Local Tip-Sample Electrostatic Force in SPM

Updated 11 January 2026

Local tip-sample electrostatic force is the electrical interaction between a scanning probe tip and a sample, defined by capacitance changes and voltage differences.
Its quantitative analysis employs models ranging from parallel-plate to sphere-plane approximations with tip dipole contributions, enabling precise mapping of surface charge and potential.
Applications in non-contact AFM, KPFM, and quantum dot microscopy demonstrate how measured force gradients and dissipative dynamics facilitate atomic-scale imaging and controlled nanofabrication.

Local tip-sample electrostatic force refers to the force arising from the electrical interactions between a scanning probe (typically an atomic force microscope or scanning tunneling microscope tip) and the surface under investigation. This force is central to the quantitative mapping of local electrical properties, imaging of charge distribution, and engineering atomic-scale contrast in scanning probe microscopy. Its physical origin, mathematical formulations, and practical implications depend critically on tip geometry, local surface electronic structure, dielectric environment, and experimental mode.

1. Physical Origin and Theoretical Formulation

The canonical model for the local electrostatic force $F_{\rm es}(z)$ between a conductive tip and sample is derived from the electrostatic energy:

$U(z) = \frac{1}{2}C(z)\bigl[V_{\rm tip} - V_{\rm sample} - V_{\rm CPD}\bigr]^2$

$F_{\rm es}(z) = -\frac{dU}{dz} = \frac{1}{2} \frac{dC}{dz}\bigl[V_{\rm tip} - V_{\rm sample} - V_{\rm CPD}\bigr]^2$

where $C(z)$ is the tip-sample capacitance, $V_{\rm tip}$ and $V_{\rm sample}$ are the tip and sample potentials, and $V_{\rm CPD}$ is the contact potential difference due to relative work functions (Harloff, 2010).

Refinements incorporate geometric and material details. For tips with a permanent dipole—arising either from apex morphology or molecular functionalization—the short-range force can be represented as:

$F(z_0) = + p_z \frac{1}{4\pi\varepsilon_0}\sum_i q_i\frac{R_i^2 - 2(z_0-z_i)^2}{[R_i^2 + (z_0-z_i)^2]^{5/2}}$

with $q_i$ site charges and $p_z$ tip dipole moment (Schneiderbauer et al., 2014).

For time-varying or resonantly driven bias, the force contains both static and dynamic components:

$F_{\rm es}(t) = \frac{1}{2} \frac{dC}{dz}\left[V_{\rm DC} + V_{\rm AC}\sin(\omega t)\right]^2$

with harmonics at $\omega$ and $2\omega$ enabling lock-in detection (Miyahara et al., 2017, Miyahara et al., 2015).

In non-contact atomic resolution imaging, short-range tip-sample interactions can be almost entirely electrostatic, as shown for polar insulators (e.g., Cu $_2$ N/Cu(100)) where atomic contrast stems from tip dipole-sample charge array interactions (Schneiderbauer et al., 2014). In scanning quantum dot microscopy (SQDM), the relevant potential at the dot is computed via the Dirichlet Green’s function kernel integrating over local surface potential (Wagner et al., 2019).

2. Dependence on Geometry, Material, and Capacitance

Accurate modeling of $C(z)$ is essential. Parallel-plate, sphere-plane, and cone-plane approximations are used depending on the separation and tip shape (Revilla, 2016, Fosco et al., 2012). For tip-sample separations $z < 200$ nm, simple parallel-plate models fail; intermediate expressions, including empirical fits $C(z) = A_1 (z+z_{\text{eff}})^{-a} + A_2$ , better capture real tip geometries with exponents varying between –2 (sphere-plane) and –1 (cone-plane):

$\text{Sphere-plane:}\ \frac{d^2C_{\rm sp}}{dz^2} \sim z^{-2}$

$\text{Cone-plane:}\ \frac{d^2C_{\rm cp}}{dz^2} \sim z^{-1}$

$\text{Empirical:}\ \frac{d^2C}{dz^2} \sim (z+z_{\text{eff}})^{-a-2}$

Experimental extraction via force-spectroscopy fits these coefficients to real AFM probes (Revilla, 2016). Local variations in sample permittivity or topography (e.g., dielectric films, oxide grains) directly modulate $C(z)$ and thus $F_{\rm es}$ , with increased $\varepsilon_r$ amplifying both capacitance and force (Gomez et al., 2016).

For dynamic or high-frequency applications, e.g., KPFM, both $\partial C/\partial z$ (force) and $\partial^2 C/\partial z^2$ (force gradient) are relevant, impacting dissipation and frequency-shift signals, respectively (Miyahara et al., 2017, Miyahara et al., 2015).

3. Electrostatic Force Imaging Modalities

Non-contact Atomic Force Microscopy (NC-AFM)

Atomic contrast achieved via tip dipole interaction with sample point charges, with contrast inversion possible through tip functionalization (e.g. CO termination) (Schneiderbauer et al., 2014).

Kelvin Probe Force Microscopy (KPFM)

Tip-sample force is decomposed into macroscopic capacitive and atomistic short-range components. Atomic-scale KPFM contrast arises from the atomistic slope $a(z) = \partial F_\mu/\partial V$ , which is essentially independent of bias over several volts (Sadeghi et al., 2012). AM-KPFM and FM-KPFM differ in contrast mechanisms, with FM-KPFM offering superior atomic resolution due to sensitivity to the force gradient.

Scanning Quantum Dot Microscopy (SQDM)

The theory computes the potential at a tip-mounted quantum dot using Green’s function boundary integrals, sensitive to nanoscale surface potential variations (Wagner et al., 2019).

Electrostatic Force Microscopy (EFM)/Surface Potential Imaging (SPI)

EFM measures the force gradient generated by electrostatic interactions, and compensates with DC tip bias to directly quantify local surface potential (Harloff, 2010). Artefacts from surface topography are interpreted as changes in the net capacitive volume sampled by the tip.

Piezoresponse Force Microscopy (PFM)

Electrostatic forces can contaminate piezoresponse measurements; reduction achieved via long, ultra-stiff cantilevers and probe design improvements (Gomez et al., 2016).

4. Quantitative and Local Effects

Local tip-induced electrostatic field strengths are empirically limited to $\sim0.5$ V/nm, corresponding to nanonewton-scale forces and Maxwell stress pressures sufficient to drive mass transport, alter surface chemistry, and mediate ionic or phase changes (Balke et al., 2016). The force is highly localized—often within nm-scale volumes directly beneath the tip apex—and is not governed by macroscopic tip radius or long-range sample conductivity (Wutscher et al., 2012). For example, bias-dependent atomic-scale force contrast in Cu $_2$ N is explained by site-specific dipole/charge interaction, with sharp decay lengths ( $\lambda \sim 60$ pm) (Schneiderbauer et al., 2014).

The "phantom force" effect in FM-AFM under STM conditions results from local voltage-drop due to tunneling current flowing through a nanometer-scale series resistance, modifying electrostatic force and image contrast at atomic resolution (Wutscher et al., 2012).

5. Dynamics, Dissipation, and Friction

The tip-sample electrostatic force not only generates static attraction but can also induce dynamic effects:

Dissipative ("friction") forces arise from the phase lag between moving tip charges and sample polarization. These tangential friction forces are several orders of magnitude smaller than normal forces in classical models, and experimentally observed dissipation likely involves other mechanisms such as dielectrophoretic heating, van der Waals friction, or patch potentials (Dedkov et al., 2017).
Resonant mechanical detection, via cantilever frequency shift or dissipation channels, provides high sensitivity to local field-induced displacements at the few-pm level (Balke et al., 2016, Miyahara et al., 2017).
For samples with finite impedance, the tip charge may lag the cantilever motion. A rigorous treatment using Lagrangian mechanics and impedance spectroscopy generalizes the force law, allowing extraction of sample resistance or conductivity (Dwyer et al., 2018).

6. Applications and Experimental Protocols

Local tip-sample electrostatics underpins diverse imaging and manipulation modalities:

Mapping local surface and interface potentials in metals, semiconductors, and dielectrics (Harloff, 2010, Sadeghi et al., 2012).
Atomic-scale contrast in polar insulators, where electrostatic forces dominate over chemical or Pauli repulsion (Schneiderbauer et al., 2014).
Controlled exfoliation of 2D materials (e.g. graphene) by electrostatic manipulation at terrace edges, as confirmed by ab-initio and MD simulations (Rubio-Verdú et al., 2016).
Surface modification and nanoscale device engineering via tip-induced gating, depletion, or charge injection—characterization of depletion radii and local fields by electron interferometry and self-consistent finite-element methods (Iordanescu et al., 2020).
Quantitative dielectric spectroscopy and extraction of local permittivity or resistance through broadband EFM protocols (Dwyer et al., 2018).

7. Limitations, Artefacts, and Mitigation Strategies

Key artefacts stem from:

Tip geometry mischaracterization, leading to incorrect capacitance and force models (Revilla, 2016, Fosco et al., 2012).
Electrostatic signals contaminating true piezoresponse (PFM) domains; mitigated by longer, stiffer cantilevers and off-resonant operation (Gomez et al., 2016).
Topography crosstalk in SPI/EFM mapping, requiring spatial averaging or subtraction protocols (Harloff, 2010).
Residual dissipative forces in AFM friction experiments, which are inherently small for metal-metal contacts but can be amplified in layered, low-conductance surfaces (Dedkov et al., 2017).

Development of accurate simulations, e.g., multiscale FD+DFT frameworks for KPFM (Sadeghi et al., 2012) and Green’s function techniques for SQDM (Wagner et al., 2019), provides necessary rigor for extracting material properties and discounting spurious signals.

The local tip-sample electrostatic force represents a fundamental and versatile interaction that can be precisely engineered, quantified, and harnessed for sub-nanometer resolution characterization of charge, potential, and ionic phenomena in solids, as well as for controlled nanofabrication and atomic manipulation across a wide spectrum of scanning probe techniques.