Applied Surface Science 140 Ž1999. 358–361
Force spectroscopy in noncontact mode
I.Yu. Sokolov
a
a,b,)
, G.S. Henderson a , F.J. Wicks
a,c
Department of Geology, UniÕersity of Toronto, 22 Russell Street, Toronto, Ontario, Canada M5S 3B1
b
Department of Physics, UniÕersity of Toronto, Toronto, Ontario, Canada M5S 1A7
c
Department of Earth Science, Royal Ontario Museum, Toronto, Ontario, Canada M5S 2C6
Received 29 June 1998; accepted 20 August 1998
Abstract
We have analyzed the possibility of using noncontact scanning force microscopy ŽNCAFM. to detect variations in
surface composition, i.e., to detect a ‘spectroscopic image’ of the sample. This ability stems from the fact that the long-range
forces, acting between the AFM tip and sample, depend on the composition of the AFM tip and sample. The long-range
force can be magnetic, electrostatic, or van der Waals forces. Detection of the first two forces is presently used in scanning
force microscopy technique, but van der Waals forces have not been used. We demonstrate that the recovery of
spectroscopic image has a unique solution. Furthermore, the spectroscopic resolution can be as good as lateral one. q 1999
Elsevier Science B.V. All rights reserved.
PACS: 07.79.L; 61.16.C; 34.20
Keywords: Atomic force microscopy; Molecular force interaction; Atomic force spectroscopy
1. Introduction
Noncontact force microscopy is able to detect
small long-range forces acting between the AFM tip
and sample surface. Analysis of these forces could,
at least theoretically, result in discrimination of variations in properties andror composition across the
sample surface. This ability arises from the fact that
the forces involved are AFM tiprsample dependent.
)
Corresponding author. Tel.: q1-416-978-0668; Fax: q1-416978-3938; E-mail: [email protected]
The long-range forces involved can be magnetic,
electrostatic, or van der Waals forces. Presently,
detection of the first two forces Žmagnetic and electrostatic. has been implemented in scanning force
microscopy techniques w1x. The van der Waals forces
have not been used. However, they too offer the
potential for determining compositional variations
across a sample surface. The difficulty with this
force is that van der Waals interactions are small and
consequently hard to detect. However, recent success
has been achieved in detecting the van der Waals
interactions under high vacuum conditions w2–5x. In
the present study, we show theoretically how van der
Waals forces may be utilized for AFM spectroscopy.
0169-4332r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 9 - 4 3 3 2 Ž 9 8 . 0 0 5 5 4 - 6
I.Yu. SokoloÕ et al.r Applied Surface Science 140 (1999) 358–361
2. The numerical model
We first consider a sample composed of different
materials. In noncontact AFM, one can measure lines
of constant van der Waals force gradient, which is a
long-range interaction, so there are contributions to
the total force from the different materials. The
problem then becomes one of whether it is possible
to recover the compositional variation based on the
scan data. Total force potential V Ž d . can be found
using a simple additive method w6x as follows:
d™
r tip
V Ž d. sC
HV
tip
™
rsample
™
d rsample ™
< rsample y™
r tip < 6
V sample
bŽ
H
.
,
Ž 1.
where V tip , V sample are volumes of the tip and
sample, C is a normalization constant, b Ž r . is a
function that takes into account screening of the
interaction inside of the material and d is the tip–
sample distance. Here we have considered the tip as
a homogeneous material and it should be noted that
we determine the sample surface topology Žvolume.
from contact scanning.
Taking into account that the tip has axial symmetry, and going to vertical Ž z . projection of the scanning force gradient, one gets:
Fz Ž ™
r0 . sC
HV
d™
rsample f z <™
rsample y™
r 0 < b Ž™
rsample . ,
ž
/
sample
Ž 2.
where f z is the z-projection of the force gradient
‘ volume density’ acting between the tip and a point
on the sample, ™
r 0 s Ž x 0 , y 0 ,d ..
To keep the force gradient Fz constant during the
scan, the distance d should be a function of the
lateral coordinates, d s dŽ x 0 , y 0 ; Fz .. Provided we
have the scan data as a set of the constant force
surfaces, i.e., a set of functions d s dŽ x 0 , y 0 ; Fz . for
different Fz , formula Ž2. can be considered as the
equations for determination of b as a function of
Ž x 0 , y 0 , z .. If one finds the function b , the distribution of the materials will be known because different
values of b are associated with different types of
materials.
Consider now the type of integral equation formula Ž2. belongs to. It is a Fredgolm equation of the
359
™
.
Ž™ ™ .
first kind w7x with the kernel K Ž™
r 0 ,r
s s f z r 0 ,y rs .
™ ™
™ ™
The kernel is symmetric K Ž r 0 ,rs . s K Ž rs ,r 0 . and,
furthermore, there is the property:
HHVV
™ 2
<
d™
r 0 d™
rs < K Ž ™
r 0 ,r
s . - `,
Ž 3.
sample
where V is the region of ™
r 0 variation Žthe set of the
.
surface images .
Inequality Ž3. is valid because both the sample
V sample and the scan image V are finite, and the
™
.
Ž
kernel K Ž™
r 0 ,r
s does not have any poles otherwise,
the tip and sample would touch each other which is
not possible in noncontact mode.. Due to the finite
size of the image area, the function b should be a
square integrated function over the image area. Property Ž3., together with the kernel symmetry, are
known to be associated with a Hilbert–Schmidt-type
kernel w7x. Therefore, Eq. Ž2. is a Fredgolm equation
of the first kind with a Hilbert–Schmidt kernel, and
from general theory w7x, such an equation has a
unique solution from the class of continuous functions, provided Fz Ž™
r 0 . g C 2 Žtwice-differential continuous functions.. The latter condition is obviously
valid because Fz Ž™
r 0 . is a constant during scanning,
and therefore, has an infinite number of derivatives.
We have also assumed here that b Ž x 0 , y 0 , z . is a
continuous function. This is justified by the fact that
in reality, we cannot detect different materials with a
resolution better than one atom Žsee later..
So far, we have only considered measurement of
an ideal case. If we include some error in the measurements, then the solution of Eq. Ž3. is no longer
unique. However, any ambiguity in the measurements is trivial. Let us assume that there are at least
two different solutions b 1Ž x 0 , y 0 , z . and b 2 Ž x 0 , y 0 , z .
for Eq. Ž3.. This means that the following is true.
HV
d™
rs f z Ž <™
rs y™
r 0 < . b 1 Ž™
rs . y b 2 Ž ™
r 0 . s 0.
Ž 4.
sample
Measurement of force gradient f with an error D f 1
means that there is an interval Žspatial resolution.
D xŽ y , z . s D frŽ f z . xŽ y , z . , inside of which the force
gradient changes cannot be detected. We can there-
1
For simplicity, we assume that the error D f includes error
connected with the definition of the sample surface V sample .
360
I.Yu. SokoloÕ et al.r Applied Surface Science 140 (1999) 358–361
fore put f constant inside that interval. Eq. Ž4. then
has a whole class of nonzero solutions w b 1Ž™
rs . y
b 2 Ž™
r 0 .x. For example, w b 1Ž™
rs . y b 2 Ž™
r 0 .x can be a
sum of any periodic functions with the periods
D xŽ y , z .rn, where n s 1,2 . . . . Nevertheless, one can
make the solution b unique if the function b is
assumed to be monotonic and smoothed, inside the
spatial resolution D xŽ y , z . . This trivial conclusion
means that it is impossible to resolve the function b
better than the spatial resolution of the force f. So,
the spectroscopic function b is a unique solution of
Eq. Ž2., provided we assume that the spatial resolution of recovery of b cannot exceed the spatial
resolution of the force measurement. This means that
we should be able, at least theoretically, to discriminate materials of different composition within the
sample surface. However, we must also address the
question of resolution. What volume of differing
material is necessary for the microscope to be able to
discriminate the differences in composition?
As noted above, b cannot exceed the force gradient spatial resolution. However, there is one further
factor that could decrease the spectroscopic resolution. The two materials should be different enough to
produce detectable force differences. Keeping this in
mind, we calculate the volume of an impregnated
material that can be detected by a regular NCAFM
set-up. To do that, we consider the tip–sample configuration shown in Fig. 1. The vertical projection of
the scan force acting in this configuration was calculated previously w6x. However, in NCAFM, since it is
much more convenient to measure the force gradient
rather than the force, we consider the derivative of
the scan force F with respect to d:
Fz s
AR
3d 3
3 Ž A y AX . V
2p R Ž d q h .
,
4
Ž 5.
where V is the volume of impregnated material, and
A, AX are the Hamaker constants of the materials of
the sample and impregnation, respectively.
To be able to detect the impregnation, the contribution to the force gradient due to the impregnated
material, the second term in Eq. Ž5., must exceed the
AFM sensitivity to force gradient d Ž F .. This condition results in
< Ž A y AX . V < G
2p R
3
4
Ž d q h. d Ž F .
f 4 = 10y2 9 Ž eV m3 .
=
ž
dqh
1 nm
4
/ž
ž
R
10 nm
/
dŽ F.
3 = 10y4 Nrm
/
,
Ž 6.
where d Ž F . s 3 = 10y4 Nrm is taken for the
NCAFM set-up used in Refs. w2,3x.
If we put <Ž A y AX .< ; 0.1 eV Žthe usual variation
for different materials w8x., we shall be able to detect
the volume V
V G 4 = 10y2 8 m3
=
Fig. 1. AFM tip–sample configuration. Dotted ball is an impregnated different material.
y
ž
ž
R
10 nm
dŽ F.
3 = 10y4 Nrm
/
/ž
.
dqh
1 nm
4
/
Ž 7.
If we take the volume of a single atom to be equal to
Ž0.2 nm. 3 s 8 = 10y30 m3 , restriction Ž7. means that
the detectable volume must contain, at least, 50
atoms, or a cube about 4 = 4 = 4 atoms. This is
about what one should expect for lateral spatial
resolution w9,10x. We can therefore say that spectroscopic resolution will be of the same order of magnitude as the lateral spatial resolution.
One further contribution to degradation of the
spectroscopic resolution is the following. As noted
above, one needs to collect a set of constant force Žor
force gradient. surfaces for each scan line Žsee an
I.Yu. SokoloÕ et al.r Applied Surface Science 140 (1999) 358–361
example in Ref. w11x.. This can be done by using the
multiple lift mode Žlift mode is a trademark of a
mode of SPM by Digital Instruments, CA, USA.,
where the surface is scanned, the tip lifted and then
rescanned; the sequence being cycled multiple times.
The problem with this approach is the stability of the
sample during these tip movements. However, provided the sample is relatively stable, then there
should be little effect on the spectroscopic resolution.
Furthermore, uncertainty in the tip shape can contribute to the calculated force gradient error. Following Ref. w6x, we estimate such uncertainty to be
; 10%. This value is not large when compared with
10% sensitivity to the force gradient d Ž F . measurement considered above.
3. Conclusion
We have shown that it is possible to distinguish
different materials using NCAFM and that the spectroscopic resolution could ultimately be as good as
the lateral spatial resolution, i.e., on the order of a
361
few nanometers. However, sample instability during
scanning, could decrease this limit of the resolution.
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