J.L. Bucaille, S. Stauss, E. Felder, J. Michler, Acta Mater. 51 (2003) 1663. [45] D.R. Lide, CRC Handbook of Chemistry and Physics, CRC Press, Boca Raton, FL, 1998. [46] B. Bhushan, X. Li, J. Mater. Res. 12 (1997) 54. [47] O. Tabata, K. Kawahata, S. Sugiyama, I. Igarashi, Sens. Actuators, A 20 (1989) 135. [48] M. Gei, D. Bigoni, S. Guicciardi, Mech. Mater. 36 (2004) 335. [49] F.P. Francois, A. Lefebvre, G. Vanderschaeve, Phys. Status Solidi, A 109 (1988) 187.
7580
M. Zhao et al. / Thin Solid Films 516 (2008) 7571–7580
[50] W. Jiang, K. Qiu, F. Liu, H. Choo, P.K. Liaw, Adv. Eng. Mater. 9 (2006) 147. [51] K.F. Yao, F. Ruan, Y.Q. Yang, N. Chen, Appl. Phys. Lett. 88 (2006) 122106. [52] J. Rabier, J.L. Demenet, Phys. Status Solidi, B 222 (2000) 63. [53] M.S. Paterson, J. Sci. Instrum. 32 (1955) 356.
[54] Y. Xiang, J.J. Vlassak, Acta Mater. 54 (2006) 5449. [55] M.F. Ashby, D.R.H. Jones, Engineering Materials I, Butterworth Heinemann, 1996. [56] J. Dolbow, M. Gosz, Mech. Mater. 23 (1996) 311.