PHYSICAL REVIEW B 75, 033406 共2007兲
Force measurement with a scanning tunneling microscope
K.-F. Braun and S.-W. Hla
Clippinger Laboratories, Ohio University, Athens, Ohio 45701, USA
共Received 19 October 2006; published 10 January 2007兲
We present a scheme to measure forces with a scanning tunneling microscope. During atomic manipulation
experiments with a scanning tunneling microscope the tip height curve is recorded. It is shown here that the
amplitude of the manipulation curve is a measure of the interaction force between the microscopes tip and a
single atom adsorbed on a surface. A simple formula is derived and tested.
DOI: 10.1103/PhysRevB.75.033406
PACS number共s兲: 68.37.Ef, 68.37.Ps, 82.37.Gk
The scanning tunneling microscope 共STM兲 has proven its
unique abilities in imaging surfaces up to atomic resolution.
Soon after its invention by Binnig and Rohrer it was realized
that, due to its close proximity to the surface atoms the STM
tip often influences and modifies the surface. This disadvantage however was used positively by modifying substrate
surfaces in a controlled way. Eigler et al.1 showed that even
single atoms can be positioned on selected adsorption sites
proving the startling possibilities to build up man-designed
functional nanostructures atom by atom. The manipulation
process itself is a subject of fundamental experiments with
single atoms. The data recorded during the positioning with
the microscopes tip yields information whether the interaction is attractive or repulsive2 and about the atoms path on
the surface.3,4 The nature of the tip-atom interaction for
metal atoms was determined recently as a chemical force,
dependant only on the tip-atom distance.5 Recent insights
into single atom processes on crystal surfaces include friction measurements with atomically sharp cantilever6 and the
dynamics of single atom switching.7 The force required to
move a single atom across a crystal surface however has not
been measured yet and we show here a simple way to measure forces with a STM. A formula is presented and tested
with simulated manipulation curves.
Atomic manipulation can be described within a simple
model where the surface is assumed to have a continuous
sinusoidal potential with the amplitude of the diffusion
barrier.3 We describe here the tip with a Morse potential and
let the atom reside in the local minimum of the combined tip
and surface potential. The scheme used here is not dependant
on the used potential, other potentials or results from firstprinciple calculations can be used too. The atom moves on a
two dimensional surface and the tip is assumed to be spherical. With these assumptions one can realistically describe
atomic movements and tip height curves once the tip-atom
force and the diffusion barrier are known. A simulation starts
by placing the tip above the atom and the height is adjusted
with closed feedback loop until the current set point is
reached. The local potential minimum of the combined potentials is determined and the atom is shifted to it. Again the
tip height is adjusted to reach the set point value of the current. These feedback loop cycles are repeated until the atom
stabilizes only then the tip is shifted laterally. Figure 1 shows
a description of the potential model where the atom always
resides in a local potential minimum of the combined potentials of tip and surface.3 In this equilibrium position the tipatom force is counterbalanced by the surface forces ex1098-0121/2007/75共3兲/033406共4兲
pressed by the migration barrier U0. With this description
topography curves or manipulation curves at constant current
can be calculated.8 The reverse, namely the tip-atom interaction forces can be determined too. They require experimental
manipulation curves. In such case first the atoms position is
determined and then the tip-atom interaction force can be
calculated. Hereby one uses that the projected lateral force
component Flat = Ftip cos ␸ needs to be equal to the surfaces
force Fsurface. By using an approximate potential for the surface one can follow a simpler way and the force can be
determined from a single formula shown in the following.
Figure 2 shows simulated tip height or manipulation
curves for a quasi-one-dimensional periodic surface potential. Experimental values for the height are typically ⬇4 Å
and the amplitudes range is ⬇0.1 Å.2 The tip moves over the
atom and changes thereby the lateral force component which
is pulling the atom across the surface plane. Once the lateral
force component exceeds the counterforces from the surface
the atom will jump to the next local potential minimum. The
recorded tip height shows a characteristic shape which can
FIG. 1. 共Color online兲 Graph showing the potential model to
describe the atomic manipulation. The adatom always resides in a
local potential minimum resulting from the sum of tip and surface
potential. In this equilibrium position the lateral component of the
tip-atom force is counterbalanced by the surface forces expressed
through the migration barrier.
033406-1
©2007 The American Physical Society
PHYSICAL REVIEW B 75, 033406 共2007兲
BRIEF REPORTS
FIG. 2. 共Color online兲 Simulated curves showing the dependence of the amplitude of the tip-height curve as a function of the
tip-atom interaction force 共a兲 and as a function of initial height with
a constant tip-atom force 共b兲.
be used to distinguish between different manipulation modes.
Graph Fig. 2共a兲 shows the pulling case where the attractive
interaction force is varied while the diffusion barrier is kept
constant apparently the stronger the force the smaller the
amplitude. The graph in Fig. 2共b兲 of the displayed manipulation curves is calculated for different initial heights and a
constant tip-atom interaction force. The curves show that the
closer the tip to the atom the smaller the amplitude since the
required lateral force component is met at the same critical
angle. We conclude that the amplitude is a function of tip
height and tip-atom interaction force.
We derive in Fig. 3 a formula which expresses the tipatom force through the amplitude and height of the manipulation curve. If one approximates the atoms potential with
a ␦-like potential then the atom does not move until a force
threshold is reached. In such a case the segments of the
tip-height profile become circular shaped. We denote this
radius with h as shown in Fig. 3共a兲. In Fig. 3共b兲 the lateral
force component FL is the projection of the radial force
FR. Both, geometrical and force quantities are related
through the angle ␸. From the amplitude a, the height z, and
the tip-atom distance h the tip-atom interaction force can
be calculated straight forward. The result is shown in
Fig. 3共c兲, here FL needs to be expressed through the
migration barrier U0 and amounts to FL = 2␲ / a0U0 with a0
the lattice constant. In good approximation one can use
z = 冑h2 − 关冑h2 − 共h − a兲2 − a0兴2 ⬇ h and then only two entities
are needed to calculate the force. We stress that this formula
is derived independent of the tip-atom interaction potential
and can be used to measure any potential forms.
We have tested this formula against simulated manipulation curves. For a quasi-one-dimensional potential surface a
range of initial tip heights were chosen and manipulation
curves simulated. From the resulting amplitudes and heights
the tip-atom interaction forces were calculated and displayed
in Fig. 4. The solid line denotes the Morse potential used for
the tip-atom interaction. The parameters of the Morse potential correspond to a binding energy of 1.0 eV and a potential
minimum at a distance of 3.0 Å, resembling the density
functional theory calculation results for single Ag atoms on a
Ag共111兲 surface.16 Taking into account an adatom diffusion
barrier U0 = 0.1 eV, the theoretical values are well reproduced. At larger tip-atom separation the calculated values are
larger than the used potential. Here the shape of the manipulation segments is no longer circular because the atoms are
pulled away from their equilibrium positions where they
would stay without the presence of the tip. Such a movement
around their equilibrium positions results in false values for
the height as compared to the static situation and causes the
shown deviation from the given force values. The direction
of the movement depends upon the specific shape of the
tip-atom interaction potential.
We conclude that the used tip-atom interaction force can
be recovered with good agreement. The evaluation of experimental data requires additionally a calibration of the tipheight with I-Z spectroscopy. The resulting error in the
height values scales nonlinear with 1 / h. On the open
fcc共111兲 surfaces which are often used for manipulation ex-
FIG. 3. 共a兲 The amplitude of the manipulation curve can be directly interpreted if one considers that the atom does not move until the
lateral force component has reached a threshold value. In such a case the segments of the tip-height profile become circular shaped with the
radius h. 共b兲 The lateral force component FL is simply the projection of the radial force FR. 共c兲 From the amplitude a, the height z and the
tip-atom distance h one can calculate the tip-atom interaction force. Here FL needs to be expressed with the migration barrier.
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PHYSICAL REVIEW B 75, 033406 共2007兲
BRIEF REPORTS
FIG. 4. The tip-atom interaction force for a single adatom adsorbed on a surface 共쎲兲 is shown together with the theoretical curve
for a Morse potential 共⫺兲 with a binding energy of 1.0 eV at 3.0 Å.
Taking into account an adatom diffusion barrier U0 = 0.1 eV, the
theoretical values are well reproduced. The deviation from a circular shape of the manipulation segments is the reason for the
deviation.
periments tip paths along the close-packed direction give tipheight curves such as in Fig. 2. Other directions yield manipulation curves which are constituted from segments with
varying amplitude and shape.4,5 The derived formula remains
valid if one is using the averaged values h̄ the mean tip
height and ␴共a兲N−1 the standard deviation of the amplitude.
These quantities need to be corrected linearly to yield the
true values for h and a depending on the tips direction.9 If
the tip path runs parallel to a closed packed 具11̄0典
direction at a distance d one can solve for the force
analytically. Here the tip atom interaction force amounts to
FR = 2FLh / 冑3冑h2 − d2 − 共z − a兲2 + d.
Tip-atom interaction forces measured in this way can be
determined over a range of a few Å between adatom and tip
atom with a sensitivity of 0.5 nN/ Å. For tip-adatom distances typically used in manipulation experiments an exponential profile is expected with values in the order of a few
hundreds pN.10,11 In the measured tip-atom distance range
London–van der Waals and metal-adhesion forces need to be
considered. In the work of Rubio-Bollinger et al.10 the interaction force between two sharp Au tips has been measured
with a mechanically controlled break junction. Following10
the crossover regime where either van der Waals forces or
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short-ranged metal-adhesion forces are dominating is about
3 Å. In an experiment by Cross et al.12 the interaction of a
three-atom terminated tungsten tip with a Au surface has
been measured by mounting a sample directly onto a cantilever. There higher forces were measured at closer distances.
If the tip is moved closer in the STM single atom experiment
of this paper typically a tip change occurs or the adatom is
transferred to the tip. We state finally that the measured force
is a many-atom force since the local environment of the tip
and surface atom i.e., their binding configuration alter their
electronic structure and therefore the atom-atom interaction.
The potential model is based on simplifying assumptions,
we argue here that the main constraints have only small influence though. The tunneling junction is simplified by taking into account current contributions from the atom to the
tip apex. A more realistic tunneling junction results in a
Gaussian appearance of single atoms. The segments in the
manipulation curves in Fig. 2 are in fact the top most portion
of such a Gaussian shape altered by the atoms movement in
the local potential minimum.13 Nevertheless with typical tipatom distances of 2 Å the surface is still 2 Å further away
making this surface contribution negligible and the simplified tunneling junction a good approximation. The vertical
force component can reduce the migration barrier or even
desorb the atom but was neglected here since the binding
energy is much larger than the migration barrier. We consider
elastic tip and surface deformation14–16 as the dominant experimental uncertainty in the measurement of the tip height
curves.
We note that the quantity which is measured within this
scheme is the force required to pull a single atom across the
surface. By pulling the atom energy is transferred to the
atom. The transferred energy dissipates during the jumps into
the new local potential minimum by electron-hole excitation
and phonon creation. The dissipating energy is transferred
from the tip to the atom by lifting the local potential minimum up until the next jump occurs. The dissipation is not
expected to show up in the manipulation curves at low temperature and small currents.
In conclusion we have measured tip-atom interaction
forces with a scanning tunneling microscope by analyzing
the amplitude of tip height data recorded in lateral manipulation experiments. The presented single atom experiment
scheme opens up new possibilities for future research on
single particle interaction present in van der Waals, ionic or
covalent bonds.
The financial support provided by the U.S. Department of
Energy Grant No. DE-FG02-02ER46012 and the NSF Grant
No. DMR 0304314 is gratefully acknowledged.
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PHYSICAL REVIEW B 75, 033406 共2007兲
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See EPAPS Document No. E-PRBMDO-75-012703 for a sequence of images showing the calculated manipulation curve
in constant current mode. The potential and the tip height
are shown. For more information on EPAPS, see http://
www.aip.org/pubservs/epaps.html.
9
K.-F. Braun and S.-W. Hla 共unpublished兲.
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G. Rubio-Bollinger, P. Joyez, and N. Agrait, Phys. Rev. Lett. 93,
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