Properties of metallic films in precise calculation of the Casimir force

0.11 MB·English

by  V. B. Svetovoy

PROPERTIES OF METALLIC FILMS IN PRECISE
CALCULATION OF THE CASIMIR FORCE
4
0
0
VITALYB. SVETOVOY
2
Transducers Science and Technology Group, EWI, University of Twente,
n
P.O. 217, 7500 AE Enschede, The Netherlands
a
J E-mail: [email protected]

7
2 Opticalpropertiesofthedepositedgoldfilmsarediscussedinconnectionwiththe
Casimir force prediction. Voids in the films and electron scattering on the grain
boundaries reduce the force at small separations on the 2 % level in comparison
]
i withthebulkmaterialprediction. Thecontributionofthepatchpotential dueto
c
polycrystallinestructureofthefilmsisshowntobesmallfortheexistingCasimir
s
forceexperiments.
-
l
r
t
m 1 Introduction
.
t
a
m Detailed investigation of the Casimir force1 between two uncharged metallic
- plates, predicted in 1948, became possible only recently with the progress in
d
microtechnologiesallowingto controlseparationbetweenbodies smallerthan
n 1 µm. The development of the experiment2,3,4,5 is characterized by steady
o
2 5
c improvementintheprecisionoftheforcemeasurementfrom5% to0.25% .
[ The progressintheoreticalpredictionof the forceis notso impressive. Start-

1 ing from the first precise calculations6,7,8,9 and up to now5 the force is cal-
v culated using the optical properties of bulk metals taken in the handbooks.
2 However, in all the experiments the force was measured between deposited
6
metallic films. Typically it was gold with the thickness 100−200 nm. It is
5
clearthatthedepositedmaterialandbulkmaterialaredifferentifwepretend
1
0 onhigh precisioncalculations. Inthis paper the questionis analyzedto what
4 extend the properties of real deposited metallic films can change the force in
0 comparisonwiththatpredictedforthebulkmaterial. Toanswerthisquestion
/
t weinevitably haveto turntothe materialscience. The stepsinthis direction
a
never been done before but without understanding ofthe used materialthere
m
will be no further progress in the prediction of the Casimir force.
- 1
d Originally the Casimir force was calculated between ideal metallic
n plates. In this case it does not include any material parameters: F =
c
o π2/240 ¯hc/a4 , where a is the separationbetween plates. At small separa-
c
: (cid:0)tions the(cid:1)(cid:0)deviati(cid:1)on of the used metal from the ideal one becomes significant
v 10
and one has to calculate the force using a more general Lifshitz formula .
i
X This formula takes into account real optical properties of the plate material
11
r via the dielectric function ε(ω), which is taken in the handbooks . Gold
a
is the best material for the force measurement since it is chemically inactive
and its low frequency behavior, where the handbook data unaccessible, can
1
be reliably predicted with the Drude model.
2 Voids in the films
Optical properties of Au films were investigated widely in 60th-70th. The re-
11
sults are collected in the handbooks . Significant deviation in the optical
data was attributed to genuine sample differences caused by different sam-
11 11
ple preparation methods . The handbook data were carefully chosen to
represent the bulk material as close as possible. A special investigation of
12
the sample preparationeffect was undertaken with the conclusion that the
mostsignificantroleplayedvoidsinthesamples. Annealedfilmsdemonstrated
larger density due to larger grain size. A single parameter model represent-
ing voids in an otherwise homogeneous medium was shown to account for
the major discrepancies in the above-band-gap (E = 2.5 eV) spectra for Au
samples prepared in different ways. It allowed to get information about the
volume fraction of voids f in different samples. For f ≪ 1 and |ε| ≫ 1
V V
the model gave the effective dielectric function hεi defined by the relation
3
hεi = ε 1− f . For the film (150 nm) evaporated with e-beam on NaCl
2 V
substrat(cid:0)e at room(cid:1) temperature with a deposition rate of 23 ˚A/s the volume
of voids was found to be larger on 4% than in the annealed samples used for
11
the handbook data . For the 50 nm thick sputtered films deposited at the
rate smaller than 13 ˚A/s the volume of voids was found to be 4 % at room
◦
temperature deposition and 1% at 250 C. The cleaved NaCl substrate was
chosenbecause the films start to growepitaxially on it. The same conclusion
13
was made in a more recent investigation .
3
In the most sensitive experiments the Au films were evaporated or
5
sputtered at room temperature, no annealing was reported. One can
12
expect that in both cases the volume of voids was of about of 4%.
The correction to the Casimir force due presence of voids was calculated
11
usingthedata forε. Thesedatawerescaledinaccordancewiththeeffective
3
dielectricfunctionhεi=ε 1− f . Calculationswereperformedintheway
2 V
similar to 7 for plate-plat(cid:0)e and sph(cid:1)ere-plate geometries. The results for the
relativecorrectiontotheforceareshowninFig. 1(a)forminimalseparations
3
in the experiments: a = 63 nm for sphere-plate and a = 260 nm for the
5
effective plate-plate geometries . Let us stress that this correction always
make the force for films smaller than that for the bulk material.
3 Scattering on the grain boundaries
It is well known that the resistivity of deposited films deviates significantly
fromthebulkresistivity. Ifthefilmisnotverythin,saythickerthan10nm,it
haspolycrystallinestructure. Thegrainboundariescontributetotheelectron
2
14 14
transport (for the references see ). In a recent paper the optical charac-
teristics of the films were treated as the grains with bulk material properties
plus electron scattering on the grain boundaries.
Thegoldfilmswiththegrainsizefrom15to45nmweredepositedonglass
14
withtwodifferentmethods . Reflectanceofthefilmswasinvestigatedinthe
wavelength range 0.3-50 µm. The grain boundaries were modelled with the
deltapotentials,scatteringfromphononsandpointdefectswereaccountedby
the relaxation time. The grain size distribution had no significant influence,
so the films were characterized only by the mean grain size D. The material
is described by the two dielectric functions longitudinal ε and transverse ε .
l t
Only ε changes because the grain boundaries are perpendicular to the film
t
surface. In the local limit for ε the following expression was found
t
3ω2 1
p 2 3 −1
ε (ω)=ε(ω)+ α−α +α ln 1+α , (1)
t
ω(ω+iω )(cid:20)2 (cid:21)
τ (cid:0) (cid:1)
−1
v R ω
F
α ≡ 1−i , (2)
Dω 1−R(cid:18) ω (cid:19)
τ τ
where ε(ω) = ε (ω) is the dielectric function of the crystalline gold, ω is
l τ
the Drude relaxation frequency, v is the Fermi velocity, R is the reflection
F
coefficientofelectronsonthegrainboundary. Thelatteristheonlyempirical
parameter, which was found in the same experiment to be R ≈ 0.65. The
reflectance of the film was expressed via the surface impedance. Comparison
with the experimental reflectance showed that the model describes well the
effect of the grain size except for the very small grains D ≈15nm when the
electron mean free path becomes comparable with D.
Weusedtheexpression(1)atimaginaryfrequenciesω =iζ andcalculated
the impedances for s and p polarizations following the Kliewer and Fuchs
15
procedure . The impedances are different due to the nonzero wave vector
q along the plates in the Lifshitz formula. The details will be reported later.
The relative correction to the Casimir force as a function of the grain size
is presented in Fig. 1(b). The typical grain size for the films 100−200 nm
16
thick deposited on the substrate at room temperature is less than 50 nm
but it can depend on the deposition details and we take as the upper limit
D <100nm.
4 Patch potential
The importance of the patch potential for the Casimir force measurement
17
was stressed in a recent paper . For a crystal the work function has dif-
ferent values for different crystallographic planes. This difference should be
compensated by the potential distribution aroundthe crystal. The deposited
filmsarepolycrystallineandforthisreasontherewillbelocalvariationofthe
3
5 2
(a) (b)
1 a=63 nm
4 % 2
2 a=260 nm a), 1.5
3 ∆on ( 1 a=63 nm
2 1 correcti 1 2 a=260 nm
e
v
1 2 Relati0.5 1
0 0
2 4 6 8 10 20 40 60 80 100 120 140
Voids fraction f, % Grain size D, nm
V
Figure 1. The relative corrections to the Casimir force ∆(a) in percents for the minimal
separationsexploredintheexperimentsa=63nm3anda=260nm5: (a)correctiondue
tovoidsinthefilm;(b)correctionduetoscatteringonthegrainboundariesasafunction
ofthemeangrainsizeD
potentialnearbythe surface. Itwillresultinanadditionalforce,electrostatic
in nature, which will be measured together with the Casimir force. A simple
17
model was proposed to estimate the effect. In this model the dipoles were
distributed on the planes. Their interaction gives the force which is defined
bythespectraldensityofthesurfacepotentialcorrelationfunctionC(k). The
model is in agreement with the independent ab initio calculation of the po-
18
tential distribution near Al crystallite . For the sphere and plate the force
was written in the form
∞
C(k)e−ka
2
Fpatch(a)=−ε0R dkk , (3)
Z sinhka
0
C(k)= d2re−ikrc(r), c(r)=hv(r)v(0)i, (4)
Z
where ε0 is the permittivity of vacuum, R is the sphere radius, v(r) is the
local potential on one of the surfaces and the potential distribution on differ-
ent surfaces was assumed to be uncorrelated. The spectral density C(k) is
17
crucial for the force estimate. To our opinion, in the original paper it was
overestimated. There was no clear idea around which value of k the spectral
density should be centered and what is the width of this function, so these
values were chosen quite arbitrary. Looking at the films surface images one
can easy guess that the spectrum should be centered around the mean grain
sizeDwiththewavelengthspresentedroughlyfromD/2to2D. Ifweconsider
the spectrum in this range as flat, the force will be the following
4
F (a)≈−4πε0σv2Rexp −2πa . (5)
patch
15a (cid:18) D (cid:19)
Here σ2 is the variance of the potential distribution that was estimated17 as
v
2
(90mV) . The relative contribution of the patch potential to the measured
Casimirforceisestimatedtobesmallerthan0.4%fortheminimalseparation
3
a = 63nm in the AFM experiment and the largest grain size D =100nm.
It decreases fast for smaller grain size or for larger separations. However, it
should be stressed that the patch force mimics the Yukawa interaction and
can be easily confused with a new force.
5 Discussion
Aboveitwasdemonstratedthatthe opticalpropertiesofdepositedgoldfilms
differfromthoseofthebulkmaterial. Themainreasonsforthedeviationsare
the voids in the films and electron scattering on the grain boundaries. Both
effects diminish the absolute value of the predicted Casimir force on the one
percent level. We define the relative correction as
F (a)−F (a)
f b
∆(a)≡ , (6)
(cid:12) F (a) (cid:12)
(cid:12) b (cid:12)
(cid:12) (cid:12)
where the indexes b and f refer(cid:12) to the bulk a(cid:12)nd film materials, respec-
3
tively. For the AFM experiment in the sphere-plate geometry the ex-
pected correction was found for the most probable values f = 4% and
V
D = 50 nm: ∆(63nm) = (1.9+0.5) % = 2.4%. The same correc-
5
tion for the MEMS experiment with the effective plate-plate geometry is
∆(260nm) = (1.0+0.9) % = 1.9%. One can see that these corrections
cannot be ignored since in both experiments the precision was better than
1%.
There is an additional effect which also reduces the absolute value of the
predicted force. At frequencies larger than ω the charge density fluctuation
p
(plasmons) can propagate in the materialdecreasing the reflection coefficient
for p-polarization. Importance of this nonlocal effect for the Casimir force at
separationsoftheorderorsmallerthanplasmawavelengthλ wasstressedfor
p
19
the first time in the paper where the correction due to plasmon excitation
3
was found. For the AFM experiment it is estimated as 3 % at a = 63 nm.
5
DuetolargerseparationthiscorrectionissmallerfortheMEMSexperiment ,
where it is 1%at a=260nm. The plasmoncorrectionwasdismissedin Ref.
5
onthebasisthattheseparationwasmuchlargerthanthepenetrationdepth
δ =λ /2π. In this connectionwe shouldnote that the separationa has to be
p
comparedwithλ butnotδ. Otherwiseonthesamebasisonecouldconclude
p
5
that the finite conductivity correction is negligible at a ∼ λ while in reality
p
7
it is of about of 50% (see Fig. 1 in the paper where the transition point at
a∼λ not at λ /2π is clearly seen).
p p
In conclusion,wedemonstratedthatitisimportanttotakeintoconsider-
ation the optical characteristics of the metallic films used in the experiments
andshowedthe wayhow the deviationfromthe bulk metalproperties canbe
estimated. Voids in the films, scattering on the grain boundaries so as non-
local effects all of them tend to reduce the force in comparison with the bulk
metal prediction. Even without nonlocal effects the reduction is expected on
the level of 2%.
Acknowledgments
This work was supported by the Dutch Technology Foundation
References
1. H. B. G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 (1948).
2. S. K. Lamoreaux, Phys. Rev. Lett. 78, 5 (1997); 81, 5475 (1998).
3. B. W. Harris, F. Chen, and U. Mohideen, Phys. Rev. A 62, 052109
(2000).
4. T. Ederth, Phys. Rev. A 62, 062104(2000);H. B. Chan, V. A. Aksyuk,
R. N. Kleiman, D. J. Bishop, and F. Capasso, Science 291, 1941 (2001);
G. Bressi, G. Carugno, R. Onofrio, and G. Ruoso, Phys. Rev. Lett. 88,
041804 (2002).
5. R.S.Decca,D.Lo´pez,E.Fischbach,andD.E.Krause,Phys. Rev. Lett.
91, 050402 (2003); R. S. Decca et. al, hep-ph/0310157 (to appear in
Phys. Rev. D).
6. M. Bostr¨om and Bo E. Sernelius, Phys. Rev. A 61 046101 (2000).
7. A. Lambrecht and S. Reynaud, Eur. Phys. J. D 8, 309 (2000).
8. G.L.Klimchitskaya,U.Mohideen,andV.M.Mostepanenko,Phys. Rev.
A 61, 062107 (2000).
9. V. B. Svetovoy,M. V. Lokhanin, Mod. Phys. Lett. A 15, 1437 (2000).
10. E.M.LifshitzandL.P.Pitaevskii,StatisticalPhysics, Part 2 (Pergamon
Press, Oxford, 1980).
11. Handbook of Optical Constants of Solids, editedbyE.D.Palik(Academic
Press, 1995).
12. D. E. Aspnes, E. Kinsbron, and D. D. Bacon, Phys. Rev. B 21, 3290
(1980).
13. Yu. Wang et al., Thing Solid Films, 313-314, 232 (1998).
14. J. Sotelo, J. Ederth, and G. Niklasson,Phys. Rev. B 67, 195106(2003).
15. K. L. Kliewer and R. Fuchs, Phys. Rev. 172, 607 (1968).
16. L. Vazquez et al., Surf. Sci. 345,17 (1996).
17. C. C. Speake and C. Trenkel, Phys. Rev. Lett. 90, 160403(2003).
6
18. C. J. Fall, N. Binggeli, and A. Baldereschi,Phys. Rev. Lett. 88, 156802
(2002).
19. R. Esquivel, C. Villarreal, and W. L. Mocha´n, Phys. Rev. A 68, 052103
(2003).
7