- Email: [email protected]
Approximations of reflection and transmission coefficients of inhomogeneous thin films based on multiple-beam interference model
Accepted Manuscript Approximations of reflection and transmission coefficients of inhomogeneous thin films based on multiple-beam interference model
Ivan Ohlídal, Jiří Vohánka, Jan Mistrík, Martin Čermák, František Vižd'a, Daniel Franta PII: DOI: Reference:
S0040-6090(19)30138-5 https://doi.org/10.1016/j.tsf.2019.03.001 TSF 37189
To appear in:
Thin Solid Films
Received date: Revised date: Accepted date:
14 September 2018 17 January 2019 1 March 2019
Please cite this article as: I. Ohlídal, J. Vohánka, J. Mistrík, et al., Approximations of reflection and transmission coefficients of inhomogeneous thin films based on multiplebeam interference model, Thin Solid Films, https://doi.org/10.1016/j.tsf.2019.03.001
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Approximations of reflection and transmission coefficients of inhomogeneous thin films based on multiple-beam interference model Ivan Ohlídala, Jiří Vohánkaa, Jan Mistríkb, Martin Čermáka, František Vižd’ac, Daniel Frantaa a
Department of Physical Electronic, Faculty of Science, Masaryk University, Kotlářká 2, 61137
Institute of Applied Physics and Mathematics, Faculty of Chemical Technology, University of
IP
b
T
Brno, Czech Republic
CR
Pardubice, Studentská 95, 532 10, Pardubice, Czech Republic Department of Mathematics and Physics, Faculty of Military Technology, University of Defence,
CE
PT
ED
M
AN
US
Kounicova 65, 662 10 Brno, Czech Republic
AC
c
ACCEPTED MANUSCRIPT
Abstract A multiple-beam interference model is used to derive approximate formulae for the reflection and transmission coefficients of inhomogeneous thin films exhibiting large gradients of refractive index profiles. It is shown that these formulae are constituted by series containing the Wentzel-Kramers-Brillouin-Jeffreys term and correction terms with increasing order
T
corresponding to number of considered internal reflections inside the films. A numerical analysis
IP
enabling us to show the influence of a degree of inhomogeneity on spectral dependencies of
CR
reflectance and ellipsometric parameters of inhomogeneous films is performed. Advantages and disadvantages of our approach compared with other approximate approaches are discussed. The
US
optical characterization of a selected non-stoichiometric silicon nitride film prepared by reactive magnetron sputtering onto silicon single crystal substrate is performed for illustration of using our
AN
formulae in practice.
AC
CE
PT
ED
M
Keywords: reflectance, transmittance, ellipsometric parameters, inhomogeneous thin films
ACCEPTED MANUSCRIPT
1.
Introduction
An enormous attention has been devoted to optics of homogeneous thin films so far. Therefore many papers dealing with problems of the optical characterization and optical synthesis of these films have been published (see e.g. [1–11]). A less attention has been devoted to optics of
T
inhomogeneous thin films exhibiting profiles of the refractive index across these films, i.e. profiles
IP
occurring in direction perpendicular to parallel plane boundaries (see e.g. [12–20]). In spite of this
CR
fact in the three last decades the investigations in optics of inhomogeneous thin films were increased because of two reasons at least. First, the design and production of layered systems were moving to those with continuously varying profiles of their refractive index. The reason of this fact
US
is that such the inhomogeneous layered systems exhibit certain better properties than those consisting of homogeneous layers with mutually different refractive indices. For example, these
AN
inhomogeneous systems with continuous refractive index profiles exhibit substantially lower light scattering caused by boundary roughness than the layered systems containing different
M
homogeneous thin films. The typical representatives of such the inhomogeneous systems are
ED
rugate filters that are utilized as the optical devices with high reflectance within certain spectra l ranges [21, 22]. Second, the development in optics industry, semiconductor technology, solar energetics, microelectronics and other modern branches of fundamental and applied researches
PT
implies a need to create thin films formed by various complex materials. These films are often more or less inhomogeneous from the optical point of view. This inhomogeneity must be
CE
characterized reliably and precisely in order to understand the film structure and properties or remove this phenomena by changing technological conditions at the film preparation. Thus, it is
AC
permanently needed to develop new efficient optical methods enabling us to characterize these complex inhomogeneous thin films. The typical examples of such the complex films frequently exhibiting the optical inhomogeneity are films of non-stoichiometric silicon nitride that are rich in silicon (see e.g. [23, 24]). At the optical characterization and design of the inhomogeneous thin films more complicated theoretical approaches must be employed in comparison with those utilized for homogeneous films. The formulae used in these complicated approaches must ensure to perform correct, precise and rapid calculations of the optical quantities of the inhomogeneous films. A
ACCEPTED MANUSCRIPT sufficient speed of the calculations is very important for achieving efficient computer algorithms utilized in procedures of the optical characterization and design of these inhomogeneous structures. This requirement of the speed of calculations is especially inevitable for real-time monitoring and feedback control of inhomogeneous films and structures at their depositions in technological arrangements (in-situ methods). The speed of the calculations is easily achieved if the refractive index profiles exhibit small gradients, i.e. if refractive indices varies slowly across
T
the films. In this case one can use simple formulae for calculating the optical quantities of the
IP
inhomogeneous thin films corresponding to the Wentzel–Kramers–Brillouin–Jeffreys (WKBJ)
CR
approximation (see e.g. [13, 25]). If gradients of the refractive index profiles are larger and the conditions of using the WKBJ approximation are not fulfilled the approximation based on
US
replacing the inhomogeneous thin films by multilayer systems with a large number of homogeneous thin films is usually used. Then the matrix formalism or recursive formalism is used
AN
to calculate the values of the optical quantities of these inhomogeneous films (see e.g. [13, 20, 26]). However, these formalisms are often slow for real- time monitoring of the deposition of the
M
inhomogeneous films or inhomogeneous layered systems. Therefore, it is necessary to derive other approximate approaches allowing to calculate the optical quantities of such the inhomogeneous
ED
thin films with sufficiently high speed. Of course, the correctness and high precision of the calculations has to be maintained within these approximate approaches.
PT
In paper [27] the sophisticated approximation of reflection coefficients of the inhomogeneous thin films based on modified recursive formalism is pre sented. For deriving the
CE
formulae of this approximation the authors utilized the recursive formula for the four- layer system. Using this formula the systematic trends were revealed for the reflections inside the multilayer
AC
systems as they subdivided the four- layer system into more and more sub-layers. In the limit for infinite number of the sub- layers the formulae for the reflection coefficients of the inhomogeneous films were obtained. These formulae contain terms modifying the formulae valid for the WKBJ approximation. The correction terms comprise the single, double and triple integrals containing gradients of the refractive index profiles in their integrands. In paper [27] it is presented that the use of this approximation allows to reduce the time of the calculation of the reflectance values between one or two orders of magnitude compared with the matrix or recursive formalism for the multilayer system if the concrete linear profile of the inhomogeneous film was taken into account (for details see [27]).
ACCEPTED MANUSCRIPT In paper [27] the approximation of the reflection coefficients of the inhomogeneous layers is also performed using the matrix formalism, i.e. using the corrections to the transfer matrix elements containing the same integrals as in the approach based on the recursive approach (for details see [27]). It should be noted that both the approximate approaches are equivalent. In this paper the new formulae for the reflection and transmission coefficients of the inhomogeneous thin films exhibiting arbitrary refractive index profiles based on multiple-beam
T
interference model are presented. It is shown that the corrections to the WKBJ formulae can be
IP
derived in a simple systematic way. Furthermore, the numerical analysis concerning the optical
CR
quantities of the inhomogeneous films with the selected refractive index profiles is performed. A practical illustration of this new approximate approach is carried out through the optical
US
characterization of a selected sample of the non-stoichiometric silicon nitride film deposited onto silicon single crystal substrate. The practical advantages of the formulae of this approximation are
Theory
M
2.
AN
also introduced.
ED
2.1. Structural model of inhomogeneous thin films The structural model of the inhomogeneous thin films considered in this paper is given by the
PT
following assumptions:
1) film and substrate consist of isotropic absorbing non-magnetic materials
CE
2) film is inhomogeneous, i.e. it exhibits some profile of the complex refractive index represented by a complex function of single variable
AC
3) boundaries of the film are smooth, flat and mutually parallel 4) boundaries are infinitely thin, i.e. transition layers at boundaries are not taken into account
5) ambient is non-absorbing, homogeneous and isotropic 6) substrate is optically homogeneous 7) no defects in volume of the film are taken into account
Figure 1: Schematic diagram of the non-absorbing inhomogeneous thin film: The symbols are explained in the text.
ACCEPTED MANUSCRIPT
The schematic diagram of this model is plotted in Fig. 1. In this figure the symbols are as follows: n0 and nS denote the refractive indices of the ambient and substrate, respectively, nU and nL represent the refractive indices of the film at the upper and lower boundaries, respectively, n( x) denotes the function describing the refractive index profile. Symbols d , 0 and S
T
denote the thickness of the film, incidence angle of light from the side of the ambient and angle of
for
the
system
depicted
in
Fig.
1,
i.e.
CR
law
IP
refraction corresponding to substrate, respectively. The following equation describes the Snell’s
n0 sin 0 = nU sin U = n( x)sin ( x) = nL sin L = nS sin S , where U , ( x) and L denote the
US
refraction angles at the upper boundary, coordinate x and lower boundary, respectively. In general all the quantities introduced above are complex except n0 , d and 0 which are always
AN
real. It is reasonable to introduce the admittances defined as follows: Yv = nv / cos v for the ppolarization and Yv = nv cos v for the s-polarization ( v = 0 , S, U, and L). Moreover, it holds that
M
Y ( x) = n( x) / cos ( x) for the p-polarization and Y ( x) = n( x) cos ( x) for the s-polarization.
Note that the convention with plus in front of imaginary parts of the complex refractive indices is
ED
utilized in this work. Furthermore, it should be noted that this schematic diagram concerns the
PT
non-absorbing inhomogeneous thin film and non-absorbing substrate.
CE
2.2. Reflection coefficients
2.2.1. Incidence of light from the ambient side
AC
Within the WKBJ approximation the following formula is valid for the reflection coefficients r0 of the inhomogeneous thin films [13, 14, 18]:
r0 =
rU rL e
ixd
1 rU rL e
ixd
,
(1)
where rU =
Y0 YU Y Y , rL = L S . Y0 YU YL YS
Symbols Y0 and YS denote the admittances of the ambient and substrate, respectively, and YU
ACCEPTED MANUSCRIPT and YL are the admittances at the upper and lower boundaries of the inhomogeneous thin film, respectively (see Fig. 1). The phase-shift angle xd is given as follows: xd =
4
d
n2 ( x) n02 sin 20 dx,
(2)
0
T
where symbol denotes the wavelength of incident light in vacuum.
IP
Figure 2: Schematic diagrams of the internal single reflections. Full lines represent reflected light
CR
and dashed lines represent transmitted light.
The formula for the first-order corrections of the WKBJ formula can be derived easily in
US
the following way. If one takes into account all the internal single reflections inside the multilayer system approximating the inhomogeneous film (see Fig. 2a) then it is possible to derive the
AN
(1) following formula expressing the reflection coefficients j1,R corresponding to these reflections
as follows:
ED
M
N 1 w (1) j1,R = R rw, w1exp ix j w=1 j =1
where
PT
rw, w1 =
Yw Yw1 , w = 1, 2, Yw Yw1
(3)
, N 1
CE
(symbol N denotes the number of the sub- layers forming the multilayer system considered).
AC
Quantities x j and R are expressed as follows: xj =
4
d j n2j n02 sin 20 ,
where n j , d j and j are the refractive index, thickness and refraction angle of the j -th sub-layer, respectively, and , R = t0,1t0,1
where t0,1 =
for the s-polarization and
2Y0 , Y0 Y1
= t0,1
2Y1 Y0 Y1
ACCEPTED MANUSCRIPT t0,1 =
c0 2Y0 , c1 Y0 Y1
= t0,1
c1 2Y1 c0 Y0 Y1
with c0 = cos 0 and c1 = cos 1 for the p-polarization. Symbol 1 is the refraction angle in the first sub- layer lying under the ambient and n0 sin 0 = n1 sin 1 , where n1 is the refractive index of represent the transmission coefficients of the uppermost this sub- layer. Symbols t0,1 and t0,1
IP
Y0 Y1 = r0,1. , r0,1 Y0 Y1
CR
r0,1 =
T
are expressed as boundary of the system whose reflection coefficients r0,1 and r0,1
The reflection coefficients of the lowest boundary are given in this way: YN YS . YN YS
US
rN , N 1 rN ,S =
AN
It holds that nN sin N = nS sin S , where nN and N are the refractive index and refraction angle of the N -th sub- layer lying above the substrate, respectively. If the number N is
M
sufficiently large the differences between the refractive indices of the adjacent sub- layers are very small. In this case the reflection coefficients rw can be expressed in this way: 1 Yw x [(x)2 ], 2Yw x
ED
rw =
PT
where d j = x = d / N is the thickness of one sub- layer and
[(x)2 ] is the residue. It holds that
Yw = Yw1 Yw .
CE
It is evident that at formulation of the corrections one must take into account all the reflections inside the multilayer system approximating the inhomogeneous thin film. This means that it is also
AC
necessary to consider the reflections from the uppermost and lowest boundaries of the system together with their combinations with the internal reflections (see Fig. 3). Thus, the reflection coefficients of the system corresponding to the first-order correction (see Fig. 2a) and a finite number of all the reflections inside this system
where
(1) 1,R
(1) 1,R
are expressed as follows:
N 1 m w = R s ( s 1) rw, w1exp ix j , w=1 s =1 j =1
(4)
ACCEPTED MANUSCRIPT
N
j =1
= r0,1rN ,S exp ix j which describes complete single passing of light between the uppermost and lowest boundaries with single reflections on these boundaries and m denotes integer (see Fig. 3).
T
Figure 3: Schematic diagram related to calculating all possible combinations of the single
IP
reflections and reflections between the uppermost and lowest boundaries of the inhomogeneous
CR
4 thin film (here s = 4 , number of all the combinations is equal to = 4 ). 1 It is apparent that one can write ( s 1)
m s =1
d m s 1 = . m d s =1 (1 ) 2
= lim
US
m
lim s
AN
After performing the limit for N going to infinity we obtain the expression for the partial (1) first-order corrections I1,R of the reflection coefficients as follows:
where (1) 1R
M
d R ix ( x ) (1) = , j1 = f ( x1 )e 1 dx1 , 2 (1 ) 0
PT
C
R d ix ( x ) (1) f ( x1 )e 1 dx1 = C1(1) R j1 , 2 (1 ) 0
ED
(1) I1,R =
x( x1 ) =
AC
CE
f ( x1 ) = 4
x1
1 dY ( x1 ) , 2Y ( x1 ) dx1 n 2 ( x) n02 sin 20 dx,
0
= r0,1rN ,Se d . ix
, where Quantity R is again given as R = t0,1t0,1
t0,1 =
2Y0 2YU = , t0,1 Y0 YU Y0 YU
for the s-polarization and t0,1 =
c0 2Y0 c 2YU = U , t0,1 cU Y0 YU c0 Y0 YU
(5)
ACCEPTED MANUSCRIPT for the p-polarization. (1)
In the same way the formula for the partial first-order correction I 1,R corresponding to internal single reflections of the type depicted in Fig. 2b is derived, i.e. (1)
(1)
(1)
(6)
I 1,R = C1,R j1 ,
(1)
d RZ 2 ix ( x ) (1) , j = f ( x1 )e 1 dx1 1 2 (1 ) 0
IP
C1,R =
T
where
ix
Z = rN ,Se d .
CR
and
US
The total corrections of the first-order r1 are then given for both the polarizations as (1) (1) r1 = I1,R I1,R .
(7)
AN
Each of the partial first-order corrections generates three partial corrections of the second
M
order which is evident from Fig. 4.
ED
Figure 4: Basic motifs of the internal reflections corresponding to the partial second-order corrections. Full lines represent the internal reflections while dashed lines represent transmitted
PT
light.
CE
In this figure the basic motifs of all the double internal reflections are plotted. Thus, the (1) (1) (1) (2) , I 2,R , corrections I1,R and I1,R generate the following partial second-order corrections I 2,R
AC
(3) (1) (2) (3) . If we take into account the mathematical procedure used to derive the I 2,R , I 2,R , I 2,R and I 2,R
first-order corrections we get the following expressions for all the partial second-order corrections:
ACCEPTED MANUSCRIPT (l ) (l ) (l ) (l ) I 2,R = C2,R j2( l ) , I 2,R = C2,R j2( l ) , l = 1, 2,3,
C
(1) 2,R
=
(1) r0,1C1,R
dd
(1) 2
, j
1
= f ( x1 ) f ( x2 )e
C
dx2dx1 ,
00
d x1
(2) 2,R
i[ x ( x1 ) x ( x2 )]
(2) 2
= ZC1,R , j
= f ( x1 ) f ( x2 )e
i[ x ( x1 ) x ( x2 )]
dx2dx1 ,
00
(1) dd ZC1,R i[ x ( x ) x ( x )] , j2(3) = f ( x1 ) f ( x2 )e 1 2 dx2dx1 , 1 00
T
(3) C2,R =
dd
1 (1) i[ x ( x1 ) x ( x2 )] C1,R , j2(1) = f ( x1 ) f ( x2 )e dx2dx1 , Z 0x
IP
(1) C2,R =
, j2(2) = f ( x1 ) f ( x2 )e
1 (1) ZC1,R
1
i[ x ( x1 ) x ( x2 )]
00
dd
, j2(3) = f ( x1 ) f ( x2 )e 00
i[ x ( x1 ) x ( x2 )]
dx2dx1 ,
dx2dx1.
AN
(3) C2,R =
r C
US
(2) C2,R =
dd
CR
1
(1) 0,1 1,R
(8) (2)
(2) (1) (3) = I 2,R = I 2,R . From Fig. 4 it is clear that the following equalities are fulfilled: I 2,R and I 2,R
M
The total second-order corrections r2 are again given as the sum of these partial corrections, i.e. (l ) (l ) r2 = I 2,R I 2,R .
ED
3
(9)
l =1
In general the partial corrections of the ( n 1 )-th order are easily derived by means of the
PT
corrections of the n -th order using the geometrical constructions presented in Figs.5-10. Each correction of n -th order gives three corrections of (n 1) -th order. When these three new
CE
corrections are derived it is necessary to distinguish two cases. In the first case the last internal reflection in the n -th order correction falls onto the upper boundary. In the second case it falls
AC
onto the lower boundary. If the last internal reflection falls onto the upper boundary, then from Fig. 5 it is implied that the partial correction of the n -th order is given as d
I
(l ) n ,R
(l ) n ,R
=C
0
n ix ( x ) f ( xq ) e 1 q =1
e
ix ( xn )
dxn
dx1 ,
where = 0, = d or = xn1 , = d . Figure 5: Schematic diagram of the n 1 internal reflections inside the film
(10)
ACCEPTED MANUSCRIPT 1) From Fig. 5 it is also obvious that d
n 1 f ( xq ) 0 q =1
d
jn(3l1 2) = 0
e
ix ( x1 )
e
ix ( xn ) ix ( xn 1 )
e
dxn 1
(11)
dx1
and ,
T
1
(12)
,3n1.
CR
where l = 1, 2,
r0,1Cn(l,R)
IP
2) Cn(3l1,R =
US
Figure 6: Schematic diagram of the n 1 internal reflections inside the film
2) If the schematic diagram in Fig. 6 is taken into account, one can derive the following
j
= 0
n 1 f ( x ) q 0 q =1
ix ( x1 )
e
ix ( xn ) ix ( xn 1 )
e
dxn 1
(13) dx1
ED
e
xn
d
M
(3l 1) n 1
AN
formulae:
and
1) Cn(3l1,R = ZCn(l,R) .
PT
(14)
CE
Figure 7: Schematic diagram of the n 1 internal reflections inside the film
AC
3) Furthermore, from Fig. 7 one can imply d
n 1 f ( xq ) 0 q =1
d
(3l ) n 1
j
= 0
e
ix ( x1 )
e
ix ( xn ) ix ( xn 1 )
e
dxn 1
(15)
dx1
and (3l ) n 1,R
C
ZCn(l,R) = . 1
Figure 8: Schematic diagram of the n 1 internal reflections inside the film
(16)
ACCEPTED MANUSCRIPT If the last internal reflection in the n -th order corrections falls onto the lower boundary, then the corresponding n -th order partial corrections is given as
n ix ( x1 ) f ( x ) e q q =1
d
(l ) n
j
= 0
e
ix ( xn )
dxn
dx1
(17)
where = 0, = d or = 0, = xn1. d
ix ( x1 )
ix ( xn ) ix ( xn 1 )
e
e
dxn 1
(3l 2) n 1,R
=
Cn(l,R) Z
(19)
AN
C
dx1
US
and
(18)
CR
0
IP
n 1 f ( xq ) x q =1 n
d
jn(3l1 2) = e
T
4) From Fig. 8 it is seen that
M
Figure 9: Schematic diagram of the n 1 internal reflections inside the film
ED
5) One can see from Fig. 9 that (3l 1) n 1
j
= 0
ix ( x1 )
PT
e
e
CE
and
d
n 1 f ( x ) q 0 q =1
d
ix ( xn ) ix ( xn 1 )
(3l 1) n 1,R
=
dxn 1
dx1
r0,1Cn(l,R)
(21)
1
AC
C
e
(20)
Figure 10: Schematic diagram of the n 1 internal reflections inside the film
6) On the basis of Fig. 10 it is clear that d
n 1 f ( xq ) 0 q =1
d
jn(3l1) = 0
e and
ix ( x1 )
e
ix ( xn ) ix ( xn 1 )
e
dxn 1
(22)
dx1
ACCEPTED MANUSCRIPT ZCn(l,R)
) Cn(3l1,R =
1
(23)
.
As mentioned above each of the partial corrections of the n -th order I n(l,R) generates three 2) 1) ) . Thus, the , I n(3l1,R partial corrections of the ( n 1)-th order, i.e. corrections I n(3l1,R and I n(3l1,R
corrections of all the orders calculated by the foregoing procedure create the tree as follows:
(1) (2) (3) 3,R 3,R 3,R
(2) I 2,R
(3) I 2,R
(4) (5) (6) 3,R 3,R 3,R
I I I
I I I
(24)
CR
I I I
(7) (8) (9) 3,R 3,R 3,R
IP
(1) I 2,R
T
(1) I1,R
the exponential factor e
ix ( x1 )
US
The same tree can be constructed for the corrections I n(,Rl ) . Of course, in corrections I n(,Rl ) must be systematically replaced by e
ix ( x1 )
and the coefficients Cn(l,R)
AN
are expressed with other formulae compared with coefficients Cn(l,R) (see e.g. the formulae for (l )
M
(l ) (l ) (l ) (l ) C2,R and C2,R in (8) and C3,R and C3,R in Table 1). Note that for the partial corrections I n ,R
the same general recursive formulae are valid as for the partial corrections I n(l,R) . Only the changes
ED
expressed in the foregoing sentence must be performed in the formulae for I n(,Rl ) . The total
rn =
3n 1
I
(l ) n ,R
I n(,Rl ) .
(25)
l =1
l
1
(l ) C3,R
(1) r0,1C2,R
1
AC
CE
PT
correction of the n -th order rn is given as follows
2
ZC
j3(l )
I 3,R (l )
(l )
j3
C 3,R
ddd
3 ix ( x ) ix ( x ) ix ( x ) (1) f ( xq ) e 1 e 2 e 3 dx3dx2dx1 r0,11C2,R 0 0 0 q =1
3 ix ( x ) ix ( x ) ix ( x ) f ( xq ) e 1 e 2 e 3 dx3dx2dx1 0 x1 0 q =1
3 ix ( x1 ) ix ( x2 ) ix ( x3 ) (1) f ( x ) e dx3dx2dxZC e e q 1 2,R 0 0 0 q =1
3 ix ( x1 ) ix ( x2 ) ix ( x3 ) f ( x ) e e dx3dx2dx1 e q 0 x1 0 q =1
d d x2
(1) 2,R
(l )
(l ) I 3,R
ddd
d d x2
ACCEPTED MANUSCRIPT 3 ix ( x ) ix ( x ) ix ( x ) f ( xq ) e 1 e 2 e 3 dx3dx2dx1 0 x1 0 q =1
d x1 d
(2) C2,R
3 ix ( x ) ix ( x ) ix ( x ) (2) f ( xq ) e 1 e 2 e 3 dx3dx2dxr10,1C2,R 1 0 0 x2 q =1
3 ix ( x1 ) ix ( x2 ) ix ( x3 ) f ( x ) e e dx3dx2dx1 e q 0 0 0 q =1
(2) r0,1C2,R
3 ix ( x1 ) ix ( x2 ) ix ( x3 ) (2) f ( x ) e e dx3dx2dx1ZC2,R e q 0 0 0 q =1
3 ix ( x1 ) ix ( x2 ) ix ( x3 ) f ( x ) e e dx3dx2dx1 e q q =1 00 0
(2) ZC2,R
3 ix ( x ) ix ( x ) ix ( x ) (2) f ( xq ) e 1 e 2 e 3 dx3dx2dx1ZC2,R 1 0 0 0 q =1
Z
d x1 d
5
1
6
1
3 ix ( x1 ) ix ( x2 ) ix ( x3 ) f ( x ) e e dx3dx2dx1 e q 0 0 x2 q =1
(3) r0,1C2,R
3 ix ( x1 ) ix ( x2 ) ix ( x3 ) (3) f ( x ) e dx3dx2dx1r0,1C2,R e e q 1 0 0 0 q =1
3 ix ( x1 ) ix ( x2 ) ix ( x3 ) f ( x ) e e dx3dx2dx1 e q 0 0 0 q =1
(3) ZC2,R
3 ix ( x ) ix ( x ) ix ( x ) (3) f ( xq ) e 1 e 2 e 3 dx3dx2dxZC 1 2,R 1 0 0 0 q =1
Z
AN
(3) C2,R
ED
1
M
ddd
8
1
PT
ddd
9
3 ix ( x ) ix ( x ) ix ( x ) f ( xq ) e 1 e 2 e 3 dx3dx2dx1 0 0 0 q =1 ddd
3 ix ( x1 ) ix ( x2 ) ix ( x3 ) (3) C2,R f ( x ) e dx3dx2dx e e q 1 Z 0 0 x2 q =1 dd d
7
d d x2
US
d x1 d
ddd
T
ddd
IP
4
3 ix ( x ) ix ( x ) ix ( x ) (1) f ( xq ) e 1 e 2 e 3 dx3dx2dx1 ZC2,R q =1 000 1
CR
3
ddd
(1) ZC2,R 1
dd d
ddd
3 ix ( x ) ix ( x ) ix ( x ) f ( xq ) e 1 e 2 e 3 dx3dx2dx1 0 0 0 q =1 ddd
CE
(l ) (l ) Table 1: The coefficients C3,R and C3,R together with integrals
j3(l ) and j3(l ) corresponding to
AC
(l ) (l ) . the partial corrections of the third order I 3,R and I 3,R
In Table 1 the expressions for the coefficients and integrals of the corrections of the third (l ) (l ) , C3,R , j3(l ) and j3(l ) , are introduced. It can easily be proved that the following order, i.e. C3,R (2) (1) (3) (4) (6) (7) (9) (8) = I3,R , I3,R = I3,R , I3,R = I3,R , I3,R = I3,R , equalities are valid: I3,R
Figure 11: Schematic diagrams corresponding to the partial corrections of the first order at the incidence of light from the non-absorbing substrate side.
ACCEPTED MANUSCRIPT The approximate expression for the reflection coefficients of the inhomogeneous thin films
rM are given as follows: M
rM = r0 rn ,
(26)
n =1
where M denotes the maximum order of the total corrections needed for achieving the required
T
accuracy of calculations of reflectances and ellipsometric quantities of the inhomogeneous thin
IP
films for reflected light in case of light incidence from the ambient side.
CR
2.2.2. Incidence of light from the substrate side
If the substrates are non-absorbing the reflection coefficients of the inhomogeneous thin films are
US
also defined for light falling onto the films from the side of these substrates. The expressions of these coefficients are derived in the same way as those presented for light incident from the
(1) and I 1,R are expressed as ambient side. In this case the partial corrections of the first order I1,R
AN
(1)
(see Fig. 11)
M
(1) (1) (1) = C1,R (1) j1(1) , I '1,R I1,R = C'1,R j1(1) ,
ED
where
PT
(1) = C1,R
CE
and
(1) 1,R
C'
(1) R e d (1) , j = j 1 1 (1 )2 ix
r2 e d = R 0,1 2 , j1(1) = j1(1) (1 ) ix
AC
and R = t N ,St N ,S . Transmission coefficients t N ,S and t N ,S obey these equations: t N ,S =
2YS 2YL , t N ,S = YS YL YS YL
for the s-polarization and t N ,S =
c 2YS cL 2YL , t N ,S = S cS YS YL cL YS YL
for the p-polarization ( cS = cos S and cL = cos L ). It is evident that
(27)
ACCEPTED MANUSCRIPT ix
2 (1) ix (1) r e dC (1) C = R e d C1,R , C'1,R = R 0,1 2 1,R . R R Z (1) 1,R
Using the corresponding schematic diagrams of the basic motifs of the reflections one obtains the formulae for the partial corrections of the second order corresponding to light incident onto the inhomogeneous thin films from the substrate side, i.e.
(l ) = C2,R (l ) j2(l ) , I 2,R (l ) = C2,R (l ) j2(l ) , l = 1, 2,3 I 2,R
T
(28)
Z (1) , j2(1) = j2(3) , C1,R 1
(2) = r0,1C1,R (1) , j2(2) = j2(1) , C2,R
(2) = C2,R C
=
(1) ZC1,R 1
, j2(1) = j2(2) ,
(29)
, j2(2) = j2(3) ,
(1) r0,1C1,R
, j2(3) = j2(1) ,
1
ED
(3) 2,R
r0,1
US
=
(1) C1,R
M
C
(1) 2,R
r0,1 (1) (2) , j2(3) = j 2 , C1,R 1
AN
(3) = C2,R
CR
(1) = C2,R
IP
where
PT
(3) = I 2,R (2) and I 2,R (2) = I 2,R (1) . It is also It is evident that the following equalities are true: I 2,R possible to derive the general recursive formulae for the partial corrections corresponding to this
CE
incidence of light. Taking into account the individual reflections inside the film analogous to those
1)
where
AC
utilized for deriving eqs. (11)-(23) the following recursive formulae are obtained for this case: l 2) l 2) (3l 2) I n(31,R = Cn(31,R jn 1
l 2) Cn(31,R = d
jn(31l 2) = 0
e
ix ( x1 )
e
Z Cn(,Rl ) , 1
d
n 1 f ( x ) q 0 q =1
ix ( xn ) ix ( xn 1 )
where = 0, = d or = 0, = xn1.
(30)
e
dxn 1
dx1 ,
ACCEPTED MANUSCRIPT 2) l 1) l 1) (3l 1) I n(31,R = Cn(31,R jn 1
(31)
where l 1) Cn(31,R = r0,1Cn(,Rl ) ,
d
e
ix ( x1 )
e
ix ( xn ) ix ( xn 1 )
e
dxn 1
dx1.
CR
3) l) l) I n(31,R = Cn(31,R jn(31l )
=
d
jn(31l ) = 0
1
d
e
ix ( xn ) ix ( xn 1 )
e
dxn 1
dx1.
ED
4)
,
n 1 f ( x ) q q =1 0
M
e
ix ( x1 )
r0,1 Cn(,Rl )
AN
C
l 2) l 2) (3l 2) I n(31,R = Cn(31,R jn 1
AC
CE
PT
where
j
l 2) Cn(31,R =
(3l 2) n 1
d
= 0
e
ix ( x1 )
e
(32)
US
where (3l ) n 1,R
IP
0
T
n 1 f ( x ) q xn q =1
d
jn(31l 1) =
Cn(,Rl ) r0,1
(33)
,
xn
n 1 f ( x ) q 0 q =1
ix ( xn ) ix ( xn 1 )
e
dxn 1
dx1 ,
where = 0, = d or = xn1 0, = d . 5) l 1) l 1) (3l 1) I n(31,R = Cn(31,R jn 1
where
(34)
ACCEPTED MANUSCRIPT l 1) Cn(31,R =
j
(3l 1) n 1
d
n 1 f ( x ) q 0 q =1
d
= 0
e
ix ( x1 )
e
Z Cn(,Rl ) , 1
ix ( xn ) ix ( xn 1 )
e
dxn 1
dx1.
6)
n 1 f ( xq ) 0 q =1
= 0
e
ix ( xn ) ix ( xn 1 )
e
dxn 1
dx1.
AN
e
ix ( x1 )
,
US
j
1
d
d
(3l ) n 1
r0,1Cn(,Rl )
CR
where l) Cn(31,R =
(35)
IP
T
l) l) I n(31,R = Cn(31,R jn(31l )
The similar recursive formulae can be derived for the corrections I n(,Rl ) . The reflection coefficients
M
of the inhomogeneous films rM for this incidence of light are given in the following way: M
ED
rM = r0 rn,
AC
and
CE
PT
where
r0 =
rn =
(36)
n =1
rL rU e
ixd
1 rL rU e
3n 1
I
(l ) n ,R
ixd
I n(,Rl ) .
l =1
2.3. Transmission coefficients The transmission coefficients for the inhomogeneous thin films placed on the non-absorbing substrates must also be considered for both the cases of incidence of light.
2.3.1. Incidence of light from the ambient side Schematic diagram of beams corresponding to the WKBJ approximation in transmitted light is
ACCEPTED MANUSCRIPT plotted in Fig. 12. In this figure the reflections between the upper and lower boundaries are plotted by the full line segments. These reflections must be completed with the beams (waves) reflected from the upper boundary and transmitted by the lower boundary (in Fig 12 such the beam is plotted as the dashed line segment). If we take into account the approximation of the inhomogeneous thin film by the multilayer system the transmission of this wave through this system is described by the
IP
N 1 N ix /2 t = tk ,k 1 e k , k =1 k =1
T
following coefficient t : (37)
CR
Figure 12: Schematic diagram of selected beams corresponding to the WKBJ approximations in
US
transmitted light.
where tk ,k 1 denotes the transmission coefficients of the internal boundaries between the
N
AN
uppermost and lowest boundary of the system. Furthermore, it holds that lim e
For the s-polarization we can write that
=e
ixd /2
.
(38)
M
N k =1
ixk /2
ED
N 1 N 1 ln tk ,k 1 = ln tk ,k 1 = k =1 k =1
N1
is the residual. Hence
AC
where
(39)
CE
PT
N 1 1 Yk 1 = ln 1 x = N k =1 2Yk x 1 N 1 1 Yk 1 = x , 2 k =1 Yk x N
N 1 N 1 ln t = ln lim k ,k 1 lim tk ,k 1 = N k =1 N k =1 d Y 1 1 dY ( x) = dx = ln U . 2 0 Y ( x) dx YL
From the foregoing it is evident that limt =
N
YU YL
e
ixd /2
.
(40)
ACCEPTED MANUSCRIPT In the same way it is obtained for the p-polarization that cU cL
limt =
N
YU YL
e
ixd /2
(41)
.
Thus, the transmission coefficients belonging to the WKBJ approximation are given as follows: ixd /2
1 r0,1rN ,Se
ixd
(42)
,
T
t0 = B
t0,1t N ,Se
YU YL
,
cU cL
YU YL
AN
B=
US
for the s-polarization and
CR
B=
IP
where
for the p-polarization. Beams (waves) corresponding to the partial corrections of the first and
M
second orders are introduced in Figs. 2 and 4. By means of these figures the partial corrections of (1) (1) the first order I1,T and I1,T are expressed in this way:
ED
(1) (1) (1) I1,T = C1,T j1
PT
where
T ix /2 (1) (1) r0,1Be d C1,R = PC1,R R
(44)
T = t0,1t N ,S .
AC
and
CE
(1) C1,T =
(43)
(1) For the correction I1,T it holds that (1) (1) (1) I1,T = C1,T j1 ,
where (1) C1,T =
T B ixd /2 (1) (1) e C1,R = QC1,R . R rN ,S
In the foregoing equations quantities P and Q are given as
(45)
ACCEPTED MANUSCRIPT P=
T B ixd /2 ix /2 r0,1Be d , Q = T e . R R rN ,S
Using Figs. 2 and 4 it is also possible to derive the formulae for the partial corrections of the second order for the inhomogeneous film in transmitted light. The achieved formulae are as follows: (1) (1) (1) (1) (1) I 2,T = C2,T j2 , C2,T = PC2,R ,
(1)
(1)
(1)
(1)
(1)
(2)
(2)
(2)
(2)
(2)
(3)
(3)
(3)
(3)
(3)
(46)
CR
I 2,T = C 2,T j 2 , C 2,T = PC 2,R ,
IP
(3) (3) (3) (3) (3) I 2,T = C2,T j2 , C2,T = QC2,R ,
T
(2) (2) (2) (2) (2) I 2,T = C2,T j2 , C2,T = QC2,R ,
I 2,T = C 2,T j 2 , C 2,T = PC 2,R ,
US
I 2,T = C 2,T j 2 , C 2,T = QC 2,R .
From the foregoing one can imply that the formulae for partial corrections for transmitted light can
AN
be inferred by using the formulae for the reflected light in an easy way. It is sufficient to multiply
M
(l ) (l ) the corresponding coefficients C2,R or C2,R by P or Q in order to obtain the correct formulae
for the partial corrections belonging to transmitted light. If the last internal reflected beam falls
ED
(l ) (l ) onto the upper boundary it is necessary to multiply the corresponding coefficients C2,R or C2,R
by P . If the last internal beam falls onto the lower boundary one must multiply the corresponding
PT
(l ) (l ) coefficients C2,R or C 2,R by Q . This enables us to state that the general recursive formulae for
the partial corrections corresponding to transmitted light are given by the formulae valid for
CE
reflected light (see eqs. (11)-(23)) with the mentioned changes in coefficients Cn(l,R) or Cn(l,R) . The
AC
approximate formula for the transmission coefficients t M of the inhomogeneous thin films for light incident from the ambient side is given as follows: M
tM = t0 tn , n =1
where tn =
3n 1
I
(l ) n ,T
l =1
2.3.2. Incidence of light from the substrate side
I n(,Tl ) .
(47)
ACCEPTED MANUSCRIPT From Figs. 2, 4 and 11 it is apparent that the formulae for the partial corrections corresponding to transmitted light at incidence of light from the substrate side can easily be formulated by means of those presented for reflection of light at the same incidence (see eqs. (27)-(35)). Thus, one can write the following equations for the corrections of the first order: (1)
(1) = C1,T (1) j1 , I1,T (1) = C1,T (1) j1(1) , I1,T
(48)
T B ixd /2 ix /2 rN ,S Be d , Q = T e , R R r0,1 1 t N ,S . , T = t0,1 B
US
B =
CR
P =
IP
(1) = PC1,R (1) , C1,T (1) = QC1,R (1) , C1,T
T
where
From the foregoing it is clear that if the last internal beam falls onto the lower boundary the
AN
coefficients Cn(,Rl ) or Cn(,Rl ) must be multiplied with P and if this beam falls onto the upper boundary these coefficients must be multiplied by Q in order to obtain the correction
M
(1) (see eqs. (29)). All the conclusions and rules valid for the corrections coefficients Cn(,Tl ) and C1,T
ED
I n(l,T) and I n(,Tl ) . are also fulfilled for the corrections I n(,Tl ) and I n(,Tl ) . The general recursive formulae for the partial corrections I n(,Tl ) and I n(,Tl ) can be easily derived using those valid for the partial
PT
corrections I n(,Rl ) and I n(,Rl ) corresponding to light reflected at incidence of light from the substrate
CE
side.
The approximate transmission coefficients for the inhomogeneous thin films at the
where
AC
incidence of light from the non-absorbing substrate obey the following formula: M
tM = t0 tn , n =1
t0 = B
t N ,Se t0,1
ixd /2
1 r0,1rN ,Se
ixd
and tn =
3n 1
I
(l ) n ,T
l =1
I n(,Tl ) .
(49)
ACCEPTED MANUSCRIPT It is possible to find the relationship between the transmission coefficients t M and t M by means of the following simple procedure. For the s-polarization these coefficients can be written as tM =
YU YL
t0,1t N ,S F ( , 0 )
(50)
t N ,S F ( , 0 ), t0,1
(51)
YL YU
IP
tM =
T
and
CR
where F ( , 0 ) represents the quantity describing the influence of all the reflections on expressing the transmission coefficients for both the directions, i.e. for the directions belonging to
US
the incidence from the ambient side and substrate side. For each path represe nting the contribution to t M there is the same path, but with opposite direction of propagation, contributing to tM .
AN
Since the mathematical representation of the contribution to F ( , 0 ) corresponding to this path is the same for t M and tM , it is evident that the functions F ( , 0 ) in eqs. (50) and (51) must be
M
, identical. From eqs. (50), (51) and the formulae for the transmission coefficients t0,1 , t N ,S , t0,1
ED
t N ,S one can infer that
(52)
PT
Y t t Y tM = U 0,1 N ,S = S . tM YL t0,1 t N ,S Y0
The same relation is true for the p-polarization. Relation (52) is correct for the approximation of
CE
arbitrary order M . Note that the results contained in the last two subsections explain why it is unnecessary to consider the influence of products of the internal transmission coefficients at
AC
derivations of individual corrections occurring in the formulae for the reflection coefficients of the inhomogeneous films. The product of B and B gives unity for every single transmission through the film with the phase-shift angle xd , i.e. for the single path: upper boundary-lower boundary-upper boundary (see equations for B and B ). If the equations for the transmittances of the p and s-polarizations are taken into account, i.e Tj =
nS cos S n cos 0 | tM |2 , T j = 0 | tM |2 ( j = p, s), n0 cos 0 nS cos S
together with eq. (52) one can easily derive that T j = T j for non-absorbing and absorbing
ACCEPTED MANUSCRIPT inhomogeneous thin films for any correction order M (symbol | | denotes the module of the complex quantity placed inside this symbol).
3.
Evaluation of the integrals using the Chebyshev polynomials
3.1. Recursive relations
IP
T
By examining the structure of the integrals jn(l ) and jn(l ) one can notice that for each order there are only two integrals in which the dependencies of the integral bounds are such that they cannot
CR
be written as products of integrals corresponding to lower orders. For the first, second and third order these integrals are given as
US
j1 = j1(1) , j1 = j1(1) , j2 = j2(2) ,
j2 = j2(1) , j3 = j3(4) , j3 = j3(2) ,
AN
where the symbols jn and jn are used to denote the indecomposable integrals for a given order
n . All the other integrals that we have encountered can be written as products of these integrals,
M
for example j3(1) = ( j1 )3 and j3(5) = j1 j2 . We will present a method that allows us to calculate the
and g n ( x1 ) . For n = 1, 2
ED
integrals jn and jn recursively. For this purpose we introduce sequences of functions g n ( x1 ) we define these functions as d
PT
g n ( x1 ) = g n 1 ( x2 ) f ( x2 )e x1
x1
g n ( x1 ) = g n 1 ( x2 ) f ( x2 )e
CE
ix ( x2 )
dx2 ,
ix ( x2 )
0
(53)
dx2 ,
and for n = 0 we define g0 ( x1 ) = g0 ( x1 ) = 1 . The integrals that we want to calculate are then
AC
obtained by substituting the values 0 and d for the variable x1 . They are given as
jn = gn (0), jn = gn (d ).
(54)
Performing the numerical quadrature for each value of the variable x1 occurring in the bounds of integrals defining the sequences of functions g n ( x1 ) and g n ( x1 ) would be impractical. A much better approach is to approximate the functions on the interval [0, d ] by polynomials, which can be then integrated analytically.
ACCEPTED MANUSCRIPT 3.2. Chebyshev polynomials In this subsection an efficient method for evaluation of the integrals in the recursive formulae (53) will be presented. This method is based on the idea that the Chebyshev interpolation can be used to approximate the integrands by the polynomials that can be then integrated analytically. Then, the speed of calculating the integrals is extremely increased.
T
The Chebyshev polynomials of the first kind Tk (t ) are defined on the interval [1,1] and
IP
non-negative integers k as [28, 29]
(55)
CR
Tk (t ) = cos(k arccos t ).
Since we are not working with functions defined on this interval but on the interval [0, d ] , it is
US
necessary to map the variable x1 to interval [1,1] as t = 2 x1 / d 1 . Let us assume that g ( x1 ) is some function defined on the interval [0, d ] for which we
AN
want to construct the polynomial approximation. The Chebyshev interpolation polynomial gˆ ( x1 ) at H 1 points is a polynomial of degree H which coincides with this function at H 1 points
M
ak given as the zeros of the Chebyshev polynomial TH 1 . These points, called the Chebyshev nodes, can be calculated as
(56)
, H . Note that we used the hat to distinguish the interpolation polynomial from the
PT
where k = 0,
d (k 12 ) cos 1 , 2 H 1
ED ak =
CE
function that it approximates. The interpolation polynomial can be written in the basis of the Chebyshev polynomials as
AC
gˆ ( x1 ) =
v0 H vkTk (2 x1 / d 1), 2 k =1
(57)
with the coefficients vk given as
vk =
2 H k ( p 12 ) g (a p ) cos . H 1 p =0 H 1
(58)
If we want to calculate the values at the Chebyshev nodes, then we can insert the representation of the Chebyshev polynomials (55) and the formula for the Chebyshev nodes (56) into (57) and write the result as
ACCEPTED MANUSCRIPT H 1 p(k 12 ) g (ak ) = v0 v p cos . 2 H 1 p =1
(59)
The formulae (58) and (59) are important since they allow us to convert between two representations of the Chebyshev interpolation polynomial, i.e. between the representation using the coefficients in the expansion in the Chebyshev polynomials and the representation using the values at the Chebyshev nodes. The representation using the expansion in the Chebyshev
T
polynomials is useful for integration of the polynomials while it is much easier to specify the
IP
function in the representation using the values at the Chebyshev nodes. Apart from some unusual
CR
normalization, the formulae (58) and (59) are the discrete cosine transforms DCT-II and DCT-III [30]. These discrete cosine transforms can be performed by algorithms that are much faster than
US
the direct evaluation of the sums in (58) and (59), therefore, they provide an efficient tool for conversion between the two representations.
AN
The integral over the interval [0, d ] of the function g ( x1 ) can be approximated by the value obtained by integrating its Chebyshev approximation gˆ ( x1 ) . The result d
0
gˆ ( x1 )dx1 =
2vk ), 2 k even ,2 k H 1 k
d (v0 2
M
(60)
ED
is known as the Clenshaw–Curtis quadrature [31]. This formula is useful if we want to calculate the definite integrals, but we also need to calculate the primitive functions. The primitive function
PT
to the polynomial gˆ ( x1 ) of degree H is a polynomial of degree H 1 . Since we want to work with polynomials of the fixed degree H , the part of the integrated polynomial proportional to the
CE
Chebyshev polynomial TH 1 will be neglected. The fact that the term proportional to TH 1 is neglected will be indicated by using the symbol
instead of the equality sign. The integrals of
AC
gˆ ( x1 ) considered as a functions of the upper and the lower bounds are then given as
x1
0
gˆ ( x2 )dx2
d Gˆ ( x1 ), gˆ ( x2 )dx2 x1
Gˆ ( x1 ),
(61)
with the polynomials Gˆ ( x1 ) and Gˆ ( x1 ) given in the same way as gˆ ( x1 ) in (57) but with the coefficients vk replaced by Vk and Vk . The zeroth coefficients are given as
ACCEPTED MANUSCRIPT V0 =
d v0 v1 H 1 (1) k vk (1) H vH , 2 2 4 k =1 1 k 2 2( H 1)
(62)
vH d v v H 1 v V = 0 1 k 2 , 2 2 4 k =1 1 k 2( H 1) 0
, H 1 and for the last coefficient we have
Vk =
d vk 1 vk 1 d vH 1 . , VH = 2 2k 2 2H
(63)
T
for k = 1,
IP
Note that since the Chebyshev polynomials are bounded by | Tk | 1 , the error that we introduce by
CR
neglecting the part proportional to TH 1 is at most | VH1 |=| vH | /(2H 2) . This should not be a
US
problem because the polynomial gˆ ( x1 ) is only an approximation of the function g ( x1 ) and if the coefficient vH is too large, then higher order of the interpolation polynomial should be used to
AN
increase the precision of the calculations.
M
3.3. Evaluations of the integrals
step we start with n = 1 ):
ED
The steps needed to calculate the integrals jn and jn can be summarized as follows (in the first
Calculate the values of the integrands in (53) at the Chebyshev nodes (56).
Use the discrete cosine transform DCT-II to calculate the coefficients (58) for the
PT
Chebyshev interpolation polynomials of these integrands. Using the formula (59) calculate the values of jn and jn .
Calculate the coefficients (62), (63). For the function g n ( x1 ) we need to calculate the
AC
CE
integral as a function of the lower bound (coefficients Vk ), while for the other function
gn ( x1 ) the integral is a function of the upper bound (coefficients Vk ).
Using the discrete cosine transform DCT-III calculate the values of the functions g n ( x1 ) and g n ( x1 ) at the Chebyshev nodes. These values will be needed to calculate the integrands in the next recursive step of (53).
Repeat the above steps for the next order.
ACCEPTED MANUSCRIPT 4.
Numerical analysis
The equalities among some corrections of the second and third orders occurring in the formulae for the reflection coefficients are presented above. They can be found using the schematic diagrams of reflections inside the films as performed above. However, for the higher orders numerical calculations must be utilized for finding these equalities. This fact implies that only certain
T
corrections must be calculated numerically within the tasks of the optical characterization and
IP
synthesis of the inhomogeneous thin films. These corrections can be called the unique corrections. The numbers of the unique corrections belonging to the individual orders in the formula for
CR
reflection and transmission coefficients rM and t M are introduced in Table 2. In this table the numbers of all the corrections are also presented for comparison. The numbers in Table 2 were
US
found using a computer program which generated all the basic motifs for a given order. The motifs which represent contributions to reflection and transmission coefficients with the same
AN
mathematical representation were then collected into groups representing unique contributions. The number of the motifs in a given group determines the multiplicity for a corresponding unique
M
contribution. Table of the numbers of unique corrections corresponding to reflection coefficient
ED
rM and transmission coefficients t M exhibit the same results.
PT
Table 2: Numbers of unique and total corrections for reflection and transmission coefficients rM and t M for a given order n .
AC
CE
n
total
(l ) unique I n ,R
(l ) unique I n ,T
1
2
2
2
2
6
5
6
3
18
13
14
4
54
27
31
5
162
59
62
6
486
110
121
7
1458
214
220
8
4374
371
398
9
13122
670
678
10
39366
1102
1161
ACCEPTED MANUSCRIPT 11
118098
1882
1886
12
354294
2979
3101
In this section a brief numerical analysis of reflectance and ellipsometric quantities in reflected light is presented for selected examples of the inhomogeneous films for the illustration of the theoretical results achieved in the foregoing sections. It is assumed that light is incident onto
IP
T
the films from the ambient side. The spectral dependencies of reflectance R =| rM , j |2 are calculated at normal incidence while spectral dependencies of associated ellipsometric parameters
CR
are calculated for oblique incidence. For non-depolarizing inhomogeneous films these associated ellipsometric parameters are expressed in the following way:
US
Is = sin(2)sin(), I c = sin(2) cos(), I n = cos(2), where azimuth and phase change are defined by this equation: rM ,p
AN
tan ei =
rM ,s
.
M
Figure 13: The spectral dependencies of reflectance R and associated ellipsometric parameters
ED
I s , I c and I n for film 1 placed onto silicon substrate.
In Fig. 13 the calculated spectral dependencies of the reflectance and associated
PT
ellipsometric parameters are plotted for the inhomogeneous thin film with thickness d = 100nm
CE
and linear refractive index profile. The boundary refractive indices are as follows: nU = 1.5 and
nL = 2.5 (film 1). Dispersion of the refractive index of the film is not taken into account for
AC
simplicity. This film is the representative of the inhomogeneous thin films with relatively large gradient of the refractive index. The film is placed on the silicon single crystal substrate whose optical constants are taken from the literature [32]. In Fig. 13 the spectral dependencies of the optical quantities mentioned above are calculated by the formulae containing corrections up to the twelfth order. These dependencies can be considered to be the true dependencies for this inhomogeneous film. Furthermore, the spectral dependencies of these optical quantities calculated by the formulae of the WKBJ approximations are plotted in this figure. One can see that there are the relatively large differences between the values of the optical quantities calculated within the WKBJ and those representing the true spectral dependencies. Thus, the WKBJ approximation is
ACCEPTED MANUSCRIPT not usable for calculating the values of the reflectance and ellipsometric parameters of this inhomogeneous thin film. In Fig. 14 the differences between the true spectral dependencies of reflectance and ellipsometric parameters on the one hand and dependencies calculated using the approximate formulae on the other hand are introduced. These differences R , I s , I c and
I n are calculated in this way: R = Rv Rtrue , Is = Is,v Is,true , I c = I c,v I c,true and
T
I n = I n,v I n,true , where v = 1, 2,3. If v = 1, 2 and 3 then the formulae for calculating the
IP
individual optical quantities contain the corrections up to the first, second and third order,
CR
respectively, are used. From Fig. 14 it is clear that all the differences decrease with increasing the order of corrections which indicates a good convergence to the true values of these optical
US
quantities. Furthermore, the small values of the differences of R , I s , I c and I n belonging to the first order indicate that the formulae containing the corrections up to the first order ensure
AN
the substantial approach of the values of R , I s , I c and I n to the true values. The influence of the corrections of the third order is very small or practically negligible (see Fig.14). This fact
M
implies that the corrections of higher orders than the third order need not be taken into account from the practical point of view. In Fig. 15 the dependencies of the same kind are plotted for the
ED
inhomogeneous thin film with thickness d = 800nm and the linear profile of the refractive index containing identical boundary values nU and nL as for the foregoing film (film 2). This film is
PT
again placed onto the silicon single crystal substrate. Such the film is the representative of the inhomogeneous films with smaller gradients of their refractive index profiles. In this case the true
CE
spectral dependencies of the optical quantities are practically identical with those calculated within the WKBJ approximation. This is also proved by the values of the differences plotted in Fig. 16.
AC
This means that the WKBJ approximation is sufficient for calculating the optical quantities of this inhomogeneous film owing to usual experimental accuracy. In Fig. 17 the spectral dependencies of reflectance and ellipsometric parameters calculated for the thin inhomogeneous film 1 situated on an aluminum substrate are introduced. The optical constants of the aluminum substrate are taken from [33]. In this figure the true spectral dependencies of the optical quantities and those calculated using the WKBJ formulae are compared. In Fig. 18 the spectral dependencies of the differences corresponding to the first, second, third and fifth orders are depicted for these optical quantities. From Fig. 17 it is seen that the differences between the true spectral reflecta nce and that corresponding to the WKBJ approximation are relatively large (even a mutual shift of the curves is
ACCEPTED MANUSCRIPT recorded). As for the ellipsometric parameters the situation is somewhat different. In the middle part of the spectral range of interest the differences between the true and WKBJ dependencies are very small while within the interval lying at the left end of this spectral range the differences are rather large. From Fig. 18 it is also seen that the corrections up to the fifth order have the substantial meaning for the calculations of the ellipsometric parameters in the interval adjacent to the left end of the spectral range. In the small interval adjacent to the right end of the spectral range
T
the corrections up to the first and second order has the small influence. In the wide middle part of
IP
the spectral range the influence of the corrections including the corrections of the first order is
CR
practically negligible which coincides with the dependencies in Fig. 17. The behavior of the spectral dependencies of R is also in coincidence with the curves presented in Fig. 17. Thus, the
US
influence of inhomogeneity exhibiting the large gradients of profiles on the calculated optical quantities in the case of the aluminum substrate is rather complicated. There is the spectral range in
AN
which this influence is strong while within the other ranges this influence is weak so that the WKBJ approximation can be used. This implies the conclusion that the influence of considerable
M
inhomogeneity of thin films on their optical quantities is generally complicated and that this influence also depends on the spectral dependencies of the optical constants of the substrates (see
ED
Figs. 13,14,17 and 18).
PT
Figure 14: The spectral dependencies of the differences of reflectance R and associated ellipsometric parameters I s , I c and I n for film 1 on the silicon substrate corresponding to
CE
the formulae containing the corrections up to the first, second and third order.
AC
Figure 15: The spectral dependencies of reflectance R and associated ellipsometric parameters
I s , I c and I n of film 2 placed on the silicon substrate. Figure 16: The spectral dependencies of the differences of reflectance R and associated ellipsometric parameters I s , I c and I n for film 2 on the silicon substrate corresponding to the WKBJ formulae and formulae containing the corrections up to the first and second order.
Figure 17: The spectral dependencies of reflectance R and associated ellipsometric parameters
ACCEPTED MANUSCRIPT I s , I c and I n for film 1 placed onto aluminum substrate. Figure 18: The spectral dependencies of the differences of reflectance R and associated ellipsometric parameters I s , I c and I n for film 1 on the aluminum substrate corresponding
T
to the formulae containing the corrections up to the first, second, third and fifth orders.
IP
If the inhomogeneous thin films exhibit complicated profiles of the refractive index the corrections of higher orders must always be used to calculate their reflectance and ellipsometric
CR
parameters. This is true for the complicated profiles exhibiting strong gradients in particular. One of the examples of the inhomogeneous films with complicated profiles is a rugate filter. In this
US
paper we deal with reflectance of the rugate filter studied in paper of Kildemo et al. [27]. This filter has ten-cycle sinusoidal modulation with matched surrounding media (no reflection from
AN
surrounding boundaries is realized) and with resonance at 550nm ( E = 2.254eV ). Sinusoidal modulation is given in this way:
M
n( x) = 2.0 0.1sin(20 x / d ),
ED
where d is the thickness of the filter ( d = 1375nm ). The spectral dependencies of reflectance of this rugate filter corresponding to the approximate formulae containing the corrections up to the seventh order are plotted in Fig. 19. It is seen that in the maximum of reflectance there are large
PT
differences between the values belonging to the first and the higher orders. Simultaneously, it is
CE
evident that the values of the reflectance calculated by the formula with the corrections of the first order are wrong in the interval of the reflectance maxima. From this figure it is also clear that the formulae containing the corrections up to the seventh-order can be considered to be sufficient for
AC
calculating whole spectral reflectance of the filter considered. Note that the corrections of the even orders are equal to zero which is apparent from the corresponding formulae presented above.
Figure 19: The spectral dependencies of the reflectance of the rugate filter at normal incidence. Numbers denote the orders of the corrections.
Note that the integrals were evaluated as described in section 3 with the degree of the Chebyshev interpolation polynomial chosen as H = 64 for the linear refractive index profiles and
ACCEPTED MANUSCRIPT H = 1024 for the rugate filter.
5.
Comparison with other approaches
It is clear that the speed of the calculations of the optical quantities of inhomogeneous thin films using our formulae depends mainly on the speed of the evaluations of the integrals occurring in our
T
approach. For a given order M there are 2M 1 such integrals and if the integration method
IP
based on the Chebyshev interpolation is used then the calculation time is proportional to this
CR
number. The similar integrals occur also in the approach of Kildemo et al. [27], therefore the speeds of these methods can easily be compared. In some cases the precision achieved by both the
US
approaches for a given order are comparable (e.g. for the linear refractive index profile) but for some complicated profiles the formulae of Kildemo et al. provide higher precision and speed of calculations at a given order (e.g. for the rugate filter). This statement was proved by numerical
AN
analysis performed using our formulae and formulae of Kildemo et al. for concrete examples of refractive index profiles. The approach developed in [27] was utilized, for example, in papers [34,
M
35].
ED
However, on the other hand our formulae enable us to estimate the precision of calculating the optical quantities of any inhomogeneous thin film more easily than the formulae of the
PT
mentioned authors. For this easy estimation we can use the formulae for the individual terms rn in the series appearing in the formulae for the reflection and transmission coefficients, i.e. for rM
CE
etc. The formulae for the reflection coefficients are given by rational functions in the approach of [27] and so the estimation of the precision of calculations is not so straightforward.
AC
The further advantage of our approach consists in the clear and simple way of derivation of the formulae for the reflection and transmission coefficients. This is especially important for the inhomogeneous thin films exhibiting large refractive index gradients. In this case the terms with higher-order integrals must be included into the formula expressing the reflection and transmission coefficients because the terms up to the third-order integrals are usually insufficient. Within the method of Kildemo et al. the derivation of these higher-order terms can be difficult, if not impossible, because of the necessity to employ the recursive formulae for the multilayer systems for this purpose (for details see [27]). In our method the terms corresponding to arbitrary high order needed for practical calculations are derived by means of the evident and systematic
ACCEPTED MANUSCRIPT mathematical procedure based on formulating the trees of the corrections occurring in the formulae. However, as shown in Table 2 the method becomes impractical for very high orders because of the very large number of contributions that must be taken into account. This impractical case corresponds to the inhomogeneous thin films with enormously complicated profiles. Fortunately, such the films occur exceptionally in practice. For the majority of the inhomogeneous
T
films encountered in practice such high orders are not needed.
IP
Table 3: Average calculation times for one point for films 1 and 2 on the silicon substrate. The
CR
columns named order and degree denote the number of internal reflections and the degree of the Chebyshev polynomial necessary to achieve the indicated absolute accuracy. The column named
US
layers is the number of subdividing layers needed to achieve the indicated accuracy if the approximation of the inhomogeneous layer by a multilayer system is used. order
degree
multilayer
time [ms]
AN
integrals
layers
time [ms]
3
8
ellipsometry
5
8
0.21
ED
reflectance
M
film 1 on the silicon substrate, absolute accuracy 103
0.35
12
0.004
16
0.005
reflectance
1
ellipsometry
2
PT
film 2 on the silicon substrate, absolute accuracy 103 32
0.23
36
0.009
32
0.45
100
0.023
ellipsometry
7
16
1.4
350
0.07
10
16
2.0
500
0.10
AC
reflectance
CE
film 1 on the silicon substrate, absolute accuracy 106
film 2 on the silicon substrate, absolute accuracy 106 reflectance ellipsometry
4
64
1.0
1000
0.19
5
64
1.3
2800
0.56
The typical calculation times for one point corresponding to a given wavelength of light and incidence angle needed to evaluate the reflectance or ellipsometric parameters with desired absolute accuracy are introduced in Table 3. The part under heading “integrals” corresponds to the
ACCEPTED MANUSCRIPT method presented in this paper while the part of the table under heading “multilayer” corresponds to the method using the approximation of the inhomogeneous film by a multilayer system of thin homogeneous films. From this table it is evident that the method using the approximation by the multilayer system is faster in all cases, but it requires relatively large number of subdividing homogeneous layers to achieve the desired accuracy. Therefore, the number of points at which the refractive index must be evaluated is much larger than in the presented method. For example, in
IP
T
order to calculate the ellipsometric parameters of the film 2 with accuracy better than 106 it is necessary to evaluate the refractive index at 2800 points in the method using the multilayer
CR
approximation but only 65 evaluations are needed in the presented method (the number of evaluations is the degree of the Chebyshev polynomial plus one). This means that the speed of the
US
method using the multilayer approximation can be negatively affected if the evaluation of the refractive index profile takes a long time while the impact on the presented method is much lower.
AN
If we look at the ratio of calculation times for the presented method and the multilayer approximation then it is evident that this ratio is lower if the films are thick and if the desired
M
accuracy is higher. It should also be noted that comparison of both the discussed methods is rather difficult because the calculation times are also dependent on the concrete implementation of these
ED
methods in the software.
Our formulae are suitable for including defects such as random roughness of boundaries.
PT
The inclusion of boundary roughness requires to calculate the statistical mean values of the reflection and transmission coefficients [36–44]. These statistical mean values are calculated more
CE
easily, precisely and efficiently if the coefficients are expressed by the series than if they are expressed by rational functions. In particular this statement is true for relatively thick
AC
inhomogeneous films exhibiting many extremes in the spectral dependencies of their optical quantities. The foregoing fact is the main advantage of our approach. The use of o ur formulae for including boundary roughness at calculating the optical quantities of the inhomogeneous thin films having this defect will be presented in our forthcoming paper. Within the multilayer approximation a sufficiently exact approach allowing to include the influence of random boundary roughness on the optical quantities of the inhomogeneous thin film has not been presented so far. Only very rough approximation has been utilized. This approximation can be partially successful only when special roughness is exhibited by boundaries of the inhomogeneous films (see e.g. [45]). In the forthcoming paper it will be shown that our
ACCEPTED MANUSCRIPT formulae presented here will enable us to describe the influence of roughness on the optical quantities of the inhomogeneous thin films in general.
6.
Experiment
In this section the optical characterization of an inhomogeneous non-stoichiometric silicon nitride
T
film deposited on silicon single crystal wafer is presented as the illustration of using the theoretical
CR
IP
approach introduced in this paper.
6.1. Preparation of the film and experimental arrangement
US
The silicon nitride film was deposited on polished side of silicon substrate by reactive magnetron sputtering [46] of silicon target in argon- nitrogen atmosphere. The refractive index profile was
AN
created by changing the nitrogen flow rate during deposition while the other deposition conditions were kept constant.
M
The spectral dependencies of the associated ellipsometric parameters were measured using the Horiba Jobin Yvon UVISEL phase modulated ellipsometer for five angle of incidence of light
6.2.1. Structural model
PT
6.2. Models of the film
ED
within the spectral range 0.6 6.3eV (197 2066nm ).
AC
subsection 2.1).
CE
The structural model of the film characterized is defined by the assumptions presented above (see
6.2.2. Dispersion model To our experiences concerning the non-stoichiometric silicon nitride films the best dispersion model of these films is given by the following expression of the imaginary parts of their dielectric function i within the spectral range of interest (see [18]):
i (E) =
N ( E Eg )2 ( Eh E ) 2 CE 2
where Eg < E < Eh and equal to zero below Eg and above Eh .
(64)
ACCEPTED MANUSCRIPT Parameters Eg and Eh represent the minimum and maximum of energy of interband transitions between the valence and conduction bands, respectively. Symbol N denotes the strength of these transitions. The normalization constant C is determined using the following equation:
E ( E )dE = N .
(65)
i
T
0
IP
The real part of this dielectric function r is calculated from the imaginary part using the
CR
Kramers–Kronig relation. The details concerning this dispersion model are presented in papers [47–49].
In our optical studies of the non-stoichiometric silicon nitride films prepared by technology
US
described above we found that the sufficient refractive index profile satisfying to these films is expressed in this way:
x d
AN
( x) = U ( L U ) ,
(66)
M
where U and L denote the complex dielectric functions at the upper and lower boundary,
ED
respectively ( U = nU2 etc.).
Thus, the dispersion parameters must be distinguished for both the boundaries, i.e. the
PT
following dispersion parameters N q , Eg,q , Eh,q and Cq must be taken into account for q = U and L . In other words, six dispersion parameters N U , N L , Eg,U , Eg,L , Eh,U and Eh,L
CE
characterize the optical properties of the film under study.
AC
6.3. Results and discussion After applying the least-square method to the experimental data it was found that the best fit was achieved using the formulae containing the corrections up the third order. The values of the thickness and dispersion parameters corresponding to this best fit are summarized in Table 4.
Table 4: The parameter values corresponding to the best fit of the experimental data with using the formulae for the reflection coefficients containing the corrections up to the third order. parameter
value
ACCEPTED MANUSCRIPT d [nm]
154.30 0.09
N U [eV 2
411 7
2.37 0.01
Eh,U [eV]
40.1 0.5
N L [eV 2
505 6
] 1.799 0.004
Eh,L [eV]
31.1 0.3
US
CR
Eg,L [eV]
IP
Eg,U [eV]
T
]
AN
Table 5: Quality of the fits expressed using quantity if the reflection formulae correspond to the WKBJ approximation (the first row), corrections up to the first order (the second row), corrections up to the second order (the third row) and corrections up to the third order (the fourth
M
row) are employed.
WKBJ
31.34
1
8.15
1+2
6.46
1+2+3
6.22
CE
PT
ED
order
The agreement between the experimental and theoretical data is good which is seen in Fig.
follows:
AC
20. This agreement is also proved by means of the value of quantity . This quantity is defined as
=
, 3N exp
Figure 20: The spectral dependencies of the ellipsometric parameters and their fits: points denote the experimental values, curves denote the theoretical data.
ACCEPTED MANUSCRIPT where is the residual sum of squares and N exp denotes number of experimental points. Number three is introduced into formula for because of using the set of three ellipsometric parameters. The values of determine the quality of fits of experimental data. In Table 5 the values of corresponding to the fits of the experimental data with using the formulae for rM belonging to the WKBJ approximation and formulae containing the corrections up to the first
T
order, up to the second order and up to the third order are presented. The best value of
IP
corresponds to the formulae containing the corrections up to the third order. If the corrections with
CR
higher orders were added in formulae for rM the corresponding values of did not change from the practical point of view. This proves that rM containing the correction up to the third order is
US
sufficiently accurate to fit the experimental data. In Fig. 21 the spectral dependencies of the refractive index and extinction coefficient of the characterized film for both the boundaries are
AN
plotted. The refractive index nL is larger than refractive index nU for all the wavelengths. The same relation is valid for the corresponding extinction coefficients. This is connected with
M
increasing the nitrogen flow during the growth of the film in the deposition process. The refractive index and extinction coefficient profiles are depicted for the selected wavelength in Fig. 22. These
ED
profiles also show that the profile gradients of the refractive index and extinction coefficient of this inhomogeneous film are rather large which coincides with the necessity to use the formulae for the
PT
ellipsometric quantities containing the corrections up to the third order in fitting experimental data. The results introduced for this film indicate that the formulae presented above are suitable for the
CE
optical characterization of the inhomogeneous thin films exhibiting relatively large gradients of
AC
profiles of their optical constants.
Figure 21: The spectral dependencies of the optical constants of the characterized film at the upper and lower boundaries.
Figure 22: Profiles of the optical constants of the characterized film for E = 2.5 eV.
7.
Conclusion
By means of the multiple-beam interference model the approximate formulae are derived for the
ACCEPTED MANUSCRIPT reflection and transmission coefficients of the inhomogeneous thin films having large gradients of the profiles of their optical constants. These formulae consist of the series whose terms consist of the first term corresponding to the WKBJ approximation and the following terms expressing the corrections of these WKBJ terms describing the influence of the internal reflections inside the films originating in consequence of the large profile gradients. The expressions for the corrections belonging to individual orders are easily and systematically formulated. It is shown that the
T
corrections constitute the trees. The advantage of the formulae presented here consists in
IP
possibility to estimate the precision of calculating the optical quantities of the inhomogeneous thin
CR
films in an easy and straightforward way. The further advantage consists in possibility to derive the corrections of arbitrary order in the clearly prescribed way. However, the main advantage of
US
the formulae presented in this article consists in possibility to include defects such as ra ndom boundary roughness into these formulae in an efficient way which will be described in our
AN
forthcoming paper. The disadvantage of our approach is smaller speed and precision of calculations of the optical quantities of inhomogeneous thin films exhibiting complicated
M
refractive index profiles in comparison with the other approximate approaches such as the multilayer approximation and approach of Kildemo et al.
ED
From the numerical analysis it is evident that the influence of the inhomogeneity with the large refractive index profile gradients on the spectral dependencies of the optical quantities is
PT
complicated. Furthermore, it is clear that the influence of this inhomogeneity is also dependent on the spectral dependencies of the optical constants of the sub strates. Moreover, it should be noted
CE
that for the inhomogeneous films with complicated profiles the corrections of rather high orders must be used in the formulae for the reflection and transmission coefficients (e.g. for rugate filters
AC
the corrections up to the seventh order must be used at least). The results of the optical characterization of the selected sample of non-stoichiometric silicon nitride film prepared by reactive magnetron sputtering onto silicon substrate are presented for illustration of usability of our formulae. Using these formulae experimental data of variable angle spectroscopic ellipsometry are processed. The spectral dependencies of the refractive indices and extinction coefficients at the upper and lower boundaries together with the profiles of both the optical constants for the selected wavelength are determined. Moreover, the determined thickness value is also presented. For finding these optical parameters the formulae containing the corrections up to the third order are used. The good fits of the experimental data support a
ACCEPTED MANUSCRIPT correctness of the results achieved. This experimental example indicates the efficiency of using our formulae within the optical characterization of the inhomogeneous thin films with large gradients of refractive index profiles in practice. Our formulae should be usable also in synthesis of inhomogeneous thin films.
T
Acknowledgments
IP
This work was supported by the project LO1411 (NPU I) funded by Ministry of Education, Youth
CR
and sports of the Czech Republic. The authors would like to thank P.Vašina and J. Ženšek for
US
preparing the samples of non-stoichiometric silicon nitride films.
References
A. Vašiček, Polarimetric methods for the determination of the refractive index and the
AN
[1]
thickness of thin films on glass, J. Opt. Soc. Am. 37 (1947) 145–153. J. M. Bennett, M. J. Booty, Computational method for determining n and k for a thin film
M
[2]
from the measured reflectance, transmittance, and film thickness, Appl. Opt. 5 (1966)
[3]
ED
41–43.
K. Vedam, W. Knausenberger, F. Lukes, Ellipsometric method for the determination of all
PT
the optical parameters of the system of an isotropic nonabsorbing film on an isotropic absorbing substrate. optical constants of silicon, J. Opt. Soc. Am. 59 (1969) 64–71. O. S. Heavens, H. M. Liddell, Influence of absorption on measurement of refractive index
CE
[4]
of films, Appl. Opt. 4 (1965) 629–630. I. Ohlídal, K. Navrátil, E. Schmidt, Simple method for the complete optical analysis of
AC
[5]
very thick and weakly absorbing films, Appl. Phys. A 29 (1982) 157–162. [6]
T. Lohner, K. J. Kumar, P. Petrik, A. Subrahmanyam, I. Bársony, Optical analysis of room temperature magnetron sputtered ITO films by reflectometry and spectroscopic ellipsometry, J. Mater. Res. 29 (2014) 1528–1536.
[7]
D. Franta, I. Ohldal, D. Petrýdes, Optical Characterization of TiO 2 Thin Films by the Combined Method of Spectroscopic Ellipsometry and Spectroscopic Photometry, Vacuum 80 (2005) 159–162.
[8]
I. Ohldal, M. Ohldal, D. Nečas, D. Franta, V. Buršková, Optical characterisation of
ACCEPTED MANUSCRIPT SiO xCyHz thin films non-uniform in thickness using spectroscopic ellipsometry, spectroscopic reflectometry and spectroscopic imaging reflectometry, Thin Solid Films 519 (2011) 2874–2876. [9]
D. Franta, I. Ohldal, V. Buršková, L. Zajčková, Optical Properties of Diamond-Like Carbon Films Containing SiO x Studied by the Combined Method of Spectroscopic Ellipsometry and Spectroscopic Reflectometry, Thin Solid Films 455-456 (2004)
M. Hacskaylo, Determination of the refractive index of thin dielectric films, J. Opt. Soc.
IP
[10]
T
393–398.
[11]
CR
Am. 54 (1964) 198–203.
J. C. Manifacier, J. Gasiot, J. P. Fillard, A simple method for the determination of the
US
optical constants n, k and the thickness of a weakly absorbing thin film, J. Phys. E: Sci. Instrum. 9 (1976) 1002–1004.
R. Jacobsson, Matching a multilayer stack to a high-refractive index substrate by means of
AN
[12]
an inhomogeneous layer, J. Opt. Soc. Am. 54 (1964) 422–423. R. Jacobsson, Light reflection from films of continuously varying refractive index, in: E.
M
[13]
Wolf (Ed.), Progress in Optics, Vol. 5, Elsevier, 1966, pp. 247–286. R. Jacobsson, Inhomogeneous and Coevaporated Homogeneous Films for Optical
ED
[14]
Applications, in: G. Hass, M. H. Francombe, R. W. Hoffman (Eds.), Physics of Thin Films,
[15]
PT
Vol. 8, Academic Press, New York, 1975, pp. 51–98. J. C. Charmet, P. G. de Gennes, Ellipsometric formulas for an inhomogeneous layer with
[16]
CE
arbitrary refractive- index profile, J. Opt. Soc. Am. 73 (1983) 1777–1784. C. K. Carniglia, Ellipsometric calculations for nonabsorbing thin films with linear
[17]
AC
refractive- index gradients, J. Opt. Soc. Am. A 7 (1990) 848–856. D. Franta, I. Ohldal, P. Klapetek, P. Pokorný, M. Ohldal, Analysis of Inhomogeneous Thin Films of ZrO 2 by the Combined Optical Method and Atomic Force Microscopy, Surf. Interface Anal. 32 (2001) 91–94. [18]
D. Nečas, D. Franta, I. Ohldal, A. Poruba, P. Wostrý, Ellipsometric characterization of inhomogeneous non-stoichiometric silicon nitride films, Surf. Interface Anal. 45 (2013) 1188–1192.
[19]
D. Li, A. Goullet, M. Carette, A. Granier, J. P. Landesman, Effect of growth interruptions on TiO 2 films deposited by plasma enhanced chemical vapour deposition, Mater. Chem.
ACCEPTED MANUSCRIPT Phys. 182 (2016) 409–417. [20]
J. Vohánka, I. Ohldal, J. Ženšek, P. Vašina, M. Čermák, D. Franta, Use of the Richardson extrapolation in optics of inhomogeneous layers: Application to optical characterization, Surf. Interface Anal. 50 (2018) 757–765.
[21]
B. G. Bovard, Rugate filter design: the modified Fourier transform technique, Appl. Opt. 29 (1990) 24–30. G. Boivin, D. St.-Germain, Synthesis of gradient-index profiles corresponding to
T
[22]
O. Debieu, R. P. Nalini, J. Cardin, X. Portier, J. Perrière, F. Gourbilleau, Structural and
CR
[23]
IP
spectralreflectance derived by inverse Fourier transform, Appl. Opt. 26 (1987) 4209–4213.
optical characterization of pure Si-rich nitride thin films, Nanoscale Res. Lett. 8 (2013) 31. A. Thomson, N. Lal, Y. Wan, Interpolating the optical properties of varied composition
US
[24]
silicon nitride, Phys. Status Solidi B 252 (2015) 2230–2235. Z. Knittl, Optics of Thin Films, Wiley, London, 1976.
[26]
A. Vašček, Optics of Thin Films, North–Holland, Amsterdam, 1960.
[27]
M. Kildemo, O. Hunderi, B. Drévillon, Approximation of reflection coefficients for rapid
M
AN
[25]
real-time calculation of inhomogeneous films, J. Opt. Soc. Am. A 14 (1997) 931–939. G. B. Arfken, H. J. Weber, Mathematical methods for physicists; 4th ed., Academic Press, San Diego, CA, 1995.
ED
[28]
A. Gil, J. Segura, N. M. Temme, Chebyshev Expansions, SIAM, 2007, Ch. 3, pp. 51–86.
[30]
N. Ahmed, T. Natarajan, K. R. Rao, Discrete cosine transform, IEEE Trans. Comput. C-23
[31]
CE
(1974) 90–93.
PT
[29]
C. W. Clenshaw, A. R. Curtis, A method for numerical integration on an automatic
[32]
AC
computer, Numerische Mathematik 2 (1960) 197–205. D. Franta, A. Dubroka, C. Wang, A. Giglia, J. Vohánka, P. Franta, I. Ohldal, Temperature-dependent dispersion model of float zone crystalline silicon, Appl. Surf. Sci. 421 (2017) 405–419. [33]
D. Y. Smith, The Optical Properties of Metallic Aluminum, in: E. D. Palik (Ed.), Handbook of Optical Constants of Solids, Vol. 1, Academic Press, New York, 1985, pp. 369–406.
[34]
M. Kildemo, Real-time monitoring and growth control of Si-gradient- index structures by multiwavelength ellipsometry, Appl. Opt. 37 (1998) 113–124.
ACCEPTED MANUSCRIPT [35]
M. Kildemo, R. Brenot, B. Drévillon, Spectroellipsometric method for process monitoring semiconductor thin films and interfaces, Appl. Opt. 37 (1998) 5145–5149.
[36]
I. Ohldal, F. Lukeš, Ellipsometric Parameters of Rough Surfaces and of a System Substrate-Thin Film with Rough Boundaries, Opt. Acta 19 (1972) 817–843.
[37]
I. Ohldal, F. Lukeš, K. Navrátil, Rough Silicon Surfaces Studied by Optical Methods, Surf. Sci. 45 (1974) 91–116. I. Ohldal, K. Navrátil, F. Lukeš, Reflection of Light by a System of Nonabsorbing Isotropic
T
[38]
IP
Film–Nonabsorbing Isotropic Substrate with Randomly Rough Boundaries, J. Opt. Soc.
[39]
CR
Am. 61 (1971) 1630–1639.
I. Ohldal, D. Franta, D. Nečas, Improved combination of scalar diffraction theory and
US
Rayleigh-Rice theory and its application to spectroscopic ellipsometry of randomly rough surfaces, Thin Solid Films 571 (2014) 695–700.
I. L. Morris, T. E. Jenkins, The effect of long-correlation- length surface roughness on the
AN
[40]
ellipsometric parameters of reflected light, Europhys. Lett. 34 (1996) 55–61. J. M. Zavislan, Angular scattering from optical interference coatings: scalar scattering
M
[41]
predictions and measurements, Appl. Opt. 30 (1991) 2224–2244. C. K. Carniglia, Scalar scattering theory for multilayer optical coatings, Opt. Eng. 18
ED
[42]
(1979) 104–115.
J. Bauer, Optical properties, band gap, and surface roughness of Si3 N 4 , Phys. Status Solidi
PT
[43]
A 39 (1977) 411–418.
M. Čermák, J. Vohánka, I. Ohldal, D. Franta, Optical quantities of multi- layer systems
CE
[44]
with randomly rough boundaries calculated using the exact approach of the Rayleigh- Rice
[45]
AC
theory, J. Mod. Opt. 65 (2018) 1720–1736. D. Franta, I. Ohldal, P. Klapetek, M. Ohldal, Characterization of thin oxide films on GaAs substrates by optical methods and atomic force microscopy, Surf. Interface Anal. 36 (2004) 1203–1206. [46]
S. Berg, T. Nyberg, Fundamental understanding and modeling of reactive sputtering processes, Thin Solid Films 476 (2005) 215–230.
[47]
D. Franta, D. Nečas, L. Zajčková, Models of dielectric response in disordered solids, Opt. Express 15 (2007) 16230–16244.
[48]
D. Franta, D. Nečas, L. Zajčková, Application of Thomas–Reiche–Kuhn sum rule to
ACCEPTED MANUSCRIPT construction of advanced dispersion models, Thin Solid Films 534 (2013) 432–441. D. Franta, D. Nečas, I. Ohldal, A. Giglia, Dispersion model for optical thin films applicable in wide spectral range, in: Optical Systems Design 2015: Optical Fabrication, Testing, and
CE
PT
ED
M
AN
US
CR
IP
T
Metrology V, Vol. 9628 of Proc. SPIE, 2015, p. 96281U.
AC
[49]
ACCEPTED MANUSCRIPT Highlights
CE
PT
ED
M
AN
US
CR
IP
T
New approximate approach for optical quantities of inhomogeneous thin films Formulae are derived using multiple-beam interference model Films can exhibit large gradients of refractive index profiles Formulae enable us to perform rapid calculations at the optical quantities Application of the new formulae for characterisation of a selected sample
AC
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11
Figure 12
Figure 13
Figure 14
Figure 15
Figure 16
Figure 17
Figure 18
Figure 19
Figure 20
Figure 21
Figure 22
Ivan Ohlídal, Jiří Vohánka, Jan Mistrík, Martin Čermák, František Vižd'a, Daniel Franta PII: DOI: Reference:
S0040-6090(19)30138-5 https://doi.org/10.1016/j.tsf.2019.03.001 TSF 37189
To appear in:
Thin Solid Films
Received date: Revised date: Accepted date:
14 September 2018 17 January 2019 1 March 2019
Please cite this article as: I. Ohlídal, J. Vohánka, J. Mistrík, et al., Approximations of reflection and transmission coefficients of inhomogeneous thin films based on multiplebeam interference model, Thin Solid Films, https://doi.org/10.1016/j.tsf.2019.03.001
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Approximations of reflection and transmission coefficients of inhomogeneous thin films based on multiple-beam interference model Ivan Ohlídala, Jiří Vohánkaa, Jan Mistríkb, Martin Čermáka, František Vižd’ac, Daniel Frantaa a
Department of Physical Electronic, Faculty of Science, Masaryk University, Kotlářká 2, 61137
Institute of Applied Physics and Mathematics, Faculty of Chemical Technology, University of
IP
b
T
Brno, Czech Republic
CR
Pardubice, Studentská 95, 532 10, Pardubice, Czech Republic Department of Mathematics and Physics, Faculty of Military Technology, University of Defence,
CE
PT
ED
M
AN
US
Kounicova 65, 662 10 Brno, Czech Republic
AC
c
ACCEPTED MANUSCRIPT
Abstract A multiple-beam interference model is used to derive approximate formulae for the reflection and transmission coefficients of inhomogeneous thin films exhibiting large gradients of refractive index profiles. It is shown that these formulae are constituted by series containing the Wentzel-Kramers-Brillouin-Jeffreys term and correction terms with increasing order
T
corresponding to number of considered internal reflections inside the films. A numerical analysis
IP
enabling us to show the influence of a degree of inhomogeneity on spectral dependencies of
CR
reflectance and ellipsometric parameters of inhomogeneous films is performed. Advantages and disadvantages of our approach compared with other approximate approaches are discussed. The
US
optical characterization of a selected non-stoichiometric silicon nitride film prepared by reactive magnetron sputtering onto silicon single crystal substrate is performed for illustration of using our
AN
formulae in practice.
AC
CE
PT
ED
M
Keywords: reflectance, transmittance, ellipsometric parameters, inhomogeneous thin films
ACCEPTED MANUSCRIPT
1.
Introduction
An enormous attention has been devoted to optics of homogeneous thin films so far. Therefore many papers dealing with problems of the optical characterization and optical synthesis of these films have been published (see e.g. [1–11]). A less attention has been devoted to optics of
T
inhomogeneous thin films exhibiting profiles of the refractive index across these films, i.e. profiles
IP
occurring in direction perpendicular to parallel plane boundaries (see e.g. [12–20]). In spite of this
CR
fact in the three last decades the investigations in optics of inhomogeneous thin films were increased because of two reasons at least. First, the design and production of layered systems were moving to those with continuously varying profiles of their refractive index. The reason of this fact
US
is that such the inhomogeneous layered systems exhibit certain better properties than those consisting of homogeneous layers with mutually different refractive indices. For example, these
AN
inhomogeneous systems with continuous refractive index profiles exhibit substantially lower light scattering caused by boundary roughness than the layered systems containing different
M
homogeneous thin films. The typical representatives of such the inhomogeneous systems are
ED
rugate filters that are utilized as the optical devices with high reflectance within certain spectra l ranges [21, 22]. Second, the development in optics industry, semiconductor technology, solar energetics, microelectronics and other modern branches of fundamental and applied researches
PT
implies a need to create thin films formed by various complex materials. These films are often more or less inhomogeneous from the optical point of view. This inhomogeneity must be
CE
characterized reliably and precisely in order to understand the film structure and properties or remove this phenomena by changing technological conditions at the film preparation. Thus, it is
AC
permanently needed to develop new efficient optical methods enabling us to characterize these complex inhomogeneous thin films. The typical examples of such the complex films frequently exhibiting the optical inhomogeneity are films of non-stoichiometric silicon nitride that are rich in silicon (see e.g. [23, 24]). At the optical characterization and design of the inhomogeneous thin films more complicated theoretical approaches must be employed in comparison with those utilized for homogeneous films. The formulae used in these complicated approaches must ensure to perform correct, precise and rapid calculations of the optical quantities of the inhomogeneous films. A
ACCEPTED MANUSCRIPT sufficient speed of the calculations is very important for achieving efficient computer algorithms utilized in procedures of the optical characterization and design of these inhomogeneous structures. This requirement of the speed of calculations is especially inevitable for real-time monitoring and feedback control of inhomogeneous films and structures at their depositions in technological arrangements (in-situ methods). The speed of the calculations is easily achieved if the refractive index profiles exhibit small gradients, i.e. if refractive indices varies slowly across
T
the films. In this case one can use simple formulae for calculating the optical quantities of the
IP
inhomogeneous thin films corresponding to the Wentzel–Kramers–Brillouin–Jeffreys (WKBJ)
CR
approximation (see e.g. [13, 25]). If gradients of the refractive index profiles are larger and the conditions of using the WKBJ approximation are not fulfilled the approximation based on
US
replacing the inhomogeneous thin films by multilayer systems with a large number of homogeneous thin films is usually used. Then the matrix formalism or recursive formalism is used
AN
to calculate the values of the optical quantities of these inhomogeneous films (see e.g. [13, 20, 26]). However, these formalisms are often slow for real- time monitoring of the deposition of the
M
inhomogeneous films or inhomogeneous layered systems. Therefore, it is necessary to derive other approximate approaches allowing to calculate the optical quantities of such the inhomogeneous
ED
thin films with sufficiently high speed. Of course, the correctness and high precision of the calculations has to be maintained within these approximate approaches.
PT
In paper [27] the sophisticated approximation of reflection coefficients of the inhomogeneous thin films based on modified recursive formalism is pre sented. For deriving the
CE
formulae of this approximation the authors utilized the recursive formula for the four- layer system. Using this formula the systematic trends were revealed for the reflections inside the multilayer
AC
systems as they subdivided the four- layer system into more and more sub-layers. In the limit for infinite number of the sub- layers the formulae for the reflection coefficients of the inhomogeneous films were obtained. These formulae contain terms modifying the formulae valid for the WKBJ approximation. The correction terms comprise the single, double and triple integrals containing gradients of the refractive index profiles in their integrands. In paper [27] it is presented that the use of this approximation allows to reduce the time of the calculation of the reflectance values between one or two orders of magnitude compared with the matrix or recursive formalism for the multilayer system if the concrete linear profile of the inhomogeneous film was taken into account (for details see [27]).
ACCEPTED MANUSCRIPT In paper [27] the approximation of the reflection coefficients of the inhomogeneous layers is also performed using the matrix formalism, i.e. using the corrections to the transfer matrix elements containing the same integrals as in the approach based on the recursive approach (for details see [27]). It should be noted that both the approximate approaches are equivalent. In this paper the new formulae for the reflection and transmission coefficients of the inhomogeneous thin films exhibiting arbitrary refractive index profiles based on multiple-beam
T
interference model are presented. It is shown that the corrections to the WKBJ formulae can be
IP
derived in a simple systematic way. Furthermore, the numerical analysis concerning the optical
CR
quantities of the inhomogeneous films with the selected refractive index profiles is performed. A practical illustration of this new approximate approach is carried out through the optical
US
characterization of a selected sample of the non-stoichiometric silicon nitride film deposited onto silicon single crystal substrate. The practical advantages of the formulae of this approximation are
Theory
M
2.
AN
also introduced.
ED
2.1. Structural model of inhomogeneous thin films The structural model of the inhomogeneous thin films considered in this paper is given by the
PT
following assumptions:
1) film and substrate consist of isotropic absorbing non-magnetic materials
CE
2) film is inhomogeneous, i.e. it exhibits some profile of the complex refractive index represented by a complex function of single variable
AC
3) boundaries of the film are smooth, flat and mutually parallel 4) boundaries are infinitely thin, i.e. transition layers at boundaries are not taken into account
5) ambient is non-absorbing, homogeneous and isotropic 6) substrate is optically homogeneous 7) no defects in volume of the film are taken into account
Figure 1: Schematic diagram of the non-absorbing inhomogeneous thin film: The symbols are explained in the text.
ACCEPTED MANUSCRIPT
The schematic diagram of this model is plotted in Fig. 1. In this figure the symbols are as follows: n0 and nS denote the refractive indices of the ambient and substrate, respectively, nU and nL represent the refractive indices of the film at the upper and lower boundaries, respectively, n( x) denotes the function describing the refractive index profile. Symbols d , 0 and S
T
denote the thickness of the film, incidence angle of light from the side of the ambient and angle of
for
the
system
depicted
in
Fig.
1,
i.e.
CR
law
IP
refraction corresponding to substrate, respectively. The following equation describes the Snell’s
n0 sin 0 = nU sin U = n( x)sin ( x) = nL sin L = nS sin S , where U , ( x) and L denote the
US
refraction angles at the upper boundary, coordinate x and lower boundary, respectively. In general all the quantities introduced above are complex except n0 , d and 0 which are always
AN
real. It is reasonable to introduce the admittances defined as follows: Yv = nv / cos v for the ppolarization and Yv = nv cos v for the s-polarization ( v = 0 , S, U, and L). Moreover, it holds that
M
Y ( x) = n( x) / cos ( x) for the p-polarization and Y ( x) = n( x) cos ( x) for the s-polarization.
Note that the convention with plus in front of imaginary parts of the complex refractive indices is
ED
utilized in this work. Furthermore, it should be noted that this schematic diagram concerns the
PT
non-absorbing inhomogeneous thin film and non-absorbing substrate.
CE
2.2. Reflection coefficients
2.2.1. Incidence of light from the ambient side
AC
Within the WKBJ approximation the following formula is valid for the reflection coefficients r0 of the inhomogeneous thin films [13, 14, 18]:
r0 =
rU rL e
ixd
1 rU rL e
ixd
,
(1)
where rU =
Y0 YU Y Y , rL = L S . Y0 YU YL YS
Symbols Y0 and YS denote the admittances of the ambient and substrate, respectively, and YU
ACCEPTED MANUSCRIPT and YL are the admittances at the upper and lower boundaries of the inhomogeneous thin film, respectively (see Fig. 1). The phase-shift angle xd is given as follows: xd =
4
d
n2 ( x) n02 sin 20 dx,
(2)
0
T
where symbol denotes the wavelength of incident light in vacuum.
IP
Figure 2: Schematic diagrams of the internal single reflections. Full lines represent reflected light
CR
and dashed lines represent transmitted light.
The formula for the first-order corrections of the WKBJ formula can be derived easily in
US
the following way. If one takes into account all the internal single reflections inside the multilayer system approximating the inhomogeneous film (see Fig. 2a) then it is possible to derive the
AN
(1) following formula expressing the reflection coefficients j1,R corresponding to these reflections
as follows:
ED
M
N 1 w (1) j1,R = R rw, w1exp ix j w=1 j =1
where
PT
rw, w1 =
Yw Yw1 , w = 1, 2, Yw Yw1
(3)
, N 1
CE
(symbol N denotes the number of the sub- layers forming the multilayer system considered).
AC
Quantities x j and R are expressed as follows: xj =
4
d j n2j n02 sin 20 ,
where n j , d j and j are the refractive index, thickness and refraction angle of the j -th sub-layer, respectively, and , R = t0,1t0,1
where t0,1 =
for the s-polarization and
2Y0 , Y0 Y1
= t0,1
2Y1 Y0 Y1
ACCEPTED MANUSCRIPT t0,1 =
c0 2Y0 , c1 Y0 Y1
= t0,1
c1 2Y1 c0 Y0 Y1
with c0 = cos 0 and c1 = cos 1 for the p-polarization. Symbol 1 is the refraction angle in the first sub- layer lying under the ambient and n0 sin 0 = n1 sin 1 , where n1 is the refractive index of represent the transmission coefficients of the uppermost this sub- layer. Symbols t0,1 and t0,1
IP
Y0 Y1 = r0,1. , r0,1 Y0 Y1
CR
r0,1 =
T
are expressed as boundary of the system whose reflection coefficients r0,1 and r0,1
The reflection coefficients of the lowest boundary are given in this way: YN YS . YN YS
US
rN , N 1 rN ,S =
AN
It holds that nN sin N = nS sin S , where nN and N are the refractive index and refraction angle of the N -th sub- layer lying above the substrate, respectively. If the number N is
M
sufficiently large the differences between the refractive indices of the adjacent sub- layers are very small. In this case the reflection coefficients rw can be expressed in this way: 1 Yw x [(x)2 ], 2Yw x
ED
rw =
PT
where d j = x = d / N is the thickness of one sub- layer and
[(x)2 ] is the residue. It holds that
Yw = Yw1 Yw .
CE
It is evident that at formulation of the corrections one must take into account all the reflections inside the multilayer system approximating the inhomogeneous thin film. This means that it is also
AC
necessary to consider the reflections from the uppermost and lowest boundaries of the system together with their combinations with the internal reflections (see Fig. 3). Thus, the reflection coefficients of the system corresponding to the first-order correction (see Fig. 2a) and a finite number of all the reflections inside this system
where
(1) 1,R
(1) 1,R
are expressed as follows:
N 1 m w = R s ( s 1) rw, w1exp ix j , w=1 s =1 j =1
(4)
ACCEPTED MANUSCRIPT
N
j =1
= r0,1rN ,S exp ix j which describes complete single passing of light between the uppermost and lowest boundaries with single reflections on these boundaries and m denotes integer (see Fig. 3).
T
Figure 3: Schematic diagram related to calculating all possible combinations of the single
IP
reflections and reflections between the uppermost and lowest boundaries of the inhomogeneous
CR
4 thin film (here s = 4 , number of all the combinations is equal to = 4 ). 1 It is apparent that one can write ( s 1)
m s =1
d m s 1 = . m d s =1 (1 ) 2
= lim
US
m
lim s
AN
After performing the limit for N going to infinity we obtain the expression for the partial (1) first-order corrections I1,R of the reflection coefficients as follows:
where (1) 1R
M
d R ix ( x ) (1) = , j1 = f ( x1 )e 1 dx1 , 2 (1 ) 0
PT
C
R d ix ( x ) (1) f ( x1 )e 1 dx1 = C1(1) R j1 , 2 (1 ) 0
ED
(1) I1,R =
x( x1 ) =
AC
CE
f ( x1 ) = 4
x1
1 dY ( x1 ) , 2Y ( x1 ) dx1 n 2 ( x) n02 sin 20 dx,
0
= r0,1rN ,Se d . ix
, where Quantity R is again given as R = t0,1t0,1
t0,1 =
2Y0 2YU = , t0,1 Y0 YU Y0 YU
for the s-polarization and t0,1 =
c0 2Y0 c 2YU = U , t0,1 cU Y0 YU c0 Y0 YU
(5)
ACCEPTED MANUSCRIPT for the p-polarization. (1)
In the same way the formula for the partial first-order correction I 1,R corresponding to internal single reflections of the type depicted in Fig. 2b is derived, i.e. (1)
(1)
(1)
(6)
I 1,R = C1,R j1 ,
(1)
d RZ 2 ix ( x ) (1) , j = f ( x1 )e 1 dx1 1 2 (1 ) 0
IP
C1,R =
T
where
ix
Z = rN ,Se d .
CR
and
US
The total corrections of the first-order r1 are then given for both the polarizations as (1) (1) r1 = I1,R I1,R .
(7)
AN
Each of the partial first-order corrections generates three partial corrections of the second
M
order which is evident from Fig. 4.
ED
Figure 4: Basic motifs of the internal reflections corresponding to the partial second-order corrections. Full lines represent the internal reflections while dashed lines represent transmitted
PT
light.
CE
In this figure the basic motifs of all the double internal reflections are plotted. Thus, the (1) (1) (1) (2) , I 2,R , corrections I1,R and I1,R generate the following partial second-order corrections I 2,R
AC
(3) (1) (2) (3) . If we take into account the mathematical procedure used to derive the I 2,R , I 2,R , I 2,R and I 2,R
first-order corrections we get the following expressions for all the partial second-order corrections:
ACCEPTED MANUSCRIPT (l ) (l ) (l ) (l ) I 2,R = C2,R j2( l ) , I 2,R = C2,R j2( l ) , l = 1, 2,3,
C
(1) 2,R
=
(1) r0,1C1,R
dd
(1) 2
, j
1
= f ( x1 ) f ( x2 )e
C
dx2dx1 ,
00
d x1
(2) 2,R
i[ x ( x1 ) x ( x2 )]
(2) 2
= ZC1,R , j
= f ( x1 ) f ( x2 )e
i[ x ( x1 ) x ( x2 )]
dx2dx1 ,
00
(1) dd ZC1,R i[ x ( x ) x ( x )] , j2(3) = f ( x1 ) f ( x2 )e 1 2 dx2dx1 , 1 00
T
(3) C2,R =
dd
1 (1) i[ x ( x1 ) x ( x2 )] C1,R , j2(1) = f ( x1 ) f ( x2 )e dx2dx1 , Z 0x
IP
(1) C2,R =
, j2(2) = f ( x1 ) f ( x2 )e
1 (1) ZC1,R
1
i[ x ( x1 ) x ( x2 )]
00
dd
, j2(3) = f ( x1 ) f ( x2 )e 00
i[ x ( x1 ) x ( x2 )]
dx2dx1 ,
dx2dx1.
AN
(3) C2,R =
r C
US
(2) C2,R =
dd
CR
1
(1) 0,1 1,R
(8) (2)
(2) (1) (3) = I 2,R = I 2,R . From Fig. 4 it is clear that the following equalities are fulfilled: I 2,R and I 2,R
M
The total second-order corrections r2 are again given as the sum of these partial corrections, i.e. (l ) (l ) r2 = I 2,R I 2,R .
ED
3
(9)
l =1
In general the partial corrections of the ( n 1 )-th order are easily derived by means of the
PT
corrections of the n -th order using the geometrical constructions presented in Figs.5-10. Each correction of n -th order gives three corrections of (n 1) -th order. When these three new
CE
corrections are derived it is necessary to distinguish two cases. In the first case the last internal reflection in the n -th order correction falls onto the upper boundary. In the second case it falls
AC
onto the lower boundary. If the last internal reflection falls onto the upper boundary, then from Fig. 5 it is implied that the partial correction of the n -th order is given as d
I
(l ) n ,R
(l ) n ,R
=C
0
n ix ( x ) f ( xq ) e 1 q =1
e
ix ( xn )
dxn
dx1 ,
where = 0, = d or = xn1 , = d . Figure 5: Schematic diagram of the n 1 internal reflections inside the film
(10)
ACCEPTED MANUSCRIPT 1) From Fig. 5 it is also obvious that d
n 1 f ( xq ) 0 q =1
d
jn(3l1 2) = 0
e
ix ( x1 )
e
ix ( xn ) ix ( xn 1 )
e
dxn 1
(11)
dx1
and ,
T
1
(12)
,3n1.
CR
where l = 1, 2,
r0,1Cn(l,R)
IP
2) Cn(3l1,R =
US
Figure 6: Schematic diagram of the n 1 internal reflections inside the film
2) If the schematic diagram in Fig. 6 is taken into account, one can derive the following
j
= 0
n 1 f ( x ) q 0 q =1
ix ( x1 )
e
ix ( xn ) ix ( xn 1 )
e
dxn 1
(13) dx1
ED
e
xn
d
M
(3l 1) n 1
AN
formulae:
and
1) Cn(3l1,R = ZCn(l,R) .
PT
(14)
CE
Figure 7: Schematic diagram of the n 1 internal reflections inside the film
AC
3) Furthermore, from Fig. 7 one can imply d
n 1 f ( xq ) 0 q =1
d
(3l ) n 1
j
= 0
e
ix ( x1 )
e
ix ( xn ) ix ( xn 1 )
e
dxn 1
(15)
dx1
and (3l ) n 1,R
C
ZCn(l,R) = . 1
Figure 8: Schematic diagram of the n 1 internal reflections inside the film
(16)
ACCEPTED MANUSCRIPT If the last internal reflection in the n -th order corrections falls onto the lower boundary, then the corresponding n -th order partial corrections is given as
n ix ( x1 ) f ( x ) e q q =1
d
(l ) n
j
= 0
e
ix ( xn )
dxn
dx1
(17)
where = 0, = d or = 0, = xn1. d
ix ( x1 )
ix ( xn ) ix ( xn 1 )
e
e
dxn 1
(3l 2) n 1,R
=
Cn(l,R) Z
(19)
AN
C
dx1
US
and
(18)
CR
0
IP
n 1 f ( xq ) x q =1 n
d
jn(3l1 2) = e
T
4) From Fig. 8 it is seen that
M
Figure 9: Schematic diagram of the n 1 internal reflections inside the film
ED
5) One can see from Fig. 9 that (3l 1) n 1
j
= 0
ix ( x1 )
PT
e
e
CE
and
d
n 1 f ( x ) q 0 q =1
d
ix ( xn ) ix ( xn 1 )
(3l 1) n 1,R
=
dxn 1
dx1
r0,1Cn(l,R)
(21)
1
AC
C
e
(20)
Figure 10: Schematic diagram of the n 1 internal reflections inside the film
6) On the basis of Fig. 10 it is clear that d
n 1 f ( xq ) 0 q =1
d
jn(3l1) = 0
e and
ix ( x1 )
e
ix ( xn ) ix ( xn 1 )
e
dxn 1
(22)
dx1
ACCEPTED MANUSCRIPT ZCn(l,R)
) Cn(3l1,R =
1
(23)
.
As mentioned above each of the partial corrections of the n -th order I n(l,R) generates three 2) 1) ) . Thus, the , I n(3l1,R partial corrections of the ( n 1)-th order, i.e. corrections I n(3l1,R and I n(3l1,R
corrections of all the orders calculated by the foregoing procedure create the tree as follows:
(1) (2) (3) 3,R 3,R 3,R
(2) I 2,R
(3) I 2,R
(4) (5) (6) 3,R 3,R 3,R
I I I
I I I
(24)
CR
I I I
(7) (8) (9) 3,R 3,R 3,R
IP
(1) I 2,R
T
(1) I1,R
the exponential factor e
ix ( x1 )
US
The same tree can be constructed for the corrections I n(,Rl ) . Of course, in corrections I n(,Rl ) must be systematically replaced by e
ix ( x1 )
and the coefficients Cn(l,R)
AN
are expressed with other formulae compared with coefficients Cn(l,R) (see e.g. the formulae for (l )
M
(l ) (l ) (l ) (l ) C2,R and C2,R in (8) and C3,R and C3,R in Table 1). Note that for the partial corrections I n ,R
the same general recursive formulae are valid as for the partial corrections I n(l,R) . Only the changes
ED
expressed in the foregoing sentence must be performed in the formulae for I n(,Rl ) . The total
rn =
3n 1
I
(l ) n ,R
I n(,Rl ) .
(25)
l =1
l
1
(l ) C3,R
(1) r0,1C2,R
1
AC
CE
PT
correction of the n -th order rn is given as follows
2
ZC
j3(l )
I 3,R (l )
(l )
j3
C 3,R
ddd
3 ix ( x ) ix ( x ) ix ( x ) (1) f ( xq ) e 1 e 2 e 3 dx3dx2dx1 r0,11C2,R 0 0 0 q =1
3 ix ( x ) ix ( x ) ix ( x ) f ( xq ) e 1 e 2 e 3 dx3dx2dx1 0 x1 0 q =1
3 ix ( x1 ) ix ( x2 ) ix ( x3 ) (1) f ( x ) e dx3dx2dxZC e e q 1 2,R 0 0 0 q =1
3 ix ( x1 ) ix ( x2 ) ix ( x3 ) f ( x ) e e dx3dx2dx1 e q 0 x1 0 q =1
d d x2
(1) 2,R
(l )
(l ) I 3,R
ddd
d d x2
ACCEPTED MANUSCRIPT 3 ix ( x ) ix ( x ) ix ( x ) f ( xq ) e 1 e 2 e 3 dx3dx2dx1 0 x1 0 q =1
d x1 d
(2) C2,R
3 ix ( x ) ix ( x ) ix ( x ) (2) f ( xq ) e 1 e 2 e 3 dx3dx2dxr10,1C2,R 1 0 0 x2 q =1
3 ix ( x1 ) ix ( x2 ) ix ( x3 ) f ( x ) e e dx3dx2dx1 e q 0 0 0 q =1
(2) r0,1C2,R
3 ix ( x1 ) ix ( x2 ) ix ( x3 ) (2) f ( x ) e e dx3dx2dx1ZC2,R e q 0 0 0 q =1
3 ix ( x1 ) ix ( x2 ) ix ( x3 ) f ( x ) e e dx3dx2dx1 e q q =1 00 0
(2) ZC2,R
3 ix ( x ) ix ( x ) ix ( x ) (2) f ( xq ) e 1 e 2 e 3 dx3dx2dx1ZC2,R 1 0 0 0 q =1
Z
d x1 d
5
1
6
1
3 ix ( x1 ) ix ( x2 ) ix ( x3 ) f ( x ) e e dx3dx2dx1 e q 0 0 x2 q =1
(3) r0,1C2,R
3 ix ( x1 ) ix ( x2 ) ix ( x3 ) (3) f ( x ) e dx3dx2dx1r0,1C2,R e e q 1 0 0 0 q =1
3 ix ( x1 ) ix ( x2 ) ix ( x3 ) f ( x ) e e dx3dx2dx1 e q 0 0 0 q =1
(3) ZC2,R
3 ix ( x ) ix ( x ) ix ( x ) (3) f ( xq ) e 1 e 2 e 3 dx3dx2dxZC 1 2,R 1 0 0 0 q =1
Z
AN
(3) C2,R
ED
1
M
ddd
8
1
PT
ddd
9
3 ix ( x ) ix ( x ) ix ( x ) f ( xq ) e 1 e 2 e 3 dx3dx2dx1 0 0 0 q =1 ddd
3 ix ( x1 ) ix ( x2 ) ix ( x3 ) (3) C2,R f ( x ) e dx3dx2dx e e q 1 Z 0 0 x2 q =1 dd d
7
d d x2
US
d x1 d
ddd
T
ddd
IP
4
3 ix ( x ) ix ( x ) ix ( x ) (1) f ( xq ) e 1 e 2 e 3 dx3dx2dx1 ZC2,R q =1 000 1
CR
3
ddd
(1) ZC2,R 1
dd d
ddd
3 ix ( x ) ix ( x ) ix ( x ) f ( xq ) e 1 e 2 e 3 dx3dx2dx1 0 0 0 q =1 ddd
CE
(l ) (l ) Table 1: The coefficients C3,R and C3,R together with integrals
j3(l ) and j3(l ) corresponding to
AC
(l ) (l ) . the partial corrections of the third order I 3,R and I 3,R
In Table 1 the expressions for the coefficients and integrals of the corrections of the third (l ) (l ) , C3,R , j3(l ) and j3(l ) , are introduced. It can easily be proved that the following order, i.e. C3,R (2) (1) (3) (4) (6) (7) (9) (8) = I3,R , I3,R = I3,R , I3,R = I3,R , I3,R = I3,R , equalities are valid: I3,R
Figure 11: Schematic diagrams corresponding to the partial corrections of the first order at the incidence of light from the non-absorbing substrate side.
ACCEPTED MANUSCRIPT The approximate expression for the reflection coefficients of the inhomogeneous thin films
rM are given as follows: M
rM = r0 rn ,
(26)
n =1
where M denotes the maximum order of the total corrections needed for achieving the required
T
accuracy of calculations of reflectances and ellipsometric quantities of the inhomogeneous thin
IP
films for reflected light in case of light incidence from the ambient side.
CR
2.2.2. Incidence of light from the substrate side
If the substrates are non-absorbing the reflection coefficients of the inhomogeneous thin films are
US
also defined for light falling onto the films from the side of these substrates. The expressions of these coefficients are derived in the same way as those presented for light incident from the
(1) and I 1,R are expressed as ambient side. In this case the partial corrections of the first order I1,R
AN
(1)
(see Fig. 11)
M
(1) (1) (1) = C1,R (1) j1(1) , I '1,R I1,R = C'1,R j1(1) ,
ED
where
PT
(1) = C1,R
CE
and
(1) 1,R
C'
(1) R e d (1) , j = j 1 1 (1 )2 ix
r2 e d = R 0,1 2 , j1(1) = j1(1) (1 ) ix
AC
and R = t N ,St N ,S . Transmission coefficients t N ,S and t N ,S obey these equations: t N ,S =
2YS 2YL , t N ,S = YS YL YS YL
for the s-polarization and t N ,S =
c 2YS cL 2YL , t N ,S = S cS YS YL cL YS YL
for the p-polarization ( cS = cos S and cL = cos L ). It is evident that
(27)
ACCEPTED MANUSCRIPT ix
2 (1) ix (1) r e dC (1) C = R e d C1,R , C'1,R = R 0,1 2 1,R . R R Z (1) 1,R
Using the corresponding schematic diagrams of the basic motifs of the reflections one obtains the formulae for the partial corrections of the second order corresponding to light incident onto the inhomogeneous thin films from the substrate side, i.e.
(l ) = C2,R (l ) j2(l ) , I 2,R (l ) = C2,R (l ) j2(l ) , l = 1, 2,3 I 2,R
T
(28)
Z (1) , j2(1) = j2(3) , C1,R 1
(2) = r0,1C1,R (1) , j2(2) = j2(1) , C2,R
(2) = C2,R C
=
(1) ZC1,R 1
, j2(1) = j2(2) ,
(29)
, j2(2) = j2(3) ,
(1) r0,1C1,R
, j2(3) = j2(1) ,
1
ED
(3) 2,R
r0,1
US
=
(1) C1,R
M
C
(1) 2,R
r0,1 (1) (2) , j2(3) = j 2 , C1,R 1
AN
(3) = C2,R
CR
(1) = C2,R
IP
where
PT
(3) = I 2,R (2) and I 2,R (2) = I 2,R (1) . It is also It is evident that the following equalities are true: I 2,R possible to derive the general recursive formulae for the partial corrections corresponding to this
CE
incidence of light. Taking into account the individual reflections inside the film analogous to those
1)
where
AC
utilized for deriving eqs. (11)-(23) the following recursive formulae are obtained for this case: l 2) l 2) (3l 2) I n(31,R = Cn(31,R jn 1
l 2) Cn(31,R = d
jn(31l 2) = 0
e
ix ( x1 )
e
Z Cn(,Rl ) , 1
d
n 1 f ( x ) q 0 q =1
ix ( xn ) ix ( xn 1 )
where = 0, = d or = 0, = xn1.
(30)
e
dxn 1
dx1 ,
ACCEPTED MANUSCRIPT 2) l 1) l 1) (3l 1) I n(31,R = Cn(31,R jn 1
(31)
where l 1) Cn(31,R = r0,1Cn(,Rl ) ,
d
e
ix ( x1 )
e
ix ( xn ) ix ( xn 1 )
e
dxn 1
dx1.
CR
3) l) l) I n(31,R = Cn(31,R jn(31l )
=
d
jn(31l ) = 0
1
d
e
ix ( xn ) ix ( xn 1 )
e
dxn 1
dx1.
ED
4)
,
n 1 f ( x ) q q =1 0
M
e
ix ( x1 )
r0,1 Cn(,Rl )
AN
C
l 2) l 2) (3l 2) I n(31,R = Cn(31,R jn 1
AC
CE
PT
where
j
l 2) Cn(31,R =
(3l 2) n 1
d
= 0
e
ix ( x1 )
e
(32)
US
where (3l ) n 1,R
IP
0
T
n 1 f ( x ) q xn q =1
d
jn(31l 1) =
Cn(,Rl ) r0,1
(33)
,
xn
n 1 f ( x ) q 0 q =1
ix ( xn ) ix ( xn 1 )
e
dxn 1
dx1 ,
where = 0, = d or = xn1 0, = d . 5) l 1) l 1) (3l 1) I n(31,R = Cn(31,R jn 1
where
(34)
ACCEPTED MANUSCRIPT l 1) Cn(31,R =
j
(3l 1) n 1
d
n 1 f ( x ) q 0 q =1
d
= 0
e
ix ( x1 )
e
Z Cn(,Rl ) , 1
ix ( xn ) ix ( xn 1 )
e
dxn 1
dx1.
6)
n 1 f ( xq ) 0 q =1
= 0
e
ix ( xn ) ix ( xn 1 )
e
dxn 1
dx1.
AN
e
ix ( x1 )
,
US
j
1
d
d
(3l ) n 1
r0,1Cn(,Rl )
CR
where l) Cn(31,R =
(35)
IP
T
l) l) I n(31,R = Cn(31,R jn(31l )
The similar recursive formulae can be derived for the corrections I n(,Rl ) . The reflection coefficients
M
of the inhomogeneous films rM for this incidence of light are given in the following way: M
ED
rM = r0 rn,
AC
and
CE
PT
where
r0 =
rn =
(36)
n =1
rL rU e
ixd
1 rL rU e
3n 1
I
(l ) n ,R
ixd
I n(,Rl ) .
l =1
2.3. Transmission coefficients The transmission coefficients for the inhomogeneous thin films placed on the non-absorbing substrates must also be considered for both the cases of incidence of light.
2.3.1. Incidence of light from the ambient side Schematic diagram of beams corresponding to the WKBJ approximation in transmitted light is
ACCEPTED MANUSCRIPT plotted in Fig. 12. In this figure the reflections between the upper and lower boundaries are plotted by the full line segments. These reflections must be completed with the beams (waves) reflected from the upper boundary and transmitted by the lower boundary (in Fig 12 such the beam is plotted as the dashed line segment). If we take into account the approximation of the inhomogeneous thin film by the multilayer system the transmission of this wave through this system is described by the
IP
N 1 N ix /2 t = tk ,k 1 e k , k =1 k =1
T
following coefficient t : (37)
CR
Figure 12: Schematic diagram of selected beams corresponding to the WKBJ approximations in
US
transmitted light.
where tk ,k 1 denotes the transmission coefficients of the internal boundaries between the
N
AN
uppermost and lowest boundary of the system. Furthermore, it holds that lim e
For the s-polarization we can write that
=e
ixd /2
.
(38)
M
N k =1
ixk /2
ED
N 1 N 1 ln tk ,k 1 = ln tk ,k 1 = k =1 k =1
N1
is the residual. Hence
AC
where
(39)
CE
PT
N 1 1 Yk 1 = ln 1 x = N k =1 2Yk x 1 N 1 1 Yk 1 = x , 2 k =1 Yk x N
N 1 N 1 ln t = ln lim k ,k 1 lim tk ,k 1 = N k =1 N k =1 d Y 1 1 dY ( x) = dx = ln U . 2 0 Y ( x) dx YL
From the foregoing it is evident that limt =
N
YU YL
e
ixd /2
.
(40)
ACCEPTED MANUSCRIPT In the same way it is obtained for the p-polarization that cU cL
limt =
N
YU YL
e
ixd /2
(41)
.
Thus, the transmission coefficients belonging to the WKBJ approximation are given as follows: ixd /2
1 r0,1rN ,Se
ixd
(42)
,
T
t0 = B
t0,1t N ,Se
YU YL
,
cU cL
YU YL
AN
B=
US
for the s-polarization and
CR
B=
IP
where
for the p-polarization. Beams (waves) corresponding to the partial corrections of the first and
M
second orders are introduced in Figs. 2 and 4. By means of these figures the partial corrections of (1) (1) the first order I1,T and I1,T are expressed in this way:
ED
(1) (1) (1) I1,T = C1,T j1
PT
where
T ix /2 (1) (1) r0,1Be d C1,R = PC1,R R
(44)
T = t0,1t N ,S .
AC
and
CE
(1) C1,T =
(43)
(1) For the correction I1,T it holds that (1) (1) (1) I1,T = C1,T j1 ,
where (1) C1,T =
T B ixd /2 (1) (1) e C1,R = QC1,R . R rN ,S
In the foregoing equations quantities P and Q are given as
(45)
ACCEPTED MANUSCRIPT P=
T B ixd /2 ix /2 r0,1Be d , Q = T e . R R rN ,S
Using Figs. 2 and 4 it is also possible to derive the formulae for the partial corrections of the second order for the inhomogeneous film in transmitted light. The achieved formulae are as follows: (1) (1) (1) (1) (1) I 2,T = C2,T j2 , C2,T = PC2,R ,
(1)
(1)
(1)
(1)
(1)
(2)
(2)
(2)
(2)
(2)
(3)
(3)
(3)
(3)
(3)
(46)
CR
I 2,T = C 2,T j 2 , C 2,T = PC 2,R ,
IP
(3) (3) (3) (3) (3) I 2,T = C2,T j2 , C2,T = QC2,R ,
T
(2) (2) (2) (2) (2) I 2,T = C2,T j2 , C2,T = QC2,R ,
I 2,T = C 2,T j 2 , C 2,T = PC 2,R ,
US
I 2,T = C 2,T j 2 , C 2,T = QC 2,R .
From the foregoing one can imply that the formulae for partial corrections for transmitted light can
AN
be inferred by using the formulae for the reflected light in an easy way. It is sufficient to multiply
M
(l ) (l ) the corresponding coefficients C2,R or C2,R by P or Q in order to obtain the correct formulae
for the partial corrections belonging to transmitted light. If the last internal reflected beam falls
ED
(l ) (l ) onto the upper boundary it is necessary to multiply the corresponding coefficients C2,R or C2,R
by P . If the last internal beam falls onto the lower boundary one must multiply the corresponding
PT
(l ) (l ) coefficients C2,R or C 2,R by Q . This enables us to state that the general recursive formulae for
the partial corrections corresponding to transmitted light are given by the formulae valid for
CE
reflected light (see eqs. (11)-(23)) with the mentioned changes in coefficients Cn(l,R) or Cn(l,R) . The
AC
approximate formula for the transmission coefficients t M of the inhomogeneous thin films for light incident from the ambient side is given as follows: M
tM = t0 tn , n =1
where tn =
3n 1
I
(l ) n ,T
l =1
2.3.2. Incidence of light from the substrate side
I n(,Tl ) .
(47)
ACCEPTED MANUSCRIPT From Figs. 2, 4 and 11 it is apparent that the formulae for the partial corrections corresponding to transmitted light at incidence of light from the substrate side can easily be formulated by means of those presented for reflection of light at the same incidence (see eqs. (27)-(35)). Thus, one can write the following equations for the corrections of the first order: (1)
(1) = C1,T (1) j1 , I1,T (1) = C1,T (1) j1(1) , I1,T
(48)
T B ixd /2 ix /2 rN ,S Be d , Q = T e , R R r0,1 1 t N ,S . , T = t0,1 B
US
B =
CR
P =
IP
(1) = PC1,R (1) , C1,T (1) = QC1,R (1) , C1,T
T
where
From the foregoing it is clear that if the last internal beam falls onto the lower boundary the
AN
coefficients Cn(,Rl ) or Cn(,Rl ) must be multiplied with P and if this beam falls onto the upper boundary these coefficients must be multiplied by Q in order to obtain the correction
M
(1) (see eqs. (29)). All the conclusions and rules valid for the corrections coefficients Cn(,Tl ) and C1,T
ED
I n(l,T) and I n(,Tl ) . are also fulfilled for the corrections I n(,Tl ) and I n(,Tl ) . The general recursive formulae for the partial corrections I n(,Tl ) and I n(,Tl ) can be easily derived using those valid for the partial
PT
corrections I n(,Rl ) and I n(,Rl ) corresponding to light reflected at incidence of light from the substrate
CE
side.
The approximate transmission coefficients for the inhomogeneous thin films at the
where
AC
incidence of light from the non-absorbing substrate obey the following formula: M
tM = t0 tn , n =1
t0 = B
t N ,Se t0,1
ixd /2
1 r0,1rN ,Se
ixd
and tn =
3n 1
I
(l ) n ,T
l =1
I n(,Tl ) .
(49)
ACCEPTED MANUSCRIPT It is possible to find the relationship between the transmission coefficients t M and t M by means of the following simple procedure. For the s-polarization these coefficients can be written as tM =
YU YL
t0,1t N ,S F ( , 0 )
(50)
t N ,S F ( , 0 ), t0,1
(51)
YL YU
IP
tM =
T
and
CR
where F ( , 0 ) represents the quantity describing the influence of all the reflections on expressing the transmission coefficients for both the directions, i.e. for the directions belonging to
US
the incidence from the ambient side and substrate side. For each path represe nting the contribution to t M there is the same path, but with opposite direction of propagation, contributing to tM .
AN
Since the mathematical representation of the contribution to F ( , 0 ) corresponding to this path is the same for t M and tM , it is evident that the functions F ( , 0 ) in eqs. (50) and (51) must be
M
, identical. From eqs. (50), (51) and the formulae for the transmission coefficients t0,1 , t N ,S , t0,1
ED
t N ,S one can infer that
(52)
PT
Y t t Y tM = U 0,1 N ,S = S . tM YL t0,1 t N ,S Y0
The same relation is true for the p-polarization. Relation (52) is correct for the approximation of
CE
arbitrary order M . Note that the results contained in the last two subsections explain why it is unnecessary to consider the influence of products of the internal transmission coefficients at
AC
derivations of individual corrections occurring in the formulae for the reflection coefficients of the inhomogeneous films. The product of B and B gives unity for every single transmission through the film with the phase-shift angle xd , i.e. for the single path: upper boundary-lower boundary-upper boundary (see equations for B and B ). If the equations for the transmittances of the p and s-polarizations are taken into account, i.e Tj =
nS cos S n cos 0 | tM |2 , T j = 0 | tM |2 ( j = p, s), n0 cos 0 nS cos S
together with eq. (52) one can easily derive that T j = T j for non-absorbing and absorbing
ACCEPTED MANUSCRIPT inhomogeneous thin films for any correction order M (symbol | | denotes the module of the complex quantity placed inside this symbol).
3.
Evaluation of the integrals using the Chebyshev polynomials
3.1. Recursive relations
IP
T
By examining the structure of the integrals jn(l ) and jn(l ) one can notice that for each order there are only two integrals in which the dependencies of the integral bounds are such that they cannot
CR
be written as products of integrals corresponding to lower orders. For the first, second and third order these integrals are given as
US
j1 = j1(1) , j1 = j1(1) , j2 = j2(2) ,
j2 = j2(1) , j3 = j3(4) , j3 = j3(2) ,
AN
where the symbols jn and jn are used to denote the indecomposable integrals for a given order
n . All the other integrals that we have encountered can be written as products of these integrals,
M
for example j3(1) = ( j1 )3 and j3(5) = j1 j2 . We will present a method that allows us to calculate the
and g n ( x1 ) . For n = 1, 2
ED
integrals jn and jn recursively. For this purpose we introduce sequences of functions g n ( x1 ) we define these functions as d
PT
g n ( x1 ) = g n 1 ( x2 ) f ( x2 )e x1
x1
g n ( x1 ) = g n 1 ( x2 ) f ( x2 )e
CE
ix ( x2 )
dx2 ,
ix ( x2 )
0
(53)
dx2 ,
and for n = 0 we define g0 ( x1 ) = g0 ( x1 ) = 1 . The integrals that we want to calculate are then
AC
obtained by substituting the values 0 and d for the variable x1 . They are given as
jn = gn (0), jn = gn (d ).
(54)
Performing the numerical quadrature for each value of the variable x1 occurring in the bounds of integrals defining the sequences of functions g n ( x1 ) and g n ( x1 ) would be impractical. A much better approach is to approximate the functions on the interval [0, d ] by polynomials, which can be then integrated analytically.
ACCEPTED MANUSCRIPT 3.2. Chebyshev polynomials In this subsection an efficient method for evaluation of the integrals in the recursive formulae (53) will be presented. This method is based on the idea that the Chebyshev interpolation can be used to approximate the integrands by the polynomials that can be then integrated analytically. Then, the speed of calculating the integrals is extremely increased.
T
The Chebyshev polynomials of the first kind Tk (t ) are defined on the interval [1,1] and
IP
non-negative integers k as [28, 29]
(55)
CR
Tk (t ) = cos(k arccos t ).
Since we are not working with functions defined on this interval but on the interval [0, d ] , it is
US
necessary to map the variable x1 to interval [1,1] as t = 2 x1 / d 1 . Let us assume that g ( x1 ) is some function defined on the interval [0, d ] for which we
AN
want to construct the polynomial approximation. The Chebyshev interpolation polynomial gˆ ( x1 ) at H 1 points is a polynomial of degree H which coincides with this function at H 1 points
M
ak given as the zeros of the Chebyshev polynomial TH 1 . These points, called the Chebyshev nodes, can be calculated as
(56)
, H . Note that we used the hat to distinguish the interpolation polynomial from the
PT
where k = 0,
d (k 12 ) cos 1 , 2 H 1
ED ak =
CE
function that it approximates. The interpolation polynomial can be written in the basis of the Chebyshev polynomials as
AC
gˆ ( x1 ) =
v0 H vkTk (2 x1 / d 1), 2 k =1
(57)
with the coefficients vk given as
vk =
2 H k ( p 12 ) g (a p ) cos . H 1 p =0 H 1
(58)
If we want to calculate the values at the Chebyshev nodes, then we can insert the representation of the Chebyshev polynomials (55) and the formula for the Chebyshev nodes (56) into (57) and write the result as
ACCEPTED MANUSCRIPT H 1 p(k 12 ) g (ak ) = v0 v p cos . 2 H 1 p =1
(59)
The formulae (58) and (59) are important since they allow us to convert between two representations of the Chebyshev interpolation polynomial, i.e. between the representation using the coefficients in the expansion in the Chebyshev polynomials and the representation using the values at the Chebyshev nodes. The representation using the expansion in the Chebyshev
T
polynomials is useful for integration of the polynomials while it is much easier to specify the
IP
function in the representation using the values at the Chebyshev nodes. Apart from some unusual
CR
normalization, the formulae (58) and (59) are the discrete cosine transforms DCT-II and DCT-III [30]. These discrete cosine transforms can be performed by algorithms that are much faster than
US
the direct evaluation of the sums in (58) and (59), therefore, they provide an efficient tool for conversion between the two representations.
AN
The integral over the interval [0, d ] of the function g ( x1 ) can be approximated by the value obtained by integrating its Chebyshev approximation gˆ ( x1 ) . The result d
0
gˆ ( x1 )dx1 =
2vk ), 2 k even ,2 k H 1 k
d (v0 2
M
(60)
ED
is known as the Clenshaw–Curtis quadrature [31]. This formula is useful if we want to calculate the definite integrals, but we also need to calculate the primitive functions. The primitive function
PT
to the polynomial gˆ ( x1 ) of degree H is a polynomial of degree H 1 . Since we want to work with polynomials of the fixed degree H , the part of the integrated polynomial proportional to the
CE
Chebyshev polynomial TH 1 will be neglected. The fact that the term proportional to TH 1 is neglected will be indicated by using the symbol
instead of the equality sign. The integrals of
AC
gˆ ( x1 ) considered as a functions of the upper and the lower bounds are then given as
x1
0
gˆ ( x2 )dx2
d Gˆ ( x1 ), gˆ ( x2 )dx2 x1
Gˆ ( x1 ),
(61)
with the polynomials Gˆ ( x1 ) and Gˆ ( x1 ) given in the same way as gˆ ( x1 ) in (57) but with the coefficients vk replaced by Vk and Vk . The zeroth coefficients are given as
ACCEPTED MANUSCRIPT V0 =
d v0 v1 H 1 (1) k vk (1) H vH , 2 2 4 k =1 1 k 2 2( H 1)
(62)
vH d v v H 1 v V = 0 1 k 2 , 2 2 4 k =1 1 k 2( H 1) 0
, H 1 and for the last coefficient we have
Vk =
d vk 1 vk 1 d vH 1 . , VH = 2 2k 2 2H
(63)
T
for k = 1,
IP
Note that since the Chebyshev polynomials are bounded by | Tk | 1 , the error that we introduce by
CR
neglecting the part proportional to TH 1 is at most | VH1 |=| vH | /(2H 2) . This should not be a
US
problem because the polynomial gˆ ( x1 ) is only an approximation of the function g ( x1 ) and if the coefficient vH is too large, then higher order of the interpolation polynomial should be used to
AN
increase the precision of the calculations.
M
3.3. Evaluations of the integrals
step we start with n = 1 ):
ED
The steps needed to calculate the integrals jn and jn can be summarized as follows (in the first
Calculate the values of the integrands in (53) at the Chebyshev nodes (56).
Use the discrete cosine transform DCT-II to calculate the coefficients (58) for the
PT
Chebyshev interpolation polynomials of these integrands. Using the formula (59) calculate the values of jn and jn .
Calculate the coefficients (62), (63). For the function g n ( x1 ) we need to calculate the
AC
CE
integral as a function of the lower bound (coefficients Vk ), while for the other function
gn ( x1 ) the integral is a function of the upper bound (coefficients Vk ).
Using the discrete cosine transform DCT-III calculate the values of the functions g n ( x1 ) and g n ( x1 ) at the Chebyshev nodes. These values will be needed to calculate the integrands in the next recursive step of (53).
Repeat the above steps for the next order.
ACCEPTED MANUSCRIPT 4.
Numerical analysis
The equalities among some corrections of the second and third orders occurring in the formulae for the reflection coefficients are presented above. They can be found using the schematic diagrams of reflections inside the films as performed above. However, for the higher orders numerical calculations must be utilized for finding these equalities. This fact implies that only certain
T
corrections must be calculated numerically within the tasks of the optical characterization and
IP
synthesis of the inhomogeneous thin films. These corrections can be called the unique corrections. The numbers of the unique corrections belonging to the individual orders in the formula for
CR
reflection and transmission coefficients rM and t M are introduced in Table 2. In this table the numbers of all the corrections are also presented for comparison. The numbers in Table 2 were
US
found using a computer program which generated all the basic motifs for a given order. The motifs which represent contributions to reflection and transmission coefficients with the same
AN
mathematical representation were then collected into groups representing unique contributions. The number of the motifs in a given group determines the multiplicity for a corresponding unique
M
contribution. Table of the numbers of unique corrections corresponding to reflection coefficient
ED
rM and transmission coefficients t M exhibit the same results.
PT
Table 2: Numbers of unique and total corrections for reflection and transmission coefficients rM and t M for a given order n .
AC
CE
n
total
(l ) unique I n ,R
(l ) unique I n ,T
1
2
2
2
2
6
5
6
3
18
13
14
4
54
27
31
5
162
59
62
6
486
110
121
7
1458
214
220
8
4374
371
398
9
13122
670
678
10
39366
1102
1161
ACCEPTED MANUSCRIPT 11
118098
1882
1886
12
354294
2979
3101
In this section a brief numerical analysis of reflectance and ellipsometric quantities in reflected light is presented for selected examples of the inhomogeneous films for the illustration of the theoretical results achieved in the foregoing sections. It is assumed that light is incident onto
IP
T
the films from the ambient side. The spectral dependencies of reflectance R =| rM , j |2 are calculated at normal incidence while spectral dependencies of associated ellipsometric parameters
CR
are calculated for oblique incidence. For non-depolarizing inhomogeneous films these associated ellipsometric parameters are expressed in the following way:
US
Is = sin(2)sin(), I c = sin(2) cos(), I n = cos(2), where azimuth and phase change are defined by this equation: rM ,p
AN
tan ei =
rM ,s
.
M
Figure 13: The spectral dependencies of reflectance R and associated ellipsometric parameters
ED
I s , I c and I n for film 1 placed onto silicon substrate.
In Fig. 13 the calculated spectral dependencies of the reflectance and associated
PT
ellipsometric parameters are plotted for the inhomogeneous thin film with thickness d = 100nm
CE
and linear refractive index profile. The boundary refractive indices are as follows: nU = 1.5 and
nL = 2.5 (film 1). Dispersion of the refractive index of the film is not taken into account for
AC
simplicity. This film is the representative of the inhomogeneous thin films with relatively large gradient of the refractive index. The film is placed on the silicon single crystal substrate whose optical constants are taken from the literature [32]. In Fig. 13 the spectral dependencies of the optical quantities mentioned above are calculated by the formulae containing corrections up to the twelfth order. These dependencies can be considered to be the true dependencies for this inhomogeneous film. Furthermore, the spectral dependencies of these optical quantities calculated by the formulae of the WKBJ approximations are plotted in this figure. One can see that there are the relatively large differences between the values of the optical quantities calculated within the WKBJ and those representing the true spectral dependencies. Thus, the WKBJ approximation is
ACCEPTED MANUSCRIPT not usable for calculating the values of the reflectance and ellipsometric parameters of this inhomogeneous thin film. In Fig. 14 the differences between the true spectral dependencies of reflectance and ellipsometric parameters on the one hand and dependencies calculated using the approximate formulae on the other hand are introduced. These differences R , I s , I c and
I n are calculated in this way: R = Rv Rtrue , Is = Is,v Is,true , I c = I c,v I c,true and
T
I n = I n,v I n,true , where v = 1, 2,3. If v = 1, 2 and 3 then the formulae for calculating the
IP
individual optical quantities contain the corrections up to the first, second and third order,
CR
respectively, are used. From Fig. 14 it is clear that all the differences decrease with increasing the order of corrections which indicates a good convergence to the true values of these optical
US
quantities. Furthermore, the small values of the differences of R , I s , I c and I n belonging to the first order indicate that the formulae containing the corrections up to the first order ensure
AN
the substantial approach of the values of R , I s , I c and I n to the true values. The influence of the corrections of the third order is very small or practically negligible (see Fig.14). This fact
M
implies that the corrections of higher orders than the third order need not be taken into account from the practical point of view. In Fig. 15 the dependencies of the same kind are plotted for the
ED
inhomogeneous thin film with thickness d = 800nm and the linear profile of the refractive index containing identical boundary values nU and nL as for the foregoing film (film 2). This film is
PT
again placed onto the silicon single crystal substrate. Such the film is the representative of the inhomogeneous films with smaller gradients of their refractive index profiles. In this case the true
CE
spectral dependencies of the optical quantities are practically identical with those calculated within the WKBJ approximation. This is also proved by the values of the differences plotted in Fig. 16.
AC
This means that the WKBJ approximation is sufficient for calculating the optical quantities of this inhomogeneous film owing to usual experimental accuracy. In Fig. 17 the spectral dependencies of reflectance and ellipsometric parameters calculated for the thin inhomogeneous film 1 situated on an aluminum substrate are introduced. The optical constants of the aluminum substrate are taken from [33]. In this figure the true spectral dependencies of the optical quantities and those calculated using the WKBJ formulae are compared. In Fig. 18 the spectral dependencies of the differences corresponding to the first, second, third and fifth orders are depicted for these optical quantities. From Fig. 17 it is seen that the differences between the true spectral reflecta nce and that corresponding to the WKBJ approximation are relatively large (even a mutual shift of the curves is
ACCEPTED MANUSCRIPT recorded). As for the ellipsometric parameters the situation is somewhat different. In the middle part of the spectral range of interest the differences between the true and WKBJ dependencies are very small while within the interval lying at the left end of this spectral range the differences are rather large. From Fig. 18 it is also seen that the corrections up to the fifth order have the substantial meaning for the calculations of the ellipsometric parameters in the interval adjacent to the left end of the spectral range. In the small interval adjacent to the right end of the spectral range
T
the corrections up to the first and second order has the small influence. In the wide middle part of
IP
the spectral range the influence of the corrections including the corrections of the first order is
CR
practically negligible which coincides with the dependencies in Fig. 17. The behavior of the spectral dependencies of R is also in coincidence with the curves presented in Fig. 17. Thus, the
US
influence of inhomogeneity exhibiting the large gradients of profiles on the calculated optical quantities in the case of the aluminum substrate is rather complicated. There is the spectral range in
AN
which this influence is strong while within the other ranges this influence is weak so that the WKBJ approximation can be used. This implies the conclusion that the influence of considerable
M
inhomogeneity of thin films on their optical quantities is generally complicated and that this influence also depends on the spectral dependencies of the optical constants of the substrates (see
ED
Figs. 13,14,17 and 18).
PT
Figure 14: The spectral dependencies of the differences of reflectance R and associated ellipsometric parameters I s , I c and I n for film 1 on the silicon substrate corresponding to
CE
the formulae containing the corrections up to the first, second and third order.
AC
Figure 15: The spectral dependencies of reflectance R and associated ellipsometric parameters
I s , I c and I n of film 2 placed on the silicon substrate. Figure 16: The spectral dependencies of the differences of reflectance R and associated ellipsometric parameters I s , I c and I n for film 2 on the silicon substrate corresponding to the WKBJ formulae and formulae containing the corrections up to the first and second order.
Figure 17: The spectral dependencies of reflectance R and associated ellipsometric parameters
ACCEPTED MANUSCRIPT I s , I c and I n for film 1 placed onto aluminum substrate. Figure 18: The spectral dependencies of the differences of reflectance R and associated ellipsometric parameters I s , I c and I n for film 1 on the aluminum substrate corresponding
T
to the formulae containing the corrections up to the first, second, third and fifth orders.
IP
If the inhomogeneous thin films exhibit complicated profiles of the refractive index the corrections of higher orders must always be used to calculate their reflectance and ellipsometric
CR
parameters. This is true for the complicated profiles exhibiting strong gradients in particular. One of the examples of the inhomogeneous films with complicated profiles is a rugate filter. In this
US
paper we deal with reflectance of the rugate filter studied in paper of Kildemo et al. [27]. This filter has ten-cycle sinusoidal modulation with matched surrounding media (no reflection from
AN
surrounding boundaries is realized) and with resonance at 550nm ( E = 2.254eV ). Sinusoidal modulation is given in this way:
M
n( x) = 2.0 0.1sin(20 x / d ),
ED
where d is the thickness of the filter ( d = 1375nm ). The spectral dependencies of reflectance of this rugate filter corresponding to the approximate formulae containing the corrections up to the seventh order are plotted in Fig. 19. It is seen that in the maximum of reflectance there are large
PT
differences between the values belonging to the first and the higher orders. Simultaneously, it is
CE
evident that the values of the reflectance calculated by the formula with the corrections of the first order are wrong in the interval of the reflectance maxima. From this figure it is also clear that the formulae containing the corrections up to the seventh-order can be considered to be sufficient for
AC
calculating whole spectral reflectance of the filter considered. Note that the corrections of the even orders are equal to zero which is apparent from the corresponding formulae presented above.
Figure 19: The spectral dependencies of the reflectance of the rugate filter at normal incidence. Numbers denote the orders of the corrections.
Note that the integrals were evaluated as described in section 3 with the degree of the Chebyshev interpolation polynomial chosen as H = 64 for the linear refractive index profiles and
ACCEPTED MANUSCRIPT H = 1024 for the rugate filter.
5.
Comparison with other approaches
It is clear that the speed of the calculations of the optical quantities of inhomogeneous thin films using our formulae depends mainly on the speed of the evaluations of the integrals occurring in our
T
approach. For a given order M there are 2M 1 such integrals and if the integration method
IP
based on the Chebyshev interpolation is used then the calculation time is proportional to this
CR
number. The similar integrals occur also in the approach of Kildemo et al. [27], therefore the speeds of these methods can easily be compared. In some cases the precision achieved by both the
US
approaches for a given order are comparable (e.g. for the linear refractive index profile) but for some complicated profiles the formulae of Kildemo et al. provide higher precision and speed of calculations at a given order (e.g. for the rugate filter). This statement was proved by numerical
AN
analysis performed using our formulae and formulae of Kildemo et al. for concrete examples of refractive index profiles. The approach developed in [27] was utilized, for example, in papers [34,
M
35].
ED
However, on the other hand our formulae enable us to estimate the precision of calculating the optical quantities of any inhomogeneous thin film more easily than the formulae of the
PT
mentioned authors. For this easy estimation we can use the formulae for the individual terms rn in the series appearing in the formulae for the reflection and transmission coefficients, i.e. for rM
CE
etc. The formulae for the reflection coefficients are given by rational functions in the approach of [27] and so the estimation of the precision of calculations is not so straightforward.
AC
The further advantage of our approach consists in the clear and simple way of derivation of the formulae for the reflection and transmission coefficients. This is especially important for the inhomogeneous thin films exhibiting large refractive index gradients. In this case the terms with higher-order integrals must be included into the formula expressing the reflection and transmission coefficients because the terms up to the third-order integrals are usually insufficient. Within the method of Kildemo et al. the derivation of these higher-order terms can be difficult, if not impossible, because of the necessity to employ the recursive formulae for the multilayer systems for this purpose (for details see [27]). In our method the terms corresponding to arbitrary high order needed for practical calculations are derived by means of the evident and systematic
ACCEPTED MANUSCRIPT mathematical procedure based on formulating the trees of the corrections occurring in the formulae. However, as shown in Table 2 the method becomes impractical for very high orders because of the very large number of contributions that must be taken into account. This impractical case corresponds to the inhomogeneous thin films with enormously complicated profiles. Fortunately, such the films occur exceptionally in practice. For the majority of the inhomogeneous
T
films encountered in practice such high orders are not needed.
IP
Table 3: Average calculation times for one point for films 1 and 2 on the silicon substrate. The
CR
columns named order and degree denote the number of internal reflections and the degree of the Chebyshev polynomial necessary to achieve the indicated absolute accuracy. The column named
US
layers is the number of subdividing layers needed to achieve the indicated accuracy if the approximation of the inhomogeneous layer by a multilayer system is used. order
degree
multilayer
time [ms]
AN
integrals
layers
time [ms]
3
8
ellipsometry
5
8
0.21
ED
reflectance
M
film 1 on the silicon substrate, absolute accuracy 103
0.35
12
0.004
16
0.005
reflectance
1
ellipsometry
2
PT
film 2 on the silicon substrate, absolute accuracy 103 32
0.23
36
0.009
32
0.45
100
0.023
ellipsometry
7
16
1.4
350
0.07
10
16
2.0
500
0.10
AC
reflectance
CE
film 1 on the silicon substrate, absolute accuracy 106
film 2 on the silicon substrate, absolute accuracy 106 reflectance ellipsometry
4
64
1.0
1000
0.19
5
64
1.3
2800
0.56
The typical calculation times for one point corresponding to a given wavelength of light and incidence angle needed to evaluate the reflectance or ellipsometric parameters with desired absolute accuracy are introduced in Table 3. The part under heading “integrals” corresponds to the
ACCEPTED MANUSCRIPT method presented in this paper while the part of the table under heading “multilayer” corresponds to the method using the approximation of the inhomogeneous film by a multilayer system of thin homogeneous films. From this table it is evident that the method using the approximation by the multilayer system is faster in all cases, but it requires relatively large number of subdividing homogeneous layers to achieve the desired accuracy. Therefore, the number of points at which the refractive index must be evaluated is much larger than in the presented method. For example, in
IP
T
order to calculate the ellipsometric parameters of the film 2 with accuracy better than 106 it is necessary to evaluate the refractive index at 2800 points in the method using the multilayer
CR
approximation but only 65 evaluations are needed in the presented method (the number of evaluations is the degree of the Chebyshev polynomial plus one). This means that the speed of the
US
method using the multilayer approximation can be negatively affected if the evaluation of the refractive index profile takes a long time while the impact on the presented method is much lower.
AN
If we look at the ratio of calculation times for the presented method and the multilayer approximation then it is evident that this ratio is lower if the films are thick and if the desired
M
accuracy is higher. It should also be noted that comparison of both the discussed methods is rather difficult because the calculation times are also dependent on the concrete implementation of these
ED
methods in the software.
Our formulae are suitable for including defects such as random roughness of boundaries.
PT
The inclusion of boundary roughness requires to calculate the statistical mean values of the reflection and transmission coefficients [36–44]. These statistical mean values are calculated more
CE
easily, precisely and efficiently if the coefficients are expressed by the series than if they are expressed by rational functions. In particular this statement is true for relatively thick
AC
inhomogeneous films exhibiting many extremes in the spectral dependencies of their optical quantities. The foregoing fact is the main advantage of our approach. The use of o ur formulae for including boundary roughness at calculating the optical quantities of the inhomogeneous thin films having this defect will be presented in our forthcoming paper. Within the multilayer approximation a sufficiently exact approach allowing to include the influence of random boundary roughness on the optical quantities of the inhomogeneous thin film has not been presented so far. Only very rough approximation has been utilized. This approximation can be partially successful only when special roughness is exhibited by boundaries of the inhomogeneous films (see e.g. [45]). In the forthcoming paper it will be shown that our
ACCEPTED MANUSCRIPT formulae presented here will enable us to describe the influence of roughness on the optical quantities of the inhomogeneous thin films in general.
6.
Experiment
In this section the optical characterization of an inhomogeneous non-stoichiometric silicon nitride
T
film deposited on silicon single crystal wafer is presented as the illustration of using the theoretical
CR
IP
approach introduced in this paper.
6.1. Preparation of the film and experimental arrangement
US
The silicon nitride film was deposited on polished side of silicon substrate by reactive magnetron sputtering [46] of silicon target in argon- nitrogen atmosphere. The refractive index profile was
AN
created by changing the nitrogen flow rate during deposition while the other deposition conditions were kept constant.
M
The spectral dependencies of the associated ellipsometric parameters were measured using the Horiba Jobin Yvon UVISEL phase modulated ellipsometer for five angle of incidence of light
6.2.1. Structural model
PT
6.2. Models of the film
ED
within the spectral range 0.6 6.3eV (197 2066nm ).
AC
subsection 2.1).
CE
The structural model of the film characterized is defined by the assumptions presented above (see
6.2.2. Dispersion model To our experiences concerning the non-stoichiometric silicon nitride films the best dispersion model of these films is given by the following expression of the imaginary parts of their dielectric function i within the spectral range of interest (see [18]):
i (E) =
N ( E Eg )2 ( Eh E ) 2 CE 2
where Eg < E < Eh and equal to zero below Eg and above Eh .
(64)
ACCEPTED MANUSCRIPT Parameters Eg and Eh represent the minimum and maximum of energy of interband transitions between the valence and conduction bands, respectively. Symbol N denotes the strength of these transitions. The normalization constant C is determined using the following equation:
E ( E )dE = N .
(65)
i
T
0
IP
The real part of this dielectric function r is calculated from the imaginary part using the
CR
Kramers–Kronig relation. The details concerning this dispersion model are presented in papers [47–49].
In our optical studies of the non-stoichiometric silicon nitride films prepared by technology
US
described above we found that the sufficient refractive index profile satisfying to these films is expressed in this way:
x d
AN
( x) = U ( L U ) ,
(66)
M
where U and L denote the complex dielectric functions at the upper and lower boundary,
ED
respectively ( U = nU2 etc.).
Thus, the dispersion parameters must be distinguished for both the boundaries, i.e. the
PT
following dispersion parameters N q , Eg,q , Eh,q and Cq must be taken into account for q = U and L . In other words, six dispersion parameters N U , N L , Eg,U , Eg,L , Eh,U and Eh,L
CE
characterize the optical properties of the film under study.
AC
6.3. Results and discussion After applying the least-square method to the experimental data it was found that the best fit was achieved using the formulae containing the corrections up the third order. The values of the thickness and dispersion parameters corresponding to this best fit are summarized in Table 4.
Table 4: The parameter values corresponding to the best fit of the experimental data with using the formulae for the reflection coefficients containing the corrections up to the third order. parameter
value
ACCEPTED MANUSCRIPT d [nm]
154.30 0.09
N U [eV 2
411 7
2.37 0.01
Eh,U [eV]
40.1 0.5
N L [eV 2
505 6
] 1.799 0.004
Eh,L [eV]
31.1 0.3
US
CR
Eg,L [eV]
IP
Eg,U [eV]
T
]
AN
Table 5: Quality of the fits expressed using quantity if the reflection formulae correspond to the WKBJ approximation (the first row), corrections up to the first order (the second row), corrections up to the second order (the third row) and corrections up to the third order (the fourth
M
row) are employed.
WKBJ
31.34
1
8.15
1+2
6.46
1+2+3
6.22
CE
PT
ED
order
The agreement between the experimental and theoretical data is good which is seen in Fig.
follows:
AC
20. This agreement is also proved by means of the value of quantity . This quantity is defined as
=
, 3N exp
Figure 20: The spectral dependencies of the ellipsometric parameters and their fits: points denote the experimental values, curves denote the theoretical data.
ACCEPTED MANUSCRIPT where is the residual sum of squares and N exp denotes number of experimental points. Number three is introduced into formula for because of using the set of three ellipsometric parameters. The values of determine the quality of fits of experimental data. In Table 5 the values of corresponding to the fits of the experimental data with using the formulae for rM belonging to the WKBJ approximation and formulae containing the corrections up to the first
T
order, up to the second order and up to the third order are presented. The best value of
IP
corresponds to the formulae containing the corrections up to the third order. If the corrections with
CR
higher orders were added in formulae for rM the corresponding values of did not change from the practical point of view. This proves that rM containing the correction up to the third order is
US
sufficiently accurate to fit the experimental data. In Fig. 21 the spectral dependencies of the refractive index and extinction coefficient of the characterized film for both the boundaries are
AN
plotted. The refractive index nL is larger than refractive index nU for all the wavelengths. The same relation is valid for the corresponding extinction coefficients. This is connected with
M
increasing the nitrogen flow during the growth of the film in the deposition process. The refractive index and extinction coefficient profiles are depicted for the selected wavelength in Fig. 22. These
ED
profiles also show that the profile gradients of the refractive index and extinction coefficient of this inhomogeneous film are rather large which coincides with the necessity to use the formulae for the
PT
ellipsometric quantities containing the corrections up to the third order in fitting experimental data. The results introduced for this film indicate that the formulae presented above are suitable for the
CE
optical characterization of the inhomogeneous thin films exhibiting relatively large gradients of
AC
profiles of their optical constants.
Figure 21: The spectral dependencies of the optical constants of the characterized film at the upper and lower boundaries.
Figure 22: Profiles of the optical constants of the characterized film for E = 2.5 eV.
7.
Conclusion
By means of the multiple-beam interference model the approximate formulae are derived for the
ACCEPTED MANUSCRIPT reflection and transmission coefficients of the inhomogeneous thin films having large gradients of the profiles of their optical constants. These formulae consist of the series whose terms consist of the first term corresponding to the WKBJ approximation and the following terms expressing the corrections of these WKBJ terms describing the influence of the internal reflections inside the films originating in consequence of the large profile gradients. The expressions for the corrections belonging to individual orders are easily and systematically formulated. It is shown that the
T
corrections constitute the trees. The advantage of the formulae presented here consists in
IP
possibility to estimate the precision of calculating the optical quantities of the inhomogeneous thin
CR
films in an easy and straightforward way. The further advantage consists in possibility to derive the corrections of arbitrary order in the clearly prescribed way. However, the main advantage of
US
the formulae presented in this article consists in possibility to include defects such as ra ndom boundary roughness into these formulae in an efficient way which will be described in our
AN
forthcoming paper. The disadvantage of our approach is smaller speed and precision of calculations of the optical quantities of inhomogeneous thin films exhibiting complicated
M
refractive index profiles in comparison with the other approximate approaches such as the multilayer approximation and approach of Kildemo et al.
ED
From the numerical analysis it is evident that the influence of the inhomogeneity with the large refractive index profile gradients on the spectral dependencies of the optical quantities is
PT
complicated. Furthermore, it is clear that the influence of this inhomogeneity is also dependent on the spectral dependencies of the optical constants of the sub strates. Moreover, it should be noted
CE
that for the inhomogeneous films with complicated profiles the corrections of rather high orders must be used in the formulae for the reflection and transmission coefficients (e.g. for rugate filters
AC
the corrections up to the seventh order must be used at least). The results of the optical characterization of the selected sample of non-stoichiometric silicon nitride film prepared by reactive magnetron sputtering onto silicon substrate are presented for illustration of usability of our formulae. Using these formulae experimental data of variable angle spectroscopic ellipsometry are processed. The spectral dependencies of the refractive indices and extinction coefficients at the upper and lower boundaries together with the profiles of both the optical constants for the selected wavelength are determined. Moreover, the determined thickness value is also presented. For finding these optical parameters the formulae containing the corrections up to the third order are used. The good fits of the experimental data support a
ACCEPTED MANUSCRIPT correctness of the results achieved. This experimental example indicates the efficiency of using our formulae within the optical characterization of the inhomogeneous thin films with large gradients of refractive index profiles in practice. Our formulae should be usable also in synthesis of inhomogeneous thin films.
T
Acknowledgments
IP
This work was supported by the project LO1411 (NPU I) funded by Ministry of Education, Youth
CR
and sports of the Czech Republic. The authors would like to thank P.Vašina and J. Ženšek for
US
preparing the samples of non-stoichiometric silicon nitride films.
References
A. Vašiček, Polarimetric methods for the determination of the refractive index and the
AN
[1]
thickness of thin films on glass, J. Opt. Soc. Am. 37 (1947) 145–153. J. M. Bennett, M. J. Booty, Computational method for determining n and k for a thin film
M
[2]
from the measured reflectance, transmittance, and film thickness, Appl. Opt. 5 (1966)
[3]
ED
41–43.
K. Vedam, W. Knausenberger, F. Lukes, Ellipsometric method for the determination of all
PT
the optical parameters of the system of an isotropic nonabsorbing film on an isotropic absorbing substrate. optical constants of silicon, J. Opt. Soc. Am. 59 (1969) 64–71. O. S. Heavens, H. M. Liddell, Influence of absorption on measurement of refractive index
CE
[4]
of films, Appl. Opt. 4 (1965) 629–630. I. Ohlídal, K. Navrátil, E. Schmidt, Simple method for the complete optical analysis of
AC
[5]
very thick and weakly absorbing films, Appl. Phys. A 29 (1982) 157–162. [6]
T. Lohner, K. J. Kumar, P. Petrik, A. Subrahmanyam, I. Bársony, Optical analysis of room temperature magnetron sputtered ITO films by reflectometry and spectroscopic ellipsometry, J. Mater. Res. 29 (2014) 1528–1536.
[7]
D. Franta, I. Ohldal, D. Petrýdes, Optical Characterization of TiO 2 Thin Films by the Combined Method of Spectroscopic Ellipsometry and Spectroscopic Photometry, Vacuum 80 (2005) 159–162.
[8]
I. Ohldal, M. Ohldal, D. Nečas, D. Franta, V. Buršková, Optical characterisation of
ACCEPTED MANUSCRIPT SiO xCyHz thin films non-uniform in thickness using spectroscopic ellipsometry, spectroscopic reflectometry and spectroscopic imaging reflectometry, Thin Solid Films 519 (2011) 2874–2876. [9]
D. Franta, I. Ohldal, V. Buršková, L. Zajčková, Optical Properties of Diamond-Like Carbon Films Containing SiO x Studied by the Combined Method of Spectroscopic Ellipsometry and Spectroscopic Reflectometry, Thin Solid Films 455-456 (2004)
M. Hacskaylo, Determination of the refractive index of thin dielectric films, J. Opt. Soc.
IP
[10]
T
393–398.
[11]
CR
Am. 54 (1964) 198–203.
J. C. Manifacier, J. Gasiot, J. P. Fillard, A simple method for the determination of the
US
optical constants n, k and the thickness of a weakly absorbing thin film, J. Phys. E: Sci. Instrum. 9 (1976) 1002–1004.
R. Jacobsson, Matching a multilayer stack to a high-refractive index substrate by means of
AN
[12]
an inhomogeneous layer, J. Opt. Soc. Am. 54 (1964) 422–423. R. Jacobsson, Light reflection from films of continuously varying refractive index, in: E.
M
[13]
Wolf (Ed.), Progress in Optics, Vol. 5, Elsevier, 1966, pp. 247–286. R. Jacobsson, Inhomogeneous and Coevaporated Homogeneous Films for Optical
ED
[14]
Applications, in: G. Hass, M. H. Francombe, R. W. Hoffman (Eds.), Physics of Thin Films,
[15]
PT
Vol. 8, Academic Press, New York, 1975, pp. 51–98. J. C. Charmet, P. G. de Gennes, Ellipsometric formulas for an inhomogeneous layer with
[16]
CE
arbitrary refractive- index profile, J. Opt. Soc. Am. 73 (1983) 1777–1784. C. K. Carniglia, Ellipsometric calculations for nonabsorbing thin films with linear
[17]
AC
refractive- index gradients, J. Opt. Soc. Am. A 7 (1990) 848–856. D. Franta, I. Ohldal, P. Klapetek, P. Pokorný, M. Ohldal, Analysis of Inhomogeneous Thin Films of ZrO 2 by the Combined Optical Method and Atomic Force Microscopy, Surf. Interface Anal. 32 (2001) 91–94. [18]
D. Nečas, D. Franta, I. Ohldal, A. Poruba, P. Wostrý, Ellipsometric characterization of inhomogeneous non-stoichiometric silicon nitride films, Surf. Interface Anal. 45 (2013) 1188–1192.
[19]
D. Li, A. Goullet, M. Carette, A. Granier, J. P. Landesman, Effect of growth interruptions on TiO 2 films deposited by plasma enhanced chemical vapour deposition, Mater. Chem.
ACCEPTED MANUSCRIPT Phys. 182 (2016) 409–417. [20]
J. Vohánka, I. Ohldal, J. Ženšek, P. Vašina, M. Čermák, D. Franta, Use of the Richardson extrapolation in optics of inhomogeneous layers: Application to optical characterization, Surf. Interface Anal. 50 (2018) 757–765.
[21]
B. G. Bovard, Rugate filter design: the modified Fourier transform technique, Appl. Opt. 29 (1990) 24–30. G. Boivin, D. St.-Germain, Synthesis of gradient-index profiles corresponding to
T
[22]
O. Debieu, R. P. Nalini, J. Cardin, X. Portier, J. Perrière, F. Gourbilleau, Structural and
CR
[23]
IP
spectralreflectance derived by inverse Fourier transform, Appl. Opt. 26 (1987) 4209–4213.
optical characterization of pure Si-rich nitride thin films, Nanoscale Res. Lett. 8 (2013) 31. A. Thomson, N. Lal, Y. Wan, Interpolating the optical properties of varied composition
US
[24]
silicon nitride, Phys. Status Solidi B 252 (2015) 2230–2235. Z. Knittl, Optics of Thin Films, Wiley, London, 1976.
[26]
A. Vašček, Optics of Thin Films, North–Holland, Amsterdam, 1960.
[27]
M. Kildemo, O. Hunderi, B. Drévillon, Approximation of reflection coefficients for rapid
M
AN
[25]
real-time calculation of inhomogeneous films, J. Opt. Soc. Am. A 14 (1997) 931–939. G. B. Arfken, H. J. Weber, Mathematical methods for physicists; 4th ed., Academic Press, San Diego, CA, 1995.
ED
[28]
A. Gil, J. Segura, N. M. Temme, Chebyshev Expansions, SIAM, 2007, Ch. 3, pp. 51–86.
[30]
N. Ahmed, T. Natarajan, K. R. Rao, Discrete cosine transform, IEEE Trans. Comput. C-23
[31]
CE
(1974) 90–93.
PT
[29]
C. W. Clenshaw, A. R. Curtis, A method for numerical integration on an automatic
[32]
AC
computer, Numerische Mathematik 2 (1960) 197–205. D. Franta, A. Dubroka, C. Wang, A. Giglia, J. Vohánka, P. Franta, I. Ohldal, Temperature-dependent dispersion model of float zone crystalline silicon, Appl. Surf. Sci. 421 (2017) 405–419. [33]
D. Y. Smith, The Optical Properties of Metallic Aluminum, in: E. D. Palik (Ed.), Handbook of Optical Constants of Solids, Vol. 1, Academic Press, New York, 1985, pp. 369–406.
[34]
M. Kildemo, Real-time monitoring and growth control of Si-gradient- index structures by multiwavelength ellipsometry, Appl. Opt. 37 (1998) 113–124.
ACCEPTED MANUSCRIPT [35]
M. Kildemo, R. Brenot, B. Drévillon, Spectroellipsometric method for process monitoring semiconductor thin films and interfaces, Appl. Opt. 37 (1998) 5145–5149.
[36]
I. Ohldal, F. Lukeš, Ellipsometric Parameters of Rough Surfaces and of a System Substrate-Thin Film with Rough Boundaries, Opt. Acta 19 (1972) 817–843.
[37]
I. Ohldal, F. Lukeš, K. Navrátil, Rough Silicon Surfaces Studied by Optical Methods, Surf. Sci. 45 (1974) 91–116. I. Ohldal, K. Navrátil, F. Lukeš, Reflection of Light by a System of Nonabsorbing Isotropic
T
[38]
IP
Film–Nonabsorbing Isotropic Substrate with Randomly Rough Boundaries, J. Opt. Soc.
[39]
CR
Am. 61 (1971) 1630–1639.
I. Ohldal, D. Franta, D. Nečas, Improved combination of scalar diffraction theory and
US
Rayleigh-Rice theory and its application to spectroscopic ellipsometry of randomly rough surfaces, Thin Solid Films 571 (2014) 695–700.
I. L. Morris, T. E. Jenkins, The effect of long-correlation- length surface roughness on the
AN
[40]
ellipsometric parameters of reflected light, Europhys. Lett. 34 (1996) 55–61. J. M. Zavislan, Angular scattering from optical interference coatings: scalar scattering
M
[41]
predictions and measurements, Appl. Opt. 30 (1991) 2224–2244. C. K. Carniglia, Scalar scattering theory for multilayer optical coatings, Opt. Eng. 18
ED
[42]
(1979) 104–115.
J. Bauer, Optical properties, band gap, and surface roughness of Si3 N 4 , Phys. Status Solidi
PT
[43]
A 39 (1977) 411–418.
M. Čermák, J. Vohánka, I. Ohldal, D. Franta, Optical quantities of multi- layer systems
CE
[44]
with randomly rough boundaries calculated using the exact approach of the Rayleigh- Rice
[45]
AC
theory, J. Mod. Opt. 65 (2018) 1720–1736. D. Franta, I. Ohldal, P. Klapetek, M. Ohldal, Characterization of thin oxide films on GaAs substrates by optical methods and atomic force microscopy, Surf. Interface Anal. 36 (2004) 1203–1206. [46]
S. Berg, T. Nyberg, Fundamental understanding and modeling of reactive sputtering processes, Thin Solid Films 476 (2005) 215–230.
[47]
D. Franta, D. Nečas, L. Zajčková, Models of dielectric response in disordered solids, Opt. Express 15 (2007) 16230–16244.
[48]
D. Franta, D. Nečas, L. Zajčková, Application of Thomas–Reiche–Kuhn sum rule to
ACCEPTED MANUSCRIPT construction of advanced dispersion models, Thin Solid Films 534 (2013) 432–441. D. Franta, D. Nečas, I. Ohldal, A. Giglia, Dispersion model for optical thin films applicable in wide spectral range, in: Optical Systems Design 2015: Optical Fabrication, Testing, and
CE
PT
ED
M
AN
US
CR
IP
T
Metrology V, Vol. 9628 of Proc. SPIE, 2015, p. 96281U.
AC
[49]
ACCEPTED MANUSCRIPT Highlights
CE
PT
ED
M
AN
US
CR
IP
T
New approximate approach for optical quantities of inhomogeneous thin films Formulae are derived using multiple-beam interference model Films can exhibit large gradients of refractive index profiles Formulae enable us to perform rapid calculations at the optical quantities Application of the new formulae for characterisation of a selected sample
AC
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11
Figure 12
Figure 13
Figure 14
Figure 15
Figure 16
Figure 17
Figure 18
Figure 19
Figure 20
Figure 21
Figure 22