Nanotechnology 24 (2013) 295702

G Cohen et al

Figure 13. Line scans of the measured (dashed line) and deconvolved (solid line) CPD across the longitudinal axis of two symmetric PbS–CdS–PbS NRs (inset KPFM image) using the Wiener filter, effective PSF and noise statistics.

at 300–400 kHz (2nd resonance frequency) was added to perform KPFM [25].

The effective AM-PSF for the KPFM measurements of the CdS–PbS nanorods is calculated using the probe in figures 5(a) and (b) and the time averaged tip–sample distances of the oscillating probe (16.7 nm) and the amplitude of the oscillations (17 nm) (see appendix). The FWHM of the effective AM-PSF is 5.5 nm. The noise statistics for each image is extracted, similar to the method presented for the calibration sample; an AWGN is observed in every image. Our reconstruction algorithm is then performed on the CdS–PbS KPFM measurements. Figures 13(a) and (b) show examples of the probe averaging effect on the CPD profile over two nanorods along their longitudinal axis. It is observed that the KPFM measurements attenuated the built-in potential within the junctions by a factor of three.

4.3. Graphene measurements

Topography and FM-KPFM measurements of graphene layers were performed simultaneously in ultrahigh vacuum (UHV) using a single pass technique (Held et al [26]). The graphene layers were grown on the Si-face of a SiC(0001) wafer by thermal decomposition, resulting in coverage by one layer of graphene and partial coverage by a second layer of graphene.

The topography was measured by means of controlling a constant frequency shift of the first cantilever resonance. The CPD was measured by minimizing the modulation of the frequency shift for the first resonance when the bias voltage was modulated. Details of sample preparation are described in [26]. For the current experiment, the amplitude of the first resonance . 100 kHz/ was held constant and the bias voltage was modulated by 200 mV at a frequency of 1 kHz [26].

The effective FM-PSFs for the KPFM measurements of the graphene layers are calculated similarly to the AM-PSF, using the probe in figures 5((a) and (b)) and by considering the minimal tip–sample distance (1 nm) and the oscillation amplitudes of the probe (3–5 nm) (see appendix). The FWHM of the effective FM-PSFs is 8.5 nm. The noise statistics for each image is extracted, similar to the method presented for the calibration sample; an AWGN is observed in all images.

Figures 14(a) and (b) show FM-KPFM measurements of single and double layers of graphene where the double layers exhibit higher CPD than the single layers with a measured contrast of about 140 mV. Figures 14(c) and (d) show the topography (green), the measured KPFM signal (blue) and the deconvolved CPD (red) along the profiles illustrated by the lines in figures 14(a) and (b), respectively. The left axes correspond to the CPD on the graphene layers, whereas the right axes correspond to the topography. It is observed that the measured KPFM and CPD profiles are virtually congruent; therefore, we conclude that the averaging effect does not play any role in such FM-KPFM measurements. Thus, we infer that the large width of CPD transitions from the single layer to double layer graphene is not due to the averaging effect of the probe in FM-KPFM. We note that the width of the CPD transition is larger in figure 14(d) (180 nm) than in figure 14(c) (120 nm). The relatively high topography step might be the reason for the large width of the CPD transition region in figure 14(d). Sadewasser et al conducted KPFM simulations for a potential step coinciding with a topography step [22]. They showed that larger width CPD transitions were obtained from high topography steps rather than flat surfaces. Moreover, the simulations showed that the step-like CPD profile was shifted towards the lower terrace, as demonstrated in figure 14(d).

We also observe that noise at high frequencies is apparent in the deconvolved CPD profile. This is due to the fact that the PSF is a LPF, which is negligible at high frequencies. Therefore, noise at high frequencies passes the Wiener filter with little attenuation (equation (3)).

5. Summary and conclusions

An algorithm to reconstruct the CPD distribution on a sample from its KPFM measurements was presented. The CPD was recovered by calculating an effective PSF for the measurement and obtaining the exact noise statistics. We have shown that measurements conducted in AM-KPFM demonstrate a very large averaging effect due to the cantilever whereas measurements conducted in FM-KPFM display CPD values close to the actual CPD with a negligible averaging effect. It should be noted that the fairest comparison should include

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