# Probing the electronic and local structure of Sr2-xLaxCoNbO6 using near-edge
and extended x-ray absorption fine structures

Ajay Kumar  Department of Physics, Indian Institute of Technology Delhi, Hauz
Khas, New Delhi-110016, India  Rishabh Shukla  Department of Physics, Indian
Institute of Technology Delhi, Hauz Khas, New Delhi-110016, India  Ravi Kumar
Atomic and Molecular Physics Division, Bhabha Atomic Research Centre, Mumbai
400085, India  R. J. Choudhary  UGC-DAE Consortium for Scientific Research,
Indore-452001, Madhya Pradesh, India  S. N. Jha  Beamline Development and
Application Section, Physics Group, Bhabha Atomic Research Centre, Mumbai
400085, India  R. S. Dhaka<EMAIL_ADDRESS>Department of Physics,
Indian Institute of Technology Delhi, Hauz Khas, New Delhi-110016, India

###### Abstract

We report the electronic and local structural investigation of double
pervoskites Sr2-xLaxCoNbO6 ($x=$ 0–1) using x-ray absorption near edge
structure (XANES) and extended x-ray absorption fine structures (EXAFS) at the
Nb, Co, and Sr $K$-edges. The $ab$ $initio$ simulations and detailed analysis
of the Nb and Co $K$-edge XANES spectra demonstrate that the observed pre-edge
features arise from the transition of 1$s$ electrons to the mixed $p-d$
hybridized states. We reveal a $z$-out Jahn-Teller (JT) distortion in the CoO6
octahedra, which decreases monotonically due to an enhancement in the JT
inactive Co2+ ions with $x$ . On the other hand, the $z$-in distortion in NbO6
octahedra remains unaltered up to $x=$ 0.4 and then decreases with further
increase in $x$. This sudden change in the local coordination around Nb atoms
is found to be responsible for the evolution of the antiferromagnetic
interactions in $x\geqslant$ 0.6 samples. Also, we establish a correlation
between the degree of octahedral distortion and intensity of the white line
feature in the XANES spectra and possible reason for this are discussed. More
interestingly, we observe the signature of KN1 double electron excitation in
the Sr $K$-edge EXAFS spectra for all the samples, which is found to be in
good agreement with the $Z$+1 approximation. Further, the Co L2,3 edge shows
the reduction in the crystal field strength and hence an increase in the
charge transfer energy ($\Delta_{ct}$) with the La substitution.

##  I  Introduction

The octahedrally-coordinated Co possesses the intriguing magnetic, transport,
optical, and electronic properties due to the small energy difference between
its various oxidation and spin-states Mabbs_book_73 ; Lloret_ICA_08 ;
Liu_PRB_18 . The strength of the crystal field (CF) is a crucial measure to
govern these properties, as a delicate competition between the Hund’s exchange
energy and CF splitting decide the electronic distribution in the CF splitted
t2g and eg states Raccah_PR_67 ; Vanko_PRB_06 ; Merz_PRB_11 . In this case,
the Co–O bond distance plays the key role in controlling the CF strength. For
example, Chen et al. have reported a correlation between the Co–O bond length
and spin-state transition of Co3+ from high spin (HS) state (3$d^{6}$;
t${}_{2g}^{4}$e${}_{g}^{2}$) to the low spin (LS) state (3$d^{6}$;
t${}_{2g}^{6}$e${}_{g}^{0}$) in SrCo0.5Ru0.5O3-δ using the pressure
(hydrostatic) dependent x-ray absorption and emission spectroscopy
Chen_JACS_14 . It was found that a Co–O bond length $>$1.93 Å favors the HS
sate, whereas a smaller value results in the most stable LS state due to the
large CF splitting in the latter Chen_JACS_14 . Also, an increase in the
Co–O–Co bond angle enhance the Co $3d$–O $2p$ orbital hybridization, which
stabilize the intermediate spin (IS) (3$d^{6}$; t${}_{2g}^{5}$e${}_{g}^{1}$)
state of Co in LaCoO3 Knizek_PRB_05 . On the other hand, the Co2+ usually
stabilizes in the HS state (3$d^{7}$; t${}_{2g}^{5}$e${}_{g}^{2}$) due to the
weaker CF as compared to Co3+ in Co based perovskite oxides Mabbs_book_73 ;
Lloret_ICA_08 ; Kumar_PRB1_20 . Thus a precise understanding of the chemical
and local coordination environment around Co and its evolution with the
external parameters like temperature, mechanical pressure, and chemical
pressure (doping) and hence CF strength is extremely important to engineer the
various physical properties of the complex oxides and their vast use in the
technological applications Tao_NM_03 ; Chakrabartty_NP_18 .

In this context, a stable crystal structure of the perovskite oxides [ABO3; A:
rare earth/alkali earth metals, B/B: transition metals (TM)] provides a
suitable matrix to accommodate the wide range of elements with tuneable CF
strength and hence degree of TM–O orbital hybridization Medarde_PRB_06 ;
Krayzman_PRB_06 ; Shukla_PRB_18 . Further, a 50% substitution of B-site
cations results in the alternating ordering of BO6 and B′(say)O6 octahedra,
where the degree of ordering depends on the ionic and valence mismatch between
two $B$-site cations Anderson_SSC_93 ; King_JMC_10 ; Vasala_SSC_15 ;
Galasso_JPC_62 . This ordering results in the additional B-O-B′-O-B exchange
interactions along with the conventional B-O-B channels, resulting in the
exotic magnetic and transport properties Niebieskikwiat_PRB_04 ;
Sanchez_PRB_02 ; Jung_PRB_07 ; Erten_PRL_11 ; Frontera_PRB_04 . For example,
the anti-site disorder driven several exchange interactions are found to play
a crucial role in determining the inverse exchange bias effect observed in
Gd2CoRuO6 Das_PRB_20 . In the double perovskite family, the Sr2CoNbO6 is one
of the most interesting candidates, where high resolution electron microscopy
(HREM) show the $B$-site ordered domains of less than 100 Å, however no such
evidence was observed in the x-ray and neutron powder diffraction measurements
due to their large coherence length as compared to the size of the ordered
domains Kumar_PRB1_20 ; Azcondo_DT_15 . This is because Sr2CoNbO6 is a border
line candidate of the ordered-disordered configurations due to the moderate
ionic and valence mismatch between Co3+ and Nb5+ ions and even a small
perturbation can transform it into either regime King_JMC_10 ; Vasala_SSC_15 .
This intriguing crystal structure of Sr2CoNbO6 is closely related to its
observed colossal dielectric properties, complex ac impedance spectroscopy,
and cluster glass like magnetic ground state Wang_AIP_13 ; Bashir_SSS_11 ;
Kumar_PRB2_20 .

Recently, we studied the doping induced spin-state transition of Co3+ from HS
to LS/IS states in Sr2-xLaxCoNbO6 ($x=$0–1) samples using the magnetization
measurements Kumar_PRB1_20 . Here it is important to note that the
substitution of each La3+ ion at Sr2+ site transform one Co from 3+ to 2+
state, which systematically reduces the CF strength and stabilize the Co2+ in
HS state Mabbs_book_73 ; Kumar_PRB1_20 ; Bos_PRB_04 . A very small tetragonal
($I4/m$) distortion was observed in the $x=$ 0 sample from the perfect cubic
($Pm\bar{3}m$) symmetry, which increases with the La substitution ($x$) up to
the $x=$ 0.4, and then transformed into the monoclinic ($P2_{1}/n$) structure
for $x\geqslant$0.6 samples Kumar_PRB1_20 . Further, an increase in the
valence mismatch between $B$-site cations ($\Delta$V= 2 $\rightarrow$ 3) with
$x$ transform the disordered parent sample to the highly ordered state in the
$x=$ 1 sample Kumar_PRB1_20 . Importantly, an abrupt enhancement in the
$B$-site ordering and evolution of antiferromagnetic (AFM) interactions were
reported for $x\geqslant$0.6 samples Kumar_PRB1_20 ; Kumar_PRB2_20 , which
indicate the strong correlation between the local crystal structure and the
complex magnetic interactions present in these samples. Also, the specific
heat measurements indicate the persistence of the discrete energy states
resulting from the crystal field splitting, spin-orbit coupling, and
octahedral distortion Kumar_PRB2_20 . However, extent of the octahedral
distortion, which can play a key role in governing the various spin-states of
Co3+, and its evolution with the La substitution have not been quantified.
Therefore, a systematic study of change in the electronic structure and local
coordination around different cations is essential to understand the origin of
complex magnetic interactions in these samples. In this direction, x-ray
absorption spectroscopy (XAS) is a versatile tool to investigate the ligand-
field, spin-state, valence state, centrosymmetry, metal-ligand overlap, and
the local coordination geometry Bunker_book_10 .

Therefore, in this paper, we use x-ray absorption spectroscopy to investigate
the element specific electronic and local structure of Sr2-xLaxCoNbO6 ($x=$
0–1), and focus on the Nb, Co and Sr $K$-edges in both the XANES and EXAFS
regions. The Nb $K$-edge XANES shows a very weak pre-edge feature in spite of
the completely unoccupied 4$d$ states, indicating the presence of Nb atoms in
the centrosymmetric octahedral environment. However, a strong pre-edge feature
is observed in case of Co $K$-edge XANES spectra, which indicates its off-
center displacement in the CoO6 octahedra. The analysis suggests that pre-edge
features in both Nb and Co $K$-edge spectra result from the transition of 1$s$
electrons to the mixed $p-d$ hybridized states. We demonstrate that the
intensity and position of the post edge feature in the Nb $K$-edge spectra
depend on the nature and arrangement of the $A$-site atoms, respectively.
Moreover, we found a strong correlation between the intensity of the white
line and degree of octahedral distortion in CoO6 and NbO6, where a higher
degree of octahedral distortion (compression as well as elongation) results in
the reduction of white line intensity of the respective $K$-edge absorption
spectra. The Co $K$-edge spectra show the monotonic reduction in the
distortion in CoO6 octahedra with the La substitution ($x$). However, the Nb
$K$-edge spectra show an abrupt change in the NbO6 octahedral distortion for
$x\geqslant$0.6 samples, causing a sudden enhancement in the degree of
$B$-site ordering, resulting in the observed AFM interactions at this critical
doping. Further, the presence of multielectronic excitation in all the samples
in Sr $K$-edge EXAFS spectra is confirmed using Z+1 approximation as well as
La $L_{3}$-edge absorption spectra. The Co L2,3 edge spectra show the presence
of Co3+ in IS/HS state and significant reduction in the crystal field strength
with $x$.

##  II  Experimental

Polycrystalline samples of Sr2-xLaxCoNbO6 ($x=$ 0–1) were synthesized by usual
solid-state route, more details and characterization can be found in ref.
Kumar_PRB1_20 . The x-ray absorption spectroscopic data were recorded at Nb,
Co and Sr $K$-edges in the transmission mode for all the samples using BL-09
(4–25 keV) beam line at Indus-2 synchrotron source (2.5 GeV, 200 mA) at Raja
Ramanna Center for Advanced Technology (RRCAT) in Indore, India. The
polycrystalline powdered samples were thoroughly mixed in the boron nitride
(BN) using a mortar pestle in such a proportion that absorption edge jump
($\Delta\mu$) lies between 0.7–1, calculated using the XAFSmass code
Klementiev_JPCS_16  and then pressed the total 100 mg powder into the circular
disc of $\sim$15 mm diameter, and sandwich between the kapton tape for the
stability. A Si(111) double crystal monochromator was used for the precise
energy selection of the incident beam with a resolving power of
$\Delta$E$/E\approx$10-4. More technical details of the beamline can be found
in Basu_JPCS_14 . We calibrate all the recorded spectra using the following
energy dependent relation Bunker_book_10 :

$E_{\rm calb}=\frac{gE_{\rm expt}}{\sqrt{(E^{2}_{\rm
expt}-g^{2})}sin(\Delta\theta)+gcos(\Delta\theta)},$ (1)

where, $E_{\rm calb}$ and $E_{\rm expt}$ are the calibrated and experimentally
recorded energies, respectively, $g\approx$1977.1 eV for the Si(111) crystal,
and $\Delta\theta$ was calculated, by recording the same absorption edge for
the reference metal foils, using the equation below:

$\Delta\theta=sin^{-1}\left(\frac{g}{E_{0}}\right)-sin^{-1}\left(\frac{g}{E_{\rm
calc}}\right),$ (2)

where, $E_{\rm calc}$ is the position of the first maxima of the first
derivative of the absorption coefficient of the reference metal foil and E0 is
its known reference energy taken at 7709 and 18986 eV for Co and Nb $K$-edges,
respectively from ref. Bunker_book_10 . Due to unavailability of the reference
metal foil, Sr K-edge spectra was calibrated by aligning the first maxima of
the first derivative for the $x=$ 0 sample at 16105 eV (for Sr2+
Sahai_JCIS_2000 ) using the above energy dependent calibration equations (1,
2) and then other samples were calibrated with reference to $x=$ 0.

A linear pre-edge and a spline post edge background were subtracted from the
XAS spectra and the resultant spectra were normalized at 800 eV above the edge
jump for all the samples. We use the Gaussian peak shape and an error step
function (accounting for the transition to the continuum) to fit the XANES
region (up to $\sim$50 eV above the edge jump) in the athena software
Ravel_JSR_05 . The position and FWHM of the step function have been decided in
such a way that it touches the rising edge region with the similar slope as
that of edge and height was adjusted such that it touches the lowest feature
of the XANES spectra under consideration. The FWHM and height of the error
function were fixed for all the samples, whereas its position was varied as
per the change in the position of rising edge region due to the valence shift.
The FEFF9.6.4 code for the $ab$ $initio$ calculations, using the full multiple
scattering within a sphere around the absorbing atom with the muffin tin
approximation in a self-consistent loop Rehr_PCCP_2010 ; Rehr_RMP_2000 , is
used to simulate the XANES spectra of Nb, Co, and Sr $K$-edges. The input
structural parameters are taken from the refinement of diffraction patterns
where the Nb and Co atoms are considered in completely ordered and disordered
states at two different B-sites for the $x=$ 0 and 1 samples, respectively
Kumar_PRB1_20 ; Bos_PRB_04 . We use Hedin-Lundqvist (HL) exchnage correlation
potential to simulate Nb and Co $K$-edges, and Dirac-Hara+HL for Sr $K$-edge
XANES spectra with a fully screened core-hole (final state rule).

The Co $L_{2,3}$-edge XAS spectra were recorded in the total electron yield
mode at beamline BL-01of the same synchrotron source. A constant energy shift
in each spectra was employed to align the center of L3 edge of the $x=$ 1
sample corresponding to the octahedrally coordinated Co2+ in HS state, as in
ref. Merz_PRB_11 , due to the unavailability of the reference sample. The Co
$L_{2,3}$-edge is simulated by the CTM4XAS software based on the atomic-
multiplet, crystal-field, and charge transfer approach Groot_PRB_90 ;
Stavitski_Micron_10 , as single electron approach is unable to reproduce the
transition metal $L$-edge XAS spectra. The simulated spectra of HS Co2+ and HS
Co3+ were arbitrarily shifted to align with the $x=$ 1 and 0.2 samples,
respectively.

Figure 1: (a) The normalized Nb $K$-edge XANES spectra of Sr2-xLaxCoNbO6 ($x=$
0–1) samples where 1, 2, 3 represent the main absorption features, and
enlarged view of features 2 and 3 are shown in the insets. (b) The first
derivative of the experimental absorption spectra of the Nb $K$-edge along the
simulated spectrum for the $x=$ 0 and 1 samples and their corresponding Nbabs
$d$-LDOS in the XANES region. The red dashed ellipse highlight the presence of
pre-edge feature in both experimental and simulated spectra, and sharp Nbabs
$d$-LDOS at this energy. All the experimental spectra have been vertically
shifted by fix 0.025 units for the clarity. (c) The best fit of XANES spectra
using the Gaussian peak shape for all the samples, and an error function
accounting for the different absorption features and the edge jump,
respectively, shown for the $x=$ 0 sample. Here, each spectrum is vertically
shifted cumulatively by 0.15 units for the clear presentation. (d, e) The
experimental and simulated Nb $K$-edge absorption spectra along the calculated
$p$-LDOS of absorbing Nb atoms for $x=$ 0 and 1 samples. The experimental
spectra have been vertically shifted by 0.3 units for the clarity. The black
dashed line represent the Fermi level and red dashed lines indicate the shift
in the Nbabs $p$-LDOS peak corresponding to the feature 3 (shaded region).

##  III  Results and discussion

###  III.1  X-ray absorption near edge structure:

The normalized XANES spectra of Nb $K$-edge is presented in Fig. 1(a) for
Sr2-xLaxCoNbO6 ($x=$ 0–1) samples to probe the electronic structure and
multiple scattering (MS) mechanism around the Nb atoms. We first discuss the
pre-edge region, where no significant intensity is evident around 18980 eV at
the first glance, in spite of the completely unoccupied Nb 4$d$ states
(4$d^{0}$ for Nb5+) in these samples. Note that the low intense pre-edge
features in the XANES spectra of TM oxides result from the transition of 1$s$
electrons to the unoccupied pure $d$ states (quadruple transitions; $\Delta
l=$2), and/or transition into the $d$ states hybridized either with their $p$
states or oxygen 2$p$ states Joly_PRL_99 ; Yamamoto_XRS_08 ; Farges_PRB_97 ;
Krayzman_PRB_06 . The probability of the pure quadruple transitions is
significantly lower and hence intensity of the pre-edge feature(s) depends
mainly on the degree of $p-d$ hybridization Farges_PRB_97 ; Yamamoto_XRS_08 .
The group theory predict the absence of the $p-d$ mixing and hence the pre-
edge feature in case of the ideal octahedral symmetry ($O_{h}$), i.e., when
absorbing atom lies in the centrosymmetric $O_{h}$ environment and intensity
of the pre-edge features increases with increase in the off-center
displacement of the absorbing atoms Yamamoto_XRS_08 ; Vedrinskii_JPCM_98 ;
Frenkel_PRB_04 . Interestingly, we observe an asymmetricity in the first
derivative of the absorption coefficient in the pre-edge region for all the
samples as highlighted by the dotted ellipse in Fig. 1(b), which indicates the
minimal off-center displacement of the Nb atoms in NbO6 octahedra. However,
the weak pre-edge feature can also result from the thermal assisted dynamic
distortion in the NbO6 at room temperature, as reported even in
centrosymmetric perovskite EuTiO3 Ravel_Ferroelectrics_98 . In order to
understand the origin of weak pre-edge feature, the Nb $K$-edge XANES spectra
have been simulated for the $x=$ 0 and 1 samples using the FEFF 9.6.4 code for
a spherical cluster of 45 atoms. Here, it is interesting to note that we have
allowed only dipole transitions in the simulations and the weak pre-edge
feature is well reproduced in the simulated spectra for both the samples, as
compared in Fig. 1(b). In fact, we find that there is no significant
enhancement in the pre-edge feature by introducing the quadruple transitions
from 1$s$ to the unoccupied $d$ states (see Fig. 1 of Kumar_XAS_SI ). This
suggest that the pre-edge feature in all the samples is predominantly due to
the dipolar transition from the 1$s$ to the $d$ states of Nb that hybridize
with its $p$ states and/or $p$-states of oxygen. To understand further, we
plot the $d$-local density of states (LDOS) for the absorbing Nb atoms (Nbabs)
in Fig. 1(b) for the $x=$ 0 and 1 samples. Interestingly, we observe a sharp
Nbabs $d$-LDOS at around 18980 eV corresponding to the weak pre-edge feature,
which suggests the key role of the Nb $d$ states in determining the pre-edge
feature in these samples. However, a slight discrepancy in intensity behavior
between Nb $d$-LDOS and the simulated spectra is evident in the pre-edge
region for the $x=$ 0 and 1 sample. The possible reason can be a change in the
degree of $p$–$d$ mixing due to different octahedral distortion in two cases,
which is also observed in the EXAFS analysis, discussed in the next
subsection. On the other hand, this small change in the pre-edge feature with
$x$ is difficult to detect in the experimental spectra.

We now discuss the rising edge (around 18990 eV) and other features (denoted
by 1, 2, 3) observed in the Nb $K$-edge, see Fig. 1(a). We note that there is
no significant shift in the rising edge position suggesting the invariance in
the valence state of Nb with the La substitution. A comparison with the
reference Nb2O5 sample confirm the pentavalent state of Nb in all the samples,
see Figs. 2(a, b) of Kumar_XAS_SI . Further, two prominent features 1 and 2,
which are separated by $\sim$ 13 eV, can be attributed to the
1$s\rightarrow$5$p$ transition and the resonance due to multiple scattering of
the photoelectrons from the neighboring atoms, respectively Ruckman_PRB_98 ;
Lytle_PRB_88 ; Marini_JPCM_16 . It is interesting to note that the intensity
of the feature 2 remains almost invariant up to the $x=$ 0.4 and then
increases monotonically with further increase in the La substitution for the
$x\geqslant$0.6 samples [see inset (a1) of Fig. 1(a)]. This is possibly due to
an abrupt change in the local coordination environment around the Nb atoms,
which can be speculated as the origin of the evolution of antiferromagnetic
interactions in the $x\geqslant$0.6 samples Kumar_PRB1_20 . We will get back
to this point during the discussion of the EXAFS part in the next subsection.
Also, a post-edge feature (marked as 3) is observed in all the samples, which
shifts towards the higher energy side (see Fig. 3 of Kumar_XAS_SI  for
clarity) with a monotonic reduction in its strength [see inset (a2) of Fig.
1(a)] with the La substitution ($x$).

To understand the quantitative change in the different spectral features with
$x$, we perform the deconvolution of all the spectra using three Gaussian
components and a steplike error function, accounting for the continuum
excitations, as presented in Fig. 1(c) and the best fit parameters are
summarized in Table I of Kumar_XAS_SI . In order to understand the origin of
the feature 3, in Figs. 1(d, e), we show the simulated Nb $K$-edge spectra,
which are in good agreement with the experimental spectra in the entire energy
range. In fact, the tendency of a doublet around 19050–19110 eV turning into
the singlet with the La substitution (highlighted by downward arrows in the
inset of Fig. 3 of Kumar_XAS_SI ) is well reproduced in the simulated spectra,
discussed in next paragraph. Note that we use a shift of –22.0 eV and an
additional broadening of 2.0 eV in the simulation to account for the
uncertainty in the Fermi level and core-hole life time, respectively.
Interestingly, we find that the feature 3 shifts towards the higher energy
with $x$ and to understand this, we plot the Nbabs $p$-LDOS in Figs. 1(d, e)
for the $x=$ 0 and 1 samples, respectively. A clear shift of around 4.5 eV in
the unoccupied density of Nb $p$ states correspond to feature 3 (shaded areas)
is observed with $x$, as highlighted by the red vertically dashed lines, which
indicate an increase in the probability of 1$s\rightarrow p$ transition at the
higher energy in case of the $x=$ 1 sample. This perturbation in the
unoccupied Nb $p$-LDOS states with La substitution is originating either from
the atomic-like electronic redistribution due to evolution of the new La
states or/and from change in the multiple scattering mechanism due to
variation in the local coordination environment around the Nb atoms. However,
these need to be further explored for complete understanding.

Figure 2: (a) The simulated Nb $K$-edge XANES spectra for the $x=$ 0 sample by
varying the number of first nearest Sr atoms. Insets (a1) and (a2) represent
the enlarged view of the spectra in different energy regimes. (b) The Nbabs
$p$-LDOS corresponding to the XANES spectra presented in (a), and the inset
shows the enlarged view of the highlighted region. (c) The comparison of
simulated Nb $K$-edge spectra of the $x=$ 0 sample for all first neighbor
$A$-site atoms as Sr and La (on the left axis) and corresponding Nbabs
$p$-LDOS (on the right axis), and the inset shows the enlarged view of the
highlighted region. (d) The Nb $K$-edge spectra for the $x=$1 sample with all
eight first neighbor $A$-site atoms as eight Sr, four each of La and Sr as
well as eight La. The insets (d1) and (d2) represent the enlarged view of the
highlighted region and the corresponding Nbabs $p$-LDOS, respectively.

Therefore, we simulate the Nb $K$-edge XANES spectra for the $x=$ 0 sample by
selectively removing the neighboring atoms from the spherical cluster. We find
a strong dependency of feature 3 on the first nearest Sr atoms located at
3.430 Å from the absorbing Nb atoms, where the intensity suppressed completely
on removal of all the eight first nearest Sr atoms, as show in Fig. 2(a) and
its inset (a1). More interestingly, the tendency of doublet turning into the
singlet in the EXAFS region is also evident on the removal of these Sr atoms,
as shown more clearly in the inset (a2) of Fig. 2(a). The corresponding Nbabs
$p$-LDOS are presented in Fig. 2(b), where inset show the enlarged view of the
highlighted (red dashed rectangle) region corresponding to feature 3. A
monotonic reduction in the Nbabs $p$-LDOS is clearly observed with the removal
of first nearest Sr atoms, causing a reduced probability of the 1$s\rightarrow
p$ transitions at this energy and hence decrease in the strength/intensity of
feature 3. This may result from the non-local hybridization between the Nb and
Sr $p$-orbitals, where an obvious reduction in the Sr $p$-LDOS is observed
around this energy (see Fig. 4 of Kumar_XAS_SI ). A non-local orbital
hybridization has been widely reported in several correlated electron systems;
for example, the presence of Pt and Ir $d$-LDOS at the different energies
cause a shift in the pre-edge peak in the Co $K$-edge spectra of La2CoPtO6 and
La2CoIrO6 due to Pt/Ir 5$d$-Co 4$p$ orbital hybridization Lee_PRB_18 .

Further, we observe a small but consistent shift in the Nb $p$-LDOS towards
the lower energy with the removal of the Sr atoms [see inset of Fig. 2(b)],
which is opposite to the observation for feature 3 in the experimental data
with $x$ (see Fig. 3 of Kumar_XAS_SI ). Therefore, in order to disentangle the
effect of change in the multiple scattering paths (structure) and evolution of
new atomic-like states with $x$, we present in Fig. 2(c), the simulated Nb
$K$-edge XANES spectra by replacing all the first nearest Sr atoms by La atoms
in the crystal structure of the $x=$ 0 sample ($I4/m$ symmetry). A clear
reduction in the intensity of feature 3, and corresponding Nb $p$-LDOS [see
inset of Fig. 2(c)] are evident with the La substitution at the first nearest
Sr site. This clearly indicate that the strength of feature 3 depends on the
atomic-like states of Sr atoms. However, no shift in the Nb $p$-LDOS is
observed around feature 3 with La substitution. Further, the doublet at the
higher energy recovered with the La substitution, indicating the local
structure as its origin. In order to further confirm the nature of feature 3,
we now use the structural parameters of $x=$1 sample ($P2_{1}/n$ symmetry) and
simulate the Nb $K$-edge XANES spectra in three configurations: keeping (i)
eight Sr, (ii) four La and four Sr, and (iii) eight La first nearest $A$-site
atoms, as shown in Fig. 2(d). We find a reduction in the strength of feature 3
with decrease in the Sr concentration [see inset (d1)], which further
indicates its dependence on the atomic like states of the Sr atoms. On the
other hand, no shift in the Nb $p$-LDOS is observed with change in the nature
of first nearest $A$-site atoms [see inset (d2)], however, as discussed above,
a shift of $\sim$4.5 eV as compared to the $x=$ 0 sample is clearly visible
(see Fig. 5 of Kumar_XAS_SI  for the direct comparison). These results suggest
that unlike amplitude, the position of feature 3 depends mainly on the
arrangement of Nb–La/Sr scattering paths.

In Fig. 3(a) we show the normalized XANES spectra of Co $K$-edge, which show
several absorption features (marked A to G) for Sr2-xLaxCoNbO6 ($x=$ 0–1)
samples. The deconvolution of the different features using seven Gaussian
components and an error function are presented in Fig. 3(b) and the best fit
parameters are given in Table I of Kumar_XAS_SI . The pre-edge feature A
around 7710 eV can be attributed to the transitions from 1$s$ to 3$d$ states
of Co, which are mixed with its 4$p$ final states and/or 2$p$ states of oxygen
atoms, where the strength of this pre-edge feature increases with the extent
of $p-d$ hybridization Farges_PRB_97 ; Wu_PRB_04 ; Yamamoto_XRS_08 . The
presence of the strong pre-edge features in the Co $K$-edge XANES spectra
suggest the higher degree of off-centre displacement of Co in the octahedra as
compared to the Nb atoms (discussed above). This difference in the local
coordination environment around the two B-site cations is also supported by
the upcoming EXAFS analysis and useful to understand their complex electronic,
magnetic, and transport properties and hence their potential use as the
multiferroics Wang_AIP_13 ; Kumar_PRB1_20 ; Ishimatsu_PRB_15 ; Lines_Book_77 .
Further, when we increase the La concentration, a significant reduction in the
pre-edge feature A is clearly observed along with a monotonic shift towards
lower energy [see inset of Fig. 3(a)]. The electron doping in Sr2-xLaxCoNbO6
transforms Co3+ (in $x=$ 0) in to Co2+ (in $x=$ 1), which brings the $p$-$d$
hybridized states of Co closer to the 1$s$ states due to increase in the
screening effect, resulting in the shift of this quadruple assisted transition
to the lower energy. Additionally, an extra electron doped in the $d$ band of
Co (3$d^{6}$ in $x=$ 0 and 3$d^{7}$ in $x=1$) results in the reduction of the
density of the unoccupied $3d$ states with $x$, which reduces the transition
probability from 1$s$ to partially unoccupied 3$d$-O(2$p$)/Co(4$p$) states and
consequently reduction in the strength of the pre-edge feature A.

Figure 3: (a) The normalized Co $K$-edge XANES spectra of Sr2-xLaxCoNbO6 ($x=$
0–1) samples. The alphabets A–G represent the absorption features resulting
from various electronic transitions (see text for the detail). The red arrow
indicate the shift in the rising edge towards the lower energy with the La
substitution. The solid black arrow and dashed red line represent the shift
and invariance in the white line for x$\leqslant$0.4 and $x\geqslant$0.6
samples, respectively. Inset shows the enlarged view of the A and B (arrow)
pre-edge features for the clarity. (b) The best fit of the spectra using the
Gaussian peak shape and an error function as shown for the $x=$ 0 sample.
Here, each spectrum is vertically shifted cumulatively by 0.2 units for the
clear presentation.

Moreover, we observe an enhancement in the intensity of feature B at around
7713 eV with the La substitution (refer table I of Kumar_XAS_SI  to see the
quantitative changes), as indicated by an arrow in the inset of Fig. 3(a),
also see Fig. 6 of Kumar_XAS_SI  for clarity. These two pre-edge features (A
and B) have also been observed in the XANES spectra of LaCoO3 and assigned as
the transition from $1s$ state to the crystal field splitted unoccupied t2g
and eg states of Co in refs. Vanko_PRB_06 ; Haas_JSSC_06 . However, in the
present case the Co2+ is more likely to exist in the HS state (3$d^{7}$;
t${}_{2g}^{5}$e${}_{g}^{2}$) as compared to Co3+ due to weaker crystal field
in the former Mabbs_book_73 ; Kumar_PRB1_20 , which advocate the reduction in
the strength of the feature B due to decrease in the density of unoccupied eg
states with the La substitution, contrary to the observation as just discussed
above. This discard the transition from 1$s$ to eg states of Co as the origin
of this second pre-edge feature B. In this line, based on the charge-transfer
multiplet calculations, Vankó $et$ $al.$ assigned this second feature B as the
dipolar transition from $1s$ to Co($4p$)-O(2$p$)-Co′(3$d$) hybridize states in
LaCoO3, where Co′ represent the next nearest Co ions Vanko_arxiv_12 . This
hypothesis was further supported by the angular independence of the intensity
of this second feature (B) in the x-ray absorption study on the epitaxial thin
films of LaCoO3 Sterbinsky_PRB_12 . Here, the enhancement in the strength of
feature B with the La substitution indicate that latter hypothesis is clearly
favored in the present case. Also, the higher transition probability from 1$s$
state to the Co($4p$)-O(2$p$)-Nb(4$d$) hybridize states as compared to
Co($4p$)-O(2$p$)-Co′(3$d$) states can be the possible reason for the
enhancement in the strength of this feature B, as alternating ordering of CoO6
and NbO6 octahedra increases with the La substitution Kumar_PRB1_20 .

The feature C around 7717 eV is attributed to the charge transfer from oxygen
to Co ions in the final state due to strong Co 3$d$–O 2$p$ orbital
hybridization, indicating the small charge transfer energy ($\Delta_{ct}$) in
these compounds, i.e., $1s^{1}\underline{c}3d^{6}L^{2}\rightarrow
1s^{1}\underline{c}3d^{7}\underline{L}^{1}$ process, where $\underline{c}$ and
$\underline{L}$ represent the core and ligand holes, respectively. For
example, Chainani $et$ $al.$ found the ground state of LaCoO3, an octahedrally
coordinated Co3+ perovskite, to be 38.5% in $1s^{1}\underline{c}3d^{6}L^{2}$,
45.4% in $1s^{1}\underline{c}3d^{7}\underline{L}^{1}$, and 14.5% even in
$1s^{1}\underline{c}3d^{8}\underline{L}^{2}$ state from their x-ray
photoemission spectroscopic measurements Chainani_PRB_92 . A small but
consistent reduction in the feature C is observed with the La substitution
[see Table I of Kumar_XAS_SI  and Fig. 3(b)], which indicates a decrease in
the tendency of charge transfer from ligand to metal. A decrease in the
valence state of Co and increase in Co–O bond distances (as evident from the
EXAFS analysis, discussed later) with the La substitution can result in the
enhancement of the $\Delta_{ct}$ and hence reduction in the intensity of
feature C. This reduction in the orbital overlapping can also be the possible
reason for decrease in the electronic conductivity of these samples with $x$
Kumar_PRB1_20 . A similar correlation between the electronic conductivity and
orbital hybridization was observed from the XAS measurements of the
Sr2-xNdxCr1+x/2Re1-x/2O6 Blasco_PRB_07 .

It is important to note that a monotonic shift in the rising edge (region
between feature C and D) is observed towards the lower energy, as indicated by
red arrow in Fig. 3(a), which indicates the conversion of Co3+ into Co2+ with
$x$. A comparison with the Co $K$-edge XANES spectra of reference LaCoO3
sample from Shukla_JPCC_21  indicate the 3+ valence state of Co in the $x=$ 0
sample, as shown in Fig. 7 of Kumar_XAS_SI . Further, the most prominent
feature D around 1725 eV is attributed to the most probable $1s\rightarrow 4p$
dipole transition. A monotonic shift in this feature to the lower energy is
observed from $x=$ 0 to 0.4, as indicated by the black arrow in Fig. 3(a),
whereas this feature remains almost invariant with the further La substitution
for $x\geqslant$0.6 samples (vertical red dashed line). Note that the position
of the white line not solely depends on the valence state but also on the
local coordination environment around the absorbing atom Ishimatsu_PRB_15 . A
structural transition from the tetragonal to the monoclinic phase was reported
in these samples for $x\geqslant$0.6 from our previous x-ray and neutron
powder diffraction measurements Kumar_PRB1_20 ; Kumar_NPD_22  . Thus, a
cumulative effect of both valence shift and structural transformation results
in the minimal shift in the feature D for the $x\geqslant$0.6 samples.
Further, the strength of this feature increases with a monotonic reduction in
the peak broadening, which indicate a decrease in the disorders in these
samples with $x$. This is because the disorders result in the statistically
varying energy states, which broadens the spectra, whereas $f$-sum rule states
that the overall integrated intensity of the absorption spectra remains
constant, irrespective of the final state Altarelli_PRB_72 , causing a
reduction in the peak height Cuartero_PRB_16 . A similar variation in the
nature of the white line has been reported in the Ti $K$-edge absorption
spectra of various complex oxides with the varying degree of the disorders in
Farges_PRB_97 , where a comparison between the crystalline, radiation damaged,
and glassy CaTiSiO5 clearly demonstrate the increase in the broadening and
decrease in the main-edge peak height with reduction in the crystallinity
Farges_PRB_97 . A variation in the white line of the Co $K$-edge XANES spectra
of Sr2-xLaxCoNbO6 is more clearly illustrated in Fig. 8 of Kumar_XAS_SI . In
the present case, reduction in the octahedral distortion (in CoO6) as well as
B-site disorders with $x$ can be the possible origin for this monotonic
enhancement in the intensity of the white line of Co $K$-edge XANES spectra.
On the other hand, the Nb $K$-edge XANES spectra show no significant variation
in the intensity of the white line up to $x=$ 0.4 [see inset (a1) of Fig.
1(a)]. Note that the influence of the B-site disorders on the main-edge
feature is expected to remain same for both Co and Nb $K$-edge spectra. This
clearly demonstrates that the intensity of white line is solely governed by
the degree of the octahedral distortion in these compounds, which is also
evident from the EXAFS analysis, discussed later.

Further, we find the decrease in strength of post edge features E, F, and G
with $x$. For example, the evolution of a valley between E and F features is
clearly observed in Fig. 3(a). Using spin-polarized DOS calculations, Pandey
$et$ $al.$ assigned the similar post edge features in the Co $K$-edge XANES
spectra of LaCoO3 as the transition of Co 1$s$ electrons to Co 4$p$ states
hybridized with the La 6$p$ and/or O 2$p$ states Pandey_JPCM_06 . In the
present case, our analysis of the FEFF simulations suggest that a decrease in
the O 2$p$ states in the post edge region causes a reduction in theses
features (particularly feature F) with the La substitution (see Fig. 9 of
Kumar_XAS_SI ). Here it is important to re-emphasize the fact that
transformation of Co from 3+ to 2+ state with the La substitution decreases
the extent of orbital overlapping between Co and oxygen atoms due to lower CF
in the latter. This reduces the delocalized nature of the Co $p$ states and
hence the strength of post-edge features in the Co $K$ edge XANES spectra with
$x$.

Figure 4: (a) The normalized Sr $K$-edge XANES spectra of Sr2-xLaxCoNbO6 ($x=$
0–1) samples. The inset (a1) shows the first derivative of the absorption
coefficient, where the vertical dashed line represents the invariance in the
rising edge position. These curves are vertically shifted for better
presentation. The inset (a2) presents the enlarged view of feature II, where
arrow shows the shift in the peak position with $x$. (b) The best fit of the
Sr $K$-edge where each spectrum is vertically shifted cumulatively by 0.2
units for the clear presentation. (c) The normalized Sr $K$\- (on top and
right axis) and La $L_{3}$-edge (on bottom and left axis) spectra for the $x=$
1 sample in 150 eV range in the vicinity of the white line. The vertical
dashed line represents the presence of the absorption feature II at the same
position with respect to the white line in both the spectra. The experimental
and simulated Sr $K$-edge absorption spectra for the (d) $x=$ 0 and (e) $x=$ 1
samples along the calculated Sr and O $p$-LDOS. A simulated Sr $K$-edge
spectrum of the $x=$ 0 sample with the structural parameters of the $x=$ 1 is
compared in (e). The experimental spectra in (d, e) have been vertically
shifted by 0.5 units for clarity. The shaded regions in (d) and (e) represent
the unoccupied Sr $p$-LDOS corresponding to the feature II.

Now we move to the discussion of the Sr $K$-edge XANES spectra [see Fig. 4(a)]
to probe the chemical environment and local structure around the $A$-site
atoms and its evolution with La substitution. There is no signature of pre-
edge peak even in the first derivative of the absorption coefficient (not
shown) for all the samples. The inset (a1) of Fig. 4(a) shows the first
derivative of the absorption coefficient in the rising edge region, where no
significant shift (as indicated by the vertical dashed line) indicates the
invariance in the valence state of Sr with $x$. Further, the most intense
feature, marked as I, is attributed to the dipole allowed $1s\rightarrow 5p$
transitions, which slightly increases with $x$. Interestingly, the intensity
of post edge feature II increases significantly with $x$, as shown more
clearly in the inset (a2) of Fig. 4(a). Also, a shift in feature II towards
lower energy is clearly observed with $x$, as shown by the arrow in inset
(a2). In Fig. 4(b), we show the the best fit XANES spectra for all the samples
and the best fit parameters are given in Table I of Kumar_XAS_SI . A similar
post edge feature in the La L3 and Sr $K$ edge XANES spectra of La2-xSrxCuO4
samples is attributed to the removal of apical oxygen around the absorbing
atoms with Sr substitution Tan_PRL_90 . However, this hypothesis was later
contradicted by the theoretical calculations by Wu $et$ $al.$ Wu_PRBR_92 . In
order to understand the origin of this feature, La L3-edge XANES spectrum has
been recorded for the $x=$ 1 sample and compared with Sr $K$-edge in Fig.
4(c). The presence of similar absorption feature above the white line in case
of La L3-edge XANES spectra [see vertical dashed line in Fig. 4(c)] discard
the atomic-like states of Sr as the origin of this feature. This suggest that
the multiple scattering from the neighboring atoms give rise to the feature II
in these samples. Further, a comparison of the La L3-edges spectra of the $x=$
1 sample with the reference La2O3 sample indicate the presence of La in the
trivalent state (see Fig. 10 of Kumar_XAS_SI ).

In order to further understand the origin of this feature II, we simulate the
Sr $K$-edge XANES spectra for the $x=$ 0 and 1 samples for a cluster of 45
atoms and present in Figs. 4(d, e), respectively. A close synchronization with
the experimental spectra is observed after shifting the simulated spectra by
-25.5 eV for both the samples. A clear enhancement in the post edge feature II
is visible in the simulated spectrum of the $x=$ 1 sample as compared to the
$x=$ 0 sample, marked by the arrows in Figs. 4(d, e). Further, we simulate the
Sr $K$-edge XANES spectrum of the $x=$ 0 sample using the structural
parameters of the $x=$ 1 sample ($P2_{1}/n$) and compare in Fig. 4(e). We find
no significant change in the intensity of the feature II, which further
confirms its structural origin. Our simulations performed by the selective
removal of the neighboring atoms indicate that the feature II in the Sr
$K$-edge XANES spectra results from the multiple scattering from the 12
nearest neighbor oxygen atoms, where any change in the Sr–O scattering paths
with the La substitution alter the oxygen $p$-LDOS at this energy, as shown in
Figs. 4(d, e) and hence intensity and position of the feature II. However, the
observed pre-edge shoulder and insignificant shift in the position of feature
II in the simulated spectra are contrary to the experimental observation for
both the samples. Here, it is important to note that the input structural
parameters for the simulation were used from the XRD/NPD data, which gives an
average crystal structure of the samples. The details of the local structure
around the absorbing atom are missing, because unlike element specific XAS
technique, diffraction techniques are site specific. For example, in case of
the $x=$ 1 sample, an average distortion in the SrO12 and LaO12 polyhedra can
be probed by the diffraction measurements Bos_PRB_04 , the spectral features
of Sr $K$-edge on the other hand depends solely on change in the local
coordination around the Sr atoms. These can be possible reasons for the
observed discrepancy between the experimental and simulated spectra in Figs.
4(d, e).

Figure 5: (a) The magnitude of the Fourier transformed $k^{3}\chi(k)$ spectra for Sr2-xLaxCoNbO6 ($x=$ 0–1) in 0–6 Å spectral range, derived from the EXAFS region of the Nb K-edge. The inset shows the comparison of simulated spectra of the $x=$ 0 sample for two different Nb–Oe–Co/Nb angles (Nb–Oa–Co/Nb=1800). (b)–(g) The enlarged view of (a) between 3.75 to 5.75 Å with cumulative increment in $x$ to clearly show the evolution of the new scattering path resulting from the Co/Nb atoms, located diagonally to the absorber, as indicated by the arrows (refer text for more details). (h–m) The fitting of the $|\chi(R)|$ and corresponding Im[$\chi(R)$] spectra in the window shown by the dotted green line, each Im[$\chi(R)$] spectrum is shifted downward for clarity in presentation. Table 1: The EXAFS fitting parameters for Sr2-xLaxCoNbO6 samples extracted from the Nb, Co, and Sr $K$-edge spectra. $x$ |  0 |  0.2 |  0.4 |  0.6 |  0.8 |  1
---|---|---|---|---|---|---
|  |  |  Nb K-edge |  |  |
R${}_{\rm Nb-O_{e}}$ (Å) |  2.024(2) |  2.025(3) |  2.038(1) |  2.028(4) |  2.020(1) |  2.023(2)
$\sigma^{2}_{\rm Nb-O_{e}}$ (x10-3Å2) |  4.1(3) |  4.5 (1) |  5.7(2) |  4.7(1) |  4.1(1) |  4.1(1)
R${}_{\rm Nb-O_{a}}$ (Å) |  1.914(2) |  1.915(3) |  1.924(4) |  1.925(1) |  1.928(1) |  1.939(2)
$\sigma^{2}_{\rm Nb-O_{a}}$ (x10-3Å2) |  0.2(2) |  0.3(2) |  0.2(2) |  0.2(4) |  0.3(2) |  0.2(3)
$\Delta_{\rm Nb}$x10-4 |  6.8(2) |  6.8(3) |  7.2(4) |  5.9(4) |  4.8(1) |  3.9(2)
RNb-Sr (Å) |  3.431(2) |  3.436(2) |  3.435(4) |  3.451(1) |  3.436(1) |  3.498(4)
$\sigma^{2}_{\rm Nb-Sr/La}$ (x10-3Å2) |  7.8(1) |  7.5(3) |  8.5(4) |  9.6(1) |  7.9(1) |  10.3(2)
RNb-La (Å) |  - |  3.439(2) |  3.480(6) |  3.460(1) |  3.426(3) |  3.481(4)
RNb-O-Nb-Nb (Å) |  3.771(2) |  3.782(3) |  3.781(1) |  3.806(4) |  3.722(3) |  3.822(6)
$\sigma^{2}_{\rm Nb-O-Nb-Nb}$ (x10-3Å2) |  5.4(1) |  4.4(3) |  6.7(4) |  4.6(2) |  6.3(5) |  4.5(5)
NNb |  3.0(1) |  2.9(0) |  2.7(1) |  1.2(1) |  0.7(0) |  0.3(1)
$S$ |  0.00(3) |  0.03(1) |  0.10(3) |  0.60(3) |  0.77(1) |  0.90(3)
R factor |  0.013 |  0.013 |  0.017 |  0.027 |  0.019 |  0.019
|  |  |  Co K-edge |  |  |
R${}_{\rm Co-O_{e}}$ (Å) |  1.925(3) |  1.932(4) |  1.935(1) |  1.968(2) |  2.020(6) |  2.023(1)
$\sigma^{2}_{\rm Co-O_{e}}$ (x10-3Å2) |  8.9(2) |  8.3(4) |  7.6(2) |  6.3(1) |  4.5(4) |  4.4(1)
R${}_{\rm Co-O_{a}}$ (Å) |  2.059(2) |  2.059(3) |  2.058(2) |  2.085(1) |  2.134(5) |  2.126(1)
$\sigma^{2}_{\rm Co-O_{a}}$ (x10-3Å2) |  5.7(2) |  4.6(4) |  2.6(2) |  1.8(1) |  1.5(2) |  2.1(1)
$\Delta_{\rm Co}$x10-4 |  10.3(3) |  9.2(4) |  8.6(2) |  7.5(2) |  6.8(6) |  5.6 (1)
RCo-Sr (Å) |  3.655(2) |  3.676(4) |  3.623(1) |  3.551(1) |  3.541(3) |  3.517(1)
$\sigma^{2}_{\rm Co-Sr/La}$ (x10-3Å2) |  11.6(2) |  10.1(5) |  18.2(3) |  18.0(6) |  18.6(3) |  17.5(1)
RCo-La (Å) |  - |  3.640(4) |  3.693(1) |  3.651(1) |  3.633(5) |  3.650(1)
RCo-O-Co-Co (Å) |  3.899(2) |  3.889(3) |  3.835(1) |  3.781(4) |  3.776(7) |  3.937(2)
$\sigma^{2}_{\rm Co-O-Co-Co}$ (x10-3Å2) |  5.4(1) |  7.4(3) |  8.3(1) |  4.9(2) |  4.2(5) |  5.0(5)
R factor |  0.007 |  0.024 |  0.017 |  0.013 |  0.029 |  0.015
|  |  |  Sr K-edge |  |  |
R${}_{\rm Sr-O_{1}}$ (Å) |  2.560(2) |  2.543(2) |  2.534(1) |  2.535(3) |  2.522(4) |  2.519(1)
R${}_{\rm Sr-O_{2}}$ (Å) |  2.668(2) |  2.670(1) |  2.671(2) |  2.618(2) |  2.639(3) |  2.563(2)
R${}_{\rm Sr-O_{3}}$ (Å) |  2.857(3) |  2.848(1) |  2.870(2) |  2.844(2) |  2.837(4) |  2.803(1)
$\sigma^{2}_{\rm Sr-O_{1}/O_{2}/O_{3}}$ (x10-3Å2) |  9.2(3) |  6.9(2) |  8.7(3) |  11.5(1) |  10.3(4) |  13.6(3)
RSr-Co/Nb (Å) |  3.315(2) |  3.327(2) |  3.347(1) |  3.415(2) |  3.451(5) |  3.453(2)
$\sigma^{2}_{\rm Sr-Sr/La}$ (x10-3Å2) |  5.8 (1) |  8.8(2) |  12.1(2) |  12.7(3) |  13.0(4) |  11.1(1)
RSr-Sr/La (Å) |  3.473(2) |  3.482(1) |  3.483(2) |  3.486(1) |  3.463(5) |  4.500(2)
$\sigma^{2}_{\rm Sr-Sr/La}$ (x10-3Å2) |  8.8(1) |  8.8 (2) |  9.1(2) |  11.4(3) |  11.8(4) |  16.2(1)
R factor |  0.012 |  0.014 |  0.008 |  0.011 |  0.019 |  0.010

###  III.2  Extended x-ray absorption fine structures:

Having discussed the electronic structure, we now focus on the EXAFS part of
XAS spectra to understand the local coordination environment around different
elements in these samples. In Fig. 5(a), we show the amplitude of the Fourier
transformed $k^{3}\chi(k)$ of high-resolution EXAFS spectra of Nb $K$-edge for
all the samples and the normalized raw data are presented in Fig. 3 of
Kumar_XAS_SI . The background-subtracted EXAFS spectra are fairly smooth up to
$k\approx$14 Å-1 (see Fig. 11 of Kumar_XAS_SI  for the $x=$ 0 sample),
allowing us to probe even the small spectral features from the higher
coordination shells with the reasonable accuracy. In Fig. 5(a), the first
intense maxima around 1.5 Å (phase uncorrected) is attributed to the Nb–O
scattering paths in the NbO6 octahedra. It is interesting to note that the
intensity of this maxima remains almost constant for $x\leqslant$ 0.4 samples
and then it increases monotonically with further increase in the La
substitution, analogous to the variation in intensity of the white line
observed in the Nb $K$-edge XANES spectra [see inset (a1) of Fig. 1(a)]. Here,
it is important to note that in the tetragonal ($I4/m$) symmetry, the four
equatorial (Oe; in $ab$ plane) and two axial (Oa; along $c$-axis) oxygen
positions lead to the two sets of different Nb–O bond distances Hudson_PRB_96
. Invariance in the peak intensity corresponding to Nb–O scattering path
indicates almost no change in the degree of NbO6 octahedral distortion for the
$x\leqslant$0.4 samples. This is also evident from the local distortion
parameter, ($\Delta_{d}$) around the Nb atom (denoted by $\Delta_{\rm Nb}$),
see Table I, quantified using below equation Zhu_PRB_20

$\Delta_{d}=\frac{1}{n}\sum_{i=1}^{n}\left(\frac{d_{n}-\langle
d\rangle}{\langle d\rangle}\right)^{2},$ (3)

where $n$ is the number of atoms coordinated to that site, $d_{n}$ is the bond
length between that site and $n$th coordinated atom, and $\langle d\rangle$ is
the average bond length of all the coordinations to that site. In the
monoclinic ($P2_{1}/n$) symmetry for $x\geqslant$0.6 samples, four Nb–Oe bonds
further split in two sets of different bond distances. However, in order to
avoid over-parameterization, we assume the identical Nb–Oe bond distances in
the EXAFS fitting, as shown in Figs. 5(h–m) for the $x=$ 0–1 samples,
respectively. Here, an increase in the intensity of the Nb–O scattering path
for $x\geqslant$0.6 samples [see Fig. 5(a)] indicate the statistically narrow
distribution of the Nb–O path lengths, i.e, decrease in the octahedral
distortion (in NbO6) for these samples (see $\Delta_{\rm Nb}$ values in Table
I). This is found to be interesting as the NbO6 octahedra accommodates the
chemical pressure exerted by the substitution of smaller La3+ ions at the Sr2+
site Shannon_AC_76  up to the $x=$ 0.4 sample. However, with further increase
in $x$, the NbO6 octahedra rotates around $x$ and $y$-axis with the same
magnitude, leading to the monoclinic distortion having b-b-c+ tilt
Kumar_NPD_22 ; Bos_PRB_04 ; Kumar_PRB1_20 , which results in the abrupt
decrease in the distortion of NbO6 octahedra for the $x\geqslant 0.6$ samples.
This is also manifested by the sudden enhancement in the B-site ordering
(refer Table I and discussion below) and hence antiferromagnetic interactions
in $x\geqslant 0.6$ samples through Co–O–Nb–O–Co 1800 superexchange path
Kumar_PRB1_20 .

Figure 6: (a) The magnitude of the Fourier transformed $k^{3}\chi(k)$ spectra
for Sr2-xLaxCoNbO6 ($x=$ 0–1) in 0–6 Å spectral range, derived from the EXAFS
spectra of Co $K$-edge. (b) The evolution of the intensity of the white line
(left axis) and amplitude of the first coordination shell (right axis) with
the La substitution for the Co $K$-edge and (c) Nb $K$-edge EXAFS spectra.
(d–i) The fitting of the $|\chi(R)|$ and corresponding Im[$\chi(R)$] spectra
in the window shown by the dotted green line, where the Im[$\chi(R)$] spectrum
in each panel is shifted downward for the clarity in the presentation.

The second and third maxima at around 2.9 and 3.4 Å are attributed to the
Nb–Sr/La and Nb–Co/Nb scattering paths, respectively. Interestingly, the
intensity of both the scattering paths suppresses with $x$, where Nb–Sr/La
path decreases more rapidly as compared to Nb–Co/Nb path [see Fig. 5(a)]. The
La substitution at Sr site results in the evolution of new Nb–La scattering
paths at a different radial distance as compared to Nb–Sr paths. Also, a
structural distortion [from tetragonal ($I4/m$) to monoclinic ($P2_{1}/n$)
phase] resulting from the La substitution causes splitting in the eight equal
Nb–Sr scattering paths, and consequently a reduction in the strength of the
corresponding feature in $|$FT[$k^{3}\chi$(k)]$|$ plot. However, the EXAFS is
unable to resolve these spatially close ($\Delta r<$0.02) Nb–Sr/La scattering
paths and such a close statistical distribution in the path lengths is
incorporated in the enhancement of Debye-Waller (DW) factor ($\sigma^{2}$)
corresponding to that path. Here, we assume the same DW factor for Nb–Sr and
Nb–La scattering paths to fit the Nb $K$-edge EXAFS spectra. A small
enhancement in the Nb–Sr/La scattering path length is also observed with $x$
[see Fig. 5(a)]. Here, a competition between decrease in the ionic radii of
$A$-site cations (r${}_{\rm La^{3+}}$=1.36 Å and r${}_{\rm Sr^{2+}}$=1.44 Å;
12 coordinated Shannon_AC_76 ) and increase in the Coulombic repulsion between
Nb5+–Sr2+/La3+ ions with $x$ govern this scattering path length and the
dominance of the latter effect results in the observed enhancement in this
scattering path length with $x$. Further, the Nb–Co/Nb scattering paths are
mediated by the oxygen atoms and hence forward scattering paths Nb–O–Nb/Co–Nb
are more favorable as compared to the single scattering (SS) Nb–Co/Nb paths.
The x-ray diffraction patterns clearly evident that structural distortion due
to La substitution causes significant deviation in the Nb–O–Co/Nb bond angle
(from $\approx$1790 in $x=$ 0 to $\approx$1580 in the $x=$ 1 sample
Kumar_PRB1_20 ), which transform Nb–O–Nb/Co–Nb forward scattering path into an
obtuse triangle with the lower scattering amplitude. This results in the
reduction in both the MS paths through Nb atoms, causing the observed
monotonic reduction in the corresponding peaks in $|$FT[$k^{3}\chi(k)]|$ plot
[see Fig. 5(a)]. In order to confirm this, we simulate the
$|$FT[$k^{3}\chi(k)]|$ spectra for the $x=$ 0 sample by varying the
Nb–Oe–Co/Nb angle from 177.7o to 157.4o, keeping all the other parameters
constant, as shown in the inset of Fig. 5(a). A significant reduction in the
spectral weight around 3.5 Å clearly indicate that the oxygen mediated MS path
is the origin of this feature. A similar change in the pseudo-radial
distribution function is reported in Sr2-xCaxYIrO6 samples with change in the
Y–O–Ir angle Marco_PRB_20 . Also, it is important to note that in a completely
B-site disordered structure, each Nb atom is statistically coordinated with 3
Co and 3 Nb first nearest B-site neighbors, whereas in case of fully ordered
structure, each Nb atom is coordinated with 6 Co atoms Das_PRB_20 ;
Bandyopadhyay_PRB_17 . Here, the coordination number of the first nearest Nb
atoms ($N_{\rm Nb}$) and the resulting order parameter, $S$=(3–$N_{\rm Nb}$)/3
from the best fit of the EXAFS spectra are listed in Table I, where $S=$ 0 and
1 represent the completely disordered and ordered structures, respectively. A
clear enhancement in the B-site ordering with the La substitution is evident
from Table I. We use the same zero-energy shift parameter ($\Delta$E0) and
many-body amplitude reduction factor (S${}_{0}^{2}$=0.8) for all the
scattering paths to fit the Nb $K$-edge EXAFS spectra of Sr2-xLaxCoNbO6
samples. Moreover, we consider a single scattering path Nb–O2 (where O2
indicate the second nearest oxygen atoms; see Fig. 12(a) of Kumar_XAS_SI ) at
around 3.8 Å causing an asymmetry in the peak at the higher radial distance,
as indicated by the arrow in Fig. 5(a). This asymmetry appears to become more
prominent as the intensity of the main peak at 3.4 Å reduces with increase in
the La substitution.

More importantly, a close inspection in the range of $\sim$4–5.6 Å clearly
indicate the evolution of new spectral features, as shown in Figs. 5(b–g)
cumulatively with $x$. Here, the feature around 4.6 Å is resulting from the
Nb–La/Sr-O2–Nb double scattering path. As discussed above, the splitting in
Nb–Sr/La scattering paths with the La substitution provide the statistically
wide range of path lengths for their corresponding MS paths, causing the
observed consistent reduction in the intensity of this peak in the
$|$FT[$k^{3}\chi(k)]|$ plot. Further, a feature around 5.3 Å is resulting from
the Nb–O3 single scattering path, as shown in Fig. 12(b) of Kumar_XAS_SI .
Another interesting observation is the evolution of a new feature around 5 Å
with the La substitution, as indicated by arrows in Figs. 5(b–g). This new
feature is resulting from the single scattering from second nearest B-site
atoms (Nb–Co2/Nb2), i.e., along $<110>$ directions of pseudocubic unit cell,
as shown in Figs. 12(c, d) of Kumar_XAS_SI . The completely random occupancy
of Co and Nb atoms at two Wyckoff positions in case of the $x=$ 0 sample and
its minimal deviation from the cubic symmetry results in the absence of this
feature in 5(b), as Nb–Co2/Nb2 SS path is mediated by the central Sr atoms in
the perfectly symmetric environment. Further, a reduction in the crystal
symmetry with the La substitution, as also speculated from the deviation in
the tolerance factor ($\tau$) from unity Kumar_PRB1_20 , results in the
evolution of this feature with $x$, as shown in Figs. 5(b–f). This is
consistent with the evolution of new Raman active modes in these samples with
the La substitution Kumar_PRB1_20 . Further, an abrupt increase in the
strength of this feature for the $x\geqslant$0.6 samples [(see Figs. 5(b–g)]
is consistent with the structural transition from tetragonal ($I4/m$) to the
monoclinic ($P2_{1}/n$) structure accompanied by the sharp enhancement in the
B-site ordering (see Table I) at this critical doping ($x=$ 0.6), which is
correlated with the degree of octahedral distortion in the NbO6 unit, as
discussed above. Though the consistent variation in these features with the La
substitution up to the radial distance of around 6 Å (phase corrected) clearly
indicate the high data quality, we fitted the EXAFS spectra upto 4 Å to
improve the reliability in the extracted parameters. It is important to note
that despite the overall enhancement in the structural distortion, the
distortion in the NbO6 octahedra decreases with the La substitution.

Table 2: The charge transfer ($\Delta$e) and charge counts in the $s$, $p$, and $d$ states of Nbabs and O atoms extracted from the simulation of the Nb $K$-edge XANES spectra of the $x=$ 0 sample by varying Nb–O bond distances. Nb–O${}_{a}({\rm\AA})\rightarrow$ |  |  1.990 |  1.980 |  1.970 |  1.960 |  1.950 |  1.940 |  1.930 |  1.920
---|---|---|---|---|---|---|---|---|---
Nb–O${}_{e}({\rm\AA})\rightarrow$ |  |  1.990 |  2.000 |  2.010 |  2.020 |  2.030 |  2.040 |  2.050 |  2.060
Nbabs |  $s$ |  0.392 |  0.392 |  0.391 |  0.390 |  0.389 |  0.387 |  0.386 |  0.385
Nbabs |  $p$ |  6.603 |  6.602 |  6.601 |  6.600 |  6.598 |  6.598 |  6.597 |  6.5967
Nbabs |  $d$ |  4.143 |  4.151 |  4.164 |  4.175 |  4.184 |  4.194 |  4.204 |  4.213
Nbabs |  $\Delta$e |  0.861 |  0.855 |  0.844 |  0.836 |  0.828 |  0.821 |  0.813 |  0.806
O |  $s$ |  1.835 |  1.836 |  1.835 |  1.835 |  1.835 |  1.834 |  1.834 |  1.834
O |  $p$ |  4.498 |  4.496 |  4.492 |  4.488 |  4.485 |  4.481 |  4.477 |  4.473
O |  $d$ |  0.074 |  0.075 |  0.076 |  0.077 |  0.078 |  0.079 |  0.080 |  0.081
O |  $\Delta$e |  -0.408 |  -0.407 |  -0.403 |  -0.400 |  -0.397 |  -0.394 |  -0.391 |  -0.387

Now we discuss the effect of La substitution on the local coordinate
environment around the Co atoms. In Fig. 6(a), we show the amplitude of the
Fourier transformed $k^{3}\chi(k)$ of Co $K$-edge EXAFS spectra for all the
samples. The normalized raw data are plotted in Fig. 13 of Kumar_XAS_SI  along
with the other details. The most prominent feature around 1.5 Å results from
the Co–O scattering paths. The intensity of this feature increases
monotonically with the La substitution, unlike Nb–O scattering path in the Nb
$K$-edge EXAFS spectra, which remains constant up to the $x=$ 0.4 [see Fig.
5(a)]. This suggest a different evolution of local coordination environment
around the Co and Nb atoms with the La substitution. Here a higher degree of
octahedral distortion results in the low intense wider peak in the $|\chi(R)|$
distribution function as compared to the Nb $K$-edge spectra. Importantly, the
increase in the intensity of this feature at 1.5 Å is consistent with the
variation in the strength of white line of Co $K$-edge XANES spectra [see Fig.
3(a)], analogous to the correlation observed in the Nb $K$-edge. The fitted Co
$K$-edge EXAFS spectra are presented in Figs. 6(d–i) and the best fit
parameters are provided in Table I. We observe the $z$-out distortion in the
CoO6 octahedra, which decreases with $x$, inspite of the reduction in the
crystal symmetry (see $\Delta_{\rm Co}$ values in Table I). This indicate the
presence of electronic instability assisted Jahn-Teller (JT) distortion in
these samples, as the intermediate spin (IS) state of Co3+
($t_{2g}^{5}$e${}_{g}^{1}$) is JT active Yamaguchi_PRB_97 ; Shukla_JPCC_21 .
This $z$-out distortion in the CoO6 octahedra is expected to cause the
observed $z$-in distortion in the neighboring NbO6 octahedra (see Table I).
Further, a decrease in the DW factor of the Co–O scattering paths is observed
with $x$, suggesting the reduction in both static as well as dynamic
distortion around the Co atoms. An enhancement in the concentration of Co2+
ions with the La substitution results in the reduction of this $z$-out JT
distortion in the CoO6 octahedra, as Co2+ is more likely to exist in the JT
inactive HS state as compared to Co3+ Mabbs_book_73 ; Kumar_PRB1_20 . However,
this simplified hypothesis based on the pure ionic picture may partially
breakdown due to the covalent character of Co–O bond, resulting from the
higher crystal field in these samples. For example, an increase in the
occupancy of 3$d$ states (more than 6 electrons for Co3+) and the partial
occupancy of eg states has been reported even for the LS configuration of
LaCoO3 Pandey_PRB_08 ; Hsu_PRB_09 . Note that the increase in ionic radius of
Co (0.545Å for LS and 0.610 Å for HS Co3+, and 0.650 Å for LS and 0.745 Å for
HS Co2+; 6 coordinated Shannon_AC_76 ) and decrease in the Coulombic
attraction between oxygen and Co atoms results in the shift of Co–O scattering
part to the higher radial distance with the La substitution, as shown by the
red arrow in Fig. 6(a). Further, a feature around 3.2 Å is attributed to the
Co–La/Sr scattering path. Here, we find that the strength of this feature as
well as the scattering path length reduces with $x$, as shown by the black
arrow in Fig. 6(a). However, note that the strength decreases, but the
scattering path length increases with $x$ in case of Nb–La/Sr, as observed and
discussed above. Here, a cumulative effect of increase in the ionic radii of
Co (Co3+ $\rightarrow$Co2+), decrease in that of $A$-site cations, and
reduction in the Coulombic repulsion between Co and $A$-site cations
(Co3+–Sr2+ for the $x=$ 0, and $<$Co2+–Sr2+ and Co2+–La3+$>$ for the $x=$ 1
sample) govern the evolution of Co–La/Sr scattering path length with the La
substitution ($x$). The dominance of the latter two effects result in the
observed reduction in this scattering path length with $x$, as shown in Fig.
6(a). Another peak at the higher radial distance of around 3.7 Å is resulting
from the nearest neighbor Co–Nb/Co scattering paths. The fitting of the Co
$K$-edge EXAFS spectra is performed in 1–4 Å range, with the same procedure as
described above for the Nb $K$-edge EXAFS analysis. Here, we use the same
degree of the B-site ordering (i.e, fraction of the first neighbor Co atoms)
as extracted from the Nb $K$-edge EXAFS spectra in order to improve the
reliability in the other fitting parameters.

Figure 7: (a) The Nb $K$-edge XANES spectra simulated for the $x=$ 0 sample by
varying the degree of distortion in NbO6 octahedra. The insets (a1) and (a2)
represent the enlarged view of the white line and variation of the white line
intensity with the distortion parameter, $\Delta_{\rm Nb}$, defined in
equation 3, respectively. The error bars in the white line intensity in inset
(a2) are accounting for the finite step size of the simulated spectra. (b) The
corresponding Nbabs $p$-LDOS simulated at different values of $\Delta_{\rm
Nb}$ for the $x=$ 0 sample. Figure 8: (a) The magnitude of the Fourier
transformed $k^{3}\chi(k)$ spectra of Sr K-edge for Sr2-xLaxCoNbO6 ($x=$ 0–1)
samples. (b) The enlarged view of the normalized Sr $K$-edge EXAFS spectra for
the $x=$ 0 sample, which shows the KN1 double-electron excitation feature and
the resulting spectra after the manual removal of this feature. (c) The
$k^{3}$ weighted $\chi(k)$ spectra and (d) its Fourier transformation for the
$x=$ 0 sample before and after the subtraction of the KN1 excitation feature,
where arrows highlight the change in the corresponding features. (e) The ratio
of the magnitude of the Sr–Sr/La and Sr–Co/Nb scattering paths as a function
of $x$. (f–k) The fitting of the $|\chi(R)|$ and corresponding Im[$\chi(R)$]
spectra in the window shown by the dotted green line, where the Im[$\chi(R)$]
spectra in each panel are shifted downward for the clarity in the
presentation. The asterisk symbol in (f) represent the peak arising from the
KN1 intra-atomic transition.

To further interpret the behavior of main feature at $\approx$1.5 Å in the
EXAFS spectra, i.e., the change in the strength of the Co–O and Nb–O
scattering paths with the La substitution, we compare them with the respective
white line intensity of Co and Nb $K$-edge XANES spectra in Figs. 6(b) and
(c), respectively. A strong correlation between the intensity of white line
and degree of the octahedral distortion can be clearly observed, irrespective
of the direction of octahedral distortion in two cases. In order to understand
this we simulate the Nb $K$-edge XANES spectra for the $x=$ 0 sample by
varying the extent of octahedral distortion in NbO6 unit, as shown in Fig.
7(a). Here, only Nb–Oa and Nb–Oe scattering path distances were varied in
order to systematically enhance the $\Delta_{\rm Nb}$, keeping all the other
structural parameters constant. We find a systematic reduction in the
intensity of the white line with increase in the octahedral distortion, as
shown in the inset (a1) and plotted quantitatively in the inset (a2) of Fig.
7(a). The corresponding intensity of Nb $p$-LDOS also show the monotonic
reduction with increase in the value of $\Delta_{\rm Nb}$ as indicated by the
arrow in Fig. 7(b). The increase in the degree of the octahedral distortion is
expected to enhance the orbital hybridization between Nb and oxygen states,
and can result into the spatially wide distribution of the Nb $p$ states. The
FEFF code simultaneously calculate the charge on the different valence
orbitals and the resulting charge transfer from the initial neutral atoms
Nesvizhskii_PRB_2000 , which can be useful to understand the electronic
structure of the underlined compounds and its evolution with the different
perturbations, as presented in Table II for the selected atoms. Here, the
$\Delta$e represent the charge transfer within the Norman radius at the time
of the self-consistent calculations, which is the difference between the total
$l$-projected charge count and number of valence electrons in the free atoms
(considered one additional electron in the valence state of absorbing Nb,
accounting for the core-hole) Modrow_PRB_03 ; Ankudinov_PRB_98 . Note that the
$\Delta$e values are much smaller than the underlined oxidation state of Nb
and O atoms in the sample, but these can be used to qualitatively understand
the variation in the electronic structure with $\Delta_{\rm Nb}$. Also, the
octahedral oxygen environment around the absorbing Nb atoms give rise to the
$sp_{3}d_{2}$ type hybridization, causing a decrease in $s$ and increase in
the $p$ and $d$ orbital electron count as compared to the free Nb and O atoms
[$s=$ 1(2), $p=$ 6(4), $d=$ 4(0) for the Nb(O)] Modrow_PRB_03 . Importantly,
the value of $\Delta$e decreases with increase in the octahedral distortion
for both Nb as well as oxygen atoms, which indicate the enhancement in the
degree of orbital overlapping between these two atoms, resulting in the
deviation from the free ion like character of Nb atom. Further, an increase in
the difference between Nb–Oe and Nb–Oa bond distances enhance the spatial
distribution of Nb $p$ states (hybridized with the O 2$p$ states) and hence
decrease the strength of the white line feature. This hypothesis suggest that
the pure $1s\rightarrow np$ transition as an origin of the white line in the
$K$-edge XANES spectra is an over simplified assumption and effect of the
molecular orbitals on the XANES features of the transition elements should
also be stressed. For example, the change in the XANES spectra of the
transition metal oxides with varying degree of electronegativity have been
well explained by Modrov $et$ $al.$ on the basis of the molecular orbital
theory Modrow_PRB_03 .

In Fig. 8(a) we present the magnitude of the Fourier transformed
$k^{3}\chi(k)$ of Sr $K$-edge spectra to investigate the local coordination
environment around the $A$-site cations. Interestingly, an intense peak is
observed at a very small radial distance of around 0.8 Å, which is smaller
even than the ionic radii of Sr2+ cation in the 12-fold coordination
environment Shannon_AC_76  and hence no scattering path is expected at this
position. These features at the low radial distances are usually not
associated with the local structural arrangement of the atoms, but arises
mainly due to the multielectronic transitions within the atoms, which can
appear well above the white line in the absorption spectra and mix with the
EXAFS oscillations Li_PRL_92 ; Angelo_PRA_93 ; Filipponi_PRA_95 ;
Angelo_PRB_96 . The inaccurate treatment of these spectral features can lead
to the significant error in structural parameters extracted from the EXAFS
analysis. In case of Sr $K$-edge absorption spectra, an intra atomic
excitation from 1$s$ to 4$s$ (KN1) states result in the sharp absorption
feature well above the white line Angelo_PRB_96 . The position of this feature
can be estimated by the Z+1 approximation, where binding energy of the Sr
($Z=$38) electrons in the different levels can be approximated by that of
Yttrium ($Z=$ 39) after the ejection of one photoelectron from the $K$ shell.
This approximation predict the KN1 double electron excitation at 43.8 eV above
the Sr $K$-edge threshold energy Angelo_PRB_96 . Interestingly a sufficiently
intense feature is clearly observed at 44.4(4) eV above the threshold energy
of Sr $K$-edge in all the samples, as shown in Fig. 8(b) for the $x=$ 0
sample, see Fig. 14 of Kumar_XAS_SI . The absence of this feature above the
white line of the La L3-edge spectra, as indicated by the black arrow in Fig.
4(c) for the $x=$ 1 sample, clearly indicate the atomic-like states of the Sr
atom as its origin rather than the local atomic arrangement around the Sr
atoms. Therefore, a careful treatment of the Sr $K$-edge EXAFS spectra is
necessary for the accurate structural probe. In order to further confirm the
origin of this low radial distant peak in Fig. 8(a), we subtract the KN1
excitation feature by a Gaussian approximation and the resulting spectrum is
presented in Fig. 8(b) for the comparison. We find a clear elimination of a
spectral feature around $k=$ 4 Å-1 as a result of the subtraction of KN1
double-excitation transition, as highlighted by arrow in Fig. 8(c) and a well
speculated sharp reduction at around 0.8 Å in the magnitude of Fourier
transformed $k^{3}\chi(k)$ spectra of Sr $K$-edge in Fig. 8(d). However, the
intensity of few other features also changes as a result of this subtraction,
possibly due to inaccuracy in the approximation of KN1 feature in the spectra
resulting from the presence of other nearby spectral features from the local
atomic arrangement. Thus, we analyze the as recorded EXAFS spectra of Sr
$K$-edge for all the samples by excluding the low radial part of the
$|\chi(R)|$ to avoid the contribution from the KN1 excitation channel. Also,
the KM2,3 and KM4,5 excitations are possible in the Sr $K$-edge EXAFS spectra
resulting from 1$s$ $\rightarrow$ 3$p$ and 1$s$ $\rightarrow$ 3$d$
transitions, respectively Angelo_PRB_96 . However, no change is observed in
the spectra at their speculated positions from the $Z$+1 approximation in any
of the samples due to dominance of the high amplitude EXAFS features resulting
from the good micro crystalline nature of the samples. However, the effect of
these transitions can not be completely neglected in the analysis of the Sr
$K$-edge EXAFS spectra and the data up to very high temperature ($>$2000 K)
are required to fairly quantify their contribution, where structural
oscillations significantly diminish due to the enhanced DW factor
Filipponi_PRA_95 .

Moreover, we observe three intense features in the Sr $K$-edge EXAFS spectra
between 1.4–3.5 Å for all the samples [see Fig. 8 (a)]. These features at
around 2, 2.6, and 3.2 Å result from the Sr–O, Sr–Co/Nb, and Sr–Sr/La
scattering paths, respectively. Also, a monotonic reduction in the intensity
of the peaks corresponding to Sr–Co/Nb and Sr–Sr/La scattering paths is
clearly visible with the La substitution [see Fig. 8(a)]. However, a small
inconsistency is present in case of the $x=$ 1 sample possibly due to decrease
in the quality of the signal because of the lower Sr concentration. In $I4/m$
symmetry, two different oxygen positions lead to three sets of Sr–O bond
distances with four bonds in each group, which further split into twelve
different bonds in $P2_{1}/n$ symmetry Kumar_NPD_22 . However, we take three
Sr–O scattering paths with the same DW factor to fit the Sr $K$-edge EXAFS
spectra in 1.4–3.5 Å range for all the samples, as shown in Figs. 8(f–k) and
best fit parameters are presented in Table I. Further, narrow spatial
distribution unable us to distinguish the Sr–Co and Sr–Nb as well as Sr–Sr and
Sr–La paths and we take single Sr–Co/Nb and Sr–Sr/La paths for the EXAFS
fitting. A reduction in the amplitude of the Sr–Co/Nb and Sr–Sr/La peaks
indicate the distortions in the corresponding scattering paths with the La
substitution, which is reflected into their corresponding Debye-Waller
factors, as presented in Table I. This effect is more prominent in case of
Sr–Co/Nb as compared to Sr–Sr/La scattering path, which is clear in the plot
of amplitude ratio of these two scattering paths [(Sr–Sr/La)/(Sr–Co/Nb)]
particularly for the $x\geqslant$0.6 samples, see Fig. 8(e). The change in
formal charge and ionic radii of $A$\- (Sr${}^{2+}\rightarrow$ La3+) and $B$\-
(Co${}^{3+}\rightarrow$ Co2+) site atoms cause the reduction in amplitude of
their respective scattering paths with $x$. Further, the different tendency of
the Nb–Sr/La (increasing) and Co–Sr/La (decreasing) SS paths with $x$ is
evident from the Nb and Co $K$-edge EXAFS spectra, respectively, results in
the spatially wide distribution and hence more rapid reduction in the
intensity of Sr–Co/Nb path as compared to Sr–Sr/La with $x$. Moreover, the two
features between 4–5 Å are attributed to the SS from the second and third
nearest oxygen atoms, located in the neighboring pseudocubic unit cells and a
feature around 5.5 Å is resulting from the Sr–Sr2/La2 SS path. A well
speculated distortion in these scattering paths cause the observed reduction
in the intensity of these features with $x$. This behavior suggests that the
chemical pressure due to La substitution induce more distortion at the
$A$-site as compared to the $B$-site(s), and results in the evolution of
complex magnetic interactions Kumar_PRB1_20 ; Kumar_PRB2_20 .

Figure 9: The Co L2,3-edge spectra (vertically offset for clarity) for
Sr2-xLaxCoNbO6 samples. The atomic multiplet based simulated spectra for HS
Co2+ and HS Co3+ are compared.

Finally, we present the Co L2,3 soft-XAS spectra in Fig. 9 where transition
from two spin-orbit splitting components of Co $2p$ final states, i.e.,
2$p_{3/2}\rightarrow 3d$ (L3) and 2$p_{1/2}\rightarrow 3d$ (L2), are observed
around 780 eV and 795 eV, respectively, for all the samples. The peak position
shifts towards lower energy (as indicated by the arrow) due to the conversion
of Co from 3+ to 2+ valence state with the La substitution. This is also
consistent with the Co $K$-edge XANES spectra, discussed above. Interestingly,
we observe the consistent reduction in the broadening with the evolution of a
hump at the shoulder of L3 edge with the La substitution, as indicted by the
vertical arrow for the $x=$ 1 sample. To understand the physical origin of
this, we simulate the $L$-edge XAS spectra for HS Co2+ and HS Co3+ using the
CTM4XAS software. A charge transfer effect from oxygen to Co atoms was also
introduced in the simulation for both the valence states of Co, resulting in
the additional $2p^{6}3d^{7}\underline{L}^{1}\rightarrow
2p^{5}3d^{8}\underline{L}^{1}$ and $2p^{6}3d^{8}\underline{L}^{1}\rightarrow
2p^{5}3d^{9}\underline{L}^{1}$ transitions in case of Co3+ and Co2+,
respectively. The Lorentzian and Gaussian line shapes with half-width of 0.5
eV and 0.45 eV were used to account for the core-hole lifetime and
instrumental broadening. The crystal field-field splitting (10Dq), on-site
Coulomb correlation energy of 3$d$ electrons ($U_{dd}$), core hole potential
($U_{pd}$), charge transfer energy ($\Delta_{ct}$), and O 2$p$-Co 3$d$
hybridization strength parameters were varied to best match the experimental
and simulated spectra. The intra-atomic interactions were approximated by
reducing the Slater integrals to 80% from their Hartree-Fock values. The best
matched parameters are 10Dq=1.3 and 0.5 eV, $U_{dd}-U_{pd}$ =3.0 and 0.2 eV,
and $\Delta_{ct}$=–6.5 and 1.4 eV for HS Co3+ and HS Co2+ spectra,
respectively. The $p-d$ hybridization strength parameter T${}_{e_{g}}$=2.0 and
0.8, and T${}_{t_{2g}}$=1.0 and 0.4 were used for Co3+ and Co2+, respectively
and the resulting spectra are presented in Fig. 9. The extracted value of 10Dq
for Co3+ is smaller than that of LaCoO3 ($\sim$2.3 eV Medarde_PRB_06 ),
indicating the higher probability of finding Co3+ in the IS/HS state, which is
consistent with the magnetization results Kumar_PRB1_20 ; Yoshii_JalCom_2000 .
Further, the large negative value of the charge transfer energy indicate the
highly covalent nature of the Co–O bonds, where the covalency character
decreases with the La substitution. Moreover, the branching ratio, defined as
(BR)= I(L3)/[I(L2)+I(L3)], where I(L2) and I(L3) represent the integrated
intensity of L2 and L3 edges, is 0.757, 0.786, 0.764, 0.763, and 0.758 for the
$x=$ 0.2–1 samples, respectively. These preliminary results of SXAS further
suggest for the mixed IS and HS states of Co3+ in Sr2-xLaxCoNbO6 samples
Thole_PRB_88 . However, high resolution SXAS measurements can be further
useful to understand the complex spin-state transition in these samples as a
function of $x$ and temperature Kumar_PRB1_20 ; Merz_PRB_11 .

##  IV  Conclusions

We have investigated the electronic and local structure of Sr2-xLaxCoNbO6
($x=$ 0–1) samples using detailed analysis of XANES and EXAFS spectra as well
as the simulation of Nb, Co and Sr $K$\- edges. A very weak pre-edge feature
in case of Nb $K$-edge XANES spectra indicate the negligible off-center
displacement of Nb atoms in the NbO6 octahedra. This pre-edge feature results
from the transition of 1$s$ electrons to the 4$d$ states of Nb mixed with Nb
and/or O $p$ states. Further, the post edge feature in the Nb $K$-edge is
found to vary sensitively with both nature as well as distance of the $A$-site
atoms. The $K$-edge XANES spectrum show that Co transform from 3+ to 2+
valence states, whereas oxidation states of Sr and Nb remains invariant with
the La substitution. Also, the intensity of pre-edge features of Co $K$-edge
spectra depends on the number of 3$d$ electrons, but analysis discard their
pure quadruple origin. Moreover, the significantly strong pre-edge feature in
case of Co $K$-edge XANES spectra as compared to Nb indicate the presence of
Co and Nb atoms in the different octahedral environment. The EXAFS
measurements evident a decrease in the local distortion around $B$ site atoms,
despite of reduction in the overall crystal symmetry with the La substitution.
The $z$-out JT distortion in CoO6 and consequently $z$-in distortion in the
NbO6 octahedra have been observed in all the samples. The distortion in the
CoO6 octahedra decreases monotonically with the La substitution due to
increase in the concentration of JT inactive Co2+ ions, whereas that in NbO6
octahedra remains almost invariant up to the $x=$ 0.4, and then decreases for
higher La concentration. This abrupt change in the local coordination
environment around the Nb atoms results in the sharp enhancement in B-site
ordering and hence evolution of AFM interactions in the $x\geqslant$0.6
samples. This is further supported by sudden enhancement in the longer
Nb–Co2/Nb2 path for the $x\geqslant$0.6 samples in the higher coordination
shell in Nb $K$-edge EXAFS spectra. Interestingly, we observe a strong
correlation between the degree of octahedral distortion in (Co/Nb)O6 units and
intensity of their respective white line features. An increase in the degree
of octahedral distortion is found to enhance the metal-ligand overlapping and
delocalized nature of the TM $p$-states, resulting in the low intense broad
white line features in their $K$-edge XANES spectra. We reveal that such a
correlation is useful to estimate the extent of octahedral distortion in the
complex compounds directly from the analysis of XANES spectra. Moreover, the
Sr $K$-edge absorption spectra show clear evidence of the KN1 double electron
excitation for all the samples, suggesting the necessity of the careful
treatment of the Sr $K$-edge EXAFS spectra. Finally, the Co $L_{2,3}$ edge
spectra indicate the decrease in the crystal field strength and tendency of O
to Co charge transfer with the La substitution ($x$).

##  V  Acknowledgments

AK and RS thank the UGC and DST-Inspire for the fellowship. We thank John J.
Rehr for very useful discussions on the XANES simulations using FEFF code.
Authors also acknowledge D. M. Phase and R. K. Sah for support and help,
respectively, during the soft XAS measurements. We acknowledge the financial
support from SERB-DST through core research grant (project reference no.
CRG/2020/003436). We also greatly acknowledge the CRS (project no. CSR-
ICISUM-36/CRS-319/2019-20/1371) for collaborative research work.

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