Casimir probe based upon metallized high Q SiN nanomembrane resonator

3.4 MB·English

by  Daniel Garcia-Sanchez

Casimir probe based upon metallized high Q SiN nanomembrane resonator
Daniel Garcia-Sanchez,1 King Yan Fong,1 Harish Bhaskaran,1 Steve Lamoreaux,2 and Hong X. Tang1
1)Department of Electrical Engineering, Yale University, New Haven, CT 06520
2)Department of Physics, Yale University, New Haven, CT 06520
(Dated: 30 January 2013)
We presentthe instrumentationand measurementscheme of a new Casimir force probe that bridges Casimir
force measurements at microscale and macroscale. A metallized high Q silicon nitride nanomembrane res-
onator is employed as a sensitive force probe. The high tensile stress present in the nanomembrane not only
3 enhancesthe qualityfactorbutalsomaintainshighflatnessoverlargeareaservingasthebottomelectrodein
1
a sphere-plane configuration. A fiber interferometer is used to readout the oscillation of the nanomembrane
0
and a phase-locked loop scheme is applied to track the change of the resonance frequency. Because of the
2
high quality factor of the nanomembrane and the high stability of the setup, a frequency resolution down to

n 2×10−9 andacorrespondingforcegradientresolutionof3 µN/mis achieved. Besidessensitivemeasurement
a
ofCasimirforce,ourmeasurementtechniquesimultaneouslyoffersKelvinprobemeasurementcapabilitythat
J
allows in situ imaging of the surface potentials.
9
2
PACS numbers: 85.85.+j, 07.10.Pz,42.50.Lc

]
l
l
a I. INTRODUCTION 1.2mm×1.2mm.
h
Here we describe the development of a new type of
-
s Casimir force probe that bridges the Casimir force mea-
The first experiment on Casimir force measurement
e surement at macroscopic scale and microscopic scale.
m was conducted in 1958 by Sparnnay1 with macroscopic
Results of the measurement of this force are presented
metal plates. In this experiment the alignment of two
. elsewhere10. In a sphere-plane geometry, the planar
t parallel plates posed great challenges. In 1997, the ac-
a plate, which serves as force transducer, is a high Q SiN
m curacy of the measurement was significantly improved
nanomembrane resonator. The sphere is a gold coated
by using a torsion pendulum balance2. The alignment
- glass sphere of millimeter size. This approach com-
d was simplified by replacing one of the plates by a sphere
bines the benefits of free space manipulability, large sur-
n characterizedby its radius of curvature R. The distance
face area and high force sensitivity of nanomembranes.
o was determined by measuring the capacitance between
c the two plates. In 2011Sushkov et al. used an improved The high force sensitivity allows precision measurement
[ of the shift of the resonance frequency of a nanomem-
version3 ofthetorsionpendulumbalancetodemonstrate
brane resonator. The shift is proportional to the gra-
1 thatthepermitivityiswelldescribedbytheDrudemodel
dient of the forces across the gap which could have
v in the range 0.7 to 7µm.
5 newtonian origin (such as electrostatic force) or non-
Moving towards smaller device sizes, in 1998 Mohin-
2 newtonian origin (such as the Casimir force). These
0 deen et al. measured the Casimir force also in a sphere- dynamic techniques8,11 have severaladvantages over the
7 plane geometry in the range of 0.1 to 0.9µm, but with static force measurement techniques which are very sus-
. an Atomic Force Microscope (AFM) as a sensitive force
1 ceptibletoenvironmentalinstabilitiessuchastheseismic
0 transducer4. An aluminium coated polystyrene sphere vibrationsandthe 1/f noise thatis inherentinthe mea-
3 was attached to an AFM cantilever. The force was mea- surement system.
1 sured between the sphere and an aluminium coated sap-
: phire plate by measuringthe deflection of the AFM can-
v
i tilever. UsingasimilarAFMapproachrepulsiveCasimir
X force wasdemonstratedina fluid medium by Munday et II. INSTRUMENTATION
r al.5.
a
A. Vacuum system
Microelectromechanical systems (MEMS) have also
been exploited to study the Casimir force. In one of
the first experiments with MEMS it was shown that a Figures 1(a) and 1(b) show a schematic illustration of
cantilever could irreversibly stick to an electrode due to the Casimir probe setup which is installed in a high vac-
the Casimir force6. A microelectromechanical torsion uum chamber of 75kg. To minimize the environmental
oscillator7 was used by Chan et al. to measure with impact, the chamber is placed on top of a granite table
high accuracy the Casimir force in the range of 0.1 to (1500kg) with pneumatic vibration isolation. Further, a
1µm and later to demonstrate non-linear oscillations in wood triangle of 2.5in in thickness is inserted between
themicromechanicaltorsionoscillatorduetotheCasimir thevacuumchamberandthegranitetabletoachieveop-
force8. In 2002 Bressi et al. measured the Casimir force timum damping of environmental vibrations. A turbo
betweenparallelplates9 in the range0.5to3µm. Inthis pump system is used to pre-pump the vacuum chamber.
experimentthe Casimirforcewasmeasuredinanareaof When the chamber pressure is below 10−6Torr, an ion
2
(a) (b)
Viewport
Nanomembrane
Gold
coated Optical
sphere Fiber
Closed-loop stage
Tilt stage
XYZ XYZ
Piezo XYZ Piezo Optical
Motors Piezo Motors fiber
XYZ feedthrough
(c) Closed-loop
stage
Sphere
holder
NNaannoo--
mmeemmbbrraannee
Ion Turbo
Pump Pump Optical
Fiber
FIG. 1. (a) Schematic of the isolation system, the vacuum chamber and the positioning system inside the vacuum chamber.
(b) Cut view of the chamber and the probe. (c) Isometric image of the sample mounting subsystem and fiber mounting
subsystem.
pump is turned on and the turbo pump is switched off B. Preparation of the spherical plate
anddetachedfromthevacuumsysteminordertoremove
the pump induced vibrations. The fiber feedthrough is
Figure 2 showsthe sphericalplate andits holder. The
constructed by modifying a Swagelok connector fitted
sphere is prepared by coating a fused silica with 200nm
with a teflon ferrule12. All electrical feedthroughs are
ofgoldusing10nmoftitaniumasadhesionlayerusingan
made ultra-high vacuum (UHV) compatible by Accu- electron beam physical vapor deposition. The sphere16
Glass Products, Inc. We finally leak-test the vacuum
hasaradiusR=4mm±2.5µm. Theholderofthesphere
system with a helium leak detector, and the leaking rate
is machinedin aluminiumandsputteredwith goldto re-
is below 10−9Torr.
duce the contact potential between the sphere and the
holder. To ensure good electrical contact between the
The positioning system inside the vacuum cham- sphere and the holder a polytetrafluoroethylene (PTFE)
ber consists of 13 motorized stages. The sphere, the stub is used to press the sphere against the holder, as
nanomembrane, and the fiber are separately mounted shown in Fig. 2(b). The PTFE piece is also used to
on three specifically designed holders, each of which is electrically isolate the sphere from the closed-loop piezo
positioned by an XYZ stage controlled by picomotors13 actuator and the vacuum chamber. Static charge might
(SeeFig.1(a)). Anadditional3-axispiezoelectricallyac- accumulate on the PTFE surface. But this is not a con-
tuated stage14 is mounted on the nanomembrane stage cern since the PTFE piece is fully electrically shielded
to achieve sample scanning and to fine adjust fiber-to- from the measurement surface.
membrane distances. The alignment of the fiber against
nanomembrane is adjusted by a manual tip-tilt stage.
The sphere is brought to approach the nanomenbrane
C. Nanomembrane fabrication
by a preloaded closed-loop piezo actuator with sub-
nanometerresolution(0.3nm)15. Allthestagesaremade
vacuumcompatibleandcarefullycleanedbeforeinstalla- In the sphere-plane configuration, the planar plate
tion. The movement of the stages can be carefully mon- used here is a nanomembrane fabricated from tensile-
itored using a telescope through the viewport at the top stressed silicon nitride which also serves as the force
of the vacuum chamber. During the measurements the transducer. Membrane resonators made of tensile-
viewportisfullycoveredwithmetalfoiltoavoidambient stressedsiliconnitrideareknowntoexhibitveryhighme-
light which is found to introduce noise in the measure- chanicalQ(intheorderof106atroomtemperature)17,18.
ments. The Q can be as high as 105 even after metallization19,
3
Gold coated (a) (b)
(a) (b)
sphere
(c) (d)
54.7º
Insulating PTFE part
Gold coated Al part (e) (f)
(d)
(c)
1 mm
1 mm
FIG. 3. Fabrication process of the nanomembrane. (a) Bare
Si wafer. (b) Deposition of 330nm LPCVD silicon nitride.
FIG. 2. Casimir sphere and its holder (a) Side-view photo- (c) CHF /O RIE of silicon nitride (d) Anisotropic KOH
3 2
graph. (b)Cross-sectionalviewofthedrawingassembly. (c) etching of silicon. (e) Ebeam evaporation of Au/Ti layer.
Perspective view. (d) Isometric view of thedrawing. (f) Optical image of thenanomembrane.
which is expected to introduce extra material loss and at a rate of 1 ˚A/s. Fig. 3(f) shows the typical optical
clamping loss. From a transducer point of view, high image of a gold coated nanomembrane. Due to the net
Q implies low thermomechanical noise and so high force tensile stress in the film, the sample remains flat across
sensitivity. Here,wemakeuseofthisadvantageousprop- the whole nanomembrane surface with RMS roughness
erty of the high Q nanomembrane to achieve precision of 3nm (measured by AFM). Figure 3 summarizes the
measurement of Casimir force. nanomembrane fabrication process.
Fabricationprocess of the nanomembrane is described
as follows. We startwith a 400µmthick, (100)-oriented,
double side polished siliconwafer (dopedp-type withre- D. Fiber interferometer and its calibration
sistivity quoted to be 10 - 20 Ωcm). A layer of 330 nm
silicon nitride is grownon both sides of the wafer by low We first describe the fiber interferometer employed to
pressure chemical vapor deposition (LPCVD). The sili- measure the resonance of the nanomembrane. The fiber
connitride Si N usedhereis notstoichiometricbuthas is cleaved after it has been introduced into the chamber
x y
a silicon-to-nitrogen ratio x : y larger than 3 : 4. When to produce a clean cleavedsurface as shownin Fig. 4(a).
deposited on a silicon substrate, the film has a built-in Figure 4(b) displays the schematic of the fiber measure-
tensile stress of 450 MPa, which is relatively lower than ment scheme. A diode laser20 (λ = 1310nm) with very
thatofastoichiomericsiliconnitride. Alayerofphotore- lowfrequencynoiseis usedasalightsource. The output
sist(Shipley S1813)wasspunonboth sidesofthe wafer. powerofthelaserisstabilizedwithacurrentandtemper-
The photoresist on the front side serves as a protective aturecontroller21andissetto9.7mW(operatingcurrent
layer. Square openings were defined in the photoresist 40mA)whichcorrespondstothesetpointwiththelowest
on the backside of the wafer using photolithography and frequency noise. Then the light is attenuated to 500µW
the silicon nitride layer was etched by reactive ion etch- and sent to the nanomembrane through a circulator22.
ing (RIE) with a mixture of CHF and O gases. After The light reflected from the nanomembrane returns to
3 2
stripping off the photoresist, an anisotropic wet etch of the circulator and is collected with a photodetector.
◦
potassium hydroxide (KOH) solution (30%, 85 C for 7 When the distance between the fiber and the
hours)wasusedtoetchthroughtheSisubstrate. During nanomembrane is changed an interference pattern is
theKOHetchingprocessaProTEK-B3(BrewerScience) obtained (see Fig. 5). In order to improve the ex-
coating was used to protect the silicon nitride layer on tinction, the fiber axis is fine adjusted by a tip-tilt
the top side of the wafer in order to ensure the high stage (see Fig. 4(b)). The maximum extinction is ob-
quality of the membrane and increase the yield. After tained when the fiber is perpendicular to the surface of
the KOH etch, the coating was removed in a bath of the nanomembrane. We record the photodetector signal
◦
N-Methyl-2-pyrrolidone (NMP) at 80 C and a subse- as the nanomembrane position is varied. The maximum
quent O plasma ashing. Finally 10nm of titanium and sensitivity is obtained where the slope is maximal in the
2
200nmofgoldwereevaporatedontopofthe membrane, interference pattern (see Fig. 5), at which point a small
4
(a)
V) 0.6
e (
g
a
olt
v
or 0.4
ct
e
et
d
o
ot
Ph 0.2
(b)
Gold
0.0
Sphere 0.0 0.2 0.4
Piezo voltage (V)
V
Au FIG. 5. Interferencepattern.
SiN
Si
High Vacuum each distance by sweeping the wavelength using a tun-
Chamber able laser23, and recording the maxima and minima of
Piezo
actuator the interferometer signal. The distance is given by
Tip-tilt
Stage −1
N 1 1
Fiber d= − (2)
2 λ λ
Interferometer (cid:18) i f(cid:19)
where d is the distance between the fiber and the
Phase-locked loop nanomembrane, N the number of peaks in the interfer-
Circulator
ence pattern in the wavelength regime, λ the starting
i
wavelength and λ the final wavelength.
f
PD
In order to collect the maximum amount of light re-
flected by the nanomembrane the fiber has to be very
close to the nanomembrane because of the relative small
(b)
Current source and numerical aperture of the single mode fiber. With fiber-
Laser Attenuator
Temperature controller membrane distance monitored in real time, the fiber
is brought to the closest yet safe distance from the
nanomembranebyoperatingthepicomotors. Duringthe
approachthe fiber is aligned severaltimes with a tip-tilt
FIG. 4. (a) Surface of the cleaved fiber. (b) Schematic dia- stage24 (see Fig. 4(b)). When the fiber is aligned the
gram of the fiber-optic interferometer and the detection and extinction of the interference pattern is maximized. By
actuation scheme.
doing this, we improve the sensitivity of the interferom-
eter and are able to approach to a very close distance.
We then move the nanocube Z stage to fine tune the
changein distance ∆d results in a changeof photodetec-
fiber-membrane distance so that maximum sensitivity is
tor signal ∆V, which is related to a change by
achieved in our fiber interferometer.
λ
∆d= ∆V (1)
Vp−p4π
F. Setup stability evaluation
whereVp−pisthepeak-to-peakvoltageoftheinterference
pattern and λ the wavelength of the laser. In order to minimize the mechanical drift of the fiber
interferometer and the apparatus the room temperature
◦
is keptat20.0±0.1 C. The bigmass ofthe vacuumsys-
E. Coarse approach betwen the fiber and the tem further filters out the variations of the temperature
nanomembrane oftheroom. Wecharacterizethestabilityofthesetupby
monitoringtheinterferometersignalovernight. Asshown
When the fiber is brought to approach the nanomem- inFig.6themeasureddriftofthefiber-to-membranedis-
brane,the distance betweenthe fiber andthe nanomem- tanceis about0.26nm/minwhichisprobablydueto the
brane is measured in real time to avoid physical contact thermal expansion of the apparatus and stages. As ex-
and breaking of the nanomembrane. We achieve this at plained in the next section one single measurement (a
5
parabola fitting) takes about 800ms which corresponds a b
toadriftof3pmwhichisnegligiblewhencomparedwith
other sources of error, such as the determination of the
distanceandthemeasurementoftheresonancefrequency
of the nanomembrane.
c d
0,040
m)
e (
c
an0,035
st
di
e f
0,030
Q
0 10 20 30 40 50
time (min)
FIG. 6. Interferometer outputfor 45 min record.
FIG. 7. (a) Optical image of a nanomembrane with a 36%
of the surface coverered by gold. (b) Optical image of a
G. Nanomembrane characterization nanomembrane with a 12% of the surface coverered by gold.
(c) Frequency response of a nanomembrane not covered by
We measured the frequency response of the SiN gold gold. (d) Frequency response of a nanomembrane fully cov-
ered by a layer of gold. (e) Resonance frequency of the
coatednanomembranes andstudied how the quality fac-
nanomembrane vs the amount of the surface covered by a
tor is modified when they are covered with gold. The
layerofgold. (f) Qualityfactorofthenanomembranevsthe
nanomembraneisexcitedwithanoscillatingvoltageV
AC percentage of gold coverage.
whichisappliedtothepiezoactuatorandtheresponseis
measured with the fiber interferometer. The amplitude-
frequency response of the nano-membrane was obtained
by measuring the oscillation amplitude of the nano- inamplifierthathasabuilt-inPLLcapability25. We use
membraneandsweepingthedrivingfrequencyexcitation. thefrequencyshiftofthenanomembranetocalculatethe
Thefrequencyresponsewasmeasuredinasetofsamples Casimir and the electrostatic forces (see section IIIA).
wheretheamountofsurfacecoveredbygoldchanges(see The temperature drift of the nano-membrane can also
Figs.7(a)and7(b)). Itgoesfromfullycoveredbyalayer shift the resonance frequency. As mentioned before the
of 200nm of gold to not covered at all. Fig. 7(c) shows temperature of the room is stabilized to 20.0 ± 0.1◦C
the response curve ofa nanomembranewhichis not cov- and also the big mass of the chamber stabilizes the tem-
eredby goldand Fig. 7(d) shows the response curve of a perature of the nanomembrane. The ambient light can
nanomembranewhichisfullycoveredbyalayerof200nm also substantially heat up the nanomembrane and shift
of gold. As we increase the amount of surface coverage the resonance frequency, therefore all the viewports of
theresonancefrequencydecreasesbecausethetotalmass the chamber are blinded. When ambient light is allowed
ofthenanomembraneincreases,thisisshowninFig.7(e). to enter into the chamber the frequency drift is about
When the nanomembrane is covered with some gold the 1Hz/min, and when the viewports are blinded the fre-
quality factor is reduced, but we have not observed a quency drift is 0.26mHz/min (see Fig. 8(a)). The fre-
deterministic trend of the quality factor with regards to quency drift is constant during a period of time much
thegoldcoverageasshowninFig.7(f). Hencewechoose longerthanatypicalmeasurement,whichisshorterthan
fully covered membrane in our following measurements. one second. Before any measurementthe frequency drift
is calibrated and the measurements are corrected ac-
cording to this calibration. By optimizing all the rel-
H. Frequency drift and resolution evant conditions, we achieve a frequency resolution of
∆f/f = 2·10−9 which allows to measure a force gra-
0
Thefrequencyofthenanomembraneistrackedusinga dient of 3µN/m (see Fig. 8(b)). As shown in the Allan
phase-lockedloop(PLL)scheme. Theoutputofthepho- deviation the drift becomes relevant above a measure-
todiode is comparedwitha referencesignalusing alock- ment time of 1s which is smaller than the measurement
6
a
z¨+2γz˙+ω2 (z−z )= ωm2 hFu,udiF (z )eiωdt
m 0 k d 0
m
+ωm2 hFu,udiF′(z )(z−z )
0 0
k
m
+ωm2 hFu,udiF′′(z)(z−z )2
0
k 2
m
+ωm2 hFu,udiF′′′(z)(z−z )3
0
k 6
m
(3)
b
where ω =2πf is the fundamental angular frequency
m m
of the nanomembrane, ω the angular driving frequency,
d
γ is the damping coefficient, k the stiffness of the
m
nanomembraneandud the eigenvectorofthe firsteigen-
mode. The product hFu,udi accounts for the overlap
between the force and the eigenmode. We can then de-
fine the effective stiffness as:
k
m
k = (4)
eff hFu,udi
FIG. 8. (a) Frequency drift with blind viewports (blue) and In the case of the first eigenmode the overlap between
withtransparentviewports(green). Theinsetshowsazoom- the force and the eigenmode will be maximum when the
in of the data. (b) Left axis shows the Allan variance vs
sphere is centered on top of the nanomembrane,and the
the integration time for several drive voltages: 500µV (red),
stiffness will be minimum from the point of view of the
10mV (black), and 50mV (blue). The right axis shows the
external force.
equivalent gradient force sensitivity.
The amplitude of motion of the resonator is set to be
in the linear regime but there is some contribution from
high order derivatives of the external force. If we con-
sider a small non-linearity the resonance frequency can
time of one parabola. be approximated by
f A2
∆f =− m F′(z )+ rmsF′′′(z )
0 0
2k 4
eff (cid:18) (cid:19)
f
m ′
=− F (z ) (5)
2k a 0
eff
′
where F is the apparent force and A the RMS am-
III. MEASUREMENT SCHEME a rms
plitude of motion of the resonator. The contribution of
the third derivative of the force to the frequency shift
can be considered as a correction and is equivalent to
A. Principle
the corrections for the roughness and the fluctuations in
plate separation that have been considered in previous
experiments3,26.
To measure the Casimir force we use the resonance
frequencyofthe nanomembraneasatransducerwhichis
verysensitivetotheforcegradient. Whenthenanomem- B. Measurement procedure
braneapproachesthesphere,itexperiencesaforcewhich
has contributions from the Casimir Force as well as the
When a difference of voltage V is applied between the
electrostatic force. This force is distributed over the
sphere and the nanomembrane an electrostatic force ap-
whole nanomembrane and the force per unit of surface
pears and is given by
can be expressed as Fe = FeFu, where Fe represents
the magnitude of the force and Fu the normalized spa- (V −V )2
m
tial distribution of the force. Now if we consider the Fe =−πǫ0R (6)
z−z
0
first3termsoftheTaylorexpansionoftheexternalforce
F(z)aboutthedistancez ,theequationofmotionofthe where ǫ is the vacuum permittivity, R the radius of the
0 0
nanomembrane becomes sphere, z−z the absolute distance between the sphere
0
7
and the nanomembrane, z the relative distance which
is set with the closed-loop piezo actuator and V the
m
residual contact potential.
AsshowninFigs.9(a)and9(b),thecontributionofthe
electrostaticforcetothefrequencyshiftcanbefittedtoa
parabola characterized by a curvature K . Therefore at
p
eachdistancebetweenthesphereandthenanomembrane
it can be found that the frequency shift is given by27
f2 =f (z)2+K (z)(V −V (z))2
0 p m
ǫ πRf2
=f (z)2− 0 m (V −V (z))2 (7)
0 k (z−z )2 m
eff 0
InthisequationK (z)(V−V (z))2 accountsfortheelec-
p m
trostatic force that can be nulled by applying a voltage
between the sphere and the nanomembrane. Note that
V is not necessarily constant with the relative position
m
between the membrane and the sphere and it has been
FIG. 9. (a) Parabolas at different target distances: 9µm
shownthatsuchavariationofV cancauseanadditional
m (blue), 4µm (red), 1µm (green), 120nm (black). (b) Fre-
electrostaticforce10,28. f2 accountsforthe Casimirforce
0 quency vs Voltage at a fix distance. The data can be nicely
and the electrostatic force that can not be cancelled. To fitted to a parabola. In this measurement χ2 = 1.05. (c)
calculatethemagnitudesKp(z),f0(z)andVm(z)thedis- ExtractedKp (curvatureoftheparabola). (d)Extrated∆f0
tance z−z0 between the sphere and the nanomembrane whichhascontributionsfromtheCasimirforceandtheresid-
is fixed and three different voltages V , V and V are ual electrostatic force10. Inset, the contact potential V vs
1 2 3 m
applied between the sphere and the nanomembrane. For thedistance with thesphere.
each voltage a frequency shift is measured: f , f and
1 2
f . With these measurements we can calculate the de-
3
siredmagnitudesandtheirdependencewiththedistance
force has the origin in the interplay between the surface
z.
patches and the curvature of the sphere.
1f2(V2−V2)+f2(V2−V2)+f2(V2−V2)
V = 3 2 1 2 1 3 1 3 2
m 2 f2(V −V )+f2(V −V )+f2(V −V )
3 2 1 2 1 3 1 3 2 f dF (z) dFel (z)
(8) ∆f0 =− m c + res (11)
2k dz dz
eff (cid:18) (cid:19)
f2(V −V )2−f2(V −V )2
f = 2 1 m 1 2 m (9)
0 s (V1−Vm)2−(V2−Vm)2 where F is the Casimir force and Fel (z) is the resid-
c res
f2−f2 ual electrostatic force10,28. To reduce the error intro-
K = 1 0 (10)
c (V1−Vm)2 ducedby the driftofthe frequencyin ∆f0, twoparabola
measurements are done, one at a very far distance and
In order to compensate for the drift in frequency of another one at the target distance d . This gives as a
0
the nanomembrane, the measured quantities f1, f2 and result f0f and f0c respectively. ∆f0 is equal to the sub-
f3 havebeencorrected. Byfitting Kp(z)totheexpected traction between these two frequencies. This procedure
model29 the quantities keff and zoff can be obtained. is repeated several times to improve the signal to noise
Withthesequantitieswecanhaveanabsolutecalibration ratio of ∆f at d . The value ∆f is the average of all
0 0 0
of the distance. Fig. 9(c) shows a typical measurement the ∆f measured at that particular distance. Fig. 9(d)
0
of Kp(z) and the fit with χ2 of 1.04. The fit gives as a shows a typical measurement of ∆f0(z), and the inset
result an effective mass of 7.5µg. The expected theoret- shows the contact potential V vs the distance with the
m
ical effective mass for this particular nanomembrane is sphere. Note that Fig. 9c and Fig. 9d are based on dif-
1.2µg. In the measurement of Fig. 9(c) the nanomem- ferent sets of data. The calibration curve of Fig. 9c is
braneappearstobeheavierbecauseinthismeasurement obtained using the displacement data of the closed loop
the sphere is not centered on top of the nanomembrane. piezo actuator. To be insensitive to the drift of the sys-
V is voltage that has to be applied to minimize the temthisdatahasbeentakenquitefast. Oncethesystem
m
electrostatic contribution to the force. This voltage may is calibrated we can measure ∆f and V by doing rep-
m
not be constant with the distance between the sphere etitions of the same measurement in order to improve
andthenanomembrane28,30. Thequantityf hascontri- the accuracy(see Fig. 9d). In this case the displacement
0
butions from the Casimir force but also from the resid- dataofthepiezoactuatorisdiscardedandthedistanceis
ual electrostatic force that can not be completely can- calculated using the measured K . This method is very
p
celedevenby setting the externalvoltageV to V . This accurate and insensitive to the mechanical drift.
m
8
C. Surface potential measurements mechanical mode of the nano-membrane and the distri-
bution of the electrostatic force (see equation (4)). The
ScanningKelvinprobeprincipleisaverypowerfulnon- effective stiffness keff is minimum, when the overlap be-
invasivetechnique31–33thatallowstomeasurethesurface tween the mechanical eigenmode and the electrostatic
contact potential ∆V between a conducting tip and a force is maximum, that is when the sphere is centered
sample. The surface contact potential ∆V is related to over the nanomembrane. When the sphere gets farther
the difference of the work function between the sphere from the center of the nanomembrane, the overlap be-
and the nanomembrane tweenthe eigenmodeandthe electrostaticforce becomes
smaller and the effective stiffness betcomes bigger. The
W −W
∆V =− 2 1 (12) spatial dependence of the effective stiffness can be used
e to center the sphere over the top of the nanomembrane.
where e is the elementary charge and W and W are Figure10(a)showsthecontourmappingofthemeasured
1 2
the work functions of the tip and the sample respec- stiffness at the center of a nano-membrane.
tively. The work function and as a consequence the sur-
face contact potential can vary along the surface due to
oxides films or adsorbed chemicals on the surface and IV. CONCLUSION
produce the patch potentials10,28. Our setup allows for
in-situ measurements of the surface potentials on the We havedesigneda Casimirforceprobethatallowsin
nanomembrane. The sphere can be moved parallel to situ measurements of the contact potential and bridges
the nanomembrane using the XYZ Piezo System in or-
the measurement of the Casimir force at microscale and
der to image the surface potential at a fix distance be-
macroscale. The Casimir force measurement is based on
tween the sphere and the nano-membrane. The move-
asphere-planegeometrywherethe planarplate isahigh
ment of the axes X and Y from the nanocube is not
Q silicon nitride nanomembrane coated with gold. The
completelyparalleltothesphere,thereforewhenwemove
probecanmeasureaforcegradientdownto3µN/m. We
the nanomembrane using the axis X and Y the distance
havealso studied the dependence of the quality factor of
between the sphere and the nanomembrane changes. In
the nanomembrane with the amount of coated gold and
ordertomeasurethecontactpotentialatafixdistancez,
we have observed that the SiN nanomembranes coated
acalibrationcurveistakenateachpoint(x,y). Thenus- with goldcan retaina reasonablyhigh quality factor. In
ingthecalibrationcurvethesphereisplacedatadistance situ scannedKevinprobeallowsdirectassessmentofcon-
z from the nanomembrane and the contact potential is
tact potentials and their distributions, which are found
measured. Fig.10(b)showsthemeasurementofthecon-
to contribute significantly to the measured position de-
tactpotentialofasamplewherethevariationsofthecon-
pendence forces in most samples we have tested.
tact potentials are large. These contact potentials play
an an important role in the precise measurement of the
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