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During conventional nanoindentation measurements, the indentation depths areusually larger than 1-10 nm, which hinders the ability to study ultra-thinfilms ($<$ 10 nm) and supported atomically thin two-dimensional (2D) materials.Here, we discuss the development of modulated \r{A}-indentation to achievesub-\r{A} indentation depths during force-indentation measurements while alsoimaging materials with nanoscale resolution. Modulated nanoindentation (MoNI)was originally invented to measure the radial elasticity of multi-wallednanotubes. Now, by using extremely small amplitude oscillations ($\ll$ 1 \r{A})at high frequency, and stiff cantilevers, we show how modulatednano/\r{A}-indentation (MoNI/\r{A}I) enables non-destructive measurements ofthe contact stiffness and indentation modulus of ultra-thin ultra-stiff films,including CVD diamond films (modulus $\sim$ 1000 GPa), as well as thetransverse modulus of 2D materials. Our analysis demonstrates that in presenceof a standard laboratory noise floor, the signal to noise ratio of MoNI/\r{A}Iimplemented with a commercial atomic force microscope (AFM) is such that adynamic range of 80 dB $-$ achievable with commercial Lock-in amplifiers $-$ issufficient to observe superior indentation curves, having indentation depths assmall as 0.3 \r{A}, resolution in indentation $<$ 0.05 \r{A}, and in normalload $<$ 0.5 nN. Being implemented on a standard AFM, this method has thepotential for a broad applicability.

$\r{A}$-Indentation for non-destructive elastic moduli measurements of supported ultra-hard ultra-thin films and nanostructures