# Dynamical screening in SrVO3: Inelastic x-ray scattering experiments and ab
initio calculations

Kari Ruotsalainen  Current affiliation Helmholtz-Zentrum Berlin, Germany,
<EMAIL_ADDRESS>Synchrotron SOLEIL, L’Orme des
Merisiers, Saint-Aubin, BP 48, F-91192 Gif-sur-Yvette, France  LSI, CNRS,
CEA/DRF/IRAMIS, École Polytechnique, Institut Polytechnique de Paris, F-91120
Palaiseau, France  Alessandro Nicolaou  Synchrotron SOLEIL, L’Orme des
Merisiers, Saint-Aubin, BP 48, F-91192 Gif-sur-Yvette, France  Christoph J.
Sahle  European Synchrotron Radiation Facility, 71 Avenue des Martyrs, 38043
Grenoble, France  Anna Efimenko  European Synchrotron Radiation Facility, 71
Avenue des Martyrs, 38043 Grenoble, France  James M. Ablett  Synchrotron
SOLEIL, L’Orme des Merisiers, Saint-Aubin, BP 48, F-91192 Gif-sur-Yvette,
France  Jean-Pascal Rueff  Synchrotron SOLEIL, L’Orme des Merisiers, Saint-
Aubin, BP 48, F-91192 Gif-sur-Yvette, France  Laboratoire de Chimie Physique –
Matière et Rayonnement, Sorbonne Université, CNRS, F-75252 Paris, France
Dharmalingam Prabhakaran  Department of Physics, Oxford University, Clarendon
Laboratory, Oxford, OX1 3PU, United Kingdom  Matteo Gatti  LSI, CNRS,
CEA/DRF/IRAMIS, École Polytechnique, Institut Polytechnique de Paris, F-91120
Palaiseau, France  European Theoretical Spectroscopy Facility (ETSF)
Synchrotron SOLEIL, L’Orme des Merisiers, Saint-Aubin, BP 48, F-91192 Gif-sur-
Yvette, France

###### Abstract

We characterize experimentally and theoretically the high-energy dielectric
screening properties of the prototypical correlated metal SrVO3. The dynamical
structure factor measured by inelastic x-ray scattering spectroscopy as a
function of momentum transfer is in very good agreement with first-principles
calculations in the adiabatic local density approximation to time-dependent
density-functional theory. Our results reveal the crucial importance of
crystal local fields in the charge response function of correlated materials:
They lead to depolarization effects for localised excitations and couple
spectra from different Brillouin zones.

##  I Introduction

SrVO3 is a cubic perovskite with a rather simple electronic structure. The V
ions are in a $d^{1}$ electronic configuration in an octahedral environment
with three $t_{2g}$ bands, well separated from other bands, crossing the Fermi
level. It is hence easily amenable to a description based on the Hubbard model
and has been long considered as a prototype strongly correlated metal Fujimori
_et al._ (1992); Martin  _et al._ (2016). Indeed, despite experimental issues
related to the surface sensitivity of early photoemission measurementsFujimori
_et al._ (1992); Inoue  _et al._ (1995); Sekiyama  _et al._ (2004); Maiti  _et
al._ (2006); Eguchi  _et al._ (2006); Laverock  _et al._ (2013) and the
formation of oxygen vacanciesBackes  _et al._ (2016), in the last fifteen
years SrVO3 has served as a testbed for the development of dynamical mean-
field theory (DMFT) implementations for real materials Pavarini  _et al._
(2004); Sekiyama  _et al._ (2004); Nekrasov  _et al._ (2005); Lechermann  _et
al._ (2006); Amadon  _et al._ (2008); Huang and Wang (2012); Tomczak  _et al._
(2012); Lee  _et al._ (2012); Sakuma  _et al._ (2013); Taranto  _et al._
(2013); Nilsson  _et al._ (2017).

The dynamical screening of the Coulomb interaction, evaluated in the random-
phase approximation (RPA), is the key physical ingredient that within Hedin’s
GW approximation Hedin (1965) describes electronic correlation beyond the
Hartree-Fock approximation. The importance of dielectric screening in order to
renormalize the electron-electron interaction has also been recognised for
using the Hubbard model in solidsAnisimov  _et al._ (1997); Aryasetiawan  _et
al._ (2004). In the recent past, the frequency dependent RPA screening has
been often employed within the DMFT framework to evaluate the on-site Hubbard
interaction U in SrVO3 Aryasetiawan  _et al._ (2004, 2006); Miyake and
Aryasetiawan (2008); Vaugier  _et al._ (2012); Casula  _et al._ (2012); Nomura
_et al._ (2012). On the other hand, on the basis of ab initio GW
calculationsGatti and Guzzo (2013), it was recently demonstrated that the
coupling with low-energy plasmon excitations is responsible for the
photoemission satellites of SrVO3. These results (see also Ref. Nakamura  _et
al._ (2016)) called for a reconsideration of correlation effects in SrVO3,
which was later confirmed by extended DMFT calculations as well Boehnke  _et
al._ (2016); Nilsson  _et al._ (2017); Petocchi  _et al._ (2020).

Both recent DMFT and GW calculations therefore indicate the crucial importance
of determining accurately the dielectric screening properties of SrVO3.
Nevertheless, to the best of our knowledge, a detailed experimental
investigation is still lacking 111Resonant inelastic X-ray scattering (RIXS)
has recently been used to study SrVO3 (see Ref. McNally  _et al._ (2019)), but
these experiments do not probe the dielectric function. Nonresonant inelastic
X-ray scattering (IXS) experiments measure the dynamic structure factor
$S(\mathbf{q},\omega)$ and can therefore provide such information Schülke
(2007). Thanks to IXS it is possible to have access to both plasmons (i.e.,
collective charge excitations) and various kinds of interband transitions,
together with their dispersion as a function of the wavevector $\mathbf{q}$.
Furthermore, IXS experiments use hard X-rays resulting in true bulk
sensitivity and reduced sample damage per unit volume. On the theoretical
side, the dynamic structure factor can be calculated within ab initio linear-
response time-dependent density-functional theory Runge and Gross (1984);
Onida  _et al._ (2002); Ullrich (2012) (TDDFT) and directly compared with
experiment.

In the present article we therefore tackle this question. By combining IXS
experiments and first-principles TDDFT calculations, we present a detailed
investigation of the high-energy electronic excitations of SrVO3 as a function
of the momentum transfer $\mathbf{q}$. An accurate knowledge of those
excitations also informs the construction of low-energy models within
DMFTAryasetiawan  _et al._ (2004): they are the charge fluctuations that are
used to calculate the Hubbard $U$222We note that, contrary to the dynamic
structure factor $S(\mathbf{q},\omega)$, the onsite interaction $U$ is an
auxiliary effective parameter and not a directly measurable quantity. Indeed,
its value depends on the choice of the one-particle orbitals in the Hubbard
model hamiltonian. Moreover, different definitions and recipes have been
proposed in literature for its calculation, see e.g. Refs. Anisimov and
Gunnarsson (1991); Aryasetiawan  _et al._ (2004); Cococcioni and de Gironcoli
(2005)..

In principle, TDDFT is an exact method to describe neutral electronic
excitations in materials. However, in practice ab initio approaches have to
adopt approximations such as the adiabatic local density approximationZangwill
and Soven (1980) (ALDA). They have been shown to provide reliable loss spectra
for $sp$ semiconductors and metalsOnida  _et al._ (2002); Botti  _et al._
(2007), but have rarely been used for strongly correlated $df$ compounds (see
e.g. Refs. Larson  _et al._ (2007); Lee  _et al._ (2010); Rödl and Bechstedt
(2012); Huotari  _et al._ (2010); Iori  _et al._ (2012); Rödl  _et al._
(2017)). Our detailed comparison of TDDFT simulations and experimental results
therefore provides an important benchmark to establish the extent to which
most widely used TDDFT approximations can be useful for describing and
understanding the physics of electronic screening also in correlated
metals333For TDDFT applications in Hubbard models see e.g. Refs. Verdozzi,
2008; Karlsson  _et al._ , 2011..

The article is organised as follows. In Sec. II we outline the experimental
and computational details. In Sec. III, on the basis of the very good
agreement between measured and calculated spectra, we analyse the origin of
main spectral features and we unravel the strong impact of the induced crystal
local fields, which turn out to be an essential aspect of the dielectric
response of SrVO3. Finally, Sec. IV summarizes our work.

##  II Methods

###  II.1 Experiment

The experiments were performed at beamline ID20 of the European Synchrotron
Radiation Facility. The data were acquired using the RIXSMoretti Sala  _et
al._ (2018) and RamanHuotari  _et al._ (2017) spectrometers. We used an
incident photon energy of 6455.93 eV. The incident flux was monochromatized
employing a Si(111) monochromator and Si(440) analyzer crystals with a 2 m
bending radius were used. The total experimental energy resolution was
approximatively 1 eV (FWHM) as read off the elastic line.

The sample was a SrVO3 single crystal grown by the floating zone technique.
The orientation was studied with the Laue method and the crystal surface was
prepared by polishing followed by removal of the damaged layers with nitric
acid. With the RIXS spectrometer we used a $\theta$-$2\theta$ scattering
geometry with q oriented along the $(100)$ direction. With the Raman
spectrometer we used a multi-analyzer setup and the momentum transfers have
components along a perpendicular reciprocal lattice vector as well. The
momentum resolution was of the order of 10% of $\mathbf{G}=(100)$ at the low
scattering angles and less than 1% at the highest. The experiments were
performed at room temperature. The tail of the elastic peak gives uncertainty
to the experimental results for $\omega\lesssim 5$ eV and, in the following,
the spectra are reported and discussed in the energy range of 5-50 eV.
Therefore, the investigation of $dd$ excitations at energies below 5 eV is
left for a future experimental work.

###  II.2 Theory

The dynamic structure factor $S(\mathbf{q},\omega)$ for a transferred
momentum444If not otherwise specified, atomic units are used throughout the
paper. $\mathbf{q}$ describes the longitudinal response of an electronic
system to an external scalar potential. Thanks to the fluctuation-dissipation
theorem, it can be related to the loss function $-\text{Im}\epsilon_{M}^{-1}$
as ($n$ is the mean electron density)Schülke (2007):

$\displaystyle S(\mathbf{q},\omega)=$
$\displaystyle-\frac{q^{2}}{4\pi^{2}n}\text{Im}\epsilon_{M}^{-1}(\mathbf{q},\omega)$
$\displaystyle=$
$\displaystyle\frac{q^{2}}{4\pi^{2}n}\frac{\text{Im}\epsilon_{M}(\mathbf{q},\omega)}{\left[\text{Re}\epsilon_{M}(\mathbf{q},\omega)\right]^{2}+\left[\text{Im}\epsilon_{M}(\mathbf{q},\omega)\right]^{2}}.$
(1)

Plasmon resonances correspond to the zeros of $\text{Re}\epsilon_{M}$ for
$\text{Im}\epsilon_{M}$ not too large, while spectral features associated to
(screened) interband transitions can be detected as peaks of
$\text{Im}\epsilon_{M}$ as well. $\epsilon_{M}$ is inversely proportional to a
diagonal element of the inverse dielectric matrixAdler (1962); Wiser (1963):

$\epsilon_{M}(\mathbf{q}=\mathbf{q}_{r}+\mathbf{G},\omega)=\frac{1}{\epsilon^{-1}_{\mathbf{G},\mathbf{G}}(\mathbf{q}_{r},\omega)},$
(2)

where $\mathbf{q}_{r}$ belongs to the first Brillouin zone and $\mathbf{G}$ is
a reciprocal-lattice vector. For $\mathbf{q}\rightarrow 0$, the expression (2)
yields the macroscopic dielectric function $\epsilon_{M}(\mathbf{q}\rightarrow
0,\omega)$ that describes the optical spectra. In any inhomogeneous material,
the off-diagonal elements $\mathbf{G}\neq\mathbf{G}^{\prime}$ of the
dielectric matrix $\epsilon_{\mathbf{G},\mathbf{G}^{\prime}}$ are not zero and
therefore contribute to the diagonal element of its inverse
$\epsilon^{-1}_{\mathbf{G},\mathbf{G}}$. By defining $M$ as the submatrix of
$\epsilon$ that does not have the row and column $\mathbf{G}$, Eq. (2) can be
rewritten as:

$\epsilon_{M}(\mathbf{q}=\mathbf{q}_{r}+\mathbf{G},\omega)=\epsilon_{\mathbf{G},\mathbf{G}}(\mathbf{q}_{r},\omega)\\\
-\sum_{\mathbf{G}_{1},\mathbf{G}_{2}\neq\mathbf{G}}\epsilon_{\mathbf{G},\mathbf{G}_{1}}(\mathbf{q}_{r},\omega)M^{-1}_{\mathbf{G}_{1},\mathbf{G}_{2}}(\mathbf{q}_{r},\omega)\epsilon_{\mathbf{G}_{2},\mathbf{G}}(\mathbf{q}_{r},\omega).$
(3)

The off-diagonal contributions due to inhomogeneities in the response
functions explicitly appear in the second row of Eq. (3): they are called
crystal local-field effects (LFE).

The inverse dielectric function is related to the density-density response
function $\chi$ via the Coulomb interaction $v_{c}$:

$\epsilon^{-1}_{\mathbf{G},\mathbf{G}^{\prime}}(\mathbf{q}_{r},\omega)=\delta_{\mathbf{G},\mathbf{G}^{\prime}}+v_{c}(\mathbf{q}_{r}+\mathbf{G})\chi_{\mathbf{G},\mathbf{G}^{\prime}}(\mathbf{q}_{r},\omega).$
(4)

Within TDDFT $\chi$ can be obtained as a solution of the Dyson-like equation
Petersilka  _et al._ (1996):

$\chi_{\mathbf{G},\mathbf{G}^{\prime}}(\mathbf{q}_{r},\omega)=\chi^{0}_{\mathbf{G},\mathbf{G}^{\prime}}(\mathbf{q}_{r},\omega)+\sum_{\mathbf{G}_{1},\mathbf{G}_{2}}\chi^{0}_{\mathbf{G},\mathbf{G}_{1}}(\mathbf{q}_{r},\omega)\times\\\
\left[v_{c}(\mathbf{q}_{r}+\mathbf{G}_{1})\delta_{\mathbf{G}_{1},\mathbf{G}_{2}}+f^{xc}_{\mathbf{G}_{1},\mathbf{G}_{2}}(\mathbf{q}_{r},\omega)\right]\chi_{\mathbf{G}_{2},\mathbf{G}^{\prime}}(\mathbf{q}_{r},\omega),$
(5)

where $\chi^{0}$ is the Kohn-Sham (KS) independent-particle response function
and $f_{xc}$ is the exchange-correlation kernel. The RPA corresponds to
setting $f_{xc}=0$ in Eq. (5). The simplest approximation beyond the RPA is
the ALDA, where a local and static approximation to $f_{xc}$ is adopted.
$\chi^{0}$ is built from the KSKohn and Sham (1965) orbitals
$\varphi_{n\mathbf{k}}$ and eigenvalues $\varepsilon_{n\mathbf{k}}$:

$\chi^{0}_{\mathbf{G},\mathbf{G}^{\prime}}(\mathbf{q}_{r},\omega)=\frac{1}{\Omega}\sum_{n,n^{\prime},\mathbf{k}}(f_{n\mathbf{k}}-f_{n^{\prime}\mathbf{k}+\mathbf{q}_{r}})\\\
\times\frac{\tilde{\rho}_{nn^{\prime}\mathbf{k}}(\mathbf{q}_{r},\mathbf{G})\tilde{\rho}^{*}_{nn^{\prime}\mathbf{k}}(\mathbf{q}_{r},\mathbf{G}^{\prime})}{\omega+\varepsilon_{n\mathbf{k}}-\varepsilon_{n^{\prime}\mathbf{k}+\mathbf{q}_{r}}+i\eta}$
(6)

where $f_{n\mathbf{k}}$ are the Fermi occupation numbers, $\Omega$ the crystal
volume, and $\eta\rightarrow 0^{+}$ gives rise to a Lorentzian broadening. The
matrix elements
$\tilde{\rho}_{nn^{\prime}\mathbf{k}}(\mathbf{q}_{r},\mathbf{G})=\int
d\mathbf{r}\varphi^{*}_{n\mathbf{k}}(\mathbf{r})e^{-i(\mathbf{q}_{r}+\mathbf{G})\cdot\mathbf{r}}\varphi_{n^{\prime}\mathbf{k}+\mathbf{q}_{r}}(\mathbf{r})$
give the oscillator strengths.

Figure 1:  Comparison between experimental IXS spectra (black) and theoretical
dynamic structure factors calculated within the ALDA (blue) and the RPA
(green). $\bf{q}_{\textrm{e(t)}}$ denotes the experimental (theoretical)
momentum transfer value in reciprocal lattice units (r.l.u.) and $\hbar\omega$
is the energy transfer ($\bf{q}_{\textrm{t}}$ is the wavevector in the
12x12x12 grid that is the closest to $\bf{q}_{\textrm{e}}$). The experimental
and theoretical spectra have been normalized to the area under the curve
between 5 eV and 40 eV. A vertical offset has been added to the spectra for
improved clarity. (a) Low $q$. (b) Intermediate $q$. (c) High $q$. The
theoretical spectra have been broadened by 0.17 (0.4) eV for panel (a)
[(b),(c)]) to match the experimental resolution.

In the RPA and when LFE are neglected, $\textrm{Im}\epsilon_{M}$ reduces to
the independent-particle picture of a sum over transitions
$(n\mathbf{k})\rightarrow(n^{\prime}\mathbf{k}+\mathbf{q}_{r})$:

$\textrm{Im}\epsilon_{M}(\mathbf{q},\omega)=\frac{4\pi^{2}}{\Omega
q^{2}}\sum_{nn^{\prime}\mathbf{k}}(f_{n\mathbf{k}}-f_{n^{\prime}\mathbf{k}+\mathbf{q}_{r}})\\\
\times\left|\tilde{\rho}_{nn^{\prime}\mathbf{k}}(\mathbf{q}_{r},\mathbf{G})\right|^{2}\delta(\omega-(\varepsilon_{n^{\prime}\mathbf{k}+\mathbf{q}_{r}}-\varepsilon_{n\mathbf{k}})),$
(7)

whereas LFE and exchange-correlation effects beyond the RPA effectively mix
all those electron-hole transitions.

In the present work, ground-state KS orbitals and energies are obtained in the
local-density approximation (LDA) within a plane-wave basis approach on a
12$\times$12$\times$12 $\Gamma$-centered $\mathbf{k}$-point grid with norm-
conserving pseudopotentials (including V $3s$ and $3p$ and Sr $4s$ and $4p$
semicore states) and an energy cutoff of 100 Hartree. To calculate $\chi^{0}$
from Eq. (6), 200 bands have been included in the sum and the matrix size has
a 245 eV cutoff. We have adopted the experimental lattice parameter of the
cubic perovskite crystal structure Lan  _et al._ (2003) $a=3.8425$ Å. Spin-
orbit coupling has a minor effect on the electronic structure of SrVO3 and has
been neglected. We have used Abinit Gonze  _et al._ (2005) for the KS ground-
state calculations and Dp dp  for the TDDFT simulations of IXS spectra.

##  III Results and analysis

###  III.1 Comparison between experiments and calculations

The low $q$ experimental IXS spectra are presented in Fig. 1(a) alongside with
comparisons to ALDA and RPA calculations. Their main characteristics are a
peak at 14.2 eV, a shoulder around 24 eV, a strong peak at 31 eV and a further
peak at 41 eV. The 14 eV peak loses relative intensity upon increasing the
momentum transfer while the high energy edge of the peak increases in
intensity with respect to the peak maximum. Intensity is suppressed in the
valley between the 14.2 eV and 24 eV peaks, and the latter increases in
intensity. The main peak at 31 eV shows broadening towards the low energy side
with increasing $q$, suggesting contributions from dispersive excitations, or
non-dispersive ones with a q dependent intensity. Finally, a non-dispersive
peak emerges at 42.6 eV.

In Fig. 1(b) we present intermediate $q$ spectra. The low energy edge of the
14.2 eV peak shows a slight suppression of intensity with increasing $q$. The
24 eV shoulder continues to develop into a clear peak. Near 31 eV, the
intensity is suppressed and the maximum is found at lower energies. The 42.6
eV peak increases in intensity as does the continuum past the peak maximum.

Finally in Fig. 1(c) we present the high $q$ spectra. One immediately observes
that the peak at 31 eV has completely merged with the continuum, and that the
24 eV peak remains. Furthermore the 41 eV peak is now the most prominent
feature of the spectrum. We also note that the intensity region from 5-15 eV
has also developed a double peak structure that was not clearly visible at low
or intermediate $q$. There is a sharp peak at 6.6 eV followed by a broad
feature extending up to 14 eV. Here the lineshapes remain roughly constant
upon increasing $q$ and the spectra overlay very well across the whole energy
range.

Overall the calculated spectra present a slight underestimation of some peak
positions and are more structured than the experimental spectra. The inclusion
of lifetime effects would damp many features, especially at high energies, and
could further improve the agreement with experiment Weissker  _et al._ (2006,
2010); Cazzaniga  _et al._ (2011); Huotari  _et al._ (2011); Seidu  _et al._
(2018). Anyway, considering the fact that in the comparison between
calculations and measurements no adjustable parameters have been used, the
agreement between ALDA and experiment is remarkable even for this supposedly
strongly correlated transition-metal oxide. Furthermore, also RPA and ALDA
results are qualitatively similar, supporting the use of the RPA for the
calculationAryasetiawan  _et al._ (2004) of the Hubbard $U$ in SrVO3 555For an
assessment of the constrained RPA downfolding procedure in model Hamiltonians
we refer e.g. to Refs. Shinaoka  _et al._ (2015); Honerkamp  _et al._ (2018)..

###  III.2 Dispersion

Figure 2: Dispersion of the calculated dynamic structure factor
$S(\mathbf{q},\omega)$ for $\mathbf{q}$ along the (100) direction. The size of
$\mathbf{q}$ ranges from q=0.083 (bottom curve) to q=4.083 (top curve) by
steps equal to 0.166 reciprocal lattice units. (a) Full calculations within
the ALDA. (b) Calculations within the RPA neglecting local field effects
(LFE).

Having verified that the calculated screening captures the most important
electronic excitations occurring in the measured loss function with very good
accuracy, we can safely use the simulated spectra to analyse more in detail
the dispersion of the measured electronic excitations as a function of
$\mathbf{q}$.

Fig. 2(a) shows the ALDA dynamic structure factor $S(\mathbf{q},\omega)$ for
$\mathbf{q}$ along the $(100)$ direction and size $q$ spanning from $(1/12)G$
to $(49/12)G$, i.e. from 0.136 to 6.677 Å-1, with $(1/6)G$ steps. Each curve,
corresponding to increasing $q$ from the bottom to the top of the figure, has
a vertical offset proportional to $q$. In other nonequivalent directions the
results are qualitatively similar and therefore not reported here for brevity.

The spectra are characterised by a rich set of features that do not display
large dispersions with $q$. Being nondispersive in reciprocal space means that
they correspond to electronic excitations that are quite localised in real
space. The shape of the spectra, instead, changes very much. Besides the
gradual development of a broad continuum at large energies and momentum
transfers that is typical of Compton lineshapesHuotari  _et al._ (2011);
Ruotsalainen  _et al._ (2015), the relative spectral feature intensities are
strongly varying with $q$. Some peaks that are weak at small $q$ increase to
become the most prominent features at large $q$. The emergence of these peaks
is even more noticeable in the spectra of Fig. 2(b) that are obtained
neglecting entirely the LFE, i.e., from the first row of Eq. (3) calculated
within the RPA. Without LFE the spectra look very different, even
qualitatively. Indeed, LFE turn out to be the crucial ingredient to reach a
good agreement with experiment. A similar strong impact of crystal local
fields was found in other materials where the electrons are analogously
localized (see e.g. Gurtubay  _et al._ (2004, 2005); Iori  _et al._ (2012);
Gatti  _et al._ (2015a); Zhu  _et al._ (2018)). By comparing Fig. 2(b) and
Fig. 2(a) we see that LFE mainly damp the most intense peaks (located at
$\sim$ 23 and 40 eV for large $q$) and redistribute their spectral weight to
higher energies.

Figure 3: Analysis of the contribution from O $2p$ valence electrons and
different semicore states to the spectra: (Left)
$\text{Im}\epsilon_{M}(\mathbf{q},\omega)$ in the independent-particle
approximation (i.e., without LFE), see Eq. (7), (center) the corresponding
loss function $-\text{Im}\epsilon_{M}^{-1}(\mathbf{q},\omega)$, and (right)
the full ALDA calculation for (upper row) a small $q$=0.083, (middle row) an
intermediate $q$=1.916, and (bottom row) a high $q$=3.916 for momentum
transfers $\mathbf{q}$ along the (100) direction.

In the following we will analyse more in detail the physical origin of the
most important peaks in the spectra and we will elucidate further the role of
LFE.

###  III.3 Origin of the spectral features

In the calculations it is possible to examine the contribution to the spectra
from the various one-particle states by selectively removing them from (or
adding them to) the sum in Eq. (6). To this end, we choose 3 representative
spectra from the series displayed in Fig. 2 for momentum transfers
$\mathbf{q}$ along the (100) direction. The outcome of this analysis is shown
in Fig. 3(a)-(c) for the small $q=(1/12)G$, in Fig. 3(d)-(f) for the
intermediate $q=(23/12)G$, and in Fig. 3(g)-(i) for the large $q=(47/12)G$. In
these figures we progressively add to the contribution from V $t_{2g}$ and O
$2p$ valence bands (green lines) the shallow core (i.e. semicore) states666We
note that O $2s$ and Sr $4p$ are also partially hybridized.: Sr $4p$ (purple
lines), O $2s$ (blue lines) and V $3p$ (red lines). The latter corresponds to
the full calculation, as V $3s$ do not contribute in the considered energy
range and Sr $4s$ (not shown) are weakly visible only at large $q$ (Sr
$4s\rightarrow 4d$ is dipole forbidden).

Fig. 3(a) displays $\text{Im}\epsilon_{M}$ calculated in an independent-
particle approximation (i.e., within the RPA without LFE), see Eq. (7). At
energies below 15 eV the excitations from the O $2p$ dominate, while the peak
at 20-25 eV is mainly associated to Sr $4p\rightarrow 4d$ transitions, with a
very small contribution from the O $2s$. Finally the peak at 40 eV has a clear
V $3p$ origin.

In the corresponding loss function $-\text{Im}\epsilon_{M}^{-1}$, see Fig.
3(b), the peaks are shifted to higher energies. The difference between the
absorption spectrum and the loss function can be understood in terms of the
long-range Coulomb interaction Onida  _et al._ (2002); Sottile  _et al._
(2005). The peaks at 13 eV and 29 eV in the calculation with all the states
[red line in Fig. 3(b)] are plasmons: they are associated to zeros of
$\text{Re}\epsilon_{M}$ (not shown). The structure at 40-44 eV is easily
identified as V $3p$ interband transition matching well the corresponding peak
of $\text{Im}\epsilon_{M}$. The semicore V $3p$ peak also partially screens
the main plasmon at smaller energy. This can be understood by comparing the
full calculation [red line in Fig. 3(b)] with the one where V $3p$ are
excluded (blue line).

It is noteworthy to remark that also the Sr $4p$ (purple line) give an
important contribution to the main valence plasmon at 29 eV, whereas the
collective charge excitation resulting from O $2p$ valence electrons only
(green line) would have a much smaller energy. This result, as found also in
other complex oxidesHuotari  _et al._ (2010), clearly indicates that the
valence charge dynamics of SrVO3 cannot be described accurately by considering
only the V $3d$ and O $2p$ electrons, but also the Sr $4p$ and O $2s$ should
be taken into account. Indeed, with only V $3d$ and O $2p$ electrons
(amounting to 19 electrons per unit cell), one would obtain a classical plasma
frequency of only $\sim$21 eV for $\mathbf{q}\rightarrow 0$ instead of 29 eV.
We can also compare SrVO3 with metallic VO2, which shares the same $d^{1}$
electronic configuration. While they have similar low-energy excitationsGatti
and Guzzo (2013); Gatti  _et al._ (2015b) (and also analogous V M2,3 edges Abe
_et al._ (1997); Gatti (2007)), the main plasmon energies are different: in
VO2 it is only 25 eVAbe  _et al._ (1997); Gatti (2007). This difference points
again towards the active role of shallow Sr core states that are obviously
present only in SrVO3. Indeed, the main plasmon energy of SrVO3 bears a closer
resemblance with SrTiO3, another cubic perovskite that is a $d^{0}$ insulator,
where the plasmon is at 30 eV Ruotsalainen  _et al._ (2015).

Besides blueshifting by less than 1 eV the two plasmon resonances, the main
effect of crystal local fields on the loss function, calculated within the
ALDA in Fig. 3(c), is to suppress the semicore V $3p$ peak, which becomes a
weak structure centred at 50 eV. Consequently the impact of V $3p$ on the
plasmon is also strongly reducedCudazzo  _et al._ (2014a). We find here the
typical depolarization effect due to the crystal local fields: the external
perturbation creates additional microscopic Hartree potentials that act as
counteracting local fields and lead to a weakening of the effective perturbing
potential. The interband transitions involving semicore states correspond to
charge excitations that are at the same time highly polarizable and localized.
Both aspects concur to strong LFEAryasetiawan  _et al._ (1994); Vast  _et al._
(2002); Cudazzo  _et al._ (2014a).

We now examine what is happening at larger $\mathbf{q}$, where the role of the
long-range Coulomb interaction is not crucial anymore Onida  _et al._ (2002);
Sottile  _et al._ (2005). As a consequence, also the loss function resembles
$\text{Im}\epsilon_{M}$, compare Fig. 3(d) and Fig. 3(e), or Fig. 3(g) and
Fig. 3(h). At this point the plasmons are completely damped by the coupling
with electron-hole excitations (Landau damping). In the loss spectra they are
replaced as most prominent features by the interband transitions that are the
main peaks of $\text{Im}\epsilon_{M}$. The largest variations for the
$\text{Im}\epsilon_{M}$ spectrum with respect to the small $\mathbf{q}$ case
in Fig. 3(a) are the progressive decrease of the oscillator strengths of the O
$2p$ contribution and the corresponding relative increase of the V $3p$ peak
with respect to the Sr $4p$. At the largest $q$, Fig. 3(g)-(h), the spectra
are dominated by the V $3p$ peak rising over a flat background given by all
the other contributions.

Investigating the loss function at larger wavevectors $\mathbf{q}$ corresponds
to probing electronic excitations that are localised in real space. The
influence of microscopic local fields therefore increases. In the intermediate
momentum range, compare Figs. 3(e) and 3(f), LFE change completely the
spectral shape, as the result of the mixing of many transitions. Even at the
largest $\mathbf{q}$, where only one $3p$ prominent peak remains, LFE
quantitatively affect the spectra, strongly reducing the intensity of the V
$3p$ peak with respect to the background [see Figs. 3(h) and 3(i)].

The measurement by IXS of these semicore excitations, which is generally
called x-ray Raman scatteringSuzuki  _et al._ (1970); Schülke (2007); Huotari
_et al._ (2017) (XRS), provides the same information as extreme ultraviolet
absorption spectroscopy on the same shallow core edges but with the advantage
of the possibility of tuning the momentum transfer, possibly disclosing
nondipolar transitions. Here we conclude that for both XRS and absorption
spectroscopies the strong LFE invalidate the simple picture of independent
electron-hole transitions [see Eq. (7)]. In order to obtain a quantitative
agreement with experiment (see Fig. 1), going beyond the independent-particle
picture and considering the mixing of electron-hole transitions through LFE
turn out to be crucial.

###  III.4 Crystal local fields beyond depolarization effects

Considering Eq. (3), LFE can be understood as the result of the coupling of
excitations with wavevectors $\mathbf{q}_{r}$ and $\mathbf{q}_{r}+\mathbf{G}$
belonging to different Brillouin zones identified by the reciprocal lattice
vectors $\mathbf{G}$. As discussed in the previous sections, commonly LFE
induce a damping and a blueshift of the peaks in the spectra, a mechanism that
is physically associated to depolarization effects. It is hence striking when
LFE instead make new features appear in the spectra. Here we will examine two
of such cases in SrVO3.

Figure 4: Loss function at q=(13/12)$|\mathbf{G}|$ along the (100) direction
calculated with or without LFE.

The first feature is the peak at $\sim$ 13 eV – see Fig. 2(a) – that displays
an intensity dependence on $q$ that is remarkably nonmonotonic and showing a
maximum in the intermediate $q$ range, i.e., qualitatively different from the
standard behavior in simple materials. A similar finding was similarly
reported for TiO2 in Ref. Gurtubay  _et al._ , 2004. In both cases, LFE are
responsible for the presence of this peak, which is here confirmed also by our
experiment (see Fig. 1). In Fig. 4 we examine the loss function for $q=13/12$
calculated with and without LFE. We see that when LFE are neglected the loss
function is featureless, whereas the inclusion of LFE induces the appearance
of a sharp peak, which is located at the same energy of the plasmon at
$q=1/12$ [see red lines in Fig. 3(b)-(c)]. Therefore we can conclude that LFE
are the key for the coupling between independent electron-hole excitations at
large $q$ and the first-Brillouin-zone plasmon. This effect is well known in
layered materials Cai  _et al._ (2006); Balassis  _et al._ (2008); Silkin  _et
al._ (2009); Hambach  _et al._ (2008); Hambach (2010); Cudazzo and Gatti
(2017); Echeverry  _et al._ (2012); Cudazzo  _et al._ (2012, 2014b); Faraggi
_et al._ (2012), which are intrinsically inhomogeneous in the direction
perpendicular to the layers, and where LFE give rise to a long-lived plasmon
that continues to exist well beyond the first Brillouin zone. In these
situations, the mechanism behind the reappearance of the first-Brillouin-zone
$\mathbf{q}_{r}$ spectra at larger momentum transfers
$\mathbf{q}_{r}+\mathbf{G}$ can be understood within a two-plasmon-band
modelOliveira and Sturm (1980); Hambach (2010).

Figure 5: Loss functions calculated at $\mathbf{q}=\mathbf{G}+\bm{\eta}$ where
$\mathbf{G}=(100)$ and $\bm{\eta}$ has a very small size $|\eta|=10^{-3}$ and
forms an angle $\alpha$ with $\mathbf{G}$ that varies from $0{\degree}$ when
$\bm{\eta}$ is along $(100)$ to $90{\degree}$ when $\bm{\eta}$ is along
$(010)$. The spectra calculated without LFE (grey curve) are independent of
$\alpha$, while spectra that include LFE are sensitive to the angle $\alpha$
with the small $\eta$ component.

The same mechanism also explains the angular anomaly (firstly discovered by
Hambach et al. in graphiteHambach  _et al._ (2008); Hambach (2010)) that is
observed here for the loss function calculated at
$\mathbf{q}=\mathbf{G}+\bm{\eta}$ where $\mathbf{G}=(100)$ and $\bm{\eta}$ has
very small size $\eta=10^{-3}$ and direction identified by the angle $\alpha$
between $\mathbf{G}$ and $\bm{\eta}$ that varies from $0{\degree}$ when
$\bm{\eta}$ is along $(100)$ to $90{\degree}$ when $\bm{\eta}$ is along
$(010)$: the loss function is found to be strongly dependent on the small
$\bm{\eta}$, see Fig. 5. Neglecting LFE, instead, all the calculated spectra
remain the same (see grey line in Fig. 5). At first sight, this result seems
even more surprising in the present case than in graphite, since SrVO3 is a
cubic material. However, a similar behavior has been also detected in
siliconHambach (2010), where it is also responsible for the presence of a
plasmon-Fano resonance in the spectra Sturm  _et al._ (1992).

By supposing that the main coupling occurs between $\mathbf{q}_{r}$ inside the
first Brillouin zone (i.e., for $\mathbf{G}=0$) and
$\mathbf{q}_{r}+\mathbf{G}$, Eq. (3) can be simplified to:

$\epsilon^{-1}_{\mathbf{G},\mathbf{G}}(\mathbf{q}_{r},\omega)=\frac{1}{\epsilon_{\mathbf{G},\mathbf{G}}(\mathbf{q}_{r},\omega)}+\\\
\frac{\epsilon_{\mathbf{G},\mathbf{0}}(\mathbf{q}_{r},\omega)\epsilon_{\mathbf{0},\mathbf{G}}(\mathbf{q}_{r},\omega)}{[\epsilon_{\mathbf{G},\mathbf{G}}(\mathbf{q}_{r},\omega)]^{2}}\epsilon^{-1}_{\mathbf{0},\mathbf{0}}(\mathbf{q}_{r},\omega).$
(8)

The first term in Eq. (8) gives rise to the loss function at
$\mathbf{q}_{r}+\mathbf{G}$ obtained neglecting LFE (grey lines in Figs. 4 and
5). The second term accounts for the LFE, showing explicitly the coupling with
the first-Brillouin-zone dielectric function
$\epsilon^{-1}_{\mathbf{0},\mathbf{0}}$, weighted by its off-diagonal
elements. When LFE are not negligible – as in the case of SrVO3 – this term
explains both the reappearance of the small $\mathbf{q}_{r}$ plasmon and the
angular anomaly. When the weighting factor is not much frequency dependent
(and the other off-diagonal terms $\mathbf{G}^{\prime}\neq\mathbf{G}$, here
neglected, are not important), from Eq. (8) we see that the spectrum at
$\mathbf{q}_{r}+\mathbf{G}$ indeed results being proportional to the spectrum
at $\mathbf{q}_{r}$. Moreover, the angular anisotropy in the spectra at
$\mathbf{G}+\bm{\eta}$ analogously stems from the off-diagonal elements
$\epsilon_{\mathbf{G},\mathbf{0}}(\bm{\eta})$ and
$\epsilon_{\mathbf{0},\mathbf{G}}(\bm{\eta})$. Indeed both depend on
$\bm{\eta}$, while the first term in Eq. (8) does not (see grey line in Fig.
5; since SrVO3 is cubic, also
$\epsilon^{-1}_{\mathbf{0},\mathbf{0}}(\bm{\eta})$ is independent of
$\bm{\eta}$).

In summary, the coupling between spectra from different Brillouin zones
associated to the LFE explains the strong dependence of the spectra on the
small $\bm{\eta}$ component, which is in contrast to what one would naively
expect 777It is also interesting to note that, contrary to graphite Hambach
_et al._ (2008); Hambach (2010), in this case LFE apparently result also from
the coupling with other $\mathbf{G}^{\prime}\neq\mathbf{G}$ elements.
Following Hambach Hambach (2010), for a more quantitative analysis one should
hence use a two-plasmon-band model..

##  IV Conclusions

We have characterised the dynamical screening properties of SrVO3 by comparing
experimental IXS spectra and first-principles TDDFT calculations. Our results
provide a useful reference for both DMFT and GW calculations. The very good
agreement between experiment and theory has allowed us to analyse in detail
the physical origin of the main electronic excitations in the 5-50 eV energy
range, together with their dispersion as a function of the wavevector
$\mathbf{q}$. Our investigation leads us to conclude that crystal local field
effects are an essential feature of the dielectric response of correlated
materials that should not be overlooked in future studies.

###### Acknowledgements.

This work has been supported by the Labex PALM (Grant No.
ANR-10-LABX-0039-PALM). Computational time was granted by GENCI (Project No.
544). We acknowledge the European Synchrotron Radiation Facility (ESRF) for
providing synchrotron radiation and Christian Henriquet for technical support
during the beamtime. DP would like to acknowledge support from the Engineering
and Physical Sciences Research Council (EPSRC-EP/T028637/1) and support from
the Oxford-ShanghaiTech collaboration project

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