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Figure 9. (a) Contact mode topography of the sample: two nickel contacts on a layer of SiO2. (b) Height profile (as indicated in (a)).

(c) CPD image of the sample. (d) CPD profile (as indicated in (c)). (e) CPD image of the sample while the left nickel contact was biased with 0.5 V and the right nickel contact was biased with 0.5 V, relative to the SiO2. (f) CPD profile (as indicated in (e)). From this profile one can observe an almost linear slope in the surface potential. A small deviation upwards is visible due to the higher CPD value on the SiO2 compared to the nickel. All KPFM measurements were conducted with AM using lift-mode, where the lift height was 5 nm.

frequency, respectively. Thus, during the measurements the cantilever oscillates only at the first mechanical resonance frequency [19]. Since this first resonance frequency is typically above 50 kHz, whereas the time constant of the Kelvin controller is around 1 ms, the effective AM-PSF (FMPSF) is calculated by minimizing the average electrostatic force (force-gradient) on the probe rather than minimizing the force (force-gradient) at each tip–sample distance [10]. A method to calculate the effective PSFs of oscillating probes is explicitly detailed in the appendix.

4. KPFM measurements

4.1. Calibration sample

We validate our reconstruction algorithm by measuring a calibration sample composed of nickel electrodes under an applied bias, and then deconvolving by using an effective PSF and the system noise statistics. 50 nm thick nickel

electrodes were deposited on an insulating layer of SiO2 using e-beam lithography, and the ambient topography and CPD measurements were conducted using a Solver PH47 (NTMDT Inc.) AFM. The sample topography was measured in contact mode, and is shown in figure 9(a) and in a line scan (white line) in figure 9(b). The CPD of the sample was measured by AM-KPFM twice, once where the substrate and the nickel electrodes shared common ground (figure 9(c)) and then when the left and right nickel contacts were biased with voltages of 0.5 V and 0.5 V, respectively (figure 9(e)). All KPFM measurements were conducted in lift-mode with a 5 nm lift height. Conductive TiN coated tips (NTMDT) with a first resonance frequency of 160 kHz were used. Line scans of the surface potentials with and without external bias are shown in figures 9(f) and (d), respectively. It is observed that due to the averaging effect of the probe, the measured KPFM contrast in the biased sample is 0.58 V instead of 1 V. If the CPD measurements were conducted with FM-KPFM the expected contrast should have been closer to 1 V. We also observe a CPD contrast of 45 mV in the unbiased sample

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Figure 10. Evaluation of the noise statistics. (a) Sections containing only clean SiO2 were considered for distinguishing the noise from the CPD signal. (b) The CPD of the SiO2 substrate taken from (a). (c) Histogram outlines the distribution of the substrate CPD (bars) and a normal fit (dashed line) with mean and variance of 0.5132 V and 3:23 10 5 V2, respectively. (d) Autocorrelation of the noise extracted from the framed area in (a).

(figure 9(d)), which stems from the lower work function of the nickel relative to the substrate.

It should be noted that the presented CPD values are -CPD since all the KPFM measurements were conducted by applying the feedback voltage .VDC/ to the probe [20]. We assume that the measured KPFM image is a convolution of the CPD with the PSF despite the topography features observed in figure 9(b). This assumption is supported by Baier et al [21] and Sadewasser et al [22], who measured AM-KPFM on flat surfaces and on surfaces with a topography step. They observed a very small change in CPD distributions for both cases. KPFM simulations carried out in our group including topography steps along with CPD variations also confirmed very little influence of topography on the averaging effect.

The system noise is evaluated by masking some features in figure 9(c) to analyze the CPD statistics of the pure SiO2 substrate (figure 10(a)). The measured KPFM signal on the substrate is shown in figure 10(b). The histogram in figure 10(c) shows the distribution of the SiO2 CPD, which is a Gaussian distribution with an expected value and variance of 0.5132 V and 3:23 10 5 V2, respectively; the noise is obtained by subtracting the mean value. Figure 10(d) presents the autocorrelation of the noise in the marked area of figure 10(a). The delta-function-like feature in the center indicates that white noise can accurately describe the system; therefore we have used an additive white Gaussian noise (AWGN) in the deconvolution process. The theoretical peak of the autocorrelation function of the AWGN is given by [16]:

S D M N Var.n/

(8)

where M and N are the numbers of pixels along the x and y axes, respectively, and Var.n/ is the variance of the noise. By substituting the values of M; N and Var.n/, we obtain a theoretical S D 0:344 27 V2, which is in excellent agreement with S D 0:344 32 V2 obtained from figure 10(d).

Deconvolution of the KPFM image shown in figure 9(e) requires the PSF of the probe calculated for the exact tip–sample distance that was used in the KPFM measurement, which is d D dtopo C dlift, where dtopo is the time averaged tip–sample distance used for recording the topography trajectory and dlift D 5 nm is the lift height distance for the KPFM measurement. Since dtopo varies for every scan, a more analytical approach is used to determine the exact tip–sample distance.

From the abrupt change of the CPD in figure 9(c), we infer that the averaging effect is not prominent when the nickel electrodes share common ground. We therefore repeated the measurement when the nickel electrodes were biased. Figure 11 shows a schematic band diagram of the biased sample, where Ef.Ni/ and Ef.tip/ indicate the Fermi levels of the nickel and tip, respectively. The local vacuum levels of the tip and the sample are indicated by LVL.tip/ and LVL.sample/, respectively. When biased, the CPD of the left and right nickel electrodes are 0.035 V and 0.965 V, respectively. 1 is defined as the CPD between the nickel and substrate when they are unbiased; from figure 9(c), we approximate 1 D 50 mV. The voltage drop along the SiO2 is not linear [23]; however, for simplicity, we assume a constant electric field between the two electrodes. From the theoretical

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Figure 11. Band diagram of the sample, where the left and right nickel electrodes are biased with 0.5 V and 0.5 V, respectively, related to the tip. Ef.Ni/ and Ef.tip/ refer to the Fermi levels of the nickel and tip, respectively. LVL(tip) and LVL(sample) are the local vacuum levels of the tip and sample, respectively, and 1 D 50 meV. The CPD is marked in red. Under bias, the CPD increases

from 0.035 V (left electrode) to 0.965 V (right electrode).

CPD shown in figure 11, we guess the CPD on the surface, VCPD (figure 12(a)). Due to a voltage drift between the first (unbiased) and second (biased) scan, we shift all CPD values by 16.5 mV. In order to determine the exact tip–sample distance, the estimated VCPD is then convolved with different PSFs calculated for several tip–sample distances (6–10 nm) by using the following expression (mathematical formation can be found in the appendix):

VDC D .VCPD Vsub/ ~ PSF C Vsub

(9)

where Vsub denotes the CPD on the SiO2 surface far away from the scanning area and the nickel electrodes. Figure 12(b) shows the measured KPFM signal with different convolution results of the theoretical CPD with PSFs generated for the probe presented in figures 5((a), (b)) at varying tip–sample distances. All profiles are along the cross-section (white line) marked in figure 12(a). The PSF generated for a tip–sample distance of 8 nm is our best estimate, since the convolution result at 8 nm bears the lowest error from VDC (in L2 norm). A good indication for our estimate is the contrast of the measured KPFM image (0.557 V), which is very similar to the summation over the PSF at 8 nm (0.544). For this PSF, we observe a FWHM of 29 nm.

Finally, after obtaining the PSF and the noise statistics, deconvolution is performed on VDC (figure 9(e)) using equation (7). Figure 12(c) shows the profiles of the measured and deconvolved KPFM images. The deconvolved CPD resembles the measured KPFM profile in shape and preserves a contrast of 1.03 V, as expected from the CPD on the surface.

4.2. CdS–PbS nanorods

CdS–PbS heterostructured nanorods (NRs) were synthesized as described in [24]. A dilute solution of the CdS–PbS nanorods (diameter of 4.0 0.2 nm and length of

Figure 12. (a) The theoretical CPD on the biased sample.

(b) Measured CPD (red), theoretical CPD (blue) and convolution of the theoretical CPD with PSFs generated for tip–sample distances of 6 nm (brown), 8 nm (black) and 10 nm (green). The inset shows that the tip–sample distance is 8 nm. (c) Measured CPD (red), theoretical CPD (black) and deconvolved CPD (blue). All CPD line scans are plotted along the white line in (a).

80 30 nm) was spin coated (Nanayakkara et al [25], NREL, USA) onto freshly-peeled highly-oriented pyrolytic graphite (HOPG). Subsequently, topography and KPFM images were simultaneously recorded in argon atmosphere using a single pass technique with an external Kelvin Probe Control Unit (Omicron, Kelvin Probe CU) and an external high-frequency lock-in amplifier (Signal Recovery, 7280 DSP) on a Veeco Dimension 5000 AFM and Nanoscope V controller system. The topography was measured in tapping mode at 50 kHz (1st resonance frequency), while an AC bias modulation

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