A comprehensive discussion of the physical origins of Kelvin probe force microscopy (KPFM) signals for charged systems is given. We extend the existing descriptions by including the open-loop operation mode, which is relevant when performing KPFM in electrolyte solutions. We define the contribution of charges to the KPFM signal by a weight function, which depends on the electric potential and on the capacitance of the tip-sample system. We analyze the sign as well as the lateral decay of this weight function for different sample types, namely, conductive samples as well as dielectric samples with permittivities both larger and smaller than the permittivity of the surrounding medium. Depending on the surrounding medium the sign of the weight function can be positive or negative, which can lead to a contrast inversion for single charges. We furthermore demonstrate that the KPFM signal on thick dielectric samples can scale with the sample size—rendering quantitative statements regarding the charge density challenging. Thus, knowledge on the weight function for charges is crucial for qualitative as well as quantitative statements regarding charges beneath the tip.

1.
M.
Nonnenmacher
,
M. P.
O'Boyle
, and
H. K.
Wickramasinghe
,
Appl. Phys. Lett.
58
,
2921
(
1991
).
2.
W.
Melitz
,
J.
Shen
,
A. C.
Kummel
, and
S.
Lee
,
Surf. Sci. Rep.
66
,
1
(
2011
).
3.
N.
Kobayashi
,
H.
Asakawa
, and
T.
Fukuma
,
Rev. Sci. Instrum.
81
,
123705
(
2010
).
4.
L.
Collins
,
J. I.
Kilpatrick
,
I. V.
Vlassiouk
,
A.
Tselev
,
S. A. L.
Weber
,
S.
Jesse
,
S. V.
Kalinin
, and
B. J.
Rodriguez
,
Appl. Phys. Lett.
104
,
133103
(
2014
).
5.
A. L.
Domanski
,
E.
Sengupta
,
K.
Bley
,
M. B.
Untch
,
S. A. L.
Weber
,
K.
Landfester
,
C. K.
Weiss
,
H.-J.
Butt
, and
R.
Berger
,
Langmuir
28
,
13892
(
2012
).
6.
K.
Umeda
,
K.
Kobayashi
,
N.
Oyabu
,
Y.
Hirata
,
K.
Matsushige
, and
H.
Yamada
,
J. Appl. Phys.
113
,
154311
(
2013
).
7.
K.
Umeda
,
K.
Kobayashi
,
N.
Oyabu
,
Y.
Hirata
,
K.
Matsushige
, and
H.
Yamada
,
J. Appl. Phys.
116
,
134307
(
2014
).
8.
U.
Zerweck
,
C.
Loppacher
,
T.
Otto
,
S.
Grafström
, and
L. M.
Eng
,
Phys. Rev. B
71
,
125424
(
2005
).
9.
M.
Bieletzki
,
T.
Hynninen
,
T. M.
Soini
,
M.
Pivetta
,
C. R.
Henry
,
A. S.
Foster
,
F.
Esch
,
C.
Barth
, and
U.
Heiz
,
Phys. Chem. Chem. Phys.
12
,
3203
(
2010
).
10.
S.
Sadewasser
,
T.
Glatzel
,
M.
Rusu
,
A.
Jager-Waldau
, and
M. C.
Lux-Steiner
,
Appl. Phys. Lett.
80
,
2979
(
2002
).
11.
C.
Barth
and
C. R.
Henry
,
Appl. Phys. Lett.
89
,
252119
(
2006
).
12.
C.
Barth
and
C. R.
Henry
,
J. Phys. Chem. C
113
,
247
(
2009
).
13.
L.
Nony
,
A. S.
Foster
,
F.
Bocquet
, and
C.
Loppacher
,
Phys. Rev. Lett.
103
,
036802
(
2009
).
14.
S.
Sadewasser
,
P.
Jelinek
,
C.-K.
Fang
,
O.
Custance
,
Y.
Yamada
,
Y.
Sugimoto
,
M.
Abe
, and
S.
Morita
,
Phys. Rev. Lett.
103
,
266103
(
2009
).
15.
S.
Kawai
,
T.
Glatzel
,
H.-J.
Hug
, and
E.
Meyer
,
Nanotechnology
21
,
245704
(
2010
).
16.
A.
Sadeghi
,
A.
Baratoff
,
S. A.
Ghasemi
,
S.
Goedecker
,
T.
Glatzel
,
S.
Kawai
, and
E.
Meyer
,
Phys. Rev. B
86
,
075407
(
2012
).
17.
L.
Gross
,
B.
Schuler
,
F.
Mohn
,
N.
Moll
,
N.
Pavliček
,
W.
Steurer
,
I.
Scivetti
,
K.
Kotsis
,
M.
Persson
, and
G.
Meyer
,
Phys. Rev. B
90
,
155455
(
2014
).
18.
F.
Mohn
,
L.
Gross
,
N.
Moll
, and
G.
Meyer
,
Nat. Nanotechnol.
7
,
227
(
2012
).
19.
B.
Schuler
,
S.-X.
Liu
,
Y.
Geng
,
S.
Decurtins
,
G.
Meyer
, and
L.
Gross
,
Nano Lett.
14
,
3342
(
2014
).
20.
L. N.
Kantorovich
,
A. I.
Livshits
, and
M.
Stoneham
,
J. Phys.: Condens. Matter
12
,
795
(
2000
).
21.
J. L.
Neff
and
P.
Rahe
,
Phys. Rev. B
91
,
085424
(
2015
).
22.
C.
Barth
,
T.
Hynninen
,
M.
Bieletzki
,
C. R.
Henry
,
A. S.
Foster
,
F.
Esch
, and
U.
Heiz
,
New J. Phys.
12
,
093024
(
2010
).
23.
T.
Hynninen
,
A. S.
Foster
, and
C.
Barth
,
e-J. Surf. Sci. Nanotechnol.
9
,
6
(
2011
).
24.
O.
Takeuchi
,
Y.
Ohrai
,
S.
Yoshida
, and
H.
Shigekawa
,
Jpn. J. Appl. Phys.
46
,
5626
(
2007
).
25.
L.
Collins
,
J. I.
Kilpatrick
,
S. A. L.
Weber
,
A.
Tselev
,
I. V.
Vlassiouk
,
I. N.
Ivanov
,
S.
Jesse
,
S. V.
Kalinin
, and
B. J.
Rodriguez
,
Nanotechnology
24
,
475702
(
2013
).
26.
L.
Kou
,
Z.
Ma
,
Y. J.
Li
,
Y.
Naitoh
,
M.
Komiyama
, and
Y.
Sugawara
,
Nanotechnology
26
,
195701
(
2015
).
27.

It is important to note that the assumption of fixed charges may not be justified in liquids that contain mobile ions.

28.
T. R.
Albrecht
,
P.
Grütter
,
D.
Horne
, and
D.
Rugar
,
J. Appl. Phys.
69
,
668
(
1991
).
29.
F. J.
Giessibl
,
Phys. Rev. B
56
,
16010
(
1997
).
30.
N.
Kobayashi
,
H.
Asakawa
, and
T.
Fukuma
,
J. Appl. Phys.
110
,
044315
(
2011
).
31.

In case of finite oscillation amplitudes A, the weight functions are obtained by averaging the numerator and denominator in Eqs. (11) and (12) as described in the  Appendix.

32.

The geometry of the full macroscopic probe is described by a tip radius of 20 nm, a cone height of 10 μm, a half opening angle of 25°, a cantilever radius of 40 μm and a cantilever thickness of 4 μm. The tip-sample distance was set to 0.5 nm. A sample thickness of 1 mm was considered in case of dielectric samples.

33.
A.
Sadeghi
,
A.
Baratoff
, and
S.
Goedecker
,
Phys. Rev. B
88
,
035436
(
2013
).
34.
As shown in Refs. 16 and 33, the infinite series of image charges can be truncated and summed up analytically. We follow the therein described approach and checked all resulting quantities for convergence with respect to the truncation cut-off. See http://www.self-assembly.uni-mainz.de/software/kpfm_conducting_sphere.py for the Python source code for all calculations.
35.

The expression in Eq. (17) is derived from Eqs. (7) and (8) by writing the sum as an integral and separating the lateral distance ranges from r ∈ [0, r0] and r ∈ [r0, rσ].

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