# Vibrational and structural properties of the $R$Fe4Sb12 ($R=$Na, K, Ca, Sr,
Ba) filled skutterudites

Juliana G. de Abrantes1, Marli R. Cantarino1, Wagner R. da Silva Neto1,2,
Victória V. Freire1, Alvaro G. Figueiredo1, Tarsis M. Germano1, Bassim
Mounssef Jr.2, Eduardo M. Bittar3, Andreas Leithe-Jasper4, Fernando A. Garcia1
1Intituto de Física, Universidade de São Paulo, 05508-090, São Paulo-SP,
Brazil  2Instituto de Química, Universidade de São Paulo, 05508-090, São
Paulo-SP, Brazil  3Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro, RJ
22290-180, Brazil  4Max Planck Institute for Chemical Physics of Solids,
D-01187 Dresden, Germany.

###### Abstract

Vibrational and elastic properties of the $R$Fe4Sb12 skutterudites are
investigated by, respectively, temperature $(T)$ dependent extended X-ray
absorption fine structure (EXAFS) and pressure $(P)$ dependent x-ray
diffraction (XRD) experiments. The Fe $K$-edge EXAFS experiments of the $R=$
K, Ca and Ba materials were performed in the $T$-interval $6<T<300$ K and XRD
experiments of the $R=$ Na, K, Ca, Sr and Ba materials were performed in the
$P$-interval $1\text{ atm }<P<16$ GPa. From EXAFS, we obtained the correlated
Debye-Waller parameters that were thus analyzed to extract effective spring
constants connected with the Fe-$Y$ (where $Y=$ either $R$, Fe or Sb)
scattering paths. Our findings suggest that in the case of the light cations,
$R=$ K or Ca, the $R$ atoms are relatively weakly coupled to the cage, in a
scenario reminiscent to the Einstein oscillators. From the XRD experiments, we
obtained the bulk modulus $B_{0}$ for all $R=$Na, K, Ca, Sr and Ba materials,
with values ranging from $77$ GPa ($R=$ K) to $R=99$ GPa ($R=$ Ba) as well as
the compressibility $\beta$ as a function of $P$. The trend in $\beta$ as a
function of the $R$ filler is discussed and it is shown that it does not
correlate with simple geometrical considerations but rather with the filler-
cage bonding properties.

##  I Introduction

Filled skutterudites are materials with chemical formula $RT_{4}X_{12}$ where
the $T_{4}X_{12}$ elements form a relatively rigid cage framework inside which
the $R$ elements are allocated [1]. In the simplest chemical bonding scenario,
the cage is structured by strong $T-X$ covalent bondings and the $R$ fillers
generically display a cationic character, donating electrons that further
stabilize the $T_{4}X_{12}$ cage. While it is well known that the description
of bonding in filled skutterudites is more involved [2], this simple picture
can be applied as a guide to discover new filled skutterudites [3] as well as
to justify their unusual vibrational dynamics [4].

Concerning the latter topic, research on the filled skutterudites is indeed
triggered mostly by the presence of an unusual vibrational dynamics that makes
the filled skutterudites good candidates for thermoelectric applications [5,
6, 7]. In this regard, earlier contributions proposed that the $R$ fillers
behave as independent, non dispersive and low energy oscillators (or
rattlers), that scatter the cage derived phonons, giving rise to a phonon
glass, which impedes the thermal conduction and raises the material
thermoelectric figure of merit $ZT$ [8].

On one account, the phonon glass scenario provided a first approach to
understand the skutterudites’ vibrational dynamics. For instance, filling the
cages with distinct fillers may cause the smaller $R$ cations to display an
unusual rattling behavior in the oversized cage, as observed [9, 10]. This
scenario, however, was soon challenged [11] and it is now well understood that
the separation between rattler and cage vibrational dynamics does not offer an
adequate description of the filled skutterudite vibrational properties [12,
13, 14]. All this understanding is particularly true in the case of the
$R$Fe4Sb12 ($R=$ Na, K, Ca, Sr, Ba, La and Yb) skutterudites [15, 16, 17, 18,
19, 20] and provides support for the design of more efficient antimony based
skutterudite materials [21, 22, 23].

On top of the good potential for thermoelectric applications, filled
skutterudites are very flexible platforms, displaying a broad range of
electronic, magnetic and thermal properties [24]. The $R$Fe4Sb12 ($R=$ Na, K,
Ca, Sr, Ba and La) materials, in particular, are itinerant magnets [25, 26,
27, 28, 15, 29] whose magnetic properties are dominated by a high density of
states due to the mixing between the Fe derived $3d$ states and Sb derived
$5p$ states forming heavy bands in the vicinity of the Fermi level [30, 31].

In this work, we focus on the vibrational and structural properties of the
$R$Fe4Sb12 ($R=$ Na, K, Ca, Sr and Ba) filled skutterudites as investigated,
respectively, by temperature $(T)$ dependent extended x-ray absorption fine
structure (EXAFS) and pressure ($P$) dependent x-ray diffraction (XRD)
experiments. EXAFS was soon recognized by Cao _et al_[32] as a suitable probe
for the site-specific vibrational dynamics in skutterudites. Later
contributions [33, 34, 35] developed the application of the technique to the
correlated nature of the filler-cage vibrational dynamics. Here, we focus on
the Fe $K$-edge EXAFS of the $R=$ K, Ca and Ba samples. In our investigation,
we find that the cage vibrational dynamics clearly depends on the filler type,
corroborating the correlated nature of the filler-cage dynamics.

Our XRD experiments were carried out for pressures typically up to $\approx
16$ GPa for all samples, except in the case of BaFe4Sb12, which was
investigated up to $24$ GPa. Overall, the $P$-dependence of all samples is
well described by the Birch-Murnaghan model [36] from which we extract the
bulk modulus $B_{0}$ for each $R$. The $B_{0}$ behavior as a function of $R$
is discussed in the context of bonding specific properties of the $R$Fe4Sb12
skutterudites. This discussion is supported by quantum chemistry calculations.
Moreover, starting at pressures about $\approx 12$ GPa, the XRD results
display broad reflections. This finding suggests that the $R$Fe4Sb12 materials
undergo a process of increasing lattice disorder triggered by pressure, which
is reminiscent of the pressure-induced partial amorphization observed for
other cage systems [37].

##  II Methods

High-quality polycrystalline samples of $R$Fe4Sb12 ($R=$Na, K, Ca, Sr, and Ba)
skutterudites were synthesized by a solid-state sintering method [25]. Fe
$K$-edge EXAFS experiments of the $R=$ K, Ca, and Ba samples were performed at
the XAFS2 [38] beamline of the Brazilian Synchrotron Light Source (CNPEM-
LNLS). The spectra of the K and Ca filled materials were measured by both
fluorescence and transmission modes, whereas the Ba filled material was
measured only in the fluorescence mode. A complete temperature scan in the
interval $300>T>6$ K was obtained for the three samples. An Fe foil, kept at
room temperature, was measured in the transmission mode as a reference
throughout the experiments. A conventional He-flow cryostat was employed to
achieve temperatures down to $T=6$ K.

The EXAFS data were analyzed by multiple scattering theory implemented by the
FEFF code [39]. The graphical Demeter platform [40] was adopted to perform the
fittings of a structural model including single and multiple scattering paths
up to $5$ Å from the Fe absorber. To decrease the number of parameters in the
fitting, the EXAFS $S_{0}^{2}$ parameter (the path amplitudes) was obtained
from _ab initio_ calculations implemented by the FEFF$8.4$ code. The _ab
initio_ calculations were performed for clusters of $226$ atoms, adopting the
Hedin-Lundqvist pseudopotential to account for the exchange interaction [31].

The powder XRD experiments were performed at the XDS [41] beamline of the
CNPEM-LNLS at room temperature. A diamond anvil cell (DAC) was employed during
experiments to achieve pressures up to $24$ GPa ($R=$ Ba), but typically up to
$16$ GPa ($R=$ Na, K, Ca, and Sr). A mixture of methanol and ethanol (in a
$1:4$ proportion) was adopted as a pressure media. The pressure was measured
by monitoring the Raman fluorescence from a ruby standard mounted in the DAC
together with the samples. In all measurements, the X-ray energy was set to
$20$ keV. The XRD powder profile fitting and background removal were performed
with the GSAS-II software [42].

The filler-cage bonding properties of the $R=$ Ca-, Sr- and Ba-filled
materials were calculated within the density functional theory (DFT)
approximation, adopting the ORCA 5.0 program package [43, 44]. A supercell
containing a total of $339$ atoms was generated for the calculations which
were implemented __ in __ an _embedded approach_ , shown to predict local
properties with good accuracy [45]. Here, a cell of $45$ atoms containing one
filler plus the cage atoms was treated at a quantum level. This cell was
segregated by $2$ atomic layers of atoms described as capped effective core
potentials (cECPs) to avoid spurious electron leakage [45]. The remaining
atoms were simulated in a molecular mechanics approach, with their charges
ascribed in agreement with the Zintl phase concept to constrain the $24$
electron rule per $\mathrm{FeSb_{3}}$ unit [3] by means of a force field. All
atomic positions were obtained from crystallographic data. For all atoms, the
BP$86$ GGA functional was used along with the Karlsruhe valence triple $\zeta$
with one set of polarization functions. Spin-orbit coupling effects were
considered but no sizable effects were observed in the constitution of most
relevant bonding MOs.

##  III Results and discussion

###  III.1 EXAFS analysis and vibrational properties

Figure 1: $(a)-(c)$ Three representations of the ternary filled skutterudite
structure (space group $Im\bar{3}$): $(a)$ the $R$Fe4Sb12 conventional unit
cell highlighting the FeSb3 distorted octahedra; $(b)$ the structural elements
considered in the EXAFS analysis; $(c)$ the vibrating elements of the 1D
correlated rattling model (see text) [46]. In all cases, the $R$, Fe and Sb
atoms are represented by, respectively, red, blue and green spheres. $(d)-(f)$
Fourier transformed EXAFS spectra of the $R=$ K, Ca and Ba materials in
$T$-interval $6\text{ K}<T<300$ K. $(g)-(i)$ Low-$T$ (upper panel) and
high-$T$ (lower panel) representative EXAFS data and data analysis (thick red
line) for the $R=$ K, Ca and Ba materials. In each panel, the inset shows the
residuals of the presented fittings.

Our EXAFS experiments focus on the Fe $K$-edge EXAFS of the $R$Fe4Sb12 ($R=$
K, Ca and Ba) materials to understand how their vibrational properties evolve
as a function of the filler. Among the $R=$ Na, K, Ca, Sr, and Ba filled
materials, the changing periodic properties concern the filler charge and the
filler-cage mass/size relation. By selecting the K and Ca filled materials, we
probe the effects of the filler charge, whereas the filler-cage mass relation
is kept nearly constant. Choosing the Ca and Ba filled cases, the filler-cage
mass/size relation is probed, whereas keeping the filler charge constant. Our
choice of $R$ thus suffices to capture the important trends of the vibrational
dynamics of the $R$Fe4Sb12 materials.

In figures 1$(a)-(c)$ we present three representations of the skutterudite
structure. The $R$ fillers, Fe atoms and Sb atoms are represented,
respectively, by red, blue and green spheres. In figure 1$(a)$, we show the
typical conventional unit cell representation, where we highlight the FeSb6
octahedra. In figure 1$(b)$, we centered the representation in a given Fe
atom, which we call the absorber, and leave only the atoms that are about $5$
Å apart from this Fe atom. In figure 1$(c)$, we show an alternative
representation, where we show the filler, the square Sb ring and the Fe cage.
This representation highlights the vibrating components of the effective 1D
model for the correlated rattling dynamics in filled skutterudites [35, 34]
which we will discuss below.

The Fourier transformed data for all $T$ values are presented in figures
1$(d)$, $(e)$ and $(f)$ for the $R=$ K, Ca and Ba materials, respectively. Our
EXAFS analysis is based upon single and multiple scattering paths within the
cluster depicted in figure 1$(b)$. This cluster includes all atoms within $5$
Å from the absorber. In this range, there are five Fe - $Y$ single scattering
paths, where $Y$ is the scattering element. The second and third Sb nearest
neighbors, denoted Sb2 and Sb3, are too close to each other, and thus we adopt
a single correlated Debye-Waller parameter
$\sigma_{\text{Fe}-\text{Sb}_{2,3}}^{2}$ to describe both paths. In total,
four $\sigma_{\text{Fe}-Y}^{2}$ are considered:
$\sigma_{\text{Fe}-\text{Sb}_{1}}^{2}$, $\sigma_{\text{Fe}-\text{R}_{1}}^{2}$,
$\sigma_{\text{Fe}-\text{Sb}_{2,3}}^{2}$ and
$\sigma_{\text{Fe}-\text{Fe}_{1}}^{2}$. Single scattering paths dominate the
EXAFS signal, but multiple scattering paths were included to improve the data
refinement. The disorder parameters of the latter were modeled in terms of the
single scattering path $\sigma_{\text{Fe}-Y}^{2}$ parameters. In its turn, the
$T$-dependence of the $\sigma_{\text{Fe}-Y}^{2}$ parameters was modeled on the
basis of the Debye model for correlated disorder for all Fe-$Y$ paths but the
Fe-$R$ paths, which were analyzed under the Einstein model, as typical for
filled skutterudite materials [32, 33]. Higher-order cumulants, which suggest
the presence of anharmonic vibrations [47], were not adopted in our models.
Representative fittings and residuals (for low and high $T$ data) for the $R=$
K, Ca and Ba cases are displayed in figures 1$(g)-(i)$, showing that the
structural model adopted in this paper offers a fair description of the data
in all $T$-interval.

Figure 2: Temperature dependence of the correlated Debye-Waller parameters
obtained from the Fe $K$-edge EXAFS analysis for the $(a)$ KFe4Sb12, $(b)$
CaFe4Sb12 and $(c)$ BaFe4Sb12 samples.

In figures 2$(a)-(c)$ we show the correlated Debye-Waller parameters obtained
from the EXAFS experiments. Whereas the Fe-Sb1 and Fe-Sb2,3 bond disorder
parameters are nearly $R$-independent, the difference between the Fe-Fe and
Fe-$R$ disorder parameters is significant when the cases of the light and
heavy fillers are compared. This concerns not only the temperature dependence
of the parameters, which is set by the relevant energy scales (either the
Debye or Einstein temperatures), but also the order of magnitude of the
parameters. In particular, one should note the relatively small disorder of
the Fe-Ba path in comparison with the Fe-K and Fe-Ca cases. For a more
quantitative analysis, we turn to the obtained energy scales. From now on, we
denote by $\theta_{D}^{R,\text{Fe-}Y}$ the Debye temperature obtained for the
Fe - $Y$ path of the $R$ filled sample and by $\theta_{E}^{\text{Fe}-R}$ the
analogous Einstein temperature. All results are compiled in table 1.

The $\theta_{D}^{R,\text{Fe-Sb}_{1}}$ parameters display a weak dependence
with $R$ and compare well with those obtained in the early study by Cao _et
al_ [32] for the $R=$ Ce and La cases ( $\approx 390$ K), suggesting that the
Fe-Sb bondings are not much affected by the filler. The
$\theta_{E}^{\text{Fe}-R}$ here obtained, however, are significantly larger
than those obtained by EXAFS of the $R=$ La and Ce cases, but are closer to
values determined from atomic displacement parameters (ADPs) for the $R=$ Ba
and Ca cases by Schnelle  _et al._ [15] (one should, of course, keep in mind
the difference between the quantities obtained by EXAFS and ADPs). As for the
overall trend in the data, $\theta_{E}^{\text{Fe-
Ba}}>\theta_{E}^{\text{Fe-K}}>\theta_{E}^{\text{Fe-Ca}}$, similar results are
obtained by heat capacity [15], even though the heat capacity results are more
indicative of $\theta_{E}^{\text{Fe-
Ba}}>\theta_{E}^{\text{Fe-K}}\approx\theta_{E}^{\text{Fe-Ca}}$. Concerning the
$\theta_{D}^{R,\text{Fe-Fe}}$ parameters, the values obtained in the $R=$ K or
Ca cases are similar to results for other skutterudites [35, 33, 32] whereas
the Fe-Fe vibrations are particularly harder in the $R=$ Ba case.

The high-temperature limit of the $\sigma^{2}$ parameters of each single
scattering path can be connected to an effective spring constant
($K_{\text{eff}}$) as discussed in detail in Refs. [35, 33]. We denote by
$K_{\text{eff}}^{R,\text{Fe-}Y}$ the effective spring constant for the Fe -
$Y$ path of the $R$ filled sample. Effective spring constants describe better
the vibrational properties than a direct comparison between the Debye
(Einstein) temperatures since they take into account the reduced mass of the
vibrating elements and the fact that the $K_{\text{eff}}^{R,\text{Fe-}Y}$ are
not linearly dependent on the Debye (Einstein) temperatures. The parameters
are listed in table 1.

Table 1: The $\theta$ (either $\theta_{D}^{R,\text{Fe-}Y}$ or $\theta_{E}^{\text{Fe}-R}$ ) and $K_{\text{eff}}$ parameters obtained from the analysis of the correlated Debye-Waller parameters.  Fe $K$-edge | KFe4Sb12 | CaFe4Sb12 | BaFe4Sb12
---|---|---|---
Path |  $\theta$ (K) |  $K_{\text{eff}}$ (eV/$\text{Å}^{2}$) |  $\theta$ (K) |  $K_{\text{eff}}$ (eV/$\text{Å}^{2}$) |  $\theta$ (K) |  $K_{\text{eff}}$ (eV/$\text{Å}^{2}$)
Fe - Sb1 | 422(15) | 8.13 | 419(14) | 8.13 | 426(17) | 8.45
Fe - R | 168(61) | 1.15 | 134(51) | 0.74 | 190(43) | 2.55
Fe - Sb2,3 | 356(26) | 3.85 | 323(23) | 3.12 | 360(15) | 3.92
Fe - Fe1 | 395(75) | 3.56 | 390(77) | 3.45 | 620(80) | 9.79

Translated to $K_{\text{eff}}^{R,\text{Fe-}Y}$, the vibrational properties of
Fe - Sb bondings for the $R=$ K, Ca and Ba filled materials are similar, with
the difference being less than $5\%$ between the parameters
$K_{\text{eff}}^{R,\text{Fe-Sb}_{1}}$. They are also very similar to the
vibrational properties of the Co - Sb bondings as investigated by EXAFS [48].
Inspection of the $K_{\text{eff}}^{R,\text{Fe-$R$}}$parameters, on the other
hand, reveals a much larger variation as a function of $R$. Indeed,
$K_{\text{eff}}^{\text{Ba},\text{Fe-\text{Ba}}}/K_{\text{eff}}^{\text{Ca},\text{Fe-
Ca}}\approx 3.5$, stating a clear distinction between the cases of heavy and
light fillers. It suggests a relative decoupling of the filler-cage
vibrational dynamics in the case of the light-weight cations, a result that is
in agreement with previous analysis of the $R$Fe4Sb12 atomic displacement
parameters (ADPs) [15]. It so appears that the light-weight cations may behave
as “rattlers” which, however, are not totally independent of the cage
vibrations. We note that the $K_{\text{eff}}^{\text{Ba},\text{Fe-\text{Ba}}}$
parameter compares well with the Ru - Ce and Pt - La cases, obtained from
EXAFS experiments of CeRu4As12 [34, 35] and LaPt4Ge12 [49] respectively, which
are other examples of heavy fillers filled skutterudites. These results,
however, are not simply expressing the filler-cage mass relation, but rather
the potential energies of the cations, as discussed by previous calculations
of the $R$Fe4Sb12 [16]. The Fe - Fe vibrations are evidenced by the analysis
of the $K_{\text{eff}}^{R,\text{Fe-Fe}}$ parameters to be distinctively harder
in the case of the Ba filled samples, suggesting stronger Fe-Fe bonding in
this case. We note that
$K_{\text{eff}}^{\text{Ba},\text{Fe-\text{Fe}}}/K_{\text{eff}}^{R,\text{Fe-
Fe}}\approx$2.8 for both $R=$ K or Ca.

The effective spring constants can be applied to the discussion of an
effective $1$D phonon model, introduced by Keiber _et al._ [35, 33], for the
qualitative description of the correlated filler-cage motion in filled
skutterudites. The model considers the three distinct elements illustrated in
figure 1$(c)$: the $R$ filler (or rattler), the Sb square ring and the Fe
metal cage, which denotes solely the Fe cubic lattice. The coupled character
of the vibrational dynamics is given by connecting the elements by four
distinct spring constants for each $R$ filled skutterudite: a rattler-square
ring ($k_{\text{rs}}^{R}$), a rattler-metal cage ($k_{\text{rc}}^{R}$), a
metal cage-metal cage ($k_{\text{cc}}^{R}$) and a metal cage-square ring
($k_{\text{cs}}^{R}$). The system dynamics is treated at a classical level and
the corresponding dynamic matrix is diagonalized to find four phonon
dispersion modes $\omega_{j}(q)$, where $q$ is the phonon wave-vector [35]. We
shall discuss two aspects of the coupled vibrations that can be connected to
our experimental results: $i)$ the filler-cage mass relation; $ii)$ the
dependence of he $k_{\text{rc}}^{R}$ constants on whether $R$ is a light or
heavy element (as suggested by the experimentally obtained
$K_{\text{eff}}^{R,\text{Fe-}R}$ parameters). Moreover, we shall fix the
parameter $k_{\text{cs}}^{R}=$1.8 eV/$\text{Å}^{2}$ for all samples, as
suggested by the experimentally obtained $K_{\text{eff}}^{R,\text{Fe-Sb}}$ and
adopt$k_{\text{cc}}^{\text{K(Ca)}}=10$ eV/$\text{Å}^{2}$and
$k_{\text{cc}}^{\text{Ba}}=25$ eV/$\text{Å}^{2}$based on the variation of the
obtained $K_{\text{eff}}^{R,\text{Fe-Fe}}$ parameters. Our experiments do not
offer an estimate about the $k_{\text{rs}}^{R}$ parameters and we chose to
adopt a range of values in reference to previous works [35, 34].

Figure 3: Effective 1D correlated rattling model for the $R=$ K (Ca) and Ba
samples. In ascending frequency, we have the acoustic mode (red) and the
first, (green), second (orange) and third (blue) optical phonon modes. The
modes are displayed as function of $qa$ where $a$ is the material lattice
constant and $q$ is the phonon wave-vector. In panels $(a)-(i)$, the mass
relation between the vibrating elements is that of the $R=$ K case and in
panels $(j)-(o)$, the mass relation between the vibrating elements is that of
the $R=$ Ba case. The adopted parameters are: $k_{\text{cs}}^{\text{R}}=1.8$
eV/$\text{Å}^{2}$ for all panels and $k_{\text{cc}}^{\text{K}}=10$
eV/$\text{Å}^{2}$, in panels $(a)$-$(i)$, and $k_{\text{cc}}^{\text{Ba}}=25$
eV/$\text{Å}^{2}$, in panels $(j)$-$(o)$, reflecting the different
$K_{\text{eff}}^{R,\text{Fe-Fe}}$ parameters. The adopted $k_{\text{rs}}^{R}$
and $k_{\text{rc}}^{R}$ values are indicated in each panel in units of
eV/$\text{Å}^{2}$.

One should keep in mind that there is no one-to-one correspondence between the
$K_{\text{eff}}^{R,\text{Fe-}Y}$ and spring constants of the 1D effective
model. Rather, the $K_{\text{eff}}^{R,\text{Fe-}Y}$ constants are adopted to
get reasonable estimates to the spring constants, in particular to their
relative values. Formally, the spring constants are a functional of the
$K_{\text{eff}}^{R,\text{Fe-}Y}$ and there is a strong functional dependence
between, for instance, $K_{\text{eff}}^{R,\text{Fe-Fe}}$ and
$k_{\text{cc}}^{R}$ or $K_{\text{eff}}^{R,\text{Fe-}R}$ and
$k_{\text{rc}}^{R}$. Results of the phonon dispersion and the exact parameters
adopted are displayed in figures 3$(a)$-$(o)$ and in the figure caption.

In figures 3$(a)$-$(i)$, the mass-relation between the $R$ cations and the
cage elements is that of the KFe4Sb12 case (which, in view of the qualitative
nature of the calculated dispersions, also describe the CaFe4Sb12 case) and is
one of the key properties determining the phonon dispersion. The other
relevant parameter is indeed the relatively small rattler-cage spring
constant. These two parameters determine, respectively, the wide separation
between the acoustic and optical modes and the conspicuous flattening of the
second optical mode (thick orange line). The latter implies a wide range of
$q$-values for which the phonon group velocity is very close to zero,
describing a rather localized phonon mode, reminiscent of the Einstein
oscillators scenario. In view of the ratio
$K_{\text{eff}}^{\text{Ca},\text{Fe-
Ca}}/K_{\text{eff}}^{\text{K},\text{Fe-K}}\approx 2/3$ (see table 1) we
propose that the CaFe4Sb12 and KFe4Sb12 vibrational dynamics are,
respectively, described by figures 3$(a)$-$(c)$ and 3$(g)$-$(i)$. The range of
values in the $k_{\text{rs}}^{R}$ leaves the acoustic and first optical modes
unchanged (red and green lines), while pushing the second optical mode
(orange) up, causing this mode to interact with the third one (blue).

The mass relation of the BaFe4Sb12 case is represented in figures
3$(j)$-$(o)$. We now observe a smaller frequency gap between the acoustic and
optical modes (due to the filler-cage mass relation) and a much smaller region
wherein the second optical mode is flat, due to the large rattler-cage
coupling. Here, the parameters’ range is similar to the one adopted in the
CeRu4As12 [35] and LaPt4Ge12 [49] cases, which allows us to pinpoint that the
filler-cage mass relation is also responsible for a larger flattening of the
optical mode, contributing to the phonon localization as recently proposed
[50]. The value of the $k_{\text{cc}}^{\text{Ba}}$ constant (reflecting the
large $K_{\text{eff}}^{\text{Ba},\text{Fe-\text{Fe}}}$) is twice the value for
the other heavy-filler filled skutterudites and it essentially increases the
frequency of the higher energy (blue) optical mode. In comparison to the K
(Ca) case, this shift of the third optical mode to a higher frequency causes
the decoupling of this mode from the second optical mode, which in the Ba case
turns out to interact mainly with the first optical mode.

###  III.2 X-ray diffraction and elastic properties

Figure 4: Representative $P$ dependent XRD experiments for all $R$Fe4Sb12
filled skutterudite, wherein we have: $(a)$ $R=$ Na; $(b)$ $R=$ K; $(c)$ $R=$
Ca; $(d)$ $R=$ Sr; and $(e)$ $R=$ Ba. The $P$ is as indicated in the panels.
The thick red lines represent the diffraction peaks of the skutterudite
structure. The blue and green arrows point to diffraction peaks not ascribed
to the skutterudite phase and are discussed in the main text and in the
supplemental material.

We now turn to the $P$ dependent XRD experiments. In figures 4$(a)-(e)$
representative data of the XRD experiments, after removing the background, are
presented for all samples. The main diffraction peaks of the skutterudite
structure are marked in red as a reference. The powder profile was compared to
the skutterudite crystal structure (the thick red lines in figures 4$(a)-(e)$
) and the $P$ dependence of the lattice parameters was extracted from the peak
positions. The blue and green arrows point to peaks that are not part of the
skutterudite phase. The peaks are, respectively, ascribed to small amounts of
non-reacted Sb and to the sample decomposition at high _P_ (see supplemental
material).

Our analysis of the XRD experiments are in figures 5$(a)-(d)$. In figure
5$(a)$, the unit cell volumes ($V$) as a function of $P$ of all samples are
presented. It is tempting to associate the skutterudite elastic properties to
the relation between the cage volume and the filler cationic size. We then
compile in figure 5$(b)$ a phenomenological parameter: the cage empty volume
fraction $f_{E}$, that we define as $f_{E}=1-V_{\text{R}}/V_{\text{C}}$, where
$V_{\text{R}}$ is the $R$ cationic volume and $V_{\text{C }}$is the Sb cage
volume, as a function of $P$. The cage is represented in the inset of figure
5$(b)$ as a reference.

The volume pressure dependence is investigated by fitting $V\times P$ data to
the Birch-Murnaghan (BM) model [36]:

$P(V)=\frac{3}{2}B_{0}[(\frac{V}{V_{0}})^{-\frac{7}{3}}-(\frac{V}{V_{0}})^{-\frac{5}{3}}]\\{1+\frac{3}{4}(B_{0}^{\prime}-4)[(\frac{V}{V_{0}})^{-\frac{2}{3}}-1]\\}$
(1)

where $V$ is the sample unit cell volume, $V_{0}$ is the sample volume at room
conditions, $B_{0}$ is the bulk modulus and $B_{0}^{\prime}$ is the bulk
modulus derivative. We keep $B_{0}^{\prime}=4$ GPa, as recently adopted to
describe other skutterudites [51, 52] in this pressure range. Values obtained
for $V_{0}$ and $B_{0}$ are compiled in table 2. The $V_{0}$ values obtained
from the fitting deviate only about $1$% from the experimentally determined
from x-ray diffraction at room conditions suggesting the adequacy of our
analysis. The $V\times P$ data and their respective fittings are in figure
5$(c)$.

Figure 5: $(a)$ The unit cell volume $V$ as function of $P$. Error bars for
$P$ are estimated from the half-width at half maximum of the ruby fluorescence
spectra and amount to about $3-5$ $\%$ in this pressure interval. $(b)$ The
cage empty volume fraction $f_{\text{E}}$ (see text) as a function of $P$. The
inset shows a representation of the skutterudite structure highlighting the Sb
cage around the filler site (the cage volume was calculated with VESTA [46]).
$(c)$ The $P$ as a function of $V$ data and their respective fittings to the
Birch-Murnaghan model. $(d)$ The compressibility $\beta$ as a function of $P$
for all samples.

The $B_{0}$ values thus obtained range from $77$ GPa ($R=$ K) to $99$ GPa
($R=$ Ba). The values compare well with those obtained for other $R$Fe4Sb12
based skutterudites, such as EuFe4Sb12 [51] and CeFe4Sb12 [53, 54]. The latter
further suggests that $B_{0}$ reflects the rigidity of the Fe4Sb12 framework
which, however, is not in total $R$ independent. The experimentally obtained
$B_{0}$ parameters are in the range of the values previously obtained by
calculations of the $R=$ Ca, Sr and Ba cases [17]. It is noteworthy, however,
that the calculated values are nearly $R$ independent and about $80$ GPa. Our
results also compare well with other Sb-based skutterudites, such as CoSb3
[52], in agreement with the understanding that the pnicogen square ring
rigidity dominates the skutterudite elastic properties. Indeed, the wider
variation in $B_{0}$ is observed when the pnicogen is changed, with P forming
characteristically rigid cages [53].

Table 2: Parameters of the Birch-Murnaghan model for all $R$Fe4Sb12 samples.  $R=$ | Na | K | Ca | Sr | Ba
---|---|---|---|---|---
$V_{0}$($\textrm{Å}{}^{3}$) | 768.3(7) | 787.8(2) | 778.7(5) | 777.6(5) | 772.0(3)
$B_{0}$(GPa) | 94(7) | 77(5) | 87(6) | 91(6) | 99(3)

In possession of the $B_{0}$ values, we then obtain the materials’
compressibility $\beta$ as a function of $P$. The results are displayed in
figure 5$(d)$. The compressibility data express well how the elastic
properties depend on $R$: it is shown that the K and Ca filled materials have
larger $\beta$ while the Ba and Na filled materials have smaller $\beta$. It
is a natural assumption to speculate that the higher $f_{\text{E }}$ more
compressible would be the material. If the whole data is inspected, this trend
is not obeyed. For instance, the Ca filled material, which has the largest
$f_{E}$, is not the most compressible. If, however, we focus on either the
$R^{1+}$ or $R^{2+}$ filled materials, an interesting picture emerges.
Restricting the analysis to the case of the $R^{1+}$ fillers, one notes that
this simple picture does not work and $\beta$ does not correlate with $f_{E}$,
with the K filled material being the more compressible. Now, if one inspects
the case $R^{2+}$ fillers case, the experimentally obtained picture obeys the
proposed trend, with the Ca filled material being the more compressible.

The case of the alkaline earth filled materials is distinct due to the
presence of the heavy Sr and Ba fillers. Here, Sr and Ba virtual atomic $4d$
and $5d$ states, respectively, are more extended in real space as well as
energetically accessible than in the case of the Ca virtual $3d$ states. Thus,
the $4d$ and $5d$ states may take part in the filler-cage bonding which, in
this scenario, is not completely ionic in character, as has been proposed for
La filled skutterudites [55]. To verify this hypothesis, we performed DFT
calculations of the filler-cage bonding properties.

We focus our analysis on the DFT results on the filler derived atomic orbitals
(AOs) contribution to the molecular orbitals (MOs) with the highest
superposition with the Sb-derived $5p$ states, since these are precisely the
MOs contributing the most to the filler–cage bonding. These MOs sit in the
vicinity of the Fermi Level (see supplemental material). Our results show that
whereas the $4d$ and $5d$ virtual AOs from Sr and Ba, respectively, take part
in the formation of these MOs, Ca only contributes its $3p$ states to the
bonding MOs. In table 3, we list the filler derived AOs contributing the most
(largest superposing orbitals) to the formation of the filler-cage bonding
MOs. The MOs symmetry classification is indicated and is made in terms of the
irreducible representations of the $T_{h}$ group, which is the point group
symmetry of the $R$ site [56, 57]. The results were obtained at room
conditions and at high $P$ ($\approx 13.6$ GPa). The $P$ effect was simulated
by adopting the results from our XRD experiments.

Table 3: The highest filler derived superposing AOs to the formation of the MOs connected to the filler-cage bonding. The nature of the AOs and the symmetry character of the MOs are indicated. The Final orbital composition analysis was performed adopting the Ros-Schuit partition method via the Multifwn program [58].  $R$ | AOs character | Room Pressure |  $\mathrm{\sim 13.6}$ GPa
---|---|---|---
Orbital Cont. (%) |  $\mathrm{E-E_{F}}$ (eV) | Orbital Cont. (%) |  $\mathrm{E-E_{F}}$ (eV)
Ca |  $3p_{x}$ $(t_{u})$ | 30.77 | $-3.23$ | 30.82 | $-3.46$
Sr |  $4d_{xy},4d_{xz},4d_{yz}$ $(t_{g})$ | 11.84 | $-1.70$ | 15.37 | $-1.82$
Ba |  $5d_{xy},5d_{xz},5d_{yz}$ $(t_{g})$ | 19.54 | $-1.60$ | 28.00 | $-1.68$

As can be observed, pressure has nearly no effect on the bonding in the case
of the Ca filler, whereas it further stabilizes and increases the $d$ states
participation in the Sr and Ba filled materials. This effect is particularly
large in the case of Ba, for which an increase of more than $\gtrsim 40\%$ of
the $5d$ orbital contribution to MOs enrolled in the filler-cage bonding is
observed in the high $P$ calculations. This result means that on top of
geometrical considerations, one must consider that $P$ induces further
stabilization to the filler-cage bonding via the available virtual $nd$
states, rendering the material more rigid. Our findings thus suggest that the
association between $B_{0}$ (or $\beta$) and $f_{\text{E}}$ in the case of the
$R^{2+}$ fillers actually reflects a deeper understanding of the problem based
upon the filler-cage bonding properties, in particular in the case of the
heavy fillers. This mechanism could also be at work in the context of the
partially filled $M_{y}$Co4Sb12 skutterudites, for which it is observed that
when $M$ is a heavy filler, in particular in the cases of $M=$ La and Yb, the
material is more rigid [52]. Moreover, our calculations suggest an association
between the participation of the Ba $5d$ states in the filler-cage bonding and
the relatively large value of the $K_{\text{eff}}^{\text{Fe-Ba}}$ parameter.

A better visualization of the filler-cage bonding, in particular of the
$d$-character of the MOs in the Sr and Ba case, can be observed in figure 6,
wherein we depict the MOs related to the results listed in table 3. The plots
are contour plots of the mentioned MOs and illustrate the real space electron
density distribution around the filler atom, which is located at the center of
the figures. As it is clear, the plotted MOs are dominated by orbitals of
$d$-character in the case of Sr- and Ba-filled materials, whereas in the case
of Ca-filled material the presence of the $3p$ lobe dominating the MOs is
clearly distinguishable.

Lastly, we comment on the significant broadening of the diffraction peaks for
high pressures that is observed. Starting at about $12$ GPa in the $R=$ Na and
K cases and at higher $P$ in the case of the alkaline earth filled
skutterudites, the peaks’ broadening amount to an increasing structural
disorder induced by $P$. The alkaline earth filled materials are suggested to
be relatively less susceptible to the disorder increase by $P$. Indeed, the
diffraction peaks can still be observed for $P$ up to $24$ GPa in the
BaFe4Sb12 case. The skutterudite structure is mainly stabilized by a
combination of geometric and charge constraints related to the filler and cage
sizes and charge distribution [2, 3]. We thus conclude that the larger
structural stability of alkaline earth filled materials is likely due to the
bonding stabilization of the extra charge offered by the $R^{2+}$ cations.

Figure 6: Contour plots for the MOs describing the filler-cage bonding in the
$(x,y)$ plane in the cases of the Ca, Sr and Ba filled materials as indicated.
The figures are formed by $120$ contours in the isovalue interval indicated in
the color scale in the right side of the figure. Ligand Orbitals are plotted
via the Gabedit program [59].

##  IV Summary and Conclusions

The site specific vibrational properties of the $R=$ K-, Ca- and Ba-filled
skutterudites $R$Fe4Sb12 were investigated by EXAFS experiments. As concluded
from the effective spring constants obtained from experiments, materials
filled with light-weight cations display weak rattler-cage couplings,
reminiscent of the independent rattler scenario, and the $R=$ Ba material
displays a characteristic strong Fe-Fe coupling. Based upon an effective $1$D
model for the skutterudite phonon dispersion, we proposed that the $R=$ K- and
Ca-filled skutterudites should display a conspicuous optical flat mode in
accordance with the Einstein oscillator scenario.

We thus introduced our pressured dependent experiments. The $V\times P$ curves
were determined and fitted to the Birch-Murnaghan model to extract the bulk
modulus $B_{0}$ and then the compressibility $\beta$. We found that the
geometric parameter $f_{\text{E}}$ cannot explain the $B_{0}$ (or $\beta$)
dependence as a function of $R$. A geometrical relation, however, was observed
to hold in the case of the $R^{2+}$ filled materials. By considering DFT
calculations, we uncovered that this association between $B_{0}$ (or $\beta$)
and $f_{\text{E}}$ actually reflects an emerging property of the filler-cage
bonding, which is particularly relevant for heavy fillers. Moreover, the
filler-cage bonding can also explain the large $K_{\text{eff}}^{\text{Fe-Ba}}$
parameter, since the contribution from Ba $5d$ states to bonding makes the Ba
more tightly coupled to the cage (no “independent” rattler in this case).

From the point of view of material design, our results suggest that
synthesizing mixed filled skutterudites, featuring light and heavy fillers, is
a good strategy to introduce localized vibrational modes (“rattlers”) in the
material’s vibrational dynamics. This speculation has some ground on our
calculations showing that the bonding scheme in skutterudites may include a
certain degree of covalency in the case of heavy fillers. Thus, introducing
light cations into materials mainly filled with heavy cations create a
situation wherein the light fillers are weakly bonded and sitting in an
oversized cage.

###### Acknowledgements.

The authors acknowledge CNPEM-LNLS for the concession of beam time (proposals
No. $20160180$, No. $20160181$, No. $20170709$ and No. $20190018$). The XAFS2
and XDS beamlines staff are acknowledged for the assistance during the
experiments. The Fundação de Amparo à Pesquisa do Estado de São Paulo
financial support is acknowledged by M.R.C. (grant No. 2019/05150-7 and grant
No. 2020/13701-0) and F.A.G. (grant No. 2019/25665-1). This study was financed
in part by CAPES - Finance Code 001. E.M.B. acknowledges Fundação Carlos
Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (grant No.
E-26/202.798/2019). A.L.J. thanks J. Grin for his steady support and interest
in this work.

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