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Correction to: The Effect of Edge Compliance on the Contact between a Spherical Indenter and a High-Aspect-Ratio Rectangular Fin

  • G. StanEmail author
  • E. Mays
  • H. J. Yoo
  • S. W. King
Correction
  • 66 Downloads

Keywords

Nanoscale contact mechanics Edge compliance Conjugate gradient method Contact resonance atomic force microscopy 

  https://doi.org/10.1007/s11340-018-0421-4

Introduction

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) [11] (with adhesion inside the contact area) and Derjaguin-Muller-Toporov (DMT) [12] (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 [13], we used intermittent contact resonance AFM (ICR-AFM) [14] 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 [15]. 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 [13] 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 [19] and Yu and Keer [20]. Their work was built around the quarter-space contact problem introduced by Hetenyi about 50 years ago [21]. As defined by Hetenyi [21], 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 [22] 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. [27] 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 [28]. 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 [29] 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 [29], now both free-edge surfaces were considered as in the work of Zhang et al. [28]. Contact adhesion was assumed as in the Maugis-Dugdale (MD) transition model [30] 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 [11] (for compliant materials with large surface energy and large contact radius) and DMT [12] (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 [30] 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 [29] 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. [27] 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

The contact stiffness measurements analyzed in this work were previously reported elsewhere [13]. They were performed by ICR-AFM on a-SiOC:H high-aspect ratio fins that are used as low-k dielectrics in advanced nanoscale Cu-based integrated electronics. The structures were defined by plasma enhanced chemical vapor deposition and subsequent plasma etching in the form of rectangular fins with width of either 20 nm or 90 nm, about 100 nm tall, and spaced at 65 nm to each other [35, 36]. In Fig. 1 an array of these fins is shown together with the contact geometry measurement and a few ICR-AFM measurements on the two examined sizes. In our previous studies [13, 18], it was demonstrated that these structures sustain serious changes in their mechanical and chemical properties during processing. Such effects are enhanced around the exposed surfaces that come into direct contact with various agents used to define and prepare the structures for embedding the Cu circuits (e.g., plasma etching, plasma ashing, wet cleans, etc.).
Fig. 1

(a) Array of high-aspect ratio low-k dielectric fins of widths of 20 nm and 90 nm; (b) and (c) schematics of measurements across the width of the fins; (d) and (e) ICR-AFM measurements across 20 nm and 90 nm wide fins, respectively. The blue curve shows the load variatiation during succesive taps and the red curve shows the ICR-AFM resonance frequency along these taps

To probe the mechanical properties of these high-aspect ratio fins, ICR-AFM [14] 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 [13] 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 [29] 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 matrix formulation developed by Zhang et al. [27] for the quarter-space and Zhang et al. [28] for finite-length line with free-lateral surfaces was used in this work. The load and stress distributions for the case of a fin normally loaded on its top facet is illustrated in Fig. 2. By following Hetenyi’s method [21], the actual problem of a limited-space fin is replaced by a series of horizontal and vertical overlapped half-spaces: three horizontally loaded half-spaces on top of the fin (Fig. 2(b)), two vertically loaded half-spaces on the left side of the fin (Fig. 2(c)), and two vertically loaded half-spaces on the right side of the fin (Fig. 2(d)). The loads associated to these half-spaces and the stresses generated by them on the facets of the fin are balanced to obtain the applied load on the top of the fin and free-lateral surfaces. The stress singularity due to edge loading is not considered in the matrix formulation of the overlapping half-spaces method [27, 28].
Fig. 2

Cross-sections of the load and stress distributions on the facets of a fin: (a) the applied load on the top facet; (b) the equivalent loads on the top half-spaces and their stress distributions on the vertical facets of the fin; (c) the equivalent loads on the left vertical half-spaces and their stress distributions on the top and right vertical facets of the fin; (d) the equivalent loads on the right vertical half-spaces and their stress distributions on the top and left vertical facets of the fin

In the matrix formulation, the region of interest is discretized in a finite number of elements, Nx = Ny = Nz = N in the current work, with uniform rectangular meshes on each side. The load and stress distributions can then be approximated as piecewise distributions, with constant values on each cell of the mesh. In vector notation, the array of the stress distribution σ generated by the load distribution P is given by
$$ \boldsymbol{\sigma} =\boldsymbol{M}\bullet \boldsymbol{P} $$
(1)
where M is the 2D array of “influence coefficients” or “reflecting coefficients” connecting the loaded patches with those on which the stress is calculated; the solution for a uniformly loaded rectangular patch was formulated by Love [22].
Following the notations of Zhang et al. [27], the stress distributions on the vertical facets of the fin due to the equivalent loads applied on the horizontal half-spaces (Fig. 2(b)) are given by
$$ {\boldsymbol{\sigma}}_{xx}^{\mathrm{H}\to \mathrm{VL}}={\mathbf{M}}_{\mathrm{H}\mathrm{C}\to \mathrm{VL}}\cdotp {\boldsymbol{p}}_{\mathrm{H}\mathrm{C}}+{\mathbf{M}}_{\mathrm{H}\mathrm{L}\to \mathrm{VL}}\cdotp {\boldsymbol{p}}_{\mathrm{H}\mathrm{L}}+{\mathbf{M}}_{\mathrm{H}\mathrm{R}\to \mathrm{VL}}\cdotp {\boldsymbol{p}}_{\mathrm{H}\mathrm{R}}\equiv {\mathbf{M}}_{\mathrm{H}\mathrm{C}}^{\mathrm{VL}}\cdotp {\boldsymbol{p}}_{\mathrm{H}\mathrm{C}} $$
(2)
$$ {\boldsymbol{\sigma}}_{xx}^{\mathrm{H}\to \mathrm{VR}}={\mathbf{M}}_{\mathrm{H}\mathrm{C}\to \mathrm{VR}}\cdotp {\boldsymbol{p}}_{\mathrm{H}\mathrm{C}}+{\mathbf{M}}_{\mathrm{H}\mathrm{L}\to \mathrm{VR}}\cdotp {\boldsymbol{p}}_{\mathrm{H}\mathrm{L}}+{\mathbf{M}}_{\mathrm{H}\mathrm{R}\to \mathrm{VR}}\cdotp {\boldsymbol{p}}_{\mathrm{H}\mathrm{R}}\equiv {\mathbf{M}}_{\mathrm{H}\mathrm{C}}^{\mathrm{VR}}\cdotp {\boldsymbol{p}}_{\mathrm{H}\mathrm{C}} $$
(3)
where the arrows indicate the connection between the facet on which the load is applied and the facet where the stress is calculated, with the following abbreviations for the fin schematically shown in Fig. 2: HC for horizontal-central, HL for horizontal-left, HR for horizontal-right, VL for vertical-left, and VR for vertical-right. In the above equations, the stress contributions from pHL (over the x < 0, −  ∞  < y < ∞ region at z = 0) and pHR (over the x > d, −  ∞  < y < ∞ region at z = 0) can be replaced in terms of stress contributions from pHC (over the x > 0, −  ∞  < y < ∞ region at z = 0) by considering the symmetry with respect to the vertical facets of the fin; d is the width of the fin along the x axis. The loads on the vertical left half-spaces, pVL (over the − ∞  < y <  ∞ ,  z > 0 region at x = 0) and pVLT (over the − ∞  < y <  ∞ ,  z < 0 region at x = 0) (with VLT for vertical-left-top), generate the following stress distributions on the top and right facets of the fin (Fig. 2(c)):
$$ {\boldsymbol{\sigma}}_{zz}^{\mathrm{VL}\to \mathrm{HC}}={\mathbf{M}}_{\mathrm{VL}\to \mathrm{HC}}\cdotp {\boldsymbol{p}}_{\mathrm{VL}}+{\mathbf{M}}_{\mathrm{VL}\mathrm{T}\to \mathrm{HC}}\cdotp {\boldsymbol{p}}_{\mathrm{VL}\mathrm{T}}\equiv {\mathbf{M}}_{\mathrm{VL}}^{\mathrm{HC}}\cdotp {\boldsymbol{p}}_{\mathrm{VL}} $$
(4)
and
$$ {\boldsymbol{\sigma}}_{xx}^{\mathrm{VL}\to \mathrm{VR}}={\mathbf{M}}_{\mathrm{VL}\to \mathrm{VR}}\cdotp {\boldsymbol{p}}_{\mathrm{VL}}+{\mathbf{M}}_{\mathrm{VL}\mathrm{T}\to \mathrm{VR}}\cdotp {\boldsymbol{p}}_{\mathrm{VL}\mathrm{T}}\equiv {\mathbf{M}}_{\mathrm{VL}}^{\mathrm{VR}}\cdotp {\boldsymbol{p}}_{\mathrm{VL}} $$
(5)
respectively. Similarly, the loads on the vertical right half-spaces, pVR (over the − ∞  < y <  ∞ ,  z > 0 region at x = d) and pVRT (over the − ∞  < y <  ∞ ,  z < 0 region at x = d) (with VRT for the vertical-right-top), generate the following stress distributions on the top and left facets of the fin (Fig. 2(d)):
$$ {\boldsymbol{\sigma}}_{zz}^{\mathrm{VR}\to \mathrm{HC}}={\mathbf{M}}_{\mathrm{VR}\to \mathrm{HC}}\cdotp {\boldsymbol{p}}_{\mathrm{VR}}+{\mathbf{M}}_{\mathrm{VR}\mathrm{T}\to \mathrm{HC}}\cdotp {\boldsymbol{p}}_{\mathrm{VR}\mathrm{T}}\equiv {\mathbf{M}}_{\mathrm{VR}}^{\mathrm{HC}}\cdotp {\boldsymbol{p}}_{\mathrm{VR}} $$
(6)
and
$$ {\boldsymbol{\sigma}}_{xx}^{\mathrm{VR}\to \mathrm{VL}}={\mathbf{M}}_{\mathrm{VR}\to \mathrm{VL}}\cdotp {\boldsymbol{p}}_{\mathrm{VR}}+{\mathbf{M}}_{\mathrm{VR}\mathrm{T}\to \mathrm{VL}}\cdotp {\boldsymbol{p}}_{\mathrm{VR}\mathrm{T}}\equiv {\mathbf{M}}_{\mathrm{VR}}^{\mathrm{VL}}\cdotp {\boldsymbol{p}}_{\mathrm{VR}} $$
(7)
respectively. The actual fin is obtained when the above half-spaces are overlapped, with the force balance conditions satisfied on each facet:
$$ -{\boldsymbol{p}}_{\mathrm{HC}}+{\boldsymbol{\sigma}}_{zz}^{\mathrm{VL}\to \mathrm{HC}}+{\boldsymbol{\sigma}}_{zz}^{\mathrm{VR}\to \mathrm{HC}}=-\boldsymbol{p} $$
(8)
$$ -{\boldsymbol{p}}_{\mathrm{VL}}+{\boldsymbol{\sigma}}_{xx}^{\mathrm{H}\to \mathrm{VL}}+{\boldsymbol{\sigma}}_{xx}^{\mathrm{VR}\to \mathrm{VL}}=\mathbf{0} $$
(9)
$$ -{\boldsymbol{p}}_{\mathrm{VR}}+{\boldsymbol{\sigma}}_{xx}^{\mathrm{H}\to \mathrm{VR}}+{\boldsymbol{\sigma}}_{xx}^{\mathrm{VL}\to \mathrm{VR}}=\mathbf{0} $$
(10)
By solving the system of equations. (8)–(10), the equivalent loads on the fin facets are obtained as a function of the actual load distribution that is applied on the top surface of the fin:
$$ {\boldsymbol{p}}_{\mathrm{HC}}={\left(\mathbf{I}-{\mathbf{M}}_{\mathrm{VL}}^{\mathrm{HC}}{\mathbf{M}}_{\mathrm{VL}\mathrm{H}}-{\mathbf{M}}_{\mathrm{VR}}^{\mathrm{HC}}{\mathbf{M}}_{\mathrm{VR}\mathrm{H}}\right)}^{-1}\cdotp \boldsymbol{p} $$
(11)
$$ {\boldsymbol{p}}_{\mathrm{VL}}={\mathbf{M}}_{\mathrm{VL}\mathrm{H}}\cdotp {\boldsymbol{p}}_{\mathrm{HC}} $$
(12)
$$ {\boldsymbol{p}}_{\mathrm{VR}}={\mathbf{M}}_{\mathrm{VR}\mathrm{H}}\cdotp {\boldsymbol{p}}_{\mathrm{HC}} $$
(13)
with
$$ {\mathbf{M}}_{\mathrm{VL}\mathrm{H}}={\left(\mathbf{I}-{\mathbf{M}}_{\mathrm{VR}}^{\mathrm{VL}}{\mathbf{M}}_{\mathrm{VL}}^{\mathrm{VR}}\right)}^{-1}\left({\mathbf{M}}_{\mathrm{HC}}^{\mathrm{VL}}+{\mathbf{M}}_{\mathrm{VR}}^{\mathrm{VL}}{\mathbf{M}}_{\mathrm{HC}}^{\mathrm{VR}}\right) $$
(14)
$$ {\mathbf{M}}_{\mathrm{VR}\mathrm{H}}={\left(\mathbf{I}-{\mathbf{M}}_{\mathrm{VL}}^{\mathrm{VR}}{\mathbf{M}}_{\mathrm{VR}}^{\mathrm{VL}}\right)}^{-1}\left({\mathbf{M}}_{\mathrm{HC}}^{\mathrm{VR}}+{\mathbf{M}}_{\mathrm{VL}}^{\mathrm{VR}}{\mathbf{M}}_{\mathrm{HC}}^{\mathrm{VL}}\right) $$
(15)

The matrices M are calculated from the Love’s solution [22] 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 top surface deformation under the indenter includes contributions from all the horizontal half-spaces but not from the vertical half-spaces due to symmetry:
$$ {\boldsymbol{u}}_z^{\mathrm{HC}}={\mathbf{N}}_{\mathrm{HC}\to \mathrm{HC}}\cdotp {\boldsymbol{p}}_{\mathrm{HC}}+{\mathbf{N}}_{\mathrm{HL}\to \mathrm{HC}}\cdotp {\boldsymbol{p}}_{\mathrm{HL}}+{\mathbf{N}}_{\mathrm{HR}\to \mathrm{HC}}\cdotp {\boldsymbol{p}}_{\mathrm{HR}}\equiv {\mathbf{N}}_{\mathrm{HC}}^{\mathrm{HC}}\cdotp {\boldsymbol{p}}_{\mathrm{HC}} $$
(16)
where the matrices N contain the influence coefficients relating the surface deflection to the applied load as in the Love’s solution [22] for uniformly loaded rectangular patches. By combining (11) and (16) a relationship between surface deformation and applied load is reached:
$$ {\boldsymbol{u}}_z^{\mathrm{HC}}={\mathbf{N}}_{\mathrm{HC}}^{\mathrm{HC}}\cdotp {\left(\mathbf{I}-{\mathbf{M}}_{\mathrm{VL}}^{\mathrm{HC}}{\mathbf{M}}_{\mathrm{VL}\mathrm{H}}-{\mathbf{M}}_{\mathrm{VR}}^{\mathrm{HC}}{\mathbf{M}}_{\mathrm{VR}\mathrm{H}}\right)}^{-1}\cdotp \boldsymbol{p} $$
(17)
The convolution incorporated in (16) between the matrices of the influence coefficients and the contact pressure distribution can be converted into direct product in the Fourier space to calculate the surface deformation. Moreover, it can be processed by a discrete convolution and fast Fourier transform (DC-FFT) algorithm introduced by Liu et al. [37]. Basically, in the DC-FFT, a linear discrete convolution is embedded into a circular discrete convolution of double length by doubling the calculation domain (e.g., the above HC area) on each dimension, zero-padding the contact pressure distribution in the extended domain, and wrapping-around (mirroring) the influence coefficients over the added area. The response is obtained as the inverse FFT (IFFT) of the direct product of the FFTs of the influence coefficient and contact pressure distribution matrices (similarly to a linear convolution in the continuous space):
$$ {\left({\overset{=}{\boldsymbol{u}}}_z^{\mathrm{HC}}\right)}_{ij}= IFFT\left( FFT{\left({\overset{=}{\mathbf{N}}}_{\mathrm{HC}}^{\mathrm{HC}}\right)}_{ij}\cdotp FFT{\left({\overset{=}{\boldsymbol{p}}}_{\mathrm{HC}}\right)}_{ij}\right),\kern1.5em i,j\in \left\{1,2,\dots, N,N+1,\dots, 2N\right\} $$
(18)
where \( {\overset{=}{\mathbf{N}}}_{\mathrm{HC}}^{\mathrm{HC}} \) is the wrap-around order of \( {\mathbf{N}}_{\mathrm{HC}}^{\mathrm{HC}} \) and \( {\overset{=}{\boldsymbol{p}}}_{\mathrm{HC}} \) is the zero-padding of pHC. The actual surface deformation matrix \( {\boldsymbol{u}}_z^{\mathrm{HC}} \) is collected from the first N × N elements of \( {\overset{=}{\boldsymbol{u}}}_z^{\mathrm{HC}} \),
$$ {\left({\boldsymbol{u}}_z^{\mathrm{HC}}\right)}_{ij}={\left({\overset{=}{\boldsymbol{u}}}_z^{\mathrm{HC}}\right)}_{ij},\kern1em i,j\in \left\{1,2,\dots, N\right\} $$
(19)

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)

Once the relationship between the surface deformation and the applied load (17) is established in the overlapping half-spaces geometry, the contact problem can be solved by an optimization algorithm like the conjugate gradient method (CGM) discretized over the region of interest. The CGM is used in general to solve numerically a system of linear equations by iteratively searching to minimalize a quadratic function whose gradient matches the system of equations. The convergence of the solution is reached in a finite number of steps, which is less than the number of equations, hence the utility of using CGM in solving large linear systems of equations as those encountered in a finite element discretization. A very efficient CGM for contact problems was first applied by Polonsky and Keer [38] to non-adhesive Hertz half-space contacts and extended recently by Bazrafshan et al. [33] and Rey et al. [34] to MD adhesive half-space contacts. In the contact problem implementations, CGM iteratively solves for the pressure distribution at the grid points over a target area by adjusting the tip-sample separation to satisfy the inequality constraints of the contact. With the selected target area divided in N × N rectangular elements, the constrains for a non-adhesive contact resume to assure a compressing stress at the grid points inside the contact area and zero outside:
$$ \left\{\begin{array}{c}{p}_{i,j}<0\kern0.5em \mathrm{at}\kern0.50em {g}_{i,j}=0\\ {}{p}_{i,j}=0\kern0.5em \mathrm{at}\kern0.50em {g}_{i,j}>0\end{array}\right.\kern0.48em \mathrm{with}\kern0.48em i,j\in \left\{1,2,\dots, N\right\} $$
(20)
where gi, j gives the tip-sample separation at the grid points over the target area and pi, j is the contact pressure distribution approximated as a piecewise function with constant values over each element of the grid; the convention here is to have negative values for compressive stress and positive for tensile stress [30]. For a MD adhesive contact, the contact pressure goes from compressive at the center of the contact to tensile at the contact edge, constant tensile stress in the adhesive region surrounding the contact (tip-sample separation less than a threshold height h0), and zero outside the adhesive region (tip-sample separations above h0):
$$ \left\{\begin{array}{c}{p}_{i,j}<{\sigma}_0\kern0.5em \mathrm{at}\kern0.50em {g}_{i,j}=0\kern2.5em \\ {}{p}_{i,j}={\sigma}_0\kern0.5em \mathrm{at}\kern0.50em 0<{g}_{i,j}<{h}_0\\ {}{p}_{i,j}=0\kern0.75em \mathrm{at}\kern0.50em {g}_{i,j}>{h}_0\kern2em \end{array}\right.\kern0.36em \mathrm{with}\kern0.36em i,j\in \left\{1,2,\dots, N\right\} $$
(21)
Furthermore, for a rigid spherical indenter of radius R brought into contact with an elastically deformable flat surface at the location (xc, yc ) of the target area, the tip-sample separation is given by the shape function of the indenter, relative approach of the two objects δ (also called indentation depth), and surface deformation uz
$$ {g}_{i,j}=\frac{1}{2R}\left[{\left({x}_i-{x}_c\right)}^2+{\left({y}_i-{y}_c\right)}^2\right]-\delta +{u}_z^{i,j}\kern0.36em \mathrm{with}\kern0.24em i,j\in \left\{1,2,\dots, N\right\} $$
(22)
Neither the contact pressure distribution, the expand of the contact, or surface deformation is known a priory but they are all determined during CGM iterations. At each iteration, the adjusted contact pressure distribution over the contact area must sum up to the applied load
$$ P=-4\varDelta x\varDelta y\sum \limits_{i=1}^N\sum \limits_{j=1}^N{p}_{i,j}\kern0.36em \mathrm{with}\kern0.36em i,j\in \left\{1,2,\dots, N\right\} $$
(23)
where 2∆x and 2∆y are the grid increments on x and y directions, respectively.

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 [29] 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) [39]. 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.

Relevant to the contact geometry of the ICR-AFM measurements on the 20 nm wide fins studied in [13], in Fig. 3 is shown an example with the stress and surface deformation profiles calculated for a spherical indenter of radius R = 70 nm pressed on the top of a fin of 20 nm wide. The elastic parameters of the material are the Young’s modulus E = 12.0 GPa and Poisson’s ratio ν = 0.3, which give the indentation modulus as E = E/(1 − ν2) = 13.2 GPa. The adhesion in CGM was considered of being MD-type with the Maugis parameter λ = 1.0 and work function w = 0.025 J/m2 . This work function corresponds to a DMT pull-off force \( {F}_{\mathrm{pull}-\mathrm{off}}^{\mathrm{DMT}}=3\pi wR/2 \) of 8.2 nN and adhesive stress σ0 = λ(2πwE∗2/(9R))1/3 = 0.34 GPa.
Fig. 3

Surface deformation (red curve with scale on the left axis) and stress (blue curve with scale on the right axis) profiles along an 20 nm fin under a load of 70 nN applied by a spherical tip of radius 70 nm in the middle of the fin. The BEM calculations were made for a MD-type adhesive contact with the Maugis parameter λ = 1.0 and work function w = 0.025 J/m2; the Young’s modulus of the fin was considered E = 12.0 GPa and its Poisson’s ratio ν = 0.3. The dotted rectangle shows the non-deformed shape of the fin

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.

From calculations like those shown in Fig. 3, the indentation depth (surface deformation at the center of the indenter) for a given applied load is also extracted and, by performing these calculations over a range of applied loads, the indentation depth versus load curve can be obtained. This in turn provides a measurable quantity related to the mechanical response of the indented structure, namely the contact stiffness in the form of the derivative of the load to the indentation depth, k = ∂P/∂δ. A set of such contact stiffness versus load curves calculated for contacts at different distances from the edges of the fin were used here to reexamine the load dependences of the contact stiffness measured by ICR-AFM previously [13] on high-aspect ratio fins. In the previous work [13], the additional contribution from the edge compliance to the contact stiffness of the structure was considered as being inversely proportional with the distance from the edge [40, 41].
$$ 1/k=1/{k}_{\mathrm{bulk}}+{C}_{\mathrm{edge}}/\left({E}^{\ast }x\right) $$
(24)
with kbulk being the contact stiffness for the bulk part (half-space) material, Cedge the edge compliance, E the reduce elastic modulus of the tip-sample contact, x the distance from the edge. In these fits, both E and Cedge were considered fit parameters. Notably, the same value of Cedge provided good fits for all the contact stiffnesses measured as a function of load at various locations from the edge of the 90 nm wide fins. Moreover, in the case of the 20 nm wide fins, two independent edge compliances were considered, one from each edge of the fin:
$$ 1/k=1/{k}_{\mathrm{bulk}}+{C}_{\mathrm{edge}}/\left({E}^{\ast }x\right)++{C}_{\mathrm{edge}}/\left({E}^{\ast}\left(d-x\right)\right) $$
(25)
with the left and right edge located at x = 0 and x = d, respectively, d being the width of the fin. In this case, it was also found that the same value of the fit parameter Cedge reproduces very well the edge dependence of the contact stiffness measured at various locations across the 20 nm wide fins [13].
The results from the current model are shown in Fig. 4 side by side with the measurements and fits from [13]. They correspond to load dependences of the contact stiffness measured by ICR-AFM across a 20 nm wide fin, with xc giving the contact position from the left edge of the fin (refer to (25) with x = xc). The BEM calculations were made for round values of the xc positions and they are symmetrical with respect to the middle of the fin (e.g position xc = 15.0 nm is equivalent to position xc = 5.0 nm from the left edge of the strip). As can be seen, both the empirical fitting and the BEM model give a satisfactory representation of the contact stiffness as a function of load and position from the edge. Both descriptions work better at loads above 30 nN and are less accurate at low loads, in the range of adhesive forces, where the tip comes in or detaches from the sample and, conceivably, some vibrations from the tip oscillations might disrupt the tip-sample necking geometry. These perturbations in the tip-sample contact formation/separation might have two potential sources, one from the low-frequency tapping motion of the scanning mode (ring-down contact oscillations of the peak-force tapping scanning mode) and one from the high-frequency CR modulation itself (the amplitude of the driven oscillation is of order of nanometer); the oscillations might be alleviated by using stiffer cantilevers. The measurements shown here are only for the retract parts of few ICR-AFM measurements and reflect the average response during scanning over few 20 nm fins. The relative uncertainty measured over the inspected thin fins was within 20% (refer to the measurement analysis presented in [13]. To the benefit of the BEM model, a more consistent trend with the measurements is observed than for the fits, in the sense that the elastic moduli from the BEM calculations show lesser dispersion across the fin than those determined from fits. A more homogeneous stiffness across the 20 nm wide fins is in accordance with the observation that changes in the mechanical and chemical properties of these low-k dielectric structures occur over a length scale of 10–20 nm from their exposed surface during the fabrication process [18]. It is worth pointing out that the elastic modulus of these 20 nm wide fins as determined from both the empirical analysis [13] and BEM (current work) is in the range of 9 GPa to 12 GPa. Such values are about 3 to 4 times larger than that of the a-SiOC:H blanket films and indicate solely the plasma-induce changes to the mechanical properties of the fins. These changes are proved after the geometrical contribution of the edge compliance was accounted for to the measured contact stiffness.
Fig. 4

Measurements and fits of contact stiffness versus load on a 20 nm wide fin. The data were converted from contact resonance frequency ICR-AFM measurements, the empirical fits were performed by considering the edge compliance and elastic modulus as fit parameters, and the BEM calculations for a MD-type adhesive contact were performed as explained in the text with the elastic modulus as the only adjustable parameter

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 Fig. 5 a series of contact stiffness values calculated at different applied loads and various locations from the edge of the quarter-space are shown. As expected, the largest deviations from the bulk limit (horizontal lines) occur near the edge of the half-space (due to a larger contribution from the edge compliance) and diminish at larger distances from the edge (due to a smaller contribution from the edge compliance). At any distance from the edge, these differences are more pronounced at higher loads and their variation with the distance from the edge is steeper as the load is increased. Accordingly, the edge compliance almost vanishes at some 25 nm from the edge at loads as small as 10 nN but, at the same distance, its contribution to the contact stiffness is still significant in contacts subjected to loads above 40 nN. It is also noted that the differences between the non-adhesive and adhesive cases are more pronounced at lower loads and further away from the edge. The calculated values of the contact stiffness by BEM were fitted by the dependence given by (24), with only one additional edge compliance contribution. For all the considered loads, the 1/x dependence provides a very good fit, with almost the same value for the edge compliance, Cedge, for both non-adhesive and adhesive cases. For each of these cases, the bulk contact stiffness was calculated accordingly with the contact model (either Hertz or MD) and at the given applied load.
Fig. 5

Load dependence of the contact stiffness as determined by BEM calculations (solid marks for non-adhesive contacts and open marks for adhesive contacts) and empirical fits (continuous lines for non-adhesive contacts and dotted lines for adhesive contacts) at various locations from the edge of a quarter-space. The bulk values of the contact stiffness at every considered load are plotted as horizontal lines (continuous lines for non-adhesive contacts and dashed lines for adhesive contacts), with the load value specified at the right side of these lines

A similar analysis was performed for the case of an elastically indented 20 nm wide fin, in which case the edge compliances from both free-edge surfaces need to be considered for the overall compliance of the structure. The results for both Hertz and MD (λ = 1.0) of BEM are shown in Fig. 6 at different loads and various locations across the fin. As can be seen, an additional decrease in the contact stiffness was calculated as a function of load and position from the edges of the fin in comparison to that of the quarter-space. Qualitatively, this can be rationalized in the fitting dependence as contributions from each edge of the fin, one at x = 0 and one at x = d, as given by (25). For each considered applied load, a satisfactory fit is obtained with the empirical dependence, but with slightly different fitting edge compliances, whereas in the above case of the quarter-space a similar fit parameter was obtained for all the considered loads. This points to the necessity of performing multiple measurements as a function of load and position across a fin when the empirical dependences are to be used to deconvolute the contribution of the edge compliance and determine the actual elastic modulus from contact stiffness measurements made around edges.
Fig. 6

Comparison between BEM calculations (solid marks for non-adhesive contacts and open marks for adhesive contacts) and empirical fits (continuous lines for non-adhesive contacts and dotted lines for adhesive contacts) across a 20 nm wide fin subjected to different loads

Conclusions

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.

Notes

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Copyright information

© This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2019

Authors and Affiliations

  1. 1.Material Measurement LaboratoryNational Institute of Standards and TechnologyGaithersburgUSA
  2. 2.Logic Technology DevelopmentIntel CorporationHillsboroUSA
  3. 3.Components ResearchIntel CorporationHillsboroUSA

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