Acknowledgements
description
Transcript of Acknowledgements
Precision Measurement of the Casimir Force For Au using a Dynamic AFM
U. MohideenUniversity of California-Riverside
AcknowledgementsExperimentC.C. ChangA.B. Banishev
R. Castillo
Theoretical ComparisonV.M. Mostepanenko
G.L. Klimchitskaya
Research Funded by: DARPA, National Science Foundation & US Department of Energy
Outline
• Measure Casimir Force for Au Sphere and Au Plate• Method (Dynamic measurement)• Force Gradient Determination• Errors• Data analysis• Results
Average Casimir Force from 30 scans
50 100 150 200 250 300 350-0.5
-0.4
-0.3
-0.2
-0.1
0.0
Experiment
Theory
Cas
imir
fo
rce
(10
-9N
)
Plate-sphere surface separation (nm)
Harris et al., Phys Rev. A, 62, 052109 (2000)
Why Need Another One?Understand the role of free carrier relaxation
Decca et al., Euro Phys J. C 51, 963 (2007)Sushkov et al. Nat Phys 7, 230 (2011) Chan et al., Science, 291, 1941 (2001)Jourdan et al., Europhys. Lett, 85,31001 (2009)
Lifshitz Formula
lk
l
lk
li
kTEk
llk
l
kll
kl
ik
TM qK
qKKr
Kq
KqKr
)(
)()(
)()(
)()()( ),(,),(
}]1)([]1)({[)1(2)(2 12,
2
0 0
12,
202
1
zqlTE
l
zqlTMllB
Cas ll eKreKrdKKqTRKzRPz
F
2/12)()(2/12 ])([,)( 2
2
2
2
KiKKqcl
kklcl
ll
R
z
R>>z
2
1
TlkB
l
2 At l=0, =0xMatsubara Freqs.
Reflection Coeffs:
Puzzles in Application of Lifshitz Formula
For two metals and for large z (or high T), x=0 term dominates
For ideal metals put e ∞ first and l, =x 0 next (Schwinger Prescription)
1),0(),0( )()(// KrKr kk
lk
l
lk
li
kk
llk
l
kll
kl
ik
qK
qKKr
Kq
KqKr
)(
)()(
)()(
)()()(
// ),(,),(
Milton, DeRaad and Schwinger, Ann. Phys. (1978)
Recover ideal metal Casimir Result
For Real Metals if use Drude
and g is the relaxation parameter
For x=0, , only half the contribution even at z≈100 mm, where it should approach ideal behavior
Get large thermal correction for short separation distances z~100 nm
Biggest problem: Entropy S≠0 as T0 (Third Law violation) for perfect lattice where g (T=0)=0 If there are impurities g (T=0) ≠0 , Entropy S=0 as T0
Bostrom & Serenelius, PRL (2000); Physica (2004)Geyer, Klimchitskaya & Mostepanenko, PR A (2003)Hoye, Brevik, Aarseth & Milton PRE (2003); (2005)Svetovoy & Lokhanin , IJMP (2003)Paris Group, Florence Group, Oklahoma group
][1)(
)(
2
e
pi
2/1
*
2
)(m
nep
0),0(,1),0( )()(// KrKr kk
Decca et al., Euro Phys J. C 51, 963 (2007)Sushkov et al. Nat Phys 7, 230 (2011)
Experimental Results
Plasma ModelDrude Model
Requirements for high precision Casimir force measurement
1) High force sensitivity system.
2) Very clean sample surface.
3) Precise, independent, and reproducible measurement of separation between two sample surfaces.
New Experimental Methodology
• Dynamic AFM• Measure Frequency Shift instead of Cantilever Deflection
Dynamic AFM Method UsedCantilever small oscillations in a force field
For small cantilever oscillations, we can take Taylor expansion of Fint at the mean equilibrium position
2
Sig
nal
frequency (Hz)
1
Band-passfilter
DC+AC
Low-passFilter
FM techniquePhase detector
(PhaseLockedLoop) Separation “d”
PID control in Q point
∆fDrive
Piezo1Piezo2
Interferometer 2 (Short coherence length)interferometer 1
Vacuum
High voltage power supplylinear voltage applied on
Piezo-tube repeatedly
d
∆V0 50 100 150 200 250 300
-0.07
-0.06
-0.05
-0.04
fringe piezocal1550 fit of sample_fringe
Applied voltage
inte
rfere
nce
signa
l (v)
0 100 200 300 400 500 600 700 800 900
-30
-20
-10
0
v1 v2 v3 v4 v5
fre
qu
en
cy s
hift
(Hz)
Z(nm) sample piezo movement (relative)
Interference signal
AC
DC10-8 Torr
0 100 200 300 400 500 600 700
-15
-10
-5
0
Au sphere - Au plate separations (nm)
fre
qu
en
cy
sh
ift
(Hz)
0.0326 V 0.0126 V 0.0026 V -0.0074 V -0.0174 V -0.0274 V -0.0374 V -0.0474 V -0.0574 V -0.0674 V -0.0874 V
-0.08 -0.04 0.00 0.04-14
-13
-12
-11
-10
-9
-8
-7
sig
na
l (H
z)
applied voltages (V)
-0.08 -0.04 0.00 0.04-4.5
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
applied voltages (V)
sig
na
l (H
z)
Determination of Au Sphere- Plate Potential Difference
Electrostatic Force Formula:
casimirFeleFF int
)(2
int0
z
F
kc
)1(cosh
)coth(cothcsc)(2
01
1
200
R
zz
nnhnVVFn
ele
STEPS1. Repeat Experiment for 12 Voltages applied to Au plate – not sequentially2. Correct separation for plate or sphere drift3. Use Parabolic dependence of force gradient on Voltage, to draw parabolas at every separation4. Vertex of Parabola, which denotes zero electrostatic force gives the residual potential
If Experiment Repeated for Same Applied Voltage to the Plate, Change in Signal is due to Drift
Correcting for Drift in Sphere-Plate Separation During Experiment- Method
separation (nm)
Sphere-Plate Separation Change in time of one Repitition
Freq
uenc
y Sh
ift
15 points
10 curves at V0
0200 sec
100 sec
separation (nm)
Sepa
ratio
n (n
m)
time (sec)
-2 -1 0 1
-6
-4
-2
freq
uen
cy s
hif
t (H
z)
Sphere-plate separation (nm)
0 200 400 600 8000.0
0.5
1.0
1.5
s
ep
ara
tio
n (
nm
)
time (sec)
Drift
<drift>=0.002 nm/sec
If Experiment Repeated for Same Applied Voltage to the Plate, Change in Signal is due to Drift
Correcting for Drift in Sphere-Plate Separation During Experiment- Data
0 100 200 300 400 500 600 700
-15
-10
-5
0
Au sphere - Au plate separations (nm)
fre
qu
en
cy
sh
ift
(Hz)
0.0326 V 0.0126 V 0.0026 V -0.0074 V -0.0174 V -0.0274 V -0.0374 V -0.0474 V -0.0574 V -0.0674 V -0.0874 V
-0.08 -0.04 0.00 0.04-14
-13
-12
-11
-10
-9
-8
-7
sig
na
l (H
z)
applied voltages (V)
-0.08 -0.04 0.00 0.04-4.5
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
applied voltages (V)
sig
na
l (H
z)
Determination of Au Sphere- Plate Potential Difference
Electrostatic Force Formula:
casimirFeleFF int
)(2
int0
z
F
kc
)1(cosh
)coth(cothcsc)(2
01
1
200
R
zz
nnhnVVFn
ele
STEPS1. Repeat Experiment for 12 Voltages applied to Au plate – not sequentially2. Correct separation for plate or sphere drift3. Use Parabolic dependence of force gradient on Voltage, to draw parabolas at every separation4. Vertex of Parabola, which denotes zero electrostatic force gives the residual potential
Overbeek et.al1971
Stability Checks Residual Potential Vo
300 400 500 600 700
-0.049
-0.042
-0.035
-0.028
-0.021
-0.014
Au sphere - Au plate separations (nm)
Res
idu
al E
lect
rost
atic
Po
ten
tial
(V
)
<V0>=-0.02750.003 V
Residual Potential Independent of Sphere-Plate Separation
No Anamalous Electrostatic Behavior
0 100 200 300 400 500 600 700
195
196
197
198
Au sphere - Au plate separations (nm)
Ave
rag
e S
epar
atio
n o
n S
ph
ere-
Pla
te C
on
tact
(n
m)
0 100 200 300 400 500 600 700
0.0114
0.0120
0.0126
Au sphere - Au plate separations (nm)
Can
tiliv
er S
pri
ng
Co
nst
ant
(N/m
)
Determination of Absolute Sphere-Plate Separation & Spring Constant
0 100 200 300 400 500 600 700
-2000
-1500
-1000
-500
0
Experiment Theory
Par
abo
la C
urv
atu
re
Au sphere - Au plate separations (nm)
<z0>=196.10. 4 nm <k>=0.012060.00005 N/m
Fit Parabola Curvature To Electrostatic Theory
Raw Experimental Data: Electrostatic+ Casimir Force
0 100 200 300 400 500 600 700
-15
-10
-5
0
Relative Au sphere - Au plate separations (nm)
fre
qu
en
cy
sh
ift
(Hz)
)]()[(2
0
int
)int
(2
zcasimirF
zeleF
k
casimirFeleFF
z
F
k
300 400 500 600 700
0.00
0.02
0.04
0.06
0.08
Ca
sim
ir F
orc
e G
rad
ien
t (m
N/M
)
Au sphere - Au plate separations (nm)
Complete Dataset – 12 applied Voltages to the Plate
)(0
2)(
zeleFk
zcasimirF
Subtract Electrostatic Force Gradient from Frequency Shift
Mean
300 400 500 600 700
0.00
0.02
0.04
0.06
0.08
Cas
imir
Fo
rce
Gra
die
nt
(mN
/M)
Au sphere - Au plate separations (nm)
300 400 500 600 700
0.00
0.02
0.04
0.06
0.08
Au sphere - Au plate separations (nm)
Ca
sim
ir F
orc
e G
rad
ien
t (m
N/M
)
300 400 500 600 700
0.00
0.02
0.04
0.06
0.08
Au sphere - Au plate separations (nm)
Ca
sim
ir F
orc
e G
rad
ien
t (m
N/M
)
300 400 500 600 700
0.00
0.02
0.04
0.06
0.08
Au sphere - Au plate separations (nm)
Ca
sim
ir F
orc
e G
rad
ien
t (m
N/M
)
Repeat ExperimentTotal of 4 x 12=48 experiments
Dataset 1Dataset 2
Dataset 3 Dataset 4
200 400 600 800 1000 1200 1400 1600
0.0
0.5
1.0
1.5
2.0
2.5
3.0
P(m
Pa
)
z (nm)
systematic error random error
Error Bars with Sphere-Plate Separation
Sphere and Plate Roughness
RMS 2.3 nm
nm %0 0.129
0.503 0.1291.006 0.1291.509 0.2582.012 2.3232.515 4.1293.018 8.3873.521 8.3874.024 11.2264.527 11.871
5.03 9.0325.533 9.4196.036 9.8066.539 5.1617.042 6.1947.545 5.1618.049 2.3238.552 2.719.055 1.8069.558 1.161
10.061 0.258
nm %0 0.296
0.532 1.4811.063 2.2221.595 4.4442.127 5.9262.658 7.556
3.19 8.4443.722 9.3334.254 11.2594.785 9.7785.317 9.635.849 7.556
6.38 4.2966.912 4.4447.444 3.5567.975 2.9638.507 3.2599.039 2.0749.571 0.741
10.102 0.59310.634 0.148
Plate Sphere
Percent vi of the surface area covered by roughness with heights hi
Sphere
Plate
RMS 2.1 nm
Roughness Effects much less than 1%
Lifshitz Theory Comparison
Proximity force approximation (PFA)
)])((1)[(2)( 2Ra
RaC
ppC
sp aREaF
3
3 1
720)(
z
czEC
pp
z
R
aIf aR
3
2 1
360)(
a
cRaF C
sp
a
aF
Ra
aEaP
Csp
CppCasimir
)()
2
1(
)()(
Generalized plasma-like permittivity : fitting by tabulated optical data
Drude-like permittivity
wp
= 8.9 eV
g = 0.035 eV
230 240 250 260 270 280 290 3000.03
0.04
0.05
0.06
0.07
0.08
Experiment Drude model Plasma model
Cas
imir
Fo
rce
Gra
die
nt
(mN
/M)
Au sphere - Au plate separations (nm)
Comparison with Theory
300 350 400 450 500
0.01
0.02
0.03
Experiment Drude model Plasma model
Ca
sim
ir F
orc
e G
rad
ien
t (m
N/M
)
Au sphere - Au plate separations (nm)
500 550 600 650 7000.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
Experiment Drude model Plasma model
Ca
sim
ir F
orc
e G
rad
ien
t (m
N/M
)
Au sphere - Au plate separations (nm)
Arbitrarily Shift Data by 3 nm to Fit Drude Model At Smallest Separation
If separation shifted by 3 nm, then also do not fit the Drude model
250 3000.02
0.04
0.06
Au sphere - Au plate separations (nm)
Ca
sim
ir F
orc
e G
rad
ien
t (m
N/M
)
Experiment Drude model Plasma model
400 450
0.005
0.010
Au sphere - Au plate separations (nm)C
as
imir
Fo
rce
Gra
die
nt
(mN
/M)
Experiment Drude model Plasma model
230 240 250 260 270 280 290 3000.03
0.04
0.05
0.06
0.07
0.08
Experiment Drude model Plasma model
Cas
imir
Fo
rce
Gra
die
nt
(mN
/M)
Au sphere - Au plate separations (nm)
Comparison with Theory
300 350 400 450 500
0.01
0.02
0.03
Experiment Drude model Plasma model
Cas
imir
Fo
rce
Gra
die
nt
(mN
/M)
Au sphere - Au plate separations (nm)
500 550 600 650 7000.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
Experiment Drude model Plasma model
Ca
sim
ir F
orc
e G
rad
ien
t (m
N/M
)
Au sphere - Au plate separations (nm)
AGREEMENT ONLY WITH PLASMA MODEL!
Even though Drude Model Describes the metal best.
Decca et al., Euro Phys J. C 51, 963 (2007)
Understanding the Patch Effectby
Electrostatic Simulation
Electrostatic simulation with COMSOL software package
Plate size = 32×32 m;Patch size = 0.6×0.6 m;Vplate=0.018 mV , Vsphere=0, Vpatches=random in [-90;90] mV, ~0.7 mV.
We solved the Poisson equation for conductive plate (variable potential Vplate) with dielectric patches on the surface (random potential distribution in [-90;90] mV) and conductive sphere on the distance z from the plate
Area filled by patches has been chosen according to condition: Surface area > Aeff=2Rd (for z=0.1 m plate size should be higher then 8 m)
Ftotal (Vplate) between
the sphere and plate = 0Vplate =V0
Aeff=2Rd
Patches
- 0r
V=0 r – relative permittivity
Electrostatic simulation with COMSOL software package
In the pictures:Sphere radius R = 100 m; Plate size = 32×32 m;Patch size = 0.3×0.3 m to 0.9×0.9 m;Vpatches=random in [-90;90] mV, ~0.7 mV.
We solved the Poisson equation for conductive plate (variable potential Vplate) with dielectric patches on the surface (random potential distribution in [-90;90] mV) and conductive sphere on the distance d from the plate
Plate
Patches
Aeff=2Rd
Sphere
Apply voltages to the plate and find voltage when electrostatic force goes to zeroThis compensating voltage (Vo) is found for different separations.
- 0r
V=0 r – relative permittivity
Simulation Results of Compensating Voltage
0.0 0.2 0.4 0.6 0.8 1.0
0.009
0.012
0.015
0.018
0.021
sphere-plate separations (m)
Re
sid
ua
l Ele
ctr
os
tati
c P
ote
nti
al (
mV
)
300 nm 300 nm 600 nm 600 nm 900 nm 900 nm
size distance between
Distance IndependentAs in Observed in Experiment –WellCompensated
Conclusions
1. Measured Casimir Force Gradient Between Au Sphere & Plate using a Dynamic AFM.
2. No anamalous Behavior of Sphere-Plate Residual Potential
3. Independent determination of Absolute Sphere-Plate separation
distance
4. The Force Gradient is in Agreement with the Plasma Model for Sphere-Plate Separations below 500 nm.
5. Verified unique curvature of the Plasma Model.
Simulation Results
Different distance between the patches
0.0 0.2 0.4 0.6 0.8 1.0
0.009
0.012
0.015
0.018
0.021
sphere-plate separations (m)
Re
sid
ua
l Ele
ctr
os
tati
c P
ote
nti
al (
mV
)
300 nm 300 nm 600 nm 600 nm 900 nm 900 nm
size distance between
Simulation Results
Different sizes of the patches
0.0 0.2 0.4 0.6 0.8 1.00.00
0.01
0.02
0.03
0.04
0.05
0.06
sphere-plate separations (m)
Res
idu
al E
lec
tro
sta
tic
Po
ten
tia
l (m
V)
distance between patches 6 m distance between patches 9 m distance between patches 12 m
Patch size 6 m
AcknowledgementsExperimentC.C. ChangA. Banishev
R. Castillo
Theoretical AnalysisV.M. Mostepanenko
G.L. Klimchitskaya
Research Funded by: DARPA, National Science Foundation & US Department of Energy
Thermal Correction Procedure
2) Measurement of each curve starts after 100 sec (50 sec for curve measurement when piezo extend and 50 sec when piezo retract). The each point of the curves should be corrected due to the thermal drift. To calculate the drift we measure the last 10 curves at the same compensate V0 voltage and calculate the drift at 15 points, then calculate average drift value.
15 points
10 curves at V0
0200 sec
100 sec
separation (nm)
Sepa
ratio
n (n
m)
time (sec)
-2 -1 0 1
-6
-4
-2
Sig
nal
(H
z)
Sphere-plate separation (nm)
0 200 400 600 8000.0
0.5
1.0
1.5
s
ep
ara
tio
n (
nm
)
time (sec)
Drift
<drift>=0.002 nm/sec
0 100 200 300 400 500 600 700
-15
-10
-5
0
Au sphere - Au plate separations (nm)
To
tal
Fo
rce
(H
z)
0.0326 V 0.0126 V 0.0026 V -0.0074 V -0.0174 V -0.0274 V -0.0374 V -0.0474 V -0.0574 V -0.0674 V -0.0874 V
-0.08 -0.04 0.00 0.04-14
-13
-12
-11
-10
-9
-8
-7
sig
na
l (H
z)
applied voltages (V)
-0.08 -0.04 0.00 0.04-4.5
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
applied voltages (V)
sig
na
l (H
z)
Experimental Forces
20 )( VVFele
...R
Z.
R
Z.
R
Z.
R.
Z
R(πε
k
ωβ
4
3
3
2
2200 4
5902003
4595922
3665711
2375222
22
piezoZZZ 0
where,
Electrostatic Force Formula:
Cantilever small oscillations in a force field
For small oscillation, we can take Taylor expansion of Fint at point Z0 corresponding to the equilibrium position
..))0(()0(0
)int
()0(int)(int2
zzozzzz
FzFzF
2
02
/0tan
2)/0(2]20
2[()0,(
)int
(2
0
Q
Q
driveAzA
z
F
ck
2
Sig
nal
frequency (Hz)
1
Proximity force approximation (PFA)
)])((1)[(2)( 2Ra
RaC
ppC
sp aREaF
3
3 1
720)(
z
czEC
pp
z
R
aIf and plate area RaS 2aR
3
2 1
360)(
a
cRaF C
sp
a
aF
Ra
aEaP
Csp
CppCasimir
)()
2
1(
)()(
1) At long distances (2.1 µm) all forces should be negligible. The signal at large separation distances of 1.8-2.1 µm was fit to a straight line. This straight line was subtracted from the measured signal measured at all sphere plate separations to correct for the effects of mechanical drift.
0
Thermal Correction Procedure
0
separation (nm) separation (nm)
2000 2050 2100
-0.3
0.0
0.3fr
equ
ency
sh
ift
(Hz)
Au sphere - Au plate separations (nm)
The Roughness
Sphere
Plate
RMS 2.3 nm
nm %0 0.129
0.503 0.1291.006 0.1291.509 0.2582.012 2.3232.515 4.1293.018 8.3873.521 8.3874.024 11.2264.527 11.871
5.03 9.0325.533 9.4196.036 9.8066.539 5.1617.042 6.1947.545 5.1618.049 2.3238.552 2.719.055 1.8069.558 1.161
10.061 0.258
nm %0 0.296
0.532 1.4811.063 2.2221.595 4.4442.127 5.9262.658 7.556
3.19 8.4443.722 9.3334.254 11.2594.785 9.7785.317 9.635.849 7.556
6.38 4.2966.912 4.4447.444 3.5567.975 2.9638.507 3.2599.039 2.0749.571 0.741
10.102 0.59310.634 0.148
PlateSphere
Percent vi of the surface area covered by roughnesswith heights hi
Sphere
Plate
RMS 2.1 nm
0 100 200 300 400 500 600 700
-15
-10
-5
0
Au sphere - Au plate separations (nm)
fre
qu
en
cy
sh
ift
(Hz)
0.0326 V 0.0126 V 0.0026 V -0.0074 V -0.0174 V -0.0274 V -0.0374 V -0.0474 V -0.0574 V -0.0674 V -0.0874 V
-0.08 -0.04 0.00 0.04-14
-13
-12
-11
-10
-9
-8
-7
sig
na
l (H
z)
applied voltages (V)
-0.08 -0.04 0.00 0.04-4.5
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
applied voltages (V)
sig
na
l (H
z)
Determination of Au Sphere- Plate Potential Difference
Electrostatic Force Formula:
casimirFeleFF int
)(2
int0
z
F
kc
)1(cosh
)coth(cothcsc)(2
01
1
200
R
zz
nnhnVVFn
ele
STEPS1. Repeat Experiment for 12 Voltages applied to Au plate – not sequentially2. Correct separation for plate or sphere drift3. Use Parabolic dependence of force gradient on Voltage, to draw parabolas at every separation4. Vertex of Parabola, which denotes zero electrostatic force gives the residual potential