# Broad-band ellipsometry study of the anisotropic dielectric response of
YAlO${}_{\text{3}}$

Laurent Bugnon  Christian Bernhard  Premysl Marsik  Fribourg University,
Fribourg, Fribourg 1700, Switzerland

###### Abstract

We present a broad band (THz to UV) ellipsometry study of the anisotropic
dielectric response of the orthorhombic perovskite YAlO${}_{\text{3}}$. The
ellipsometric measurements have been performed on YAlO${}_{\text{3}}$ crystals
with three different surface cuts and for six high symmetry configurations of
the crystal axes with respect to the plane of incidence of the photons. The
obtained data are presented in terms of the Mueller Matrix elements $N$, $C$,
and $S$ and their features are analyzed and discussed with respect to the
anisotropy of the dielectric response tensor. In particular, in the infrared
range we have identified all 25 infrared active phonon modes that have been
predicted from theoretical studies. We also discuss a negative refraction
effect that naturally occurs in the vicinity of an anisotropic longitudinal-
optical phonon. Moreover, we have determined the temperature dependence of the
phonon parameters between 10 and $330\text{\,}\mathrm{K}$. The dielectric
response above the phonon range, from about $0.16.5\text{\,}\mathrm{eV}$, is
shown to be featureless and characteristic of an insulator with a large band
gap above $6.5\text{\,}\mathrm{eV}$ and is well described by anisotropic
Cauchy model.

##  I Introduction

Yttrium Aluminate (YAlO${}_{\text{3}}$, also known as YAP – Yttrium Aluminum
Perovskite) can be readily grown as large high quality single crystals. It is
transparent in the visible range with a moderate refractive index (of about
1.93) and a small birefringence. The phonon spectrum in the far-infrared range
is fairly rich and exhibits a clear anisotropy. YAlO${}_{\text{3}}$ is also a
hard material (Mohs hardness 8.6) with a rather high thermal conductivity. For
some of those qualities, it has been proposed as a material for hosting lasing
dopants [1, 2, 3], as alternative for the well-known yttrium-aluminum-garnet
Y${}_{\text{3}}$Al${}_{\text{5}}$O${}_{\text{12}}$ (YAG).

YAlO${}_{\text{3}}$ is also frequently used as a substrate for the epitaxial
growth of thin films from unconventional oxides. In particular, the lattice
parameters of YAlO${}_{\text{3}}$ are well matched with those of several
interesting perovskite manganites and their anisotropy can be utilized for the
tailored growth of strain-engineered samples [4, 5]. The aim of the present
work is to lay the ground for the optical spectroscopy of such thin films and
heterostructures which require a detailed a-priori knowledge of the
anisotropic response of the YAlO${}_{\text{3}}$ substrate.

The Raman response of YAlO${}_{\text{3}}$ has been already studied in great
detail [6, 7, 8, 9, 10]. Theoretical studies of the IR and Raman phonon
spectra of YAlO${}_{\text{3}}$ have also been reported [11, 7, 12, 10].
However, the corresponding detailed optical studies of the IR dielectric
tensor and the anisotropy of the phonon response are still missing.

Infrared spectroscopic ellipsometry has proven to be an excellent technique
for studies of the phonon response of ionic crystals [13], especially of those
with a strong optical anisotropy. For isotropic bulk materials, ellipsometry
allows one to directly measure the complex dielectric function. Unlike IR
reflectivity measurements which probe only the amplitude of the reflected
signal (but not the phase), it does not require a Kramers-Kronig
transformation and the related extrapolation of the data to zero and infinite
frequency.

For anisotropic crystals, the interpretation of the ellipsometry spectra is
more complicated since the probing light interacts with the sample under
general angle of incidence and polarization state. In general, the measured
spectra can be affected by all the elements of the dielectric tensor. In such
cases, several ellipsometry measurements must be performed on different
surfaces and for various orientations of the crystal axes with respect to the
plane of incidence and the combined data need to be analyzed with an
anisotropic model that allows for finite diagonal and off-diagonal elements of
the dielectric tensor. For materials with tetragonal or orthorhombic symmetry,
this can be done on high symmetry cuts [14, 15, 16, 17] or with a sequence of
different orientations of single low-symmetry cut crystal [18].

In the following, we present a broad-band (THz to UV) ellipsometry study of
the dielectric tensor of orthorhombic YAlO${}_{\text{3}}$. The measurements
are performed on crystals with high symmetry surface cuts and for high
symmetry orientations of the crystal axes with respect to the plane of
incidence of the photon beam. We present a detailed analysis of ellipsometric
data of all 6 high symmetry configurations and document the temperature
dependence of the YAlO${}_{\text{3}}$ phonon parameters.

Figure 1:  a) Sketch of the distorted perovskite structure of
YAlO${}_{\text{3}}$ with the unit cell indicated by the red lines and the
atomic position adapted from [7]. b) Sketch of a YAlO${}_{\text{3}}$ crystal
(light blue) with a wedge angle, $\alpha$, that is mounted with silver paint
(gray) on a copper sample holder (orange). Also shown is the coordinate system
of the ellipsometer with the plane of incidence of the photon beam defined by
the $x$\- and $z$-axes and the incident and reflected beam at the angle
$\varphi$ with respect to the surface normal. The beam reflected from the
backside is deviated by the angle $\delta$. The side of the sample is covered
by aluminium tape (violet) to further reduce scattered and unwanted light. c)
Wedge angle along the plane of incidence that is required for a deviation of
$\delta=$5\text{\,}\mathrm{\SIUnitSymbolDegree}$$ (red lines) or for a total
internal reflection (black lines) calculated as a function of the refractive
index and the incidence angle $\varphi$ ranging from
$60\text{\,}\mathrm{\SIUnitSymbolDegree}$ (thick lines) to
$80\text{\,}\mathrm{\SIUnitSymbolDegree}$ (indicated by the arrows). d) Same
as (c) but with the crystal rotated by
$90\text{\,}\mathrm{\SIUnitSymbolDegree}$, i.e. with the wedge perpendicular
to the plane of incidence. Horizontal dashed lines mark a wedge angle of
$2.3\text{\,}\mathrm{\SIUnitSymbolDegree}$.

##  II Samples and Experiment Setups

YAlO${}_{\text{3}}$ crystalizes in a distorted perovskite structure with
orthorhombic symmetry as displayed in Fig. 1a. Using the Pbnm notation, the
orthorhombic lattice parameters are
$a=$5.18\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$,
$b=$5.33\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ and
$c=$7.375\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ [19, 20]. Throughout the
manuscript, we will use the notation $a$-, $b$-, and $c$-axis for the crystal
axes [100], [010], [001], respectively, and $a$-cut, $b$-cut and $c$-cut for
the (100), (010), (001) surface cuts, respectively. The coordinate system $x$,
$y$, $z$ will be used to describe the ellipsometry setup and the orientation
of the crystal surface with respect to the plane of incidence of the incoming
and reflected phonon beam, as shown in Fig. 1b.

The ellipsometry experiments have been performed on a set of
YAlO${}_{\text{3}}$ crystals with high symmetry surface cuts along (100),
(010), and (001) that have been purchased from a commercial vendor
(SurfaceNet). For the FIR to UV ranges we used crystals with dimensions of
$5\times 5\times$0.5\text{\,}{\mathrm{mm}}^{3}$$, in the THz range we used two
additional, larger crystals with $10\times
10\times$0.5\text{\,}{\mathrm{mm}}^{3}$$.

The $5\times 5\times$0.5\text{\,}{\mathrm{mm}}^{3}$$ samples have been wedged
by mechanical polishing the backside surface such that it is inclined by an
angle of $\alpha=$2.3\text{\,}\mathrm{\SIUnitSymbolDegree}$$ with respect to
front surface (see Fig. 1b). The wedge has been orientated for the $a$-cut
sample in the $ab$-plane; for the $b$-cut sample in $bc$-plane; and for the
$c$-cut sample in the $ca$-plane. This wedging causes a deviation between the
primary beam that is reflected from the front surface and the one reflected
from the back surface by an angle of $\delta$ (indicated in Fig. 1b). This
ensures that the light reflected from the back surface does not reach the
detector, in the frequency ranges for which YAlO${}_{\text{3}}$ is fully
transparent. Moreover, for a large enough angle of incidence of the photons,
$\varphi$, the beam reflected from the backside cannot even exit the sample,
due to the total internal reflection at the front surface. Fig. 1c shows the
critical wedge angle for this total internal reflection effect (black lines)
and for a deviation of the beam by
$\delta=$5\text{\,}\mathrm{\SIUnitSymbolDegree}$$ (red lines). The latter
value corresponds to the maximal acceptance angle of the focusing mirrors of
the infrared ellipsometer.

Note that the wedging also helps to avoid unwanted backside reflection if the
sample is rotated by $90\text{\,}\mathrm{\SIUnitSymbolDegree}$ around the
$z$-axis – which is the case when measuring with the plane of incidence along
the other high symmetry axis of the crystal. As shown in Fig. 1d, a much
larger wedging angle is required here to completely suppress the backside-
reflected beam due to the total internal reflection. This is because the wedge
angle lies in a plane perpendicular to the plane of incidence and therefore
does not add up to the incidence angle. Nevertheless, as long as the
refractive index is large enough, i.e. for $n\geq 1.5$, the wedge angle of
$2.3\text{\,}\mathrm{\SIUnitSymbolDegree}$ leads to deviation of
$\delta>$5\text{\,}\mathrm{\SIUnitSymbolDegree}$$ out of the plane of
incidence that ensures that the beam from the backside does not reach the
detector.

The ellipsometry measurements in the infrared range from about
$405000\text{\,}{\mathrm{cm}}^{-1}$ have been performed with a home-built
setup that is attached to a Bruker 70v FTIR spectrometer and has the infrared
beam-path in a rough vacuum of about ${10}^{-1}\text{\,}\mathrm{mbar}$. A more
detailed description of the ellipsometer can be found in [21]. It is equipped
with a He-flow cryostat (CryoVac) for measurements at temperatures from
$10350\text{\,}\mathrm{K}$.

In the far-infrared range from $40700\text{\,}{\mathrm{cm}}^{-1}$, we used a
Hg arc lamp, a solid Si beam-splitter, and a $1.2\text{\,}\mathrm{K}$
bolometer (Infrared Laboratories) as detector. The ellipsometric measurements
have been performed in rotating analyser mode (RAE) with a fixed polarizer (P)
in front of the sample (S) and a rotating analyser (A${}_{\text{R}}$) after
the sample (P-S-A${}_{\text{R}}$ configuration). The polarizers made from a
metal wire grid on polyethylene substrate have been purchased from a
commercial vendor (Tydex). For some measurements, we also used an optional,
static compensator (C) that is based on the total internal reflection in a
prism of made from undoped Si (P-C-S-A${}_{\text{R}}$ configuration). Since
the raw RAE data can be affected by imperfections of the polarizers, and a
polarization sensitivity of the detector, the measured data have been treated
with a correction algorithm similar to that described in [22]. The details
will be discussed elsewhere, together with the calibration procedure for the
offsets of the polarizers and the retarder.

In the mid-infrared range, from about $5005000\text{\,}{\mathrm{cm}}^{-1}$, we
used a Globar source, a Germanium coated KBr beam-splitter, and a
$4.2\text{\,}\mathrm{K}$ bolometer (Infrared Laboratories) as detector. For
the latter, the Winston cone has been replaced with a cold focusing optics as
to reduce the polarization sensitivity. The polarizer in front of the sample
consisted of a high-quality free standing gold wire grid (InfraSpecs). The
polarization state after the reflection from the sample has been analysed with
a rotating compensator (C${}_{\text{R}}$) followed by a static analyser
(P-S-C${}_{\text{R}}$-A configuration). The rotating compensator is based on
the internal reflection in a ZnSe prism that is mounted together with three
gold mirrors in a four-bounce configuration that can be rotated without
changing the beam path to the detector, similar as in [23]. The static
analyzer (typically at A=0) consisted of a tandem of KRS5 based grid
polarizers (Specac).

All the far- and mid-infrared data have been collected with a resolution of
$2\text{\,}{\mathrm{cm}}^{-1}$ and at an incidence angle of
$\varphi=$75\text{\,}\mathrm{\SIUnitSymbolDegree}$$. The mid-infrared RCE and
far-infrared RAE spectra have shown a very good overlap such that we could
simply merge the data, either at $585\text{\,}{\mathrm{cm}}^{-1}$ or at
$540\text{\,}{\mathrm{cm}}^{-1}$. The upper frequency limit of the far-
infrared data was determined by the multi-phonon absorption band of Si (beam-
splitter and the optional static compensator) between
600-$620\text{\,}{\mathrm{cm}}^{-1}$. The lower limit of the MIR data was
determined by the phonon absorption in the ZnSe prism which sets in below
$520\text{\,}{\mathrm{cm}}^{-1}$.

The high energy spectra from $0.56.5\text{\,}\mathrm{eV}$
($403352\,423\text{\,}{\mathrm{cm}}^{-1}$), spanning the near-infrared,
visible, and UV ranges, have been measured with an IR-extended J. A. Woollam
VASE instrument, at four angles of incidence (55, 65, 75 and
$85\text{\,}\mathrm{\SIUnitSymbolDegree}$), with a spectral resolution of
$0.05\text{\,}\mathrm{eV}$, in an ambient atmosphere.

The low energy spectra in the THz range from $0.12.5\text{\,}\mathrm{THz}$
($383\text{\,}{\mathrm{cm}}^{-1}$) have been obtained with a home-built time-
domain THz ellipsometer, as described in [24]. This instrument has been
operated in a rotating-analyzer (RAE) mode at an incidence angle of
$\varphi=$75\text{\,}\mathrm{\SIUnitSymbolDegree}$$. The phase sensitive time-
domain detection technique enables one to determine the ellipsometric angles
$\Psi$ and $\Delta$, the latter in the full
$360\text{\,}\mathrm{\SIUnitSymbolDegree}$ range without the need of
additional compensator. The time-resolved detection scheme also allows one to
distinguish between the signals that arise from the reflections on the front
surface and the back-side that are delayed in time. Accordingly, the
contribution of the latter back-side reflection can be removed by cutting the
data above a certain time delay. We have used a time window of
$15.5\text{\,}\mathrm{ps}$ which results in a spectral resolution of about
$1.5\text{\,}{\mathrm{cm}}^{-1}$. To minimize artefacts that may arise from
diffraction effects that become increasingly important towards longer
wavelength [25], we have used the larger 10x10x$0.5\text{\,}{\mathrm{mm}}^{3}$
crystals for the THz ellipsometry measurements.

Corresponding broad-band (THz to UV) spectroscopic ellipsometry experiments
using the above described combination of ellipsometers have been previously
reported for various thin films on isotropic substrates [26, 24, 27, 28] and
for anisotropic samples [29, 30].

##  III Data analysis

The orthorhombic unit cell of YAlO${}_{\text{3}}$ (see Fig. 1a) contains 4
formula units with 20 atoms and in total 60 degrees of freedom. From all the
possible displacement modes, besides the acoustic, Raman-active or silent
ones, 25 modes are reported to be infrared-active [11, 12]. The orthorhombic
symmetry yields a diagonal dielectric tensor $\varepsilon$ with the principal
axes oriented along the $a$, $b$ and $c$ lattice directions and 9 modes in the
$\varepsilon_{a}$ component, 9 modes in $\varepsilon_{b}$, and 7 modes in
$\varepsilon_{c}$.

The samples were measured in high symmetry configurations for which the $x$,
$y$, and $z$ axes, that define the plane of incidence of the ellipsometer and
the measured surface of the YAlO${}_{\text{3}}$ crystal (see Fig. 1b),
coincide with the $a$, $b$ and $c$ axes directions of the YAlO${}_{\text{3}}$
crystals for any of the 6 possible permutations. For these high symmetry
configurations the birefringence does not give rise to a mixing of the $p$\-
and $s$\- components. In the following, we describe the reflection and
refraction in the framework of the $x$\- and $z$-components of the
wavevectors. For the sake of brevity, we use the geometrical components,
$\boldsymbol{\kappa}$, that are related to the physical wavevector,
$\boldsymbol{k}$, according to $\boldsymbol{k}=\boldsymbol{\kappa}\omega/c$.
The incident wavevector (in vacuum) is denoted as
$\boldsymbol{\kappa}_{inc}=(\xi,0,q)$, with $\xi=\sin\varphi$ and
$q=\cos\varphi$. The $x$-component $\xi$ is conserved for the reflected as
well as the refracted beams in $p$\- and $s$-polarization. Accordingly, the
reflection problem can be treated as stationary in $x$ and independent of the
$y$-component [31]. For the $z$-components of the two refracted beams denoted
as $\kappa_{p}$ and $\kappa_{s}$, it follows that:

$\kappa_{p}=\sqrt{\varepsilon_{x}\left(1-\frac{\xi^{2}}{\varepsilon_{z}}\right)},\quad\kappa_{s}=\sqrt{\varepsilon_{y}-\xi^{2}},$
(1)

where $\varepsilon_{x}$, $\varepsilon_{y}$ and $\varepsilon_{z}$ are the
elements of the diagonal dielectric tensor, i.e. one of the permutations of
$\varepsilon_{a}$, $\varepsilon_{b}$ and $\varepsilon_{c}$. The Jones matrix
thus acquires a diagonal shape, with the following Fresnel coefficients
$r_{p}$ and $r_{s}$ on the diagonal:

$r_{p}=\frac{\kappa_{p}-\varepsilon_{x}q}{\kappa_{p}+\varepsilon_{x}q},\quad
r_{s}=\frac{q-\kappa_{s}}{q+\kappa_{s}}.$ (2)

Note that we employ Fresnel’s convention for $r_{p}$ [32].

The ellipsometric response is traditionally written in terms of the complex
ratio $\rho=r_{p}/r_{s}=\tan\Psi\exp(i\Delta)$, with the ellipsometric angles
$\Psi$ and $\Delta$. In RAE or RCE mode, the directly accessible quantities
are the components of the normalized Muller matrix, which in the high symmetry
configuration with diagonal Jones matrices acquires the following block-
diagonal form,

$\hat{M}=\begin{bmatrix}1&N&0&0\\\ N&1&0&0\\\ 0&0&C&S\\\
0&0&-S&C\end{bmatrix},$ (3)

The elements of this so-called NCS-form of the Mueller matrix are related to
the ellipsometric angles by $N=-\cos 2\Psi$, $C=\sin 2\Psi\cos\Delta$, and
$S=\sin 2\Psi\sin\Delta$. Effects from depolarization are not considered here.

In the simplest case of RAE in P-S-A${}_{\text{R}}$ configuration, with the
polarizer set to $P=$45\text{\,}\mathrm{\SIUnitSymbolDegree}$$, the modulation
of the intensity at the detector, as a function of the analyzer azimuth A, is
given by $I(A)=I_{0}(1+N\cos 2A+C\sin 2A)$, i.e. the $N$ and $C$ components
are measured directly. In P-C-S-A${}_{\text{R}}$ configuration, with an ideal
$\lambda/4$-retarder as a static compensator and
$P=$45\text{\,}\mathrm{\SIUnitSymbolDegree}$$, the intensity at the detector
follows $I(A)=I_{0}(1+N\cos 2A+S\sin 2A)$, hence the remaining element $S$ of
the sample NCS Mueller matrix is determined. The far-infrared data presented
below have been obtained using this approach.

With a rotating compensator (RCE), in P-S-C${}_{\text{R}}$-A configuration, as
it has been used in the mid-infrared range, the modulation at the detector is

$\displaystyle I(B)=$ $\displaystyle I_{0}(1+\frac{N}{2+N}\cos
4B+\frac{C}{2+N}\sin 4B$ $\displaystyle-\frac{S}{2+N}\sin 2B),$

where $B$ is the azimuth of the rotating compensator. Therefore, RCE provides
access to all three $N$, $C$, $S$ elements, with
$P=$45\text{\,}\mathrm{\SIUnitSymbolDegree}$$ and $A=0$, see e.g. [33].

The detection of amplitude and phase by the time-domain THz spectroscopic
technique allows measuring directly the complex ratio $\rho=\tan\Psi\exp
i\Delta$, [34, 35], or, in a general sense, the components of the complex
Jones matrix. To facilitate the comparison with the infrared data, the
measured spectra in the THz range are also shown in terms of the Mueller
matrix elements $N$, $C$, $S$.

The three diagonal elements of dielectric tensor of YAlO${}_{\text{3}}$,
$\varepsilon_{a}$, $\varepsilon_{b}$ and $\varepsilon_{c}$, are calculated in
the infrared range with the following harmonic-oscillator approximation (HOA):

$\varepsilon_{j}(\omega)=\varepsilon_{\infty
j}+\sum^{N_{j}}_{i=1}\frac{A_{ji}\omega^{2}_{\mathrm{TO}ji}}{\omega^{2}_{\mathrm{TO}ji}-\omega^{2}-i\omega\gamma_{ji}},$
(4)

where $j=a,b,c$; $\varepsilon_{\infty j}$ is the high-frequency dielectric
constant, $N_{j}=9$ for $j=a,b$ and $N_{j}=7$ for $j=c$. The parameters
$A_{ji}$, $\omega_{\text{TO}ji}$, $\gamma_{ji}$ account for the strength
$A=\Delta\varepsilon(\omega=0)$, the transversal optical resonance frequency
$\omega_{\text{TO}}$, and the broadening $\gamma$ of the individual
oscillators. The corresponding longitudinal optical phonon frequencies,
$\omega_{\text{LO}ji}$, coincide with the poles of $1/\varepsilon_{j}(\omega)$
in the limit $\gamma_{ji}\rightarrow 0$. From this dielectric tensor, the
z-components of $\kappa_{p}$ and $\kappa_{s}$ (Eq.(1)) have been calculated,
for a given angle of incidence, $\varphi$, and the particular high symmetry
orientation. Subsequently, the Fresnel coefficients (Eq.(2)), their ratio
$\rho$, and finally the Mueller matrix elements have been derived and fitted
to the experimental NCS data, by varying the parameters of the phonon
oscillator parameters in Eq.(4).

As a curiosity, we also discuss how the negative refraction effects, which can
occur in the vicinity of the longitudinal optical phonon modes
($\omega_{\text{LO}}$) of such anisotropic crystals, show up in the
ellipsometric spectra. According to the original work of Veselago [36], a
negative refractive index can occur if both the real parts of the permittivity
$\varepsilon$ and the permeability $\mu$ are negative. Of the two solutions of
the complex square-root, $N=n+ik=\sqrt{\varepsilon\mu}$, the physically
relevant one requires $k>0$ and thus $n<0$, such that the plane waves
propagate in a direction that is opposite to the one of the energy flow (or
the Pointing vector). This is equivalent to the statement that whenever the
product of generally complex values of $\varepsilon$ and $\mu$ acquires a
negative imaginary part, the physically correct solution of the square root is
the one with positive imaginary part ($k>0$), eventually leading to $n<0$
[37].

A similar situation can appear in the expression of the p-polarized reflection
from an anisotropic material. The expression for $\kappa_{p}$ in Eq. (1)
reveals that negative refraction can occur if the real parts of the dielectric
function fulfil the condition that $\varepsilon_{x}>0$ and $\varepsilon_{z}<0$
[38]. This occurs naturally in an anisotropic material in the vicinity of
$\omega_{\text{LO}}$. However, this is complicated by the fact that
$\varepsilon_{x}>0$, $\varepsilon_{z}<0$ will lead to negative refraction in
terms of Poynting vector $\boldsymbol{S}$ , but positive (forward-wave)
solution for $\kappa_{p}$. Vice versa, negative $\kappa_{p}$, characterized by
backward wave propagation (negative refractive index), will lead to “positive”
refraction in terms of energy flow, when the condition is reversed,
$\varepsilon_{x}<0$, $\varepsilon_{z}>0$ [39].

Indeed, such a negative refraction effect has been observed for several
crystals, e.g. Quartz [40] or MgF${}_{\text{2}}$ [41] and it appears also in
the anisotropic effective response of dielectric heterostructures [42] and
their ellipsometric spectra [43]. We will discuss the corresponding features
in the ellipsometric data of the orthorhombic YAlO${}_{\text{3}}$.

##  IV Results and discussion

Figs. 2a, 3a and 4a show the infrared ellipsometry spectra at room temperature
measured on the three different high symmetry surfaces for 6 high symmetry
configurations. Fig. 2a shows the spectra for the $b$-cut and $c$-cut surfaces
with the $a$-axis parallel to the $x$-axis (plane of incidence) of the
ellipsometer ($abc$, $acb\leftrightarrow xyz$). Fig. 3a displays the
corresponding spectra with the $b$-axis parallel to $x$ for the $a$-cut and
$c$-cut surfaces ($bac$, $bca\leftrightarrow xyz$) and Fig. 4a with the
$c$-axis axis parallel to $x$ for $a$-cut and $b$-cut surfaces ($cab$,
$cba\leftrightarrow xyz$).

###  IV.1 Pseudo-isotropic response

In addition to the measured spectra (coloured symbols), Figs. 2a, 3a and 4a
also show the best fits with the anisotropic model (coloured lines) and, for
comparison, the simulated ellipsometric spectra (using the fit parameters
derived with the anisotropic model) for the fictive scenario of an isotropic
response for $\varepsilon_{a}$, $\varepsilon_{b}$, and $\varepsilon_{c}$,
respectively (grey lines). These isotropic spectra serve to highlight spectral
ranges where the anisotropy has a clear effect on the experimental data. They
make it apparent that in large parts the ellipsometric data are dominated by
the dielectric tensor component along the axis $x$ direction. This
circumstance is also evident from Eq. (2), where $r_{p}$ depends explicitly on
$\varepsilon_{x}$, due to the matching condition of the in-plane components of
the electromagnetic field on the interface. Moreover, in Eq. (1) the value of
$\varepsilon_{z}$ appears only in the denominator of $r_{p}$ such that
$\kappa_{p}\approx\sqrt{\varepsilon_{x}}$ if $\varepsilon_{z}\gg 1$.

Figure 2:  a) Far-infrared ellipsometry spectra of the Mueller matrix elements
$N$, $C$, $S$ of YAlO${}_{\text{3}}$ at $300\text{\,}\mathrm{K}$ with angle of
incidence $\varphi=$75\text{\,}\mathrm{\SIUnitSymbolDegree}$$ measured on
$b$-cut (green points) and $c$-cut (blue points) surfaces with the $a$-axis
parallel to the plane of incidence. Sketches of the measurement configurations
are shown in the inset of the middle panel. Also shown are the best fits with
the anisotropic model (blue and green lines) and, for comparison, the spectra
for a fictive sample with an isotropic response that have been calculated with
the parameters of $\varepsilon_{a}$ (gray lines). Marked by arrows on the
bottom panel are the positions of $\omega_{\text{TO}}$ (black arrows) and
$\omega_{\text{LO}}$ (gray arrows) of $\varepsilon_{a}$. Dashed vertical lines
highlight some of the pronounced anisotropy features in the vicinity of
$\omega_{\text{LO}}$ in the $b$-axis response (green dashed line) and the
$c$-axis response (blue dashed line). b) Calculated geometric $z$-component of
the refracted wavevector for p-polarized light, Re{$\kappa_{p}$} (thick lines)
and Im{$\kappa_{p}$} (thin lines), with the colours matching the configuration
of the experiment. c) Calculated absolute values of the Fresnel coefficients
$|r_{p}|$ (thick lines) and $|r_{s}|$ (thin lines). d) Calculated argument of
the complex Fresnel coefficients $\arg(r_{p})=\Delta_{p}$ (thick lines) and
$\arg(r_{s})=\Delta_{s}$ (thin lines).

Nevertheless, the measured far infrared spectra also reveal some clear
deviations from the isotropic model that occur in the vicinity of the LO
frequencies of the response functions in the $y$ and $z$ directions (marked by
vertical dashed lines). These anisotropy features will be discussed further
below. At first, we discuss the pseudo-isotropic behaviour of the NCS spectra.
The frequencies of the TO and LO phonon modes in the spectra of
$\varepsilon_{a}$, $\varepsilon_{b}$, and $\varepsilon_{c}$ (as fitted with
the anisotropic model) are marked by black and grey arrows in the bottom
panels of Figs. 2a, 3a and 4a, respectively. Note that our definition of
$N=-\cos 2\Psi$ differs in sign with respect to the convention used in most
textbooks ($N=\cos 2\Psi$). This sign change has the advantage that i) the
value of $N$ equals the Mueller matrix element $N=m_{21}=m_{12}$, ii) the
shape of the $N$-spectra resembles that of $\Psi$, or $\tan\Psi$ (which is
widely used for presenting ellipsometric data) and iii) their shape is also
reminiscent of the normal incidence reflectivity, $R$. Accordingly, the so-
called Reststrahlen bands are clearly visible in the spectra of $N$ as maxima
that span the region of the TO-LO splitting of the phonon modes, at which
$N\rightarrow 0$ (or
$\Psi\rightarrow$45\text{\,}\mathrm{\SIUnitSymbolDegree}$$ and $R\rightarrow
1$). The Mueller matrix element $S$ is complementary to $N$, since its value
approaches zero ($S\rightarrow 0$) between the Reststrahlen bands, that is,
for range between LO and the following TO frequency. The Mueller matrix
element $C$ has a characteristic zigzag shape with maxima at the TO and minima
at the LO frequencies. From there it can be inferred that ellipsometric data
in NCS form are equally well represented by the dielectric function
$\varepsilon$ as they are by its inverse $1/\varepsilon$, suggesting the use
of the factorized dielectric function, as introduced by Gervais and Piriou
[44] and discussed in the context of ellipsometry by Schubert et al. [15] and
references therein. However, we do not see any indication of asymmetric phonon
line-shapes (different broadening of the LO and TO modes) and we parameterized
the dielectric functions with harmonic oscillator approximation as in Eq.(4).
Indications of a slightly enhanced broadening have only been observed at the
highest LO edge and will be discussed further below.

Figure 3:  a) Far-infrared response along the $b$-axis of YAlO${}_{\text{3}}$
at $300\text{\,}\mathrm{K}$ with
$\varphi=$75\text{\,}\mathrm{\SIUnitSymbolDegree}$$ measured on $a$-cut (red
points) and $c$-cut (blue points) surfaces (see the sketches in the inset of
the middle panel) and expressed in terms of the Mueller matrix elements $N$,
$C$, $S$. Also shown are the best fits with the anisotropic model (red and
blue lines) as well as the simulated response of a fictive isotropic sample
with the $\varepsilon_{b}$ parameters as obtained from the anisotropic model
(gray lines). Arrows on the bottom panel mark the positions of
$\omega_{\text{TO}}$ (black arrows) and $\omega_{\text{LO}}$ (gray arrows) of
$\varepsilon_{b}$. Dashed vertical lines highlight some of the pronounced
anisotropy features in the vicinity of $\omega_{\text{LO}}$ in the $a$-axis
response (red dashed line) and the $c$-axis response (blue dashed line). b)
Calculated geometric $z$-component of the refracted wavevector for
$p$-polarized light, Re{$\kappa_{p}$} (thick lines) and Im{$\kappa_{p}$} (thin
lines), with colours matching the configuration of the experiment. c)
Calculated absolute values of the Fresnel coefficients $|r_{p}|$ (thick lines)
and $|r_{s}|$ (thin lines). d) Calculated argument of the complex Fresnel
coefficients $\arg(r_{p})=\Delta_{p}$ (thick lines) and
$\arg(r_{s})=\Delta_{s}$ (thin lines).

Towards very low frequencies in the THz regime, the measured spectra start to
exhibit significant deviations from those of the model curves that are caused
by diffraction effects [25]. In particular, the spectra of $S$ exhibit a steep
downturn below approximately $100\text{\,}{\mathrm{cm}}^{-1}$. The spectra of
$N$ and $C$ are less strongly affected by the diffraction and show clear
deviations only below $20\text{\,}{\mathrm{cm}}^{-1}$. Besides these
diffraction artefacts (which prevail below $100\text{\,}{\mathrm{cm}}^{-1}$),
the $a$-cut $b$-axis data of the element $S$, in Fig. 3a, are slightly
deviating from the expected model values below
$250\text{\,}{\mathrm{cm}}^{-1}$. Likely, this experimental artefact is caused
by a backside reflection or light that is otherwise scattered (but still
reaches the detector). Otherwise, the model curves show excellent agreement
with the measured data and thus allows us to identify and interpret the
observed features.

Figure 4:  a) Far-infrared response along the c-axis of YAlO${}_{\text{3}}$ at
$300\text{\,}\mathrm{K}$ with
$\varphi=$75\text{\,}\mathrm{\SIUnitSymbolDegree}$$ as measured on $a$-cut
(red points) and $b$-cut (green points) surfaces and shown by the Mueller
matrix elements $N$, $C$, $S$. Also displayed are the best fits with the
anisotropic model (red and green lines) and the response of a fictive
isotropic sample simulated with the $\varepsilon_{c}$ parameters (gray lines).
The inset of the middle panel shows a comparison of the modelled spectra of
the real part of dielectric functions $\varepsilon_{a}$, $\varepsilon_{b}$,
and $\varepsilon_{c}$. Arrows on the bottom panel mark the positions of
$\omega_{\text{TO}}$ (black arrows) and $\omega_{\text{LO}}$ (gray arrows) of
$\varepsilon_{c}$. Dashed vertical lines highlight some of the pronounced
anisotropy features in the vicinity of $\omega_{\text{LO}}$ in the $a$-axis
response (red dashed line) and the $b$-axis response (green dashed line). b)
Calculated geometric $z$-component of the refracted wavevector for
$p$-polarized light, Re{$\kappa_{p}$} (thick lines) and Im{$\kappa_{p}$} (thin
lines), with colours matching the configuration of the experiment. c)
Calculated absolute values of the Fresnel coefficients $|r_{p}|$ (thick lines)
and $|r_{s}|$ (thin lines). d) Calculated argument of the complex Fresnel
coefficients $\arg(r_{p})=\Delta_{p}$ (thick lines) and
$\arg(r_{s})=\Delta_{s}$ (thin lines).

###  IV.2 Anisotropy features

With the help of the above-described model, we can now discuss the signatures
of the anisotropy in the ellipsometry spectra of Fig. 2 to 4. These are marked
by the vertical dashed lines across the NCS panels in Figs. 2a, 3a, and 4a.
They highlight that the anisotropy effects generally occur in the vicinity of
the LO modes of those crystal axes that are oriented along the $y$\- and
$z$-direction of the coordinate system of the ellipsometer (see Fig. 1b). Note
that only those LO modes have been marked that lead to the strongest
anisotropy effects in the data.

Up to approximately $600\text{\,}{\mathrm{cm}}^{-1}$ the spectra follow mostly
the isotropic model, the anisotropy only gives to some rather weak and
localized features near the LO frequencies. This changes above
$600\text{\,}{\mathrm{cm}}^{-1}$ where the spectra reveal strong anisotropy
features, in particular, in the ranges from
$600650\text{\,}{\mathrm{cm}}^{-1}$ and from
$800900\text{\,}{\mathrm{cm}}^{-1}$. The response in the range from
$600650\text{\,}{\mathrm{cm}}^{-1}$ is somewhat complicated by the overlap of
the TO and LO modes from all three directions, with relatively strong LO modes
but rather broad and thus weak TO modes that are superimposed on a weak
dielectric background. Notably, the strong anisotropy features around
$850\text{\,}{\mathrm{cm}}^{-1}$ appear well above the highest LO frequencies
that are similar for all three axes and occur at
750-$770\text{\,}{\mathrm{cm}}^{-1}$ (see Fig. 4e, inset in the middle panel
of Fig. 4a). Such features, above the highest LO frequency, are commonly
observed in the ellipsometric spectra of orthorhombic and tetragonal systems
[15, 16]. They occur at the end of the Reststrahlen band, which in normal-
incidence reflectivity experiment coincides with $\omega_{\text{LO}}$ (where
$\varepsilon\leftarrow 0$), but for an oblique angle of incidence, $\varphi$,
is shifted to the frequency at which $\varepsilon\sim\sin^{2}\varphi$, i. e.
to the point where the $z$-component of the wavevector vanishes (see Eq. (1)).
In the literature this frequency is commonly denoted as
$\omega_{\text{LO}^{*}}$. The shift of the Reststrahlen band edge with the
angle of incidence occurs also for isotropic materials, but without the
anisotropy effects.

The strong features that occur around the $\omega_{\text{LO}^{*}}$ point in
the response of anisotropic samples arise from the difference between the
$\omega_{\text{LO}^{*}}$ frequencies of the dielectric tensor components that
are aligned with the $x$, $y$, and $z$ directions. Specifically, the strong
peaks and dips appear around the frequencies at which the dielectric functions
$\varepsilon_{a}$, $\varepsilon_{b}$ or $\varepsilon_{c}$ are crossing the
value of $\sin^{2}\varphi$ ($\equiv\xi^{2}$ in Eq. (1)). For the presented
data, which have been obtained for an angle of incidence
$\varphi=$75\text{\,}\mathrm{\SIUnitSymbolDegree}$$, this corresponds to the
condition $\sin^{2}\varphi=0.933$ that has been marked with a horizontal line
in Fig. 4e and yields estimates of
$\omega_{a,\text{LO}^{*}}\approx$834\text{\,}{\mathrm{cm}}^{-1}$$,
$\omega_{b,\text{LO}^{*}}\approx$829\text{\,}{\mathrm{cm}}^{-1}$$, and
$\omega_{c,\text{LO}^{*}}\approx$841\text{\,}{\mathrm{cm}}^{-1}$$. Also marked
are the frequencies at which the dielectric functions reach unity and thus
equal that of the ambient. This matching with the dielectric constant of the
ambient occurs at $842\text{\,}{\mathrm{cm}}^{-1}$,
$836\text{\,}{\mathrm{cm}}^{-1}$ and $848\text{\,}{\mathrm{cm}}^{-1}$ for
$\varepsilon_{a}$, $\varepsilon_{b}$ and $\varepsilon_{c}$, respectively. At
these frequencies, the Fresnel coefficients acquire their extremal values, as
well as their ratio $\rho=r_{p}/r_{s}$, and subsequently $\Psi$, $\Delta$, or
$N$, $C$, $S$. The strong spectral features that originate from the contrast
between $r_{p}$ and $r_{s}$, are governed here by the divergence of the
$\xi^{2}/\varepsilon_{z}$ term in Eq. (1) which strongly impacts $r_{p}$ (but
not $r_{s}$.). This is unlike the pseudo-isotropic parts of the spectra below
$600\text{\,}{\mathrm{cm}}^{-1}$ for which $r_{p}$ is dominated by
$\varepsilon_{x}$. Note that in both cases the $y$-response solely affects
$r_{s}$.

As a result, in the vicinity of $\omega_{\text{LO}^{*}}$ it is the contrast
between $\varepsilon_{z}$ and $\varepsilon_{y}$ that gives rise to the
anisotropy features in the ellipsometry spectra that appear in the same
frequency range but show opposite trends (minima versus maxima) for the pairs
of different configurations ($abc$ versus $acb$ in Fig. 2a, $bac$ versus $bca$
in Fig. 3a and $cab$ versus $cba$ in Fig. 4a. Note that this mutually opposite
behaviour is a general characteristic for all the anisotropy features of the
infrared spectra.

This effect can be understood as follows: Fig. 2b displays the $z$-component
of the geometrical wave-vector of the $p$-polarized light, $\kappa_{p}$, for
the two configurations $abc$ and $acb$, calculated using the values of the
best fit with the anisotropic model. For the moment we skip the feature
occurring at the actual $\omega_{\text{LO}}$, which will be discussed later.
In general, the position of $\omega_{\text{LO}^{*}}$ coincides with the
minimum of the absolute value of $\kappa_{p}$. For the $b$-cut surface
($b\parallel z$) the minimum of $|\kappa_{p}|$ thus occurs at
$\omega_{b,\text{LO}^{*}}$ and for the $c$-cut sample ($c\parallel z$) at
$\omega_{c,\text{LO}^{*}}$. The corresponding $s$-component, $\kappa_{s}$ (not
plotted), has an overall similar shape with a minimum at the
$\omega_{\text{LO}^{*}}$ of the $y$-axis response, but does not contain the
features related to $\omega_{\text{LO}}$. The magnitudes of the corresponding
complex Fresnel coefficients $|r_{p}|$ and $|r_{s}|$ are plotted in Fig. 2c,
and their phases $\Delta_{p}$ and $\Delta_{s}$ on Fig. 2d.

The minima of $|\kappa_{s,p}|$ at $\omega_{\text{LO}^{*}}$ also give rise to
characteristic features in spectra phase shifts, $\Delta_{p}$ and
$\Delta_{s}$. For the $b$-cut surface, $\Delta_{p}$ and $\Delta_{s}$ have
fairly sharp maximum at
$\omega_{b,\text{LO}^{*}}=$829\text{\,}{\mathrm{cm}}^{-1}$$ and
$\omega_{c,\text{LO}^{*}}=$841\text{\,}{\mathrm{cm}}^{-1}$$, respectively. For
the $c$-cut surface, this order is inverted, i.e. the maxima of $\Delta_{p}$
and $\Delta_{s}$ occur at
$\omega_{c,\text{LO}^{*}}=$841\text{\,}{\mathrm{cm}}^{-1}$$ and
$\omega_{b,\text{LO}^{*}}=$829\text{\,}{\mathrm{cm}}^{-1}$$, respectively.
These splitting of the maxima of the phase shifts $\Delta_{p}$ and
$\Delta_{s}$, for which the order is inverted between $b$\- and $c$-cut
surfaces, gives rise to pronounced features (dips and kinks) in the
ellipsometric angle Delta ($\Delta=\Delta_{p}-\Delta_{s}$), and consequently
in the Mueller matrix element $S$ in the lower panel of Fig. 2a. For the
latter, the extrema are slightly shifted, i.e. they occur slightly below
$\omega_{b,\text{LO}^{*}}=$829\text{\,}{\mathrm{cm}}^{-1}$$ and slightly above
$\omega_{c,\text{LO}^{*}}=$841\text{\,}{\mathrm{cm}}^{-1}$$.

For the interpretation of the corresponding feature in $N$, we recall that $N$
is closely related to the ellipsometric angle $\Psi$ ($N=-\cos 2\Psi$), and
$\tan\Psi=|r_{p}|/|r_{s}|$. The extremum in $N$ coincides here with the minima
of $|r_{p}|$ and $|r_{s}|$ (Fig. 2c) which occur at the point at which the
contrast between dielectric constant of the sample and the ambient vanishes.
For $|r_{s}|$ this occurs as the $y$-axis response crosses unity, i.e. for the
$c$-cut surface ($abc$) at $836\text{\,}{\mathrm{cm}}^{-1}$ for
$\varepsilon_{b}$ and for the $b$-cut surface (acb) at
$848\text{\,}{\mathrm{cm}}^{-1}$ for $\varepsilon_{c}$. Note that a finite
absorption broadens these $|r_{s}|$ minima and shifts them to slightly higher
frequencies.

The coefficient $|r_{p}|$ depends in general on both the $x$-axis and $z$-axis
responses (see Eq. 1, 2). However, since in the relevant range both of them
are close to unity, the minimum of $|r_{p}|$ is more sensitive to the $z$-axis
response and appears slightly above the point where the $z$-axis response
crosses unity. Note that this $|r_{p}|$ minimum also represents one of the
solutions for the general Brewster condition [38].

For the sake of completeness, we have also plotted the corresponding
$\kappa_{p}$, $|r_{p}|$ and $|r_{s}|$ spectra for the other sample
orientations and measurement configurations in Fig. 3b-d and 4b-d. The
resulting anisotropy effects in the NCS spectra are analogous to the ones that
we have discussed above for the $abc$ and $acb$ configurations that are
displayed in Fig. 2a. They simply involve different permutations of the $a$-,
$b$\- or $c$-components of the relevant critical frequencies.

The above discussion about the large anisotropy features in the ellipsometric
response in the range from $800900\text{\,}{\mathrm{cm}}^{-1}$ can be
summarized as follows: The features in the Mueller matrix element $N$ (or in
the ellipsometric angle $\Psi$) are related to the anisotropy of the minima in
$|r_{p}|$ and $|r_{s}|$ that occur if the dielectric function of the sample
matches that of the ambient (i.e. $\varepsilon\rightarrow 1$ in case of vacuum
ambient). The corresponding features in the Mueller matrix element $S$ (or in
$\Delta$) depend on the actual anisotropy of $\omega_{\text{LO}^{*}}$
($|\kappa|\rightarrow 0$, onset of total reflection), which at
$\varphi=$75\text{\,}\mathrm{\SIUnitSymbolDegree}$$ yields the condition
$\varepsilon=0.933$. Note that both kind of features get broadened and
slightly shifted by the nonzero imaginary parts of the response functions.
Moreover, since these critical points tend to be rather close in frequency,
the different contributions to the anisotropy effects in NCS are partially
merged, especially if the angle of incidence is large with $\xi^{2}\rightarrow
1$. Measurements at a lower angle of incidence would allow one to better
separate the features appearing at $\omega_{\text{LO}^{*}}$ and at
$\varepsilon\rightarrow 1$. In both cases, however, these anisotropy features
are directly related to the differences of the z-axis and y-axis response
functions ($\varepsilon_{z}$ and $\varepsilon_{y}$). Notably, near
$\omega_{\text{LO}^{*}}$ these features are almost independent of
$\varepsilon_{x}$ – although at lower frequencies, the $x$-axis response
dominates the ellipsometry spectra.

Note that the smaller anisotropy effects that appear near the
$\omega_{\text{LO}}$ frequencies in the range below
$600\text{\,}{\mathrm{cm}}^{-1}$ (marked by the vertical dashed lines in Figs.
2a, 3a, 4a) have the same origin as described above for the
$800900\text{\,}{\mathrm{cm}}^{-1}$ range. The main difference concerns the
separation between $\omega_{\text{LO}}$, $\omega_{\text{LO}^{*}}$ and the
point of $\varepsilon\rightarrow 1$ that becomes much smaller.

###  IV.3 Negative refraction

Another interesting feature concerns an anomaly in the geometrical
$z$-component of the refracted wave-vector of the p-polarized beam,
$\kappa_{p}$, that occurs in the vicinity of the highest LO mode, i.e. near
the zero crossing of the real part of $\varepsilon$ around
$750\text{\,}{\mathrm{cm}}^{-1}$ (see Figs. 2b, 3b and 4b, see also inset Fig.
4e). In the experimental data, the anisotropy of this highest LO mode with
$\omega_{b,\text{LO}}<\omega_{a,\text{LO}}<\omega_{c,\text{LO}}$ gives rise to
a small drop in $N$ (or $\Psi$) and corresponding feature in $C$ and $S$.

Notably, this feature appears well below $\omega_{\text{LO}^{*}}$ and thus
within the Reststrahlen band, for which in case of an isotropic sample only an
evanescent wave (but not a refracted wave) could exist. Such a refracted wave
is however possible for the anisotropic case of YAlO${}_{\text{3}}$ for which
the expression for $\kappa_{p}$ in Eq. (1) has a non-evanescent solution in
the range where $\varepsilon_{x}$ and $\varepsilon_{z}$ have opposite signs.
For the condition $\varepsilon_{x}<0$, $\varepsilon_{z}>0$, there exists a
strongly attenuated, but propagating wave that has an unconventional
refraction. The real part of the $z$-component of the wave vector
$\kappa_{p}$, which determines the direction of the phase propagation, has a
negative value such that it describes a backward travelling wave (the
$x$-component, $\xi$, of the wave vector remains positive). Nevertheless, for
the energy flow the refraction remains positive since the $x$\- and the
$z$-components of the Poynting vector are positive.

For the opposite case of $\varepsilon_{x}>0$ and $\varepsilon_{z}<0$, there is
negative refraction effect: the $x$-component of the Poynting vector becomes
negative, although the real part of $\kappa_{p}$ is positive and the waves are
travelling forward from the interface to the bulk [39]. The Figs. 2b, 3b, 4b
show the $\kappa_{p}$ spectra calculated with the parameters of the best-fit
model. In Fig. 2b, the solution for the $abc$-configuration has a positive
Re{$\kappa_{p}$} around $770\text{\,}{\mathrm{cm}}^{-1}$ – leading to a
forward wave but a negative energy refraction. The $acb$-configuration shows a
negative Re{$\kappa_{p}$} around $750\text{\,}{\mathrm{cm}}^{-1}$ –
corresponding to a backward wave with positive energy flow. In Fig. 3b, the
$bac$\- and $bca$-configurations have positive Re{$\kappa_{p}$} and negative
refraction. Finally, in Fig. 4b, both the $cab$\- and $cba$-configurations
have negative Re{$\kappa_{p}$} and positive refraction in the discussed
spectral range.

The above discussion raises the question whether based on the ellipsometric
data one can tell the difference between the cases of negative and positive
refraction. The answer is that the ellipsometry data do not allow on to
distinguish between these two cases. Indeed, it turns out that the change in
the sign of the real part of $\kappa_{p}$ does not give rise to any
characteristic feature in $r_{p}$ and the resulting Mueller matrix elements
$N$, $C$, $S$ that could serve as indicator. This evident from the comparison
of the real part of the $\kappa_{p}$ curve for the bac configuration in Fig.
3b (with positive values) with that of the cab-configuration in Fig. 4b (with
negative values) and an inspection of the resulting NCS spectra in Figs 3a and
4b which exhibit no characteristic difference in the vicinity of the high
$\omega_{\text{LO}}$.

Figure 5:  Diagonal elements of the dielectric tensor along the three main
crystal axes of orthorhombic YAlO${}_{\text{3}}$. The spectra represent the
best fit with an anisotropic model to the ellipsometry data taken at room
temperature for six different configurations with respect to the surface cut
and the mutual orientation of the crystal axes. Panels a), b), and c) show the
far-infrared spectra of $\varepsilon_{a}$, $\varepsilon_{b}$ and
$\varepsilon_{c}$, respectively. Thin lines represent the real part of the
dielectric function, $\varepsilon_{1}$, thicker lines the imaginary part,
$\varepsilon_{2}$. The positions of the weak phonons are marked with the pin
symbols. Panels d), e), and f) show the extension of the model in the
MIR/NIR/VIS/UV ranges on a logarithmic energy scale. Lines represent an
anisotropic Cauchy model. Symbols show the result of an anisotropic point-by-
point fit.

###  IV.4 Dielectric response along the main crystal axes of
YAlO${}_{\text{3}}$

Figs. 5a, b and c display the far-infrared dielectric response along the main
crystal axes of orthorhombic YAlO${}_{\text{3}}$ that represent the diagonal
elements, $\varepsilon_{a}$, $\varepsilon_{b}$ and $\varepsilon_{c}$, of the
dielectric tensor. They have been obtained from a simultaneously fitting with
an anisotropic model to the room temperature ellipsometric data for six
different configurations of the surface cuts and relative orientations of the
crystal axes. The corresponding data and model curves have been presented in
Figs. 2a, 3a, and 4a. The model contains a sum of Lorentzian oscillators that
describe the response of the infrared active phonons, as given in Eq.(4). With
this model, we identified 9 phonons in the $a$-axis response,
$\varepsilon_{a}$, 9 phonons in the $b$-axis response, $\varepsilon_{b}$, and
7 phonons in the $c$-axis response, $\varepsilon_{c}$, as predicted from
theory [12, 10]. Several of the TO phonons shown in Fig. 5 are rather weak,
their position has therefore been highlighted with the pin symbols.

The positions, $\omega_{\text{TO}}$, broadenings, $\gamma$, and oscillator
strengths, $\Delta\varepsilon$, of the line-shapes were fitting parameters of
the model. The obtained best-fit parameters for the room temperature spectra
are listed in Table 1. The additional fit parameter $\varepsilon(\infty)$
represents the high-frequency dielectric constant. The LO mode frequencies,
$\omega_{\text{LO}}$, have been derived from the poles of $1/\varepsilon$ when
$\gamma\rightarrow 0$, and $\varepsilon(0)$ represents the static dielectric
constant calculated from the sum of $\varepsilon(\infty)$ and the oscillator
strengths of the phonons, $\Delta\varepsilon$.

Except for the highest LO mode, we have found that it is not necessary to
introduce an asymmetric broadening, e.g. by using different values for the TO
and LO-modes ( $\gamma_{\text{TO}}\neq\gamma_{\text{LO}}$) as described by
Gervais and Piriou [44]. For practical reasons, we have accounted for the
asymmetric broadening of the highest LO mode by introducing instead a nonzero
imaginary part of $\varepsilon(\infty)$. This additional parameter allowed us
to obtain a good fit of the shapes of the anisotropic features at the highest
$\omega_{\text{LO}^{*}}$. While such a procedure is somewhat unphysical, since
it violates the Kramers-Kronig consistency, we found that it has a negligible
impact on the spectra except for the anisotropy features at the highest
$\omega_{\text{LO}^{*}}$.

The obtained value of the TO phonon positions listed in Table 1 are in rather
good agreement with the previous theoretical predictions [12, 10]. Most of the
TO eigenfrequencies are within $3\text{\,}{\mathrm{cm}}^{-1}$ of the predicted
values. In some cases, the predictions differ up to
$10\text{\,}{\mathrm{cm}}^{-1}$ from each other, while the experimental value
lies in between. Only in few cases the experimental values differ by up to
$10\text{\,}{\mathrm{cm}}^{-1}$ from theory. The phonon at
$432.5\text{\,}{\mathrm{cm}}^{-1}$ in the c-axis response is extremely weak,
in the experimental data it is only evident in the $S$-element (bottom panel
of Fig. 4a, and thus could only be identified with the help of the theory.

Overall, we conclude that YAlO${}_{\text{3}}$ has very stable and predictable
lattice dynamical properties: A single model describes very well the
experimental data that have been obtained on three samples with different
surface cuts. Moreover, the older and newer theoretical calculations (of TO
phonon positions) agree reasonably well and match the data to high accuracy.

Table 1:  Infrared model parameters of the anisotropic dielectric response of YAlO${}_{\text{3}}$ at room temperature. |  $a$-axis
---
$\omega_{\mathrm{TO}}(\mathrm{cm^{-1}})$ | $\gamma(\mathrm{cm^{-1}})$ | $\Delta\varepsilon$ | $\omega_{\mathrm{LO}}(\mathrm{cm^{-1}})$
169.7 | 2.9 | 0.60 | 171.9
240.3 | 3.0 | 7.01 | 303.3
367.8 | 3.5 | 0.12 | 369.5
394.7 | 6.7 | 0.29 | 397.8
435.5 | 6.7 | 2.55 | 464.5
480.5 | 5.7 | 1.04 | 500.0
504.7 | 5.2 | 0.37 | 580.2
581.6 | 10 | 0.02 | 642.6
659.7 | 12 | 0.10 | 756.7
$\varepsilon(\infty)$ | 3.58 (+i0.037) |
$\varepsilon(0)$ | 15.67 |

|  $b$-axis
---
$\omega_{\mathrm{TO}}(\mathrm{cm^{-1}})$ | $\gamma(\mathrm{cm^{-1}})$ | $\Delta\varepsilon$ | $\omega_{\mathrm{LO}}(\mathrm{cm^{-1}})$
163.8 | 2.8 | 0.70 | 166.8
255.9 | 4.2 | 5.75 | 307.1
347.0 | 5.3 | 0.26 | 351.6
380.0 | 3.1 | 0.27 | 383.7
426.6 | 6.8 | 0.39 | 428.3
444.5 | 4.2 | 3.50 | 555.6
560.7 | 5.2 | 0.08 | 614.5
633.5 | 14 | 0.15 | 676.0
676.4 | 6.9 | 0.002 | 749.9
$\varepsilon(\infty)$ | 3.54 (+i0.048) |
$\varepsilon(0)$ | 14.65 |

|  $c$-axis
---
$\omega_{\mathrm{TO}}(\mathrm{cm^{-1}})$ | $\gamma(\mathrm{cm^{-1}})$ | $\Delta\varepsilon$ | $\omega_{\mathrm{LO}}(\mathrm{cm^{-1}})$
226.3 | 3.0 | 7.12 | 265.5
268.0 | 2.9 | 0.17 | 298.7
432.5 | 6.9 | 0.04 | 432.7
448.0 | 5.1 | 3.08 | 479.2
486.2 | 3.6 | 0.56 | 602.2
604.9 | 11 | 0.008 | 616.7
656.9 | 11 | 0.27 | 766.5
$\varepsilon(\infty)$ | 3.46 (+i0.058) |
$\varepsilon(0)$ | 14.70 |

Figs. 5d, e and f show the model dielectric functions $\varepsilon_{a}$,
$\varepsilon_{b}$ and $\varepsilon_{c}$ for an extended energy range that
spans the mid-, near-infrared, visible and ultraviolet (up to
$6.5\text{\,}\mathrm{eV}$). Here, we employed two different modelling
approaches. For the first type of fitting, the whole MIR to UV range has been
represented by an anisotropic Cauchy dispersion (model lines in Figs. 5d, e,
f), while the parameters describing the phonons in the FIR range have been
fixed and the FIR data have not been included in the fit. The imaginary part
of $\varepsilon(\infty)$ that was used in the far-infrared fitting has been
omitted. These fits have been performed simultaneously on the mid-infrared
data, measured in the range $0.10.5\text{\,}\mathrm{eV}$ at
$\varphi=$75\text{\,}\mathrm{\SIUnitSymbolDegree}$$, and on the NIR-UV data in
the range from $0.56.5\text{\,}\mathrm{eV}$ measured with a Woollam
ellipsometer (VASE) at variable angles of incidence of
$\varphi=$55\text{\,}\mathrm{\SIUnitSymbolDegree}$$,
$65\text{\,}\mathrm{\SIUnitSymbolDegree}$,
$75\text{\,}\mathrm{\SIUnitSymbolDegree}$ and
$85\text{\,}\mathrm{\SIUnitSymbolDegree}$. As for the phonons in the FIR-
range, the fitting has been performed on an extended data set for all the 6
high symmetry configurations.

The surface roughness, which is known to affect especially the data in the
VIS/UV region, has been accounted for by introducing an isotropic 50%
Bruggeman effective medium (BEMA) layer which mixes the actual $x$-axis
response of the YAlO${}_{\text{3}}$ sample (depending on orientation) with the
ambient. The obtained thickness values of the BEMA layers are
$2.8\text{\,}\mathrm{nm}$, $1.9\text{\,}\mathrm{nm}$ and
$4.4\text{\,}\mathrm{nm}$ for the $a$-cut, $b$-cut and $c$-cut surfaces,
respectively.

The best-fit parameters of the Cauchy model are listed in table II. The
optimized Cauchy constants $\varepsilon_{a0}$, $\varepsilon_{b0}$ and
$\varepsilon_{c0}$ differ from the values of $\varepsilon(\infty)$ listed in
Table 1. While for an ideal case these parameters should be identical, we
could not achieve perfect match of model and data for the entire THz to UV
range.

For the second fitting approach, we have fixed the thicknesses of the
effective roughness layers, (to the value derived from the Cauchy plus BEMA
fit) and fitted the same extended set of ellipsometry data in a point-by-point
manner with an unconstrained orthorhombic model (fitting independently the
real and imaginary parts of $\varepsilon_{a}$, $\varepsilon_{b}$ and
$\varepsilon_{c}$). The obtained dielectric functions are shown by symbols in
Fig. 5d, e and f. It is evident that the result of the point by-point fit
agrees well with the one derived from the fitting with the parameterized model
functions. Notably, no sign of energy gap edge is detected up to
$6.5\text{\,}\mathrm{eV}$.

Table 2:  Near-infrared to UV Cauchy model parameters of YAlO${}_{\text{3}}$, room temperature. Calculated anisotropic dielectric constant and refractive index at $2\text{\,}\mathrm{eV}$. |  $a$-axis
---
$\varepsilon_{a0}$ | 3.700 |
$\varepsilon_{a1}$ | 0.0150 | $\mathrm{eV^{-2}}$
$\varepsilon_{a2}$ | 0.00057 | $\mathrm{eV^{-4}}$
$\varepsilon_{a}$(2eV) | 3.764 |
$n_{a}$(2eV) | 1.940 |

|  $b$-axis
---
$\varepsilon_{b0}$ | 3.677 |
$\varepsilon_{b1}$ | 0.0133 | $\mathrm{eV^{-2}}$
$\varepsilon_{b2}$ | 0.00056 | $\mathrm{eV^{-4}}$
$\varepsilon_{b}$(2eV) | 3.735 |
$n_{b}$(2eV) | 1.933 |

|  $c$-axis
---
$\varepsilon_{c0}$ | 3.625 |
$\varepsilon_{c1}$ | 0.0142 | $\mathrm{eV^{-2}}$
$\varepsilon_{c2}$ | 0.00048 | $\mathrm{eV^{-4}}$
$\varepsilon_{c}$(2eV) | 3.685 |
$n_{c}$(2eV) | 1.920 |

###  IV.5 Temperature dependence of the phonons YAlO${}_{\text{3}}$

Finally, Fig. 6 shows the temperature dependence of the TO phonon positions
between $10\text{\,}\mathrm{K}$ and $330\text{\,}\mathrm{K}$. The full set of
fit parameters is available in the supporting information [45]. The
temperature dependent measurements were performed on the 3 different surface-
cut samples, but in single orientation on each: the $a$-axis response was
measured on the $c$-cut sample, the $b$-axis response on the $a$-cut sample
and $c$-axis one on the $b$-cut sample. Once the anisotropic model was
optimized on the full set of data from 6 orientations at room temperature, the
reduced set of 3 orientations has been found to be sufficient to reliably
model and fit the temperature dependence of the phonon parameters. Since the
temperature dependent measurements have only been performed in the far-
infrared part of the spectra below $700\text{\,}{\mathrm{cm}}^{-1}$, the
highest $\omega_{\text{LO}^{*}}$ feature is not included.

As there is no instability or phase transition predicted or observed for
YAlO${}_{\text{3}}$, the temperature behaviour of the phonon positions is
rather conventional. Only the high frequency modes, likely related to oxygen
octahedra stretching, show some anomalous temperature dependence. Discussion
about origin of the anomalies is beyond the scope of this work.

Figure 6:  Temperature dependence of the eigenfrequency $\omega_{0}$ of the
infrared-active TO phonons of YAlO${}_{\text{3}}$: a) $a$-axis response b)
$b$-axis response c) $c$-axis response.

##  V Summary and Conclusions

We have performed broad-band (THz to UV) spectroscopic ellipsometry
experiments to determine the full dielectric tensor of the orthorhombic
perovskite YAlO${}_{\text{3}}$. This material is widely used as substrates for
the thin film growth of complex oxides. The detailed knowledge of its
anisotropic dielectric response function is thus a precondition for a correct
analysis of the thin film response.

The ellipsometry measurements have been performed for 6 different orientations
of the plane of incidence of the photons and the sample surface with respect
to the principal axes of the orthorhombic YAlO${}_{\text{3}}$ crystal. Wedged
crystals have been used to avoid artefacts that can arise from photons that
are reflected from the backside surface in the frequency ranges where
YAlO${}_{\text{3}}$ is fully transparent, i.e. below about
$100\text{\,}{\mathrm{cm}}^{-1}$ ($200\text{\,}{\mathrm{cm}}^{-1}$) at room
temperature (at low temperature) and well above the range of the multi-phonon
excitations, i.e. at $\omega>$1500\text{\,}{\mathrm{cm}}^{-1}$$.

The measured spectra are presented in terms of the Mueller matrix elements
$N$, $C$, and $S$. It is shown that their analysis with an anisotropic model
provides robust and unambiguous results that capture all the essential
spectroscopic features. In particular, it yields all the infrared-active
phonon modes with eigenfrequencies (positions) that agree well with previous
reports and theoretical predictions. We also discuss the spectral features
that arise from the anisotropy of the dielectric response, including a
negative refraction effect that occurs in the vicinity of the highest
anisotropic LO-mode, $\omega_{\text{LO}}$.

From the 6 possible high-symmetry orientations, we have compared pairs of NCS
datasets taken on different crystal surfaces, but with the same crystal axis
oriented along the $x$-axis of the instrument coordinate system (intersection
of the plane of incidence and the sample surface). This comparison revealed a)
quasi-isotropic character of the ellipsometric spectra in the ranges away from
the LO frequencies – they are dominated by the response function of the
crystal axis aligned with the $x$ coordinate of the ellipsometer, and b) the
anisotropic effects, occurring near the $\omega_{\text{LO}}$ frequencies of
the remaining two crystal axes, are mutually opposite when the $y$\- and
$z$-axis responses are interchanged.

Last but not least, we have studied the temperature dependence of the phonon
parameters. Whereas no anomalous behaviour has been observed, we expect that
the detailed knowledge of the temperature dependent optical response of
YAlO${}_{\text{3}}$ will be useful for future studies of various thin films of
complex oxides that are grown on YAlO${}_{\text{3}}$ substrates.

###### Acknowledgements.

We acknnowledge funding by the Swiss National Science Foundation (SNSF)
through projects No. 200020-172611 and 200021-214905.

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##  Appendix A materials of ”Broad-band ellipsometry study of the anisotropic
dielectric response of YAlO${}_{\text{3}}$”

Figure 7:  FIR response along the a-axis measured on c-cut YAlO${}_{\text{3}}$
at angle of incidence $\varphi=$75\text{\,}\mathrm{\SIUnitSymbolDegree}$$, at
temperature from $10330\text{\,}\mathrm{K}$ : in a) the $N$ element, in b) the
$C$ element and in c) the $S$ element of the Mueller Matrix. The polyethylene
window of the detector and the polyethylene substrate of the polarizers are
not transparent in the spectral region $700750\text{\,}{\mathrm{cm}}^{-1}$. To
obtain the $S$ data in c), we used a Si-prism as a compensator. In this case,
the Si material are not transparent in the spectral region
$590630\text{\,}{\mathrm{cm}}^{-1}$. Figure 8:  FIR response along the b-axis
measured on a-cut YAlO${}_{\text{3}}$ at angle of incidence
$\varphi=$75\text{\,}\mathrm{\SIUnitSymbolDegree}$$, at temperature from
$10330\text{\,}\mathrm{K}$ : in a) the $N$ element, in b) the $C$ element and
in c) the $S$ element of the Mueller Matrix. The polyethylene window of the
detector and the polyethylene substrate of the polarizers are not transparent
in the spectral region $700750\text{\,}{\mathrm{cm}}^{-1}$. To obtain the $S$
data in c), we used a Si-prism as a compensator. In this case, the Si material
are not transparent in the spectral region
$590630\text{\,}{\mathrm{cm}}^{-1}$. Figure 9:  FIR response along the c-axis
measured on b-cut YAlO${}_{\text{3}}$ at angle of incidence
$\varphi=$75\text{\,}\mathrm{\SIUnitSymbolDegree}$$, at temperature from
$10330\text{\,}\mathrm{K}$ : in a) the $N$ element, in b) the $C$ element and
in c) the $S$ element of the Mueller Matrix. The polyethylene window of the
detector and the polyethylene substrate of the polarizers are not transparent
in the spectral region $700750\text{\,}{\mathrm{cm}}^{-1}$. To obtain the $S$
data in c), we used a Si-prism as a compensator. In this case, the Si material
are not transparent in the spectral region
$590630\text{\,}{\mathrm{cm}}^{-1}$. Figure 10:  Temperature dependence of
YAlO${}_{\text{3}}$ a-axis infrared-active TO phonons in terms of
eigenfrequency $\omega_{0}$, broadening $\gamma$ and strength
$A=\Delta\varepsilon$, at temperatures from $10330\text{\,}\mathrm{K}$. Figure
11:  Temperature dependence of YAlO${}_{\text{3}}$ b-axis infrared-active TO
phonons in terms of eigenfrequency $\omega_{0}$, broadening $\gamma$ and
strength $A=\Delta\varepsilon$, at temperatures from
$10330\text{\,}\mathrm{K}$. Figure 12:  Temperature dependence of
YAlO${}_{\text{3}}$ c-axis infrared-active TO phonons in terms of
eigenfrequency $\omega_{0}$, broadening $\gamma$ and strength
$A=\Delta\varepsilon$, at temperatures from $10330\text{\,}\mathrm{K}$.