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