Bottom-actuated GHz-ultrasound subsurface atomic force microscopy (AFM) can visualize nanoscale features buried micrometers deep — far beyond the reach of classical elasticity-based subsurface AFM — because the contrast is dominated by scattering and diffraction of acoustic waves at the buried features. However, the measured surface-displacement amplitude and phase maps are near-field interference patterns that, as the source work (van Es et al., 2020) states explicitly, cannot be directly interpreted in terms of the buried pattern, and the supporting forward simulations are two-dimensional approximations that omit the water coupling layer. The open problem is to make this modality quantitative and predictive: build validated full three-dimensional wave-propagation models of the complete acoustic stack (including the sub-micrometer coupling layer and finite-size buried scatterers), and solve the associated inverse problem of recovering buried-feature geometry and material contrasts from multi-frequency surface-displacement data with quantified accuracy, so the technique can serve as non-destructive metrology for optically opaque multilayer stacks such as semiconductor 3D-integration devices. Reviews in 2021, 2023, and 2025 consistently identify the quantitative inversion of subsurface AFM signals as a major open bottleneck; the forward problem is comparatively mature.

Background

Atomic force microscopy (AFM) maps a sample's properties by scanning a sharp tip mounted on a micromachined cantilever across its surface. Since the 1990s, ultrasound-assisted AFM variants — ultrasonic force microscopy (UFM), atomic force acoustic microscopy (AFAM) and ultrasonic AFM (UAFM), heterodyne force microscopy (HFM), and scanning near-field ultrasound holography (SNFUH) — have tried to extend it below the surface. In most of these modalities the contrast arises because the effective stiffness of the tip–sample contact changes when the tip, whose stress field penetrates only a few nanometers, passes over a buried region of different visco-elasticity; by Hertzian contact mechanics this elasticity-based contrast can only reach a depth of roughly three times the tip–sample contact radius, i.e. about 10 nm for a soft cantilever. Seeing much deeper requires a different contrast mechanism. When the ultrasound carrier frequency is raised into the GHz range, the acoustic wavelength in solids and polymers shrinks to the micrometer scale, and scattering and diffraction of acoustic waves by buried features become significant: subsurface imaging with GHz AFAM showing scattering contributions was reported by Hu, Su and Arnold (2011), and the relative roles of wave scattering versus contact-stiffness modulation were debated for years (Sharahi et al., 2017). The TNO group developed a bottom-actuated configuration — a piezoelectric transducer beneath the sample launches ~GHz compressional waves through a silicon delay line, a sub-micrometer water coupling layer, and the sample stack — first for shallow elasticity-based mapping (van Es et al., 2018) and then with optimized acoustic coupling (van Neer et al., 2019). The source work (van Es et al., 2020; arXiv preprint, TNO) demonstrated unambiguous scattering contrast in this configuration: an AFM tip in contact with the top surface of a sample consisting of 5 µm of PMMA (poly(methyl methacrylate), a standard polymer) on silicon, with a matrix of square pit-like features buried at the PMMA–silicon interface, detects the local surface displacement (amplitude and phase, obtained by demodulation of the tip response at the difference frequency). Features buried 5 µm deep — far beyond the ~10 nm elasticity range — were imaged. Four observations established acoustic scattering as the dominant contrast: diffraction ripples with a sinc-like envelope outside the feature matrix; strong changes of the observed pattern with carrier frequency (the acoustic wavelength near 1.9 µm is comparable to the 1.6 µm feature pitch, placing the surface in the near field); thickness-resonance signatures of the acoustic stack matching a KLM model (a one-dimensional piezoelectric-transducer transmission-line model due to Krimholtz, Leedom and Matthaei); and quantitative agreement between measured surface-displacement patterns and two-dimensional finite-element method (FEM) simulations without fitting parameters. Yet the same physics creates the interpretation problem this task is about. The measured map is an interference pattern of the incident and scattered wavefields, not an image of the buried geometry: the paper's Discussion states that the observed pattern cannot be directly interpreted in terms of the buried pattern. The supporting simulations are two-dimensional (effectively infinitely long trenches); finite-size pillar- or pit-like scatterers were emulated by rotating a single-feature 2D result into a point-spread function (PSF, the response to a point-like scatterer) and convolving it with the feature-array pattern, an approximation that does not capture true three-dimensional scattering and diffraction. The simulations also omit the water coupling layer, so the carrier frequencies of optimum contrast differ between simulation and experiment. Finally, the technique's motivation — metrology for three-dimensional semiconductor stacks such as 3D NAND (a flash-memory architecture built from many vertically stacked device layers), where the number of layers keeps growing and optical methods fail through opaque layers — demands calibrated, quantitative recovery of feature geometry and material contrast rather than qualitative contrast images. Defining terms used below: FEM (finite-element method) is the numerical solution of the elastic-wave equations on a meshed geometry; a forward model predicts the measured signal for a given buried structure, while the inverse problem recovers the buried structure from measured signals.

Problem Statement

Make scattering-contrast GHz-ultrasound subsurface atomic force microscopy a quantitative, predictive measurement technique. Concretely, given measured surface-displacement amplitude and phase maps of the top surface of an opaque layered sample, acquired with a bottom-actuated GHz-ultrasound AFM at a set of carrier frequencies, together with the acoustic stack geometry (transducer, delay line, coupling layer, sample layers): (1) construct and validate a full three-dimensional forward model of elastic-wave propagation in the complete stack — including the sub-micrometer water coupling layer and realistic finite-size buried scatterers such as pits, pillars, and voids — that predicts measured carrier-frequency spectra and surface-displacement patterns without free fitting parameters; and (2) solve the associated inverse problem, recovering buried-feature geometry (lateral position and extent, depth, shape) and material contrasts from the multi-frequency surface-displacement data, with quantified accuracy and robustness. These deliverables serve a single overarching objective: converting qualitative scattering-contrast subsurface AFM into quantitative non-destructive metrology. Validation of the reconstruction against known structures is the expected evidence of success: the validation ground truth must be independent of the model being inverted — measured data, or an independent forward model or simulator other than the solver's own — and synthetic data generated and inverted with the same model may be used for development only, not as the evidence of success. Calibration and sensitivity analyses (detection and resolution limits as functions of depth, material contrast, and carrier frequency) are valuable supporting context for this objective, but are not a mandatory deliverable or acceptance condition. No particular algorithm or answer form is prescribed; the field's criterion for quantitative is agreement with ground-truth geometry and material parameters within stated uncertainties.

References

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