Optics Communications 271 (2007) 148–153
www.elsevier.com/locate/optcom
On the angular dependence of gaps in 1-D Si/SiO2 periodic structures
Herman Högström
b
a,*
,
Carl G. Ribbing
b
a
Division of Solid state physics, The Ångström Laboratory, Uppsala Box 534, SE-751 21 Uppsala, Sweden
Department of Functional Materials, Swedish Defence Research Agency; Box 1165;SE-581 11 Linköping, Sweden
Received 24 April 2006; received in revised form 21 September 2006; accepted 5 October 2006
Abstract
A multilayer of silicon and silicon dioxide was used to study the angular dependence of reflectance maxima originating from interference and bulk optical properties. Silicon dioxide has a lattice resonance in the infrared causing an interval of high reflectance for wavelengths around 9 lm. The multilayer was designed such that the interference maxima do not overlap/interact with the material related
reflectance maximum. In this way the different angular behavior for the two types of reflectance maxima can be studied simultaneously.
Experimental and calculated reflectance spectra for s- and p-polarized light for angles of incidence between 0 and 90 collected for every
5 are presented. The reflectance features caused by interference generally move to shorter wavelengths with increasing angle of incidence, and the materials related peak is widened for (s-polarized light) and excitation of the longitudinal modes was observed for p-polarized light.
2006 Elsevier B.V. All rights reserved.
1. Introduction
In a previous study we have pointed out the possibility
to use Si/SiO2 double layers, to suppress the thermal emission in both thermal atmospheric windows 3–5 and
8–13 lm [1]. This double function is based on the combined
effect of two optical reflectance maxima, giving low radiative emittance in the respective windows. The emittance
reduction in the short wavelength range is obtained by
the choice of layer thickness to obtain a photonic gap
which we shall name ‘‘structural’’. In the conventional
nomenclature it is the result of optical interference between
two thin film materials with different refractive indices. In
contrast, the long wavelength window is partly covered
by the Reststrahlen band of the oxide material, SiO2. By
coincidence this material-related polaritonic gap is widened
by another interference maximum which improves the performance. It is thus the combined effect of two basically
different physical mechanisms that ensures an average thermal emittance in the range 0.3–0.4 in the two windows [1].
*
Corresponding author. Tel.: +46 184716215; fax: +46 18500131.
E-mail address: [email protected] (H. Högström).
0030-4018/$ - see front matter 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.optcom.2006.10.083
The spectral results reported in [1] concerned near normal incidence reflectance measurements. Angular dependence was only represented by spectrally integrated
curves recorded with heat-cameras, filtered to sum the
emission in each window. The technical application in
focus for that work was signature reduction for military
platforms. In this application, non-normal excitance is also
highly relevant. In this contribution we shall report experimental and calculated spectra showing the angular variation of the optical reflectance from a 3 · Si/SiO2-layer.
This periodic 6-layer structure is a multilayer for which
the nomenclature of 1D photonic crystals can be used since
the dominating spectral features are present, even though it
is not an infinite periodic structure. In this detailed and
more basic study we shall not use the particular case, discussed above, which gives maximum signature reduction.
The reason for this choice is the coincidence mentioned
that the two kinds of gaps (at normal incidence) have
merged into one broad feature. This maximized signature
reduction gives an added complexity which confuses the
basic differences we wish to investigate. Our ambition is
on the contrary to separate the two kinds of features in
order to observe their respective angular dependence.
H. Högström, C.G. Ribbing / Optics Communications 271 (2007) 148–153
For interference controlled, non-dispersive, dielectric
multilayers a rule of thumb is that spectral features move
to shorter wavelengths when the angle of incidence is
increased because of change in optical depth. This can be
understood most simply from the case of a single dielectric,
non-dispersive film of thickness d with refractive index n at
wavelength k. If the angle of incidence is /, Snell’s law
gives the refracted angle inside the film:
sin /
/1 ¼ arcsin
ð1Þ
n
which gives the optical phase difference between the reflected beams
d ¼ 2p
nd
cos /1 :
k
ð2Þ
Therefore, if the angle of incidence, /, is increased then
also /1, will increase which will cause d to decrease. The
reflectance interference maxima occur when the path length
in the film is [2]:
d cos /1 ¼ ð2p þ 1Þ
k
4n
ð3Þ
with p is an integer. Therefore, in this simplest case of a
free-standing, dielectric non-dispersive film, the interference-features will move to shorter wavelengths if / is
increased. This simple mathematical argument is generalized into the rule mentioned above. It is important to note
that argument is based on n = const. This is not generally
valid, and if not, the rule does not apply.
A fortiori, this will not be valid for the present results,
which include a reflectance maximum in the infrared called
the Reststrahlen band. For a bulk material in air, this feature is determined by the dielectric function of the material
and the Fresnel equations. Using a single oscillator model,
the dielectric function as a function of angular frequency x
can be approximated with a simple four parameter model
[3]:
x2 x2
eðxÞ ¼ e1 1 þ 2 L 2 T
;
ð4Þ
xT x ixC
where e1 is the high frequency real screening parameter,
(typically n2 in the visible range), xT and xL are the zero
wave vector transverse- and longitudinal optical phonon
frequencies, and C a damping parameter expressing
absorption by induced oscillations in the ionic lattice. Primarily, it is the transverse optical phonon mode, that couples to the incoming transverse radiation, which is the
source of strong dispersion. Generally it is only the transverse mode that can be excited, because of the transverse
nature of light, but we will find that also the longitudinal
mode can be excited. In the region xT < x < xL the dielectric function is strongly negative and if C xT the material is highly reflective in this Reststrahlen band. The
concept of polaritons (the quasi particle causing the Reststrahlen band) is mostly used for crystalline materials but it
can also be used for amorphous materials and gases since it
149
originates from molecular properties, which are clearly related [3].
It is interesting to compare the optical properties of an
ionic solid in the Reststrahlen range, with those of a
highly conducting metal in the visible and infrared wavelength ranges. The underlying physical mechanisms are
completely different: polarization of ionic pairs and
mobile conduction electrons, respectively. In both cases,
however, the result is e1 0, i.e. in terms of the refractive
index, k n. As a consequence of this, the bulk normal
reflectance is high, and only weakly wavelength dependent. Because of the high k-value, the extinction-length
is much smaller than the wavelength. If the sample is
not bulk, interference patterns will be seen on the edges
of the band, but the band itself is relatively unchanged
down to surprisingly small thicknesses [3]. This weak
thickness dependence can be understood as yet another
consequence of k n, in contrast to the dielectric case,
i.e. Eq. (2) above. The total phase shift will be dominated
by the interface passage contribution given by k, and relatively insensitive to the term in (2) which includes the
d-dependence.
The Reststrahlen band exhibits an angular dependence
which is markedly different from that of interference features. In a study made on hexagonal beryllium oxide it
was shown that the width of the Reststrahlen band
increases because the short wavelength edge moves to
higher frequencies than xL [4]. The transformation in
the p-polarized spectra of the weak, short-wavelength
shoulder at normal incidence to a resolved peak close
to xL, at high incidence angles, is well-known from early
studies. The basic explanation is that p-polarized light at
non-normal incidence will have a component normal to
the surface. This component will create a longitudinal
mode which under two circumstances is absorbed. Berreman demonstrated that this will occur because of the
boundary conditions, if the sample has cubic structure
and is a film thinner than the wavelength [5]. This could
explain the experimental results in Figs. 2 and 3, since the
SiO2-layers are only 1.85 lm thick. However, it cannot
alone explain that we observe similar behavior also in
the calculated curves for a bulk sample – see Fig. 1
below. In this case, we either have to invoke microroughness that will cause depolarization or a weak, separate oscillator. In the first case the peak will always stay
within the Reststrahlen region, and the detailed position
is determined by the geometrical shape of the roughness
[6]. Scrutinizing the literature values of the optical constants [7] we conclude that they indicate a 2nd weak oscillator around 8.7 lm. There are no good reasons to
assume that glass sample used in [7] was in fact microrough. The optical parameters for crystalline a-quartz,
that is anisotropic, include two k A2-oscillators, the average of which is very close to 8.7 lm (1160 cm1) [8]. The
long-wavelength edge of the band does not move. As just
mentioned, the overall shape of the band is polarization
dependent. In Fig. 1 we show bulk reflectance spectra
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H. Högström, C.G. Ribbing / Optics Communications 271 (2007) 148–153
2. Sample preparation and analysis
Fig. 1. Calculated bulk reflectance spectra for SiO2 glass showing the
Reststrahlen band for unpolarized light at normal and 55 incidence as
indicated. Notice the growth of the short wavelength peak in the 55spectrum. The optical constant where taken from Refs. [7,8].
for normal and 55 incidence for glassy SiO2, to be used
in this study, calculated from published optical constants
[7,9].
One notices the increased width of the band by the
growth of a secondary peak at the short wavelength edge.
From the more detailed results below we shall find that this
is mainly, but not only, a contribution from the spectrum
for p-polarized light.
It is interesting to note that a recent work considers the
same system [10]: a 1D Si/SiO2 photonic crystal to be
used as a pass- and stop-band filter in a thermophotovoltaic (TPV) system. The TPV generator only uses the fraction of NIR-radiation with shorter wavelengths than
1.8 lm from 1100 to 1500 K thermal source. The efficiency is therefore improved if the unused radiation is
rejected and reflected back to the source. Such a filter
would be positioned close both to the source and detector, and therefore also the behaviour at high angles of
incidence is relevant, just as in our case. A major difference, however, is that the design is only intended for
NIR-application in the region 0.8–3.3 lm, where it should
be reflecting. In this case both components are dielectric
with a high index contrast and the polaritonic behaviour
of SiO2 is not considered.
In the remaining part of this report we shall deal with
the case of a 3 · Si/SiO2-multilayer designed such that both
types of resolved reflectance peaks are present in the resulting spectra. We shall investigate, by calculations and experiments, the angular dependence of the spectral structures.
As usual in this more complex case, we shall find that
s- and p-polarization must be studied separately for a
meaningful interpretation of the results. Before the results
are given and discussed, we shall describe briefly the procedures used for the calculations, sample preparation and
reflectance measurements.
The 3 · Si/SiO2-multilayer was made with two standard
microelectronic CVD-processes on a (1 0 0) 550 lm Siwafer [11]. Si layers (poly-Si) were grown from decomposition of silane (SiH4) gas and SiO2 layers from hydrolyzation of tetra-ethyl-ortho-silicate (TEOS). The layers were
deposited, one at a time with the temperatures and deposition rates: 650 C and 10.4 nm/min for the poly-Si process
and 710 C and 5.6 nm/min for the SiO2 process. The
thicknesses of the layers were 1.20 and 1.85 lm, respectively. We chose to have a structure with Si as the outermost material because if SiO2 would be placed on top, it
would dominate the optical properties of the structure in
and above the polaritonic wavelength region around
9 lm. More details about the fabrication and the optical
response with SiO2 on top were given in [12], where a comparison between Figs. 2 and 3 shows that only when the Silayer is on top does the thickness variation affect the polaritonic gap and there are intereference variations at longer
wavelengths.
The multilayer calculations were made with a matrix
formalism computer program which can be used for direct
calculation of R-, T- and A-spectra for a given multilayer,
as well as reverse fitting calculations to approach a target
performance with a limited number of given optical thin
film materials [9]. Not all the optical constants required
for calculations in the 2.5–15 lm range are included in
the data-basis of [9] therefore additional data from [7] were
used.
Fig. 2. Experimental and calculated normal incidence reflectance spectra
for the multilayer 3 · Si(1.20 lm)/SiO2(1.85 lm) on a 550 lm Si-wafer.
Note that the reflectance peak presented in Fig. 1 around 9 lm is
unaffected by interference with the surrounding Si-layers. The redshift
between the experimental and calculated interference reflectance peaks
(k < 8 lm) is because the deposited layers are somewhat thicker than the
thicknesses used in the calculations. This does not, however, affect the
overall results.
H. Högström, C.G. Ribbing / Optics Communications 271 (2007) 148–153
Fig. 3. Experimental and calculated reflectance spectra for (a) s- and (b) ppolarized light at 55 of incidence on the same sample-structure as in
Fig. 2.
The specular reflectance spectra for the samples were
measured with a Perkin-Elmer 983 IR spectrophotometer
equipped with a ‘‘Variable angle specular reflectance accessory 186–0445’’ permitting measurements at incidence
15–75 for s- and p-polarized light. The spectra were
recorded using a gold-mirror as reference for both polarizations and each angle of incidence.
3. Results and discussion
To establish the initial situation and to illustrate the
level of agreement between the calculated and the experimental spectra we cite in Fig. 2 our results for normal incidence from [12]. Considering that no fitting has been
performed, and that the optical data are taken from independent measurements in other laboratories, the agreement
between the experimental and calculated curves is very satisfactory and confirms that our sample preparation is
under good control with regard to purity and thickness.
151
Comparing Figs. 1 and 2 we notice that the Reststrahlen
band is virtually identical for normal incidence in the bulk
case and in the periodic structure. The thickness, 1.85 lm is
much less than the wavelength, but near the resonance, the
extinction is strong enough to give high reflectance, like for
metallic films in the visible range. For our continued analysis it is important to distinguish the polaritonic reflectance
peak around 9 lm (hm 0.138 eV) from the most prominent interference peaks at 6.1 lm (0.202 eV), 4.5 lm
(0.275 eV) and 3.30 lm (0.375 eV). The redshift between
the experimental and calculated interference reflectance
peaks (k < 8 lm), seen in Figs. 2 and 3, is because the
deposited layers are somewhat thicker than the thicknesses
used in the calculations. The deposited layers have a varying thickness over the wafer surface and a variation
between the layers. On the average the experimental layers
are thicker than the thicknesses used in the calculations.
This does not, however, affect the overall results and the
scope of this report.
In Fig. 3 we show the first results for a high angle of incidence: 55, and it is then necessary to separate according to
the polarization of the incident light.
Comparing first the Reststrahlen bands in Figs. 2 and 3
for increased angle of incidence, we notice that for s-polarized light the peak grows in intensity, and to some extent in
width, by the growth of a shoulder on the short wavelength
side. The main p-peak is reduced in intensity but instead
the secondary peak at short wavelengths, around 7.9 lm,
develops. These changes are the background to the differences seen between the two spectra in Fig. 1 which are averages for s- and p-polarized light. It is an important
distinction to make that the changes observed between normal and 55 incidence should not be described as a shift of
the peak position, but rather a widening of the entire peak
[4].
In contrast to this, the two interference peaks at 6.1 and
3.3 lm have both clearly shifted towards shorter wavelengths for both polarizations, although more so in the spolarized case. This behavior is in agreement with the
rule-of-thumb for interference peaks of dielectric multilayers, mentioned in the introduction. However, just as for the
Reststrahlen band, the intensity of the s-peak has grown
and the p-peak has shrunk.
In order to have a more detailed and systematic representation of the angular variation than can be obtained
with individual spectra, we have introduce a compound
diagram below. The experimental and calculated reflectance spectra for the two polarizations and different
angles of incidence were recorded and stored as colorgraphs in which the color from blue to red scales with
the reflectance from 0 to 1.0 as indicated. Fig. 4a contains
19 calculated (0–90) spectra and Fig. 4b, 13 experimental
spectra, which were collected for every 5. In the nomenclature of photonic crystals, photon energy intervals with
100%, or high reflectance for a finite structure, are named
‘‘gaps’’. In order to bring out the analogy between our
‘‘gap-structure’’ with photonic band structures, we there-
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H. Högström, C.G. Ribbing / Optics Communications 271 (2007) 148–153
fore convert the spectral wavelengths to photon energies
and rotate them to the y-axis. A red horizontal range will
thus indicate a gap. However, since the input to our diagram represent vacuum states for light; the dispersion
inside the multilayer does not influence the numerical values. The boundaries between high and low reflectance are
consequently not dispersion curves, and the gap-structure
is not a photonic band structure. Primarily the x-axis is
the angle of incidence /, which is used as an independent
variable both in the experiments and in the multilayer calculations. To bring out the analogy between our gapstructure and a photonic band structure, we shall use an
x-scale that is proportional to sin/. This is because kk,
the component of the wave vector k that is parallel to
the interface and is conserved when the light enters the
multilayer, is proportional to sin/. This is identical to
the x-axis used in the 1D projected band structure-plot
presented by Winn et al. [13]. The latter includes, however, the true dispersive behavior inside the multilayer,
although the bands merge into areas of permitted states
because of the ky-projection.
In Fig. 4a and b we show the calculated and experimental reflectance spectra for the 6-layer structure used for the
two angles represented in Figs. 2 and 3.
In the diagrams, the E(kk) = E(kk)-symmetry has been
used to include both polarizations as indicated.
First we can use the two diagrams to confirm the agreement between calculated and experimental curves. The
overall quantitative agreement is satisfactory and it should
be noted that the experimental colorgraph does not include
all angles of incidence (only between 15 and 75) presented
in the calculated graph. The importance of 4, however, is
that they provide a detailed and complete picture of the
theme of this work, i.e. the angular variation of optical features. We shall concentrate on the reflectance maxima, i.e.
the red and yellow bands, but the corresponding analysis
can equally well be carried out for the minima, i.e. blue
bands. The polaritonic reflectance peak for p-polarized
light, at 0.138 eV for normal incidence, does indeed split
into a weaker second peak at kk-values above 0.75. This
splitting comes from the excitation of the LO mode.
P-polarized light will have a longitudinal component which
increases with the angle of incidence and therefore it will
cause this excitation if the film thickness of the polar material is thin enough [5]. The typical behavior for p-polarized
light, the bulk reflectance decreases with angle to a minimum before it goes up again, is also noted. However, as
noted above, in the case of glassy SiO2, this behaviour is
complicated by the presence of a secondary oscillator.
For s-polarized light the peak is both widened and
strengthened for kk > 0.7.
Concerning the interference structure in general, we
notice that the s-polarized peaks widens at high angles,
while the p-peak narrows. The boarder lines for two of
the three interference peaks, identified above at 0.202
and 0.375 eV for normal incidence, are upwards concave
in agreement with the rule that interference structures
move to higher photon energy with increasing angle of
incidence. There is one exception: the lower s-edge of
the 0.202-band is a horizontal line. Possibly because this
Fig. 4. (a) Calculated reflectance spectra for the 3 · Si/SiO2 multilayer on a silicon wafer for angles of incidence in the range 0–90. Reflectance scale and
polarization as indicated. (b) Experimental reflectance spectra for the 3 · Si/SiO2 multilayer on a silicon wafer with angles of incidence in the range 15–75.
The white zone in the middle represents the lower limit, 15 to the angle of incidence, that could be used.
H. Högström, C.G. Ribbing / Optics Communications 271 (2007) 148–153
153
is so close to the polaritonic resonance that the dispersion
of SiO2 is anomalous and strong enough to invalidate the
conditions for the general rule [2]. The intermediate peak
at 0.275 eV (4.5 lm) does not follow the general rule – it
vanishes by weakening (p-pol) or splitting (s-pol) at high
angles of incidence in both the calculated and experimental cases. However, the detailed agreement between measured and calculated data is not so good as elsewhere in
this case.
Acknowledgements
4. Summary and conclusions
References
We have studied the angular variation of infrared reflectance structure for the 3 · Si/SiO2 multilayer on a silicon
wafer. This combination of materials ensures that there
are structures of two different origins: optical interference
and the ionic lattice resonance in the SiO2-layer Reststrahlen band. These two sources exhibit a distinct angular
behavior. The interference features in general follow the
rule and move to shorter wavelengths with increasing angle
of incidence. The features for s-polarized light grow with
angle, while the p-related shrink. The exceptions to this
are most likely to be caused by dispersion. The Reststrahlen band, or the polaritonic gap, is widened in the same
process and the peak for p-polarized light grows at the
short wavelength edge. Again the peak for s-polarized light
grows with angle, while the p-peak is weakened before it
finally increases for high angles. The case of glassy SiO2
includes the particular complication of a second weak
oscillator.
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[8] E.D. Palik, Handbook of Optical Constants of Solids III, Academic
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[13] J.N. Winn, Opt. Lett. 15 (1998) 1573.
We thank Anders Heljestrand at the Micro Structure
Laboratory, Ångström Laboratory, Uppsala University
for help with the preparation of the sample. Dr George
Dobrowolski, NRC of Canada, is acknowledged for fruitful mail-discussions. The Swedish Research Council, The
Swedish Defence Research Agency and The Swedish Defence Nanotechnology programme for funding.