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Measurement of the gold–gold bond rupture force at 4 K in a single-atom chain using photonmomentum-based force calibration
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2015 Meas. Sci. Technol. 26 025202
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Measurement Science and Technology
Meas. Sci. Technol. 26 (2015) 025202 (8pp)
doi:10.1088/0957-0233/26/2/025202
Measurement of the gold–gold bond rupture
force at 4 K in a single-atom chain using
photon-momentum-based force calibration
D T Smith and J R Pratt
National Institute of Standards and Technology, Gaithersburg, MD 20899-8520, USA
E-mail: douglas.smith@nist.gov
Received 7 August 2014, revised 12 November 2014
Accepted for publication 24 November 2014
Published 22 December 2014
Abstract
We present instrumentation and methodology for simultaneously measuring force and
displacement at the atomic scale at 4 K. The technique, which uses a macroscopic cantilever as
a force sensor and high-resolution, high-stability fiber-optic interferometers for displacement
measurement, is particularly well-suited to making accurate, traceable measurements of force and
displacement in nanometer- and atomic-scale mechanical deformation experiments. The technique
emphasizes accurate co-location of force and displacement measurement and measures cantilever
stiffness at the contact point in situ at 4 K using photon momentum. We present preliminary
results of measurements made of the force required to rupture a single atomic bond in a gold
single-atom chain formed between a gold flat and a gold tip. Finally, we discuss the possible use
of the gold–gold bond rupture force as an intrinsic force calibration value for forces near 1 nN.
Keywords: atomic force, bond strength, interferometry, force metrology, length metrology
(Some figures may appear in colour only in the online journal)
1. Introduction
the specimen surface and the interaction force between the tip
and the surface. This interaction is typically attractive when
the tip is close to, but not yet in contact with, the surface and
becomes repulsive if the tip continues to advance and penetrate the surface.
As these instruments improve their ability to test smaller
and smaller specimens, the forces and displacements involved
shrink accordingly, and accurate measurement of these quantities becomes increasingly difficult. Measurement of force in
this regime is often based on the deflection of some elastic
mechanical transducer, and accurate determination of force
requires measurement of both transducer deflection and transducer stiffness at the location where the force is being applied.
Many experiments that study forces at the molecular and
atomic scale use the deflection of a cantilever to measure that
force. Cantilevers, particularly those with low stiffness, can
be extremely sensitive force transducers, and it is not unusual
for cantilever-based transducers to have force sensitivity in
the piconewton range. However, both deflection and stiffness
must be measured with an accuracy equal to, or better than,
the desired accuracy in force, and displacements well below
As microelectromechanical systems and nanoelectromechanical systems become increasingly common in technology
applications, designers of these systems need reliable mechanical property data on the wider and wider range of materials
used in fabrication. Those mechanical property measurements
often must be made on structures and components that are
only tens of micrometers or smaller in size, either because the
material cannot be fabricated in larger sizes or, when larger
test specimens are available, tests on those larger specimens
will not accurately predict the mechanical performance of
micrometer-scale components. As a result, a range of test
instruments are now commercially available to measure the
mechanical properties of very small volumes of material. The
two most common instrument classes are low-force instrumented indenters [1] and atomic force microscopes (AFMs)
[2]. Both techniques are based on measuring the mechanical
interaction of a probe tip, which is part of the test instrument,
with the surface of a specimen, and both characterize that
interaction by measuring the displacement of the tip relative to
0957-0233/15/025202+8$33.00
1
© 2015 IOP Publishing Ltd Printed in the UK
D T Smith and J R Pratt
Meas. Sci. Technol. 26 (2015) 025202
1 nm must be measured, both to monitor the motion of the
probe tip relative to the specimen surface, and also to monitor
the deflection of the force sensor.
Interferometry techniques exist that achieve the requisite displacement resolution, but the most sensitive of those
use displacement modulation methods that do not provide
position or displacement information with the long-term stability necessary for quasistatic mechanical property measurement. Accurate force measurement is even more problematic.
Calibration traceable to the International System of Units (SI)
[3] using deadweight masses extends only down to a force of
5 µN, although for forces in the micronewton and nanonewton
range, techniques have been developed recently for realizing
force traceably using electrostatic methods involving the units
of capacitance, length and time [4–8]. Even when an accurate
realization of force in the micronewton range or below is available, however, using a calibrated force to determine a cantilever
stiffness is not always straightforward, because of issues related
to locating loading points and the fact that the contact between a
probe and a cantilever typically involves friction and other contact mechanics complications that can change the mechanical
boundary conditions of the cantilever and introduce additional
sources of uncertainty in the stiffness calibration [9–14].
In response to these issues surrounding force calibration at
the micronewton level and below, we consider the possibility
of identifying and using reproducible, intrinsic forces that
occur in nature as potential ‘reference forces’ for the calibration of very-low-force transducers. In this work, we present
an experimental observation of the force required to rupture a
gold–gold atomic bond in a single-atom chain (SAC) of gold
atoms in a break junction experiment [15] using a cantilever
whose stiffness was measured in situ using a non-contact
method based on photon momentum.
ferrule at the face closest to the cantilever, taking care not to
get glue on either the end of the fiber or the tip of the Au wire.
The glue wicks into the space between the two ferrule bores
and the Au wire and optical fiber to form a stiff mechanical
assembly, ensuring that the motion of the Au wire tip is accurately measured by the motion of the cleaved optical fiber end.
The fiber/wire/ferrule assembly is mounted to a nanopositioning stage [17] that is compatible with use both in vacuum
and at cryogenic temperatures. A second optical fiber is positioned behind the cantilever (the right side in figure 1). This
fiber is used both as part of a second interferometer system
to measure force on the break junction via cantilever deflection and also, by changing laser sources, for making an in situ
measurement of cantilever stiffness using photon momentum,
as described below. Both the wire tip and the rear optical
fiber are aligned with the longitudinal axis of the cantilever,
and wire tip and both optical fibers are positioned such that
they all are the same distance (approximately 7 mm) from the
clamped base of the cantilever. This positioning ensures that
cantilever and tip displacement will be measured at the same
distance from the cantilever base that forces between the Au
tip and cantilever are occurring, and the alignment of the tip
with the longitudinal axis of the cantilever prevents unwanted
torque on the cantilever. The two optical fibers are deliberately
offset laterally to prevent the possibility of (very weak) optical
cross-talk through the Au-coated glass cantilever.
The entire assembly in figure 1(b) is suspended from
springs in a vacuum-cryogenic chamber capable of operation
between 4 and 400 K. Braided straps of very-fine-gauge Cu
wire are connected between the suspended assembly and the
chamber’s main sample cooling platform, both to damp spring
motion and to connect the experiment thermally to the platform. The experimental assembly is thermally isolated from
the room-temperature walls of the vacuum chamber by two
layers of radiation shields. The chamber, which is supported
by a pneumatic vibration isolation system, is cooled using a
continuous flow of liquid He in a ‘split-flow’ configuration
that allows independent cooling and temperature control
of the experimental stage and the radiation shields. In performing an experiment, the vacuum chamber is first pumped
down to a pressure of ≈10−4 Pa (≈10−6 Torr), and then the radiation shields and experimental area are cooled, with care being
taken to ensure that the radiation shields always cooled faster,
so that any residual gas in the chamber condenses on them
rather than on the components of the experiment system. Once
temperatures in the experimental chamber approach 4 K, a gate
valve between the chamber and the vacuum pumping system
is closed and the pumps are turned off, to reduce mechanical
vibrations near the chamber. The entire experiment is housed
in a metrology laboratory with 0.1 K temperature control
located 12 m underground. This environment improves the
stability of the interferometer and ancillary electronic equipment and reduces the effects of building-born vibrations.
2. Experimental design
2.1. Basic break junction and force sensor design
The basic arrangement of the break junction instrumentation
is shown schematically in figures 1(a) and (b). At the heart
of the system is a vertically mounted, Au-coated glass cantilever approximately 2 mm wide, 8 mm long and 100 µm thick,
clamped at its base in an electrically insulating mount. The
Au coating, which is deposited on both sides of the cantilever,
is 100 nm thick, and a thin (10 nm) Cr layer is used between
the glass and Au to ensure good adhesion of the Au to the
cantilever. On the front side of the cantilever (the left side in
figure 1), a nanopositioning stage holds both a Au wire tip
and an optical fiber with a cleaved end facing the cantilever,
where it forms a Fabry–Perot (FP) interferometer cavity [16]
between the cleaved fiber end and the cantilever. The fiber and
Au wire are held parallel by feeding them through a doublebore glass ferrule. The Au tip and fiber end are positioned
such that when the Au tip touches the cantilever, the length
of the FP cavity formed between the cleaved fiber end and the
cantilever is approximately 100 µm. After setting the relative
positions of the Au wire and optical fiber, a very small amount
of cyanoacrylate glue is applied to the end of the double-bore
2.2. Fiber optic interferometer
The design and performance of the fiber optic interferometer
system has been described in detail elsewhere [18], but its basic
2
D T Smith and J R Pratt
Meas. Sci. Technol. 26 (2015) 025202
Figure 1. (a) Schematic representations of the entire Au break junction experimental design. Two complete fiber interferometer systems are
employed, one to measure the position of the Au probe tip relative to the cantilever surface, and the other to measure the deflection of the
cantilever, for the determination of force. The bias voltage source and current amplifier are located outside of the cryostat. (b) An enlarged
side view of the cantilever assembly. The cantilever is Au coated on both sides and held at its base with an electrically insulating clamp. The
entire assembly is suspended from damped springs in a vacuum chamber, and is cooled to 4 K via thermal anchoring to a continuous-flow
liquid helium cryostat.
features will be presented here. It is essentially a simple fiberoptics-based homodyne interferometer designed to measure
changes in length of a low-finesse Fabry–Perot cavity formed
between the cleaved end of an optical fiber and an opposing
parallel reflecting surface. The configuration is in general the
same as that described in Rugar et al [19], but with the emphasis
placed on maximizing long-term stability in low-frequency
and dc applications. The laser source is a wavelength-tunable
infrared (IR) laser with a center wavelength of 1550 nm and
a tuning range of approximately ±50 nm about 1550 nm. This
tuning capability is critical in our application; for maximum
sensitivity, the FP cavity must be operated near a quadrature,
or inflection, point of an interference fringe, either by adjusting
cavity length or laser wavelength. Although we set the approximate length of both FP cavities when assembling the instrument (cavity lengths are typically in the range 50–200 µm), we
cannot fine-tune the length of either cavity in situ during an
experiment, and therefore must tune the wavelength for each
interferometer so as to optimize sensitivity. This requires two
separate tunable laser sources for the experiment.
The interferometer system described above has previously
been applied successfully to the study of atomic-scale break
3
D T Smith and J R Pratt
Meas. Sci. Technol. 26 (2015) 025202
4 K, cantilever dimensions change, clamping conditions at the
base of the cantilever may change, and even the elastic moduli
of the cantilever materials change. We therefore make use of
features of the experimental configuration to measure directly
k in situ during an experiment.
One common method for in situ calibration of cantilever
stiffness that is particularly convenient for AFM cantilevers
is based on the equipartition theorem and involves measuring
the vibrational spectrum of a cantilever that results purely
from the thermal motion of the cantilever, independent of any
external vibrational driving forces [24, 25]. When employed
in an AFM system, this thermal method makes use of the
AFM’s cantilever deflection measurement method (whether it
be optical lever, interferometer, or another) to measure the frequencies and amplitudes of cantilever resonances (primarily
the fundamental, or lowest, resonance) that are being driven
purely by the thermal energy of the cantilever. We made these
measurements on our cantilever, the fundamental resonant
frequency of which was approximately 1.15 kHz at 4 K, but
the oscillation amplitude of the fundamental resonance was
significantly larger than expected from thermal energy, and
the resulting calculated stiffness was 3.1 N m−1, or more than
ten times lower than we knew the approximate stiffness to be
from simple calculations based on cantilever dimensions. This
error was almost certainly due to the fundamental resonance
being driven by other forces in addition to thermal energy, most
likely seismic vibration transferred from the floor through the
pneumatic support system and from acoustic room vibrations
coupled from the chamber walls through the mechanical support for the experimental platform in the chamber. In order
for thermal calibration methods to be accurate, it must be the
case that the only forces exciting the cantilever modes are
thermal fluctuations. In our system, this was not the case and
as a result, we were unable to use this method for in situ stiffness calibration.
However, the fact that we had two optical fibers directed
at the cantilever, both at the same distance out from the cantilever base as the location where the Au contact was being
made, gave us another method of in situ stiffness calibration,
based on photon momentum [26]. To implement this method,
we used the interferometer on the front (Au break junction)
side of the cantilever to measure its deflection, but temporarily
replaced the tunable IR laser system that comprised the rear
interferometer with a fiber-optic-coupled incoherent superluminescent diode (SLD) light source with center wavelength
1550 nm. The output power of this light source was measured
independently using an Agilent 81634B power sensor with a
stated total uncertainty of ±4.5%, and could be modulated by
an external voltage. This enabled us to apply an oscillating
force of known amplitude to the rear of the cantilever, through
change in momentum of reflected photons, while simultaneously measuring the amplitude of the resulting cantilever
oscillation. For this case, the optical force applied to the cantilever for normal photon incidence is simply:
junctions, although without the inclusion of a force sensor. In
that work [20], a Au tip and optical fiber were mounted in a configuration similar to the left side of arrangement in figure 1, but a
rigidly mounted Au flat was used in place of the cantilever. The
interferometer signal was used to measure the motion of the Au
tip relative to the Au flat, but also was used in a feedback-control
loop to the nanopositioner that canceled any drift in separation
of tip and flat, in what was termed a feedback-stabilized break
junction (FSBJ) configuration. Total noise in the interferometer
system corresponded to an equivalent noise in cavity length of
2 pm (measurement bandwidth 0.1 Hz to 1 kHz), and changes
in cavity length of 5 pm, initiated by changes in the setpoint
of the feedback control loop, were clearly visible. This same
arrangement is used in the current work to measure and control the position of the Au tip relative to the cantilever. Absolute
drift in the interferometer and feedback system was checked
by operating two completely independent interferometer systems simultaneously against a common Au flat. The use of one
interferometer to lock the position of the Au flat resulted in the
complete cancellation of any observed drift in the second, independent interferometer. In previous work, this level of stability
allowed the creation of individual Au SACs that could be maintained for 30 min or longer while electron transport through
them was studied in detail [20, 21].
3. Experimental results
3.1. In situ measurement of cantilever stiffness
In order to obtain accurate force data for the Au break junction, the stiffness of the cantilever at the location of the junction must be known. The selection of an optimal stiffness, or
spring constant, k, for the cantilever used in the break junction
experiment represents a trade-off between mechanical stability and force sensitivity. Force resolution is determined by
the resolution with which its displacement can be measured;
force sensitivity will suffer if an unnecessarily stiff cantilever
is used. However, in the presence of attractive forces between
the Au tip and the Au-coated cantilever, the cantilever must
be stiffer than the Au contact; otherwise, the system will be
mechanically unstable, with the tip snapping into contact on
approach when the gradient of the attractive interaction is
greater that the cantilever stiffness, and pulling off unstably
on retraction when the contact breaks under tensile force.
We estimated that a cantilever force sensor with a stiffness
in the approximate range k = 10 to 100 N m−1 would provide
the required mechanical stiffness without unnecessarily sacrificing sensitivity, and this range became our target stiffness for
the cantilever at the Au point of contact. This range was based
on experimental observations [22, 23] that the stiffness of a
Au SAC is expected to of the order of 1 N m−1.
To ensure that we know the correct value of k for the cantilever during an experiment, we align the Au tip and back FP
cavity as described above. We could measure k at the same distance from the base at room temperature before beginning an
experiment. However, we do not want to assume that that stiffness is correct when we are performing a Au break junction
experiment at 4 K. In cooling down from room temperature to
Fphoton = 2PR / c,
(1)
where P is the optical power, R is the reflection coefficient
of the Au-coated cantilever surface and c = 3.0 × 108 m s−1 is
4
D T Smith and J R Pratt
Meas. Sci. Technol. 26 (2015) 025202
3.5
3.0
25 Hz
Cantilever Displacement (pmrms)
2.5
2.0
1.5
50 Hz
1.0
100 Hz
0.5
0.0
200 Hz
0
50
100
150
200
Time (s)
Figure 2. The response of the cantilever to a photon driving force modulated at the frequencies shown and applied to the back side of the
cantilever via the same optical fiber used for the rear FP interferometer. These data were used to characterize photothermal effects resulting
from photon driving of the cantilever. Photothermal effects were observed to drop approximately as 1/frequency, but represented a small
source of error which had to be included in our photon momentum calibration of cantilever stiffness.
the speed of light. R for 1550 nm light incident on a smooth
Au surface is taken to be R ≈ 0.97 [27]. The assumption of
normal incidence neglects the fact that the beam exiting the
cleaved fiber end is diverging slightly, but for the approximately 100 µm spacing between the fiber and the cantilever,
we have estimated that the cosine error in momentum transfer
will be less than 1%.
The fact that R for our system is 0.97 and not unity means
that 3% of the optical power incident on the cantilever is being
absorbed and converted to heat. The SLD source was modulated such that the optical power being delivered to the cantilever varied sinusoidally between a minimum of 1.66 mW
and a maximum of 12.45 mW. This resulted in an equivalent
steady-state incident power of 7.06 mW, an incident rootmean-square (rms) ac oscillation amplitude of 3.81 mW, and
0.21 mW of heat delivered to the cantilever. This heating
resulted in detectable but unquantifiable changes in the
average cantilever deflection, as evidenced by drifts in the
average position of the cantilever, and hence average or quasistatic cantilever deflection was not used to calculate cantilever
stiffness. Instead, only the ac component of the optical drive,
Prms, is used to calculate stiffness: the applied rms ac optical
power, in units of mW, is related to the rms amplitude of the
ac displacement response, in pm, as measured by the front
interferometer system using a lock-in amplifier locked to the
SLD modulation signal.
Even when only the ac component of the optical driving
force is considered, at low frequencies the cantilever still displayed a frequency-dependent heating response, in addition to
quasistatic bending. However, this component of the thermal
response drops rapidly with increasing drive frequency as the
cantilever loses its ability to respond in phase to the thermal
oscillations. This behavior is shown in figure 2, where the rms
amplitude of the displacement oscillation, d, as measured by
the front interferometer, is seen to drop steeply as the modulation frequency is increased from 25 to 200 Hz. (At 400 Hz
and above, the drive frequency began to interact weakly with
the cantilever’s lowest resonance at 1.15 kHz, and d began to
rise.) The drop in the ac thermal response for frequencies up to
200 Hz allowed us to quantify and correct for it, and we were
able to determine that at 200 Hz, the cantilever response for an
incident ac optical power of 3.81 mW in the absence of an ac
thermal response was 0.57 pm. Using equation (1), this then
gives a cantilever stiffness, k:
k = Fphoton / d rms = 2PrmsR /(c ⋅ d rms) = 43 N m −1 ± 3 N m −1
(2)
at the break junction location at 4 K. Here, the uncertainty of
3 N m−1 is dominated by the uncertainty in the absolute accuracy of the power sensor, as stated by the manufacturer, as
noted above.
3.2. Au–Au bond rupture force
The electronic and mechanical properties of SACs have been
of great interest for many years [15], due in no small part to
the fact that they are one of the closest approximations to a
true 1D physical system that is experimentally accessible. The
5
D T Smith and J R Pratt
Meas. Sci. Technol. 26 (2015) 025202
Conductance Quantum, G0
quantization of ballistic electron transport through a variety
of SACs has been predicted theoretically [28–30], observed
and studied in detail experimentally [31], and calculated using
a variety of ab initio computational methods [32–35]. The
quantum of conductance, G0, is given by 2e2/h, where e is
the charge on the electron and h is Planck’s constant. This
conductance is 77.5 µS, which corresponds to a resistance of
12.9 kΩ. In addition, several experimental groups have used
AFM-based techniques to measure chain breaking forces [22,
36, 37], and a particularly thorough set of density-functionaltheory (DFT) calculations have predicted [34] that the bond
rupture force for a Au contact that has been drawn down to
a SAC has a remarkably consistent value of 1.54 ± 0.06 nN
at temperatures near 0 K, independent of the initial crystallographic orientation or tensile axis along which the SAC
is drawn. This remarkably narrow range of force makes the
rupture of a gold SAC a promising candidate for a reproducible intrinsic force calibration point in the nanonewton range
at low temperatures. Therefore, as one application of the
experimental system described in this manuscript, we use it
to measure the tensile force required to rupture a 1D chain of
Au atoms in a break junction, e.g. the force associated with
breaking a single Au–Au bond in a chain, at 4 K.
A typical measurement begins by bringing the Au tip into a
firm, low resistance contact with the flat Au-coated cantilever
surface. A constant 5.0 mV bias voltage is maintained across
the junction. This low value of bias voltage is used to keep the
energy of electrons passing through the junction low, thereby
maintaining transport that is predominantly ballistic by minimizing the probability of electron-phonon interactions. Motion
of the Au tip relative to the Au flat is measured using the FP
interferometer on the front side of the cantilever; that signal
can, if desired, be used to lock the tip position relative to the
flat under servo control, as described earlier in the discussion
of the FSBJ work. At the same time, the deflection of the cantilever, relative to the base where it is clamped, is measured by
the second FP, or the rear of the cantilever, and provides the
determination of the tip/flat interaction force using cantilever
deflection dc, and stiffness k. We then retract the Au tip, while
monitoring the current through the junction, until electrical
continuity across the junction is lost. This process is repeated
many times. Initially, we often find that junction does not form
a SAC with conductivity 1G0, but instead breaks abruptly
from a high-conductivity state (10G0 or more). In those cases,
a large-area contact has formed that is stiffer than the cantilever, and the system is mechanically unstable. However, by
repeatedly working or ‘training’ the junction [38], we routinely
are able to ‘sharpen’ the contact region such that we can draw
SACs with conductivity at, or just under, 1G0, and can then
measure the tensile force at which they break.
Typical data from such an experiment are shown in figure 3,
for three discrete breaking events. For each event, both the
conductivity of the junction, in units of G0, and the quasistatic
cantilever deflection, are shown as a function of time. In each
of these events, a high-conductivity junction (G > 3G0) was
formed and then drawn out until a long plateau at 1G0 was
observed, indicating the formation of a SAC that then broke.
At each break, a clear relaxation of the cantilever position is
4
a
3
2
1
0
0
5
0
5
10
15
10
15
Cantilever Position (pm)
120
100
80
60
40
20
0
Conductance Quantum, G0
Time (s)
4
b
3
2
1
0
0
2
4
0
2
4
6
8
10
12
6
8
10
12
Cantilever Position (pm)
200
150
100
50
0
Conductance Quantum, G0
Time (s)
4
c
3
2
1
Cantilever Position (pm)
0
0
5
0
5
10
15
10
15
100
50
0
Time (s)
Figure 3. Three typical observations (out of more than 30) of Au SACs
breaking from the 1G0 (single-atom-chain) quantized conduction state
at 4 K. Abrupt changes in cantilever position of 30–50 pm are observed
when the chain breaks. Cantilever stiffness is measured in situ
at 4 K at the location of the junction, and is found to be 43 N m−1.
6
D T Smith and J R Pratt
Meas. Sci. Technol. 26 (2015) 025202
observed. Not all junctions broke from a SAC configuration;
some breaks occurred when the conductivity was greater than
1G0. While these observations are interesting, it is assumed
that in those cases the observed breaking force resulted from
the rupture of more than one Au bond (and in fact the observed
breaking force was higher in those cases), and they are not
included in the present analysis.
For those cases where the junction broke following the creation of an extended 1G0 plateau like those shown in figure 3,
we observe an abrupt relaxation in the cantilever position of
41 ± 10 pm (one standard deviation in the experimentally
observed values) when the SAC breaks. Using the value of
k = 43 N m−1 for the cantilever stiffness at the contact point,
this implies a breaking force of 1.8 ± 0.4 nN, an average value
that is slightly higher than, but not inconsistent with, earlier
experimental values and DFT calculations.
Just after the SAC breaks, there may still be a residual
force interaction between the tip and cantilever, particularly
given that the cantilever position relaxes by only ≈50 pm or
less. We believe, however, that this possible interaction does
not affect the accuracy of our breaking force measurement, for
several reasons. First, if we continue to withdraw the tip after
the chain breaks, we typically do not see any further relaxation of the cantilever position. Second, when a chain breaks
after the drawing of a SAC, we believe that the resulting gap
is significantly larger than the 50 pm cantilever relaxation distance because the Au atoms that had been in the SAC collapse
back onto either the tip or flat. This belief is reinforced by our
observation that, after breaking a SAC, we typically have to
move the tip forward again, toward the cantilever, by at least
1 nm before we reestablish electrical contact. And finally, even
if we did have some long-range force interaction between the
tip and flat that we are not able to detect or quantify, it is reasonable to assume that the abrupt change in cantilever position when the SAC breaks still represents the chain breaking
force, since a long-range force would not be expected to vary
significantly as a result of a 50 pm cantilever movement.
movements of the Au tip at the front of the cantilever, but is
a limiting factor when measuring 40 pm changes in cantilever
deflection. Raising the finesse of the rear FP cavity by coating
the cleaved end of the optical fiber with reflective dielectric
multilayer coatings can raise the sensitivity of the rear FP
cavity dramatically. The trade-offs are that (1) cavity alignment must be more exacting as finesse goes up, and (2) the
total range of travel that can be measured is reduced, but neither of these represent significant obstacles in this application.
Second, correction for the thermal effects that result from
the Au coating absorbing 3% of the SLD power when performing the stiffness calibration limits the accuracy of that
calibration. Here again, multilayer dielectric coatings offer
a significant potential benefit. Dielectric coatings are commercially available that have reflectivity greater than 99.9%
at 1550 nm, significantly reducing the amount of heat being
pumped into cantilever during the stiffness calibration. A
next-generation instrument could use a cantilever with a Au
coating on the break junction side, and a high-reflectivity dielectric coating on the back.
Finally, it will be easy to replace the single fiber in the
single-bore ferrule behind the cantilever with two parallel
fibers in a double-bore ferrule like the one used on the front
side of the cantilever to hold a fiber and Au wire. It would then
be possible to use one rear fiber in a high-finesse FP interferometer while simultaneously driving the cantilever with a SLD
source. Not only would this facilitate the cantilever stiffness
calibration, and improve its precision, but it would also provide the possibility of using damping, or ‘feedback cooling’,
to reduce undesired vibrations of the cantilever through the
application of phase-shifted optical force feedback derived
from measured cantilever velocity.
Acknowledgments
The authors wish to thank Paul Wilkinson for useful discussions on optical power and photon momentum measurements,
and George Jones for assistance with laboratory instrumentation. Certain commercial equipment, instruments, or materials
are identified here to specify the experimental procedure adequately. Such identification does not imply recommendation
or endorsement by NIST, nor is it intended to imply that the
equipment, instruments or materials identified are necessarily
the best available for the purpose.
4. Discussion
We have developed an instrument and associated methodology for measuring the Au–Au bond breaking force that uses
photon momentum to enable in situ calibration of the force
sensor, and have used it to obtain a value for that breaking force
that is consistent both with previous experimental work and
DFT calculations. There are, however, several ways in which
both the accuracy and the precision of the technique could be
improved, and we plan to continue to make refinements.
First, the resolution with which cantilever deflection, and
hence force, can be measured can be improved by increasing
the finesse of the FP cavity that measures that deflection. In our
current system, both FP cavities are formed between a cleaved
glass fiber surface and a Au surface, and are essentially lowfinesse cavities, since the finesse of a FP cavity is primarily
determined by the surface with the lowest reflectivity (4% for
a glass-vacuum interface). The noise floor of our low-finesse
systems (≈2 pm) is adequate for measuring nanometer-scale
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