Supplemental Information
Optical Constants Evaluation
In the following we have described the method to estimate the optical constants of
transparent optical films deposited on optically transparent and flat substrates. The
methodology was originally developed by Manifacier et. al [1].
The Envelope Method
Figure 1 shows a single layer insulating film on a transparent substrate. There are
two measurable quantities. The transmittance T (≡ IT/I0 ) and reflectance R (≡ IR /I0 )
, where I0, IT and IR
are the intensities of the incident, transmitted and reflected
radiations respectively, for a given wavelength λ. From these, three unknown quantities
i.e refractive index (n), extinction coefficient (k) as a function of wavelength (λ ) and film
thickness (d) are to be determined. In case of multilayered film the problem is more
complicated as there are 3m (m = no. of layer) unknown quantities to be determined as a
function of λ from measured T and R. Here in case of a single layer film we have
assumed that the refractive index of the substrate (ns ) is known and it is fully transparent
to the incident light.
For a single layer weakly absorbing film on a transparent substrate of known
refractive index n, Manifacier et al. [1] developed a technique known as “Envelope
Method” for deriving the optical constants. By this method from the transmission
spectrum alone (i.e T as a function of λ) the refractive index (n) extinction coefficient (k)
as a function of λ and the film thickness can be determined.
Figure 2 shows a typical transmittance and reflectance spectra. The transmission
spectrum can be subdivided into three distinct regions. Weak absorption region (T ≥ :0.6)
medium absorption (0.6 ≥ T ≥ 0.4 ) and strong absorption region (T ≤ 0.4). The envelop
method
is
applicable
only
in
weak
and
medium
absorption
regions.
The transmittance of a film, which is homogeneous, uniform in thickness and deposited
on a sapphire and completely transparent substrate is described by Heavens et al [2].
T' =
Ax
(B-Cx-Dx 2 )
(1)
where
x = cxp(-d)
A = 16n s n 0 (n 2 + k 2 )
B= [(n 0 + n) 2 + k 2 ][(n + n s ) 2 + k 2 ]
C =2[(n 2 - n 02 + k 2 )(n 2 - n s2 + k 2 ) - 4k 2n s ]cos() - 2[2kn s (n 2 - n 02 + k 2 )
+ 2k(n 2 - n s2 + k 2 )] sin()
D = [(n - n 0 ) 2 + k 2 ] [(n - n s ) 2 + k 2 ]
here α is the absorption coefficient of the film and equal to 4πnd/λ, d is the film thickness
k is the extinction coefficient and n, n0, ns are the refractive indices of the film, air and
substrate respectively. Now, incase of film with weak absorption, Manfacier made the
allowing assumptions to derive the equations.
(i) n 2 >> k 2
(ii) (n- n 0 ) 2 >> k 2 and (n - n s ) 2 >> k 2
the first assumption yields
A = 16 n s n 0 n 2
B= (n 0 + n) 2 (n + n s ) 2
near a maximum or minimum for T we can neglect the sin(γ) term of the expression
describing C and also in most practical case n > n0 and n > ns, hence
4 n s n 0 < (n + n 0 )(n + n s )
from assumption (ii)
k 2 << (n - n 0 )( n - n s )
4 n s n 0 k 2 << (n 2 -n 02 )( n 2 - n s2 )
so
C = 2[(n 2 -n 02 )( n 2 - n s2 )] cos()
and
D =(n - n 0 ) 2 ( n - n s ) 2
Hence equation 7.1 becomes; taking n0 = 1
16 n s n 2 x
T=
[(1 + n) 2 (n + n s ) 2 -2x[(n 2 - 1)(n 2 - n s2 )] cos() + (n -1)2 (n- n s2 )x 2 ]
alternatively T can also be written as
T=
16 n 2 n 0 n s x
[C12 + C22exp(-8kd/) - 2C1C2 xcos(4nd/)]
(2)
where,
x = exp(-4nd/)
C1 = [(n+n 0 )(n s +n)]
C2 = [(n - n 0 )(n - n s )]
Film absorption coefficient α is related to k (extinction coefficient) and wavelength λ
= 4k/
(3)
the cosine in the denominator of equation 6.1 oscillates between +1 and -1 so that
Tmax
l6 n 0 n s n 2 x
=
(C1 - C2 x) 2
(4)
l6 n 0 n s n 2 x
(C1 + C2 x) 2
(5)
Tmin =
Light waves from air-film and film substrate interfaces interfere giving rise to
interference fringes. A satisfactory free hand envelope curve is drawn through the
maxima and minima points of interference fringes. Then the graph is digitized and
polynomial best fits Tmax (λ) and Tmin (λ) are obtained. In other words the transmittance
spectrum is enveloped by Tmax (λ) and Tmin (λ) best fitted curves. Figure 2 shows the %
T, % R. spectra along with the best fitted Tmax (λ) and Tmin (λ) curves. Now from equation
3 and 4 one can solve
x = exp(-d)
= [(n + 1)(n + n s )[(Tmax /Tmin )0.5 - 1)]/ [(n - 1)(n - n s )[(Tmax /Tmin )0.5 +1)]
(6)
(7)
and
n = [N ' + (N '2 - n s ) 0.5 ]0.5
(8)
where
N ' = 0.5(1 + n s2 ) + [ 2n s [(Tmax -Tmin ) / Tmax Tmin ]]
The refractive index (ns) of sapphire substrate as a function of photon wavelength (λ ) can
be calculated using the following empirical dispersion relation
n s2 =1+ [A  2 / ( 2 - 12 )] + [B 2 / ( 2 -  22 )] + [C  2 / ( 2 -  32 )]
(9)
Where, A=1.023798, B=1.058264, C=5.280792, λ1=0.00377588, λ2=0.0122544,
λ3=321.3616 and λ is in m.
Hence using values of maxima and minima in the transmittance spectrum one can obtain
the refractive index of the film as a function of .
2.1.1.1 Evaluation of the refractive index (n) of the film
The refractive index (n) of the film as a function of the incident photon wavelength () is
calculated using Eqn. 8. The data of the refractive n(λ) as a function of wavelength (λ)
can be fitted to a Sellmeier type dispersion equation, assuming that the material is
composed of individual dipole oscillators which are set to force vibrations by incident
light. The dispersion relation can be expressed as [3].
Si i2
n() 2  1   [
]
2
i 1  ( i /  )
(10)
Where Si is the strength of the individual dipole oscillator and λi is the oscillator
wavelength. For pure material the wavelength dependence of optical constants was
treated by Lorentz [4]. The theory assumes that the material is composed of a series of
independent oscillators which are set to forced vibrations by the incident radiation.
Domenico et al. [3] proposed that for semiconducting and insulting materials the lowest
energy oscillator as the largest contributor to ‘n’ and a single term Sellmeier relation
could adequately describe the wavelength dispersion of refractive index. Therefore, Eqn.
(10) can be written as:
S0  02
n ( )  1 
[1  ( 0 / ) 2 ]
2
(11)
where S0 is an average oscillator strength and λ0 is an average oscillator wavelength. A
plot of (n2-1)-1 versus (1/λ)2 yields a straight line and the parameters So and λo can be
obtained from the slope (1/So) and intercept (1/Soλo2) of the line. Knowing the values of
So and λo, the energy of the oscillator Eo=hc/λo, (where h = Planck’s constant, c is the
velocity of light) and the refractive index dispersion parameter (Eo/So) are determined.
2.1.1.2 Determination of the film thickness
To determine the thickness (d) of the film, first a trial value of thickness d‫ ׳‬is calculated
using the following relation [5]
d' 
1 2
(2[n(1 ) 2  n( 2 )1 ])
(12)
where n(λ1) and n(λ2) are the refractive indices at two adjacent maxima (or minima) at λ1
and λ2. A number of d' values are thus obtained. Their average (d)' is calculated. Now the
well known formula for interference fringes is
2nd '  m'
(13)
Here m' is an integer for maxima and a half integer for minima.Using (d)', m' is obtained
from Eqn.13. It is rounded off to the nearest integer for maxima or half integer for
minima to get the modified m. Now again from Eqn. 12 using these m, λ and n, a set of
accurate d is obtained. The thickness of the film is the average of d i.e df.
2.1.1.3 Evaluation of the absorption coefficient () and extinction coefficient (k)
Knowing x from Eqn. 7 and thickness (d) as described above, we have calculated  from
Eqn. 6 and k from Eqn.3.
2.2.2 Calculation of optical constants near optical band gap
Near the absorption edge, the refractive index can be calculated using the following
relation [6]
n
(1  R 0.5 )
(1  R 0.5 )
(14)
The transmittance value here is less than 40%. The absorption coefficient (α) has been
calculated from the following relation [7].
  1 d f ln{(1  R)[1  (n  n s ) 2 /(n  n s ) 2 ]  [1  (n s  1) 2 /(n s  1) 2 ] / T}
(15)
where R and T are the refectance and transmittance at a particular wavelength, n and ns
are the refractive indices of film and substrate, respectively, and d is the film thickness.
For a direct band gap material the absorption coefficient as a function of photon energy
can be expressed as [8].
(hc / )2  constant. {(hc) /(  Eg )}
(16)
where α is the absorption co-efficient, (hc/λ) is the incident photon energy and Eg is the
band gap energy. By plotting (αhc/λ)2 vs. (hc/λ), Eg can be evaluated from the
extrapolated linear portion of the plot.
2.2.2.1 Evaluation of the packing fraction
The packing fraction (f) of each film was calculated using the effective medium
approximation (EMA). In a heterogeneous medium, the effective dielectric constant is
related to the dielectric constant of each component according to the effective medium
approximation (EMA) described by Bruggeman [9]. The packing fraction (f) of each film
was calculated using the effective medium approximation (EMA). Note that the dielectric
constant of PLT15 is high compared to CFO, and also CFO weight fraction is kept low in
PLT15 matrix. Therefore, we have tacitly assumed these films to be a single
phase dielectric material containing some amount of porosity as second phase. Hence
f
nb2  n 2
(1  n 2 )

(1

f
)
0
nb2  2n 2
(1  2n 2 )
(17)
where nb is the refractive index of bulk ferroelectric phase and f is the fractional porosity.
References
[1] J.C. Manifacier, J. Gasiot, J.P. Fillard, J. Phys. E: Sci. Instrum 9, 1002 (1976).
[2] O. S. Heavens, Optical Properties of Thin Solid Films, Dover, New York,
Chapter 4, (1955).
[3] M. Didomenico, Jr, S. H. Wemple, J. Appl. Phys. 40, 720 (1969).
[4] Dekker, A. J. (1985), Solid State Physics, Macmillan India Ltd.
[5] C.H. Peng and S.B. Desu, J. Am. Ceram. Soc. 77, 929 (1994).
[6] W. Chen, X. F. Chen, Z. H. Wang, W. Zhu, and O. K. Tan, J. Mater. Sci. 44, 4939
(2009).
[7] S.J. Kang and Y. S. Yoon, Jpn. J. Appl. Phys. 36, 4459 (1997).
[8] U. Pal, S. Saha, A. K. Chaudhary, V. V. Rao and H. D. Banerjee J. Phys. D 22, 965
(1989).
[9] D. A. G. Bruggeman, Ann. Phys (‚eipzig) 24,636 (1935).
List of Figures
Figure 1 Reflection and transmission of light by a single film deposited on a weakly
absorbing transparent substrate.
Figure 2 Transmittance and reflectance spectra and the Tmax and Tmin envelopes for
undoped PLT film on a completely transparent sapphire (0001) substrate.
Fig. 1 Subhasis Roy et. al
Fig. 2 Subhasis Roy et. al