Variable-Temperature KPFM

• Variable-temperature KPFM is a scanning probe technique that measures local contact potential differences and electromechanical responses under controlled thermal conditions.
• It employs both amplitude-modulation and contact-mode methods to differentiate temperature-driven phase transitions from electrostatic and ionic effects.
• The approach enhances insights into ferroelectric switching, local chemical potential variations in graphene, and thermodynamic behavior in diverse material systems.

Searching arXiv for papers on variable-temperature Kelvin probe force microscopy. Variable-temperature Kelvin probe force microscopy (KPFM) comprises Kelvin-probe measurements performed under controlled temperature variation in order to track local electrostatic or electromechanical observables as a function of temperature. In the reported implementations, the method spans cryogenic amplitude-modulation KPFM (AM-KPFM) for mapping local chemical potential in hBN-encapsulated monolayer graphene, and elevated-temperature contact KPFM (cKPFM) for probing ferroelectric switching in heterogeneous BaTiO$_3$ thin films with sub-10 nm sampling (Lee et al., 14 Jul 2025, Schmitt et al., 2023). Across these regimes, temperature is not merely an environmental parameter: it is used to discriminate mechanisms, quantify phase evolution, and stabilize or contextualize the interpretation of local Kelvin signals.

1. Physical basis of variable-temperature KPFM

KPFM measures the local contact potential difference (CPD) between tip and sample, which reflects the difference in work functions and surface charge state. The fundamental relation is

$$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$

For a tip-sample system described by a capacitance $C(z)$, the electrostatic force and its gradient are

$$F_{\mathrm{es}}=\frac{1}{2}\frac{\partial C}{\partial z}(V-V_{\mathrm{CPD}})^2, \qquad k_{\mathrm{es}}=\frac{\partial F_{\mathrm{es}}}{\partial z} =\frac{1}{2}\frac{\partial^2 C}{\partial z^2}(V-V_{\mathrm{CPD}})^2.$$

In AM-KPFM, a DC bias $V_{\mathrm{dc}}$ is applied to the tip and an AC bias $V_{\mathrm{ac}}\sin(\omega t)$ to the sample, so that

$$U(t)=V_{\mathrm{dc}}+V_{\mathrm{ac}}\sin(\omega t)-V_{\mathrm{CPD}},$$

and the first-harmonic electrostatic force is

$$F_{\omega}=\frac{\partial C}{\partial z}(V_{\mathrm{dc}}-V_{\mathrm{CPD}})V_{\mathrm{ac}}.$$

The Kelvin feedback nulls $F_{\omega}\to 0$, yielding the Kelvin condition $V_{\mathrm{dc}}=V_{\mathrm{CPD}}$ (Lee et al., 14 Jul 2025).

In cKPFM, the tip remains in mechanical contact, and the measured signal arises from an interplay of electrostatic modulation of the contact stiffness via $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$0, electromechanical deformation under bias, and band-excitation (BE) detection around the contact resonance. In the reported ferroelectric implementation, the normal cantilever deflection is recorded while applying controlled DC write and read bias sequences, and the response is analyzed as a nonlinear function of read bias. The BE transfer function is fit to a single harmonic-oscillator model,

$$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$1

where $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$2 and $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$3 are the contact resonance and quality factor, respectively. For ferroelectrics, the temperature dependence of polarization is described conceptually by a Landau-type free-energy density,

$$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$4

so that as $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$5 approaches $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$6, spontaneous polarization decreases, hysteresis loops collapse, and the response becomes quasi-linear (Schmitt et al., 2023).

A central distinction therefore separates two variable-temperature KPFM lineages. In AM-KPFM, the observable is the nulling bias corresponding to CPD. In cKPFM, the observable is a spectroscopic electromechanical response whose temperature evolution helps distinguish ferroelectric switching from electrostatic or ionic effects. This distinction is essential when comparing variable-temperature data across quantum materials and ferroelectric oxides.

2. Instrumental architectures and thermal operating windows

Recent variable-temperature KPFM implementations differ markedly in mechanical design, signal transduction, and thermal regime. One realization is a cryogen-free, GM-cooler-based system built around a Montana Instruments S200 cryostat and intended for stable, sensitive operation from cryogenic to ambient temperatures. The other is a heating-stage cKPFM implementation on an Asylum Research Cypher for elevated-temperature studies of ferroelectric phase evolution (Lee et al., 14 Jul 2025, Schmitt et al., 2023).

Implementation	Core hardware	Reported thermal regime
Cryogen-free AM-KPFM	Montana Instruments S200 cryostat, SmarAct cryogenic xyz coarse positioners, Attocube ANSxyz100 scanner, fiber interferometer, passive isolator	Demonstrated at $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$7, $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$8, and room temperature
Variable-temperature cKPFM	Asylum Research Cypher, heating stage, Cr/Pt-coated Budget Sensors Multi75E-G probe	$$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$9, then back to $C(z)$0

The cryogenic AM-KPFM system combines FM-AFM for topography with one-pass AM-KPFM for CPD. Its cantilever resonance is $C(z)$1 with $C(z)$2 at cryogenic temperatures. The phase detector is an SRS SR865A lock-in amplifier configured with a 3rd-order low-pass filter and $C(z)$3. The phase-locked loop tracks resonance in real time with $C(z)$4 bandwidth, frequency resolution $C(z)$5, and $C(z)$6 tunable range. Passive vibration isolation is implemented by inserting a soft mechanical spring with $C(z)$7 and a soft copper braid-based heat sink between the cryostat cold plate and the AFM/KPFM head, reducing tip-sample vibration from $C(z)$8 RMS to $C(z)$9 RMS in the 3–300 Hz band (Lee et al., 14 Jul 2025).

The elevated-temperature cKPFM implementation uses conductive Cr/Pt-coated Multi75E-G probes with nominal free resonance $$F_{\mathrm{es}}=\frac{1}{2}\frac{\partial C}{\partial z}(V-V_{\mathrm{CPD}})^2, \qquad k_{\mathrm{es}}=\frac{\partial F_{\mathrm{es}}}{\partial z} =\frac{1}{2}\frac{\partial^2 C}{\partial z^2}(V-V_{\mathrm{CPD}})^2.$$0, spring constant $$F_{\mathrm{es}}=\frac{1}{2}\frac{\partial C}{\partial z}(V-V_{\mathrm{CPD}})^2, \qquad k_{\mathrm{es}}=\frac{\partial F_{\mathrm{es}}}{\partial z} =\frac{1}{2}\frac{\partial^2 C}{\partial z^2}(V-V_{\mathrm{CPD}})^2.$$1, and tip radius $$F_{\mathrm{es}}=\frac{1}{2}\frac{\partial C}{\partial z}(V-V_{\mathrm{CPD}})^2, \qquad k_{\mathrm{es}}=\frac{\partial F_{\mathrm{es}}}{\partial z} =\frac{1}{2}\frac{\partial^2 C}{\partial z^2}(V-V_{\mathrm{CPD}})^2.$$2. Measurements are made in contact mode around a contact resonance $$F_{\mathrm{es}}=\frac{1}{2}\frac{\partial C}{\partial z}(V-V_{\mathrm{CPD}})^2, \qquad k_{\mathrm{es}}=\frac{\partial F_{\mathrm{es}}}{\partial z} =\frac{1}{2}\frac{\partial^2 C}{\partial z^2}(V-V_{\mathrm{CPD}})^2.$$3, with BE bandwidth $$F_{\mathrm{es}}=\frac{1}{2}\frac{\partial C}{\partial z}(V-V_{\mathrm{CPD}})^2, \qquad k_{\mathrm{es}}=\frac{\partial F_{\mathrm{es}}}{\partial z} =\frac{1}{2}\frac{\partial^2 C}{\partial z^2}(V-V_{\mathrm{CPD}})^2.$$4 centered around $$F_{\mathrm{es}}=\frac{1}{2}\frac{\partial C}{\partial z}(V-V_{\mathrm{CPD}})^2, \qquad k_{\mathrm{es}}=\frac{\partial F_{\mathrm{es}}}{\partial z} =\frac{1}{2}\frac{\partial^2 C}{\partial z^2}(V-V_{\mathrm{CPD}})^2.$$5, BE measurement amplitude $$F_{\mathrm{es}}=\frac{1}{2}\frac{\partial C}{\partial z}(V-V_{\mathrm{CPD}})^2, \qquad k_{\mathrm{es}}=\frac{\partial F_{\mathrm{es}}}{\partial z} =\frac{1}{2}\frac{\partial^2 C}{\partial z^2}(V-V_{\mathrm{CPD}})^2.$$6 AC, and $$F_{\mathrm{es}}=\frac{1}{2}\frac{\partial C}{\partial z}(V-V_{\mathrm{CPD}})^2, \qquad k_{\mathrm{es}}=\frac{\partial F_{\mathrm{es}}}{\partial z} =\frac{1}{2}\frac{\partial^2 C}{\partial z^2}(V-V_{\mathrm{CPD}})^2.$$7 pulse duration. Temperature control is provided by a heating stage with a $$F_{\mathrm{es}}=\frac{1}{2}\frac{\partial C}{\partial z}(V-V_{\mathrm{CPD}})^2, \qquad k_{\mathrm{es}}=\frac{\partial F_{\mathrm{es}}}{\partial z} =\frac{1}{2}\frac{\partial^2 C}{\partial z^2}(V-V_{\mathrm{CPD}})^2.$$8 hold at each temperature to equilibrate the setup; thermal drift requires re-registration of nominally identical regions after each temperature change (Schmitt et al., 2023).

A recurrent misconception is that variable-temperature KPFM is defined by a single feedback topology. The reported systems show instead that the relevant architecture is modality-dependent: cryogenic CPD metrology benefited from off-resonant AM-KPFM combined with FM-AFM topography and strong vibration isolation, whereas variable-temperature ferroelectric mapping used contact-mode spectroscopy, BE detection, and temperature-cycled hysteresis analysis.

3. Spectroscopic protocols and spatial mapping

In the ferroelectric cKPFM protocol, the essential unit of measurement is per-pixel spectroscopy. For each pixel, the BE response amplitude and phase spectrum over a $$F_{\mathrm{es}}=\frac{1}{2}\frac{\partial C}{\partial z}(V-V_{\mathrm{CPD}})^2, \qquad k_{\mathrm{es}}=\frac{\partial F_{\mathrm{es}}}{\partial z} =\frac{1}{2}\frac{\partial^2 C}{\partial z^2}(V-V_{\mathrm{CPD}})^2.$$9 bandwidth is acquired for every DC write/read step and repeated four times. The DC write sequence comprises 32 steps between $V_{\mathrm{dc}}$0 and $V_{\mathrm{dc}}$1 using three ramps $V_{\mathrm{dc}}$2 with step width $V_{\mathrm{dc}}$3. After each write pulse, an 11-step DC read sequence between $V_{\mathrm{dc}}$4 and $V_{\mathrm{dc}}$5 with step width $V_{\mathrm{dc}}$6 is applied. The read window is shifted by $V_{\mathrm{dc}}$7 relative to the write sequence to compensate polarization imprint. The sequence is repeated four times per pixel, giving $V_{\mathrm{dc}}$8 measurements per pixel and a total time per pixel of $V_{\mathrm{dc}}$9 (Schmitt et al., 2023).

Spatial mapping in this cKPFM study was carried out at two distinct scales. High-resolution maps covered $V_{\mathrm{ac}}\sin(\omega t)$0 areas with $V_{\mathrm{ac}}\sin(\omega t)$1 points, corresponding to $V_{\mathrm{ac}}\sin(\omega t)$2 pixels and a total acquisition time of about 8 h for 2500 points. Repeatability tests used $V_{\mathrm{ac}}\sin(\omega t)$3 areas with $V_{\mathrm{ac}}\sin(\omega t)$4 points at the same $V_{\mathrm{ac}}\sin(\omega t)$5 pixel size. Variable-temperature maps used $V_{\mathrm{ac}}\sin(\omega t)$6 areas with $V_{\mathrm{ac}}\sin(\omega t)$7 points and $V_{\mathrm{ac}}\sin(\omega t)$8 pixels; the larger pixel size was chosen to preserve tips across repeated temperature cycles. Topography recorded after cKPFM mapping confirmed no surface modification (Schmitt et al., 2023).

The cryogenic AM-KPFM workflow is structurally different. Topography is tracked at $V_{\mathrm{ac}}\sin(\omega t)$9 via FM-AFM, while the Kelvin loop nulls the electrostatic AM signal at an excitation frequency $$U(t)=V_{\mathrm{dc}}+V_{\mathrm{ac}}\sin(\omega t)-V_{\mathrm{CPD}},$$0 detuned from $$U(t)=V_{\mathrm{dc}}+V_{\mathrm{ac}}\sin(\omega t)-V_{\mathrm{CPD}},$$1 by about $$U(t)=V_{\mathrm{dc}}+V_{\mathrm{ac}}\sin(\omega t)-V_{\mathrm{CPD}},$$2. The applied AC modulation is $$U(t)=V_{\mathrm{dc}}+V_{\mathrm{ac}}\sin(\omega t)-V_{\mathrm{CPD}},$$3. AFM feedback keeps $$U(t)=V_{\mathrm{dc}}+V_{\mathrm{ac}}\sin(\omega t)-V_{\mathrm{CPD}},$$4 in the tens-of-nanometers range, and a resolution analysis uses $$U(t)=V_{\mathrm{dc}}+V_{\mathrm{ac}}\sin(\omega t)-V_{\mathrm{CPD}},$$5 across 30 nm top hBN to quantify CPD sensitivity. Demonstrated scans include $$U(t)=V_{\mathrm{dc}}+V_{\mathrm{ac}}\sin(\omega t)-V_{\mathrm{CPD}},$$6 maps at $$U(t)=V_{\mathrm{dc}}+V_{\mathrm{ac}}\sin(\omega t)-V_{\mathrm{CPD}},$$7 and $$U(t)=V_{\mathrm{dc}}+V_{\mathrm{ac}}\sin(\omega t)-V_{\mathrm{CPD}},$$8 local spot grids; specific line rates are not reported (Lee et al., 14 Jul 2025).

These protocols illustrate two complementary uses of temperature control. In the ferroelectric case, temperature cycling is integrated directly into spectroscopic loop acquisition and phase discrimination. In the graphene case, temperature-dependent CPD is combined with local gate referencing to extract local chemical potential and compressibility.

4. Data reduction, clustering, and thermodynamic inference

For heterogeneous ferroelectrics, data reduction proceeds from BE spectra to compact loop descriptors. The amplitude and phase spectra are fit with a single harmonic oscillator; phase offsets are corrected; and the resulting cKPFM response magnitude as a function of read voltage forms the feature vector used for clustering. K-means clustering is implemented via Python scikit-learn version 0.23.2 KMeans. The number of clusters is selected by the elbow method using KneeLocator, and hierarchical clustering dendrograms provide a secondary check. In the reported BaTiO$$U(t)=V_{\mathrm{dc}}+V_{\mathrm{ac}}\sin(\omega t)-V_{\mathrm{CPD}},$$9 maps, the optimal choice was $$F_{\omega}=\frac{\partial C}{\partial z}(V_{\mathrm{dc}}-V_{\mathrm{CPD}})V_{\mathrm{ac}}.$$0 across all studied regions; adding a fourth cluster degraded reproducibility and assignment clarity (Schmitt et al., 2023).

The physical interpretation of the three clusters is explicitly phase-related. C1 is a quasi-linear, non-hysteretic response assigned to hexagonal BaTiO$$F_{\omega}=\frac{\partial C}{\partial z}(V_{\mathrm{dc}}-V_{\mathrm{CPD}})V_{\mathrm{ac}}.$$1, which is nonpolar at room temperature. C2 and C3 are distinct hysteretic responses indicating out-of-plane ferroelectric polarization in tetragonal BaTiO$$F_{\omega}=\frac{\partial C}{\partial z}(V_{\mathrm{dc}}-V_{\mathrm{CPD}})V_{\mathrm{ac}}.$$2, with different amplitudes likely arising from grain size and orientation variations. In a $$F_{\omega}=\frac{\partial C}{\partial z}(V_{\mathrm{dc}}-V_{\mathrm{CPD}})V_{\mathrm{ac}}.$$3, $$F_{\omega}=\frac{\partial C}{\partial z}(V_{\mathrm{dc}}-V_{\mathrm{CPD}})V_{\mathrm{ac}}.$$4 point map, the reported fractions were C1 $$F_{\omega}=\frac{\partial C}{\partial z}(V_{\mathrm{dc}}-V_{\mathrm{CPD}})V_{\mathrm{ac}}.$$5, C2 $$F_{\omega}=\frac{\partial C}{\partial z}(V_{\mathrm{dc}}-V_{\mathrm{CPD}})V_{\mathrm{ac}}.$$6, and C3 $$F_{\omega}=\frac{\partial C}{\partial z}(V_{\mathrm{dc}}-V_{\mathrm{CPD}})V_{\mathrm{ac}}.$$7. The cluster maps are consistent with the coexistence of tetragonal and hexagonal nanocrystals identified independently by sliding FFT with non-negative matrix factorization of TEM images and by Raman spectroscopy, including a hexagonal $$F_{\omega}=\frac{\partial C}{\partial z}(V_{\mathrm{dc}}-V_{\mathrm{CPD}})V_{\mathrm{ac}}.$$8 mode at $$F_{\omega}=\frac{\partial C}{\partial z}(V_{\mathrm{dc}}-V_{\mathrm{CPD}})V_{\mathrm{ac}}.$$9 (Schmitt et al., 2023).

For graphene heterostructures, the primary reduction is not clustering but differential electrostatic referencing to the local charge neutrality point (CNP). At each point one identifies $F_{\omega}\to 0$0, the back-gate and CPD values at the local CNP, and then defines $F_{\omega}\to 0$1 and $F_{\omega}\to 0$2. The local chemical potential and carrier density are

$F_{\omega}\to 0$3

The inverse compressibility $F_{\omega}\to 0$4 is then fit with the reported interaction-renormalized expression using $F_{\omega}\to 0$5, $F_{\omega}\to 0$6, and $F_{\omega}\to 0$7, yielding $F_{\omega}\to 0$8 and $F_{\omega}\to 0$9 (Lee et al., 14 Jul 2025).

Two methodological clarifications follow from these workflows. First, absolute tip work-function calibration is not universally required in variable-temperature KPFM: it is not part of the reported cKPFM ferroelectric protocol, and in the graphene study differential referencing to the local CNP removes unknown absolute offsets from $V_{\mathrm{dc}}=V_{\mathrm{CPD}}$0 and $V_{\mathrm{dc}}=V_{\mathrm{CPD}}$1 extraction. Second, in contact-mode ferroelectric measurements, loop classification and temperature evolution are part of the physical inference, not merely post-processing conveniences.

5. Temperature-dependent behavior across material classes

In the nanocrystalline BaTiO$V_{\mathrm{dc}}=V_{\mathrm{CPD}}$2 thin film, temperature-dependent cKPFM was used to follow the change in polarization pattern as the Curie temperature was approached and crossed. The film was 18 nm thick and consisted of BaTiO$V_{\mathrm{dc}}=V_{\mathrm{CPD}}$3 nanocrystallites on a Sr-silicate interface with Si(100), with grain sizes of about 5–20 nm. At room temperature, tetragonal BaTiO$V_{\mathrm{dc}}=V_{\mathrm{CPD}}$4 and hexagonal BaTiO$V_{\mathrm{dc}}=V_{\mathrm{CPD}}$5 coexist; the former is ferroelectric, whereas the latter is nonpolar at room temperature and exhibits ferroelectric behavior only at low temperatures around 74 K, with $V_{\mathrm{dc}}=V_{\mathrm{CPD}}$6 at 5 K. In variable-temperature maps acquired at $V_{\mathrm{dc}}=V_{\mathrm{CPD}}$7 and $V_{\mathrm{dc}}=V_{\mathrm{CPD}}$8, then back to $V_{\mathrm{dc}}=V_{\mathrm{CPD}}$9, the fraction of the nonpolar-like cluster C1 increased steadily from 25 to $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$00, while C2 and C3 decreased to zero at $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$01; after cooling, ferroelectric clusters reappeared, with four pixels classified as nonpolar. The transition temperature $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$02 for the film therefore lay between 75 and $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$03, lower than bulk BaTiO$$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$04 at about $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$05, consistent with nanograin size effects (Schmitt et al., 2023).

In the graphene heterostructure, temperature primarily modulates the local chemical-potential landscape and the compressibility peak near the Dirac point. At cryogenic temperatures, including $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$06 and $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$07, thermal excitation near the Dirac point is suppressed, allowing clear observation of the interaction-driven increase of $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$08 at low carrier density. At room temperature, the $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$09 peak broadens because $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$10 increases and thermal excitations dominate more strongly. The measurements are consistent with the linear dispersion $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$11, with $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$12 in the non-interacting approximation, and with interaction-driven renormalization of the Fermi velocity near the CNP (Lee et al., 14 Jul 2025).

Temperature thus serves different inferential roles in the two cases. In cKPFM on ferroelectrics, the disappearance and recovery of hysteresis across temperature is used as a robust discriminator of ferroelectric switching relative to electrostatic or electrostrictive contributions. In cryogenic AM-KPFM on graphene, temperature dependence enters the quantitative extraction of local chemical potential, carrier density, and inverse compressibility. A plausible implication is that variable-temperature KPFM should be understood less as a single materials technique than as a temperature-enabled framework for mechanism separation in local probe measurements.

6. Performance metrics, limitations, and application domains

The reported performance metrics differ by modality. In cKPFM, sub-10 nm probing was achieved with an $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$13 lateral step size, even though the tip radius was about $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$14; marked differences in electromechanical loops were observed between positions separated by only $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$15. The reported interpretation is that oversampling combined with robust clustering mitigates local contact variability and noise. Repeatability was high for $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$16 across repeated maps. Large-area thermal maps used $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$17 pixels to preserve the tip across cycles, and occasional artifacts at map corners or sides were attributed to tip contamination events that were self-corrected in subsequent contacts (Schmitt et al., 2023).

In the cryogenic AM-KPFM system, AFM topography reached about $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$18 lateral resolution, whereas CPD mapping was limited to about $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$19 by stray electric-field broadening through the 27 nm top hBN and the effective capacitive footprint. The CPD noise floor was about $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$20 at $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$21 under $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$22, corresponding to a chemical-potential sensitivity of about $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$23. Charge puddles with spatially varying $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$24 on the order of several $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$25 were resolved, while long-duration cryogenic scans benefited from the passive isolation and high-$$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$26 PLL/AGC architecture (Lee et al., 14 Jul 2025).

The main limitations are also modality-specific. For contact-mode ferroelectric measurements, sources of error include tip wear or contamination, local contact variability, imprint, electrostrictive contributions, and electrochemical reactions under high DC bias. The implemented mitigations were short pulse durations, modest AC amplitude, imprint compensation through a $$V_{\mathrm{CPD}}=\frac{\phi_{\mathrm{tip}}-\phi_{\mathrm{sample}}}{e}.$$27 read offset, BE/HO fitting, clustering, temperature holds, and post-map topography verification. For cryogenic AM-KPFM, lateral resolution is limited by stray capacitance through dielectrics, and the manuscript does not report explicit numerical values for vacuum pressure, cooldown time, or temperature stability. In both regimes, drift remains important: exact same pixels could not be guaranteed in the heated cKPFM sequence, and quantitative drift figures were not reported for the cryogenic setup (Schmitt et al., 2023, Lee et al., 14 Jul 2025).

Application domains follow directly from these demonstrated strengths. The cKPFM workflow is suited to mapping the polar domain distribution of spatially highly heterogeneous ferroelectric materials and, as envisioned in the ferroelectric study, to complex polar textures such as curled or swirled textures, vortices, and skyrmions in epitaxial heterostructures or nanocomposites. The cryogenic AM-KPFM platform functions as a local probe of quantum phases in van der Waals heterostructures, including graphene moirés, transition-metal dichalcogenides, correlated or topological states, edge phenomena, and spatially inhomogeneous phases. More generally, the reported studies indicate that variable-temperature KPFM is most powerful when temperature cycling is integrated into the measurement logic itself: as a way to null CPD accurately under cryogenic conditions, to track thermodynamic observables, or to verify that a nonlinear nanoscale response genuinely collapses at a phase transition.