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The low-frequency limit is indicative of the conventional FMM mode. Here, the ratio of the tip deflection (experimentally measured quantity) to the sample oscillation amplitude is only sensitive to the surface mechanical properties if the contact stiffness is on the order of the cantilever spring constant. Hence, if the cantilever is "soft" enough to measure nanoscale topography, it will only be capable of qualifying the spatial dependence of the surface modulus for soft materials. Consequently, FMM is primarily a tool used for polymeric or soft materials. The high-frequency limit is more intriguing from the viewpoint of nanomechanics. At high frequencies, the cantilever cannot follow the driving frequency (superresonant motion), and the driven tip oscillation amplitude decreases as a square of the driving frequency. However, the prefactor of this amplitude depends linearly on the ratio of the surface contact stiffness to the cantilever spring constant. In other words, large deflection implies large, local contact stiffness. Imaging in this mode can provide a signal directly proportional to the surface Young's modulus. Since the contact area of the tip is still on the order of the radius of curvature of the pyramidal tip (~5-20 nm), this mode (christened SLAM by Burnham for scanning local acceleration microscopy) is truly capable of imaging the mechanical properties of a wide range of z

materials. Care is required in this technique as the superres-onant frequency must fall within the bandwidth of the photodiode detector of the SFM. Also, the expression [Eq. (7)] is valid only in the limit of amplitude-independent contact stiffness, that is, a linear force-displacement response. This is not strictly true in the case of the aforementioned JKRS or DMT models due to the force-dependent area of the contact (even in the case of a flat-end punch) [17, 18, 20]. Consequently, the calibration of SLAM images requires detailed knowledge of the true contact stiffness so that a region of model validity can be determined. However, the simplicity of the model is a major advantage in data interpretation and image artifact identification.

A variation of this approach was investigated several years earlier by Yamanaka et al. [33-34]. Similar to SLAM, an out-of-plane surface vibration was applied to the sample. This approach considered ultrasonic frequencies beyond the response of the photodiode, so the direct oscillation of the cantilever was not observed. However, at high oscillation amplitudes (typically >0.4 nm), a nonzero dc tip deflection was observed. This observation implied that the nonlinear force-displacement curve was being probed, as expected from the models presented above. Moreover, the use of such frequencies reduced the tip oscillation amplitude dramatically, rendering it effectively immobile. As such, the tip constituted (on the high-frequency time scale) an infinitely stiff surface against which the sample periodically is indented [34]. Ideally, then, the force-displacement curve for the tip-sample indentation should precisely correspond to the models presented above. The appearance of a nonzero dc displacement of the tip implies that a nonlinear region of the force-displacement curve is being probed. The oscillation amplitude at which this nonlinearity appears is then related to the slope of the linear portion of the F versus h curve, that is, the contact stiffness. This imaging mode has been referred to as ultrasonic force microscopy or UFM.

The critical concept for mechanical imaging via UFM is the inherently nonlinear interaction of a scanning-probe can-tilevered tip in contact with a surface that is undergoing out-of-plane ultrasonic vibrations at a frequency far exceeding the resonant frequency of the cantilever [33]. In such a case, the cantilever is inertially damped and, on the ultrasonic time scale, effectively rigid [34]. Hence, the surface rigidity of materials with contact stiffness orders of magnitude higher than the cantilever spring constant can be quantitatively measured. To illustrate how surface nanomechan-ical rigidity can be extracted via UFM, Figure 5 shows a schematic of a prototypical force-displacement curve for a rigid nanoprobe tip in contact with a surface. Negative displacement (indentation) of the sample by the tip results in a strong repulsive force. For positive displacement, an attractive force exists between the tip and sample until the sample is pulled away from the tip. This attractive force may result from van der Waals forces, adsorbed fluid layers, or other types of adhesive interactions.

For a scanning probe tip in static contact with a surface, the relative tip-sample distance is ho and the tip-sample contact force is Fo. Apply an ultrasonic out-of-plane oscillation Ah(t) = ho + A cos(wt). If A is small, the average of the force excursion AF is approximately equal to Fo, that is,

Force F(h)

\

Fo

'AF

\

' f

ho

\

Indentation (h) Pull-off

Ah=ho+Acos(a>t)

Figure 5. Schematic force-displacement curve for nanoprobe tip and a sample surface. F(h) is the applied force. h is the relative displacement between the tip and the undeflected sample surface.

Figure 5. Schematic force-displacement curve for nanoprobe tip and a sample surface. F(h) is the applied force. h is the relative displacement between the tip and the undeflected sample surface.

the tip undergoes no average displacement. If the oscillation amplitude is increased, the nonlinear region of the curve must be included to correctly calculate the average force on the cantilever tip [35]:

where kc and zc are the cantilever spring constant and tip deflection, respectively. A simulation of this AF as a function of oscillation amplitude A is shown in Figure 6 using the JKRS model [17-18]. The inset in Figure 6 highlights a critical feature of this average force. There is a "kink" or slope discontinuity in the averaged force, which appears for oscillation amplitudes exceeding a certain threshold value Ath, for which the tip approaches pull off. This slope discontinuity in (F(ho,A)) translates into a sudden increase in steady-state tip deflection [19]. The threshold amplitude is directly related to the slope of the force-response curve, defined as the contact stiffness between the sample surface and tip

Figure 6. Simulated average tip-sample force-displacement curves for increasing values of A using Eq. (9) and the JKRS model (normalized units). Inset details the discontinuous slope resulting from oscillation amplitudes approaching or exceeding tip pull off.

In the repulsive regime of the force-displacement curve, S is related to the reduced Young's modulus of the tip and the sample. The details of this relationship depend on the complexity of the model used to describe the tip-surface interaction. For a simple Hertzian contact [10-11]

F is the average force, and R is the scanning probe tip radius. For a more realistic approach, such as the JKRS model, the relationship is more complex, and can be extracted using a forward modeling approach based on curves similar to those in Figure 6.

As noted above, it is critical for the oscillation frequency to significantly exceed the fundamental cantilever resonance frequency (typically hundreds of kilohertz). In this frequency domain, the cantilever is inertially damped, not responding quickly enough to the oscillation to be deflected. For cantilever probes designed for use with conventional SPMs, this superresonant frequency typically exceeds 1 MHz [33]. The indentation of such tips at the forces of interest are totally elastic, and do not damage the sample. Frequencies as high as 80 MHz have been used, although higher order cantilever flexural resonance effects must be considered or instabilities may arise [36-37].

Since Ath depends monotonically upon the sample contact stiffness, its spatial variation will provide a map of the surface elastic response. Periodic ramping of the vibration amplitude beyond Ath as the tip is scanned across the sample provides elastic maps in the same fashion as topographic images. Figure 7 shows a simulated average tip deflection resulting from the application of a triangular-modulated vibration amplitude to the tip-sample displacement [36]. In the half cycle containing the ramp, note the discontinuity at Ath. Experimentally, this periodic tip deflection is extracted for modulation frequencies that are large compared to the tip-displacement feedback frequency in a normal topography mode, but small compared to the response frequency of the SPM photodiode detector used to measure tip deflection (typically, a few kilohertz). Experimental results are shown in Figure 8 for UFM tip deflection responses from aluminum and polymer substrates [38]. The differentiation of threshold voltages between the two materials is clearly

Modulated Ultrasonic Waveform

Modulated Ultrasonic Waveform

Time

Figure 7. Simulated deflection response of a cantilevered tip (bottom) to a modulated ultrasonic vibration (top). Note that the onset of tip deflection requires a threshold amplitude.

Time

Time (sec)

Figure 8. Experimental ultrasonic tip-deflection curves obtained from aluminum (upper curve) and polymer (lower curve) surfaces. Arrows denote the onset of tip deflection, estimating threshold amplitudes. The ramped modulation envelope for the ultrasonic sample vibration is shown at the bottom of the figure.

Time (sec)

Figure 8. Experimental ultrasonic tip-deflection curves obtained from aluminum (upper curve) and polymer (lower curve) surfaces. Arrows denote the onset of tip deflection, estimating threshold amplitudes. The ramped modulation envelope for the ultrasonic sample vibration is shown at the bottom of the figure.

resolved. To translate these tip-deflection curves into pixilated mechanical image maps, lock-in amplification is used to translate the area under each curve into a raster-scan pixel intensity.

Determination of the sample modulus from these data can be carried out by mapping the amplitude response of the tip deflection to extract the functional dependence of the threshold amplitude versus Fo [33, 35, 39]. Extracting the amplitude variation of the "kink" (Fig. 7) provides the necessary information to determine the static or zero-amplitude contact stiffness from which the Young's modulus can be calculated. Figure 9 displays the ultrasonic tip-deflection response as a function of applied tip force Fo for a spin-on polymer film. The inset in Figure 9 plots the variation of threshold amplitude against increasing force. Assuming a Hertzian response for the contact stiffness, the reduced Young's modulus was directly calculated to be 4.8 GPa using a known tip radius and the absolute vibration amplitude (determined from an optical vibrom-eter). Using this methodology, Young's modulus measurements have been undertaken for materials ranging from nanoporous silicates (Er ~ 2 GPa) to Si3N4 (Er ~ 300 GPa). Application of this technique by Dinelli et al. reported relatively large sample-to-sample variations of modulus measured using this method [35]. This variation is significantly reduced for measurements carried out in vacuum following a low-temperature bake to remove surface-adsorbed water

Time(s)

Figure 9. Variation of ultrasonic tip deflection as a function of applied tip force Fo for a polymeric film. The variation of threshold amplitude is apparent. Inset plots this variation as a function of applied force.

Time(s)

Figure 7. Simulated deflection response of a cantilevered tip (bottom) to a modulated ultrasonic vibration (top). Note that the onset of tip deflection requires a threshold amplitude.

Figure 9. Variation of ultrasonic tip deflection as a function of applied tip force Fo for a polymeric film. The variation of threshold amplitude is apparent. Inset plots this variation as a function of applied force.

films. For silicon oxide films, recent work has demonstrated an absolute error for modulus determination of <25%. Relative local variations on a single sample (point to point) are less than 1% [40].

Burnham et al. described this mode as the "mechanical diode" mode in that the deflection response to an imposed vibration was not symmetric [31]. UFM has been used by that group and a host of others to characterize multiple nanoscale systems to investigate the nanomechanical variations of materials. One of the earliest applications of UFM was by Yamanaka et al. to image subsurface edge dislocation lines in highly oriented pyrolytic graphite [34, 41]. Their work revealed near-surface edge dislocations in the UFM scan as low-contrast lines against a bright field. No associated features were evident in the topography. The subsurface sensitivity of UFM (as for FMM) is attributed to the elastic deformation region in the vicinity of the tip/sample contact. Using a simple static picture, it is reasonable to assume that defects in the near-surface region would alter the local effective modulus. This is the picture commonly used for nanoindentation (usually for a much larger tip/surface contact area). A more accurate picture for the UFM case is one of wave propagation. In the reference frame of the sample, the tip impinges with ultrasonic frequency, and the average displacement of the tip depends upon the surface boundary conditions and the transfer of acoustic energy [42]. A localized defect below a solid surface will affect the average tip displacement if it is within the volume probed by the evanescent acoustic wave excited in the sample. Initial estimates placed these defects within approximately 10 nm of the surface. More recent work by the same authors have not only used UFM as a probe for such subsurface edge dislocation lines, but have also used the pressure field below the tip to induce dislocation motion. By observing the dislocation as a function of load, they were able to displace a single dislocation by 20 nm for a load increase of 80 nN. Reduction of this load permitted the defect to move to its original location. The observation of such dislocation motion is a truly remarkable achievement in nanomechanics. Burnham et al. also used UFM to image subsurface mechanical variations. In that case, a thin polyamide film was applied over a GaAs grating (250 nm period). The polyamide thickness was not revealed in [31] so the depth sensitivity could not be inferred.

Geisler et al. specifically considered sensitivity of UFM to large subsurface voids in Si [43]. The experiment consisted of ion milling angled voids in the cross section of a single-crystal Si wafer. A void ceiling height varied linearly, providing a controlled void depth. From the point of view of mechanics, the deflection of the Si as the tip was scanned across the region resulted from deflection rather than deformation (compression). The contact stiffness of the surface was modeled using finite-element analysis (FEA) to obtain an effective deflection map. Initial investigations assigned a maximum detectable microvoid depth of ~200 nm. However, the large contact forces required in this method often resulted in significant wear of the tip [43].

Altemus and co-workers similarly utilized the UFM to map the deflections of suspended microstructures (can-tilevered mirrors) to investigate oscillation-induced hardening of the Au-Cr-Si trilayer beam [44]. After prolonged oscillation (~109 cycles), slight gradations in the UFM intensity were observed that presumably resulted from highamplitude induced wear. The gradations most likely reflect variations in the effective high-frequency contact stiffness as subresonant effects would most likely not be observable via UFM.

The capability of UFM to investigate variations of the surface elasticity of higher modulus materials was demonstrated for Ge quantum dots grown (QDs) grown on a silicon surface. Kolosov and co-workers carried out a UFM investigation of such a system [45]. QDs were deposited following Kaimins [46]. Their data showed a clear UFM contrast difference between the Si (Ebulk = 164 GPa) and Ge (Ebulk = 121 GPa). More recently acquired data on the same structures are shown in Figure 10 [47]. As in [45], the quantum dots possess a relatively unimodal diameter distribution peaked near 100 nm. The QD height was typically 15 nm (~50 Ge monolayers). Closer inspection of the UFM image in the region between the large Ge QDs reveals lower gradations of elastic contrast. When compared to the topographic image, these areas represent plateaus approximately 0.4 nm in height. We associate these regions with a single or double Ge monolayer [48-49]. The ability of UFM to apparently distinguish the mechanical response of only a few monolay-ers of Ge is impressive, and an important demonstration of its utility for nanomechanical imaging. Since no comparable studies have been published regarding SLAM on similar structures, it is difficult to gauge the comparative sensitivity of the two.

The application of UFM for other nanoparticle films has recently been examined. An investigation of the nucleation of thin Cu films was carried out to investigate the ability of UFM to distinguish between nanoscale Cu clusters and an underlying TaN film. The Young's modulus for Cu varies between 110 and 120 GPa, depending upon processing [50-51]. E for crystalline Ta is approximately 189 GPa, while for TaN, E can exceed 300 GPa, although typical vapor-phase processed TaN can exhibit mechanical moduli variations of 40% below that maximum [52]. Consequently, it would be expected that UFM imaging of Cu nanoparticles on a TaN substrate would appear as low-contrast islands on a bright background. Figure 11 shows a series of images for

Figure 10. 1 fxm x 1 fxm AFM scans (left) and UFM scans (right) of Ge QDs on Si. The bright background of the UFM scan corresponds to the higher modulus Si substrate. The gray contrast gradations between the dark 100 nm diameter Ge QDs are tentatively identified as monolayer islands of Ge. One such area is highlighted in both scans by the white circle.

Figure 10. 1 fxm x 1 fxm AFM scans (left) and UFM scans (right) of Ge QDs on Si. The bright background of the UFM scan corresponds to the higher modulus Si substrate. The gray contrast gradations between the dark 100 nm diameter Ge QDs are tentatively identified as monolayer islands of Ge. One such area is highlighted in both scans by the white circle.

Figure 11. 0.4 fxm x 0.4 fxm AFM and UFM scans of Cu grains nucleated on a TaN substrate. (a) AFM and (b) UFM of a 25 nm thick (nominal) Cu film. The UFM contrast is dominated by variations within grains. The bright regions at grain boundaries result from an increased tip/sample contact area. (c) AFM and (d) UFM scans of a 5 nm thick (nominal) Cu film. In the UFM image, the Cu grain appears as a dark area on a bright substrate. The thickness of the outlined grain in (c) and (d) is approximately 5 nm. The medium contrast regions above this grain correspond to Cu regions ~1.5 nm in thickness.

Figure 11. 0.4 fxm x 0.4 fxm AFM and UFM scans of Cu grains nucleated on a TaN substrate. (a) AFM and (b) UFM of a 25 nm thick (nominal) Cu film. The UFM contrast is dominated by variations within grains. The bright regions at grain boundaries result from an increased tip/sample contact area. (c) AFM and (d) UFM scans of a 5 nm thick (nominal) Cu film. In the UFM image, the Cu grain appears as a dark area on a bright substrate. The thickness of the outlined grain in (c) and (d) is approximately 5 nm. The medium contrast regions above this grain correspond to Cu regions ~1.5 nm in thickness.

Cu nanograins on a TaN substrate (images have undergone no postimage processing, including flattening). The UFM image clearly separates Cu nanograins from the substrate. A 5 nm thick grain highlighted in Figure 11 has a clear contrast differential from the TaN substrate [53]. From topographic analysis, the medium-contrast areas correspond to Cu as thin as 1.5 nm (~five monolayers). Larger Cu grains from thicker (~25 nm films) are also shown in Figure 11. Here, the entire surface has been covered by Cu, and the UFM contrast largely results from variation of tip/surface contact area at grain boundaries or slight variations of contact stiffness within individual grains.

Although Cu can be considered soft compared to the Si3N4 tips typically used for UFM, such may not be the case for materials like TaN, as described above. The limits of applicability for UFM have been examined in detail for such high-stiffness materials. Dinelli et al. considered modulus extraction from UFM analysis of polished silicon and sapphire surfaces [35]. Using the JKRS model to carry out simulations of the tip-sample force-displacement response curves acquired from UFM scans, extracted moduli were in good agreement with expectations if the tip-surface contact area remained relatively constant. This work specifically notes complexities introduced into the UFM force-displacement response curve analysis of the tip radius of curvature variation as a function of applied force. This variation is above and beyond the simple elastic deformation predicted by a simple Hertzian model. For example, Dinelli et al. note that, for investigations of the contact stiffness of silicon, the effective tip radius of curvature may increase from an effective value of 6 nm for a low-load-force (~70 nN) to an effective value of 25 nm for a load force of 280 nN. Such challenges are not unexpected for tip and sample moduli of the same order.

However, despite system-dependent complexities in quantitative modulus extraction, the relative modulus sensitivity of UFM is well established. Kolosov and co-workers examined an impressively wide range of material systems via UFM to explore the capabilities of the technique [39]. In addition to the Ge QD work referenced above, UFM investigations of carbon fiber composites, polymer-polymer composites, two-phase Langmuir-Blodgett films, polymer-glass laminates, and immobilized proteins have been undertaken. In each case, UFM imaging revealed nanoscale mechanical variations associated with nanoscale composition. In the case of thin-film laminates, UFM revealed areas of delam-ination between the thin film and the substrate. This was demonstrated for thin silicon oxide films deposited on PET (polyethylene terephthalate) polymer substrates [54]. Similar phenomena have been observed very recently for the delamination of metal films on polymers. Figure 12 shows the stress-induced delamination of a 50 nm thick Ta film on a polyaromatic polymer substrate (SiLK® polymer, Dow Chemical Corporation). The topography scan clearly reveals a large hillock feature associated with buckling of the Ta film. Corresponding UFM measurements reveal a clear drop in contrast at the edge of the raised region, denoting a dramatic increase in mechanical compliance of the Ta film. Since no compositional variation of the Ta film was evident via spectroscopic analysis, the compliance loss was assumed to result from delamination. An interesting feature of this UFM scan is the high-contrast region at the crest of the raised Ta buckle. It is possible that the UFM tip is sampling a local increase in contact stiffness due to a micromechanical

II ill llll

Figure 12. 20 ^m x 20 ^m AFM (left) and UFM (right) scans of a 50 nm Ta film deposited on Dow Chemical's SiLK polymer. The AFM scan highlights a buckled section of the Ta film. The corresponding UFM scan reveals the dramatically increased compliance of this region. The high-contrast line running along the center of the delaminated region may result from a micromechanical "arch" effect. The striations at the upper right quadrant of both images correspond to topographic and mechanical compliance corrugations, respectively, presaging the onset of delamination.

arch effect of the Ta film. Investigations into this phenomena are continuing.

As the above discussion notes, surface contact stiffness is affected by a wide array of phenomena related to both the material under investigation as well as the particulars of the nanoprobe used. Consequently, a lower estimate of the sensitivity of UFM to pure modulus variation is of significant interest. Dinelli and co-workers carried out such an investigation using cross-sectional UFM imaging of a GaAs-AlxGa1-x As supper lattice (lattice constant ~200 nm) [35]. Their work showed that relative changes in the elastic modulus as low as 0.08% (corresponding to a Ax of Al as low as 0.1) were observable in UFM. This is a truly remarkable sensitivity.

Geer and co-workers have recently investigated the use of UFM for integrated-circuit (IC) and nanomaterial characterization [54-57]. The surface mechanical variation of IC interconnect test structures consisting of inlaid (damascene) Al lines in a low-dielectric-constant polymer (ben-zocyclobutene) was investigated with respect to variations in the polymer modulus resulting from IC processing. The damascene technique used in IC processing consists of the formation of recessed trenches and vias in a dielectric layer via reactive plasma processing (so-called reactive ion etching) [58]. In this manner, an IC interconnect pattern can be "transferred" into a dielectric layer. The recessed features are coated with a TiN barrier layer (to inhibit Al diffusion in the polymer), and subsequently filled with Al metal through vapor deposition [59]. The excess Al is removed via a chemical mechanical planarization (CMP) process resulting in an extremely flat inlaid pattern of transistor interconnect wires [60]. This process is repeated until a multilevel metal interconnect circuit is fabricated that completes the functioning IC circuit. Polymeric dielectrics have been under intense investigation in IC fabrication recently, owing to their low dielectric constant and associated reduction in RC losses during chip operation. However, the dramatic mismatch of mechanical properties between metal lines (in some cases < 100 nm wide) and polymer dielectrics places limitations on the internal stresses that can be tolerated during the thermal excursions of chip operation. Consequently, to accurately model chip stability, it is important to know the variation in mechanical properties of such dielectrics during their deposition. The reactive ion etch process that is used to "dig" the trenches in the dielectric can also result in changing local polymeric reaction properties [61]. UFM analysis of these structures (Fig. 13) revealed that the reactive ion etch effectively "hardened" the polymer near the exposed surface [54]. This hardening resulted from selective removal by the ion etch of carbon, leaving an SiO2-rich polymer at the surface of the recessed feature [62]. No such characterization currently used in IC processing has been able to provide such direct evidence of process-induced mechanical variation of materials.

Moreover, UFM has also shown the potential for imaging the surface damage of polymeric materials. Figure 14 shows UFM and topography scans for a region of an Al/polymer IC test structure after CMP [63]. In the UFM image, the lateral lines are Al-filled trenches in the polymer matrix. There is an associated topography (Al lines are slightly depressed relative to the polymer). The UFM also reveals striations

Figure 13. 2.0 fim x 2.0 fim AFM (left) and UFM (right) plan-view scans Al lines (dark regions in topography) in a polymer matrix (bright regions in topography). In the UFM scan, the contrast is reversed, as expected for an elastic image map. Note the local variation of the contrast within the polymer at the edges of the trench walls. This increase in elastic modulus result from silicon-oxide polymer formation in the polymer after exposure to a reactive-ion etch plasma.

Figure 13. 2.0 fim x 2.0 fim AFM (left) and UFM (right) plan-view scans Al lines (dark regions in topography) in a polymer matrix (bright regions in topography). In the UFM scan, the contrast is reversed, as expected for an elastic image map. Note the local variation of the contrast within the polymer at the edges of the trench walls. This increase in elastic modulus result from silicon-oxide polymer formation in the polymer after exposure to a reactive-ion etch plasma.

crisscrossing the trench field. These striations look similar to scratches, not unexpected after CMP processing. However, the "scratches" have no corresponding feature in the topography image, and only occur in the polymer region. Preliminary investigations associate these features in the UFM image with mechanical stressing in the polymer that results in cross-link bond breakage in the polymer, resulting in a lower local Young's modulus [64]. The reduction of cross linking in these areas also reduces the local shear modulus, and permits relaxation of topographical variations, leading to the featureless topography scan.

Similar behavior relating to the nanomechanical imaging of surface-deformed polymers was carried out by Iwata and co-workers [65]. An AFM tip was used to locally deform or scratch the surface of a polycarbonate film. The local stresses induced the organization of the local carbon nanobundles [66]. In this work, UFM demonstrated

Figure 14. 40 fim x 40 fim AFM (left) and UFM (right) plan-view scans of an Al trench field (horizontal lines) in a polymer matrix. The sample was planarized via chemical mechanical planarization. In the UFM, vertically oriented striations are evident. These striations are not evident in the AFM image, and are believed to result from local bond breaking in the polymer due to the action of the CMP abrasive. Although not visible at this magnification, the striations are not continuous across the Al regions.

Figure 14. 40 fim x 40 fim AFM (left) and UFM (right) plan-view scans of an Al trench field (horizontal lines) in a polymer matrix. The sample was planarized via chemical mechanical planarization. In the UFM, vertically oriented striations are evident. These striations are not evident in the AFM image, and are believed to result from local bond breaking in the polymer due to the action of the CMP abrasive. Although not visible at this magnification, the striations are not continuous across the Al regions.

sufficient sensitivity for the observation of slight elastic variations of the modified PC surface. The bundles grew in vertical extent with repeated scan scratching, yet the associated bundle mechanical response decreased. The associated contrast of the UFM images of individual bundles decreased, consistent with the threshold behavior of the cantilever response. It was suggested by the authors that the increase of the mechanical bundle compliance is indicative of the microcracks or voids due to the local tip-induced stresses associated with repeated scratching.

Recently, Kolosov has expanded the applicability of UFM by demonstrating so-called waveguide UFM or W-UFM. In such a mode, the ultrasonic vibration is introduced to the cantilever, and not the sample. If a higher order flexural vibration is excited, the tip of the sample will oscillate as an antinode for the standard vibrating beam problem. However, the suppression of the fundamental resonance mode will still restrict the overall cantilever rigidity, resulting in the indentation of the sample against the tip, similar to conventional UFM. In fact, the tip-sample force-displacement response curves for the two techniques are quite similar [67].

Ultrasonic force microscopy offers exceptional sensitivity to surface contact stiffness. Although no quantitative comparison has been carried out, preliminary work by Shekhawat et al. implies that it may be superior to SLAM from the perspective of overall sensitivity. However, the difficulty of quantitative analysis for high-stiffness materials via UFM resulting from contact area variations demands further technique development. A parallel technique developed for nanomechanical imaging offers a different route for the qualitative and quantitative analysis of surface contact stiffness and modulus, so-called acoustic atomic force microscopy (AAFM). Early use of AAFM was reported by Arnold and co-workers [68-70]. It used a slightly different experimental apparatus for mapping the surface elastic response. However, for all intents and purposes, it is essentially the same as UFM when the sample oscillation amplitude is large enough to probe the nonlinear region of the tip-sample force-displacement response curve. The notable difference is the utilization of AAFM in the linear regime for extraction of the contact stiffness. Linear regime AAFM (also referred to as ultrasonic AFM by Quate and co-workers [71]; the tendency of each group to define a unique acronym for their particular experimental configuration is confusing for newcomers to the field) is similar to Burnham's SLAM. However, Burnham modeled the AFM cantilever tip as a simple spring, and hence one resonance. The groups of both Arnold and Quate took a more involved approach, and solved the full dynamical equation of a can-tilevered tip in contact with the surface. As the surface load of the tip is varied, the resonant frequency spectrum of the cantilever is modified. Hence, the varying surface contact stiffness probed by a cantilevered tip as it is scanned across a sample surface will result in a modified resonance spectra for the cantilever. At a given point on the sample surface, determination of the shift in the cantilever resonance is used to extract the contact stiffness and determine, if desired, the Young's modulus of the surface.

Assuming a linear response, the relation between the surface contact stiffness k and a resonant frequency spectra k2n = 4k212p/(b2E) [72], the surface-modified cantilever

frequency spectrum is given by [69]

— (cosh knLt sinknLt — sinhknLt cos knLt) kt x (1 + cos knL cosh knL) kc + (cosh knL sin knL — sinh knL cos knL') x (1 — cos knLx coshknLx) (k L )3

Here, L is the total cantilever length, b is the cantilever thickness, L1 is the base-to-tip length of the cantilever, L' is the tip-to-end length of the cantilever, and E is the Young's modulus of the cantilever material. For cantilever width a, the cantilever spring constant kc is given by kc =

Eb3a 4L\

Rabe and co-workers used the shift in resonance to qualitatively map the elastic response of a surface [69]. For a given point on a sample, the contact stiffness was approximated. Crozier et al. [71] and Yaralioglu et al. [73] extended this analysis with the development of an iterative algorithm to fit the shifts of multiple resonant peaks of the cantilever spectrum at points on the sample to determine, with high precision, the calibrated contact stiffness. Application of this technique to the film thickness measurement represented an impressive achievement for nanomechanical characterization.

The culmination of this approach for ultrasonic-based scanning probe microscopy has recently been reported by Yamanaka's group [74]. They have modified the conventional AAFM configuration to include a feedback circuit to maintain cantilever resonance as the tip is scanned across the sample with a minimal increase in image acquisition time. Hence, a true nanoscale "map" of the cantilever resonance frequency can be obtained and used for conversion to contact stiffness data (although quantitative conversion has yet to be done). That technique demonstrated some of the highest resolution images to date regarding subsurface dislocations in graphite, and promises a bright future for the direct nanoelastic imaging of materials.

A variety of other related techniques for nanomechanical imaging have been reported; however, the majority represent variations of those discussed with respect to the introduction of lateral oscillation or combinations of high and low frequencies to the tips and samples [75-79]. A unique, and possibly powerful, technique for probing the elastic AND viscoelastic properties of materials at the ultrasonic time scale was recently put forth by Kolosov and Briggs [80-81]. This technique combines W-UFM and UFM. The heterodyne interference of the two vibrations produces a low-frequency amplitude modulation that traverses the nonlinear portion of the force response curve in much the same fashion as the externally modulated ultrasonic vibration in conventional UFM. However, the phase coherence of the two vibrations yields a temporally calibrated phase image that can resolve time shifts on the nanosecond time scale.

Hence, viscoelastic (dynamic) nanomechanical imaging is possible through so-called heterodyne UFM (HFM). In this mode, two ultrasonic vibrations are applied to the measuring system. One ultrasonic vibration is applied to the tip, which acts as a mechanical waveguide, while a second ultrasonic vibration is applied to the sample (as in conventional UFM). This results in coherent oscillations of the tip and sample surfaces at the local point of contact [80]

Here, the tip vibration waveform acts as a temporal reference. The temporal phase delay associated with the surface vibration waveform is designated by . A linear force-displacement relationship between the tip and sample would result in no average cantilever deflection. However, a nonlinear force-displacement curve yields a heterodyne coupling between the two oscillations. This is seen by expanding the tip-sample force in a Taylor's series to second order in the tip-sample separation, and calculating the average force responsible for cantilever deflection kc (ztip(wtip> WS> ^tip^s))

1 fT

T J (*1(*tip - zs) + X2(ztip - Zs)2 + ••• ) dt (16)

Xi and Xi denote first- and second-order "spring constants," respectively, associated with the force-displacement curve. The cantilever deflections resulting from both oscillations [Eq. (15)] are substituted into Eq. (16) and time averaged over the response time t of the SPM photodiode detector (hundreds of kilohertz). Only the nonlinear term is nonva-nishing. Likewise, the high-frequency terms are replaced by their rms values, and the remaining time dependence corresponds to the heterodyne or difference frequency term

The deflection of the tip is now dynamically linked to the vis-coelastic response of the sample to the ultrasonic vibration. The phase term represents the dissipative lag/lead in the surface response with respect to the tip reference frequency. In the vicinity of an interface, this dissipation is directly linked to a local loss tangent, indicative of the adhesive or interfacial bonding strength. For continuous materials, the dissipation is due to a classical viscosity. Extracting the spatial dependence of this phase term provides image contrast indicative of the dynamics of materials, material interfaces, and defect structures with the same nanometer spatial resolution as UFM.

Experimentally, the two oscillations are applied to the tip and sample by two matched piezocrystals attached to the tip substrate and the base of the sample, respectively, and driven by a separate waveform with a mixer/filter circuit providing the heterodyne frequency to an RF lock-in amplifier for amplitude (x2) and phase (faS) extraction. Although the amplitude calibration is more complex than that for

UFM, the phase can be imaged directly, as displayed in Figure 15 for an integrated-circuit test structure consisting of square 5 /m polymer pads inlaid within an Al layer [60, 82]. The right panel data of Figure 15 were acquired with both ultrasonic vibration amplitudes above threshold for Al and the polymer. The left panel data of Figure 15 were acquired with both amplitudes below threshold. No phase variation is seen in the latter since the response is linear, validating the simple model presented above. The phase difference between the Al and polymer regions ~10 ns. Data were taken at a carrier frequency of 3 MHz. Recent efforts in our laboratory have demonstrated HFM operation up to 80 MHz. The 1° relative phase resolution of our HFM system provides a temporal (phase lag) resolution less than 40 ps.

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