Extended Data Fig. 5: 3D numerical simulations.
From: In-plane dielectric constant and conductivity of confined water
a, Schematic (not to scale) of the full-3D numerical model used to fit the experimental data. b, Calculated absolute value of |dC/dz| versus tip–surface distance z above the centre of the hBN spacer for empty channels (black dashed line) and bulk-water-filled channels (red-coloured symbols) at various frequencies. Channel thickness: h = 1.5 nm. Water channel parameters: ε// = 80, σ// = 0.1 S m−1, ε⊥ = 2, σ⊥ = 0, l = 3 μm, wc = 200 nm, ws = 800 nm. Upper plot, difference (‘Error’) between calculated curves for water-filled channels and those for empty channels. c–e, Corresponding calculated electric potential for three representative cases: low frequency (10 kHz) and purely dielectric channels (ε// = 80 and σ// = 0) (c); same low frequency but conductive channels (ε// = 80 and σ// = 0.1 S m−1) (d). e, Same as d but at f beyond conductivity relaxation. f, Potential along the Z direction (counted from the AFM tip) for the tip placed at zscan = 20 nm above the centre of the water channel (red and dashed blue lines) and above the centre of the spacer region (cyan line). Frequencies 10 kHz and 1 GHz, colour-coded. g, Same as f but along the Y direction at Z = 20 nm measured from the AFM tip (over the top layer). h, Same as g but along the X direction. Other parameters: R = 100 nm, θ = 25°, Hcone = 6 μm, Hcantilever = 3 μm, Lcantilever = 0 μm, Htop = 50 nm, H = 100 nm, ε//hBN = 5, ε⊥hBN = 3.
