We now discuss the linear dielectric properties of a composite medium constituting a host and some inclusions. For the sake of simplicity, we mostly deal with only a two-component system with two constituent materials, which are homogeneous and isotropic. This can be generalized to a multicomponent system very easily. We assume the average size of inclusions to be much smaller than the wavelength of light for quasi-static approximation to hold. We use the mean field approximation to calculate the effective dielectric response of the composite. The local fieldEL on the dipole is given by
EL=E0+Ed+ES+En, (5.39) with E0 as the applied electric field and Ed as the depolarizing field due to the charges induced on the surface of the sphere. The superpositionE0+Ed
gives the homogenous macroscopic (Maxwell) field averaged over the entire volume of the material. In terms of the macroscopic polarizationPthe field Ed can be given by
Ed =−P ǫ0
. (5.40)
In Eq. (5.39)ES is the field due to the charges induced on the Lorentz sphere (of radius R) and En is the field caused by other dipoles inside the sphere.
The fieldES can be evaluated following Ref. [3], and is given by ES= P
3ǫ0. (5.41)
If now we choose the homogenous medium to have cubic crystal structure, then the contribution fromEn goes to zero due to inherent symmetry of the lattice structure. Thus for the local field we have
EL=E+ P 3ǫ0
. (5.42)
The single molecule polarizabilityαcan be related to the macroscopic polar- izationPas follows:
P=N αEL=N α
E+ P 3ǫ0
, (5.43)
where N denotes the average number of molecules per unit volume. Using Eq. (5.43),D can be written as
D=ǫ0 1 + N α/ǫ0
1−N α3ǫ0
!
E=ǫ0ǫrE. (5.44)
Optical properties of dielectric, metal and engineered materials 79 Rewriting the relative permeability of the mediumǫrin terms ofαyields the famous Clausius-Mossotti relation as
N α 3ǫ0
=ǫr−1
ǫr+ 2, (5.45)
leading to the expression forαas α=3ǫ0
N
ǫr−1 ǫr+ 2
. (5.46)
Eq. (5.45) relates the microscopic quantityαto a macroscopic propertyǫ. We will now extend this to spherical particle inclusions of relative permittivityǫ1
embedded in a host medium with relative permittivity ǫh. Let the effective dielectric function of the composite be denoted byǫef f. The Clausius-Mossotti relation then reduces to
N α 3ǫhǫ0
= ǫef f−ǫh
ǫef f + 2ǫh
, (5.47)
and the polarizabilityαtakes the form α= 3ǫ0ǫh
f N
ǫ1−ǫh
ǫ1+ 2ǫh, (5.48)
with f as the volume fraction of the inclusion. Note that 1/N in Eq. (5.48) is now replaced byf /N to account for the volume occupied by the inclusion.
Replacingαin Eq. (5.48), we get ǫef f−ǫh
ǫef f + 2ǫh
=f ǫ1−ǫh
ǫ1+ 2ǫh
. (5.49)
The mediumǫef f is then given by ǫef f =ǫh
1 + 2f
ǫ1−ǫh
ǫ1+2ǫh
1−f
ǫ1−ǫh
ǫ1+2ǫh
. (5.50)
5.5.1 Maxwell-Garnett theory
For smallf, Eq. (5.50) can be approximated by ǫef f =ǫh+ 3f ǫh ǫ1−ǫh
ǫ1+ 2ǫh
+O(f2). (5.51)
This is known as the Maxwell-Garnett (MG) formula. The same formula can be derived using a different method that is consistent with Eq. (5.51) for f ≪1 [9]. Whenǫ1+ 2ǫh= 0 we have resonance, which is possible only when one of the components is a metal, since metals can have a large negative real part of the dielectric function (e.g., Re(ǫ1)<0). Such resonances are known as localized plasmon resonances. Note that ǫef f given by the MG formula cannot predict the percolation threshold, which is overcome in Bruggeman theory, discussed next.
5.5.2 Bruggeman theory for multicomponent composite medium
We will now derive the effective dielectric function for the n-component composite medium. Letǫj denote the dielectric functions of thej-th compo- nent with volume fractionfj in a host with dielectric functionǫh. Eq. (5.47) gets modified as
ǫef f−ǫh
ǫef f + 2ǫh
=N1α1
3ǫ0ǫh
+N2α2
3ǫ0ǫh
+· · ·+Njαj
3ǫ0ǫh
+· · ·+Nnαn
3ǫ0ǫh
, (5.52)
whereαj, (j= 1,2,· · ·, n) is given by αj =3ǫ0ǫh
Nj
fj
ǫj−ǫh
ǫj+ 2ǫh
, (5.53)
andNj denotes the average number of molecules of thej-th species per unit volume. After substitutingαj in Eq. (5.52), we have
ǫef f −ǫh
ǫef f + 2ǫh
=f1 ǫ1−ǫh
ǫ1+ 2ǫh
+f2 ǫ2−ǫh
ǫ2+ 2ǫh
+· · ·+fj ǫj−ǫh
ǫj+ 2ǫh
+· · ·+fn ǫn−ǫh
ǫn+ 2ǫh
. (5.54) For ann-component effective composite medium, we have
f1+f2+· · ·+fj+· · ·+fn= 1, (5.55) and ǫh cannot be distinguished from the effective ǫef f (ǫef f =ǫh); the left- hand side of Eq. (5.54) reduces to zero and we arrive at the Bruggeman formula
n
X
j=1
fj ǫj−ǫef f
ǫj+ 2ǫef f = 0, with
n
X
j=1
fj= 1. (5.56)
Note that the derivation for the n-component system presented here does not differentiate the inclusion and host as in the Maxwell-Garnett formula.
The present approach treats both inclusions and host on an equal footing.
The effective medium dielectric function ǫef f for a two-component (n = 2) composite simplifies to
f1
ǫ1−ǫef f
ǫ1+ 2ǫef f +f2
ǫ2−ǫef f
ǫ2+ 2ǫef f = 0, with f1= 1−f2. (5.57) Eq. (5.57) represents a quadratic equation inǫef f having roots
ǫef f = 1
4{(3f1−1)ǫ1+ (3f2−1)ǫ2
± q
[(3f1−1)ǫ1+ (3f2−1)ǫ2]2+ 8ǫ1ǫ2}. (5.58)
Optical properties of dielectric, metal and engineered materials 81
2.25 2.3 2.35
0 0.05 0.1
2.25 2.55 2.85 3.15
0 2 4 6 8
400 500 600 700 800 400 500 600 700 800
400 500 600 700 800 400 500 600 700 800 400 500 600 700 800 400 500 600 700 800 f = 0.005
f = 0.005 f = 0.04
f = 0.04
f = 0.2
f = 0.2
0 2 4 6
0 0.5 1 ( mIεeff)( eRεeff)
λ (nm) λ (nm) λ (nm)
(a) (b) (c)
(e)
(d) (f)
FIGURE 5.4: [(a), (b) and (c)] Real and [(d), (e) and (f)] imaginary parts of ǫef f as function ofλfor a gold-silica composite evaluated using the Bruggeman (solid curve) and the Maxwell-Garnett (dash-dot) formula for three different volume fractions of gold inclusions, namely,f = 0.005 [(a) and (d)],f = 0.04 [(b) and (e)] andf = 0.2 [(c) and (f)]. The dielectric function of gold is taken from Johnson and Christie [13], whileεh= 2.25.
The proper sign in Eq. (5.58) is to be chosen such that Im (ǫef f)>0 to ensure causality.
We now compare the estimates ofǫef f as predicted by the Maxwell-Garnett (MG) and Bruggeman formulas. As mentioned earlier, MG formula is valid only for lower values off, as can be seen from Fig. 5.4. Fig. 5.4 shows the real and imaginary parts ofǫef f as a function of wavelengthλfor different values of volume fraction of gold inclusions in silica. The solid (dash-dot) curve in Fig. 5.4 corresponds to the MG (Bruggeman) formula. It is evident that for lower values off (for example,f = 0.005), both the formulas match very well, but for higher values off they differ drastically. Thus due attention is to be paid when dealing with higher values off of the metal inclusions sinceǫef f
can be completely different as estimated by the two approaches. Throughout the derivation we have considered the inclusions to be spherical, but this can be generalized to other geometries in a straightforward manner. In a similar vein, anisotropic character of the inclusions can be incorporated.