In-plane dielectric fixed and conductivity of confined water

Device fabrication

We fabricated our units utilizing procedures much like these described in ref. ²². A free-standing silicon nitride (SiN) membrane was used as a substrate for our van der Waals meeting that concerned 4 atomically flat crystals (Extended Data Fig. 1). We began with etching an oblong aperture of roughly 3 × 20 μm² within the SiN membrane. This aperture was required to later function a water inlet from a reservoir positioned behind the wafer. Using the dry switch methodology, we transferred a graphite crystal (thickness roughly 10 nm) to seal the aperture. The graphite later served as the underside electrode. It was etched by way of from the again aspect utilizing reactive ion etching, which projected the aperture into graphite. Then a comparatively thick hBN crystal (H ≈ 50–200 nm, to function the underside layer) was transferred on prime of the graphite and once more dry-etched by way of from the again. Next we chosen a second hBN crystal (thickness h ≈ 1–60 nm), known as spacer, and patterned it into parallel stripes (spaced by about 200 nm) utilizing electron-beam lithography and reactive ion etching. We transferred the spacer layer on prime of the underside hBN utilizing moist switch procedures and aligning the stripes perpendicular to the oblong aperture. Finally, the third hBN crystal was transferred on prime of the spacer layer utilizing the dry switch methodology. This sealed the nanochannels in addition to the aperture. The thickness of the highest hBN (H_prime ≈ 20–80 nm) was fastidiously chosen to carry the AFM tip as shut as doable to the water but in addition to make sure that the highest layer reveals some sagging into empty channels with out blocking them fully²² (Extended Data Fig. 2a,d). After every switch, we annealed the meeting in Ar/H₂ at 300 °C for 3 h after which at 400 °C for five h to take away polymer residues and different contamination. As the ultimate step, we made {an electrical} contact with the graphite utilizing photolithography or, alternatively, utilizing silver paint to reduce the variety of cleanroom processes. Because the highest layer turned flattened (now not sagged)²² after liquid water crammed the channels (see Extended Data Fig. second–g), this allowed us to make sure that, no matter the dielectric response of water, the investigated nanochannels had been absolutely full of water (by evaluating AFM topography photographs earlier than and after filling; Extended Data Fig. 2e,f). Note that, given the massive in-plane conductivity of water discovered on this work for all thicknesses of channel, we may additionally confirm the water filling by taking atomic drive microscopy (AFM) photographs within the intermittent-contact enticing mode utilizing low-f utilized voltages (a number of okayHz right down to DC). In this case, the acquired distinction over the channels reversed from destructive to constructive on filling the water (Extended Data Fig. 2b,c) owing to the onset of robust electrostatic forces related to the conductivity of water, because the AFM suggestions adjusted the z-scanner to compensate for these forces throughout the scan. We emphasize that the distinction showing in Extended Data Fig. 2c (exhibiting roughness of a number of nanometres) displays variations within the electrical properties of water and hBN (therefore the pink/blue color scale) and never the precise topography, which is proven individually in gray in Extended Data Fig. 2a. When water crammed the channels, the sagging disappeared and the floor turned atomically flat with residual roughness of lower than 3 Å independently of the channel thickness, as described beforehand²². This flattening is demonstrated right here utilizing true topography photographs in Extended Data Fig. 2e,f and their profiles (Extended Data Fig. 2g).

As one other enchancment with respect to our earlier research, earlier than filling with water, we usually uncovered the underside aspect of the units to low-power Ar/O₂ plasma (8 W, 16 sccm Ar and eight sccm O₂ circulation) for 1 s. This made SiN extra hydrophilic and in addition cleared the entrances of channels from doable contamination. This process has proved useful for getting water inside. We additionally discovered that our units tended to delaminate if contamination remained trapped between van der Waals layers. To stop this from taking place, nice care was taken to examine for cleanliness of the units at every fabrication step (utilizing optical and AFM imaging). For instance, if dark-field optical photographs confirmed bubbles trapped between layers, such units had been discarded. Representative optical photographs of the studied units with clear interfaces are supplied in Extended Data Fig. 1a–c.

Local broadband dielectric imaging and spectroscopy

All SDM^10,41 measurements had been carried out utilizing a industrial atomic drive microscope (Nanotec Electronica with WSxM software program⁴²) operated at RT in dry ambiance. SDM was carried out within the amplitude-modulated electrostatic-force detection mode^43,44, adapting the strategy described in ref. ¹⁰. Briefly, we utilized an AC voltage between the AFM tip and the underside electrode. By detecting mechanical oscillations of the cantilever, we measured the electrostatic drive and, due to this fact, the primary by-product of the tip–pattern capacitance, dC/dz. Its worth depends upon each dielectric and conductive properties of investigated samples. SDM has beforehand been used for native measurements of each floor and sub-surface dielectric properties of varied supplies in numerous frequency regimes, from quasi-static to GHz (refs. ^45,46,47,48). The identical strategy has additionally been extensively used to review near-surface electron transport in strong samples^49,50,51,52, exhibiting the sensitivity of the approach to native conductivity. The force-sensing strategy was preferable in our case to the opposite current-sensing and microwave-sensing AFM strategies that may additionally study native impedance^53,54 as a result of the latter are typically much less delicate and extra complicated to implement.

To perform dielectric spectroscopy over the required very large bandwidth (100 Hz to 1 GHz), we mixed two beforehand reported force-sensing detection strategies. Both had been carried out right here utilizing conductive diamond-coated AFM probes (CDT-CONTR or CDT-FMR from Nanosensors) with spring fixed okay of a number of Nm⁻¹ and resonance frequency within the vary 20–60 okayHz. At low f as much as the cantilever resonance frequency, we used the twoω-detection strategy, as described in ref. ²² for measurements of the ε_⊥ of water, during which ω = 2πf is the angular frequency. Briefly, we utilized an AC voltage with amplitude v_AC = 4 V and measured the amplitude D_2ω(z) of the ensuing mechanical oscillations of the cantilever at double the utilized frequency, utilizing an exterior lock-in amplifier (Zurich Instruments HF2LI). The AFM tip–pattern capacitance gradient was calculated as |dC/dz| = D_2ω(z)4okay/v_AC². Despite the comparatively massive AC excitation, the response remained inside the linear regime, as verified by systematic amplitude-dependence measurements on reference samples and validated utilizing comparatively thick channels that exhibited the anticipated bulk water properties. For f greater than the cantilever resonance frequency, we used the heterodyne detection approach demonstrated in ref. ⁵⁵. To this finish, we utilized a high-f (provider) sign modulated by low frequency (f_mod = 1 okayHz, amplitude v_AC = 0.5 V) utilizing an exterior RF/GHz sign generator (Rohde & Schwarz SMA100B). We detected the cantilever mechanical oscillations at f_mod utilizing our exterior lock-in amplifier and, from these measurements, obtained the capacitance gradient on the provider f as |dC/dz| = D_mod(z)8okay/(gv_AC²), during which g is the f-dependent achieve of the exterior circuit and D_mod(z) is the amplitude of the cantilever mechanical oscillations at f_mod. This strategy allowed retrieval of |dC/dz| variations at f greater than the cantilever resonance frequency and as much as GHz frequencies⁵¹. Note that such measurements yield solely the amplitude of dC/dz and never its section, so the obtained spectra mirror the modulus of the complicated dielectric fixed (see the ‘Analysis of local dielectric spectra’ part).

To decrease systematic errors, we fastidiously calibrated {the electrical} achieve g for every system and atomic drive microscopy tip at every measurement frequency. This is crucial at excessive frequencies (> 1 MHz), at which each utilized and measured alerts present robust f-dependent variations as a result of the cable impedance will not be matched to the native pattern impedance and the long-range impedance between the AFM tip and the pattern. We used a calibration process much like that in ref. ⁴⁷, which depends on buying |dC/dz| curves as a operate of the tip–floor distance z over a tool area with f-independent impedance. We decided g by evaluating |dC/dz|(z) curves at every frequency in opposition to a low-f reference curve (2 okayHz), for which no achieve or loss is anticipated. To keep away from potential modifications in g in numerous areas of the system, these curves had been taken over the hBN spacer close to the water-filled channel (Extended Data Fig. 3a). We then scaled them to match the 2-kHz curve utilizing g and the offset as the 2 becoming parameters (Extended Data Fig. 3c). Note that, no matter f, the offset is all the time current in dC/dz curves and is often subtracted earlier than evaluation, as a result of it’s unbiased of native electrical properties (see, for instance, refs. ^10,56). We discovered that, at excessive f (10 MHz to 1 GHz), the offsets modified considerably however had been an identical above water channels and hBN spacers. This confirmed that, for all the frequencies, the offsets originated from long-range impedance contributions. Notably, though g is crucial for precisely evaluating {the electrical} properties of water from the measured alerts, offsets play little function on this examine as a result of we analysed the relative modifications in |dC/dz| alongside the system floor–water channels versus hBN spacers (see the ‘Dielectric images and dielectric spectra acquisition’ and ‘Numerical modelling and data analysis’ sections). We validated the calibration utilizing reference samples (hBN on doped Si), acquiring the anticipated f-independent dielectric spectra as much as GHz frequencies (Extended Data Fig. 3e,f). Furthermore, massive channels full of water successfully served as one other reference, yielding the anticipated spectral behaviour (flat response at excessive f) and the dielectric fixed of bulk water. To stop electrostatic crosstalk in calibration curves taken above hBN spacers from water contained in the channels, we used channels separated by about 800 nm, a spacing chosen to be sufficiently massive. Numerical simulations confirmed that, inside our experimental accuracy (roughly 1 zF nm⁻¹), the calibration curves had been unaffected by electrostatics from water within the channels (Extended Data Figs. 5b and 7f). We emphasize that, if the separation had been too small, such crosstalk may result in underestimation of the dielectric fixed of water.

Dielectric photographs and dielectric spectra acquisition

The dielectric photographs in Fig. 1 and Extended Data Fig. 4 had been taken at fixed top z_scan (sometimes about 15–25 nm) from the highest layer, as in ref. ²², utilizing the constant-height dual-pass mode SDM developed earlier^10,41. In this mode, the primary move information the topography with out utilized voltage and the second move acquires the dielectric picture with the AFM suggestions loop disabled and the z-scanner mounted (the scanner was fastidiously aligned parallel to the pattern airplane in order that z_scan remained fixed alongside the horizontal path). To keep away from piezo creep results, we allowed ample settling time at the start of every line for the scanner to succeed in the goal top. This strategy ensures most management of z_scan, as that is measured with sub-nanometre accuracy by buying DC and AC deflection strategy curves at scan line edges (right here over the hBN spacer areas)¹⁰. The DC deflection curve permits measuring the tip–floor distance z with sub-nanometre accuracy, whereas the AC deflection curve gives |dC/dz| as a operate of z, permitting additional validation of z_scan and correction of z-scanner drifts by matching with the |dC/dz| scan line (Extended Data Fig. 3b,c). Because the second-pass scan maintains the z-scanner mounted, potential artefacts from the z-scanner movement that may come up in the usual dual-pass ‘lift’ mode are prevented. Small variations in z_scan owing to the DC electrostatic drive are additionally measured and brought into consideration by recording the DC deflection throughout the second move. The photographs as a operate of f reported in Fig. 2b had been taken at each fixed z_scan and fixed y place (y-axis is alongside the nanochannels size). We sometimes recorded 25 strains at every f. Standard AFM picture processing consisting of flattening and Gaussian filtering was utilized.

The dielectric spectra in Fig. 2a and Extended Data Fig. 10 had been obtained for fixed z_scan (about 15–20 nm) and fixed y however after buying your complete 3D dataset |dC/dz|(x,z,f), akin to that proven in Extended Data Fig. 3d. This set consists of many |dC/dz|(x,z) photographs (Extended Data Fig. 3a), which had been obtained at completely different f by scanning the AFM tip throughout the channels (alongside the x-axis) at fixed y and approaching the floor in steps. This refined process was mandatory right here to reduce errors in z_scan owing to z-scanner drifts throughout lengthy measurements throughout the entire frequency sweep. Although the spectral behaviour doesn’t change with scan top, the amplitude of |dC/dz| does (Extended Data Fig. 6g,h), making it important to keep up the identical z_scan at every f for prime accuracy in our spectral evaluation. When taking |dC/dz|(x,z) photographs, the AFM tip was approached from bigger distances in the direction of the floor right down to the minimal distance of sometimes about 15 nm, past which tip collapse occurred owing to long-range enticing electrostatic forces, as anticipated underneath our measurement situations (utilized voltage, RT, comfortable cantilevers and huge AFM tip radii). At every step, we acquired dozens of strains, which had been then averaged to acquire the corresponding profile (Extended Data Fig. 3b). This notably improved the signal-to-noise ratio. By taking small steps in the direction of the floor, we may then reconstruct the dielectric spectra (Extended Data Fig. 10) through the use of the worth measured in the course of nanochannels on the identical z_scan (±1 nm) for every f. This 1-nm uncertainty has negligible influence on the extracted electrical properties proven in Fig. 3, as demonstrated in Extended Data Fig. 7g,h (see additionally the ‘Numerical modelling and data analysis’ part). We be aware that all the spectra had been obtained on the nearest doable distances achievable earlier than tip collapse to make sure most sign power and measurement stability. Larger z_scan had been prevented, because the |dC/dz| sign decays quickly with z, compromising measurement accuracy, significantly for small channels (Extended Data Fig. 6g,h).

As properly because the scanning top z_scan, the spectral values trusted geometry/dimension of our units and the AFM tip parameters (Extended Data Fig. 10). All of the required geometric parameters of the studied units had been measured by taking their topography photographs, whereas the AFM tip parameters (its radius R and half-angle θ) had been decided by becoming |dC/dz|(z) curves taken straight above the graphite backside layer, as in ref. ²², following the procedures described in refs. ^56,57. For instance, the tip radii had been discovered to be within the vary 50–200 nm, in good settlement with the values specified for industrial diamond-coated ideas. This enabled quantitative evaluation of the ‘absolute-value’ spectra (proven in Extended Data Fig. 10) versus simply relative variations with f, by way of full-3D numerical simulations that account for the detailed geometry of the system (see the ‘Numerical modelling and data analysis’ part). We emphasize that the values of |dC/dz| within the spectra that we analysed to extract {the electrical} parameters of water are the height values over the centre of the water channel relative to the values measured over the centre of the hBN spacers on the identical scan top, as in ref. ²². This differential strategy makes our evaluation strong in opposition to uncertainties in geometric parameters and eliminates the affect of the long-range geometry of the system, as described beneath.

To facilitate direct comparability of {the electrical} response of water throughout completely different units and experiments and improve readability of presentation, we normalized our consultant spectra for numerous water thicknesses in Fig. 2a. This normalization used the low-f plateau as a reference level, dividing every spectrum by this worth. In this low-f regime, the measured sign turns into successfully unbiased of each {the electrical} properties of water and its thickness (see Extended Data Figs. 6b and 8b and the ‘Analysis of local dielectric spectra’ part). As the sign on this regime is primarily decided by the AFM tip radius and its distance from the channel, normalizing with respect to the low-f worth successfully eliminates the geometric contributions, permitting for extra direct comparability between completely different experiments. We emphasize that various normalization utilizing high-f values wouldn’t be useful, because the sign on this regime strongly depends upon each the channel thickness and its dielectric properties.

Numerical modelling and knowledge evaluation

The noticed dielectric spectra had been fitted to full-3D finite-element numerical calculations carried out utilizing COMSOL Multiphysics 5.4a (AC/DC electrostatic module), not different simplified fashions offered within the manuscript. These 3D calculations had been primarily based on the electrostatic mannequin beforehand utilized in ref. ²², tailored right here to simulate the frequency-dependent drive appearing on the tip as a operate of ε and σ of water when an AC electrical discipline was utilized. They compute absolute |dC/dz| values whereas absolutely accounting for the precise geometry and dimensions of the system, together with the tip and system, in addition to the conductive and dielectric properties of water and their anisotropy, thereby eliminating potential geometric and electrostatic artefacts that might come up within the knowledge evaluation utilizing simplified fashions.

A schematic of the mannequin is proven in Extended Data Fig. 5a. Following ref. ²², the AFM tip was modelled as a truncated cone with the half-angle θ terminated with a tangent hemispherical apex of radius R. The values of θ and R utilized in our simulations had been measured for every AFM tip, as mentioned above. The AFM cone top H_cone was restricted to six μm and the cantilever was omitted to scale back the computational time (until said in any other case). We checked that these approximations had no influence on the simulated outcomes. The simulated nanochannel consisted of two insulating slabs of hBN separated by a lossy water slab of top h and width w (measured from topography for every system). We modelled every slab explicitly with its personal respective dielectric fixed and conductivity in response to the overall definition of anisotropic, complicated dielectric fixed

during which ω is the angular frequency, ε₀ is the dielectric permittivity of vacuum, ε_⊥ and ε_// are the dielectric constants perpendicular and parallel to the channel, respectively, and σ_⊥ and σ_// are the corresponding conductivities, respectively. Note that, for the water slab, the imaginary time period in equation (1), represented by σ_⊥,//, accounts for all doable losses, together with these from cost transport and dipolar leisure, whereas ε_⊥,// could comprise ionic contributions owing to ion–water and ion–ion correlations, in addition to the purely dielectric response of water^58,59. Both ε_⊥,// and σ_⊥,// of water had been handled right here as frequency-independent, as this approximation adequately reproduces the noticed dispersion, as defined beneath and in addition within the ‘Analysis of local dielectric spectra’ part. For hBN, σ_⊥,// is zero, so its ({varepsilon }_{perp ,//}^{* }) reduces to its recognized actual half, ε_⊥hBN = 3.5 and ε_//hBN = 5.5, that are fixed inside our experimental bandwidth. To mannequin doable contributions from close by nanochannels, the system was modelled as three parallel water nanochannels of size l = 3 μm and measured spacing w_s inside the surrounding hBN dielectric matrix, with dimensions matching the measured prime, backside and spacer hBN layers (size l = 3 μm, width W = 3 μm and top H_prime + h + H).

We numerically solved the Poisson’s equation within the frequency area for every system, calculated the drive appearing on the AFM probe and, from that, obtained |dC/dz| by integrating the built-in Maxwell stress tensor on the floor of the probe. The simulations used the identical field dimension and boundary situations as in ref. ²². Examples of calculated dielectric spectra for consultant geometrical parameters of our units and ε and σ of water are proven in Extended Data Fig. 6. These simulations intently match the experimental spectra, reproducing the noticed Debye-type frequency dispersion.

Our modelling strategy is equal to the unique evaluation by Maxwell and Wagner^60,61, which led to the Debye-like Maxwell–Wagner (MW) formalism generally used to interpret dielectric relaxations arising from DC conductivity in macroscale spectra of heterogeneous lossy dielectrics^62,63,64 (see the ‘Alternative phenomenological Debye-like MW analysis’ part). Using the complicated dielectric constants outlined in equation (1), Maxwell and Wagner confirmed that the efficient capacitance of a planar layered system, during which every layer has fixed ε and σ, acquired a frequency dependence much like a Debye-type leisure^62,63. In our case, the AFM tip replaces the highest electrode of their derivation, precluding an actual analytical answer and necessitating 3D numerical simulations. Nevertheless, the underlying electrostatic downside is identical, due to this fact no additional frequency dependence of ε and σ in equation (1) for the water slab is required to breed the noticed dispersion, until an additional leisure course of is current. In our case, no such additional leisure must be assumed or is anticipated to happen, primarily based on current understanding of strongly confined water. Consistent with this, the experimental spectra exhibited solely a single Debye-like leisure at decrease frequencies arising from DC conductivity (for extra particulars, see the ‘Analysis of local dielectric spectra’ part).

Experimental |dC/dz| spectra had been fitted with the ε_// and σ_// of water as the one becoming parameters. All of the opposite parameters wanted for the simulations had been decided experimentally, as detailed above. ε_⊥ was set to the values measured in ref. ²² and σ_⊥ was set to the measured worth for our bulk water (σ_bulk = 2 × 10⁻⁴ S m⁻¹). The extracted ε_// and σ_// values had been discovered to rely little on actual values of ε_⊥ or σ_⊥, indicating that our experimental geometry was slightly insensitive to the out-of-plane traits of water, and the affect was negligible for our smallest channels (Extended Data Figs. 6e,f and 8e,f). This is as a result of the impedance of the nanochannel within the out-of-plane path is far smaller than the sequence impedance from the hBN layers and AFM tip–floor capacitances, whereas the impedance of the nanochannel within the in-plane path is far bigger (see the ‘Electrical modelling’ part). From these fitted values of ε_// and σ_//, we inferred the interfacial dielectric fixed, ε_//int, and conductivity, σ_//int, of the confined water layer utilizing the three-capacitor mannequin, as defined in the principle textual content, with out introducing ε_//int and σ_//int straight into the electrostatic downside. This strategy minimized the complexity of our numerical calculations.

For brevity, in all the figures, |dC/dz| refers back to the relative dielectric distinction (until said in any other case), that’s, the response relative to the hBN spacer in order that |dC/dz| ≡ dC(z_scan,ε_⊥,//,σ_⊥,//)/dz − dC(z_scan,ε_⊥,//hBN,0)/dz. Accordingly, we computed absolutely the worth of |dC/dz| over the centre of the water channel as a operate of f and subtracted the corresponding values for the case of the tip positioned over the centre of the hBN spacer, matching the experimental knowledge processing. This differential strategy avoids systematic errors. As talked about above, it reduces the influence of uncertainties in geometric parameters and permits us to mannequin solely the native geometry, because the long-range geometric contributions don’t have an effect on |dC/dz| variations relative to the spacer. Furthermore, every system was modelled utilizing its precise dimensions (spacer and channel width and top, prime and backside hBN heights) and with the measured radius of the AFM tip utilized in that experiment. This ensured that the 3D calculations reproduced the life like electric-field distribution between the tip and the confined water, yielding dependable electrical properties of water with minimal approximations.

The thickness H of the hBN backside layer determines whether or not the measurements are largely delicate to ε_// or ε_⊥. Extended Data Fig. 7 illustrates this for the case of channel thickness h = 5 nm at excessive f, past the conductivity leisure regime (see the ‘Analysis of local dielectric spectra’ part) and for a big AFM tip radius (100 nm), as utilized in our experiments. When the channel is within the instant proximity to a metallic floor (Extended Data Fig. 7a), as in ref. ²², the |dC/dz| sign is unbiased of ε_// and, consequently, the extracted ε represents the out-of-plane element, ε_⊥. Furthermore, measurements are delicate solely to comparatively small values of ε_⊥ (as much as about 20 for h = 5 nm), because the sign saturates with additional will increase in ε_⊥ (for bigger channel thicknesses, the sensitivity extends to more and more bigger values of ε_⊥, as proven in ref. ²²). Conversely, when the channel is on a thick hBN layer (H = 200 nm; Extended Data Fig. 7b), the |dC/dz| sign tremendously will increase for ε_// > 20, turning into dominated by the in-plane element. Therefore, for comparatively massive values of ε_// starting from about 80 to about 1,000, as measured on this work, the sign at such H reveals little dependence on ε_⊥. Extended Data Fig. 7c reveals how the sign evolves as a operate of H. In these simulations, we elevated the cone top and cantilever size of the AFM tip to their nominal values (H_cone = 12 μm and L_cantilever = 20 μm). This allowed us to incorporate electrostatic contributions from longer-range elements of the atomic drive microscopy probe⁶⁵, which we discovered to be comparatively small however nonetheless non-negligible for very thick backside hBN (H > 500 nm). The outcomes present that the |dC/dz| sign and its sensitivity to ε_// peaks between 50 and 500 nm (Extended Data Fig. 7c). For thinner hBN layers, the ε_// contribution is comparatively small or negligible and the sign is dominated by ε_⊥. For hBN thicker than 500 nm, the |dC/dz| sign decreases, turning into much less delicate to the investigated native electrical properties, in line with earlier outcomes⁵⁶. This is anticipated as a result of the AFM tip is moved distant from the underside electrode. In this work, we used backside hBN layers of thickness as much as 200 nm. This limitation was dictated by constraints within the fabrication of our units, as a result of thicker hBN crystals had been stiffer and supplied poor adhesion, resulting in delamination on filling the channels with water.

We emphasize that, as a result of the amplitude of the dielectric spectra depends upon the scan top, knowledge evaluation may be affected by this parameter. Specifically, though the worth of σ_// is unbiased of z_scan (because the cut-off frequency stays unchanged with scan top; Extended Data Fig. 6g,h), the extracted ε_// could also be influenced, as it’s given by the amplitude of the high-f plateau. As described above, on this examine, we used particular procedures to regulate z_scan and decide its worth with most doable accuracy, estimated at ±1 nm. Despite the steep improve of |dC/dz| with reducing z, such 1-nm experimental uncertainty has negligible influence on the extracted ε_//, even for our smallest channels. This is as a result of our evaluation depends on the measurement of variations of |dC/dz| relative to the hBN spacer area on the identical z_scan. This differential strategy makes the evaluation strong in opposition to small uncertainties in z_scan. To illustrate this robustness, Extended Data Fig. 7h reveals 3D simulated curves for the best-fit z_scan worth (equivalent to the info in Extended Data Fig. 10c) and for best-fit z_scan ± 1 nm. The three curves almost overlap, with experimental knowledge factors scattered round them, in line with the said ±1 nm experimental uncertainty. The obtained 30% error within the extracted ε_// = 900 incorporates this uncertainty, together with the usual deviation for knowledge factors on the high-f plateau. Notably, qualitative experimental proof permits us to discard massive uncertainties in z_scan instead clarification for the improved ε_// noticed in our smallest water channels. Indeed, to justify the massive values of |dC/dz| noticed at excessive f with bulk-like ε_// would suggest extraordinarily small scan heights (⪝5 nm). This is proven in Extended Data Fig. 7g, during which we plot the sensitivity curve of |dC/dz| to ε_// at numerous values of z_scan for 1.5-nm channels. However, such small values of z_scan are bodily unattainable in our constant-height scan mode, as beneath about 15 nm, the tip collapses onto the floor (see the ‘Dielectric images and dielectric spectra acquisition’ part). Also, the implied small values of z_scan would result in a lot bigger |dC/dz| values than noticed for the low-f spectral plateau, which is extremely delicate to the scan top (see the ‘Analysis of local dielectric spectra’ part). In different phrases, though the high-f plateau may theoretically be reproduced by assuming a a lot smaller scan top (roughly 5 nm for the case of the consultant system with h ≈ 1.5 nm in Extended Data Fig. 10c), such a simulation would fail to breed the low-f plateau, as proven in Extended Data Fig. 7h. These observations rule out substantial errors in z_scan and the massive ε_// reported right here for our smallest units could be defined by inaccuracies within the scan top.

We stress that the electrical discipline above the hBN spacer varies little with {the electrical} properties of confined water (Extended Data Fig. 5c–h) and that our differential evaluation relative to the hBN spacer area doesn’t introduce systematic errors related to such electric-field variations. This is demonstrated in Extended Data Fig. 5b, which reveals full-3D numerical simulations of absolutely the worth of |dC/dz| versus tip-surface distance z calculated with the tip above the centre of the hBN spacer at completely different f, evaluating water-filled and empty channels, for our smallest channels (h = 1.5 nm). All curves virtually overlap, with deviations at intermediate and excessive f not exceeding 0.1 zF nm⁻¹, an order of magnitude smaller than our experimental uncertainty (about 1 zF nm⁻¹). Only on the lowest f do the deviations improve notably due to the in-plane conductivity of water however nonetheless stay inside experimental uncertainty. Furthermore, regardless of their minimal influence, these small variations are absolutely accounted for in our evaluation, which includes the life like electric-field distribution and system geometry. This additionally reveals that, even when these discipline variations weren’t accounted for—for instance, if utilizing the simplified analytical mannequin described beneath—the errors can be negligible, significantly at GHz frequencies at which we extract ε_//.

Analytical modelling

Because of the non-uniform distribution of electrical discipline throughout nanochannels and the complicated geometry of AFM probes, the electrostatic downside can’t be solved precisely utilizing analytical fashions. Hence, to suit the experimental knowledge and procure the outcomes proven in Fig. 3, we used the 3D numerical simulations described above. Nonetheless, it’s informative to offer an analytical approximation that may substantiate our numerical outcomes and provide additional bodily perception into the noticed spectral behaviour that’s comparatively unbiased of the main points of experimental geometry. Such an approximation may very well be used for semi-quantitative estimates. To this finish, we used the point-charge mannequin during which the AFM tip was changed by a degree cost Q positioned at distance z = z_scan + R from the highest hBN floor (Extended Data Fig. 8a). The units had been modelled as a stack of various slabs, infinite within the in-plane path. The slabs equivalent to the hBN layers had been positioned at 0 < z < −H_prime and −(H_prime + h) < z < −(H_prime + h + H) and modelled as insulators with zero conductivity. The water layer at −H_prime < z < −(H_prime + h) was described by anisotropic dielectric constants ε_//,_⊥ and conductivities σ_//,_⊥. The boundary situation of zero voltage was imposed at z = −(H_prime + h + H) to mannequin the extremely conducting floor electrode. By making use of the in-plane translational invariance, the Laplace equation for the Fourier remodel electrostatic potential within the in-plane path ϕ_q(z) turns into

during which q = (q_x,q_y) is the wave vector within the in-plane path and the position-dependent dielectric constants are denoted as ε_//(z) and ε_⊥(z), respectively. The dielectric fixed of the hBN slabs was, for simplicity, assumed to be isotropic with ε_//(z) = ε_⊥(z) = ε_hBN = 4. The Laplace equation was solved inside every slab utilizing exponential features. The options had been matched at every interface by imposing the next boundary situations: continuity of ϕ_q(z) and

during which η is an infinitesimal fixed. An identical relation holds on the point-charge location, z_scan + R, however the right-hand aspect in equation (3) turns into equal to Q. The potential in actual area ϕ(r,z) was then obtained by performing the Fourier remodel of ϕ_q(z) within the in-plane path. The capacitance was estimated as C = Q/ϕ(0,z_scan) and, from the latter, the capacitance gradient |dC/dz| was calculated. Examples of the calculated dielectric spectra for consultant parameters of our units and ε and σ of water are given in Extended Data Fig. 8, during which, once more for brevity, |dC/dz| signifies its worth relative to the case of the heterostructure absolutely fabricated from hBN, |dC/dz| = dC(z_scan,ε_⊥,//,σ_⊥,//)/dz − dC(z_scan,ε_hBN,0)/dz. The calculated spectra present that the mannequin captures all the fundamental options of the noticed behaviour as a operate of varied parameters and agrees properly with our numerical simulations in Extended Data Fig. 6. Note, nevertheless, that, as a result of the analytical mannequin assumes an infinite water layer and doesn’t account for a finite w, the transition between low-f and high-f plateaus turns into much less pronounced (Extended Data Fig. 8d). The analytical outcomes agree with the numerical simulations just for w a lot bigger than the tip radius R (Extended Data Fig. 6d). Accordingly, if solely the analytical mannequin had been used to suit the reported experimental knowledge, this might end in systematic underestimation of each ε_// and σ_//. In explicit, the values extracted for quasi-2D water layers (h ⪝ 2 nm) can be underestimated by an element of about 5, though all the developments with altering h would stay right.

Analysis of native dielectric spectra

The noticed dispersion arises from DC conductive losses of water inside the channels, not from its dipolar orientational (Debye) leisure⁶⁶. The latter in bulk water happens solely at f ≈ 10 GHz (ref. ⁶⁷), properly above the frequencies examined right here. Furthermore, the spectra exhibit very excessive low-f plateaus, which, if fitted with Debye-like fashions, would yield bodily implausible values for the ε_// of water (see the ‘Alternative phenomenological Debye-like MW analysis’ part). Instead, these plateaus are precisely reproduced by our mannequin utilizing life like values of σ_// of water within the conductive time period of equation (1). An identical state of affairs is encountered within the evaluation of macroscale dielectric spectra of heterogeneous lossy dielectrics measured by broadband dielectric spectroscopy⁶² (for instance, colloidal suspensions and composite liquid/strong supplies akin to water confined in porous media^68,69), during which MW interfacial polarization relaxations^60,61 emerge. In such complicated heterogeneous methods, phenomenological Debye-like fashions develop into mandatory, as a result of the measured response represents the efficient dielectric fixed of your complete system and express modelling of every dielectric element is precluded by geometric complexity. Although these approximations can match macroscale spectra, they sometimes yield unrealistically excessive ε at low frequencies, usually reaching very massive values (>10⁴). Such excessive obvious permittivities, additionally reported for water confined in porous media⁶⁸, don’t symbolize intrinsic dielectric constants however, slightly, mirror DC conductive losses^68,70. By distinction, our three-layer nanochannel system allows a ‘first-principles’ description during which every dielectric layer is modelled explicitly with its personal frequency-independent ε and σ, as in equation (1). This strategy permits ε_// and σ_// to be decided independently from the spectra with out introducing unphysical parameters or approximations.

No additional frequency dependence of ε_// and σ_// of water was required on this evaluation, because the measured spectra present solely a single Debye-like leisure at low f arising from DC conductivity, with no indication of extra relaxations. This is clear from the high-f plateaus, which stay flat and correspond to bulk-like or enhanced values of ε_//. If in our smallest channels (<4 nm) the intrinsic dipolar leisure of water had been shifted to decrease f by the geometric confinement, we might anticipate both an additional dielectric plateau or broadening of the Debye-like leisure transition area, each resulting in decreased high-f values of ε_//. Instead, we persistently noticed excessive, flat plateaus at excessive f throughout all of our units, ruling out additional dipolar relaxations within the in-plane path. The identical reasoning applies to anomalous Debye-like dipolar relaxations of the hydrogen-bond community reported for confined water⁶⁹ within the 100 okayHz to 10 MHz vary, which might additionally scale back each the low-f and the high-f plateaus, an impact not noticed right here. Although a weak dipolar leisure from hydrogen-bond restructuring within the out-of-plane path can’t be totally dominated out in our largest channels, that is unlikely for our smallest channels, which present very low ε_⊥ at low frequencies (okayHz), as beforehand reported²². In any case, variations in ε_⊥ have negligible influence on the extracted ε_// and σ_// and, due to this fact, wouldn’t alter our conclusions.

The motive for a DC conductivity contribution to our spectra could be understood by noticing that we measured the modulus of dC/dz, that’s, the modulus of the complicated dielectric fixed (varepsilon _{perp ,//}^{* }(omega )|). As a end result, the dielectric response is decided by each ε and σ and, relying on frequency, both the primary or the second time period in equation (1) turns into dominant. At sufficiently low f, the conductivity time period all the time dominates (varepsilon _{perp ,//}^{* }(omega )|), much like the case of macroscale measurements utilizing commonplace broadband dielectric spectroscopy⁶². To this finish, it’s helpful to recall that, at low frequencies, the spectrum of deionized water at macroscale is thought⁶⁷ to be dominated by σ_bulk. Therefore, the spectrum is successfully divided into two areas separated by the conductivity leisure frequency, f_r,bulk, at which the behaviour modifications. In the conduction-dominated area (f < f_r,bulk), the dielectric response decreases with growing f, whereas for f > f_r,bulk, it’s fixed and relies upon solely on ε_bulk. The worth of f_r,bulk is given by⁶²

which follows from equation (1) if we use |ε_⊥,//| = ε_bulk and |σ_⊥,//| = σ_bulk. Note that this frequency is analogous to the cut-off frequency for methods that may be modelled by a easy RC circuit (see the ‘Electrical modelling’ part). For the majority water utilized in our experiments (ε_bulk ≈ 80, σ_bulk ≈ 2 × 10⁻⁴ S m⁻¹), equation (4) yields roughly 45 okayHz. This worth agrees properly with f_r,bulk obtained from each numerical and analytical modelling mentioned above for the particular experimental geometry. Indeed, the analyses proven in Extended Data Figs. 6b and 8b yielded the purely dielectric behaviour characterised by high-f plateaus beginning at f ⪞ 45 okayHz for all of our nanochannels, no matter their top h. The additional plateau present in our simulations at low f displays the presence of non-lossy dielectrics (air hole between the AFM tip and the system; the highest and backside hBN layers). These dielectrics could be represented as additional capacitances in sequence to the contribution of water (see the ‘Electrical modelling’ part). Accordingly, for f beneath the cut-off frequency f_c,bulk, the modelled response turns into purely dielectric, reflecting the sequence capacitances. For f_c,bulk < f <f_r,bulk, the response is dominated by the σ of water, whereas above the relief frequency f_r,bulk, it’s once more purely dielectric however now dominated by the ε of water. For different related values of σ and ε of water, the spectra exhibit related behaviour (Extended Data Figs. 6e,f and 8e,f): there’s a low-f plateau as much as the cut-off frequency f_c, above which the response decreases with growing f, and the high-f plateau develops above the conductivity leisure frequency f_r.

The onset of the low-f plateau with reducing f is decided by the in-plane conductivity of water σ_//, whereas no info could be inferred about σ_⊥ from our experimental knowledge as a result of the measurement geometry is insensitive to the latter conductivity. This could be seen in Extended Data Figs. 6e and 8e, during which the low-f plateau shifts in frequency with various σ_//, however not σ_⊥, and fully disappears if σ_// = 0 and σ_⊥ ≠ 0. Qualitatively, the stronger dielectric response at low f displays the truth that the electrical potential drops largely between the AFM tip and the conductive water layer. We used our numerical simulations for instance this impact (Extended Data Fig. 5). At low f and σ_// ≠ 0, the electrical potential decreases virtually totally throughout the air and the highest hBN layer with little voltage drop left beneath the water layer (Extended Data Fig. 5f). Also, the potential distribution extends alongside the size of the nanochannel for a number of micrometres (Extended Data Fig. 5d). By distinction, at excessive f > f_r, the potential drop happens throughout your complete thickness of our units, much like the case during which the AFM tip is positioned above hBN spacers (Extended Data Fig. 5f–h). The potential drop at excessive f additionally extends almost equally in all the lateral instructions across the tip apex (Extended Data Fig. 5e), much like the case in Extended Data Fig. 5c for non-conducting water layer (σ_// = 0). The efficient screening by the conductivity of water alongside the channel size explains the noticed massive low-f response. Note that absolutely the worth of the low-f response is unbiased of σ_// however depends upon geometric parameters, specifically the channel width w, tip radius R (Extended Data Fig. 6c) and the bottom-layer thickness H (Extended Data Figs. 7c and 8c). This makes simulations important for correct analysis of the magnitude of modifications on the dispersion curves.

Although the worth of σ_// doesn’t have an effect on the low-f response, it controls the cut-off frequency f_c, which shifts to greater f proportionally to σ_// (Extended Data Figs. 6e and 8e) however independently of ε_// (Extended Data Figs. 6f and 8f). This behaviour could be described by

during which α is the geometrical parameter that may be estimated analytically (see the ‘Electrical modelling’ part). For extra correct outcomes, α was obtained utilizing numerical simulations (Extended Data Fig. 10), which yielded α ≈ 2.8 × 10⁻³, 6.2 × 10⁻⁴ and 9.2 × 10⁻⁵ for our three consultant units mentioned in the principle textual content, with h ≈ 30, 5 and 1.5 nm, respectively. These values of α counsel that f_c ought to shift comparatively little (from about 10 okayHz to about 500 Hz) if water filling the channels exhibited the majority properties (Extended Data Figs. 6b,c and 8b,c). This shift is far smaller than that noticed experimentally and, in actual fact, happens in the other way, in the direction of decrease f for smaller h, in robust distinction to the experimental behaviour (Fig. 2a). This reiterates the truth that the noticed improve in f_c with reducing h can’t be related to modifications in geometry however comes from an enormous improve within the σ_// of water for stronger confinement.

At excessive f, past the conductivity leisure regime, water behaves as a purely dielectric media and, accordingly, the relative top of the high-f plateau is now not dependent of σ_// however depends upon the ε of water, the geometry of the channel (specifically its top h) and the AFM tip radius (Extended Data Fig. 6c). The top of the plateau permits us to extract ε_// utilizing numerical simulations. Notably, for small channels and huge in-plane dielectric fixed (ε_// ≥ 80), the contribution of ε_⊥ turns into negligible (Extended Data Figs. 6f and 8f). Furthermore, we discover that f_r shifts to greater frequencies proportionally to σ_// and inversely proportionally to ε_// and is given by

This equation is unbiased of the system geometry and presents an equal of equation (4) legitimate on the macroscale. Therefore, as soon as σ_// is thought, ε_// could be straight obtained from f_r utilizing equation (6), supplied that f_r is properly separated from f_c, as within the case of our smallest channels. Also, if α is thought, each σ_// and ε_// may very well be readily estimated from the 2 attribute frequencies f_c and f_r with out the necessity for simulations, merely utilizing the equations

which comply with straight from equations (5) and (6). Note that the above issues and equations (5)–(8) can be useful for the evaluation of dielectric spectra obtained utilizing different scanning probe approaches⁴⁴, together with those who study greater derivatives of |dC/dz| and scanning impedance/microwave microscopy that straight investigates the native impedance.

Electrical modelling

It is instructive to make use of an equal impedance circuit to explain the noticed dielectric dispersion. However, due to long-range contributions from numerous AFM cantilever elements and the complicated geometry of our nanochannel units that end in a non-uniform distribution of the electrical discipline, an equal circuit needs to be so difficult that it’s unrealistic to explain our spectra quantitatively. Below, we offer a simplified mannequin that goals to clarify the physics behind and assist our numerical outcomes (Extended Data Fig. 9). To this finish, the AFM tip–nanochannel interplay could be modelled by the capacitance, C_tip, that—for simplicity—accounts for each tip–air and top-hBN-layer capacitances and, to a primary approximation, could be calculated as C_tip = 2πε₀Rln(1 + R(1 − sinθ)/(z_scan + H_prime/ε_⊥hBN)) utilizing the system described in ref. ⁵⁷. We neglect the stray capacitances related to the AFM cantilever and take into account solely the tip apex capacitance that’s anticipated to offer the dominant contribution. As for the water-filled nanochannel, we take into account it as a distributed RC community proven in Extended Data Fig. 9a. It consists of two elementary RC circuits describing in-plane and out-of-plane impedances Z_//(ω) = R_///(1 + iωR_//C_//) and Z_⊥(ω) = R_⊥/(1 + iωR_⊥C_⊥), respectively. R_⊥,// and C_⊥,// could be estimated as C_// = ε₀ε_//wh/Δl, R_// = Δl/(σ_//wh) for the in-plane path and C_⊥ = ε₀ε_⊥wΔl/h and R_⊥ = h/(σ_⊥wΔl) for the out-of-plane path, during which Δl is the size of the circuit component alongside the channel. We additionally thought-about one other capacitance C_b in sequence to Z_⊥, to mannequin the impact of the underside hBN layer between the water channel and the bottom. Plugging within the experimental values related to our units, we are able to readily discover that Z_⊥(ω) ≪ 1/(ωC_b) for all frequencies, which means that the contribution of Z_⊥(ω) is negligible in our experiments, in settlement with the above numerical and analytical calculations. The distributed community can then be simplified additional and described by {the electrical} circuit proven in Extended Data Fig. 9b, during which the water impedance is modelled by a single RC unit within the in-plane path, that’s, Z_ch(ω) = R_///(1 + iωR_//C_//), during which C_// = ε₀ε_//wh/l*, R_// = l*/(σ_//wh) and l* is the efficient size of the nanochannel contributing to electrostatic interactions with the AFM tip. With reference to Extended Data Fig. 5d, l* notably exceeds the tip diameter and, with out lack of generality, could be assumed to be on the order of a number of micrometres. The complete equal impedance between the AFM tip and the bottom is then given by

during which C_geom = C_tipC_b/(C_tip + C_b) is the capacitance that depends upon geometric parameters however not on {the electrical} properties of water. Extended Data Fig. 9c reveals |Z| as a operate of f for the three consultant units. The efficient capacitance of the modelled circuit is given by 1/ωZ(ω) and reveals the identical qualitative dependence on f, σ and ε of the capacitance gradient, |dC/dz|. Extended Data Fig. 9d plots |1/ωZ(ω)| that certainly reveals each low-f and high-f plateaus characterised by frequencies f_c and f_r, in good settlement with the experiment and numerical simulations. Despite its simplicity, {the electrical} mannequin additionally reproduces properly the modifications within the high-f plateau with various ε_// of water (Extended Data Fig. 9f) and modifications in f_c with various σ_// (Extended Data Fig. 9e).

Using this mannequin, we are able to additionally corroborate the expressions for f_c and f_r given by equations (5) and (6). Indeed, the pole of equation (9) yields the relief frequency as f_r = 1/(2πR_//C_//) and, plugging in R_// and C_// when it comes to σ_// and ε_//, leads to equation (6). The zero of equation (9) yields the cut-off frequency in order that f_c ≅ 1/(2πR_//C_geom), for which we keep in mind that C_geom is bigger than C_//. Accordingly, f_c is proportional to σ_// ∝ 1/R_// and depends upon the measurement geometry (by way of C_tip and C_b) however is unbiased of ε_//, in settlement with our numerical simulations. Expressing C_tip, C_b and R_// when it comes to geometric and electrical parameters as outlined above and taking C_b = ε₀ε_hBNwl*/H, we acquire equation (5), during which, within the case of our geometry, the geometric issue α could be approximated to α ≅ (hw/l*)(1/(2πR) + H/(ε_hBNwl*)). Using the particular parameters for our consultant units (h ≈ 30, 5 and 1.5 nm) and the efficient channel size l* = 3 μm, this yields α of about 4 × 10⁻³, 1 × 10⁻³ and a pair of × 10⁻⁴, respectively, in cheap settlement with the numerically simulated values of α. Thus, equations (7) and (8) with α calculated analytically, as obtained from the simplified electrical mannequin proven in Extended Data Fig. 9b, additionally enable for semi-quantitative estimates of σ_// and ε_//, that are discovered to vary by solely an element of lower than 2 from the values extracted by way of our extra quantitative, numerical evaluation.

Alternative phenomenological Debye-like MW evaluation

For completeness, we exhibit that making use of the MW formalism sometimes used for macroscale dielectric spectra^62,63,64 would yield the identical ε_// and σ_// values as reported in Fig. 3, though in a extra convoluted manner. In this various framework, the anisotropic, complicated dielectric operate in equation (1) for the water slab is changed with the next expression:

during which τ_⊥ and τ_// are attribute Debye-type leisure occasions and ε_lf⊥ (ε_hf⊥) and ε_lf// (ε_hf//) are the low-f (high-f) dielectric constants of the water slab, respectively, in every path. No broadening parameters are required, as a single, perfect Debye-type leisure describes our spectra. We omit the express conductivity time period and additional simplify equation (10) to:

as a result of the Debye time period in equation (10) inherently accounts for conductivity contributions at finite frequencies. This is clear by rewriting equation (11) when it comes to its actual and imaginary components:

This equation reproduces the noticed spectral behaviour. However, as a result of the imaginary half vanishes as ω → 0, becoming the noticed low-f plateau requires assuming unrealistically massive in-plane ε_lf//, as normally occurs within the MW evaluation of macroscale methods^68,70. Extended Data Fig. 11 reveals outcomes from implementing equation (10) in our 3D numerical simulations and becoming simulated |dC/dz| to the experimental spectra for our consultant nanochannels with thicknesses h = 30, 5 and 1.5 nm. Consistent with the fittings in Extended Data Fig. 10, the simulated spectra are largely insensitive to ε_⊥ or σ_⊥ and the matches yield the identical in-plane dielectric constants at excessive f as our mannequin, that’s, ε_// = ε_hf// (Extended Data Fig. 10 and Fig. 3a). They correspond to intrinsic dielectric constants of water, extracted past the MW leisure regime (≫10 MHz), during which the conductivity of water now not dominates. This reveals that the extracted ε_// are unbiased of the particular evaluation used. At low f, nevertheless, the match utilizing equation (10) yields very massive dielectric constants (ε_lf// ≈ 10⁴–10⁶) as much as the cut-off frequency f_c. These values usually are not bodily dielectric constants and shouldn’t be mistaken with the values of ε_// reported in Fig. 3a. They merely mirror the in-plane DC conductivity of the water, which—on this formalism—could be estimated as σ_// = ε₀ε_lf///τ_//, yielding the identical σ_// values as obtained from equation (1) (Extended Data Fig. 10 and Fig. 3b). Alternatively, utilizing the fitted high-f dielectric fixed, ε_hf//, and the relief frequency, which—on this approximation—is given by f_r = σ_///(2πε₀ε_hf//), we once more get well the identical σ_// values.

Although this phenomenological approximation in the end offers an identical ε_// and σ_// to our mannequin, it has notable drawbacks. First, as mentioned above, it yields unphysically massive ε_lf// that don’t have any bodily which means. Second, not like macroscale measurements, during which the MW approximation could be fitted on to the measured efficient dielectric operate, our nanoscale spectra nonetheless require full-3D numerical simulations to account for the system geometry, such because the AFM tip geometry, the scan top and the nanochannels construction. Moreover, on this framework, extracting σ_// depends upon both ε_lf// or ε_hf//, each of which might solely be decided from 3D numerical becoming of the low-f or high-f plateaus, respectively. By distinction, in our mannequin, σ_// is straight proportional to the cut-off frequency f_c by way of the geometric issue α (equation (5)), which could be estimated analytically, permitting extraction of σ_// with out additional numerical simulations.

We emphasize that, regardless of similarities, there are additionally key variations between our native spectra and people measured on the macroscale. Therefore, warning is required when making use of the MW approximation to our native measurements. First, a number of combos of ε_lf// and τ_// can match the low-f plateau equally properly, as a result of neither parameter has bodily which means. In truth, in our native spectra, the amplitude of the low-f plateau is about by the geometric capacitances in sequence with the channels—decided primarily by the AFM tip dimension, the channel thickness and its distance from the underside gate—slightly than by the conductivity of water. As a end result, greater values of σ_// don’t result in greater low-f plateaus in our spectra, as we would naively anticipate. Instead, it solely shifts the MW leisure to greater frequencies, as described by equation (5). We additionally be aware that, in our ‘first-principles’ mannequin, the complicated dielectric constants outlined in equation (1) diverge at zero frequency for a finite conductivity, however this divergence is truncated by the system geometry straight handled within the modelling, producing a Debye-like plateau unbiased of the ε_// or σ_// of water. Finally, the attribute frequency 1/(2πτ_//) of the Debye-like leisure in equation (10) doesn’t coincide with the cut-off frequency f_c. Again, it is because, in our native measurements, f_c depends upon the system geometry. In the MW formalism, f_c = α_MW/(2πτ_//), during which α_MW is a unique system-dependent parameter that isn’t purely geometric however associated to the purely geometric parameter α launched in our mannequin by α_MW = αε_lf//, during which ε_lf// is extracted from numerically becoming the low-f plateau. These observations spotlight that, though the Debye-like MW approximation in the end produces the identical outcomes as these obtained utilizing equations (1) and (5)–(8), it’s much less easy, relies upon extra closely on simulations and introduces parameters with out bodily which means.