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Finite-element modeling of nanoindentation for evaluating mechanical properties of MEMS materials
Surface and Coatings Technology 103–104 (1998) 268–275
Finite-element modeling of nanoindentation for evaluating mechanical properties of MEMS materials J.A. Knapp a,*, D.M. Follstaedt a, S.M. Myers a, J.C. Barbour a, T.A. Friedmann a, J.W. Ager IIIb, O.R. Monteiro b, I.G. Brown b a Sandia National Laboratories, Albuquerque, NM 87185-1056, USA b Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
Abstract We have developed procedures based on finite-element modeling of nanoindentation data to extract mechanical properties from thin, hard films and ion-beam-modified layers on softer substrates. The method accurately deduces the yield stress, Young’s modulus, and hardness from indentations as deep as 50% of the layer thickness. We use these procedures to evaluate two hard layers potentially useful for reducing friction and wear of components in micro-electromechanical systems (MEMS): Ni implanted with Ti and C, and diamond-like carbon layers. We show that the modeling works well even for materials whose hardness approaches that of diamond, a case where commonly used analytical methods for deducing the modulus fail. © 1998 Elsevier Science S.A. Keywords: Modeling; Mechanical properties; MEM; Nanoindentation
1. Introduction As materials applications such as micro-electromechanical systems (MEMS ) use smaller structures and thinner coatings, characterization of the mechanical properties of the materials has become more difficult. One method that has become widely accepted is depthsensing indentation at low loads, or ‘‘nanoindentation’’. However, as the thickness of the materials continues to decrease, obtaining a substrate-independent measure of the mechanical properties of layers becomes increasingly complicated due to the influence of the underlying material. The problems are particularly severe for application of nanoindentation to implantation-modified layers, since the properties of a layer may vary through its depth. For films with hardnesses that may approach that of bulk diamond, the deformation and possible yielding of the indenter tip are additional complications. Since analytical derivation of the mechanical properties is often not feasible for these cases, we have developed procedures based on finite-element modeling to reliably extract mechanical properties for thin, hard films on * Corresponding author. Tel: +1 505 844 2305; Fax: +1 505 844 7775; e-mail: [email protected] 0257-8972/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S 02 5 7 -8 9 7 2 ( 9 8 ) 0 0 42 2 - 8
softer substrates [1–9]. We developed PC-based software to generate meshes and depth-dependent material property descriptions specific to each sample, while the simulations of the response of the materials to nanoindentation are performed on a UNIX workstation using the commercial, large-strain, finite-element code ABAQUS/Standard (ABAQUS Version 5.6, Hibbitt, Karlsson and Sorensen, Inc., Pawtucket, RI ). The results are transferred back to the PC for analysis. Blunting of the indenter tip, friction between tip and surface, and pre-existing stress in the layers can all be included in the modeling. Two parameters for the layer are extracted from the fit of simulation to experiment: the yield stress Y (defined at a plastic strain of 0.002) and Young’s elastic modulus E. An additional simulation using the yield stress and elasticity of the layer for a hypothetical bulk ‘‘sample’’ then deduces the intrinsic hardness of the layer material, where the hardness H is defined as force/(projected contact area) during contact. These methods have been applied to several hard materials with potential application for reducing friction and wear in MEMS components: Fe, Ni, and Ni–Fe alloys implanted with Ti and C [4–8], deposited ‘‘diamondlike’’ carbon layers with hardnesses of 70–90 GPa [4,9], and AlO layers formed by ECR plasma deposition or x pulsed laser deposition [1–4].
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2. Modeling of nanoindentation The nanoindentation measurement consists of pushing a three-sided Berkovich-shaped diamond indenter into the sample under tight load control. (All of the nanoindentation tests were performed at Nano Instruments, Inc., Knoxville, TN.) The measured force as a function of depth during loading is characteristic of the resistance of the material to deformation, including in general both elastic and plastic responses. During unloading of the indenter, the decrease of force with depth is controlled by the elastic response of the material. The initial rate of change with depth of the decrease in force (i.e. the slope) is defined as the sample stiffness. In a standard nanoindentation measurement, the indenter is inserted to some specified depth and then extracted, giving the loading force over the whole depth range but giving a value for stiffness only at the deepest portion of the test. For a bulk sample or a relatively thick film, analytical methods have been developed to deduce H and E from the data [10], but since the analysis requires both the force and the stiffness, H and E are deduced only at the deepest penetration. For a thin layer, such an analysis gives an H and E that are due to the combined response of the layer and substrate. One of the primary advantages of finite-element modeling is the ability to separate the properties of a layer from those of the substrate, and an example is given here for an ion-implanted sample. Another advantage is the ability to account for distortions of the indenter by very hard materials. Our second example is an application to a DLC layer whose hardness approaches that of diamond, where conventional analysis fails to deduce E correctly, even for a relatively thick film. This second example will also demonstrate modeling for another type of indentation measurement, termed a continuous stiffness measurement (CSM ) [10]. To model each set of nanoindentation measurements, we first construct a mesh that reflects as accurately as possible the structure of the sample ( layer thicknesses) and the shape of the diamond indenter. This includes the area vs. depth function of the indenter, as determined separately with indents of fused silica [10], so that any blunting of the indenter near the tip is accounted for. The sample and indenter were approximated by twodimensional axisymmetric meshes for all the work presented here. We have compared three-dimensional calculations to the two-dimensional approximation and have confirmed that the two approaches give results within a few per cent of each other, but with greatly increased computational efficiency for the latter [1–3]. Materials parameters for all the known materials such as the substrate and the indenter are specified, as well as the proposed properties of the unknown layer. The work-hardening rate for the unknown layer (i.e. the slope of the stress–strain curve above the yield stress) is
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fixed at a value characteristic of the material being studied, as taken from the literature for similar materials. For amorphous or non-metallic materials, the hardening is assumed to be zero. The indentation and modeling thereof are not very sensitive to the work hardening rate, since at any point in time, the test induces a wide range of stress in different parts of the sample. Poisson’s ratio for the unknown material is also set using the value for similar materials. All materials are modeled as isotropic, elastic–plastic solids, using the classical metal plasticity model in ABAQUS with a Mises yield surface and associated plastic flow [11]. This and the use of a two-dimensional axisymmetric mesh comprise the main simplifying assumptions for the modeling. For each simulation, an input file is constructed of the mesh description and material definitions, along with definitions of the loading and unloading history. This file is used as input for running ABAQUS on a desktop workstation (Digital AlphaStation 250 4/266), with a typical run time of 30–40 min. The output of the simulation is transferred to a spreadsheet for analysis. Typically, three or four simulations are performed with values for the unknown material Y and E that bracket the observed material response. The fit of each of these simulations to the experimental force at one or more fixed depths and the stiffness at the start of unloading are quantified, and the results used to extrapolate to optimized values of Y and E that should give a good fit to the data. The extrapolation assumes an approximately linear dependence of both the force and the stiffness on both Y and E, an assumption that works well for small deviations. By using this technique, which has been presented in more detail elsewhere [2–4], a very good fit of the simulation to the experiment can be obtained with five or six simulations. Statistical analysis of the fits provides a measure of the uncertainty in the derived values for Y and E, including errors in extrapolation and in the statistical spread in the experimental data. If the fit is done using values of the force at multiple depths, the error also reflects the overall fit to the experimental curve. The errors quoted in this paper use this approach. If the force at only one depth is used for the fit, as we have done in earlier work [1–4], the error does not include the quality of fit over the entire forcedepth range of the data. A commonly used parameter for comparing the strengths of materials is hardness, since it is deduced directly from indentation experiments on bulk materials and it provides a convenient measure of large-strain plastic response. The hardness, defined as the force divided by the projected area of contact, is available in the simulations at any step by outputting the force and the radial position of the edge of the contact between the indenter and the top surface. For the simulations of a thin layer, this is the combined hardness of layer+substrate, even though the fit has deduced Y and
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E for the layer alone. To obtain the hardness of the layer material alone, we do one additional simulation with a semi-infinite bulk ‘‘sample’’ described with the Y and E values deduced for the unknown layer material. The ‘‘intrinsic’’ hardness obtained from this final simulation can then be compared directly to measurements of bulk materials.
3. Examples 3.1. Ti+C implanted Ni The first example is a pure Ni sample implanted with 2×1017 Ti cm−2 at 180 keV, followed by 2×1017 C cm−2 at 45 keV [5–8]. Rutherford backscattering analysis (RBS) showed a peak Ti concentration of 18 at.% at the surface, decreasing with depth to ~180 nm, and a C concentration at the surface of 16 at.% and a peak of 22 at.% near 60 nm. Crosssectional transmission electron microscopy ( TEM ), shown in Fig. 1a, demonstrated that the ion-treated region was divided into three distinct regions: (1) a region of mixed crystalline and amorphous phases extending from the surface to 30 nm in depth, with a thin oxide on top, (2) a buried, fully amorphous layer from ~30 nm to ~80 nm depth, and (3) a highly dislocated crystalline region from ~80 nm to ~180 nm. Additional details of the microstructure are given elsewhere [7]. The amorphous layer in region 2 is the material of interest: it is similar to amorphous alloy layers formed in pure Fe and Fe-based alloys by ion implantation of
Ti and C, a treatment that results in a hard, low-friction layer [12–14]. The amorphous phase in this sample was thus expected to be harder than the other ion-modified layers and substantially harder than the substrate, but separating its properties from those of the two surrounding layers and the substrate required detailed modeling. The sample was modeled using a two-dimensional mesh 15 mm in radius and depth, with a 15-mm-thick indenter. Fig. 1b shows the central portion of the mesh, plotted during a simulation at the point of deepest indenter penetration, 70 nm. The diamond indenter was modeled as an isotropic, purely elastic solid with Poisson’s ratio n =0.100 and E =1160 GPa. Poisson’s i i ratio for pure Ni was used for the substrate and all modified layers, while Y and E for the substrate under the ion-modified layers were set at values determined by separate indentations and modeling for untreated Ni. This modeling had shown a near-surface increase in hardness for the pure Ni [6 ], so the yield stress for the substrate just under the ion-treated zone was set at 0.65 GPa, decaying to the bulk Ni value of 0.15 GPa at a depth of 900 nm. E was set at 220 GPa for the entire substrate. The experiment being simulated consisted of 10 separate indents to a depth of 70 nm, each in a fresh spot on the sample. The circles in Fig. 2a show the force vs. depth data from one of these indents, and dashed lines indicate the upper and lower standard deviations from the average for the 10 indents. The data show an inflection in slope at ~25 nm that is characteristic of thin hard layers on soft substrates. The shallowest portion of the indent is most sensitive to the hardness near the surface, whereas at greater depths, the substrate properties dominate the response. This difference as a function of depth allows us to extract depth information
Fig. 1. (a) Bright-field, cross-section TEM image of Ni implanted with Ti and C, with the f.c.c. Ni in diffraction contrast. (b) Axisymmetric mesh used to model the sample in (a), shown during a simulation with the indenter at maximum penetration. The gray areas mark the three layers with different assigned properties in the simulations.
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Fig. 2. (a) Force vs. depth for experimental and simulated indentations into the Ni( Ti,C ) sample of Fig. 1. Circles show a representative experimental result, and dotted lines show ±1 standard deviation for the set of 10 experimental indents. The dashed-dot line is a best-fit simulation using a single layer to model the ion-treated region. The solid line is the best fit using the three layer model. (b) Differences of the simulated data from the average of the 10 experimental curves ( loading only). The dashed-dot line is the single-layer model, and the solid line is the three-layer model. Dotted lines are again ±1 standard deviation for the experiment.
about the mechanical properties, using careful evaluations of the fitting of simulations to experiment. Since TEM shows that the ion-treated layer has three distinct regions, we modeled the sample with three layers, each of whose properties could be set individually. The grayed areas in Fig. 1b indicate the three regions. In the starting mesh, the regions were 30, 50 and 100 nm thick, to match the observed layers. Fig. 2b illustrates the sensitivity of the simulation fits to the partitioning of properties between the three regions. The difference between the simulated force and the average experimental force is plotted versus depth for the best-fit simulation using two different models. In one, shown by the dashed-dot line, all three layers were given the same properties for every simulation, i.e. the ion-treated material was modeled as a single layer. The best-fit modeled force is clearly too high in the 10–20-nm range and then too low from 30 nm on. The best-fit calculated force vs. depth curve for the single layer model is also plotted directly in Fig. 2a, where the deviation from experiment is not as clearly evident. The fitting for the single layer model deduced values for the ion-treated region of Y=4.1±0.3 GPa and E= 345±68 GPa. The above fitting results suggest that a more reason-
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able partitioning of the layer properties might be a softer surface layer, followed by a hard layer and then another relatively softer layer (although still harder than the substrate). The second modeling result in Fig. 2 was obtained using such properties for the three layers, and illustrates the improvement that can be obtained. The substrate properties were fixed as before, but the properties of the three layers were varied independently. A number of combinations were tried [6 ], and the partitioning that gave the best overall fit is shown in Fig. 2. The fit was not very sensitive to the partitioning of E between the layers but was reasonably sensitive to the choices of Y. The best results were obtained with the first layer’s Y fixed at the substrate value plus half the value for the second layer. Different values of Y and E for the third layer were chosen, and then for each, a series of simulations were performed that deduced the corresponding best values for Y and E for the second layer. The overall fits were then compared. Fig. 3 illustrates this process. Fig. 3a is a plot of the deduced yield strength Y for layer 2 as a function of the yield 2 strength Y chosen for layer 3. Each point shows the 3 value of Y that gave a best fit for the given Y ; multiple 2 3 points for the same value of Y represent fits using 3 different combinations of E. Fig. 3b is the corresponding overall fitting error for each of the points in the lower panel. Clearly, the overall fit is optimized with
Fig. 3. (a) Y for layer 2, as a function of Y chosen for layer 3. Each 2 3 square represents the best-fit value for Y from a series of simulations 2 with Y fixed. Y for layer 1 in each case was pinned midway between 3 1 Y and Y for the substrate. Multiple points at a single value of Y 2 3 represent the results using different choices for partitioning E between the three layers. (b) The overall fitting error (evaluated at 20, 40 and 60 nm) for each point in (a). The minimum at Y =2.7 GPa indicates 3 the best combination of Y2 and Y3.
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Y =2.7 GPa for the third layer, which gives 3 Y =4.7 GPa for the second layer. Of course, these 2 values are dependent on the choice for Y of the first layer and thus not uniquely determined, but varying Y for the first layer showed that this additional uncertainty is at most 10%. The values for the amorphous layer that were deduced by the above procedure were Y=4.7±0.13 GPa and E=416±104 GPa, which were obtained using Y= 2.57 GPa and E=318 GPa for the first layer and Y= 2.7 GPa and E=318 GPa for the third layer. The force vs. depth curve and the difference from the average experiment are shown in Fig. 2a and b, respectively. The three-layer simulation is seen to match the experimental data more closely over the entire depth range of the indent. A separate ‘‘bulk’’ simulation using the properties of the second, amorphous layer alone gave an intrinsic hardness for the amorphous material of 13.7±2 GPa. The properties of the amorphous layer are of most interest here and, whereas the above procedure gives values that are more uncertain than a fit to a single monolithic layer would have been, the inferred hardness and elasticity for the material are still expected to be accurate to within ±15%. More details on the properties of this sample and the modeling results are given in Refs. [7,8]. 3.2. Pulsed laser deposited DLC on Si The second example is of a simpler structure, a uniform deposited layer of diamond-like carbon (DLC ) on Si. The layer was deposited by pulsed laser deposition (PLD), with an in-situ annealing protocol that results in a very hard, nearly stress-free film [9]. The layer in this example is so thick at 1.2 mm that conventional wisdom would suggest that modeling is not needed: a simple analysis of the indentation data at <100 nm should give an H and E representative of the layer material without substrate effects. However, the modeling shows that for this very hard DLC material, the analytical derivation of H and E from nanoindentation data does not give an answer free of substrate effects. Fig. 4 shows a representative experimental force vs. depth curve and the best-fit simulation. The 10 indents done for this sample were essentially identical; the standard deviation of the force at 150 nm was only 0.33 out of 19 mN. The indentations were modeled by treating the layer as an elastic–plastic material with no work hardening. The residual compressive stress of 200 MPa, as measured by wafer curvature, was included in the modeling. We again used a two-dimensional mesh 15 mm in radius, depth, and indenter thickness. The depthindependent properties of the substrate were set to those determined by previous nanoindentation and modeling of Si [1]. The diamond indenter was again modeled as an isotropic, purely elastic solid with Poisson’s ratio
Fig. 4. Experimental force vs. depth curve for a representative nanoindentation of DLC/Si and the best-fit simulation using Y=77.6 GPa and E=1100 GPa for the layer.
n =0.100 and E =1160 GPa. A Poisson’s ratio of n= i i 0.100 was also used for the DLC layer. Treating the diamond in the indenter as purely elastic was justified by the reproducibility of the data and separate checks of the indenter shape before and after indenting the DLC layers. Since no changes were observed, the indenter was not yielding during these experiments. The modeling shown in Fig. 4 deduced properties for the layer of Y=77.6±2.4 GPa and E=1101±18 GPa, for which the separate ‘‘bulk’’ simulation gave an intrinsic hardness of 88.1±2.2 GPa. Conventional analysis of the experimental data, using the Oliver–Pharr method [10], gives H=80.2±1.2 GPa and E=551±3 GPa. As noted above, since the indent depth is only 13% of the layer thickness, the Oliver–Pharr analysis might be expected to give H and E for the layer which are relatively independent of the substrate. Whereas the value of H from analysis is within 10% of that derived by modeling, the value of E is substantially lower. To investigate further whether the discrepancy for E is a problem in the modeling or in the conventional analysis, we modeled an additional set of nanoindentation data that was collected from this sample using a different approach, the continuous stiffness measurement (CSM ) [10]. In a CSM experiment, the indenter is loaded as before, but an AC modulation is superimposed on the loading. By using the AC response of the sample, its stiffness can be derived for all depths, rather than just at the end of the loading. Analysis of the force and stiffness data, again using the Oliver–Pharr method [10], then gives both H and E as a function of depth. Typically, 10 indents are made and the results averaged. The open circles in Fig. 5a and b show the force and stiffness, respectively, obtained from the 1.2-mm DLC layer on Si using CSM. We modeled the CSM experi-
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Fig. 6. Hardness vs. depth from the CSM experiment and from the simulation.
Fig. 5. (a) Experimental and simulated force vs. depth from a CSM indentation of the DLC/Si sample. (b) Experimental and simulated stiffness vs. depth for the same CSM indentation. The simulation was done with Y=77.6 GPa and E=1100 GPa for the DLC layer.
ment by adding a 0.5-nm unload at 5-nm intervals to the loading history in the computer. Each short unload gives a measurement of the combined stiffness at that depth, which can then be compared to the experiment along with the force. The solid lines in Fig. 5a and b are the results of a single simulation using the properties of the DLC already derived by modeling conventional indentations. That is, there is no fitting involved in this case—the CSM experiment is being simulated using Y= 77.6 GPa and E=1101 GPa for the layer. The agreement with the experiment is very good for both force and stiffness, indicating that our modeling approach for the CSM method is valid and that the Y and E already derived for the layer are accurate. Once the force and stiffness are measured, the experimental hardness is derived from them using the Oliver–Pharr method [10], which deduces the area of contact from the measured stiffness. The open circles in Fig. 6 show H as a function of depth derived in this way from the experimental CSM data. The solid line, however, is H taken directly from the simulation, calculated as the force divided by projected contact area. Remember that in this case H is the combined H for layer+substrate. Since the layer is so thick, H is essentially equal to that of a ‘‘bulk’’ simulation. As in Fig. 5, the agreement between H derived from experiment and H taken directly from the simulation is good, suggesting that the analytical derivation of H is working well. This also implies that the contact area determined by the
Oliver–Pharr method agrees reasonably well with the calculated contact area. The final step in the analysis of experimental CSM data is to use the contact area derived from the Oliver–Pharr method with the measured stiffness to deduce the modulus of elasticity. The method is based on Sneddon’s analysis of the elastic contact between a rigid, axisymmetric punch and an elastic half space [15,16 ]. Pharr et al. [17] have shown that the solution is independent of the geometry of the punch, leading to the following equation for the modulus: 1 Ep S E= , r b 2 EA
(1)
where S is the stiffness and A the contact area. b is a small correction for the non-axisymmetric indenter shape (1.034 for a Berkovich tip) [18]. The derivation of Eq. (1) uses several assumptions, but the most important is that of vertical displacements prescribed by the original indenter shape, i.e. a rigid indenter. To account for a non-rigid indenter, the modulus derived by Eq. (1) is assumed to be a reduced modulus E defined by: r 1−n2 1−n2 −1 i s , (2) E= r E E i where E and n are Young’s modulus and Poisson’s ratio s for the sample, and E and n are Young’s modulus and i i Poisson’s ratio for the indenter. The underlying approximation is that Sneddon’s solution for a rigid indenter can be used, with the reduced modulus accounting for any deformation in the indenter. For most bulk materials, this seems to work very well, but for indenting very hard materials, such as the DLC layer shown here, the approximation may be inadequate. The open circles in Fig. 7 show E as a function of depth derived from the experimental CSM data using
C
D
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4. Conclusions
Fig. 7. Elastic modulus vs. depth derived from the CSM data and from the simulated data using Sneddon’s solution. The modulus actually used in the simulation is marked by a dashed line.
Eqs. (1) and (2). The solid line is the same derivation of E using Eqs. (1) and (2), but with the calculated area and stiffness from the simulation. The dashed line at 1100 is the value of E actually used for the DLC layer in the simulation. The agreement between E derived from experimental and simulated data simply confirms that the modeling is working well, but both are much lower than the value of E used as an input to the simulation. The problem with the analytical derivation of E is at least partly due to a much stronger substrate effect than would normally be expected for an indent that is only 13% of the layer thickness. The downward slope of E at deeper depths in the CSM experimental result suggests a substrate effect, as might be expected due to the extreme hardness of the DLC material relative to Si. However, another simulation of this CSM experiment using a ‘‘bulk’’ sample of the DLC material still gives an analytically derived value of E that is somewhat lower than the input to the simulation. Since there is no substrate effect for this latter simulation, this suggests that the approximations underlying the analysis based on Sneddon’s solution may be failing for this extremely hard material, as discussed above. Examination of the mesh in the simulation of Figs. 5–7 supports this interpretation: at 160-nm depth, the indenter is dramatically deformed, being compressed by 64 nm at its center. Such a distortion of the tip is not unexpected since the layer material is approaching both the hardness and the elasticity of pure diamond. As a check, a simulation of CSM data obtained from fused silica, which is much softer, does not exhibit the same problem with the derivation of E; Sneddon’s solution applied to the simulated results gives the same E that is used as an input to the simulation. In that case, the indenter deformation is minimal, consistent with our hypothesis.
We have used finite-element modeling to extract yield stress, Young’s modulus and intrinsic hardness for thin films and implanted structures in a variety of systems, with many of direct application to coatings for MEMS. The examples shown here were Ni implanted with C and Ti, and very hard DLC layers deposited by pulsed laser deposition. The modeling extends the applicability of nanoindentation to a number of problems where accurate measurement of the properties would be difficult, if not impossible, without it. These include multilayer samples, samples whose properties vary with depth, layers with pre-existing stress, very thin layers, and samples of very hard materials. In the latter case, modeling may be required even for bulk samples. Our results indicate that the conventional analyses based on the Oliver–Pharr method [11] and Sneddon’s solution [15–17] may be inadequate for properly deriving the elasticity of materials that are approaching the hardness of the diamond in the indenter. Further investigation of the possible limits of the analysis based on Sneddon’s solution for such hard materials is underway. The limitations of the modeling include the inability to model penetration of a layer, an assumption of perfect bonding between layers, and the inability to model any time-dependent behavior, such as creep. The absolute uncertainty of the evaluated mechanical properties is judged to be in the order of 10% or better in most cases.
Acknowledgement Technical assistance by K.G. Minor, G.A. Petersen and M.P. Moran is gratefully acknowledged. Careful nanoindentation measurements by Barry Lucas at Nano Instruments are appreciated. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000. This work was supported in part by the BES Center for Synthesis and Processing and in part by the SNL Laboratory Directed Research and Development Program.
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J.A. Knapp et al. / Surface and Coatings Technology 103–104 (1998) 268–275 [5] S.M. Myers, D.M. Follstaedt, J.A. Knapp, T.A. Christenson, Mater. Res. Soc. Symp. Proc. 444 (1997) 99. [6 ] S.M. Myers, J.A. Knapp, D.M. Follstaedt, M.T. Dugger, J. Appl. Phys. 83 (1998) 1256. [7] D.M. Follstaedt, S.M. Myers, J.A. Knapp, M.T. Dugger, T.A. Christenson, Surf. Coat. Technol. this issue. [8] S.M. Myers, J.A. Knapp, D.M. Follstaedt, M.T. Dugger, T.A. Christenson, Surf. Coat. Technol. this issue. [9] T.A. Friedmann, J.P. Sullivan, J.A. Knapp, D.R. Tallant, D.M. Follstaedt, D.L. Medlin, P.B. Mirkarimi, Appl. Phys. Lett. 71 (1997) 3820. [10] W.C. Oliver, G.M. Pharr, J. Mater. Res. 7 (1992) 1564. [11] F.A. McClintock, A.S. Argon ( Eds.), Mechanical Behavior of Materials, Addison-Wesley, Reading, MA, 1966, pp. 276–279.
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Finite-element modeling of nanoindentation for evaluating mechanical properties of MEMS materials J.A. Knapp a,*, D.M. Follstaedt a, S.M. Myers a, J.C. Barbour a, T.A. Friedmann a, J.W. Ager IIIb, O.R. Monteiro b, I.G. Brown b a Sandia National Laboratories, Albuquerque, NM 87185-1056, USA b Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
Abstract We have developed procedures based on finite-element modeling of nanoindentation data to extract mechanical properties from thin, hard films and ion-beam-modified layers on softer substrates. The method accurately deduces the yield stress, Young’s modulus, and hardness from indentations as deep as 50% of the layer thickness. We use these procedures to evaluate two hard layers potentially useful for reducing friction and wear of components in micro-electromechanical systems (MEMS): Ni implanted with Ti and C, and diamond-like carbon layers. We show that the modeling works well even for materials whose hardness approaches that of diamond, a case where commonly used analytical methods for deducing the modulus fail. © 1998 Elsevier Science S.A. Keywords: Modeling; Mechanical properties; MEM; Nanoindentation
1. Introduction As materials applications such as micro-electromechanical systems (MEMS ) use smaller structures and thinner coatings, characterization of the mechanical properties of the materials has become more difficult. One method that has become widely accepted is depthsensing indentation at low loads, or ‘‘nanoindentation’’. However, as the thickness of the materials continues to decrease, obtaining a substrate-independent measure of the mechanical properties of layers becomes increasingly complicated due to the influence of the underlying material. The problems are particularly severe for application of nanoindentation to implantation-modified layers, since the properties of a layer may vary through its depth. For films with hardnesses that may approach that of bulk diamond, the deformation and possible yielding of the indenter tip are additional complications. Since analytical derivation of the mechanical properties is often not feasible for these cases, we have developed procedures based on finite-element modeling to reliably extract mechanical properties for thin, hard films on * Corresponding author. Tel: +1 505 844 2305; Fax: +1 505 844 7775; e-mail: [email protected] 0257-8972/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S 02 5 7 -8 9 7 2 ( 9 8 ) 0 0 42 2 - 8
softer substrates [1–9]. We developed PC-based software to generate meshes and depth-dependent material property descriptions specific to each sample, while the simulations of the response of the materials to nanoindentation are performed on a UNIX workstation using the commercial, large-strain, finite-element code ABAQUS/Standard (ABAQUS Version 5.6, Hibbitt, Karlsson and Sorensen, Inc., Pawtucket, RI ). The results are transferred back to the PC for analysis. Blunting of the indenter tip, friction between tip and surface, and pre-existing stress in the layers can all be included in the modeling. Two parameters for the layer are extracted from the fit of simulation to experiment: the yield stress Y (defined at a plastic strain of 0.002) and Young’s elastic modulus E. An additional simulation using the yield stress and elasticity of the layer for a hypothetical bulk ‘‘sample’’ then deduces the intrinsic hardness of the layer material, where the hardness H is defined as force/(projected contact area) during contact. These methods have been applied to several hard materials with potential application for reducing friction and wear in MEMS components: Fe, Ni, and Ni–Fe alloys implanted with Ti and C [4–8], deposited ‘‘diamondlike’’ carbon layers with hardnesses of 70–90 GPa [4,9], and AlO layers formed by ECR plasma deposition or x pulsed laser deposition [1–4].
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2. Modeling of nanoindentation The nanoindentation measurement consists of pushing a three-sided Berkovich-shaped diamond indenter into the sample under tight load control. (All of the nanoindentation tests were performed at Nano Instruments, Inc., Knoxville, TN.) The measured force as a function of depth during loading is characteristic of the resistance of the material to deformation, including in general both elastic and plastic responses. During unloading of the indenter, the decrease of force with depth is controlled by the elastic response of the material. The initial rate of change with depth of the decrease in force (i.e. the slope) is defined as the sample stiffness. In a standard nanoindentation measurement, the indenter is inserted to some specified depth and then extracted, giving the loading force over the whole depth range but giving a value for stiffness only at the deepest portion of the test. For a bulk sample or a relatively thick film, analytical methods have been developed to deduce H and E from the data [10], but since the analysis requires both the force and the stiffness, H and E are deduced only at the deepest penetration. For a thin layer, such an analysis gives an H and E that are due to the combined response of the layer and substrate. One of the primary advantages of finite-element modeling is the ability to separate the properties of a layer from those of the substrate, and an example is given here for an ion-implanted sample. Another advantage is the ability to account for distortions of the indenter by very hard materials. Our second example is an application to a DLC layer whose hardness approaches that of diamond, where conventional analysis fails to deduce E correctly, even for a relatively thick film. This second example will also demonstrate modeling for another type of indentation measurement, termed a continuous stiffness measurement (CSM ) [10]. To model each set of nanoindentation measurements, we first construct a mesh that reflects as accurately as possible the structure of the sample ( layer thicknesses) and the shape of the diamond indenter. This includes the area vs. depth function of the indenter, as determined separately with indents of fused silica [10], so that any blunting of the indenter near the tip is accounted for. The sample and indenter were approximated by twodimensional axisymmetric meshes for all the work presented here. We have compared three-dimensional calculations to the two-dimensional approximation and have confirmed that the two approaches give results within a few per cent of each other, but with greatly increased computational efficiency for the latter [1–3]. Materials parameters for all the known materials such as the substrate and the indenter are specified, as well as the proposed properties of the unknown layer. The work-hardening rate for the unknown layer (i.e. the slope of the stress–strain curve above the yield stress) is
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fixed at a value characteristic of the material being studied, as taken from the literature for similar materials. For amorphous or non-metallic materials, the hardening is assumed to be zero. The indentation and modeling thereof are not very sensitive to the work hardening rate, since at any point in time, the test induces a wide range of stress in different parts of the sample. Poisson’s ratio for the unknown material is also set using the value for similar materials. All materials are modeled as isotropic, elastic–plastic solids, using the classical metal plasticity model in ABAQUS with a Mises yield surface and associated plastic flow [11]. This and the use of a two-dimensional axisymmetric mesh comprise the main simplifying assumptions for the modeling. For each simulation, an input file is constructed of the mesh description and material definitions, along with definitions of the loading and unloading history. This file is used as input for running ABAQUS on a desktop workstation (Digital AlphaStation 250 4/266), with a typical run time of 30–40 min. The output of the simulation is transferred to a spreadsheet for analysis. Typically, three or four simulations are performed with values for the unknown material Y and E that bracket the observed material response. The fit of each of these simulations to the experimental force at one or more fixed depths and the stiffness at the start of unloading are quantified, and the results used to extrapolate to optimized values of Y and E that should give a good fit to the data. The extrapolation assumes an approximately linear dependence of both the force and the stiffness on both Y and E, an assumption that works well for small deviations. By using this technique, which has been presented in more detail elsewhere [2–4], a very good fit of the simulation to the experiment can be obtained with five or six simulations. Statistical analysis of the fits provides a measure of the uncertainty in the derived values for Y and E, including errors in extrapolation and in the statistical spread in the experimental data. If the fit is done using values of the force at multiple depths, the error also reflects the overall fit to the experimental curve. The errors quoted in this paper use this approach. If the force at only one depth is used for the fit, as we have done in earlier work [1–4], the error does not include the quality of fit over the entire forcedepth range of the data. A commonly used parameter for comparing the strengths of materials is hardness, since it is deduced directly from indentation experiments on bulk materials and it provides a convenient measure of large-strain plastic response. The hardness, defined as the force divided by the projected area of contact, is available in the simulations at any step by outputting the force and the radial position of the edge of the contact between the indenter and the top surface. For the simulations of a thin layer, this is the combined hardness of layer+substrate, even though the fit has deduced Y and
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E for the layer alone. To obtain the hardness of the layer material alone, we do one additional simulation with a semi-infinite bulk ‘‘sample’’ described with the Y and E values deduced for the unknown layer material. The ‘‘intrinsic’’ hardness obtained from this final simulation can then be compared directly to measurements of bulk materials.
3. Examples 3.1. Ti+C implanted Ni The first example is a pure Ni sample implanted with 2×1017 Ti cm−2 at 180 keV, followed by 2×1017 C cm−2 at 45 keV [5–8]. Rutherford backscattering analysis (RBS) showed a peak Ti concentration of 18 at.% at the surface, decreasing with depth to ~180 nm, and a C concentration at the surface of 16 at.% and a peak of 22 at.% near 60 nm. Crosssectional transmission electron microscopy ( TEM ), shown in Fig. 1a, demonstrated that the ion-treated region was divided into three distinct regions: (1) a region of mixed crystalline and amorphous phases extending from the surface to 30 nm in depth, with a thin oxide on top, (2) a buried, fully amorphous layer from ~30 nm to ~80 nm depth, and (3) a highly dislocated crystalline region from ~80 nm to ~180 nm. Additional details of the microstructure are given elsewhere [7]. The amorphous layer in region 2 is the material of interest: it is similar to amorphous alloy layers formed in pure Fe and Fe-based alloys by ion implantation of
Ti and C, a treatment that results in a hard, low-friction layer [12–14]. The amorphous phase in this sample was thus expected to be harder than the other ion-modified layers and substantially harder than the substrate, but separating its properties from those of the two surrounding layers and the substrate required detailed modeling. The sample was modeled using a two-dimensional mesh 15 mm in radius and depth, with a 15-mm-thick indenter. Fig. 1b shows the central portion of the mesh, plotted during a simulation at the point of deepest indenter penetration, 70 nm. The diamond indenter was modeled as an isotropic, purely elastic solid with Poisson’s ratio n =0.100 and E =1160 GPa. Poisson’s i i ratio for pure Ni was used for the substrate and all modified layers, while Y and E for the substrate under the ion-modified layers were set at values determined by separate indentations and modeling for untreated Ni. This modeling had shown a near-surface increase in hardness for the pure Ni [6 ], so the yield stress for the substrate just under the ion-treated zone was set at 0.65 GPa, decaying to the bulk Ni value of 0.15 GPa at a depth of 900 nm. E was set at 220 GPa for the entire substrate. The experiment being simulated consisted of 10 separate indents to a depth of 70 nm, each in a fresh spot on the sample. The circles in Fig. 2a show the force vs. depth data from one of these indents, and dashed lines indicate the upper and lower standard deviations from the average for the 10 indents. The data show an inflection in slope at ~25 nm that is characteristic of thin hard layers on soft substrates. The shallowest portion of the indent is most sensitive to the hardness near the surface, whereas at greater depths, the substrate properties dominate the response. This difference as a function of depth allows us to extract depth information
Fig. 1. (a) Bright-field, cross-section TEM image of Ni implanted with Ti and C, with the f.c.c. Ni in diffraction contrast. (b) Axisymmetric mesh used to model the sample in (a), shown during a simulation with the indenter at maximum penetration. The gray areas mark the three layers with different assigned properties in the simulations.
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Fig. 2. (a) Force vs. depth for experimental and simulated indentations into the Ni( Ti,C ) sample of Fig. 1. Circles show a representative experimental result, and dotted lines show ±1 standard deviation for the set of 10 experimental indents. The dashed-dot line is a best-fit simulation using a single layer to model the ion-treated region. The solid line is the best fit using the three layer model. (b) Differences of the simulated data from the average of the 10 experimental curves ( loading only). The dashed-dot line is the single-layer model, and the solid line is the three-layer model. Dotted lines are again ±1 standard deviation for the experiment.
about the mechanical properties, using careful evaluations of the fitting of simulations to experiment. Since TEM shows that the ion-treated layer has three distinct regions, we modeled the sample with three layers, each of whose properties could be set individually. The grayed areas in Fig. 1b indicate the three regions. In the starting mesh, the regions were 30, 50 and 100 nm thick, to match the observed layers. Fig. 2b illustrates the sensitivity of the simulation fits to the partitioning of properties between the three regions. The difference between the simulated force and the average experimental force is plotted versus depth for the best-fit simulation using two different models. In one, shown by the dashed-dot line, all three layers were given the same properties for every simulation, i.e. the ion-treated material was modeled as a single layer. The best-fit modeled force is clearly too high in the 10–20-nm range and then too low from 30 nm on. The best-fit calculated force vs. depth curve for the single layer model is also plotted directly in Fig. 2a, where the deviation from experiment is not as clearly evident. The fitting for the single layer model deduced values for the ion-treated region of Y=4.1±0.3 GPa and E= 345±68 GPa. The above fitting results suggest that a more reason-
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able partitioning of the layer properties might be a softer surface layer, followed by a hard layer and then another relatively softer layer (although still harder than the substrate). The second modeling result in Fig. 2 was obtained using such properties for the three layers, and illustrates the improvement that can be obtained. The substrate properties were fixed as before, but the properties of the three layers were varied independently. A number of combinations were tried [6 ], and the partitioning that gave the best overall fit is shown in Fig. 2. The fit was not very sensitive to the partitioning of E between the layers but was reasonably sensitive to the choices of Y. The best results were obtained with the first layer’s Y fixed at the substrate value plus half the value for the second layer. Different values of Y and E for the third layer were chosen, and then for each, a series of simulations were performed that deduced the corresponding best values for Y and E for the second layer. The overall fits were then compared. Fig. 3 illustrates this process. Fig. 3a is a plot of the deduced yield strength Y for layer 2 as a function of the yield 2 strength Y chosen for layer 3. Each point shows the 3 value of Y that gave a best fit for the given Y ; multiple 2 3 points for the same value of Y represent fits using 3 different combinations of E. Fig. 3b is the corresponding overall fitting error for each of the points in the lower panel. Clearly, the overall fit is optimized with
Fig. 3. (a) Y for layer 2, as a function of Y chosen for layer 3. Each 2 3 square represents the best-fit value for Y from a series of simulations 2 with Y fixed. Y for layer 1 in each case was pinned midway between 3 1 Y and Y for the substrate. Multiple points at a single value of Y 2 3 represent the results using different choices for partitioning E between the three layers. (b) The overall fitting error (evaluated at 20, 40 and 60 nm) for each point in (a). The minimum at Y =2.7 GPa indicates 3 the best combination of Y2 and Y3.
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Y =2.7 GPa for the third layer, which gives 3 Y =4.7 GPa for the second layer. Of course, these 2 values are dependent on the choice for Y of the first layer and thus not uniquely determined, but varying Y for the first layer showed that this additional uncertainty is at most 10%. The values for the amorphous layer that were deduced by the above procedure were Y=4.7±0.13 GPa and E=416±104 GPa, which were obtained using Y= 2.57 GPa and E=318 GPa for the first layer and Y= 2.7 GPa and E=318 GPa for the third layer. The force vs. depth curve and the difference from the average experiment are shown in Fig. 2a and b, respectively. The three-layer simulation is seen to match the experimental data more closely over the entire depth range of the indent. A separate ‘‘bulk’’ simulation using the properties of the second, amorphous layer alone gave an intrinsic hardness for the amorphous material of 13.7±2 GPa. The properties of the amorphous layer are of most interest here and, whereas the above procedure gives values that are more uncertain than a fit to a single monolithic layer would have been, the inferred hardness and elasticity for the material are still expected to be accurate to within ±15%. More details on the properties of this sample and the modeling results are given in Refs. [7,8]. 3.2. Pulsed laser deposited DLC on Si The second example is of a simpler structure, a uniform deposited layer of diamond-like carbon (DLC ) on Si. The layer was deposited by pulsed laser deposition (PLD), with an in-situ annealing protocol that results in a very hard, nearly stress-free film [9]. The layer in this example is so thick at 1.2 mm that conventional wisdom would suggest that modeling is not needed: a simple analysis of the indentation data at <100 nm should give an H and E representative of the layer material without substrate effects. However, the modeling shows that for this very hard DLC material, the analytical derivation of H and E from nanoindentation data does not give an answer free of substrate effects. Fig. 4 shows a representative experimental force vs. depth curve and the best-fit simulation. The 10 indents done for this sample were essentially identical; the standard deviation of the force at 150 nm was only 0.33 out of 19 mN. The indentations were modeled by treating the layer as an elastic–plastic material with no work hardening. The residual compressive stress of 200 MPa, as measured by wafer curvature, was included in the modeling. We again used a two-dimensional mesh 15 mm in radius, depth, and indenter thickness. The depthindependent properties of the substrate were set to those determined by previous nanoindentation and modeling of Si [1]. The diamond indenter was again modeled as an isotropic, purely elastic solid with Poisson’s ratio
Fig. 4. Experimental force vs. depth curve for a representative nanoindentation of DLC/Si and the best-fit simulation using Y=77.6 GPa and E=1100 GPa for the layer.
n =0.100 and E =1160 GPa. A Poisson’s ratio of n= i i 0.100 was also used for the DLC layer. Treating the diamond in the indenter as purely elastic was justified by the reproducibility of the data and separate checks of the indenter shape before and after indenting the DLC layers. Since no changes were observed, the indenter was not yielding during these experiments. The modeling shown in Fig. 4 deduced properties for the layer of Y=77.6±2.4 GPa and E=1101±18 GPa, for which the separate ‘‘bulk’’ simulation gave an intrinsic hardness of 88.1±2.2 GPa. Conventional analysis of the experimental data, using the Oliver–Pharr method [10], gives H=80.2±1.2 GPa and E=551±3 GPa. As noted above, since the indent depth is only 13% of the layer thickness, the Oliver–Pharr analysis might be expected to give H and E for the layer which are relatively independent of the substrate. Whereas the value of H from analysis is within 10% of that derived by modeling, the value of E is substantially lower. To investigate further whether the discrepancy for E is a problem in the modeling or in the conventional analysis, we modeled an additional set of nanoindentation data that was collected from this sample using a different approach, the continuous stiffness measurement (CSM ) [10]. In a CSM experiment, the indenter is loaded as before, but an AC modulation is superimposed on the loading. By using the AC response of the sample, its stiffness can be derived for all depths, rather than just at the end of the loading. Analysis of the force and stiffness data, again using the Oliver–Pharr method [10], then gives both H and E as a function of depth. Typically, 10 indents are made and the results averaged. The open circles in Fig. 5a and b show the force and stiffness, respectively, obtained from the 1.2-mm DLC layer on Si using CSM. We modeled the CSM experi-
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Fig. 6. Hardness vs. depth from the CSM experiment and from the simulation.
Fig. 5. (a) Experimental and simulated force vs. depth from a CSM indentation of the DLC/Si sample. (b) Experimental and simulated stiffness vs. depth for the same CSM indentation. The simulation was done with Y=77.6 GPa and E=1100 GPa for the DLC layer.
ment by adding a 0.5-nm unload at 5-nm intervals to the loading history in the computer. Each short unload gives a measurement of the combined stiffness at that depth, which can then be compared to the experiment along with the force. The solid lines in Fig. 5a and b are the results of a single simulation using the properties of the DLC already derived by modeling conventional indentations. That is, there is no fitting involved in this case—the CSM experiment is being simulated using Y= 77.6 GPa and E=1101 GPa for the layer. The agreement with the experiment is very good for both force and stiffness, indicating that our modeling approach for the CSM method is valid and that the Y and E already derived for the layer are accurate. Once the force and stiffness are measured, the experimental hardness is derived from them using the Oliver–Pharr method [10], which deduces the area of contact from the measured stiffness. The open circles in Fig. 6 show H as a function of depth derived in this way from the experimental CSM data. The solid line, however, is H taken directly from the simulation, calculated as the force divided by projected contact area. Remember that in this case H is the combined H for layer+substrate. Since the layer is so thick, H is essentially equal to that of a ‘‘bulk’’ simulation. As in Fig. 5, the agreement between H derived from experiment and H taken directly from the simulation is good, suggesting that the analytical derivation of H is working well. This also implies that the contact area determined by the
Oliver–Pharr method agrees reasonably well with the calculated contact area. The final step in the analysis of experimental CSM data is to use the contact area derived from the Oliver–Pharr method with the measured stiffness to deduce the modulus of elasticity. The method is based on Sneddon’s analysis of the elastic contact between a rigid, axisymmetric punch and an elastic half space [15,16 ]. Pharr et al. [17] have shown that the solution is independent of the geometry of the punch, leading to the following equation for the modulus: 1 Ep S E= , r b 2 EA
(1)
where S is the stiffness and A the contact area. b is a small correction for the non-axisymmetric indenter shape (1.034 for a Berkovich tip) [18]. The derivation of Eq. (1) uses several assumptions, but the most important is that of vertical displacements prescribed by the original indenter shape, i.e. a rigid indenter. To account for a non-rigid indenter, the modulus derived by Eq. (1) is assumed to be a reduced modulus E defined by: r 1−n2 1−n2 −1 i s , (2) E= r E E i where E and n are Young’s modulus and Poisson’s ratio s for the sample, and E and n are Young’s modulus and i i Poisson’s ratio for the indenter. The underlying approximation is that Sneddon’s solution for a rigid indenter can be used, with the reduced modulus accounting for any deformation in the indenter. For most bulk materials, this seems to work very well, but for indenting very hard materials, such as the DLC layer shown here, the approximation may be inadequate. The open circles in Fig. 7 show E as a function of depth derived from the experimental CSM data using
C
D
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4. Conclusions
Fig. 7. Elastic modulus vs. depth derived from the CSM data and from the simulated data using Sneddon’s solution. The modulus actually used in the simulation is marked by a dashed line.
Eqs. (1) and (2). The solid line is the same derivation of E using Eqs. (1) and (2), but with the calculated area and stiffness from the simulation. The dashed line at 1100 is the value of E actually used for the DLC layer in the simulation. The agreement between E derived from experimental and simulated data simply confirms that the modeling is working well, but both are much lower than the value of E used as an input to the simulation. The problem with the analytical derivation of E is at least partly due to a much stronger substrate effect than would normally be expected for an indent that is only 13% of the layer thickness. The downward slope of E at deeper depths in the CSM experimental result suggests a substrate effect, as might be expected due to the extreme hardness of the DLC material relative to Si. However, another simulation of this CSM experiment using a ‘‘bulk’’ sample of the DLC material still gives an analytically derived value of E that is somewhat lower than the input to the simulation. Since there is no substrate effect for this latter simulation, this suggests that the approximations underlying the analysis based on Sneddon’s solution may be failing for this extremely hard material, as discussed above. Examination of the mesh in the simulation of Figs. 5–7 supports this interpretation: at 160-nm depth, the indenter is dramatically deformed, being compressed by 64 nm at its center. Such a distortion of the tip is not unexpected since the layer material is approaching both the hardness and the elasticity of pure diamond. As a check, a simulation of CSM data obtained from fused silica, which is much softer, does not exhibit the same problem with the derivation of E; Sneddon’s solution applied to the simulated results gives the same E that is used as an input to the simulation. In that case, the indenter deformation is minimal, consistent with our hypothesis.
We have used finite-element modeling to extract yield stress, Young’s modulus and intrinsic hardness for thin films and implanted structures in a variety of systems, with many of direct application to coatings for MEMS. The examples shown here were Ni implanted with C and Ti, and very hard DLC layers deposited by pulsed laser deposition. The modeling extends the applicability of nanoindentation to a number of problems where accurate measurement of the properties would be difficult, if not impossible, without it. These include multilayer samples, samples whose properties vary with depth, layers with pre-existing stress, very thin layers, and samples of very hard materials. In the latter case, modeling may be required even for bulk samples. Our results indicate that the conventional analyses based on the Oliver–Pharr method [11] and Sneddon’s solution [15–17] may be inadequate for properly deriving the elasticity of materials that are approaching the hardness of the diamond in the indenter. Further investigation of the possible limits of the analysis based on Sneddon’s solution for such hard materials is underway. The limitations of the modeling include the inability to model penetration of a layer, an assumption of perfect bonding between layers, and the inability to model any time-dependent behavior, such as creep. The absolute uncertainty of the evaluated mechanical properties is judged to be in the order of 10% or better in most cases.
Acknowledgement Technical assistance by K.G. Minor, G.A. Petersen and M.P. Moran is gratefully acknowledged. Careful nanoindentation measurements by Barry Lucas at Nano Instruments are appreciated. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000. This work was supported in part by the BES Center for Synthesis and Processing and in part by the SNL Laboratory Directed Research and Development Program.
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