Extended Data Fig. 3: Measurement and calibration procedures for local broadband dielectric spectroscopy.
From: In-plane dielectric constant and conductivity of confined water
a, Representative |dC/dz|(x,z) image taken at 2 kHz for channels with h ≈ 5 nm. The image shows three nanochannels and was acquired by measuring |dC/dz| at various constant heights and a fixed y position. The tip approached the surface from above using small steps. b, Absolute values of |dC/dz| for scanning along the lateral (x) axis at constant y and various heights. The profiles were averaged over about ten lines using images in a. The signal became sensitive to the presence of water inside the channels at z ≲ 30 nm. c, Gain-calibrated |dC/dz|(z) curves at different frequencies taken over the centre of the spacer region (colour-coded lines). White symbols indicate the values of |dC/dz| and zscan corresponding to the profiles shown in b. Inset, same curves without calibration of the electrical gain g and the offsets (colour-coded). We used the curve at 2 kHz as a reference to calibrate g at high frequencies by matching the reference curve with the curves taken at higher f, using only g and an offset as fitting parameters. d, Schematic representation of our 3D AFM datasets |dC/dz|(x,z,f) obtained by acquiring subsequent images |dC/dz|(x,z) for many frequencies from kHz to GHz. The black dashed arrows in a and d indicate the position at which the |dC/dz|(z) curves in c were acquired before taking such 3D images. e,f, Example of broadband dielectric imaging spectroscopy on a test sample of hBN on doped silicon using the same g calibration procedure: topography image (grey coloured) of 21-nm-thick hBN crystal on doped Si and corresponding calibrated dielectric images (cyan-coloured images) at frequencies 6 MHz, 240 MHz and 1.1 GHz at a constant height (about 45 nm). The plotted values of |dC/dz| are variations after subtracting the |dC/dz| values measured on the Si substrate. Scale bars, 1 μm. The curves in f are gain-calibrated |dC/dz|(z) curves taken above the Si substrate (solid curves) and the hBN layer (dotted) at various f. The gain g for each f was obtained by scaling the curves measured on doped Si to the low-f reference curve at 2 kHz using only g and an offset as fitting parameters. By fitting the calibrated curves on hBN using the procedure and analytical formula in ref. 72, we obtained the dielectric constant of the hBN layer as a function of f (inset). Its average value across all of the f up to GHz was found to be ε⊥hBN = 2.9 ± 0.2, which is in good agreement with the expected bulk value for hBN in the perpendicular direction (ε⊥hBN ≈ 3, dashed line). Bars, standard error defined as 95% confidence interval of the fitting, not shown when smaller than the symbol. Other fitting parameters: AFM tip R = 102 nm and θ = 25°, obtained by fitting the curve taken on doped Si at 2 kHz.
