Nanotechnology 24 (2013) 295702

G Cohen et al

Figure 14. (a) and (b) show raw CPD measurements of single layers and double layers of graphene. (c) and (d) show line scans of the topography (green line), measured CPD (blue line), and deconvolved CPD (red line) along the profiles illustrated in (a) and (b), respectively. The left axes correspond to the CPD, whereas the right axes correspond to the topography. The deconvolved CPD values were calculated using the Wiener filter, effective PSF and noise statistics obtained beforehand.

measurements of the same sample both in AM and in FM in the same instrument, the same tip–sample distance and oscillation amplitude, etc. A full comparison is outside the scope of this paper.

In addition, we have analyzed the influence of the cantilever on the measured CPD. We showed that for both AMand FM-KPFM the cantilever does not change the measurement resolution. However, it was shown that the cantilever has a profound influence on the absolute measured CPD value in AM-KPFM whereas in FM-KPFM the cantilever hardly changes the measured CPD. We also studied the tip–sample distance effect in KPFM measurements and found that for conventional tip–sample distances (1–50 nm) there is almost no attenuation when performing FM-KPFM, however in AM-KPFM measurements conducted above 10 nm the measured KPFM signal is derived mainly from the substrate and not from the feature beneath the tip apex.

Based on our simulations, we conclude that FM-KPFM is preferred to AM-KPFM in terms of the measurement resolution, contrast and insensitivity to tip–sample distance; however, the signal-to-noise ratio is much better in AMKPFM.

The presented reconstruction algorithm is valid only for flat surfaces or surfaces which exhibit small topography variations, since it is based on the fact that the system response is invariant to the probe position. A possible extension to this work is to consider a real sample with a rough surface and correlate the surface potential to the KPFM signal while taking into consideration the sample topography. An additional improvement might be a development of an improved reconstruction algorithm by designing more sophisticated deconvolution filters instead of the Wiener filter.

Acknowledgment

The research at Tel-Aviv University was supported by the Israel Science Foundation (grant number 498/11).

Appendix

A.1. AM-PSF extrapolation

Since integration of AM-PSFs does not converge to 1 within the scanned area, AM-PSFs must be extrapolated beyond the measurement boundaries in order to obtain the full system response. Away from the origin, the two-dimensional AM-PSF illustrated in figure 4(a) converges to the expression:

8jxj 50 nmV PSF.x; y/ D A1 x P1

8jyj 50 nmV PSF.x; y/				(A.1)
D	(A3	y P3	;	y < 0
	A2	y P2	;	y > 0

where A1 D A1.y/; P1 D P1.y/ are positive numbers which vary for each y line and A2 D A2.x/; A3 D A3.x/; P2 D P2.x/; P3 D P3.x/ are positive numbers which vary for each x line. A2 and P2 are related to the cantilever side whereas A3 and P3 are related to the opposite side.

Considering the above expressions, we calculated all the parameters in equation (A.1) using the dependence of log.PSF/ versus log.x/ and log.PSF/ versus log.y/, as shown in figure A.1(a). Once all the parameters are obtained, we extrapolated the PSF for all x and y lines, as illustrated by

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Nanotechnology 24 (2013) 295702

G Cohen et al

Figure A.1. (a) Two-dimensional Loglog plot of the AM-PSF; a linear relation is observed far from the origin. (b) Expanding the AM-PSF to an infinite area. First, the x and y lines are extrapolated (illustrated by blue arrows). Next, extrapolation is performed on the diagonal direction (illustrated by red arrows). In the final step, the remaining pixels on the plane (black domains) are interpolated using 2 pixels which were previously extrapolated. An example for this interpolation is marked with a green area.

the blue arrows in figure A.1(b). Next, we observe that near the corners the PSF converges to the expression:

8jrj 70 nmV

(

PSF.r/	D	Ac r Pc ;			y > 0
		At r Pt ;			y < 0
where r D q					s.t. jxj D jyj:
			x2 C y2			(A.2)

Ac; At; Pc; Pt are positive numbers and r is the distance from the origin in a diagonal direction, as illustrated by the red squares in figure A.1(b). By calculating the parameters in equation (A.2), as demonstrated earlier, we extrapolate the PSF to the corners of the image. The PSF components of the remaining areas of the plane (black domains) are interpolated by including two pixels which were extrapolated in the preceding steps. An example of such interpolation is illustrated by the green area in figure A.1(b). The two pixels used for this interpolation are marked with red and blue filled squares.

A.2. Effective PSF

In the single pass technique, the Kelvin probe controller minimizes the oscillations of the cantilever’s second resonance, leaving it to oscillate only in the first mechanical resonance [19]. We define d as the average tip–sample

Figure A.2. (a) Representation of the probe in the y–z axes. is the angle between the cantilever and the horizontal axis. Distance d is the averaged tip–sample distance. (b) Representation of the probe in the y0–z0 axes. A is the oscillation amplitude.

distance and A as the amplitude of the cantilever oscillation, as illustrated in figures A.2(a) and (b). Since the cantilever is inclined towards the surface by degrees, it is more convenient to analyze its motion of the probe in the rotated axis system y0–z0, as shown in figure A.2(b). Neglecting the influence of the tip on the cantilever oscillation, the vertical deformation along the y0 axis as a function of time is given by [27]:

Z0.y0; t/ D

2

cos L

0

cosh

L

0 C sin

L

0

A

By

By

By

sinh

By0

cos

2 t

(A.3)

LT0

where B D 1:875; D 0:7341; L is the cantilever length and T0 is the oscillation period. Thus, the deflection of the tip is:

Z0.y0 D L; t/ D A cos		2T0		:
		t

Considering the tip position of the cantilever unaffected by the sample (free cantilever) in figure A.2(a) to be at .y; z/ D

.0; d/, the tip position in the rotated system will be:

.y0; z0/ D . d sin. /; d cos. //:

Therefore, when oscillating, the tip position over time is:

.y0.t/; z0.t// D	d sin. /; d cos. / A cos		2T0		:
			t

Using the rotation matrix, we describe the oscillations in the y–z system:

" # y.t/

z.t/

cos. / sin. /

d sin. /

D "

sin. / cos. /#

2d cos. /

A cos

2 t

3

T0

4

5

2

A cos

2 t

sin. /

3

D

T0

:

(A.4)

6d

A cos

2 t

cos. /7

T0

6

7

4

5

The CdS–PbS AM-KPFM measurement was performed using tapping mode, therefore we bound minfzg D 0 for the tip

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