The experimental dielectric function of porous anodic alumina in the infrared region; a comparison with the Maxwell-Garnett model

Article abstract of DOI:10.1088/0953-8984/8/23/019

The infrared reflectance from thin alumina films on metal substrates has a deep minimum for p-polarized light at oblique incidence. This originates from absorption when light couples with a longitudinal optical (LO) phonon mode with k-vector zero. The absorption band is wide for amorphous alumina and is shifted to longer wavelengths for porous oxides compared to non-porous ones. Anodic alumina, prepared in phosphoric acid, with a pore volume fraction of 0.3, has been investigated. The s- and p-polarized reflectance has been measured for selected angles of incidence between 15° and 75°, and the dielectric function has been determined from these measurements. The effective dielectric function has been calculated using Maxwell-Garnett effective-medium theory for a two-component anisotropic medium consisting of air-filled cylindrical pores perpendicular to the surface in an alumina matrix with optical constants of non-porous evaporated alumina. The theoretical and experimental results are in good agreement, which shows that the redshift of the LO mode absorption for p-polarized light can be explained by the presence of pores.

Full text of DOI:10.1088/0953-8984/8/23/019

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The experimental dielectric function of porous anodic alumina in the infrared region; a
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J. Phys.: Condens. Matter 8 (1996) 4289–4299. Printed in the UK
The experimental dielectric function of porous anodic
alumina in the infrared region; a comparison with the
Maxwell-Garnett model
Ewa W a¨ ckelg a˚ rd
Solid State Physics, Department of Technology, Box 534, Uppsala University, S-751 21,
Uppsala, Sweden
Received 23 November 1995, in final form 14 March 1996
Abstract. The infrared reflectance from thin alumina films on metal substrates has a deep
minimum for p-polarized light at oblique incidence. This originates from absorption when light
couples with a longitudinal optical (LO) phonon mode with k-vector zero. The absorption band
is wide for amorphous alumina and is shifted to longer wavelengths for porous oxides compared
to non-porous ones. Anodic alumina, prepared in phosphoric acid, with a pore volume fraction
of 0.3, has been investigated. The s- and p-polarized reflectance has been measured for selected
◦
◦
angles of incidence between 15 and 75 , and the dielectric function has been determined
from these measurements. The effective dielectric function has been calculated using Maxwell-
Garnett effective-medium theory for a two-component anisotropic medium consisting of air-filled
cylindrical pores perpendicular to the surface in an alumina matrix with optical constants of non-
porous evaporated alumina. The theoretical and experimental results are in good agreement,
which shows that the redshift of the LO mode absorption for p-polarized light can be explained
by the presence of pores.
1. Introduction
Alumina is used in a variety of technical applications, which has made it the focus of
rigorous investigations covering many aspects, including its optical characteristics. The
most complete set of optical infrared measurements has been performed on single-crystal
α-alumina [1]. From dispersion analyses using the Lorentz oscillator model, the frequencies
were found for the transverse optical phonon modes (the TO modes) at wavevector zero
with the electric field perpendicular or normal to the c-axis of the crystal. In total six
modes exist. The longitudinal optical phonon (LO mode) frequencies were taken from the
zeros of the real part of the dielectric function. Other crystalline phases of alumina, such
as γ -alumina for which four optically active modes have been determined, have also been
investigated by dispersion analysis [2].
In the case of crystalline bulk dielectrics, the interaction between the electromagnetic
field and the optical phonon modes is manifested by a high reflectance in the frequency range
between the transverse and longitudinal optical phonon modes. This is called the reststrahlen
band. The high reflectance originates from the inability of waves to propagate in the
forbidden bandgap between the TO and LO frequencies. The infrared optical properties of
amorphous alumina, prepared by thermal oxidation, anodization, evaporation or sputtering,
have also been studied. Amorphous alumina which is thick enough to be considered to have
bulk optical properties in the infrared, has a much lower reflectance in the reststrahlen band
0
953-8984/96/234289+11$19.50 ꢀc 1996 IOP Publishing Ltd
4289

4290
E W a¨ ckelg a˚ rd
than crystalline alumina [3]. This can be explained by the large energy dissipation caused
by phonon scattering in a system with only short-range order.
However, if the film is thinner, less then 1 µm, which is common in many of the
reports on amorphous alumina, a reststrahlen band is not observed since waves can actually
propagate through matter with a thickness much smaller than the wavelength of the wave [4].
Instead, the transmittance in this wavelength range is high, except at the wavelengths around
the TO and LO modes respectively, where light is absorbed [4] because of phonon scattering
processes. LO modes can only be excited for p-polarized light, at non-normal angles of
incidence, which has been used to determine the frequencies of the LO modes directly
from transmittance or reflectance spectral measurements on thin alumina films [5–9]. The
LO absorption has a large bandwidth for amorphous alumina so it is impossible to resolve
more than the highest-frequency LO mode which causes the strongest absorption. The
location of TO modes has also been determined from s-polarized transmittance or reflectance
measurements on samples with thicknesses that are small compared to the wavelength, but
not thinner than about 0.5 µm [6, 5, 8]. It should be mentioned that the reflectance
measurements must be made on samples on a metallic substrate, otherwise the thin-film
approximation referred to [4] is not valid. Dispersion analysis has also been used for thin
amorphous alumina films [10, 2] whereby optical constants have been determined using the
Fresnel formalism for both evaporated [11] and anodized [12, 13] amorphous alumina.
From the reports referred to above, one can conclude that amorphous alumina has two
−
1
optically active infrared modes. The stronger one is reported at about 650 cm (15.4 µm),
except on those occasions when dispersion analysis has been used. In these two cases it
−
1
−1
has been reported at 720 cm (13.9 µm) [2] and 850 cm (11.8 µm) [10]. In contrast,
the frequency of the stronger LO mode is in surprisingly good agreement, regardless of the
measuring technique or the data processing. In this case the different values seem to be
related to the type of amorphous alumina. Evaporated, sputtered, thermal and barrier anodic
−
1
oxides each have the maximum in the LO mode absorption reported at around 950 cm
10.5 µm) [6, 9, 5, 8], but in anodic alumina formed in sulphuric acid, the maximum has
shifted to a lower frequency, 920 cm (10.9 µm), whilst when phosphor anodized, it has
(
−1
−1
shifted to 880 cm (11.4 µm) [7]. There is a significant morphological difference which
could explain a shift of the LO mode: anodic alumina formed in sulphuric and phosphoric
acid has pores perpendicular to the surface, with phosphoric acid having a larger pore
fraction compared to alumina prepared in other acids [14, 15]. The diameter of the pores
can be as large as 200 nm and constitute up to 40% of the total volume [16]. Therefore it can
be hypothesized that the pores play a crucial role in the redshift of the LO mode. Another
suggestion is that the LO mode is shifted because of differences between the chemical bonds
of the non-porous and porous alumina. In the work presented here, theoretical modelling
based on effective-medium theory has been used to explain the redshift of the LO mode
frequency for porous alumina. The optical constants for evaporated alumina have been
taken for these calculations to obtain the effective optical constants for porous oxide using
the Maxwell-Garnett effective-medium model. The results are then compared with the
experimentally determined optical constants for phosphor-anodized alumina.
2. Sample preparation and sample characterization
Rolled aluminium sheet plated with a 3 µm thick aluminium layer has been used as the
substrate. The samples were anodized in 2.5 M phosphoric acid at a constant dc voltage of
◦
1
5 V for 15 minutes at a temperature of 19.0 ± 0.5 C. The current density was typically
−2
about 3.5 mA cm . The oxide film thickness and surface roughness were then measured

The dielectric function of anodic alumina in the IR region
with an Alpha-Step 200 stylus mechanical profilometer.
4291
A distinct step from the oxide surface to the aluminium substrate was produced by
selectively dissolving the oxide on part of the sample area in chromic phosphoric acid [15].
The oxide film thickness was found to be 0.50 ± 0.02 µm. The average surface roughness
was 0.02 µm, both for the substrate and for the oxide surface.
The porous structure was investigated with scanning electron microscopy, SEM. Cracks
in the oxide, made by bending the sample, revealed the cross-sectional structure. In common
with previously published micrographs [17], [14], the micrographs showed channels of a
fibrous nature in the oxide perpendicular to the surface.
The pore size was determined from transmission electron microscopy, TEM, pictures.
Some aluminium oxide was stripped from the aluminium metal by amalgamation [15].
These oxide fragments were then rinsed in distilled water, placed on a copper wire grid
and mounted in a TEM instrument. The pores in the oxide are electron transparent since
there is only a thin barrier layer below the pore. The pores are not straight uniform
cylinders perpendicular to the surface, which makes the estimation of pore size from these
micrographs somewhat uncertain. Nevertheless, by measuring the number of bright spots
on the micrographs and their width, the volume fraction of pores was found to be about
30% with an average diameter of 30 nm.
From grazing-incidence x-ray diffraction measurements it is clear that the aluminium
oxide is amorphous. Some low-intensity peaks reveal some crystallization, although the
peaks could not be identified as belonging to any one of the known phases of aluminium
oxide or crystalline aluminium hydroxides or oxyhydroxides. It is known that the poorly
◦
crystalline pseudo-boehmite is formed during anodization [18], but after heating to 400 C
it is transformed into aluminium oxide. Since the diffraction peaks disappeared after the
◦
sample was heated to 400 C, these peaks could belong to low concentrations of pseudo-
boehmite.
3. Optical measurements
Specular reflectance was measured in the wavelength range 2–50 µm with a double-beam
Perkin–Elmer 983 Infrared Spectrophotometer. The s- and p-polarized reflectance was
◦
measured in steps of 5 using a variable-angle-of-incidence accessory for angles between
1
◦ ◦
5 and 75 . The measurements were repeated for each angle with a reference sample of
evaporated aluminium on glass replacing the actual samples. The reflectance data for the
samples were divided by the reference spectrum and multiplied by the calculated reflectance
of aluminium using optical constants from the literature [19].
−
1
−1
A scan mode was chosen with a resolution of 7 cm at 1150 cm . The stray light,
−
1
as specified in the instrument manual, was less than 1% between 400 and 180 cm , and
was lower in the rest of the range measured. An upper limit of 0.01 was set for the absolute
error in the reflectance throughout this range, regardless of the value of the reflectance. The
◦
angle setting for the variable-angle accessory could not be more accurate than within 0.5 .
The samples are highly specular; the diffuse reflectance was measured at 2.5 µm in
a Beckman UV 5240 photospectrometer equipped with an integrating sphere. The diffuse
component was 0.03 and the total reflectance 0.92. Therefore the reflectance can only be
considered to be specular for longer wavelengths, as in the infrared wavelength range.

4292
E W a¨ ckelg a˚ rd
Figure 1. Specular reflectance of porous alumina for three different angles of incidence, in the
s- and p-polarized cases.
4. Experimental results from optical measurements
The reflectance measurements are presented in figure 1 for s- and p-polarized light at angles
◦
◦
◦
of incidence of 25 , 50 and 75 . There is a broad reflectance minimum at 11.4 mm
(
−
1
880 cm ), which is strongly angle dependent, and, since it appears only for p-polarized
light, it is identified as absorption by longitudinal optical phonons in the aluminium oxide.
The wavelength of the peak absorption is the same as has been observed before for phosphor-
anodized alumina [7]. A weak absorption band corresponding to the wavelength range of the
TO mode appears as a shoulder on the long-wavelength side of the LO absorption. Both the
LO and TO absorption bands of amorphous structures are much broader than for crystalline
3
−
2−
4−
structures. Anions from the electrolytes PO , HPO4 and H2PO absorb in the region
9
4
−1
–10 µm (1100–1000 cm ) and are responsible for the absorption on the short-wavelength
side of the reflectance minimum of the LO mode [20].
The pseudo-boehmite can be identified from absorption bands in the infrared wavelength
−
1
range: the AlO–H stretching mode introduces a broad band centred at 3600 cm (2.8 µm),
−1
the H–O–H bending, one at 1600 cm (6.2 µm), and Al–OH bending forms a narrow band
at 1040 cm (9.6 µm). The latter cannot be resolved from the LO mode absorption band
of the oxide [18].
−
1
From these infrared optical measurements it is evident that the main constituent of the
material is amorphous aluminium oxide.
5. Determining the optical constants for porous alumina
Optical constants have been determined before for alumina films in the infrared wavelength
region [11–13]. The method used for electron-beam-evaporated alumina [11] was to measure
normal transmittance with oxide evaporated onto an infrared transparent substrate, and
near-normal reflectance with a highly reflecting substrate. Anodic aluminium oxide can be
detached from the aluminium substrate for transmittance measurements [12].

The dielectric function of anodic alumina in the IR region
4293
The phosphor-anodized alumina is anisotropic because the pore structure has pores
preferentially oriented perpendicular to the surface. A uniaxial symmetry perpendicular to
the surface can be described by two sets of complex refractive indices, one set parallel and
the other perpendicular to the surface.
Optical constants can be derived solely from reflectance measurements. In this case,
when four measurements are needed, one can use a combination of s- and p-polarized
reflectance for two angles of incidence, or just p- or s-polarized reflectance for four different
angles of incidence. The advantage of the reflectance method is that the alumina films do
not need to be detached from the aluminium substrate. The disadvantage is that it can
only be used for wavelengths with a significant difference in reflectance between the angles
of incidence used [21]. This rules out s-polarized reflectance altogether since the angular
dependence of the reflectance is very weak, as demonstrated in figure 1. In contrast, the
p-polarized reflectance is strongly angle dependent in the region of the broad LO absorption
band. Thus the combination of four p-polarized reflectance measurements was tried to
obtain a fit to the data for the calculated reflectance of an anisotropic thin film on an
aluminium substrate [22]. The film thickness was fixed to the value determined from the
stylus profilometer measurement and the optical constants for the aluminium substrate were
taken from the literature [19]. A Newton–Raphson iterative numerical method was used
and the fitting was limited to the wavelength range 5–30 µm. However, the calculations
did not converge, or converged to unphysical values, which showed that four variables was
too many.
The number of variables was then reduced by taking the Fresnel equations for an
isotropic film on an absorbing substrate into account. On using p-polarized light at high
angles of incidence, the isotropic constants obtained for the film should be relatively close to
the component perpendicular to the surface of the proper anisotropic set of optical constants.
◦
◦
The combination Rp(75 )/Rp(50 ) was used rather than one with angles even closer
◦
◦
together, such as Rp(75 )/Rp(70 ), as this gave greater uncertainty in the refractive index
since the reflectance for these two angles does not differ substantially.
To examine the validity of the optical constants determined with the chosen combination,
the reflectance was also calculated for the remainder of the measured angles of incidence
using the Fresnel formulae for isotropic films on aluminium. The result is shown in figure 2
for three selected angles. The calculated p-polarized reflectance is in good agreement with
that measured, except for at the lowest angles. The reflectance for s-polarized light is high
for all angles of incidence and it is seen that the calculated reflectance is systematically
about 1–2% higher for the measured reflectance. There is also a small LO absorption in the
experimental data which should not be present for pure s-polarized light.
6. Theory
The dielectric function calculated from the experimental complex refractive index is
presented in figure 3. Error bars are drawn for some experimental points to show the
level of uncertainty in different intervals. These were obtained for each point from the
exp
◦
combination of R ± 0.01, θ ± 0.5 and the thickness ±0.02 µm which gave the largest
deviation in the refractive index. Error bars for shorter wavelengths are smaller than the
size of the experimental points in the plot.
As mentioned above, experimental data do exist [11] for the dielectric function of
amorphous evaporated alumina; these are also plotted in figure 3 for comparison. The
dielectric functions of the non-porous and porous alumina have the same curvature, but
differ in three important respects.

4294
E W a¨ ckelg a˚ rd
◦
◦
◦
Figure 2. Specular reflectance for angles of incidence 25 , 45 and 70 , experimental and
calculated using the experimentally derived refractive index. The experimental curves are drawn
as solid lines.
(i) The real part of the dielectric function is reduced and the function has no zero for
the porous alumina.
ii) The maximum of the imaginary part for the porous oxide is about two thirds of the
(
value for the non-porous oxide, but there is no shift of the peak located at 15.9 µm. The
half-width–maximum ratio is similar for the two types of alumina. The second maximum
at 26 µm is not clearly seen from the porous oxide data because of the large errors in
this wavelength region. However, a linear fit to the data actually indicates that this second
maxima is also present for the porous oxide.
(iii) The maximum of the dielectric loss function representing the LO mode is shifted
from 10.6 µm for the non-porous oxide to 11.4 µm for the porous oxide. Again, the peak
value of the porous oxide is about two thirds of that of the non-porous one and the peak is
broader.
A model has been proposed in which the amorphous system has a Gaussian distribution
of oscillators, each oscillator having a small Lorentzian damping [10]. This model has
been used to fit reflectance data from thin alumina films on silicon, and the result was
considerably better than that obtained using a classical Lorentz model. The same model can
00
be applied to these data since the shape of ε in figure 3 for both porous and non-porous
oxides is not Lorentzian, but, rather, is more Gaussian in form.
The main features of the dielectric function in the reststrahlen band depend on the
phonon modes, their damping and the electronic contribution to the dielectric function.
By definition [23] wavelengths of longitudinal and transverse optical phonon modes are
0
0
00
determined from the zero and pole of the dielectric function ε = ε + iε for a system with
no damping. For a heavily damped system, as in this case, there is neither a zero nor a pole
for the dielectric function for real wavelengths [23]. The LO mode can be approximately

The dielectric function of anodic alumina in the IR region
4295
Figure 3. The dielectric function and energy-loss function of porous alumina. Line fits are
drawn as a guide to the eye. For comparison the dielectric function of evaporated non-porous
alumina [1] is presented in the figure.
defined as the minimum of |ε| and the TO mode as the maximum of |ε| [23], or as the
0
0
maximum of Im(1/ε) and maximum of ε , respectively [1].
According to Berreman’s thin-film approximations [4] the p-polarized reflectance of a
dielectric film on a metal substrate exhibits a minimum at the maximum of the energy-loss
function, Im(1/ε), which means that for a slightly damped system the LO modes can be
determined from this minimum of reflectance. For amorphous alumina the damping is so
strong that the minimum of |ε| is shifted to a shorter wavelength compared to the maximum
of Im(1/ε), so the reflectance minimum should not be identified as the accurate wavelength
of the LO phonon frequency. The shift of the minimum in the p-polarized reflectance for
porous relative to non-porous alumina cannot be ascribed to a shift in the actual LO mode
frequency only from such measurements. A shift in the LO mode frequency could be the

4296
E W a¨ ckelg a˚ rd
result of changes in the chemical bonds; thus a shift should also be expected for the TO mode
00
frequency. According to the results above there is no wavelength shift for ε corresponding
to a TO mode shift. Based on these results, it is expected that the chemical structure is
similar for both porous and non-porous alumina. This is also confirmed by calculations of
the radial distribution function (RDF) from x-ray diffraction data for amorphous alumina
films prepared by evaporation (non-porous) [24], and anodization (porous) [25], with only
small differences in nearest-neighbour distances and coordination numbers [26, 27]. The
amorphous short-range order can be described as aluminium located predominantly in
tetrahedral sites in a close-packed oxygen random network.
If a change in chemical bonding is not the main reason for the LO mode shift, the
actual presence of the pores could be. Since the pores have dimensions much less than
the wavelength, an effective medium can be used to describe the porous alumina when
interacting with infrared light.
The Maxwell-Garnett and Bruggemann effective-medium models [28, 29, 30] for a
two-component inhomogeneous material were used to calculate the dielectric function for
air-filled pores. The basic definition of an effective medium is that a random unit cell,
when embedded in the effective medium, should not be detectable in an experiment using
electromagnetic radiation confined to a specific wavelength range. Effective-medium theory,
EMT, is valid for inhomogeneous two-component materials when the size of at least one
of the components is less than the wavelength of the light used in the measurement. This
condition is fulfilled in this context since the pore dimension is less than the infrared
wavelength. The Maxwell-Garnett model of the inhomogeneous medium is closest to
the morphology of the anodic film, with well separated grains, the pores, embedded in
a continuous host of alumina. Calculations were also executed with the Bruggemann model
for comparison. This model is better suited for aggregate structures in which the two
components appear in equal distributions to form a space-filling random mixture. The
results appeared to be independent of the model in the wavelength range analysed, and
for that reason only the calculations made using the Maxwell-Garnett effective dielectric
function are presented here since it has an analytic form. The effective dielectric function is
a function of the pore filling factor, the dielectric function of the constituents, and the shape
of the pores—which determines the depolarization field of the pores and the surrounding
alumina.
The Maxwell-Garnett effective dielectric function for elliptical particles can be expressed
as [31]
pore
pore
alumina
εalumina(1 − Lx,z + Lx,z )εpore + f (1 − L
)(εpore − εalumina)
x,z
alumina
ε = εalumina
.
pore
pore
εalumina(1 − Lx,z ) + Lx,z εpore − f L
(εpore − εalumina)
x,z
The symbol Lx,z denotes the depolarization factor in the direction of the incoming
electromagnetic radiation, the x-direction in the surface plane and the z-direction
perpendicular to the surface. For simplicity, the depolarization factors are set equal for
both pores and alumina, a condition which is valid for f close to unity, where f is the
filling factor of the total pore volume relative to the total volume. The shapes of the pores
are simplified to the geometry of long cylinders perpendicular to the surface. The dielectric
function was calculated for the two directions of highest symmetry of the cylinder: εz,
parallel, and εx, perpendicular to the long axis of the cylinder. The z-component here is
parallel to the surface normal. The depolarization factors in these directions are 0 and 0.5
respectively. The dielectric function of the electron-beam-evaporated aluminium oxide [11]
was used for alumina. The volume fraction of air-filled pores found from TEM, 0.3, was
used. The equations for the effective dielectric function for the Maxwell-Garnett theory

The dielectric function of anodic alumina in the IR region
4297
Figure 4. The effective dielectric function, obtained using Maxwell-Garnett effective-medium
theory, is compared with the experimental dielectric function. The filling factor is 0.3 and the
pores are regarded as cylinders perpendicular to the surface. The figure shows the components
of the theoretical dielectric function and energy-loss function parallel and perpendicular to the
surface.
with the appropriate numerical values for the depolarization and filling factors were
εz = 0.3εpore + 0.7εalumina
εx = εalumina(0.65εpore + 0.35εalumina)/(0.35εpore + 0.65εalumina).
Figure 4 shows the experimental and calculated dielectric functions and demonstrates that
there is good agreement, especially with εz, which was expected since high-angle p-polarized
reflectance has been used to determine the dielectric function. For the energy-loss function,
however, the computed peak for εz has shifted to longer wavelengths compared to the
−
1
energy-loss function of the plain oxide, from 10.6 to 10.9 µm (i.e., 943 to 917 cm ).
The theoretical peak is higher and narrower than the experimental one, but the integrated
energy loss of the wavelength range of figure 4 is the same for both. To obtain the larger

4298
E W a¨ ckelg a˚ rd
experimental shift, the pore fraction must be as high as 0.45 which is not realistic according
to the TEM measurements.
7. Conclusion
Substantial absorption of electromagnetic radiation occurs for wavelengths where the
radiation couples with the longitudinal optical phonon. The absorption is nearly 100%
for a 0.5 µm alumina film on an aluminium substrate for oblique incidence. The method of
using reflectance measurements to determine optical constants is rather limited, but it can be
used in the reststrahlen band wavelength range for thin amorphous alumina films using p-
polarized light at two high angles of incidence. Anodic porous alumina is anisotropic since
the pores are developed as channels perpendicular to the surface. The dielectric function
derived from high-angle p-polarized reflectance should then be similar to the element in
the dielectric tensor of an anisotropic medium which is perpendicular to the surface. The
dielectric function perpendicular to the surface calculated from Maxwell-Garnett effective-
medium theory and the experimental dielectric function are in good agreement. The peak
0
0
−1
in ε at 15.9 µm (630 cm ) is not shifted for an effective medium composed of air-
filled pores in alumina from its position for non-porous alumina. This implies that the
differences between porous and non-porous alumina originate mainly from the pores, not
from differences in chemical bonds. This conclusion is also confirmed by the wavelength
shift of the peak in the energy-loss function found both theoretically and experimentally for
porous alumina compared to non-porous alumina, although the theoretical shift is smaller.
The wavelength shift is also seen in the reflectance minima for p-polarized light at a non-
normal angle of incidence.
Acknowledgments
Dr Gunnar Niklasson is acknowledged for good advice and fruitful discussions, research
engineer Rein Kalm for assisting with SEM and TEM measurements and Fil. Lic. Nils-Olov
Ersson for the x-ray diffraction analyses. This work was financed by the Swedish Council
for Building Research (BFR) and the Marcus and Amalia Wallenberg Foundation.
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