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# Investigation of large-scale extended Granger causality (lsXGC) on synthetic
functional MRI data

Axel Wismüller  Department of Electrical and Computer Engineering, University
of Rochester, NY, USA  Department of Imaging Sciences, University of
Rochester, NY, USA  Department of Biomedical Engineering, University of
Rochester, NY, USA  Faculty of Medicine and Institute of Clinical Radiology,
Ludwig Maximilian University, Munich, Germany  M. Ali Vosoughi  Department of
Electrical and Computer Engineering, University of Rochester, NY, USA  Adora
DSouza  Department of Electrical and Computer Engineering, University of
Rochester, NY, USA  Anas Z. Abidin  Department of Biomedical Engineering,
University of Rochester, NY, USA

###### Abstract

It is a challenging research endeavor to infer causal relationships in
multivariate observational time-series. Such data may be represented by
graphs, where nodes represent time-series, and edges directed causal influence
scores between them. If the number of nodes exceeds the number of temporal
observations, conventional methods, such as standard Granger causality, are of
limited value, because estimating free parameters of time-series predictors
lead to underdetermined problems. A typical example for this situation is
functional Magnetic Resonance Imaging (fMRI), where the number of nodal
observations is large, usually ranging from $10^{2}$ to $10^{5}$ time-series,
while the number of temporal observations is low, usually less than $10^{3}$.
Hence, innovative approaches are required to address the challenges arising
from such data sets. Recently, we have proposed the large-scale Extended
Granger Causality (lsXGC) algorithm, which is based on augmenting a
dimensionality-reduced representation of the system’s state-space by
supplementing data from the conditional source time-series taken from the
original input space. Here, we apply lsXGC on synthetic fMRI data with known
ground truth and compare its performance to state-of-the-art methods by
leveraging the benefits of information-theoretic approaches. Our results
suggest that the proposed lsXGC method significantly outperforms existing
methods, both in diagnostic accuracy with Area Under the Receiver Operating
Characteristic (AUROC = $0.849$ vs. $[0.727,0.762]$ for competing methods,
$p<\\!10^{-8}$), and computation time ($3.4$ sec vs. [$9.7$, $4.8\times
10^{3}$] sec for competing methods) benchmarks, demonstrating the potential of
lsXGC for analyzing large-scale networks in neuroimaging studies of the human
brain.

###### keywords:

Resting-state MRI (fMRI), Large-Scale Extended Granger Causality (lsXGC),
functional connectivity, graph learning, relation discovery, machine learning

##  1 INTRODUCTION

For studying the underlying pathophysiology of neurological and psychiatric
disease, functional connectivity among various cortical regions has been
identified as an important research subject [1]. By providing images with
sufficiently high spatial resolution and the hemodynamic response dynamics
over a temporal axis, fMRI studies have demonstrated a tremendous potential to
serve as a biomarker for neurologic and psychiatric disease [2, 3, 4, 5].
Currently, most of the diagnosis in brain-related disorders relies on clinical
symptom evaluations, such as based on neuropsychological testing. However,
there is a need for more objective biomarkers. To this end, more recently,
studies have investigated, if such diagnostic information can be extracted
non-invasively from brain activity data. Despite promising results, there is
scope for improvement, specifically with regards to using more meaningful
connectivity analysis approaches [6].

Here, biomarkers from brain imaging using resting-state functional MRI (rs-
fMRI) can be derived using Multi-Voxel Pattern Analysis (MVPA) techniques [7].
MVPA is a machine-learning framework that extracts differences in patterns of
brain connectivity for discriminating between connectivity profiles of
individuals with neurological disorders and healthy individuals. Results of
such studies demonstrate that clinically meaningful information can be learned
from fMRI data. However, the simple conventional approach of using cross-
correlation between time-series does not measure directed connectivity. Hence,
more relevant information in the fMRI data may be captured by more
sophisticated connectivity measures which retain information about the
direction of interdependence between time-series.

Various methods have been proposed to obtain such directional relationships in
multivariate time-series data, including transfer entropy [8] and mutual
information [9]. However, in high-dimensional systems, estimating the system’s
underlying density function becomes computationally expensive [10]. Under the
Gaussian assumption, transfer entropy is equivalent to Granger causality [11].
However, the computation of multivariate Granger causality for short time
series in large-scale problems is challenging [12, 13, 14, 15, 2]. Addressing
this challenge, we recently proposed large-scale Extended Granger Causality
(lsXGC), which is a method that combines the advantages of dimensionality
reduction with the augmentation of a conditional source time-series adopted
from the original space. The lsXGC method uses predictive time-series modeling
for estimating directed causal relationships among fMRI time-series [2], which
makes it a suitable method to quantify the uncertainty associated with the
stochastic process. This work extends our previous investigation on lsXGC and
evaluates its ability to accurately recovering directional information from
synthetic rs-fMRI data with known ground truth (GT). Specifically, we compare
the performance of lsXGC to conventional multivariate Granger causality (GC)
[16], mutual information (MI) [9], and transfer entropy (TE) [8]. Based on
extensive simulations, we investigate the performance of lsXGC at correctly
unveiling the underlying dynamics of the synthetic fMRI dataset, even at a
significantly lower computational expense when compared to competing methods.
Our results qualify lsXGC as a promising candidate for serving as a potential
biomarker for brain disease in clinical fMRI studies.
This work is embedded in our group’s endeavor to expedite artificial
intelligence in biomedical imaging by means of advanced pattern recognition
and machine learning methods for computational radiology and radiomics, e.g.,
[17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54,
55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73,
74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89].

##  2 METHODS

###  2.1 Large-scale Extended Granger Causality (lsXGC)

Large-scale Extended Granger Causality (lsXGC) has been developed based on 1)
the principle of original Granger causality, which quantifies the causal
influence of time-series $\mathbb{x_{s}}$ on time-series $\mathbb{x_{t}}$ by
quantifying the amount of improvement in the prediction of $\mathbb{x_{t}}$ in
presence of $\mathbb{x_{s}}$. 2) the idea of dimension reduction, which
resolves the problem of the tackling a under-determined system, which is
frequently faced in fMRI analysis, since the number of acquired temporal
samples usually is not sufficient for estimating the model parameters [84,
83].

Consider the ensemble of time-series $\mathcal{X}\in\mathbb{R}^{N\times T}$,
where $N$ is the number of time-series (Regions Of Interest – ROIs) and $T$
the number of temporal samples. Let
$\mathcal{X}=(\mathbb{x_{1}},\mathbb{x_{2}},\dots,\mathbb{x_{N}})^{\mathsf{T}}$
be the whole multidimensional system and $x_{i}\in\mathbb{R}^{1\times T}$ a
single time-series with $i=1,2,\dots,N$, where
$\mathbb{x_{i}}=(x_{i}(1),x_{i}(2),\dots,x_{i}(T))$. In order to overcome the
under-determined problem, first $\mathcal{X}$ will be decomposed into its
first $p$ high-variance principal components
$\mathcal{Z}\in\mathbb{R}^{p\times T}$ using Principal Component Analysis
(PCA), i.e.,

$\mathcal{Z}=W\mathcal{X},$ (1)

where $W\in\mathbb{R}^{p\times N}$ represents the PCA coefficient matrix.
Subsequently, the dimension-reduced time-series ensemble $\mathcal{Z}$ is
augmented by one original time-series $\mathbb{x_{s}}$ yielding a dimension-
reduced augmented time-series ensemble $\mathcal{Y}\in\mathbb{R}^{(p+1)\times
T}$ for estimating the influence of $\mathbb{x_{s}}$ on all other time-series.

Following this, we locally predict $\mathcal{X}$ at each time sample $t$, i.e.
$\mathcal{X}(t)\in\mathbb{R}^{N\times 1}$ by calculating an estimate
$\hat{\mathcal{X}}_{\mathbb{x_{s}}}(t)$. To this end, we fit an affine model
based on a vector of $m$ vector of m time samples of
$\mathcal{Y}(\tau)\in\mathbb{R}^{(p+1)\times 1}$($\tau=t-1,t-2,\dots,t-m$),
which is $\mathbb{y}(t)\in\mathbb{R}^{m.(p+1)\times 1}$, and a parameter
matrix $\mathcal{A}\in\mathbb{R}^{N\times m.(p+1)}$ and a constant bias vector
$\mathbb{b}\in\mathbb{R}^{N\times 1}$,

$\hat{\mathcal{X}}_{\mathbb{x_{s}}}(t)=\mathcal{A}\mathbb{y}(t)+\mathbb{b},~{}~{}t=m+1,m+2,\dots,T.$
(2)

Now $\hat{\mathcal{X}}_{\setminus{\mathbb{x_{s}}}}(t)$, which is the
prediction of $\mathcal{X}(t)$ without the information of $\mathbb{x_{s}}$,
will be estimated. The estimation processes is identical to the previous one,
with the only difference being that we have to remove the augmented time-
series $\mathbb{x_{s}}$ and its corresponding column in the PCA coefficient
matrix $W$.

The computation of a lsXGC index is based on comparing the variance of the
prediction errors obtained with and without consideration of $\mathbb{x_{s}}$.
The lsXGC index $f_{\mathbb{x_{s}}\xrightarrow{}\mathbb{x_{t}}}$ , which
indicates the influence of $\mathbb{x_{s}}$ on $\mathbb{x_{t}}$, can be
calculated by the following equation:

$f_{\mathbb{x_{s}}\xrightarrow{}\mathbb{x_{t}}}=\log{\frac{\mathrm{var}(e_{s})}{\mathrm{var}(e_{\setminus
s})}},$ (3)

where $e_{\setminus s}$ is the error in predicting $\mathbb{x_{t}}$ when
$\mathbb{x_{s}}$ was not considered, and $e_{s}$ is the error, when
$\mathbb{x_{s}}$ was used. In this study, we set $p=1$ and $m=2$.

##  3 Simulation on synthetic fMRI datasets

Quantitative evaluation of lsXGC and competing algorithms was performed on
semi-realistic data of rs-fMRI brain recordings [90, 91], in which noise and
the hemodynamic response were included in the data generation process. These
data were adopted from Netsim [90], and the network ’sim3’ with $N=15$ was
used in our simulations. The hemodynamic responses were generated to model the
time-delayed interactions among different brain regions, and, as a result, a
smooth signal was generated that resembles recorded fMRI brain signals with an
inherent signal-to-noise ratio (SNR) of 20 $dB$. The advantage of the
simulated fMRI is that the ground truth (GT) for network connectivity is
known, which is not available for real fMRI datasets. The graph and GT are
shown in Fig. 1. Hence, the test results can be used to quantitatively compare
different algorithms in the presence of noise hemodynamic response condition,
which is not possible for real-world datasets. In line with the literature, we
use the Area Under the Receiver Operating Characteristic (AUROC) as a
comparison metric. Here, the task is to infer the underlying connectivity
between 15 brain regions across 50 network simulation instances.

Figure 1: Netsim network ’sim3’ adopted from [90] and the ground truth of one
of the networks is shown in this figure. The network simulations model
hemodynamic response and noise, which resembles real-world resting-state fMRI
data, with signal-to-noise ratio of 20 $dB$. The network has 15 nodes, and 50
different realizations of the network were used in our evaluations.

Results are shown in Fig. 2 and Table. 1. In Fig. 2, the AUROC of the proposed
lsXGC algorithm significantly outperforms the competing methods, affirming its
ability at accurately detecting the underlying directed network connectivity
structure in simulated fMRI signals under conditions of noise and smoothing
based on hemodynamic response, resembling the real-world situation. Simulation
time and diagnostic accuracy results are listed in Table. 1.

Figure 2: Performance of various algorithms, namely mutual information (MI) [9], transfer entropy (TE) [8], multivariate Granger causality (GC) [16], and our proposed method (lsXGC, [2]) on the Netsim dataset [90]. Areas under the Receiver Operating Characteristics (AUROC) are shown in the figure, where each column represents a different algorithm (see titles). Boxplots represent [Q1, Q3]=[0.25, 0.75] quartiles and median, and the cap lines represent [minimum, maximum]=[Q1-1.5$\times$(Q3-Q1), Q3+1.5$\times$(Q3-Q1)]. As can be seen, the proposed algorithm (lsXGC) significantly outperforms other methods from the literature, suggesting robustness of the method against noise and hemodynamic response effects encountered in fMRI data. Wilcoxon test $p$-values of the proposed lsXGC method as compared to MI, TE, and GC are less than $\\{<\\!10^{-9},<\\!10^{-9},<\\!10^{-8}\\}$, correspondingly. Table 1: Simulation time and performance on Netsim dataset [90] in identical conditions, except TE and MI, which were performed on Nvidia GeForce 1080-Ti GPU, with the rest based on CPU operations. All simulations were performed in Python 3.8. | Time (seconds) | AUROC
---|---|---
MI | 4753 | 0.728 $\pm$ 0.010
TE | 2806 | 0.727 $\pm$ 0.006
GC | 9.7 | 0.762 $\pm$ 0.038
lsXGC | 3.4 | 0.849 + 0.032

##  4 CONCLUSIONS

This paper investigates a novel method, large-scale Extended Granger Causality
(lsXGC), for discovering relations in high-dimensional dynamical systems
involving observational data. The lsXGC method addresses the curse of
dimensionality in large-scale dynamic systems by combining dimensionality
reduction with data augmentation, namely by augmenting data from the
conditional source time-series in the original input space onto a
dimensionally reduced state-space representation of the multidimensional time-
series system, thus improving prediction quality in a Granger causality
setting. Here, we focus on numerical results for learning meaningful
representations from synthetic resting-state fMRI data, although there may be
a wide scope of other application domains, ranging from neuroscience to
climatology and finance. Our results suggest competitive performance of the
proposed method in comparison to other widely used techniques from the
literature, namely conventional Granger causality, transfer entropy, and
mutual information. Most importantly, our method infers network connectivity
at a significantly lower computational expense, while accurately identifying
causal representations from multidimensional time-series data. We conclude
that lsXGC exhibits important advantages over competing methods from the
existing literature for inferring directed causal network connectivity in
large-scale systems with limited observational data. Specifically, our results
suggest that lsXGC may qualify as a promising candidate for serving as a
potential biomarker for brain disease in clinical fMRI studies.

###### Acknowledgements.

This research was partially funded by the American College of Radiology (ACR)
Innovation Award “AI-PROBE: A Novel Prospective Randomized Clinical Trial
Approach for Investigating the Clinical Usefulness of Artificial Intelligence
in Radiology” (PI: Axel Wismüller) and an Ernest J. Del Monte Institute for
Neuroscience Award from the Harry T. Mangurian Jr. Foundation (PI: Axel
Wismüller). This work was conducted as a Practice Quality Improvement (PQI)
project related to American Board of Radiology (ABR) Maintenance of
Certificate (MOC) for A.W.

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# Unconventional multiple plasmons generation in Oxygen-rich Strontium Niobate
mediated by Local Field Effects

Tao Zhu  Department of Physics, National University of Singapore, Singapore
117542, Singapore  Singapore Synchrotron Light Source, National University of
Singapore, 5 Research Link, Singapore 117603, Singapore  Paolo E. Trevisanutto
Department of Physics, National University of Singapore, Singapore 117542,
Singapore  Singapore Synchrotron Light Source, National University of
Singapore, 5 Research Link, Singapore 117603, Singapore  Graphene Research
Centre and CA2DM, National University of Singapore, 117542, Singapore  Teguh
Citra Asmara  Singapore Synchrotron Light Source, National University of
Singapore, 5 Research Link, Singapore 117603, Singapore  NUSNNI-NanoCore,
National University of Singapore, Singapore 117411, Singapore  Lei Xu
Department of Physics, National University of Singapore, Singapore 117542,
Singapore  Yuan Ping Feng  Department of Physics, National University of
Singapore, Singapore 117542, Singapore  Graphene Research Centre and CA2DM,
National University of Singapore, 117542, Singapore  Andrivo Rusydi
<EMAIL_ADDRESS>Department of Physics, National University of Singapore,
Singapore 117542, Singapore  Singapore Synchrotron Light Source, National
University of Singapore, 5 Research Link, Singapore 117603, Singapore  NUSNNI-
NanoCore, National University of Singapore, Singapore 117411, Singapore  NUS
Graduate School for Integrative Sciences and Engineering, National University
of Singapore, Singapore 117456, Singapore

###### Abstract

Recently it has been experimentally shown that a peculiar behaviour has
occurred in Strontium Niobate Oxide: under O-enrichment, SrNbO3.4, a spectral
weight transfer in the plasmon spectra has generated multiple new plasmons in
the visible to ultraviolet ranges from an initial single plasmon. Herewith, we
theoretically investigate the origin of plasmons generation by means of first
principle calculations. We infer that the spectral weight can be ascribed to
the dielectric function local fluctuations called Local Field Effects. These
are created by the presence of extra O atoms in the atomic structure. The
Local Field Effects are unusually preponderant with formation of plasmons not
affected by the theoretical chosen structure of SrNbO3.4 in the calculations.
Our result shows a new path to generate multiple plasmons in the ultraviolet-
deep ultraviolet regime.

##  I Introduction

The Sr1-xNb1-yO3+δ family of Oxides are known to have rich structures and
electronic properties that vary with O atoms content. For example, perovskite
SrNbO3 was conducting with an Nb $4d^{1}$ SrTiO3 ; SrTiO3_2 ; SrTiO3_3 ;
SrTiO3_4  electronic structure, and Sr1-xNbO3 SrTiO3_2  has been found to
potentially be a good photocatalyst in water splitting applications SrTiO3_3 .
On the other hand, O atoms-rich SrNbO3.4 and SrNbO3.5, derived by
interspersing the perovskite lattice of SrNbO3 with extra O atoms planes along
the [101] direction at certain periodic intervals SrTiO3_4 ; SrTiO3_5 , were
quasi-one-dimensional (1D) conductor SrTiO3_4 ; SrTiO3_5 ; SrTiO3_6 ; SrTiO3_7
; SrTiO3_8  and ferroelectric insulatorSrTiO3_8 ; SrTiO3_9 , respectively.

Between these properties, very recently, multiple formation of plasmons,
collective excitations, has been observed in SrNbO3.4 Asmara . The
experimental findings have shown that these plasmons have multiple frequencies
and are low loss in the visible-ultraviolet range, not obtainable from the
classical single frequency bulk plasmon models in metals. Interestingly, the
Drude tail in metals and in doped materials is displayed for SrNbO3 whereas it
is absent in SrNbO3.4. More obscure, these plasmons have been only observed in
the enriched SrNbO3.4 and they are transformed into a single intense plasmon
as detected in the loss function of SrNbO3.

These features may be connected to a possible metal-insulator transition. In a
recent work Katsensoln , in the dynamical mean-field theory (DMFT) framework
for impurities in materials Georges ; kotliar  it has been demonstrated that,
taking also into account the particle-hole interactions by means of the Bethe
Salpeter Equation (BSE), strongly correlated systems display an analogous
spectral weight transfers in the calculated loss function with respect to the
Independent Particle (IP) Random Phase Approximation (RPA) loss function: the
final result is a generation of new different plasmon peaks accompanied by a
suppression of the Drude response.

Nevertheless, SrNbO3 has been broadly considered a weakly correlated material
and it remains an interesting subject whether it can have a Mott transition
when it is enriched with O atoms. In this letter, we investigate the loss
function of SrNbO3+δ (with $\delta=0$ and $\delta=0.4$) by using first
principle calculations. The experimental findings Asmara  are compared with
our RPA loss function with and without Local Field Effects (LFEs). The
importance of LFEs, the rapid variations of the dielectric function, on loss
spectra has been widely studied in various materials LFE-Louie ; LFE-Daling ;
LFE-Waidmann ; LFE-Arya ; LFE-Cudazzo ; LFE-Vast  in which LFE strongly
modified the shape and intensity of plasmon located at high energies where
semicore local electrons were evolved LFE-Vast . In particular, we show that
the RPA loss function calculations reproduce the observed conventional plasmon
in SrNbO3. The nature of this plasmon is determined by the interplay between
the inter and intra band transitions. In the IP approximation, this plasmon
peak is still present in the enriched SrNbO3.4. Interestingly, the spectral
weight transfer and the creation of unconventional plasmon peaks in the
SrNbO3.4 have to be attributed to LFEs which become determinant to
theoretically describe experimental findings Asmara .

##  II Computational details

The structure and physical properties of Strontium Niobates are known to be
strongly dependent on their O atoms content SrTiO3_3 ; SrTiO3_4 ; SrTiO3_9 ;
SrTiO3_5 .The simplest SrNbO3 compound is a red d1 Oxides with cubic
perovskite structure. Besides, a homologous series SrnNbnO3n+2 family can be
derived from the 3D network of the SrNbO3 perovskite structure by separating
the NbO6 octahedra parallel to the [101] planes SrTiO3_4 ; SrTiO3_9 ; SrTiO3_5
. These O atom-rich series have strong anisotropic structures which exhibit
extra O planes separating the neighbouring layers along one crystal
orientation.

A typical compound of SrNbO3+δ (with $\delta=0$ and $\delta=0.4$) are chosen.
We start with PBE-DFT ground state calculation using plane wave basis set with
500 eV cutoff energy and the PAW method as implemented in Vienna ab initio
simulation package (VASP) Vasp1 ; Vasp2 . The used pseudopotentials are taken
from the VASP library. The Sr-4s4p5s, Nb-4p5s4d, and O-2s2p orbitals are
treated as valence states. The geometry structures are fully relaxed with
$14\times 14\times 14$ $\Gamma$ centered k-point mesh for cubic SrNbO3 and
$9\times 7\times 1$ $\Gamma$ centered k-point mesh for layered SrNbO3.4
supercell. The equilibrium lattice parameters are very close to that reported
in the preceding works SrTiO3_4 ; SrTiO3_5 . In order to deal with the loss
function, we have calculated the microscopic dielectric function
$\varepsilon({\bf q},\omega)_{{\bf G},{\bf G}^{\prime}}$ (where ${\bf q}$ and
${\bf G}$ are the reciprocal vectors in the first (IBZ) and full Brilloun
zone, respectively, and $\omega$ is the frequency) in the RPA approach
Onida_RMP . The macroscopic loss function is determined at the limit ${\bf
q}\to 0$ as:

$\lim_{{\bf q}\to 0}\Im(\varepsilon({\bf q},\omega)_{{\bf G},{\bf
G}^{\prime}})^{-1}_{{\bf G}={\bf G}^{\prime}=0}.$ (1)

The off-diagonal elements of the dielectric function $\varepsilon({\bf
q},\omega)_{{\bf G},{\bf G}^{\prime}}$ in the matrix inversion of equation (1)
are responsible for the LFEs and they become important in the inhomogeneous
systems specially in compounds where the localization of the atomic orbitals
like in the 3d and 4d transition metals plays a major role. In this
calculations we have compared the RPA loss function spectrum with LFE and
without them (i.e. inverting only the head (${\bf G}={\bf G}^{\prime}=0$) of
the dielectric function in equation (1), also called Independent Particle
Approximation (IPA)). The LFEs are included taking into account the dielectric
matrix with the energy cutoff for ${\bf G}$ vectors equal to $150$ eV. In our
analysis, we have also included the intraband transitions in the RPA
susceptibility as implemented in dp-code Cazzaniga  in order to quantify the
effects on the Drude tail and on the plasmon peak nature.

##  III Results

Figure 1: From top to bottom, the macroscopic SrNbO3 RPA dielectric function
Real $\varepsilon_{1}$, Imaginary $\varepsilon_{2}$ part, and loss function.
The Black circle experimental results form Ref. Asmara , red and green shaded
lines are, respectively, the results for only interband transitions RPA with
and without LFE. The purple solid line is the RPA $+$ LFE, inter and intra
bands. The horizontal dashed line indicates that the zero of $\varepsilon_{1}$
and it is the point set for placing the plasmon peak.  Figure 2: From top to
bottom, the macroscopic SrNbO3.4 RPA dielectric function Real
$\varepsilon_{1}$, Imaginary $\varepsilon_{2}$ part, and loss function. The
Black circle experimental results form Ref. Asmara , red and green shaded
lines are with and without LFE.

In Fig. (1), the complex dielectric function ($\varepsilon$) and Loss function
for SrNbO3 are shown. The upper and central panels show the Real part
($\varepsilon_{1}$) and Imaginary part ($\varepsilon_{2}$), respectively. The
lower panel is the loss function at long wavelength limit (${\bf q}\to 0$).
Experimental values (black circle line) are from the spectroscopic
ellipsometry measurements of Sr1-xNb1-yO3+δ film which deposited on $[001]$
LaAlO3 substrates under low O atom pressure (labelled as $lp-SNO$ in Ref.
Asmara ).

In the IP approximation, the most pronounced peak (dashed green line) of the
loss functions is located at 0.87 eV, which coincides with zero of the real
part and with a vanishing imaginary part of the dielectric function. This
indicates the peak is a collective excitation of electrons which involves the
plasmon resonance determined by the interbands transition of Nb 4d orbital
near the Fermi energy to O 2p orbitals. Another broad peak shows up at about
$3$ eV with the local minima of the $\varepsilon_{1}$. It is worth to noticing
that this broad peak is another form of collective excitation which is not
conventional plasmon as the later requires zero of real part dielectric
function. In SrNbO3, LFEs (dashed red line) do not play an important role and
the loss function is almost unchanged. Nevertheless, when adding the intraband
transitions (of Nb 4d orbitals), the total Drude tail shifts the plasmon peak
up to $2.34$ eV in the loss function. The position of new plasmon is about
$0.44$ eV higher with respect to the experimental findings. The shape and
intensity of the loss function are in good agreement with the experimental
loss function. From these calculations, we infer that the plasmon at 1.9 eV in
SrNbO3 are originated by cumulative effects of interband and intraband
transitions.

The situation in the SrNbO3.4 is very different. In Fig. 2. we shows the same
calculations for SrNbO3.4. The direction of the electric field is parallel to
the $[001]$ axis. This choice is originated by the experimental spectrum which
shows the insulting phase of Sr1-xNb1-yO3+δ film grown under high O atom
pressure (labelled as $hp-SNO$ in Ref. Asmara ). SrNbO3.4 has a strong
anisotropy similar to a quasi-1D material (which a layered structure shows in
$[001]$ direction) whereas in the other directions it maintains perovskite-
like structures supp .

Figure 3: From top to bottom, the macroscopic SrNbO3.4 (n = 4, 6,…) RPA
dielectric function Real $\varepsilon_{1}$, Imaginary $\varepsilon_{2}$ part,
and loss function. The Black circle experimental results form Ref. Asmara ,
red and green shaded lines are, respectively, the results for only interband
transitions RPA with and without LFE.

The experimental findings (black dotted line) display a different physics. The
Drude tail is absent, indication of an insulator behaviour in such direction.
The main Plasmon peak in SrNbO3 is not present and, through spectral weight
transfer, new peaks are created at 1.7 eV, 3.0 eV, and 4.0 eV. On the other
hand, the IP-RPA calculations of the loss function (green dashed line) the
intense plasmon peak is still present in SrNbO3 but red shifted in energy up
to 1.2 eV. The Drude tail is absent demonstrating that the ground state DFT
calculations are able to reproduce the insulating phase in this polarization
direction. The overall dielectric function $\varepsilon$ does not reproduce
the experiments in the energy range between 0.5 eV and 4.0 eV . In the LFE-RPA
picture (red line), two main plasmon peaks at 1.9 eV and 2.9 eV are found in
good agreement with the experimental peaks. The third peak at 4 eV is instead
not present. With respect to the experimental findings, the peaks are more
intense. More important, differently to the experimental case, the peaks
correspond to the zeros of the real part of the dielectric function.

Since SrNbO3.4 belongs to n = 5 members of the layered series SrnNbnO3n+2
family, in which the layers are n NbO6 octahedra thick. We repeat the same
calculations with a new structure of SrNbO3.4 supp  which involves well-
ordered stacking sequences of layers with different thickness, namely, the
structure with sequence n = 4, 6, 4, 6,… This configuration shares the same
total electron density as n = 5 SrNbO3.4 and hence reveals the role of
microscopic electron distribution in determining the theoretical dielectric
properties. In Fig. 3, the IP-RPA loss function is almost unchanged with
respect to SrNbO3.4. In contrast, in the RPA-LFE results three main peaks
appear in the loss function at 1.0 eV, 2.0 eV, and 3.3 eV. They are less
intense and the real part of the dielectric function becomes more similar to
the experimental one. The shift between the imaginary part of the dielectric
function and the loss function, indicative of plasmon features, is absent: in
this configuration these peaks are rather collective excitations. Furthermore,
it has to be considered that in the experiment Asmara  the sample under study
may contain different oxygen content and configurations. Therefore, in
principle, these collective modes can become multiple genuine plasmons. In
summary, we have analyzed the dielectric function and loss function of two
typical Strontium Niobate compounds: SrNbO3 and SrNbO3.4. Through fully ab
initio calculations, we proved that most of the behaviour of experimental
plasmons generation can be reproduced within RPA approach when the LFEs are
taken into account. Our results have shown that in the isotropic metallic
SrNbO3 the plasmon determined by the coexistence of both inter and intra band
transitions, dominates the collective excitation in this material. On the
other hand, in SrNbO3.4 due to the layered anisotropic structure, LFEs have a
fundamental impact on the loss spectrum and reshape completely the loss
function. The spectral weight transfer is stable and it is not affected by
different O atom configurations.

###### Acknowledgements.

This work is supported by the Ministry of Educations (MOE2015-T2-2-065,
MOE2015-T2-1-099, and MOE2015-T2-2-147), Singapore National Research
Foundation under its Competitive Research Funding (NRF-CRP 8-2011-06,
R-398-000-087-281 and NRF-CRP15-2015-01) and under its Medium Sized Centre
Programme (Centre for Advanced 2D Materials and Graphene Research Centre), NUS
YIA, and FRC (R-144-000-379-114 and R-144-000-368-112).

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