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Contribution of the Numerical Approach to Kelvin Probe Force Microscopy on the Atomic-Scale

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Kelvin Probe Force Microscopy

Part of the book series: Springer Series in Surface Sciences ((SSSUR,volume 48))

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Abstract

The goal of this chapter is to gather and detail recent numericaldevelopments addressing the issue of atomic-scale measurements in Kelvin Probe Force Microscopy (KPFM). It is argued why the problem requires the combination between the atomistic description of the distance- and bias voltage-dependent force field occurring between the tip and the surface, as well as an accurate numerical implementation of the complex noncontact atomic force microscopy and KPFM setup. When combining these tools, it is possible to draw conclusions regarding the origin of the atomic-scale KPFM contrast and its connections with usual physical observables such as the surface potential and the local work function. These aspects are discussed with respect to the surface of a bulk ionic crystal.

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Notes

  1. 1.

    In- and off-phase components of the LIA are essentially defined upon the structure of its input signal, i.e., here the driving signal of the cantilever which is arbitrarily generated out of a sinus waveform.

  2. 2.

    On the experimental level, the AM-KPFM setup requires that the bandwidths of the PSD and of the preamplifier are large enough to allow for the detection of the 2nd eigenmode without attenuation.

  3. 3.

    The expressions of the DC value of the bias voltage that maximizes Δf(z, V b) and the force F(z, V b) with V mod = 0, i.e., within the framework of spectroscopic curves in FM- and AM-KPFM, respectively are given by: \({\left.(\partial \Delta f(z,{V }_{\mathrm{b}})/\partial {V }_{\mathrm{dc}})\right \vert }_{{V }_{\mathrm{mod}}=0} = 0\)(1) and \({\left.(\partial F(z,{V }_{\mathrm{b}})/\partial {V }_{\mathrm{dc}})\right \vert }_{{V }_{\mathrm{mod}}=0} = 0\)(2), respectively. The expression of Δf is derived from the approach by Giessibl [71]: \(\Delta f(z,{V }_{\mathrm{b}}) \propto {\int \nolimits \nolimits }_{0}^{{T}_{0}}F(z,{V }_{\mathrm{b}})\sin ({\omega }_{0}t)\mathrm{d}t\) with: \({V }_{b} = {V }_{\mathrm{dc}} - {V }_{\mathrm{CPD}} + {V }_{\mathrm{mod}}\sin ({\omega }_{\mathrm{mod}}t)\). If the force has the usual quadratic-like form: \(F(z,{V }_{\mathrm{b}}) = h(z) \times {V }_{\mathrm{b}}^{2}\) (e.g., \(F = \frac{1} {2}\partial C/\partial z{V }_{\mathrm{b}}^{2}\)), then conditions (1) and (2) yield equivalently to V dc = V CPD. However, if the force has a less usual fully polynomial form, as this is the case when dealing with SRE forces [19,  36]: \(F(z,{V }_{\mathrm{b}}) = h(z) \times [A(z){V }_{\mathrm{b}}^{2} + B(z){V }_{\mathrm{b}} + C(z)]\), then conditions (1) and (2) give: \({V }_{\mathrm{dc}} = {V }_{\mathrm{CPD}} - I\prime/(2J\prime)\) (with \(I\prime ={ \int \nolimits \nolimits }_{0}^{{T}_{0}}B(z)h(z)\sin ({\omega }_{0}t)\mathrm{d}t\) and \(J\prime ={ \int \nolimits \nolimits }_{0}^{{T}_{0}}A(z)h(z)\sin ({\omega }_{0}t)\mathrm{d}t\)) and \({V }_{\mathrm{dc}} = {V }_{\mathrm{CPD}} - B(z)/(2A(z))\), respectively. Therefore the maxima of both spectroscopic methods differ. The main reason is that the force is dynamically z-dependent (which makes the compensated CPD z-dependent as well, as seen with the above equation), while Δf is averaged over the oscillation cycle, hereby averaging the force as well.

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Acknowledgements

LN, FB, and CL wish to thank E. Meyer, T. Glatzel and S. Kawai from the Department of Physics of the University of Basel for stimulating discussions and acknowledge support from the ANR with the PNANO project MolSiC (ANR-08-P058-36). ASF wishes to thank L.N. Kantorovich for useful discussions and acknowledges support from the Academy of Finland and ESF FANAS programme.

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Authors and Affiliations

  1. IM2NP, Centre scientifique de Saint-Jérôme, Aix-Marseille Université, Avenue Escadrille Normandie-Niemen, Case 151, 13397, Marseille, CEDEX 20, France

    Laurent Nony, Franck Bocquet & Christian Loppacher

  2. CNRS, IM2NP (UMR 6242), Marseille-Toulon, France

    Laurent Nony, Franck Bocquet & Christian Loppacher

  3. Department of Physics, Tampere University of Technology, 692, 33101, Tampere, Finland

    Adam S. Foster

  4. Department of Applied Physics, Aalto School of Science, 1100, 02015, Helsinki, Finland

    Adam S. Foster

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  1. Laurent Nony
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  2. Franck Bocquet
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  4. Christian Loppacher
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Correspondence to Laurent Nony .

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Editors and Affiliations

  1. Laboratory, International Iberian Nanotechnology, Avda. Mestre José Veiga s/n, Braga, 4715-330, Portugal

    Sascha Sadewasser

  2. Inst. Physik, Universität Basel, Klingelbergstr. 82, Basel, 4056, Switzerland

    Thilo Glatzel

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Nony, L., Bocquet, F., Foster, A.S., Loppacher, C. (2012). Contribution of the Numerical Approach to Kelvin Probe Force Microscopy on the Atomic-Scale. In: Sadewasser, S., Glatzel, T. (eds) Kelvin Probe Force Microscopy. Springer Series in Surface Sciences, vol 48. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22566-6_5

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