The consistent application of Maxwell–Garnett effective medium theory to anisotropic composites
I.L.Skryabin,A.V.Radchik,P.Mos,and G.B.Smith a)
Department of Applied Physics,University of Technology,Sydney,Broadway,N.S.W.2007,Australia
͑Received27December1996;accepted for publication24February1997͒
The Maxwell–Garnett class of effective medium model applies if a reprentative cell can be found
who polarization vanishes upon inrtion in the effective medium.For an anisotropic composite
with randomly distributed ellipsoidal particles aligned along the principal axis,this leads to
electrostatic restraints on the shape of such cells.It is shown that the cell boundary must have
different depolarization factors to the inclusion within the cell.Practice is to equate them.A new
physically correct ellipsometric modeling routine still with only twofitting parameters is
demonstrated.©1997American Institute of Physics.͓S0003-6951͑97͒00517-2͔
This article eks to refine fundamental ideas ud in
electrostatics and some optical problems which involve ar-
rays of polarizable inclusions.The simplest models u ei-
ther the classical Lorentz cavity approach or the related con-
cept of a reprentative cell containing one inclusion which
when embedded in the medium produces no additional po-
larization͑or scattering͒.For random arrays,the term ran-
dom unit cell͑RUC͒was introduced for the latter
approach.1,2The issue of how to handle such problems when
either the particles or the array are anisotropic has not been
rigorously addresd,although the systems are often mod-
eled using various simple but unjustified assumptions.3–5
We ek to correct a common practice in effective me-
dium models for ellipsoidal inclusions.Composites contain-
ing inclusions which are nonspherical,will display either
uniaxial or biaxial optical respon.The complex refractive
index NϭnϪik will be a3ϫ3tensor in the materials,if
the effective medium approach is applicable.The composite
behaves like a single component anisotropic material in its
optical respon.
It is possible,in principle,to extract the components of
N using spectroscopic ellipsometry.However,in practice,
a multiparameter model͑either polynomial or oscillator type͒
for each principle component of N introduces manyfitting
parameters.Nevertheless inversion or modeling of ellipso-
metric or spectrophotometric data to obtain
Nϭ(N x,N y,N z͒has been attempted.5–7Examples of impor-tant materials where such respons occur include columnar
thinfilms,8metal/insulator composites ud for glare and en-
ergy control,micropolarizing materials,9and nanostructures.
Some of the exhibit strong absorption anisotropy leading
to asymmetric transmittance for light incident in the same
plane but opposite sides of the normal.8If reliable and sim-
ply implemented effective medium models and some struc-
tural information is available the number of arbitraryfitting
parameters in the models is reduced and they have a physical
basis.Various simple effective medium models for aniso-
tropic composites exist and are widely ud in practice.
Some of the formulations,however,neglect underlying
physical restraints.Unfortunately,it is the that em to
have been more widely ud to invert ellipsometric data.6,7
Our purpo in this article is two-fold.One is to estab-lish a broad principle which dictates the link between the particle shape and the additional‘‘shape parameters’’in simple Maxwell–Garnett type models.‘‘Envelopes’’in such theories may be either reprentative cells containing one particle1,2or a Lorentz cavity containing many inclusions. Our cond aim is to refine the practical application of an-isotropic Maxwell–Garnett models,in ellipsometric model-ing routines.
We consider an anisotropic composite as a dielectric ma-trix͑the host͒with embedded ellipsoidal particles͑the guest͒aligned along the optical axis͑Fig.1͒.Two extreme limits of the ellipsoids are either spheres or cylinders so this approach covers all possible degrees of anisotropy,which are described by the t of guest depolarization factors L j g(jϭx,y,or z͒.The extended Maxwell–Garnett effective medium theory1is usually expresd as
⑀j MGϪ⑀h
⑀hϩL j h͑⑀j MGϪ⑀h͒ϭf
⑀gϪ⑀h
⑀hϩL j g͑⑀gϪ⑀h͒,͑1͒where⑀j MG is an effective dielectric constant;f is a volume
a͒Electronic mail:
G.Smith@uts.edu.au
FIG.1.An anisotropic composite with ellipsoidal inclusions and two pos-
sible basic cells for u in deriving the MG model.Cell with surface geo-
metrically similar to its inclusion is on the left and one in which it is
confocal is on the right.
fraction of the guest particles;superscripts g and h denote
guest and host matrix,respectively.In all examples,we have
en to date the practical realization of Maxwell–Garnett
theory͑1͒in ellipsometric procedures is simplified by the
assumption6,7,10L gϭL h.For an ellipsoid of rotation͑Fig.1͒with mi axes a,bϭc,the depolarization factors of this el-
lipsoid are given by11L zϭ(1/3)2F1(1,1,5/2,1Ϫ2),L x,y ϭ1/2(1ϪL z͒,whereϭa/b͑anisotropic ratio͒and2F1is a hypergeometric function.12For an ellipsoid with all three mi-axes,different͑a b c͒more complex expressions for L j apply but conclusions are the same.L gϭL h means gϭh.This assumption of geometrical similarity in shapes is demonstrated in Fig.1lower left͑particle in its cell͒.Al-though intuitive structurally,it is not correct.
To derive the Maxwell–Garnett Eq.͑1͒for ellipsoidal
inclusions,the Laplace’s equation for a coated ellip is
solved1with the following necessary assumption:the coating
or cell surface is confocal with the ellipsoidal particle as en
in Fig.1,lower right.Therefore geometrical similarity be-
tween the particle and shell,would explicitly lead to a dif-
ferent effective medium model than that of Maxwell–
Garnett.The choice L gϭL h in͑1͒,though adopted widely, contradicts the following simple electrostatic arguments.A
metallic ellipsoid within a confocal dielectric is placed in an
external uniform electrostaticfield.The shell yielding a
Maxwell–Garnett approximation reprents the surface of a
shell who dimensions depend on thefill factor f of guest
material.Now let us vary f͑i.e.,shrink or expand the inner
ellipsoid͒in two different ways.
͑1͒The shrunken ellipsoid remains confocal with the outer shell.As both surfaces are still equipotentials,the shrinking of the inner ellipsoid would not perturb the internalfield and therefore the polarization of the outer shell stays the same.In other words,the shrinking pro-cess will be reflected in͑1͒only by the change of thefill factor.The surfaces of the inner and outer ellipsoid are coordinate surfaces in the same frame and can be con-sidered as electrostatically similar.
͑2͒Geometrically similar shrinking has the shrunken inner ellipsoid miaxes proportionally smaller in relation to the outer ones.The decrea in size shifts the focal po-sitions,so the inner and the outer ellipsoids are not equi-potentials of the same coordinate frame.Thefield lines are distorted by this shrinking so the polarization of the outer shell changes.It is obvious that this change in po-larization cannot be reprented simply by varying the fill factor in͑1͒so a Maxwell–Garnett effective medium model is invalid.
To summarize,for a general Maxwell–Garnett type of
effective medium theory,the cell can be constructed only to
be electrostatically similar with the inclusion surface.The
shape of this electrostatic unit cell͑EUC͒must also produce
a uniform electricfield within the cell.Only a sphere,ellip-
soid,and a‘‘flat capacitor’’13are known to satisfy this cri-
teria.To construct an EUC to be confocal with the ellipsoidal
guest particle,we proceed as follows.1,14The surface
of the cell is given by the equation of an ellipsoid ͓(x2ϩy2)/(b h)2]ϩ͓z2/(a h)2͔ϭ1with mi axes:
b hϭͱkϩ͑b g͒2
a hϭͱkϩ͑a g͒2.
k is dependent on thefill-factor fϭ͓a g(b g)2/a h(b h)2͔and could be found as a real positive root of the cubic equation:
͓kϩ͑a g͒2͔͓kϩ͑b g͒2͔2ϭ
͑a g͒2͑b g͒4
f2
.͑3͒The depolarization factors of the cell boundary are thus functions offill factor.For smallfill factors,where the Maxwell–Garnett model usually applies all cell depolariza-tion factors approach L hϭ1/3,which is significantly different from the inclusion.The spherical surface is the limiting equi-pote
ntial of any ellipsoidal coordinate frame.If the compos-ite is den and the Maxwell–Garnett approach is still valid, the cell surface must cloly remble the surface shape of a particle.The depolarization factors of particle and the shell are thus then clo in magnitude and approach L z gϽ1/3or L x gϾ1/3,which can be very different for elongated particles. Thus from the degenerate value Lϭ1/3at fϭ0,the cell de-polarization factors ri or fall smoothly with f until they equal tho of the inclusion at fϭ1.Calculations using͑2͒and͑3͒shows smooth change with most variation between fϭ0and0.4.
Ellipsometric modeling subject to the constraints͑2͒and ͑3͒can now be implemented.Thefitting parameters are the fill-factor f and the degree of anisotropy of the inclusion g.Other parameters in͑1͒are then determined via the con-straints.We now compare predicted measurable ellipsomet-ric parameters⌿and⌬when L hϭL g with tho from the correct confocal model.The host material is Al2O3with sil-ver inclusions.Dielectric constants are from Palik,15guest ellipsoids are aligned perpendicular to the sample surface, fϭ0.2and the particle shape is given by the ratiogϭ1.5.⌿and⌬calculated assuming geometric similarity are com-pared with tho using electrostatic similarity in Fig.2.The difference between the two models is much larger than typi-cal experimental uncertainty.
Figure3shows excellent agreement between thinfilm experimental data obtained on a Jobin–Yvon Variable Angle Spectroscopic Ellipsometer and curves obtained with ourfit-ting procedure,for an angle of incidence of40°.Angles of incidence data from30°to80°in10°steps were ud simul-taneously tofit the data.The model structure consists of
bulk FIG.2.Ellipsometric parameters calculated for the same material using existing practice͑dashed lines͒and the suggested model͑solid lines͒.
Al 2O 3and ellipsoidal voids.Fill factor,anisotropic ratio,and film thickness were ud as fitting parameters.Best fit was obtained for the following parameters:thickness ϭ6998Å,a g /b g ϭ0.26,f ϭ0.22.
In conclusion,there are two electrostatic constraints on the surface of a cell yielding a Maxwell–Garnett model ͑i ͒it must be spherical,ellipsoidal,or a ‘‘flat capacitor’’and ͑ii ͒it must reprent equipotentials of the coordinate frame where the inclusion surface is also an equipotential.Two depolar-ization factors can then be defined for anisotropic compos-
ites.A new ellipsometric fitting procedure results using an-isotropic ratio and fill factor f as independent fitting parameters.
The authors would like to express our gratitude to S.Dligatch for the assistance with providing suitable experi-mental ellipsometric results.
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FIG.3.Measured ͑solid lines ͒and fitted ͑dashed lines ͒ellipsometric spectra for thin film Al 2O 3with voids,deposited onto quartz with rf planar magne-tron sputtering.Angle of incidence was 40°for this data.
