Optical properties of thin metal films
Optical constants of metallic thin films made from: Ag, Au, Hf, Ir, Mo, Nb, Os, Pd, Pt, Re, Rh, Ru, Ta, W, Zr were determined on the basis of measured index of refraction in region of wavelength λ = 241216 Å. Two types of relations were used for the calculation. Some of them were obtained, with tak...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
1999
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| Цитувати: | Optical properties of thin metal films / S.A. Kovalenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 1999. — Т. 2, № 3. — С. 13-20. — Бібліогр.: 25 назв. — англ. |
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irk-123456789-1198822017-06-11T03:02:31Z Optical properties of thin metal films Kovalenko, S.A. Optical constants of metallic thin films made from: Ag, Au, Hf, Ir, Mo, Nb, Os, Pd, Pt, Re, Rh, Ru, Ta, W, Zr were determined on the basis of measured index of refraction in region of wavelength λ = 241216 Å. Two types of relations were used for the calculation. Some of them were obtained, with taking into account that refractive index of absorbing medium can be presented in the form ñ = n ± iæ. Other were obtained from Maxwell boundary condition. Both approaches give rise to very close results for æ, however the dependences n = f(λ) for λ > 200 Å are essentially different. The reasons of such differences are discussed. 1999 Article Optical properties of thin metal films / S.A. Kovalenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 1999. — Т. 2, № 3. — С. 13-20. — Бібліогр.: 25 назв. — англ. 1560-8034 PACS: 78.66; 78.20.C http://dspace.nbuv.gov.ua/handle/123456789/119882 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Optical constants of metallic thin films made from: Ag, Au, Hf, Ir, Mo, Nb, Os, Pd, Pt, Re, Rh, Ru, Ta, W, Zr were determined on the basis of measured index of refraction in region of wavelength λ = 241216 Å. Two types of relations were used for the calculation. Some of them were obtained, with taking into account that refractive index of absorbing medium can be presented in the form ñ = n ± iæ. Other were obtained from Maxwell boundary condition. Both approaches give rise to very close results for æ, however the dependences n = f(λ) for λ > 200 Å are essentially different. The reasons of such differences are discussed. |
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Kovalenko, S.A. |
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Kovalenko, S.A. Optical properties of thin metal films Semiconductor Physics Quantum Electronics & Optoelectronics |
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Kovalenko, S.A. |
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Kovalenko, S.A. |
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Optical properties of thin metal films |
| title_short |
Optical properties of thin metal films |
| title_full |
Optical properties of thin metal films |
| title_fullStr |
Optical properties of thin metal films |
| title_full_unstemmed |
Optical properties of thin metal films |
| title_sort |
optical properties of thin metal films |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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1999 |
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http://dspace.nbuv.gov.ua/handle/123456789/119882 |
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Optical properties of thin metal films / S.A. Kovalenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 1999. — Т. 2, № 3. — С. 13-20. — Бібліогр.: 25 назв. — англ. |
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Semiconductor Physics Quantum Electronics & Optoelectronics |
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AT kovalenkosa opticalpropertiesofthinmetalfilms |
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2025-07-08T16:50:24Z |
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2025-07-08T16:50:24Z |
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1837098277377409024 |
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13© 1999, Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
Semiconductor Physics, Quantum Electronics & Optoelectronics. 1999. V. 2, N 3. P. 13-20.
PACS: 78.66; 78.20.C
Optical properties of thin metal films
S.A. Kovalenko
Institute of Semiconductor Physics, NAS of Ukraine
45, prospect Nauki, 252650 Kyiv, Ukraine
Abstract. Optical constants of metallic thin films made from: Ag, Au, Hf, Ir, Mo, Nb, Os, Pd, Pt, Re,
Rh, Ru, Ta, W, Zr � were determined on the basis of measured index of refraction in region of wave-
length λ = 24�1216 Å. Two types of relations were used for the calculation. Some of them were ob-
tained, with taking into account that refractive index of absorbing medium can be presented in the form
ñ = n ± iæ. Other were obtained from Maxwell boundary condition.
Both approaches give rise to very close results for æ, however the dependences n = f(λ) for λ > 200 Å are
essentially different. The reasons of such differences are discussed.
Keywords: thin layers, refraction coefficient, transmission coefficient, refraction index, absorption index
Paper received 02.09.99; revised manuscript received 29.09.99; accepted for publication 18.10.99.
1. Introduction
Investigations of optical properties of solids, including these
of thin films, are aimed on: first, to obtain data for ascer-
taining their band structure; second, to determine such im-
portant parameters as refractivities and absorptivities nec-
essary for designing devices and their elements.
Thin films of transparent and light absorbing materials
are widely used in manufacturing interferential [1,2] and
absorption [3,4] filters, in production of multi-layer mirrors
[5,6], polarizers [7,8], light detectors [9,10], in making anti-
reflection coatings [11,12]. In all enumerated cases one needs
to have exact data upon optical constants, n and æ, of con-
sidered materials. For many years respective calculations
were based on relationships that took into account repeated
reflections and interference in a thin layer, and measured in
experiment were intensities of light beams reflected and
transmitted by the thin layer.
Formulae for intensities of a light wave reflected and
transmitted by the layer include Fresnel coefficients of re-
flection and transmission for separate interfaces, namely:
air (vacuum) � thin layer and thin layer � substrate.
In the case of absorbing media a refraction index is
complex value n + iæ. Therefore, transition from transpar-
ent substance to absorbing one is usually postulated by the
change ñ → n + iæ, which cannot be considered as fully
justified. It is this change that leads to meaningless results.
For example, when a thickness of metal [13-15] and semi-
conductor [16-18] films tends to zero value, the refraction
index increases to infinity, though it must approach to unity.
Such monotonic thickness dependence of the refraction in-
dex for thin absorbing films all the more cast doubt, be-
cause it must have at least one resonance maximum on the
curve n = f (d) that corresponds to a wedgeshaped layer
thickness of which changes from zero to the value when
this layer becomes absolutely transparent. Calculations ful-
filled in accordance with above formulae [19] do not give
such a maximum.
As it is known, investigations of metal optical proper-
ties are based on using ellipsometric method and measuring
reflection coefficients for different angles of light incidence
on mirror surface of a bulk sample. In these cases work
formulae for n and æ estimations are usually obtained by
changing ñ → n + iæ. As a result, obtained values of n in
wide ranges of the optical spectrum are less than unity. As an
example, we can give data [20,21] for gold. Authors of the
first paper considered the range of wavelength 0.5-1 µm
where n monotonically decreases from 0.84 to 0.19. In the
second paper, measurements were performed in the range of
1.0-4 µm that supplements the previous interval. n-values
less than unity, were obtained in the region of 1.0-2.5 µm.
With increasing wavelength, these raise from 0.224 to 0.82.
Thus, almost in the whole visible part of the spectrum, and
in a considerable part of the infra-red region joining to it,
the refraction index of gold is less than unit. We shall not
give more numeric data by other authors for metals, semi-
conductors and dielectrics where values of n < 1 are fixed
in wide ranges of the optical spectrum. Moreover, in this
paper we cite plots for n = f (λ) dependences from [22] for
16 metals, and each of them has respective ranges of the
spectrum.
The inequality n < 1 is not alone to generate surprise. It
S.A. Kovalenko: Optical properties of thin metal films
14 SQO, 2(3), 1999
is difficult to imagine why it is intrinsic for wide spectral
intervals while minimum n-values are usually met in the
narrow range of frequencies corresponding to band half-
widths of electron or vibration transitions, that is, in the
range of anomalous dispersion. As known, in this region,
the group velocity with which light wave energy is trans-
fered, is connected with refraction index dispersion through
known relationship:
λ
λ
ϑ
⋅−
=
d
dn
n
c
gr , (1)
were c – is the light velocity in vacuum. Since a phase
velocity ncph /=ϑ , then at frequencies where the sign of
dn/dλ is changed, i.e., dn/dλ = 0,
n
c
gr == ϑϑ ph > c , (2)
taking into consideration that n < 1. This result contradicts
the special theory of relativity. By the way, if one takes for
example the n-value of silver for the wavelength close to
0.5 µm that is equal to 0.05 then the respective group velocity
must exceed the light velocity in vacuum by 20 times! It is
a fantastic result, there is nothing to say.
One example more [23] where authors cast doubt on
legitimateness of using known Fresnel formulae. The authors
had the task to measure depth of X- ray penetration into on
oxide film on a silicon surface. Thicknesses of such films
are small, so for increasing losses of an energy of an X- ray
beam at the cost of absorption and scattering, authors
directed it at a grazing angle of incidence, and the depth of
penetration was determined by the formula:
−= 0
2cosIm2
1 θεω
cD
. (3)
Here, ω is an angular frequency, c is the light velocity
in vacuum, ε is the dielectric permittivity of a SiO2 – layer,
θ is an angle of grazing incidence. A large deviation of theo-
retically estimated and measured D-values is explained by
authors as stemmed from non-Fresnel character of Si-SiO2
system behavior.
In our opinion, above examples are enough to motivate
a necessity of searching new relationships which could ful-
fil the same functions as Fresnel formulae did for many
years. The first steps along this line are made by
T.O.Kudykina [25] who, using boundary conditions by
Maxwell, obtained analogs to Fresnel formulae. Using these
formulae and experimental data upon transmission and
reflection by thin layers of atomic semiconductors obtained
earlier [16-18], authors of [25] calculated thickness depen-
dencies of optical constants n and æ. As it was expected,
each curve has a resonance maximum, and n- values ap-
pear to tend to unity when d goes to zero.
It would seem that such brilliant confirmation of the
Kudykina formulae workability might interpose all on their
places. However, as it will be seen below in discussing the
problem, with these new opportunities some new questions
arose.
In this paper, using reflection spectra of thin films for
16 metals optical data of which in the range of λ = 24 -
1216 Å were obtained in [22], we calculated n- and æ- values
in accordance to old [19] and new [25] theories.
Investigated in [22] metal films were prepared by elec-
tron-beam sputtering in vacuum ~ 5·10-6 Torr on silicon
substrates temperature of which was kept to be equal to
300 ºC. The thickness of the films was approximately
1000 Å.
2. Work formulae
Fig.1 represents a scheme of a sample like those investigated
in [22]. Let us consider that the substrate and air are
transparent for wavelengths used in the work, i.e. æ1 = æ3=0.
It is also accounted that at thicknesses close to 1000Å the
film−substrate boundary and, all the more, the substrate−
air one give negligable contribution into total−reflectivity.
It is in such conditions this new theory is used for calculation
n and æ. Necessary relationships see below. These have the
following appearance:
′−
+=
′−
−
=
,
1
,
1
1
*34
2
*34
*14
*
*34
34
14
RRR
TR
RR
T
RRR
R
T
(3)
where T14, R14 are transmission and reflection coefficients
of the system shown in Fig.1, respectively,
2
3
3
34 1
1
+
−
=
n
n
R
is the Fresnel reflection coefficient of the substrate−air
boundary;
R* = C/A;
Ŕ* = B/A;
AnnT /)(16 2
2
2
23* ea+= ;
γγγδσγρτ 222 sin2cos2)exp()exp( ntnsA ++−+= eaea ;
γγγδτγσρ 2222 sin2cos2)exp()exp( nrnqB −+−+= eaea ;
r34
n4
t 344
d2 t12
t 23
r23
r12
n 3
n 2 22
n1
3
1
Fig.1.
S.A. Kovalenko: Optical properties of thin metal films
15SQO, 2(3), 1999
γγγσργδτ 22 sin2cos2)exp()exp( nrnqC ++−+= eaea ;
q n n n n n= + + − + − −( æ )( ) ( æ ) æ2
2
2
2
3
2
2
2
2
2 2
3
2
3 2
21 4 ;
ρ = + +( ) æn1
2
2
21 ;
s n n n n n= + + − + − +( æ )( ) ( æ ) æ2
2
2
2
3
2
2
2
2
2 2
3
2
3 2
21 4 ;
σ = − +( ) æn n2 3
2
2
2 ;
r n n n= − + +2 12 3
2
2
2
2
2
3æ ( ) (æ ) ;
τ = + +( ) æn n2 3
2
2
2 ;
t n n n= + + −2 12 3
2
2 2
2
3æ ( ) (æ );
γ = 4πd2/λ,
where λ is the light wavelength in vacuum.
Both the traditional and the new theory start from the
same relationships for reflection and transmission coeffi-
cients. These are as follows:
ψαα
ψαα
cos)exp(2)2exp(1
cos)exp(2)2exp(
2312
2
23
2
12
2312
2
23
2
12
drrdrr
drrdrr
R
−−−+
−−−+= , (4)
ψαα
α
cos)exp(2)2exp(1
)exp(
2312
2
23
2
12
2312
drrdrr
dTT
T
−−−+
−= , (5)
where α
π
λ
= 4 2æ
, (6)
,
4
2dn
λ
πψ = (7)
Contrary to the old theory, in the new one, coefficients
of reflection and transmission for separate boundaries air−
film and film−substrate do not include absorption indices
at the normal angle of light beam incidence:
23
23
23
12
12
12 ;
nn
nn
r
nn
nn
r
+
−
=
+
−
= , (8)
23
1
12
1 2
23
2
12 ;
nn
n
nn
n
tt ++ == , (9)
32
3
21
2 2
32
2
21 ;
nn
n
nn
n
tt ++ == , (10)
It is worth to note that R12= (r12)2 , R23 = (r23)2 , T12 = t12t21,
T23= t23t32.
Substituting Zi = exp (-αd), instead of Eqs (4) and (5),
one can obtain quadratic equations relatively to Zi :
ii
ii
ZrrZrr
ZrrZrr
R
⋅−+
⋅−+=
ψ
ψ
cos21
cos2
2312
22
2312
2312
22
23
2
12 , (11)
ψcos21 2312
22
23
2
12
2312
ii
i
ZrrZrr
ZTT
T
−+
= . (12)
Solving Eq. (12) yields roots for Z:
Z b a
b
a
T
a1 2
2
22
4, / ,= ± − (13)
where
cos2
,
23122312
2
23
2
12
rTrTTb
rTra
+=
=
ψ.
Substituting Zi into Eq. (11), one can find n2- value as a
root of the function
F R
r r Z r r Z
r r Z r r Z
i i
i i
= −
+ − ⋅
+ − ⋅
12
2
23
2 2
12 23
12 23
2 2
12 23
2
1 2
cos
cos
ψ
ψ
. (14)
Zi – value must satisfy the condition 0 < Zi < 1. The sign
before the root of the solution (13) is chosen in accordance
with it.
After calculation of the n2- value we can find the æ2-
value using the next relation:
æ
d
Z
ñ
i
π
λ
4
ln
2
⋅−= , (15)
which follows from (6) accounting the change Zi = exp(-ad).
Spectral dependencies of n and æ calculated by using
both the traditional formulae and those of the new theory
are shown in Figs 2-17.
3. Discussion
The analysis of curves testifies that only for short wave-
lengths taken from the range of λ = 24 - 100Å refractive
index values calculated using both theories practically co-
incide. At λ > 100 Å these diverge, moreover, the new theory
leads to considerably larger n- values. For gold, silver, iridi-
um, platinum and rhenium a maximum difference corresponds
to wavelength taken from the interval λ = 530 - 550 Å. But
for molibdenium, osmium, palladium, rhutenium and rhod-
ium it is shifted up to λ ≈ 600 Å. The largest shift can be
observed in cases of hafnium, tantalum (λ ≈ 1000Å), tung-
sten, zirconium and titanium (λ ≈ 1100 - 1200Å). It is most
frequently at these wavelength that the new theory describes
the n- maximum which can be explained by the quantum-
sized resonance.
As for absolute values of refraction indices, the new
theory leads only to n > 1 in all range of wavelength used.
At the same time, the old one leads to n < 1 for all metals in
wide ranges of the spectrum without any exception.
If we take into account that in the range of wavelengths
λ = 24 - 1200Å quantum transitions realize between inner
electron shells polarizabilities of which are considerably less
than those of the outer ones, then n- values exceeding unity
can cast doubts.
A very interesting result was obtained for absorption
indices. These almost coincide in all range of wavelength
used, moreover, their absolute meanings do not cast doubts.
Coincidence can be explained by respective estimations. If
æ2/n2 = 0.05 then there is no place for any divergence be-
tween old and new theories, but when æ2/n2 ≥ 0.25 such
difference will be considerable.
S.A. Kovalenko: Optical properties of thin metal films
16 SQO, 2(3), 1999
Au B
D
n
2.5
2.0
1.5
1.0
0.5
Ag
C
E
0 200 400 600 800 1000 1200 1400
0
5
0
Ag B
D
n
2.0
1.5
1.0
æ
1.0
0.5
0.0
Au C
E
æ
1.0
0.5
0.0
Hf B
D
Hf
C
E
B
D
Ir Ir C
E
0 200 400 600 800 1000 1200 1400
λ λ λ λ λ , , , , , Å
0 200 400 600 800 1000 1200 1400
λ λ λ λ λ , , , , , Å
3.0
2.5
2.0
1.5
1.0
0.5
n
3.0
2.5
2.0
1.5
1.0
0.5
n
1.5
1.0
0.5
0.0
æ
1.5
1.0
0.5
0.0
æ
Figs 2-5.
S.A. Kovalenko: Optical properties of thin metal films
17SQO, 2(3), 1999
Mo B
D
Mo C
E
Nb B
D
Nb C
E
B
D
Os C
E
Os
Pd B
D
Pd C
E
0 200 400 600 800 1000 1200 1400
λ λ λ λ λ , , , , , Å
0 200 400 600 800 1000 1200 1400
λ λ λ λ λ , , , , , Å
2.5
2.0
1.5
1.0
0.5
n
2.5
2.0
1.5
1.0
n
3.0
2.5
2.0
1.5
1.0
0.5
n
2.5
2.0
1.5
1.0
0.5
n
1.0
0.5
0.0
æ
1.0
0.5
0.0
æ
1.0
0.5
0.0
æ
1.0
0.5
0.0
æ
Figs 6-9.
S.A. Kovalenko: Optical properties of thin metal films
18 SQO, 2(3), 1999
B
D
Pt C
E
Pt
B
D
Re
C
E
Re
B
D
Rh
C
E
Rh
B
D
Ru C
E
Ru
0 200 400 600 800 1000 1200 1400
λ λ λ λ λ , , , , , Å
2.5
2.0
1.5
1.0
0.5
n
2.5
2.0
1.5
1.0
0.5
n
2.5
2.0
1.5
1.0
0.5
n
2.5
2.0
1.5
1.0
0.5
n
1.0
0.5
0.0
æ
1.0
0.5
0.0
æ
1.0
0.5
0.0
æ
1.0
0.5
0.0
æ
0 200 400 600 800 1000 1200 1400
λ λ λ λ λ , , , , , Å
Figs 10-13.
S.A. Kovalenko: Optical properties of thin metal films
19SQO, 2(3), 1999
B
D
Zr
B
D
Ta
C
E
Ta
B
D
Ti C
E
Ti
B
D
W
C
E
W
C
E
Zr
0 200 400 600 800 1000 1200 1400
λ λ λ λ λ , , , , , Å
2.5
2.0
1.5
1.0
0.5
n
3.0
2.5
2.0
1.5
1.0
0.5
n
n
2.5
2.0
1.5
1.0
0.5
n
3.5
3.0
2.5
2.0
1.5
1.0
0.5
1.5
1.0
0.5
0.0
æ
1.0
0.5
0.0
æ
1.5
1.0
0.5
0.0
æ
1.0
0.5
0.0
æ
0 200 400 600 800 1000 1200 1400
λ λ λ λ λ , , , , , Å
Figs 14-17.
S.A. Kovalenko: Optical properties of thin metal films
20 SQO, 2(3), 1999
Conclusions
1. Optical constants n and æ for 16 metals in the spectral
region of λ = 24 - 1216Å are estimated. Experimental de-
pendences of reflection coefficients on the wavelength are
considered using two theories: the traditional one, where
energetic coefficients of transmission and reflection on the
boundary between two substances are calculated using sub-
stitutions ñ → n + iæ in the Fresnel formulae for transpar-
ent media, and the new theory in which the same coefficients
are determined using formulae obtained from the Maxwell
boundary conditions.
2. It is shown that coincidence of results takes place only
for absorption indexes and for all metals in the whole spectral
region. It is reasonable to believe that this coincidence has
a random character and can be explained by a small æ-value
in comparison with an n-value.
3. The accordance of n = f (λ) curves with predictions of
the new theory points on noticeable increase in n- values
with wavelength. However, an absolute n- magnitude does
not correlate with low polarizability of inner electron shells
of atoms in the range of deep ultraviolet and X-rays.
Therefore, the new theory predicts higher n-values.
4. Contrary to the previous conclusion, the traditional
theory leads to lower n-values. There are wide spectral
regions where n < 1, i.e., one has the result deprived of real
physical sense.
The author would like to emphasize his gratitude to
Academician M.P.Lisitsa who is the supervisor and the
author of the main idea of this work.
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