Frequency-dependent force between ac-voltage-biased plates in electrolyte solutions.

We analyze the frequency dependence of the force between ac-voltage-biased plates in electrolyte solutions. To this end we solve analytically the Poisson-Nernst-Planck transport model in the dilute concentration and low voltage regime for a 1:1 symmetric electrolyte with blocking electrodes under a dc+ac applied voltage. The total force, which is the resultant of the electric and osmotic forces, shows a complex dependence on plate separation, frequency, ion concentration, and compact layer properties, different from that predicted from electrostatic current models or equivalent circuit models, due to the relevance of the osmotic force contribution in almost the whole range of frequencies. For the total dc force, we show that it decays at fixed ion concentration, linearly with plate separation for separations larger than a few times the Debye screening length. This linear dependence is due to the assumption about the conservation of the number of ions in the system. Moreover, the 1ω and 2ω ac harmonics of the total force show a broad peak at intermediate frequencies; it is centered at about the inverse of the charging time of the double layer capacitance, and covers the frequency range between the inverse of the diffusion time and the inverse of the electrolyte dielectric relaxation time. Finally, the 1ω ac harmonic component attains its high frequency asymptotic value at frequencies much higher than the inverse of the electrolyte dielectric relaxation time due to the very slow relaxation of the osmotic 1ω harmonic component at high frequencies. The derived analytical expressions for the total force remain valid up to voltages of the order of the thermal voltage, as has been assessed by means of numerical calculations. The numerical calculations are also used to explore the onset of higher force harmonics for larger applied voltages. Understanding the frequency dependence of the force acting on voltage-biased plates in electrolyte solutions can be of relevance for electrical actuation strategies in microelectromechanical systems and for the interpretation of some emerging electric scanning probe force microscopy techniques operating in electrolyte solutions.

Sizing single nanoscale objects from polarization forces.

Sizing natural or engineered single nanoscale objects is fundamental in many areas of science and technology. To achieve it several advanced microscopic techniques have been developed, mostly based on electron and scanning probe microscopies. Still for soft and poorly adhered samples the existing techniques face important challenges. Here, we propose an alternative method to size single nanoscale objects based on the measurement of its electric polarization. The method is based on Electrostatic Force Microscopy measurements combined with a specifically designed multiparameter quantification algorithm, which gives the physical dimensions (height and width) of the nanoscale object. The proposed method is validated with ~50 nm diameter silver nanowires, and successfully applied to ~10 nm diameter bacterial polar flagella, an example of soft and poorly adhered nanoscale object. We show that an accuracy comparable to AFM topographic imaging can be achieved. The main advantage of the proposed method is that, being based on the measurement of long-range polarization forces, it can be applied without contacting the sample, what is key when considering poorly adhered and soft nanoscale objects. Potential applications of the proposed method to a wide range of nanoscale objects relevant in Material, Life Sciences and Nanomedicine is envisaged.

Anomalously low dielectric constant of confined water.

The dielectric constant ε of interfacial water has been predicted to be smaller than that of bulk water (ε ≈ 80) because the rotational freedom of water dipoles is expected to decrease near surfaces, yet experimental evidence is lacking. We report local capacitance measurements for water confined between two atomically flat walls separated by various distances down to 1 nanometer. Our experiments reveal the presence of an interfacial layer with vanishingly small polarization such that its out-of-plane ε is only ~2. The electrically dead layer is found to be two to three molecules thick. These results provide much-needed feedback for theories describing water-mediated surface interactions and the behavior of interfacial water, and show a way to investigate the dielectric properties of other fluids and solids under extreme confinement.

Nanoscale dielectric microscopy of non-planar samples by lift-mode electrostatic force microscopy.

Lift-mode electrostatic force microscopy (EFM) is one of the most convenient imaging modes to study the local dielectric properties of non-planar samples. Here we present the quantitative analysis of this imaging mode. We introduce a method to quantify and subtract the topographic crosstalk from the lift-mode EFM images, and a 3D numerical approach that allows for extracting the local dielectric constant with nanoscale spatial resolution free from topographic artifacts. We demonstrate this procedure by measuring the dielectric properties of micropatterned SiO2 pillars and of single bacteria cells, thus illustrating the wide applicability of our approach from materials science to biology.

Nanoscale electric polarizability of ultrathin biolayers on insulating substrates by electrostatic force microscopy.

We measured and quantified the local electric polarization properties of ultrathin (∼5 nm) biolayers on mm-thick mica substrates. We achieved it by scanning a sharp conductive tip (<10 nm radius) of an electrostatic force microscope over the biolayers and quantifying sub-picoNewton electric polarization forces with a sharp-tip model implemented using finite-element numerical calculations. We obtained relative dielectric constants εr = 3.3, 2.4 and 1.9 for bacteriorhodopsin, dioleoylphosphatidylcholine (DOPC) and cholesterol layers, chosen as representative of the main cell membrane components, with an error below 10% and a spatial resolution down to ∼50 nm. The ability of using insulating substrates common in biophysics research, such as mica or glass, instead of metallic substrates, offers both a general platform to determine the dielectric properties of biolayers and a wider compatibility with other characterization techniques, such as optical microscopy. This opens up new possibilities for biolayer research at the nanoscale, including nanoscale label-free composition mapping.

Nanoscale imaging of the growth and division of bacterial cells on planar substrates with the atomic force microscope.

With the use of the atomic force microscope (AFM), the Nanomicrobiology field has advanced drastically. Due to the complexity of imaging living bacterial processes in their natural growing environments, improvements have come to a standstill. Here we show the in situ nanoscale imaging of the growth and division of single bacterial cells on planar substrates with the atomic force microscope. To achieve this, we minimized the lateral shear forces responsible for the detachment of weakly adsorbed bacteria on planar substrates with the use of the so called dynamic jumping mode with very soft cantilever probes. With this approach, gentle imaging conditions can be maintained for long periods of time, enabling the continuous imaging of the bacterial cell growth and division, even on planar substrates. Present results offer the possibility to observe living processes of untrapped bacteria weakly attached to planar substrates.

Quantitative electrostatic force microscopy with sharp silicon tips.

Electrostatic force microscopy (EFM) probes are typically coated in either metal (radius ∼ 30 nm) or highly-doped diamond (radius ∼ 100 nm). Highly-doped silicon probes, which offer a sharpened and stable tip apex (radius ∼ 1-10 nm) and are usually used only in standard atomic force microscopy, have been recently shown to allow enhanced lateral resolution in quantitative EFM and its application for dielectric constant measurement. Here we present the theoretical modelling required to quantitatively interpret the electrostatic force between these sharpened tips and samples. In contrast to a sphere-capped cone geometry used to describe metal/diamond-coated tips, modelling a sharpened silicon tip requires a geometry comprised of a cone with two different angles. Theoretical results are supported by experimental measurements of metallic substrates and ∼10 nm radius dielectric nanoparticles. This work is equally applicable to EFM and other electrical scanned probe techniques, where it allows quantifying electrical properties of nanomaterials and 3D nano-objects with higher resolution.

Finite-size effects and analytical modeling of electrostatic force microscopy applied to dielectric films.

A numerical analysis of the polarization force between a sharp conducting probe and a dielectric film of finite lateral dimensions on a metallic substrate is presented with the double objective of (i) determining the conditions under which the film can be approximated by a laterally infinite film and (ii) proposing an analytical model valid in this limit. We show that, for a given dielectric film, the critical diameter above which the film can be modeled as laterally infinite depends not only on the probe geometry, as expected, but mainly on the film thickness. In particular, for films with intermediate to large thicknesses (>100 nm), the critical diameter is nearly independent from the probe geometry and essentially depends on the film thickness and dielectric constant following a relatively simple phenomenological expression. For films that can be considered as laterally infinite, we propose a generalized analytical model valid in the thin-ultrathin limit (<20-50 nm) that reproduces the numerical calculations and the experimental data. Present results provide a general framework under which accurate quantification of electrostatic force microscopy measurements on dielectric films on metallic substrates can be achieved.

Calibrated complex impedance and permittivity measurements with scanning microwave microscopy.

We present a procedure for calibrated complex impedance measurements and dielectric quantification with scanning microwave microscopy. The calibration procedure works in situ directly on the substrate with the specimen of interest and does not require any specific calibration sample. In the workflow tip-sample approach curves are used to extract calibrated complex impedance values and to convert measured S11 reflection signals into sample capacitance and resistance images. The dielectric constant of thin dielectric SiO2 films were determined from the capacitance images and approach curves using appropriate electrical tip-sample models and the εr value extracted at f = 19.81 GHz is in good agreement with the nominal value of εr ∼ 4. The capacitive and resistive material properties of a doped Si semiconductor sample were studied at different doping densities and tip-sample bias voltages. Following a simple serial model the capacitance-voltage spectroscopy curves are clearly related to the semiconductor depletion zone while the resistivity is rising with falling dopant density from 20 Ω to 20 kΩ. The proposed procedure of calibrated complex impedance measurements is simple and fast and the accuracy of the results is not affected by varying stray capacitances. It works for nanoscale samples on either fully dielectric or highly conductive substrates at frequencies between 1 and 20 GHz.

Quantification of the dielectric constant of single non-spherical nanoparticles from polarization forces: eccentricity effects.

We analyze by means of finite-element numerical calculations the polarization force between a sharp conducting tip and a non-spherical uncharged dielectric nanoparticle with the objective of quantifying its dielectric constant from electrostatic force microscopy (EFM) measurements. We show that for an oblate spheroid nanoparticle of given height the strength of the polarization force acting on the tip depends linearly on the eccentricity, e, of the nanoparticle in the small eccentricity and low dielectric constant regimes (1 < e < 2 and 1 < ε(r) < 10), while for higher eccentricities (e > 2) the dependence is sub-linear and finally becomes independent of e for very large eccentricities (e > 30). These results imply that a precise account of the nanoparticle shape is required to quantify EFM data and obtain the dielectric constants of non-spherical dielectric nanoparticles. Experimental results obtained on polystyrene, silicon dioxide and aluminum oxide nanoparticles and on single viruses are used to illustrate the main findings.

Theory of amplitude modulated electrostatic force microscopy for dielectric measurements in liquids at MHz frequencies.

A theoretical analysis of amplitude modulated electrostatic force microscopy (AM-EFM) in liquid media at MHz frequencies, based on a simple tip-sample parallel plate model, is presented. The model qualitatively explains the main features of AM-EFM in liquid media and provides a simple explanation of how the measured electric forces are affected by: the frequency of the applied voltage, the tip-sample distance, the ionic concentration, the relative dielectric constant of the solution, and the relative dielectric constant and thickness of the sample. These results provide a simple framework for the design of AM-EFM measurements for localized dielectric characterization in liquid media.

Nanoscale measurement of the dielectric constant of supported lipid bilayers in aqueous solutions with electrostatic force microscopy.

We present what is, to our knowledge, the first experimental demonstration of dielectric constant measurement and quantification of supported lipid bilayers in electrolyte solutions with nanoscale spatial resolution. The dielectric constant was quantitatively reconstructed with finite element calculations by combining thickness information and local polarization forces which were measured using an electrostatic force microscope adapted to work in a liquid environment. Measurements of submicrometric dipalmitoylphosphatidylcholine lipid bilayer patches gave dielectric constants of ε(r) ~ 3, which are higher than the values typically reported for the hydrophobic part of lipid membranes (ε(r) ~ 2) and suggest a large contribution of the polar headgroup region to the dielectric response of the lipid bilayer. This work opens apparently new possibilities in the study of biomembrane electrostatics and other bioelectric phenomena.

Quantifying the dielectric constant of thick insulators by electrostatic force microscopy: effects of the microscopic parts of the probe.

We present a systematic analysis of the effects that the microscopic parts of electrostatic force microscopy probes (the cone and cantilever) have on the electrostatic interaction between the tip apex and thick insulating substrates (thickness > 100 µm). We discuss how these effects can influence the measurement and quantification of the local dielectric constant of the substrates. We propose and experimentally validate a general methodology that takes into account the influence of the cone and the cantilever, thus enabling us to obtain very accurate values of the dielectric constants of thick insulators.

Quantitative dielectric constant measurement of thin films by DC electrostatic force microscopy.

A simple method to measure the static dielectric constant of thin films with nanometric spatial resolution is presented. The dielectric constant is extracted from DC electrostatic force measurements with the use of an accurate analytical model. The method is validated here on thin silicon dioxide films (8 nm thick, dielectric constant approximately 4) and purple membrane monolayers (6 nm thick, dielectric constant approximately 2), providing results in excellent agreement with those recently obtained by nanoscale capacitance microscopy using a current-sensing approach. The main advantage of the force detection approach resides in its simplicity and direct application on any commercial atomic force microscope with no need of additional sophisticated electronics, thus being easily available to researchers in materials science, biophysics and semiconductor technology.

Nanoscale electrical conductivity of the purple membrane monolayer.

Nanoscale electron transport through the purple membrane monolayer, a two-dimensional crystal lattice of the transmembrane protein bacteriorhodopsin, is studied by conductive atomic force microscopy. We demonstrate that the purple membrane exhibits nonresonant tunneling transport, with two characteristic tunneling regimes depending on the applied voltage (direct and Fowler-Nordheim). Our results show that the purple membrane can carry significant current density at the nanometer scale, several orders of magnitude larger than previously estimated by macroscale measurements.

Electron transport through supported biomembranes at the nanoscale by conductive atomic force microscopy.

We present a reliable methodology to perform electron transport measurements at the nanoscale on supported biomembranes by conductive atomic force microscopy (C-AFM). It allows measurement of both (a) non-destructive conductive maps and (b) force controlled current-voltage characteristics in wide voltage bias range in a reproducible way. Tests experiments were performed on purple membrane monolayers, a two-dimensional (2D) crystal lattice of the transmembrane protein bacteriorhodopsin. Non-destructive conductive images show uniform conductivity of the membrane with isolated nanometric conduction defects. Current-voltage characteristics under different compression conditions show non-resonant tunneling electron transport properties, with two different conduction regimes as a function of the applied bias, in excellent agreement with theoretical predictions. This methodology opens the possibility for a detailed study of electron transport properties of supported biological membranes, and of soft materials in general.

Nanoscale capacitance imaging with attofarad resolution using ac current sensing atomic force microscopy.

Nanoscale capacitance imaging with attofarad resolution (∼1 aF) of a nano-structured oxide thin film, using ac current sensing atomic force microscopy, is reported. Capacitance images are shown to follow the topographic profile of the oxide closely, with nanometre vertical resolution. A comparison between experimental data and theoretical models shows that the capacitance variations observed in the measurements can be mainly associated with the capacitance probed by the tip apex and not with positional changes of stray capacitance contributions. Capacitance versus distance measurements further support this conclusion. The application of this technique to the characterization of samples with non-voltage-dependent capacitance, such as very thin dielectric films, self-assembled monolayers and biological membranes, can provide new insight into the dielectric properties at the nanoscale.

Shot noise in linear macroscopic resistors.

We report on direct experimental evidence of shot noise in a linear macroscopic resistor. The origin of the shot noise comes from the fluctuation of the total number of charge carriers inside the resistor associated with their diffusive motion under the condition that the dielectric relaxation time becomes longer than the dynamic transit time. The present results show that neither potential barriers nor the absence of inelastic scattering are necessary to observe shot noise in electronic devices.