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Imaging Static Charge Distributions: A Comprehensive KPFM Theory

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Kelvin Probe Force Microscopy

Part of the book series: Springer Series in Surface Sciences ((SSSUR,volume 65))

Abstract

We analyze Kelvin probe force microscopy (KPFM) for tip-sample systems that contain static charges by presenting a rigorous derivation for the respective KPFM signal in all common KPFM modes, namely amplitude modulation, frequency modulation, or heterodyne detection in the static, open-loop or closed-loop variant. The electrostatic model employed in the derivation is based on a general electrostatic analysis of an arbitrary tip-sample geometry formed by two metals, and which can include a static charge distribution and dielectric material in-between. The effect of the electrostatic force on the oscillating tip is calculated from this model within the harmonic approximation, and the observables for each of the above KPFM modes are derived from the tip oscillation signal. Our calculation reveals that the KPFM signal can for all modes be written as a weighted sum over all charges, whereby each charge is multiplied with a position-dependent weighting factor depending on the tip-sample geometry, the KPFM mode, and the oscillation amplitude. Interestingly, as the weight function does not depend on the charges itself, the contribution of the void tip-sample system and the charge distribution can be well-separated in the KPFM signal. The weight function for charges allows for a detailed understanding of the KPFM contrast formation, and enables to trace the dependence of the KPFM signal on different parameters such as the tip-sample geometry or the oscillation amplitude.

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Notes

  1. 1.

    We distinguish here between influence as the effect of shifting electron density within or between conductors due to an external field and polarisation as the result of generating a displacement field in a dielectric material.

  2. 2.

    The three quantities \(F_\text {el,a}\), \(F_\text {el,b}\) and \(F_\text {el,c}\) might firsthand appear to be spectral components of \(F_\text {el}\) —they would represent a static component (at zero frequency), a first harmonic (at frequency \(\nu _\text {el}\)) and a second harmonic (at frequency \(2\nu _\text {el}\)). However, it is important to remember that during dynamic AFM and dynamic KPFM measurements, the tip-sample distance \(z_\text {ts}\) is also a function of time. Therefore, considering \(F_\text {el,a}\), \(F_\text {el,b}\) and \(F_\text {el,c}\) to be spectral components of \(F_\text {el}\) is only reasonable under the assumption that \(z_\text {ts}\) is fixed.

  3. 3.

    In a FM-AFM experiment, the demodulated deflection signal is typically available as the excitation frequency or the frequency shift relative to a reference frequency.

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Acknowledgements

The authors are much obliged to Lev Kantorovich (King’s College London), Ralf Bechstein and Angelika Kühnle (both from University of Mainz) for fruitful discussions. P.R. gratefully acknowledges financial support by the German Research Foundation (DFG) via grant RA2832/1-1. H.S. is a recipient of a DFG-funded position through the Excellence Initiative (DFG/GSC 266).

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Author notes

  1. Hagen Söngen

    Present address: Physical Chemistry I, Bielefeld University, Universitätsstraße 25, 33615, Bielefeld, Germany

Authors and Affiliations

  1. Fachbereich Physik, Universität Osnabrück, Barbarastrasse 7, 49076, Osnabrück, Germany

    Philipp Rahe

  2. Institut für Physikalische Chemie, Johannes Gutenberg-Universität Mainz, Duesbergweg 10-14, Mainz, Germany

    Hagen Söngen

  3. Graduate School Materials Science in Mainz, Staudinger Weg 9, 55128, Mainz, Germany

    Hagen Söngen

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Correspondence to Philipp Rahe .

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Editors and Affiliations

  1. International Iberian Nanotechnology Laboratory, Braga, Portugal

    Sascha Sadewasser

  2. Department of Physics, University of Basel, Basel, Switzerland

    Thilo Glatzel

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Rahe, P., Söngen, H. (2018). Imaging Static Charge Distributions: A Comprehensive KPFM Theory. In: Sadewasser, S., Glatzel, T. (eds) Kelvin Probe Force Microscopy. Springer Series in Surface Sciences, vol 65. Springer, Cham. https://doi.org/10.1007/978-3-319-75687-5_6

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