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