Enhanced sensitivity to dielectric function and thickness of absorbing thin films by combining

total internal reflection ellipsometry with standard ellipsometry and reflectometry

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IOP PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 46 (2013) 105501 (13pp) doi:10.1088/0022-3727/46/10/105501

Enhanced sensitivity to dielectric functionand thickness of absorbing thin films bycombining total internal reflectionellipsometry with standard ellipsometryand reflectometry

A Lizana1, M Foldyna1, M Stchakovsky2, B Georges3, D Nicolas3 andE Garcia-Caurel1

1 LPICM–CNRS (UMR 7647), Ecole Polytechnique, 91128, Palaiseau, France2 Horiba Scientific, 5 Avenue Arago, 91380, Chilly-Mazarin, France3 Saint-Gobain Recherche, 93303, Aubervilliers Cedex, France

E-mail: enric.garcia-caurel@polytechnique.edu

Received 24 September 2012, in final form 11 December 2012

Published 8 February 2013

Online at stacks.iop.org/JPhysD/46/105501

Abstract

High sensitivity of spectroscopic ellipsometry and reflectometry for the characterization of

thin films can strongly decrease when layers, typically metals, absorb a significant fraction of

the light. In this paper, we propose a solution to overcome this drawback using total internal

reflection ellipsometry (TIRE) and exciting a surface longitudinal wave: a plasmon-polariton.

As in the attenuated total reflectance technique, TIRE exploits a minimum in the intensity of

reflected transversal magnetic (TM) polarized light and enhances the sensitivity of standard

methods to thicknesses of absorbing films. Samples under study were stacks of three films,

ZnO : Al/Ag/ZnO : Al, deposited on glass substrates. The thickness of the silver layer varied

from sample to sample. We performed measurements with a UV–visible phase-modulated

ellipsometer, an IR Mueller ellipsometer and a UV–NIR reflectometer. We used the

variance–covariance formalism to evaluate the sensitivity of the ellipsometric data to different

parameters of the optical model. Results have shown that using TIRE doubled the sensitivity

to the silver layer thickness when compared with the standard ellipsometry. Moreover, the

thickness of the ZnO : Al layer below the silver layer can be reliably quantified, unlike for the

fit of the standard ellipsometry data, which is limited by the absorption of the silver layer.

(Some figures may appear in colour only in the online journal)

1. Introduction

Spectroscopic ellipsometry is often used for determining the

dielectric functions of different substrates or multi-layer mate-

rials. Standard ellipsometric and reflectometric measurements

are commonly performed in external configuration, which

means that a light beam propagating in air is reflected by or

transmitted through a sample and then it propagates again in the

air before arriving at the detector. In the external configuration,

if the beam is not tightly focused, it can be roughly associated

with a propagating plane wave. The standard ellipsometry canmeasure simultaneously the relative modulus and the phase oflinearly polarized components of a light beam. The sensitivityof phase measurements to thin film thickness has its roots inan interferometric effect, in which the light reflected by thefirst interface of a layer present in the sample, interferes withthe light reflected by the second face. The scheme is validalso for a multi-layer. For absorbing layers, the light reflectedfrom the back face is reduced, which reduces also the inter-ference effect and diminishes the sensitivity of the technique.

0022-3727/13/105501+13$33.00 1 © 2013 IOP Publishing Ltd Printed in the UK & the USA

J. Phys. D: Appl. Phys. 46 (2013) 105501 A Lizana et al

In the limit of high absorption or non-negligible thickness of

a particular absorbing thin film, the light that penetrates the

film is absorbed before it can be reflected back and interfere

with the light reflected from the first film interface. Therefore,

all the information concerning the thickness and the layer(s)

below the absorbing one is lost. This drawback can be min-

imized by generating electromagnetic waves having a spatial

distribution and a propagation scheme substantially different

from waves figuring under standard external conditions. This

is the case of surface waves called plasmon-polaritons, which

are non-radiative photon–electron oscillations that propagate

along the surface of the sample. A complete review on phys-

ical properties, applications and practical methods of exciting

plasmon-polaritons, can be found in books of Agranovich [1]

and Raether [2]. In cases in which the thin absorbing layer is

sandwiched between thin films, the field distribution and the

propagation characteristics of plasmon-polaritons are strongly

influenced by the thickness and the refractive index of the

layer(s) which are in contact with the metallic one. The field as-

sociated with the plasmon polariton can be well spread above,

below and even inside of the absorbing layer (see for instance

figure 6(b)). In consequence, optical measurements involv-

ing plasmon-polaritons can provide information about layers

below the absorbing one in contrast to measurements under

standard conditions. The excitation of a plasmon-polariton

with an optical wave requires that the wave-vector of the plas-

mon and that of the optical wave matches. In general, the

excitation of plasmon-polaritons in plane surfaces by optical

beams is not possible using the external configuration scheme.

The reason for that is that the wave-vector of light impinging

the sample from air is smaller than that of plasmon-polaritons.

In order to satisfy the plasmon-polariton boundary condition,

the light must reach the sample with the absorbing layer from

a medium with a refractive index higher than that of the air. In

practice, this can be achieved in many ways [1, 2], for instance,

coupling optically a prism to the sample as depicted schemat-

ically in figure 1. Henceforth, measurements performed using

a prism will be referred to as measurements in the internal

reflection configuration as opposed to the standard (external)

configuration in which the light arrives directly from the air or

vacuum. Measurements in external and internal configurations

represent fundamentally different (complementary) illumina-

tion configurations.

Standard optical characterization techniques, such as

reflectometry or ellipsometry, can be easily adapted to

measurements in the internal reflection configuration. In

particular, the reflectometry, called also surface plasmon

resonance (SPR) spectroscopy, has become extremely popular

for studies involving relative or qualitative measurements in

biology and medicine [3]. In contrast, ellipsometry is used in

total internal reflection configuration only sporadically. The

first ellipsometric measurements in internal reflection mode

were performed by Abeles [4] and Lopez-Rios [5] in the 1970s.

After those pioneering works, maybe because of technical

difficulties to build optical set-ups or maybe because of the lack

of computing facilities to treat the data (simulation and fitting

of parametric models), the technique fell into disuse. Thirty

years later, the field started to regain interest as documented

Figure 1. (a) Sketch of the ZnO : Al/Al/ZnO : Al stack on a glasssubstrate. (b) Sketch of internal reflection or TIR configuration.

in recent reviews [6, 7]. In spite of the scarce number of

works, the technique has received different designations such

as the surface electromagnetic waves ellipsometry [4], the SPR

enhanced ellipsometry [8], the differential surface plasmon

ellipsometry [9] and the total internal reflection ellipsometry

(TIRE) [6]. In agreement with the arguments given in [6], we

find that TIRE is the most adequate term. The term TIRE is

useful because it avoids confusion with infrared ellipsometry,

sometimes called IRE. Most published works involving the

TIRE technique involve very simple structures consisting

of either single or double layers deposited on semi-infinite

substrates. Moreover, there are only a few reports focusing on

the analysis of the sensitivity of the technique [6, 10].

When it comes to study complex stacks of layers, the

use of the TIRE may reduce optical contrast and sensitivity

to layers located between the metal layer and the coupling

media. For that reason, the best option is to combine the

information given by TIRE with that provided by standard

ellipsometry. In an earlier study, Fontana et al [11] shown

that the combination of reflectivity in internal configuration

with the ellipsometry in external configuration was efficient

to determine optical constants and thicknesses of dielectric-

coated metallic mirrors. In this work we would like to go a

step further and show that the combination of ellipsometric

measurements in external and internal configuration together

with external reflectometry, can lead to an enhancement of the

accuracy and sensitivity in the determination of the thickness

and dielectric functions of absorbing layers, and the layers

2

J. Phys. D: Appl. Phys. 46 (2013) 105501 A Lizana et al

neighbouring them. We have evaluated the sensitivity of

measurements to different sample parameters and we have

used it to compare the accuracy of the TIRE and external

ellipsometry. We have carried out our experiments on a model

structure composed of a tri-layer stack of a thin silver film

surrounded by two layers of aluminium-doped zinc oxide

(ZnO : Al) deposited on a glass substrate. Similar structures

have been used in numerous works, in which spectroscopic

ellipsometry or reflectometry in external configuration was

used for the determination of dielectric function of silver

[12, 13]. Our choice is also motivated because these stacks

are used in industrial applications as back-reflectors for solar

cells or low-emissivity coatings for architectural windows

or automobile panels [13, 14]. The stacks chosen for this

paper are already integrated in existing commercial products

by Saint-Gobain company, the production of samples is well

controlled and standardized, and it is the simplest benchmark

structure to study the optical–electrical properties of thin

silver layers. Zinc oxide is not the only dielectric used for

low emissivity windows; sometimes it is replaced by silicon

nitride or titanium oxide. The stacks used for this work

incorporate zinc oxide because the required deposition process

and deposition conditions are well known and are compatible

with the rest of used materials used (silver and glass), and also

with the facilities available in the production lines of Saint-

Gobain, which provided the samples. Our approach can be

easily implemented either in an in situ or ex situ fabrication

or quality control process and can provide a very accurate

determination of the silver and dielectric layer thickness (in

the order of 1 nm). Our method allows us to determine the

value of the dielectric function of all materials present in the

sample in a wide spectral range (from the ultraviolet to the mid-

infrared). It is an invaluable source of information to get an

insight into different physical properties of materials (atomic

ordering, conductivity of the metallic layer, transparency of

the dielectrics, etc) providing useful criteria for the quality

control. In summary, the choice for our samples obeys to a

double purpose: the use of a simple and adequate benchmark to

show that the accurate determination of optical properties of an

absorbing film (such as silver) is possible with the combination

of external and TIR ellipsometry, and also, the development of

an optical, non-contact procedure to analyse and to guarantee

the quality of the coatings which are integrated in commercial

products.

In the following we introduce the detailed experimental

background and sample descriptions. After that, we will

show in detail dielectric functions of the glass substrate, the

silver and the ZnO : Al layers. The next section is devoted

to show the sensitivity enhancement to the determination of

thickness and dielectric functions by measurements in internal

configuration. We will also show the sensitivity of the optical

data to different physical parameters of layers in the coating.

The comparison of the sensitivity of the standard ellipsometry

and TIRE is presented in section 5. In section 6 we show

the validation with independent techniques. The first one

is a comparison between the conductivity deduced from our

optical measurements and the conductivity measured using a

standard electrical characterization method. The second test is

the comparison between the infrared emissivity of the samples

deduced from our method and the emissivity measured directly

according to ISO international standardization rules.

2. Sample description and experimental background

In the first part of this section we provide a description of

the deposition process of ZnO : Al/Ag/ZnO : Al multi-layer

samples studied in this work. The second part is devoted to the

description of optical properties of the glass substrate, ZnO : Al

and silver by means of parametric analytical functions.

2.1. Sample description/thin film deposition conditions

We have studied multi-layer stacks of aluminium-doped

ZnO : Al and silver layers forming the ZnO : Al/Ag/ZnO : Al

structure with thin silver films of the following nominal

thicknesses: 8, 10, 12, 14, 16, 20, 30 and 50 nm. From now on

samples will be labelled Ag8, Ag10, Ag12, Ag14, Ag16, Ag20,

Ag30 and Ag50, respectively. Nominally 20 nm thick ZnO : Al

layers were produced by an inline magnetron sputtering reactor

with a ZnO : Al target (deposition was carried out in a reactive

O2/Ar atmosphere with the pressure fixed to 2 µbar and Al

forming 2% of the total weight of the target). Silver layers

were also produced by the magnetron sputtering using a pure

silver target (in a pure argon atmosphere under the pressure

of 8 µbar). Sputtering was used as deposition technique for

ZnO : Al and silver layers for practical reasons related to the

existing deposition facilities. All the samples were deposited

on iron-free float glass substrates. Nominal thicknesses of

films were controlled by the velocity with which samples

were being moved through the deposition chamber. Samples

were prepared one after each other in order to ensure similar

conditions and deposition reproducibility. For the sake of

clarity, the structure of prepared sample series is sketched in

figure 1(a).

2.2. Standard spectroscopic optical measurements

Optical characterization of silver-based samples was per-

formed with two different techniques: spectroscopic ellipsom-

etry and infrared reflectometry.

Infrared reflectometry measurements were taken with

a commercial (Bruker) FTIR Spectrophotometer in the

spectral wavelength range from 4 to 30 µm. Spectroscopic

ellipsometry measurements in the ultraviolet to the near-

infrared (270–1600 nm) were performed with a commercial

phase-modulated ellipsometer UVISEL (HORIBA Jobin-

Yvon). Ellipsometric data in the mid-infrared were measured

with a patented ‘in-house’ prototype which has been described

elsewhere [15]. The angle of incidence was the same, 68.5◦,

for the UVISEL and the infrared ellipsometer in order to have

continuous data.

2.3. Optical measurements in TIR configuration

Internal reflection ellipsometric measurements were per-

formed using the UVISEL spectroscopic ellipsometer because

the most prominent spectral features appear at visible and

3

J. Phys. D: Appl. Phys. 46 (2013) 105501 A Lizana et al

ultraviolet wavelengths (due to the silver). The measurement

configuration, sketched in figure 1, consists of a BK7 prism

optically matched to the substrate using an oil designed for

this purpose. The prism was cut at an angle of 57.6◦. Mea-

surements were performed from the substrate side of samples

as depicted in the figure 1. This experimental arrangement cor-

responds to a modified Otto configuration [16, 17], also called

Sarid configuration [18]. The angle of incidence of light was

set to 57.6◦ to make the beam arrive perpendicularly to the

prism face. Since at normal incidence there is no difference in

the transmission of the TM and TE polarized waves through

the air-BK7 and BK7-air interfaces, they do not affect mea-

sured ellipsometric values, which simplifies the modelling of

the experimental data.

Ellipsometry measures the complex ratio of the

Fresnel coefficients for wave polarization parallel ‘rp’ and

perpendicular ‘rs’ to the plane of incidence. The ellipsometric

ratio is expressed in terms of the ellipsometric angles �

and � as

rp/rs = tan(�) exp(i�). (1)

In general, ellipsometers do not measure � and � directly, but

rather variables from which � and � can be easily calculated.

In the special case of phase-modulated ellipsometers there are

three measured magnitudes called Ic′, Is and Ic or also N , S

and C. Relations among them and the ellipsometric angles are

as follows: Ic′ = N = cos(2�), Is = S = sin(2�) sin(�)

and Ic = C = sin(2�) cos(�). As these values are described

by sine and cosine functions, they are bounded between −1

and 1. Numerical computations carried on Is, Ic and Ic′ are

more stable than that carried on � and �. For the figures

shown in this work, we prefer to show � and �, as they are

directly related to Fresnel coefficients.

3. Parametrized dielectric functions and opticalmodel of the multilayered samples

The simulation of the optical response of a sample in terms of

reflectivity or ellipsometric angles using a theoretical model

based on Maxwell equations can be understood as a direct

problem. The ellipsometric practice is based on solving the

inverse problem, i.e. to extract model or material parameters

from a given ensemble of experimental data. In general,

the theoretical model depends on a series of parameters

characteristic of the sample that must be adjusted in order to

make the theoretical data to ‘fit’ the measured data. A common

model for a stack of layers includes the thickness and dielectric

functions of layers. The dielectric function of the substrate

must be considered as well [19]. The quality of the fit is usually

evaluated with a figure of merit function and it is used during

the fit process to guide the numerical algorithm which searches

for the best-fitted values of the model parameters. According

to [19] we used the following definition for the figure of merit:

χ2 =1

N − M − 1

∑

k

(IsTk − Is

Expk )2

σ 2Is

+(IcT

k − IcExpk )2

σ 2Ic

+(Ic′ T

k − Ic′ Expk )2

σ 2Ic′

+(RT

k − RExpk )2

σ 2R′

(2)

where N refers to the total number of data points and M

is the total number of fitted parameters. The superscripts

‘T’ and ‘Exp’ refer to the theoretical and experimental data,

respectively. The summation is done over all the spectral

data points. The sigmas in denominators correspond to

the estimated measurement uncertainty of the corresponding

experimental values. Those uncertainties are related to the

accuracy and precision of the instruments used. In our case,

the sigmas were estimated to be around 0.01 for the whole

spectral range.

The spectral dependence of the dielectric function of solids

is determined by different interaction mechanisms between

light and the electrons inside the solid. At optical frequencies

(ultraviolet to mid-infrared) there are three main mechanisms

that contribute to the dielectric function: interband excitations

(mainly in the UV), intraband excitations created by free

conduction electrons (from the visible to infrared for metals

and in the infrared for doped semiconductors) and phonon

excitation (mid-infrared). A proper model of those phenomena

needs a rigorous ab initio approach, which implies complex

numerical calculations and that is impossible to implement

for practical applications related to fitting procedures. A

practical alternative to circumvent this technical drawback

is to use analytical functions that depend on a reduced

number of parameters [19, 20]. In general, those functions

are derived from phenomenological arguments, asymptotic

approximations or empirical observations and therefore, their

domain of application is limited to a given spectral range or

a class of materials. Not surprisingly, decades of research in

material science using optical methods have produced dozens

of parametrized model functions. Therefore, the choice of

a particular expression to model a particular phenomenon in

a given material is not unique and it is not completely free of

debate. After a careful review of the literature we have selected,

for each material, the function that it is commonly accepted to

be the most appropriate and that produced the best fit. In the

first part of this section, we provide a detailed description of

parametric optical functions used for the glass substrate, the

silver and the ZnO : Al. At the end of the section we will give

an overview of the data analysis procedure and we will detail

the two models used to analyse the experimental data. Optical

models of layered samples are currently built with the help

of commercial or ‘in-house’ software packages. For this study

we used a commercial package, DeltaPsi2, designed by Horiba

Scientific and an ‘in-house’ package developed under Matlab.

3.1. Glass substrate

Glass substrate is an amorphous dielectric, transparent in

the visible and in the near infrared parts of spectrum. It

is absorbing in the ultraviolet due to interband electronic

transitions and in the mid-infrared due to the excitation of the

vibrational modes (phonons) of the different chemical bonds

of the elements present in the amorphous atomic network.

The dielectric function of the glass substrate in the infrared

can be well described with a combination of four Lorentz

oscillators (OS). The absorption in the ultraviolet spectrum was

represented using the Tauc–Lorentz (TL) model [21], which

4

J. Phys. D: Appl. Phys. 46 (2013) 105501 A Lizana et al

is well adapted for amorphous semiconductors and dielectrics,

and it takes into account the optical band gap in the interband

region. The substrate that we used is a float glass. During the

production of this kind of glasses one side of the substrate is in

contact with a surface of liquid tin and the other surface is in

contact with the air. Due to the physical contact between the

substrate and the metal, some tin diffuses inside the glass up to

a thickness of about 100 nm. Because of the tin, the dielectric

constant of the glass is slightly increased. We accounted for

this effect using two dielectric functions. The first one for the

bulk and the second one for the tin side [22]. The parametric

expression for the dielectric function ε(ω), corresponding to

the bulk and to the tin side can be written as follows:

εGlass(ω) = ε∞ +

4∑

j=1

εOS(fj , ω0j , γ ) + εTL(Eg, f, ω0, Ŵ),

εOS =fjω

20j

ω20j − ω2 + iγjω

,

Im(εTL)

1

ω

f ω0Ŵ(ω − ωg)2

(ω − ω0)2 + Ŵ2ω2for ω > ωg

0 for ω < ωg

;

Re(εTL) = KKR(Im(εTL)), (3)

where the parameters fj , ω0j and γj are related to the Lorentz

oscillator formula and the parameters f , ω0, Ŵ and Eg are

related to the Tauc–Lorentz approximation. The parameter fj

is the oscillator strength, ω0j is the resonant energy and γj

is related to the oscillator bandwidth. Similarly, in the TL

expression, f is the oscillator strength, ω0 is the resonance

position and the parameter Ŵ gives the damping. The real

constant ε∞ accounts for contributions of optical transitions not

represented by oscillators in equation (3). The real part of the

dielectric constant corresponding to the Tauc–Lorentz model

is calculated via a Kramers–Kronig transformation (KKR) of

the imaginary part.

3.2. ZnO : Al

We refer as ZnO : Al to zinc oxide doped with aluminium. The

amount of aluminium doping was carefully controlled during

deposition of the films to compensate a slight excess of oxygen

in order to obtain a highly electrical resistive material. In

consequence, the ZnO : Al we worked with can be considered

as a wide gap dielectric. Optically, ZnO : Al films show a

strong ultraviolet absorption due to interband transitions and

we observed the onset of absorption at a photon energy of

ωg = 2.8 eV with no absorption in the measured infrared

spectrum. Interband transitions were modelled with a TL

formula considering two oscillators. Our model expression

for the dielectric function has the following form

εZnO : Al(ω) = ε∞ + εint(ω) = ε∞ +

2∑

k=1

εTL(ωg, fk, ω0k, Ŵk).

(4)

The physical meaning of parameters fk , ω0k and Ŵk are the

same as those in equation (3).

3.3. Silver thin film

Silver is a noble metal which shows a noticeable absorption in

the whole spectral region from the infrared to the ultraviolet.

Similarly to other noble metals, silver has a characteristic

frequency, called plasma frequency, which determines its

optical behaviour. Above the plasma frequency the absorption

is governed by interband transitions, and below it by intraband

absorptions due to free carriers. Therefore, the parametric

model built for the silver dielectric function includes two terms,

for interband transitions (εinterband) and for the free carriers

(εFC), respectively. The choice of our model function for the

interband transitions is based on the electronic band structure

of silver calculated from first principles [24, 25] and reference

optical data taken from various commonly used bibliographic

sources [26–28]. Reference spectroscopic data show the onset

of interband transitions at 3.8 ± 0.05 eV. Below this energy,

the behaviour of the reference dielectric function is smooth

and it can be easily modelled with the well-known Drude

expression. Above the gap, the main contribution to the

absorption can be attributed to direct transitions from the upper

band below the Fermi energy to a first conduction band which

crosses the Fermi level and from the first conduction band

to the second conduction band. Those bands are labelled 5,

6 and 7, respectively, in [25]. By analogy between the band

structures of silver and crystalline semiconductors, we used the

same parametrized function that is commonly used to model

direct transitions of E1 type in semiconductors. A necessary

condition to assume transitions of E1 type is that the density

of states, as a function of the energy, could be considered as a

constant within the energy region where transitions take place,

which is the case for silver around the Fermi energy [29]. The

parametric dielectric formula for silver has the following form

εAg(ω) = ε∞ −ω2

p

−ω2 + iωŴD

+ εE1 + εTL(Eg, f, ω0, Ŵ)

Im(εE1(E01, B1, Ŵ1)) = −B1

ξ 2ln(1 − ξ 2); ξ 2 =

E01 + iŴ1

E1

Re(εE1(E01, B1, Ŵ1)) = KKR(Im(εE1(E01, B1, Ŵ1))), (5)

where the ωp and ŴD parameters are the plasma frequency and

damping factor of the Drude model, respectively. Eg gives the

energy gap for the TL formula. The parameters B1 and Ŵ1

are related to the oscillator force and damping, respectively.

Finally, ε∞ accounts for high-frequency contributions not

represented by the expression in equation (5).

4. Modelling strategies and fitting results

4.1. Fit of the optical properties of Glass and ZnO : Al

Following the example given by Synowicki et al [22], we

performed ellipsometry and reflectometry measurements on

both sides of an uncoated glass substrate. The sample was

modelled as a thick layer surrounded by two thin layers. One

layer, with a characteristic thickness of 2 nm, represented the

roughness and the second one represented the portion of glass

containing tin impurities. As previously mentioned, we used

two dielectric functions, one for the bulk glass and the second

for the layer containing tin. The only difference between

5

J. Phys. D: Appl. Phys. 46 (2013) 105501 A Lizana et al

Figure 2. (a) Experimental (grey dots) and best-fitted values (solidlines) for the ellipsometer angles � and � corresponding to the airside of the glass substrate. (b) Refractive index, n, and absorptioncoefficient, k, corresponding to bulk glass (grey lines) and glasswith tin (black lines) calculated with equation (3) and parameters intable 1.

the parameters corresponding to the bulk and to the tin face

dielectric functions is the value of ε∞. parameter. The main

differences between our work and [22] are as follows: (i) we

did not use measurements in transmission, (ii) we did not

detect stress-induced birefringence and (iii) our measurements

covered a wider spectral range (from the ultraviolet to the

mid-infrared). In figure 2, we provide a comparison between

best-fitted data and experimental values obtained for the glass

substrate at the air side. In addition, the values of the refractive

index and absorption coefficient deduced from the best-fitted

parameters are also plotted in figure 2. The best-fitted values

for parameters of the model dielectric function of the glass

substrate are given in table 1.

Concerning the characterization of optical properties of

ZnO : Al, we worked with a sample consisting of a glass

substrate coated with a 30 nm layer of ZnO : Al. The ZnO : Al

layer was deposited under the same conditions as layers being

part of the tri-layers. The glass substrate was the same as the

one described above. The optical model of the sample was

identical to that of the glass substrate but with an additional

layer representing the ZnO : Al film. The addition of a second

Table 1. Values of the parameters corresponding to the Lorentzoscillators and to the TL expression in equation (3) used tocharacterize the glass substrate. Results are accurate up to the lastdigit shown in the table.

Term Parameter Fitted value

ε∞ ε∞BULK 2.20

ε∞ ε∞TIN-SIDE 2.51

IR Oscillator 1 f1 0.150ω01 0.130γ1 0.006

IR Oscillator 2 f2 0.419ω02 0.123γ2 0.009

IR Oscillator 3 f3 0.113ω03 0.093γ3 0.007

IR Oscillator 4 f4 0.056ω04 0.140γ4 0.008

Tauc–Lorentz f 0.02ω0 8.1Ŵ 0.23

layer to account for the roughness of the ZnO : Al film did not

improve the quality of fits, therefore we did not considered it.

Topographic measurements of the surface of the ZnO : Al film

made with an atomic force microscope gave an RMS roughness

of about 1 nm, which confirmed the flatness of the sample. In

figure 3, we provide a comparison between the best-fitted and

the experimental data obtained for the ZnO : Al sample. The

features visible in � and � spectra at infrared wavelengths

larger than 8 µm are due to the substrate and are visible because

the ZnO : Al is transparent in this region, which also confirms

the absence of free electron related transitions in our material.

In figure 3 we also show the values of the refractive index

and the absorption coefficient deduced from the best fitted

parameters. The best-fitted values for the parameters of the

ZnO : Al dielectric function are provided in table 2.

4.2. Typical dielectric function of silver

Unlikely to glass and ZnO : Al, it was not possible to

characterize single silver layers deposited on the glass because

silver oxidizes quickly when exposed to open air, and its

optical response changes significantly. In order to work with

silver, we had to use the complete ZnO : Al/Ag/ZnO : Al tri-

layer samples. We built the optical model representing the tri-

layered samples adding three homogeneous layers to the model

structure used to represent the substrate. In our data analysis,

we assumed that the dielectric function of the glass and

the ZnO : Al was identical to the values acquired previously.

Accordingly, what was left to be fitted from measured data

were the dielectric function of silver and the thicknesses of all

three layers.

The dielectric function of the Ag is governed by intraband

transitions due to free electrons and interband transitions.

Whereas intraband transitions are related to kinetic motion of

free electrons, interband transitions are related to electrostatic

6

J. Phys. D: Appl. Phys. 46 (2013) 105501 A Lizana et al

Figure 3. (a) Experimental (grey dots) and theoretical (solid lines)values for the ellipsometric angles � and � corresponding to theAZO sample. (b) Refractive index, n, and absorption coefficient, k,calculated with equation (4) and parameters in table 2.

Table 2. Values of the parameters corresponding to the two TLformulae in equation (4) used to characterize the AZO thin film.Results are accurate up to the last digit shown in the table.

Term Parameter Fitted value

Eps inf. ε∞ 3.811

TL 1 f1 0.095ω01 3.52Ŵ1 0.36

TL 2 f2 1.12ω02 3.33Ŵ2 1.76

environment of electrons, which is determined by the chemical

composition. On a previous study [30], a similar set of

tri-layer samples (with the same structure, composition and

deposition conditions) was studied using EELS and electronic

microscopy. One of the conclusions of this study was that

the chemical composition and the atomic structure of the

silver layers remained unchanged regardless of the thickness

of the films. Accordingly, we assumed that the parameters

describing interband transitions should not suffer noticeable

changes with the thickness and we did not fit them from

sample to sample. Considering that the mean free path

Table 3. Values of the parameters corresponding to the Drudemodel and to the direct transitions used to characterize the opticalresponse of the silver thin films. Results are accurate up to the lastdigit shown in the table.

Term Parameter Value

Direct type E1 E01 4.0 eVB1 1.66Ŵ1 0.15 eV

TL Eg 3.8 eVF 0.52ω0 4.7 eVγ0 1.42 eV

Drude model ωp 9.2 eVŴD 0.038 eV

(MFP) of electrons in bulk silver is about 40 nm, it is not

surprising that once dimensions of the layer become of the

same order or even smaller than MFP, interfaces of the film

have a non-negligible impact on the kinetics of free electrons.

In the framework of the Drude approximation there are two

parameters describing the behaviour of intraband transitions:

the plasma frequency and the Drude damping factor. The

plasma frequency depends on the volume density of electrons,

whereas the Drude damping factor is related to the inverse of

the MFP. According to [30], if the chemical composition of the

silver does not change with thickness, the amount of electrons

per unit of volume and therefore the plasma frequency should

be constant. In consequence, the only parameter that should

change noticeably from sample to sample is the Drude damping

factor. As shown in [30] the silver in our samples was

polycrystalline, so a dielectric function similar but not exactly

equal to that in the literature was expected. We started our

analysis by fitting values of the dielectric function given in

[25], which are valid for monocrystalline silver, in order to

have a first guess on values of parameters governing interband

transitions. The resulting best-fitted value for the plasma

frequency was 9.2 ±0.2 eV, in accordance with the theoretical

and experimental estimations found in the literature. This

result partially confirmed the validity of expression for the

dielectric function in equation (5). To refine the dielectric

function we performed a second fit with the sample having the

thickest silver film (50 nm nominal). Because of its thickness,

the silver layer was opaque, thus the optical response of the

sample did not depended on the ZnO : Al layer below the

silver layer and the interface between them, which simplified

our model. Parameters to be fitted were the thickness of the

ZnO : Al layer and the optical dielectric function of the silver

layer. The resulting best-fitted parameters are shown in table 3.

As expected, those values were quite close to those of the

monocrystalline silver. However, the value of the damping

factor, ŴD, strongly depends on the silver thickness and

therefore, it varies from sample to sample.

4.3. Fitting of the tri-layer system

Taking in consideration the above discussion the fitting

parameters that may reasonably change from sample to sample

are the thickness of the silver layer, dAg, the thickness of the

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J. Phys. D: Appl. Phys. 46 (2013) 105501 A Lizana et al

ZnO : Al layer in contact with air, dZnO up, the thickness of

the ZnO : Al layer in contact with the glass, dZnO down, and

the Drude damping factor ŴD. It was not possible to fit the

ensemble of those parameters when we used ellipsometric

data in external configuration or reflectometry data only. The

reason was the importance of numerical correlations between

parameters. For instance, we observed a particular coupling

between parameters dZnO up and ŴD and also between dZnO down

and ŴD. By numerical correlation between two parameters,

say A and B, we mean that for a given set of experimental data

there exist a multitude of pairs of values for the parameters A

and B that give fits of similar quality. In the presence of strong

numerical correlations it is not possible to fit parameters A

and B simultaneously. One of the two parameters must be

fixed using independent information in order to fit the second

one [19]. Two (or more) parameters are numerically correlated

because the fitted experimental data are insensitive to their

individual contributions. For this reason an additional strategy

to reduce parameter correlations is to fit simultaneously data

obtained from different sources [19].

In our case, the combination of ellipsometric data

in external configuration with reflectometry proved to be

successful to provide acceptable values of the silver layer

thickness and the ŴD for samples having relatively thin silver

layers (<20 nm). However, the correlation between dZnO up

and dZnO down with ŴD remained high. In consequence, values

of dZnO up and dZnO down were kept fixed to their nominal value

of 27 nm. For samples with relative thick layers (>20 nm)

the combined sensitivity of ellipsometry and reflectometry

to dAg was very low because of the lack of transparency

of silver layers. In consequence, the correlation between

dAg and ŴD was very high, thus preventing us from working

with such samples. The best-fitted values are summarized in

table 5. Figure 4 shows the experimental and best-fitted data

for samples Ag08 and Ag16. In the short wavelength region

(∼510 nm), we observe systematic discrepancies between

the experimental and fitted values. Those discrepancies

are especially visible for the � spectra. For wavelengths

around 500 nm, the silver layer is semi-transparent and optical

data are sensitive to the thicknesses of layers beneath it.

Systematic differences between fitted and measured spectra

appear because thicknesses of ZnO : Al layers had to be kept

fixed. On the other hand, in the long wavelength region, where

ZnO : Al layers are transparent, the quality of the fit is excellent.

In order to go beyond those limitations, we have

performed ellipsometric measurements in TIR configuration

and combined them with ellipsometric and reflectometry

measurements in external configuration. As will be detailed

in the following section, this approach allowed enhancing the

sensitivity of measurements, to obtain acceptable results for

samples with silver layer thickness up to 50 nm and also to fit

the values of dZnO up and dZnO down.

The best-fitted values for the whole set of samples are

summarized in table 6. In figure 5 we have represented the

experimental and the best-fitted ellipsometric data for three

representative samples. The fact of being able to fit both

ZnO : Al layers thicknesses (dZnO up and dZnO down) without

any ambiguity improved the quality of the fit in the short

Figure 4. Experimental (circles) and best-fitted (lines) � and �spectra corresponding to samples Ag08 and Ag16.

wavelength region (compare figures 5 and 4) and provided a

more precise characterization of the tri-layers. A study of the

impact of TIRE measurements on the sensitivity enhancement

for the determination of the optical parameters of samples is

given in the next section.

In figure 5(c) we provide the n and k values for the silver

layers. These values can be acquired using equation (5) with

parameters shown in tables 3 and 6 (for specific values of ŴD).

The data in figure 5(c) show that the refractive index and the

absorption increase with the silver layer thickness. This is

in accordance with the Drude theory of free electrons which

predicts that the electrical conductivity is proportional to the

optical absorption [20]. As we will show in section 6, the

electrical conductivity of layers is increased when the layer

thickness becomes thicker.

5. Impact of TIR configuration on the determinationof physical parameters of interest

In this section, we present an evaluation of the sensitivity

of experimental data to parameters of the optical model.

Sensitivities of the standard and the TIRE sets of data

to the optical model parameters will be compared for

different thicknesses of silver. The phenomenon of SPRs

on metal/dielectric interfaces is very well known for a long

time. The study of the effect of the resonance on the

measured reflectivity and spectral ellipsometry data has also

been reported in multiple occasions. In particular, the most of

works concentrate on the position of the sharp spectral feature

in the spectral response due to a decreased reflectivity of a TM

polarized light.

The spectral position of the plasmon peak depends

strongly on the angle of incidence and the metal/dielectric

interface, which has been successfully exploited in our work

for enhancing the sensitivity to model parameters.

SPRs have high penetration depth which leads to the

interaction of the light with the structure beneath it, as shown in

figure 6. In the case of internal configuration, figure 6(b), The

real components of longitudinal electric field of TM polarized

incident waves show an electric field penetration through

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J. Phys. D: Appl. Phys. 46 (2013) 105501 A Lizana et al

Figure 5. Experimental (circles) and best-fitted (lines) � and �spectra corresponding to samples Ag08 (dashed), Ag16 (black) andAg30 (grey). (a) Data in TIR configuration. (b) Data in externalconfiguration. (c) Spectral dependence of n and k values of thestudied silver layers. Notice that the ordinate axis of the graphicrepresenting n values is plotted in log scale and that the axis of thegraphic representing k values in broken to improve visualization.

the 30 nm silver layer and a non-negligible field intensity

within the ZnO : Al layers. On the other hand, in the case

of external reflection, figure 6(a), only a very small portion

of the transversal electric field of TE polarized incident wave

is transmitted through the silver layer. The particular special

Figure 6. Distribution of the real part of transversal andlongitudinal electromagnetic field for (a) TE and (b) TM polarizedincident wave inside the ZnO : Al/Ag/ZnO : Al structure in the TIRconfiguration with the nominal silver thickness of 30 nm. Surfaceplasmon field strongly interacts with the bottom ZnO : Al layer. Thefield on the top layer corresponds to the prorating wave from theincident and reflected beams. This figure shows that thanks to thesurface wave there is a non-negligible amount of field distributedwithin all layers of the sample. In both figures, (a) and (b), the blacklines indicate the position of the three layers.

distribution of the electromagnetic field associated with surface

plasmons is at the origin of an increased sensitivity of the TIR

configuration to dielectric functions and thicknesses inside the

tri-layer stacks.

5.1. Sensitivity of TIR and standard ellipsometry to silver

layer thicknesses

The improved sensitivity of TIRE with respect to standard

ellipsometry is based on the plasmonic enhancement of the

TM polarized electromagnetic field as it was shown before.

Although, it is not immediately obvious how much the

sensitivity can be improved when a wide spectral range is

considered. In order to evaluate the sensitivity of the data to

fitted observables, i.e. silver film thickness,we have calculated

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J. Phys. D: Appl. Phys. 46 (2013) 105501 A Lizana et al

Table 4. Parameter uncertainties evaluated for standard ellipsometry and TIRE models at angles of incidence of 70◦ and 57.6◦, respectively.Table includes uncertainties of the silver layer (σAg), the top ZnO : Al layer (σZnO1) and the bottom ZnO : Al layer (σZnO2) thicknesses as wellas the uncertainty for ŴD damping factor (σŴD

). All uncertainties are evaluated for all nominal thicknesses of the silver layer.

SE at AOI of 70◦ TIRE at AOI of 57.6◦

σAg σZnO1 σZnO2 σŴDσAg σZnO1 σZnO2 σŴD

(nm) (nm) (nm) (eV) (nm) (nm) (nm) (eV)

Ag 8 0.18 0.78 1.90 0.047 0.095 0.56 0.94 0.020Ag 12 0.25 0.56 1.83 0.047 0.11 0.35 0.83 0.018Ag 16 0.32 0.44 1.73 0.048 0.14 0.26 0.73 0.017Ag 20 0.37 0.37 1.60 0.049 0.17 0.21 0.64 0.016Ag 30 0.48 0.24 1.42 0.042 0.29 0.14 0.53 0.013Ag 50 1.75 0.19 3.44 0.035 0.82 0.16 0.47 0.013

variance from

Var(X) = σ 2(X) = E{[X − E(X)]2}, (6)

where E(X) denotes the mean value of the observable X.

For the more straightforward estimation of the parameters

sensitivity, we have calculated by how much the observed

parameter has to change in order to create a significant

difference in the spectral response. The significant response

was defined as changing the value of the merit function in

the point of minima by 1% with a constant spectral precision

estimated to be w = 0.01. The sensitivity of the observable

to a free parameter can be described by a derivative of the

observable X by the parameter p, which corresponds to a

vector in the case of only one free parameter and a Jacobian

matrix in the case of multiple parameters:

Jp =∂X

∂p. (7)

In order to estimate the parameter uncertainty corresponding to

the proposed change in the spectral response, we can calculate

an inverse of an approximate Hessian matrix as follows:

H−1p = (J T

p Jp/n)−1, (8)

where n is the number of the spectral data points. The

estimation of the parameter uncertainty can be expressed using

equations (7) and (8) as follows:

σp =

√

w2H−1p . (9)

Accordingly, we have estimated the uncertainty in the

determination of different model parameters. Calculations are

carried out for both the standard and the TIRE set of data

separately, and results are summarized in table 4.

In figure 7 we have represented uncertainties for

thicknesses of each one of the three layers. In general, we

observed that TIRE measurements are about two times more

sensitive (uncertainties less than one half) than measurements

in the standard external configuration. We can observe two

characteristic trends in figure 7 with the increasing thickness

of the silver layer: the uncertainty of the thickness of the

top ZnO : Al layer decreases uniformly and the uncertainty

of the thickness of the silver layer increases uniformly. The

uncertainty of the top ZnO : Al layer decreases as the increasing

Figure 7. Parameter uncertainty σp as a function of the nominalthickness of the Ag layer.

Ag layer thickness also increases the contrast between the two

materials. The increase in the uncertainty of the silver layer

thickness follows a linear trend corresponding to relative errors

of 1.7% and 1% for the external and the internal configuration,

respectively. Relative errors increase when the silver layer

reaches value of 50 nm due to its significantly reduced

transparency. The evolution of the uncertainty of the bottom

ZnO : Al layer for data taken in standard external configuration

shows a dual behaviour: the uncertainty decreases with the

silver layer thickness smaller than 30 nm (as it is the case for

top ZnO : Al layer), whereas it is increased for 50 nm thick

silver layer due to its opacity. The reason for the reduction

in the uncertainty with the silver layer thickness is the same

that the one given by of the top ZnO : Al layer, increased

optical contrast. In contrast, once the silver layer reaches

30 nm, it becomes practically opaque. For thicknesses larger

than 30 nm , the amount of light reaching the bottom ZnO : Al

layer is considerably reduced, and therefore its effect on the

ellipsometric data. Not surprisingly, sensitivity calculations

show that for TIRE measurements the uncertainty in the

determination of the bottom ZnO : Al layer does not have

a dual behaviour. Uncertainty in thickness keeps decreases

uniformly even for silver layers thicker than 30 nm. This fact

is clearly illustrated by the electrical field distribution depicted

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J. Phys. D: Appl. Phys. 46 (2013) 105501 A Lizana et al

Table 5. Fitted results for silver samples obtained using the standardmodel. Columns represent: the nominal thickness of the silver layer,dAg,nom, the fitted thickness of the silver layer, dAg,fit , the sheetresistance, RSQR, ρc,ele (deduced from the RSQR values) the dampingfactor ŴD and ρc,opt (deduced from optical data). Uncertainties areshown in table 4.

dAg,nom dAg,fit RSQR ρc,elec ŴD ρc,opt

Samp. (nm) (nm) (�) (µ� cm) (eV) (µ� cm)

Ag8 8 9.0 6.52 5.22 0.065 5.71Ag10 10 10.7 5.24 5.24 0.057 5.01Ag12 12 13.1 3.89 4.66 0.052 4.57Ag14 14 14.9 3.20 4.48 0.047 4.13Ag16 16 18.1 2.37 3.78 0.046 4.01Ag20 20 22.5 1.78 3.55 0.039 3.43Ag30 30 — 1.11 3.34 — —Ag50 50 — 0.58 2.90 — —

in figure 6(b). In figure 6(b) it is shown that even for the thickest

silver layer, there is an important amount of electric field

intensity within the bottom ZnO : Al layer, which prevents an

increase in the uncertainty. These results clearly demonstrate

advantages of TIRE measurements for the characterization and

metrology of thin films buried under absorbing layers.

6. Validation of results

6.1. Comparison between optical and electrical conductivity

From values of the plasma frequency ωp and the damping factor

ŴD, obtained from the optical fitting, the optical resistivity

ρc,opt is calculated using the following relation [31]:

ρc,opt =ŴD

ε0ω2p

, (10)

where ε0 is the vacuum permittivity. Using the four-probe

point method we measured the sheet resistance (RSQR) of all

the samples. RSQR is related to the electrical resistivity ρc,ele

by the following relation:

ρc,ele = RSQRdAg, (11)

where dAg is the thickness of the silver thin film. Results

obtained using the best-fitted parameters with the standard and

the combined set of data are summarized in tables 5 and 6,

respectively.

To perform a qualitative interpretation of those results, it

is useful to represent the product of the resistivity by the silver

thickness (i.e. dsilver · ρc) as a function of dsilver, [32].

According to [33], the dAg · ρc product can be rewritten

in terms of a linear function, where the slope is related to the

intrinsic resistivity of the silver layer (ρ0) and the crossing

with the y-axis gives information about the influence of the

scattering of electrons by the surfaces of the layer [33]:

ρcdAg = ρ0dAg + K, (12)

Figure 8 shows the dAg · ρc product as a function of dAg. Grey

lines correspond to values deduced from electrical data. Black

lines correspond to values deduced from the best fitted values

Table 6. Fitted results for silver samples obtained using thecombined model. Columns represent: the nominal thickness of thesilver layer, dAg,nom, the fitted thickness of the silver layer, dAg,fit , thesheet resistance, RSQR, ρc,ele (deduced from the RSQR values),thedamping factor ŴD and ρc,opt (deduced from optical data).Uncertainties are shown in table 4.

dAg,nom dAg,fit RSQR ρc,elec ŴD ρc,opt

Samp (nm) (nm) (�) (µ� cm) (eV) (µ� cm)

Ag8 8 9.7 6.52 5.22 0.075 6.59Ag10 10 12.0 5.24 5.24 0.072 6.32Ag12 12 14.5 3.89 4.66 0.064 5.62Ag14 14 16.9 3.20 4.48 0.062 5.45Ag16 16 19.4 2.37 3.78 0.054 4.74Ag20 20 24.0 1.78 3.55 0.046 4.04Ag30 30 35.7 1.11 3.34 0.039 3.43Ag50 50 58.7 0.58 2.90 0.038 3.34

Figure 8. Electrical (grey) and the optical (black) results in terms ofthe dsilver · ρc product. Solid lines are obtained by linear regressionof experimental data. Dashed lines correspond to the 95%confidence interval. The data are obtained using (a) the standardmodel and (b) the combined model.

using either the standard set (figure 8(a)) or the combined set

(figure 8(b)). The linear regression slope corresponding to the

electrical data is (2.1 ± 0.9) µ�cm.

The values of the slope corresponding to the optical data

in figures 8(a) and (b) are (2.1 ± 0.4) µ� cm and (2.1 ±

0.3) µ� cm, respectively. The value of the bulk resistivity

for the three sets of data is slightly higher than the value for

bulk monocrystalline silver at room temperature 1.59 µ� cm.

This is because the silver in our samples was polycrystalline.

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J. Phys. D: Appl. Phys. 46 (2013) 105501 A Lizana et al

Figure 9. Experimental emissivity as a function of the silver layerthickness for the whole set of samples. The two other curvesrepresent theoretical predictions using either the standard or thecombined model.

The electrical potential barriers existing between silver grains

create an extra scattering effect that increases the resistivity of

the material.

For both sets of optical data, standard and combined, the

agreement with the electrical data is excellent. However, as a

consequence of the improved sensitivity provided by including

TIR ellipsometry measurements, the agreement is extended to

the sample with 50 nm thick silver layer. Also note that error

bars related to optical base data are significantly smaller than

those related to electrical data.

6.2. Infrared integrated emissivity

The normal thermal emissivity of an absorbing body can be

deduced from reflectivity and transmission measurements in

quasi-normal incidence according to the following relation:

ε = 1 − R − T , (13)

where R and T are the reflectance and the transmittance,

respectively. For low-emissivity glasses, the ISO European

norms (EN12898 and ISO10292) define the normal

emissivity as

ε = 1 −1

N

N∑

n=1

R(λn), (14)

where R(λn) correspond to the reflectance values correspond-

ing to N different wavelengths sampled in the spectral range

and specified in the ISO norm. The transmission does not ap-

pear in equation (14) because the glass is completely opaque

at infrared thermal wavelengths (>8 µm).

In figure 9, we represent the simulated emissivity values

as a function of the silver layer thickness, obtained using the

standard model and the combined model. The experimental

emissivity is also provided from experimental measurements

of the reflectance taken with an infrared spectrophotometer.

An excellent prediction of the emissivity is achieved using

developed optical models and when TIR ellipsometry is

applied, the agreement extends to the 50 nm thick silver layer.

7. Conclusions

We have characterized a tri-layer structure in which a silver

layer was stacked between two ZnO : Al layers. Due to

sensitivity limitations by silver absorption, the standard

ellipsometry method appears to be not sensitive enough for

studying silver layers with thickness higher than 30 nm.

The combined method allowed pushing this limit even

beyond 50 nm.

We have demonstrated advantages of using TIRE for

determination of thicknesses and dielectric functions. The

enhanced sensitivity is mediated by surface plasmons. Thanks

to them, the electromagnetic field has a non-negligible

intensity within the tri-layer stack, thus allowing achieving

high accuracy in the determination of model parameters

(silver layer thickness, damping factor) used to interpret

the measured optical data. In our study, the refraction

index and absorption coefficients of silver layers have been

measured over a wide spectral range. Moreover, the variation

of the optical indices with the silver layer thickness has

been measured, analysed and discussed. The validation of

the method using independent measurements of emissivity

and resistivity has confirmed the clear interest of combining

standard spectroscopic ellipsometry with TIRE because of

their non-destructive nature, and sensitivity beyond other

standard optical techniques. The method itself is very general

and can be applied to any multi-stack including a highly

absorbing layer, or, in some cases, even to stacks made of

multiple absorbing layers.

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