n n

jxj res and jyj res 2 2

otherwise

Nanotechnology 24 (2013) 295702

G Cohen et al

bottom position. Meaning, A D cosd. / . To take into account the cantilever oscillations we sampled the position of the oscillating tip 25 times in one oscillation period:

	2 d tan. / cos					n			3
y.n/						2
						N
"z.n/# D	6 d	1		cos	2 n				7
	6				N				7
	4						5

where N D 25; n increases from 1 to 25, D 12 and d D 16:7 nm. Since this first mechanical resonance frequency is around 50 kHz, whereas the time constant of the Kelvin controller is 1 ms, the effective AM-PSF is calculated by minimizing the average electrostatic force on the probe rather than minimizing the force at each tip–sample distance [10].

The graphene CPD images were measured by FM-KPFM using a minimal tip–sample distance of 1 nm, therefore we bound minfzg D 1 nm in equation (A.4) to find the exact

	position of the oscillating tip in time:
		2		A cos	t			sin. /		3
	y.t/				2
					T0
	"z.t/# D	61	C	A cos. /	1			cos	2 t	7
									T0
		6								7
		4								5

where D 13 and A D 3–5 nm were used in the measurements. The effective FM-PSF was then calculated in the same manner as the AM-PSF, by minimizing the averaged force-gradient instead of the force at each tip–sample distance.

A.3. Convolution and deconvolution

Let PSF1 be the physical, continuous point spread function of the probe. The Kelvin voltage (DC voltage) on the probe satisfies:

VDC D VCPD ~ PSF1
D .VCPD Vsub C Vsub/ ~ PSF1:	(A.5)
Since sSamplePSF1 ds D 1 we obtain:
VDC Vsub D .VCPD Vsub/ ~ PSF1:	(A.6)

Let PSF be the discrete point spread function calculated by Matlab. PSF is cropped to a specific area:

PSF.x; y/

(

PSF1.x; y/;

D

0;

(A.7)

where n is the number of simulated surface points in x- and y-axes and res is the scan resolution. If there are no features outside the scan area we can approximate .VCPD.x; y/ Vsub/ 0 for x and y outside the scan area. Since PSF1.x; y/ values for jxj; jyj > n2 res are significantly smaller than the values close to the origin, we estimate:

.VCPD Vsub/ ~ PSF .VCPD Vsub/ ~ PSF1: (A.8)

And from equation (A.6) we get:

VDC Vsub D .VCPD Vsub/ ~ PSF:

(A.9)

Since deconvolution in Matlab requires a normalized PSF, we

P divide and multiply the above expression by PSF:

VDC Vsub

|{z }

Deconvolution input

D h.VCPD Vsub/ XPSF i ~						PSF	:	(A.10)
						PSF
|		{z		}		P
	Deconvolution output

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