Correction to: The Effect of Edge Compliance on the Contact between a Spherical Indenter and a High-Aspect-Ratio Rectangular Fin
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KeywordsNanoscale contact mechanics Edge compliance Conjugate gradient method Contact resonance atomic force microscopy
One of the common methods for testing mechanical properties of materials at the small scale is that of indentation with direct measurements of the applied load and relative deformation between an indenter and the material tested. The technique was initially developed at the micrometer scale as instrumented indentation [1, 2, 3, 4] and afterwards extended to the nanoscale as atomic force microscopy (AFM)-based indentation [5, 6, 7, 8]. A common quantity sought to be determined in these measurements is the Young’s modulus of the material tested based on the determination of the contact stiffness of the elastically deformed tip-sample contact. As such, the measured applied load and deformation of the tip-sample contact are used in a contact mechanics model that best describes the contact geometry and contact deformation during the test. The common choice of contact models for interpreting indentation tests at small scale are those with simple analytical closed-form solutions, either the Hertz model [9, 10] for frictionless non-adhesive contacts or Johnson-Kendall-Roberts (JKR)  (with adhesion inside the contact area) and Derjaguin-Muller-Toporov (DMT)  (with adhesion outside the contact area) models for frictionless adhesive contacts. However, these contact models are valid for contacts between two elastically homogeneous bodies with continuous curved surfaces. When applied to the contact between an indenter and a half-space surface, the contact must be placed far away from any edge, as the models don’t account for edge compliances.
It is the purpose of this work to develop and test a boundary element method (BEM) for small scale indentation on samples with edge geometries in the presence of adhesion. Specifically, the specimens are in the form of high-aspect ratio fins for which both edges contribute to the compliance of the structure. In previous work , we used intermittent contact resonance AFM (ICR-AFM)  to measure the contact stiffness as a function of applied load and position from the edge of the fin for fabricated low-k dielectric fins with width in the range of tens of nanometers. Such structures are incorporated in today’s electronics to form low-k dielectric patterns in Cu circuits . As the density of circuits steadily increased in the last decade, the size of low-k dielectric structures was reduced to the point where the surface to volume ratio substantially affects the material properties of the fabricated structure [16, 17, 18]. It has been shown that significant changes in the mechanical properties of these nanoscale low-k dielectric fins are due to plasma exposure during processing. The semi-analytic model used in  incorporates the edge compliance as a fit parameter to interpret the contact stiffness measurements as a function of applied load and distance from the edges of the fin. In this work, we developed a numerical model to eliminate this fit parameter and recover the edge compliance directly from measurements for the given measurement conditions and contact geometry.
The effect of edge compliance on the contact mechanics between an indenter and a tested quarter-space was investigated theoretically from the perspective of real applications like rolling-element bearings and rail wheels by Hanson and Keer  and Yu and Keer . Their work was built around the quarter-space contact problem introduced by Hetenyi about 50 years ago . As defined by Hetenyi , a reflection iterative method of two overlapped half-spaces can be used to resolve the elastic response of a quarter-space under the action of a concentrated normal load. The resulting integral equations are solved by means of numerical discretization based on the Love’s solution  for stress and strain generated by uniformly loaded patches. Over several years various improvements and developments were made in terms of the numerical procedure [23, 24, 25, 26]. Recently, the quarter-space contact problem was solved by Zhang et al.  in a matrix formulation with a direct solution for Hetenyi’s infinite iteration of the overlapped half-spaces. Moreover, this method was extended also to the case of a loaded rectangular fin geometry that includes the contributions of both edges to the compliance of the structure . In all the above work, the contact interaction between the indenter and the loaded surface was considered frictionless and non-adhesive.
To provide more versatility and applicability to indentation tests performed at low applied forces on fins, we added here the contribution of adhesive forces to the contact between a rigid spherical indenter and a rectangular fin. It extends our previous work  on the effect of edge compliance to the adhesive contact between a spherical indenter and a quarter-space. Unlike our previous work on a quarter-space , now both free-edge surfaces were considered as in the work of Zhang et al. . Contact adhesion was assumed as in the Maugis-Dugdale (MD) transition model  to account for broad range of adhesive contact interactions. In the MD model, the contact stress underneath the indenter goes from compressive in the middle of the contact area to tensile at the edge of the contact area. It also assumes a constant tensile stress in the immediate adhesive region outside the contact area up to a critical distance of the tip-sample separation. The merit of the MD model is that it provides a continuous transition between the limiting cases of JKR  (for compliant materials with large surface energy and large contact radius) and DMT  (for stiff materials with low surface energy and small contact radius) when the MD adhesive parameter λ is varied from zero (DMT limit) to infinity (JKR limit). Commonly, the JKR and DMT limits are invoked in the contact mechanics analysis of AFM measurements depending on the nature of the material tested and contact geometries. In particular, the MD model  can provide a selective material analysis for heterogeneous polymers, as the adhesive properties change with the material [31, 32]. Recently, the MD model was numerically reproduced for a spherical indenter in contact with a half-space homogenous material by using the conjugate gradient method (CGM) [33, 34]. We extended  this numerical implementation of the MD model to the adhesive contact problem of an indented quarter-space based on the matrix formulation of Zhang et al.  and here we further examine the MD model for the case of a fin with two free-edge surfaces. Comparisons of the numerical results with empirical formulations used for indentations nearby edges will be examined for both the adhesive and non-adhesive contact geometries on fins.
Materials and Measurements
To probe the mechanical properties of these high-aspect ratio fins, ICR-AFM  was used due to its unique ability of providing high spatial resolution, load-dependent measurements, and intermittent contact interaction with the sample during scanning. In Fig. 1d and e, the ICR-AFM resonance frequency shows the sample response at locations mapped across the 20 nm and 90 nm wide fins, respectively. It can be seen that, as the AFM probe was brought into and out of contact during scanning, the resonance frequency varied as a function of load, position, and fin. Specifically, although the maximum applied load in every tap was 60 nN (blue curve), the maximum values reached by the ICR-AFM resonance frequency during these taps deacrease around the edges of the 90 nm fin and across the 20 nm wide fin. These variations contain contributions from both contact geometry and material properties. It is through an appropiate contact mechanics analysis that the measured contact resonance is converted into contact stiffness and used to examine possible variations in the elastic modulus of the structure tested. The most particular contribution here is from the edges of the structure in the form of edge compliances, a topic not fully addressed in indentation models and rarely used in interpreting measurements.
The contribution from the edge compliance to the contact stiffness of the fins was considered previously  on the basis of empirical derivation with respect to the half-space (also referred to as bulk) contact stiffnes of the consitutent material. This was done by reproducing correctly the load-dependence of the contact stiffness measured in the middle (far way from the edges) of the 90 nm fin, around the edges of the 90 nm wide fins, and across the 20 nm wide fins. All of the fits were performed for the same measurement parameters, namely loads up to 60 nN and AFM tip radius about 70 nm. It was found that all fits work well for the same edge compliance parameter, which confirms the correct usage of the empirical description for the contribution of the edge compliance to the contact stiffness of the structures. As mentioned above, the mathematical framework for frictionless adhesive contacts near the edge of a quarter-space was devloped elsewhere  and can be used to resolve the edge dependence observed in the contact stiffness measurements on the 90 nm wide fins. In this work, we focus exclusively on the BEM analysis of the contact stiffness measurements across the 20 nm fins, with edge compliance contributions from both edges of the fin. In contrast to the 90 wide fins, the 20 nm wide fins exhibit almost homogeneous material properties across their widths [13, 18]. This is due to the fact that plasma induced changes during processing occur over a length scale of 10–20 nm, namely these changes in structure and material properties extend over the entire volume of the 20 nm wide fins. It is worth pointing out that once the edge compliance is properly accounted for to the measured contact stiffness of the fins, the determined elastic modulus does reflect the increase in stiffness sustained by material during processing, which is about three to four times bigger than that of the blanket films [13, 18].
The Matrix Formulation
The matrices M are calculated from the Love’s solution  for the given geometrical dimensions of the fin and chosen discretization mesh. These are used back in the equations (11)–(13) to calculate the equivalent load distributions pHC, pVL, and pVR for a given applied load p on the top of the fin. The equivalent loads can then be used to calculate the stresses and deformations that they generate within their half-spaces.
The DC-FFT algorithm is fast and minimizes the errors that would otherwise be introduced by the non-periodic boundary conditions of a contact problem.
The Conjugate Gradient Method (CGM)
The programming details of the CGM algorithm as applied to non-adhesive and adhesive contacts on half-spaces can be found in the references mentioned above [33, 34, 38]. In addition to these, in the case of a contact nearby to free edges (one edge for contact on a quarter-space and two edges for contact on a fin), the contributions of the edge compliances need to be considered. As detailed in the previous section, this can be done by including contributions from the reflexive half-spaces in the matrix formulation for non-adhesive contacts on either a quarter-space or a fin [27, 28]. Based on these previous works, the matrix formulation of the CGM was extended to MD adhesive contacts on quarter-space  and is investigated here for MD adhesive contacts on fin geometries.
Results and Discussion
In the current matrix formulation, the CGM calculations for a MD adhesive contact between a spherical indenter and a fin were performed with a custom programming code in Mathematica 10.3 (Wolfram Research, 2015) . The discretization mesh used in calculations consisted of 128 × 128 × 128 points equally distributed over the 20 nm × 20 nm × 20 nm volume of the fin, with the y axis along one of the top-left edge of the fin and the x axis bisecting the top facet of the fin (refer to Fig. 2). The indents were considered along the x axis and normal to the xy plane. For a given load, the surface deformation and contact pressure distributions were calculated. The convergence of the numerical solution was defined when the load given by the calculated contact pressure distribution was within 1% of the considered load.
As can be seen, the CGM calculation reproduced the characteristic MD profiles for stress and deformation inside and outside the contact area. Inside the contact area, the stress goes from compressive in the middle to a tensile at the edge of the contact, where the value of the adhesive stress σ0 is reached. Outside the contact area, the stress continues to be tensile over the adhesive region at the same σ0 value and zero beyond the adhesive region. The effect of the tensile stress within the adhesion region is also observed in the surface deformation in the form of a kink with respect to its non-adhesive profile. The fin sustains a significant deformation across its entire top surface, going from about 0.9 nm in the middle (same as the accounted indentation depth) to about 0.4 nm at each of the edges of the fin.
Based on the above findings, a working case will be analyzed to further detail the correspondence between the BEM calculations and empirical formulae used for the contact stiffness of structures with edges. In this study both non-adhesive and adhesive contacts between a spherical rigid indenter of radius R = 20 nm and either a quarter-space or a fin were investigated. The considered elastic parameters for these structures were Young’s modulus of E = 12.0 GPa and Poisson’s ratio ν = 0.3 (indentation modulus E∗ = E/(1 − ν2) = 13.2 GPa). For the adhesive contacts, the adhesion was assumed to be MD-type with the Maugis parameter λ = 1.0 and work function w = 0.025 J/m2. The BEM calculations were done by using the above described CGM algorithm in the matrix formulation. The mesh used consisted of 128 × 128 × 128 points equally distributed over a 30 nm × 30 nm × 30 nm volume for the quarter-space and a 20 nm × 20 nm × 20 nm volume for the fin. For both selected meshed volumes, the half-space solutions of the considered non-adhesive and adhesive contacts were in excellent agreement with their analytical counterparts. As before, the coordinate system was chosen with the y axis along the left edge of the specimen and the x axis bisecting the top facet of the specimen. The contact stiffness values were calculated as k = ∂P/∂δ from the load versus depth indentation BEM curves.
In this work, the CGM matrix formulation was used in the form of a BEM model for non-friction adhesive contacts on fin geometries. The model provides numerical calculations of the stress and strain fields for a fin structure subjected to indentation. As a direct application to indentation tests, the contact stiffness of the tip-sample contact was calculated for some testing geometries as a function of the applied load and position from the edges of a narrow fin. It was found that the model results reproduce the measurements very well, without any fit parameters. Moreover, for both quarter-space (with one free edge) and fin geometry (with two free edges), the BEM results confirmed the empirical 1/x dependence of the edge compliance to the contact stiffness of the structures. However, the fitting procedure requires adjacent measurements as a function of distance from the edge/edges of the structure to determine the unknown fit parameter for a given contact interaction whereas the BEM model is free of any fit parameter and includes the proper (non-adhesive or adhesive) contact interaction.
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