Extended Data Fig. 7: Sensitivity to the in-plane and out-of-plane dielectric constants of water and calibration curves using the 3D numerical simulations.
From: In-plane dielectric constant and conductivity of confined water
a,b, Calculated |dC/dz| at 1 GHz as a function of the ε// and ε⊥ of water (with σ// = σ⊥ = σbulk = 2 × 10−4 S m−1) at scan height zscan = 20 nm. h = 5 nm and the bottom hBN thicknesses H = 0 (metallic substrate) (a) and H = 200 nm (b). c, Calculated |dC/dz| as in a and b but as a function of H for different combinations of ε// and ε⊥. AFM tip cone height Hcone = 12 μm and cantilever length Lcantilever = 20 μm (nominal parameters) to include long-range range effects that play a role for very thick H. d, Cross-sections from a as a function of the ε⊥ of water for different values of ε//. e, Cross-sections from b as a function of ε// for different ε⊥. f, Calculated absolute value of |dC/dz| versus tip–surface distance z above the centre of the hBN spacer, as illustrated in the inset, for empty channels (black dashed line) and bulk-water-filled channels (cyan-coloured symbols) at various frequencies. The channel thickness is h = 5 nm and the bottom hBN thickness is H = 100 nm. Upper panel, error between the calculated curves for water-filled channels and for empty channels. g, Same as in e but for h = 1.5 nm and different zscan. The dotted red line corresponds to the typical |dC/dz| value on our smallest channels yielding ε// ≈ 1,000 at zscan estimated to be about 20 nm. The light-shaded region indicates the typical experimental uncertainty in |dC/dz| of about 1 zF nm−1. Other parameters used in a–g (unless stated otherwise in the legends): R = 100 nm, θ = 25°, Hcone = 6 μm, Hcantilever = 3 μm, Lcantilever = 0 μm, zscan = 20 nm, l = 3 μm, wc = 200 nm, ws = 800 nm, Htop = 50 nm. h, Calculated dielectric spectra (lines) corresponding to the experimental data (symbols) for our smallest device in Extended Data Fig. 10 (and in Fig. 2 before normalization) for different scan heights. Bars indicate the random noise level as measured at each f. The red solid curve is the best fit with the ε// and σ// of water as the only fitting parameters (values shown as labels) and the experimentally estimated zscan = 19 nm (other experimental and simulated parameters are the same as in Extended Data Fig. 10). The light-shaded region indicates the uncertainty in the extracted values of ε// and σ//. The red dashed curves indicate 3D numerical simulations for the same device assuming the properties of bulk water (εbulk = 80, σbulk = 2 × 10−4 S m−1). The pink and orange curves are calculated values for zscan = 18 and 20 nm, respectively, corresponding to the experimentally estimated uncertainty on zscan of ±1 nm. The black curves are the 3D simulations for the dielectric constant of bulk water using zscan as the fitting parameter to match the high-frequency plateau, yielding zscan = 5 nm, with σ// = σbulk (black dashed line) and σ// = 1.7 S m−1 (black solid line). All of the plotted |dC/dz| curves in panels a–e, g and h are after subtracting the |dC/dz| values at the centre of the hBN spacer region.
