# Emergence of half-metallic ferromagnetism in transition metal substituted
Na0.5Bi0.5TiO3

Chandan Kumar Vishwakarma  Department of Physics, Indian Institute of
Technology, Hauz Khas, New Delhi 110016, India  B. K. Mani
<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology, Hauz Khas, New Delhi 110016, India

(August 28, 2024)

###### Abstract

The multifunctional materials with prominent properties such as electrical,
ferroelectric, magnetic, optical and magneto-optical are of keen interest to
several practical implications. In the roadmap of designing such materials, in
the present work, using density functional theory based first-principles
calculations, we have investigated the functional properties of transition
metal substituted-NBT. Our calculations predict the emergence of half-metallic
ferromagnetism in the system. A nonzero magnetic moment of 1.49 $\mu_{\rm
B}/{\rm f.u.}$ is obtained for 25% concentration of Ni. Our data on optical
properties for pure NBT is in excellent agreement with available theory and
experiments. For Ni-NBT, we observed a diverging nature of static dielectric
constant, which could be attributed to the induced metallic character in the
material. Our simulations on MOKE predict a significant Kerr signal of 0.7∘
for 6.25% Ni-concentration.

††preprint: APS/123-QED

##  I Introduction

The development of multifunctional materials with two or more properties, such
as magnetic, ferroelectric, piezoelectric, and optical, has received a lot of
interest in recent years Du  _et al._ (2014); Wang  _et al._ (2005); Tian  _et
al._ (2013). These materials have the potential to revolutionize various
industry applications, including healthcare, energy and electronics Lee  _et
al._ (2020); Zheng  _et al._ (2021); Huang  _et al._ (2020). In the search for
such materials, sodium bismuth titanate, Na0.5Bi0.5TiO3 (NBT), has received a
remarkable attention than any other lead-free ferroelectrics due to its
tendency to show multifunctionality by various mechanisms Vakhrushev  _et al._
(1985); Jones and Thomas (2002); Siny  _et al._ (1995). NBT is a complex
perovskite oxide with two cations (Na+ and Bi3+) on the A-site and one cation
(Ti2+) on B-site with a rhombohedral symmetry at room temperature Smolenskii
_et al._ (1960). It exhibits various anomalous properties associated with
site-specific substitutions, including improved ferroelectricity and
piezoelectricity, magnetism and optoelectronic properties Mishra  _et al._
(2019); Zannen  _et al._ (2015).

The presence of Ti at the B site provides a strategy to introduce
ferromagnetism by substituting transition-metal (TM) at B-site. In
experimental studies, Refs. Jain Ruth  _et al._ (2020) and Wang  _et al._
(2009), ferromagnetism at room temperature was observed for Fe and Co-doped
NBT, respectively. In a similar experimental work by Dung et al., a room-
temperature ferromagnetism was reported for Ni-doped NBT Dung  _et al._
(2020). The maximum magnetization value reported was around $0.91$ $\mu_{\rm
B}$/Ni for 9% of Ni concentration at 5 K. Moreover, it was also observed in
the same study that the optical bandgap decreases with Ni-concentration.
However, in a different experimental study, Pradhan et al., the optical band
gap was observed to increase with Ni concentrations Pradhan and De (2018). The
contradictory trend of experimental data suggest the lack of understanding for
optical behavior of TM-doped NBT. In addition, to the best of our knowledge,
there are no data from theory simulations on probing magnetism in TM-doped
NBT. It can thus be surmised that there is a need for a systematic theoretical
study to understand the underlying mechanism behind the multifunctional
properties in TM-doped NBT.

The present study aims to probe, with the help of the state-of-the-art of
first-principles calculations, the electronic, magnetic, optical and magneto-
optical properties of NBT and TM-substituted NBT. More precisely, we aim is to
address the following questions: i) What is the impact of Ni substitution on
the electrical and optical properties of NBT? ii) Assimilate the mechanism
behind the advent of magnetic degrees of freedom in Ni-substituted NBT iii)
How this introduced ferromagnetism couples with the dielectric properties of
NBT? To assess the coupling between magnetic and optical degrees of freedom,
we have examined the linear magneto-optic Kerr effect in the polar geometry,
in which the spin and incident photons are perpendicular to the sample
surface. This configuration of the Kerr effect is the most favorable way to
trace the magneto-optical properties experimentally Bastjan  _et al._ (2008);
Xu  _et al._ (2009); Malakhovskii  _et al._ (2012).

The texts in the paper are organized in four sections. In Section II, we
provide a brief description of the computational methods used in our
calculations. In Section III, we present and analyze our results on electronic
structure, magnetic, optical, and magneto-optical properties for NBT and Ni-
substituted NBT. The summary of our findings is presented in the last Section.

##  II Computational Methodology

Probing transition metal-substituted NBT structure and emerging properties
requires an accurate treatment of interstitial effects in the material at
atomic scale. For this, we have performed ab-initio spin-polarized
calculations using density functional theory (DFT) as implemented in the
Vienna ab-initio simulation package (VASP)Kresse and Furthmüller (1996, 1996).
To account for the exchange correlation among electrons, we used Perdew-Burke-
Ernzerhof (PBE) Perdew  _et al._ (1996) variation of generalized-gradient
approximation pseudopotential. And, to account for the strongly correlated
$3d$-electrons of Ni we have incorporated the Hubbard U correction Anisimov
_et al._ (1991) in our calculation. The value, 11.57, of U is computed self-
consistently using density functional perturbation theory (DFPT) employing the
cococcioni’s et al. Cococcioni and de Gironcoli (2005) approach. A
rhombohedral supercell of size $2\times 2\times 2$ with 80-atoms is used to
incorporate various concentrations of Ni. All the structures were optimized
using full relaxation calculations up to $10^{-4}$ eV Å-1 force tolerance. For
this, we used conjugate gradient algorithm with Monkhorst-Pack Monkhorst and
Pack (1976) $k$-mesh of $5\times 5\times 5$. For the self-consistent-field
(SCF) calculations, the Brillouin zone was sampled with $9\times 9\times 9$
$k$-mesh. The energy convergence criterion is maintained at $0.001$ meV,
whereas the plane wave energy cutoff used was $600$ eV. The real and imaginary
parts of the dielectric function is calculated using DFPT as implemented in
VASP.

##  III Results and Discussions

Figure 1: Crystal structures of Na0.5Bi0.5TiO3 (panel (a)) and
Na0.5Bi0.5(Ti0.75Ni0.25)O3 (panel (b)).

###  III.1 Crystal Structure

The structural parameters for pure NBT were taken from the experimental
crystal structure (space group $R3c$) data Jones and Thomas (2002) and
optimized further through the full relaxation calculations to achieve a
minimum energy configuration. Our computed lattice parameters and Wyckoff
positions are given in Table 1, along with the data from the literature for
comparison Benyoussef  _et al._ (2022) As we observed from the table, our
computed lattice parameter $5.65$ is in good agreement with the experimental
value 5.51 Jones and Thomas (2002). The reason for slightly larger value could
be attributed to the use of GGA functional our calculation Stampfl and Van de
Walle (1999).

To incorporate various Ni concentrations in Na0.5Bi0.5[Ti1-xNix]O3 (Ni-NBT),
we used optimized NBT structure and created a $2\times 2\times 2$ supercell.
And, we investigated the properties for x $=0.0625,0.125,0.1875$, and $0.25$
concentrations of Ni. The Ni-NBT structures were fully optimized again using
the force tolerance up to the $10^{-5}$ eV Å-1. From our simulations we find
that all Ni-NBT structures crystallize in rhombohedral (R$3$m) phase. In Fig.
2, we have shown the crystal structures for NBT (panel(a)) and 0.25Ni-NBT
(panel(b)). The optimized lattice parameters for the chosen concentrations of
Ni are given in Table 1. To the best of our knowledge, there are no
experimental or other theory data for lattice parameters for Ni-NBT available
in the literature for comparison.

Table 1:  Computed lattice parameters and Wyckoff positions for NBT. Values given in the parenthesis are the data from the experiment Benyoussef  _et al._ (2022). | Lattice Parameters |  |
---|---|---|---
|  $a=b=c=5.647(5.51)$ (Å) |  |
| $\alpha=\beta=\gamma=58.758(59.803)(^{\circ})$ |  |
| Wyckoff Positions |  |
Na1 (1a) | 0.25644 | 0.25644 | 0.25644
Bi1 (1a) | 0.77324 | 0.77324 | 0.77324
Ti1 (1a) | -0.00056 | -0.00056 | -0.00056
Ti2 (1a) | 0.48900 | 0.48900 | 0.48900
O1 (3b) | 0.23013 | 0.72562 | -0.76987
O2 (3b) | 0.72771 | 0.21016 | -0.27229
Figure 2: Calculated electronic band structure of Na0.5Bi0.5TiO3 (panels (a),
(b)) and Na0.5Bi0.5(Ti0.75Ni0.25)O3 (panels (c), (d)) for majority and
minority spin electrons.

###  III.2 Electronic Structure and Ferroelectric Properties

In Fig. 2 we have shown the spin-polarized electronic band structures of NBT
(panels (a) and (b)) and 25Ni-NBT crystals (panels (c) and (d)). We chose to
report the data for the highest concentration of Ni as it has the largest
effect on the computed properties. The corresponding data for other
concentrations are, however, provided in the supplementary material. As we
observed from the panels (a) and (b) of the figure, NBT exhibits a direct band
gap electronic structure at $\Gamma$ point for both the spin channels. The
calculated band gap, 2.57 eV, is in good agreement with the other theoretical
value, 2.82, reported in Lü  _et al._ (2014). The observed wide band gaps for
both spin channels suggests the semiconducting nature of the NBT crystal, and
is consistent with the data reported in the literature Benyoussef  _et al._
(2022); Zeng  _et al._ (2010); Behara  _et al._ (2020). For 25Ni-NBT, however,
we observe an asymmetry in the majority and minority spin channels (panels (c)
and (d)). For majority spin, the Fermi level lies in the valence band and
shows a metallic nature. Whereas, for minority spin sub band, a large band gap
of $\sim 2.56$ eV is obtained, which resembles the electronic structure of NBT
shown in panel (b). This mixed nature of electronic structure indicates a
half-metallic character of 25Ni-NBT. A similar electronic structures we also
obtain for other concentrations of Ni.

To get further insight into the half-metallicity in Ni-NBT, we examined the
atom-projected and orbital-projected electronic structures of NBT and 25Ni-
NBT. The data from this for bands and density of states (DoS) are shown in
Figs. 3 and 4, respectively. For NBT, as discernible from the panels (a) and
(b) of Fig. 4, the valence band for both the spin channels have dominant
contributions from O, where $2p$-electrons contribute the most. For the
conduction band, however, the most significant contribution comes from the
$3d$-electrons of Ti. This observed nature of the electronic structure of NBT
is consistent with the reported trend in Refs Zeng  _et al._ (2010).

For 25Ni-NBT, for majority spin channel, the bands around the Fermi energy are
of mixed O and Ni character, with O contribution more prominent than Ni at the
Fermi energy (panels (a), (c) in Fig. 3). This is also consistent with the
atom-projected DoS shown in Fig 4(b). At Fermi energy, O contributes $\approx$
70% which mainly comes from $2p$-electron, whereas the contribution from Ni
(mostly from $3d$-electrons) is about $\approx$ 20% of the total value. And
like NBT, the conduction band is dominated by the $3d$-electrons of Ti. For
minority spin sub band, a significant contribution of O in the valance band
below the Fermi level is observed, and comes from the $2p$-electrons. However,
unlike majority spin sub band, there is a negligible contribution from Ni in
the bands below the Fermi level. Using the number of electronic states at
Fermi level for both spin channels we calculated the spin polarization, which
comes out to be 100%. This non-zero electronic states at the Fermi level for
majority spin and a wide band gap for minority spin confirms the half-
metallicity in Ni-substituted NBT. A similar trend of half-metallicity and
spin polarization is also observed for Fe-NBT.

Figure 3: The atom-projected band structure of Na0.5Bi0.5(Ti0.75Ni0.25)O3 for
O and Ni atoms for majority (panels (a), (b)) and minority (panels (c), (d))
spin electrons.

Next, we examine the ferroelectric properties in NBT and Ni-NBT. For this, we
calculated the remanent polarization for NBT and Ni-NBT. The electronic
contribution to polarization was computed using the Berry phase approach King-
Smith and Vanderbilt (1993); Resta (1994). NBT is a well known lead-free
ferroelectric material with experimentally reported remanent polarization as
38 $\mu{\rm C}/{\rm cm}^{2}$ Smolenskii  _et al._ (1960), 32 $\mu{\rm C}/{\rm
cm}^{2}$ Yu and Ye (2008) and 42.4 $\mu{\rm C}/{\rm cm}^{2}$ Mohanty  _et al._
(2017) along [001] pseudo-cubic direction. From theory calculations, the
reported value of spontaneous polarization, $P_{\rm S}$, is 26 $\mu{\rm
C}/{\rm cm}^{2}$ Niranjan  _et al._ (2013). Our computed value of $P_{\rm S}$
is 49 $\mu{\rm C}/{\rm cm}^{2}$. The reason for the difference from experiment
could be attributed to the fact that the reported experimental polarizations
are at room temperature, whereas the computed values are at 0 K. The
spontaneous polarization deceases after the Ni-substitution. We obtained 33.7
and 29.7 $\mu{\rm C}/{\rm cm}^{2}$ of $P_{\rm S}$ for 12 and 25% of Ni,
respectively. The reason for this decease could be attributed to the
increasing metallic nature due to Ni-substitution.

Figure 4: The density of states and orbital-projected density of states for
Na0.5Bi0.5TiO3 (panels (a) and (b)) and Na0.5Bi0.5(Ti0.75Ni0.25)O3 (panels (c)
and (d)).

###  III.3 Optical Properties

Figure 5: The real and imaginary components of the diagonal permittivity of
NBT (panels (a) and (b), respectively) and Ni-substituted NBT (panels (c) and
(d), respectively). The experimental data in panels (a) and (b) are from the
Ref. Andriyevsky  _et al._ (2009).

Next, we investigate the optical properties of NBT and Ni-NBT. For this, we
calculated the frequency-dependent complex dielectric function using
Ehrenreich and Cohen’s equation Ehrenreich and Cohen (1959),
$\epsilon(\omega)=\epsilon_{1}(\omega)+i\epsilon_{2}(\omega)$, where
$\epsilon_{1}$ and $\epsilon_{2}$ are the real and imaginary parts,
respectively. The complex dielectric function of a material is a key parameter
and could be useful in probing several fundamental properties of the material.
The imaginary part of the dielectric function could be calculated using the
linear response theory Gajdoš  _et al._ (2006) as

$\epsilon_{2}(\omega)=\frac{2\pi
e^{2}}{\epsilon_{0}\Omega}\sum_{k,v,c}\delta(E_{k}^{c}-E_{k}^{v}-\hbar\omega)\;|\bra{\Psi_{k}^{c}}({\mathbf{\hat{n}.r}})\ket{\Psi_{k}^{v}}|^{2}.$
(1)

Here, the indices $k$, $v$ and $c$ represent the wave vector, valence and
conduction bands, respectively. The states $|\Psi_{k}^{v}\rangle$ and
$|\Psi_{k}^{c}\rangle$ are the wavefunctions associated with valence and
conduction bands, respectively, and, $E_{k}^{v}$ and $E_{k}^{c}$ are the
corresponding energies. The constants, $e$, $\Omega$ and $\epsilon_{0}$ are
the charge of the electron, volume of the cell and the permittivity of the
free space, respectively. The operator $\hat{\mathbf{n}}$ represents the
direction of the applied electric field. The real component of the dielectric
constant can be derived from the imaginary component using the Kramers-Kronig
relation Gajdoš  _et al._ (2006); Kreibig (1970)

$\epsilon_{1}(\omega)=1+\frac{2}{\pi}P\int_{0}^{\infty}\frac{\epsilon_{2}(\omega^{\prime})\omega^{\prime}d\omega^{\prime}}{\omega^{\prime
2}-\omega^{2}-i\eta},$ (2)

where P is the principal value and $\eta$ is an infinitesimal broadening
associated with the adiabatic switching of the dielectric perturbation.

The $\epsilon_{1}(\omega)$ and $\epsilon_{2}(\omega)$ for NBT and Ni-NBT from
our calculations along with the available experimental data are shown in Fig.
5. For NBT, as discernible from panels (a) and (b), our calculated real and
imaginary components of dielectric function are in good agreement with the
experiment Andriyevsky  _et al._ (2009). The slight deviation could be
attributed to the temperature effects in experiment, as the reported
experimental data are at room temperature. Inspecting the real component more
closely, the value of static dielectric constant, $\epsilon(0)$, is 6.5. This
is consistent with the value reported in previous theory calculation K.
Vishwakarma and Mani (2023) and experiment Andriyevsky  _et al._ (2009). This
relatively higher value of $\epsilon(0)$ suggests NBT as a potential candidate
for light-harvesting applications Sudha Priyanga and Thomas (2018). The other
important characteristic of real spectrum is, negative values at higher
energies. This trend is consistent with the previously reported theoretical
data Zeng  _et al._ (2010), and suggests NBT as a potential candidate for
plasmonic applications Shabani  _et al._ (2021).

Figure 6: The absorption coefficient of NBT and Ni-substituted NBT (panels (a)
and (b), respectively) and optical band gap as function of Ni-concentrations
(panel (c)). Refs. [a], [b], [c] and [d] correspond to the works K.
Vishwakarma and Mani (2023), Zeng  _et al._ (2010), Andriyevsky  _et al._
(2009) and Dung  _et al._ (2020), respectively.

Examining the imaginary component more closely, we observe one preeminent peak
at $\sim 4.2$ eV and four low-intensity secondary peaks at $\sim 5.7$, $\sim
6.4$, $\sim 7.7$, and $\sim 9.0$ eV energies. The primary peak originates from
the interband transitions from O-$2p$ to Ti-$3d$ and Bi-$6p$ states. The
secondary peaks, however, embed major contributions from the O-$2p$ to
Na-$2s/2p$ transitions. Our calculated $\epsilon_{2}(\omega)$ spectrum is in
qualitative agreement with the reported theoretical data Zeng  _et al._
(2010). The onset of $\epsilon_{2}(\omega)$ spectrum suggests the optical band
gap of NBT as $\approx 2.57$ eV, which is consistent with the direct
electronic bandgap discussed in previous section.

Considering the case of Ni-NBT, as discernible from the panels (c) and (d),
$\epsilon_{1}(\omega)$ and $\epsilon_{2}(\omega)$ show a similar trend as NBT
at higher energies (above 2.5 eV). And, as can be inferred from Fig. 4(d), the
reason for the observed peaks is attributed to the same interband transitions,
O-$2p$ to Ti-$3d$ and Bi-$6p$. In the low energy regions (below 0.5 eV),
however, we observe a diverging nature of $\epsilon_{1}(\omega)$. The reason
for this could be attributed to the half-metallic nature of Ni-NBT. The value
of static dielectric constant is observed to increase with Ni concentration,
and the highest value of $\approx 61$ is obtained for 25$\%$ concentration.
Consistent with the trend of $\epsilon_{1}(\omega)$, $\epsilon_{2}(\omega)$
shows sharp peaks below 0.5 eV, with increasing amplitudes with Ni-
concentration.

To get more insight and compare with experimental observations, we have
examined the absorption coefficient, $\alpha$, for NBT and Ni-NBT. In
addition, we have extracted the optical bandgap, $E_{g}$, for minority spin
channel of Ni-NBT at different concentrations using the Tauc’s plot Tauc  _et
al._ (1966). The data from this are shown in Fig. 6. As discernible from the
panel (a) of the figure, for NBT, our simulation is in good agreement with the
experiment, with a slight shift in the onset of the peak. The reason for this
shift could be attributed to the temperature effects in experimental data. For
Ni-NBT, on contrary, we observe nonzero peaks in the IR region, which is
consistent with the electronic structure data suggesting the half-metallic
nature Ni-NBT. Panel (c) shows the optical bandgap for different Ni-
concentrations. Consistent with the trend in experiment our computed $E_{g}$
decreases with Ni concentration. This could be explained in terms of inverse
relationship between static dielectric constant and $E_{g}$ using the Penn
model Penn (1962). The increased $E_{g}$ for 25% concentration could be
attributed the decrease in $\epsilon_{1}(0)$ to 61 from 81 for 18%. Our
computed $E_{g}$ for NBT is close to the previous calculations K. Vishwakarma
and Mani (2023). The experimental values, Refs. Andriyevsky  _et al._ (2009);
Dung  _et al._ (2020), are however on the higher side as they are at finite
temperatures.

###  III.4 Magnetic Properties

Next, as a probe to magnetic degrees of freedom introduced in the system, we
examined the magnetic moments of NBT and Ni-NBT. And, to find the actual
ground state magnetic configuration of the system, we probed both
ferromagnetic (FM) and antiferromagnetic (AFM) orientations of magnetic
moments. From our calculations, we find FM phase as the actual ground state
for all concentrations Ni. This is evident from the relative energies of FM
and AFM phases given in Table 2 for 25% of Ni, where the AFM energy is
observed to be larger $\approx$ 25 meV. In Fig. 7, we have shown total
magnetic moment as function of Ni-concentration. Consistent with literature,
and as to be expected due to pure ferroelectric nature, we obtained a zero
magnetic moment for NBT. For Ni-NBT, however, we observed a trend of
increasing (nonzero) magnetic moments as function of Ni-concentrations. The
maximum magnetic moment observed is $\leavevmode\nobreak\ 1.48$
$\mu_{B}$/f.u., for the highest concentration of 25%. The increase in the
magnetic moment with concentration could be attributed to the increasing
ferromagnetic exchange between neighboring Ni ions at higher concentrations.
Our calculated magnetic moment, 0.76 $\mu_{B}$/f.u., for concentration 12.5%
is in good agreement with the experimental value 0.91, reported for 9 % of Ni
Dung  _et al._ (2020).

Table 2: Relative energies of FM and AFM configurations, and the atom resolved magnetic moments (in $\mu_{B}$/atom) for 25Ni-NBT. Contribution from the orbital magnetic moment is given in the parenthesis. EFM (meV/f.u.) | 0
---|---
EAFM (meV/f.u.) | 24.9
$m_{\rm s}^{\rm Ni}$ ($m_{\rm o}^{\rm Ni}$) | 1.539 (0.02)
$m_{\rm s}^{\rm O}$ ($m_{\rm o}^{\rm O}$) | -0.122 (-0.006)
$m_{\rm s}^{\rm Ti}$ ($m_{\rm o}^{\rm Ti}$) | 0.036 (0.0)
$m_{\rm s}^{\rm Bi}$ ($m_{\rm o}^{\rm Bi}$) | 0.012 (-0.001)
$m_{\rm s}^{\rm Na}$ ($m_{\rm o}^{\rm Na}$) | 0.001 (0.0)
mTot($\mu_{B}$/f.u.) | 1.479
Figure 7: (a) Magentic moment as function of Ni concentration in Ni-NBT. (b)
Octahedral filling of electrons in $d$-orbitals. The experimental value in
panel (a) is from the Ref. Dung  _et al._ (2020).

To get more insight into the origin of nonzero magnetic moments in Ni-NBT, we
examined separate contributions from each ion. The data from this is tabulated
in Table 2 for the highest concentration of 25%. The values listed in the
parenthesis are the contributions from orbital magentic moment. As to be
expected, Ni contributes dominantly, with $\approx$ 105% of the total magnetic
moment. The spin magentic moment originates from the unpaired $3d$-electrons
in $e_{g}$ states (panel (b)). The obtained value of spin magentic moment,
1.54 $\mu_{B}$/atom, is however smaller than the expected theory value of 2.83
$\mu_{B}$/atom. The reason for this could be attributed to the strong
hybridization between O $2p$ and Ni $3d$ orbitals. Ni is also observed to
display a small orbital magnetic moment of 0.02 $\mu_{B}$/atom parallel to the
spin contribution through SoC. The second dominant contribution is from the O
ions. They contribute $\approx$ -9% of the total value. The opposite
contribution from O leads to a decrease in the total magentic moment. Like Ni,
O also has a small parallel contribution from the orbital magentic moment.
Among the other ions, Bi contributes about 2% of total value, whereas Na and
Ti has less than 1% contributions.

###  III.5 Magneto-optical Properties

Figure 8: Real and imaginary components of magneto-optical permittivity
(panels (a) and (b), respectively), Kerr rotation angle (panel (c)) and Kerr
ellipticity (panel (d)) as function of energy.

The presence of magnetic degrees of freedom in Ni-NBT leads to an anisotropy
in the dielectric tensor due to the breaking of time-reversal symmetry. The
dielectric tensor for a magnetized material could be written as

$\epsilon_{ij}=\epsilon^{(0)}_{ij}+\epsilon^{(1)}_{ij},$ (3)

where $\epsilon^{(0)}_{ij}$ is dielectric tensor in absence of magnetization
and $\epsilon^{(1)}_{ij}$ represents the contribution due nonzero
magnetization. Within linear in magnetization $M$, $\epsilon^{(1)}_{ij}$ could
be expressed as $\epsilon^{(1)}_{ij}=K_{ijk}M_{k}$, where $K$ is the magneto-
optical coefficient. To examine the magneto-optical properties of Ni-NBT, we
computed MOKE spectra in the polar configuration (Fig. 8(a)), where both the
incident linearly-polarized wave and magnetization are considered
perpendicular to the surface. The polar configuration is one of the most
common setups used to trace the magneto-optical properties experimentally. The
Kerr rotation angle, $\theta_{k}$, and Kerr ellipticity, $\eta_{k}$, can be
extracted from the diagonal and off-diagonal dielectric response as Picozzi
_et al._ (2006); Sangalli  _et al._ (2012)

$\theta_{k}+i\eta_{k}=-\frac{K}{\sqrt{\epsilon^{(0)}}(1-\epsilon^{(0)})},$ (4)

where $\epsilon^{(0)}$ is the diagonal component of the dielectric tensor in
the absence of magnetization. Separating real and imaginary components in Eq.
(4), we can derive

$\displaystyle\theta_{k}$ $\displaystyle=$
$\displaystyle-\frac{[K_{1}^{2}+K_{2}^{2}]^{1/2}}{[\epsilon_{1}^{2}+\epsilon_{2}^{2}]^{1/4}[(1-\epsilon_{1})^{2}+\epsilon_{2}^{2}]^{1/2}}\cos\Theta{\;\;\;\rm
and}$ (5) $\displaystyle\eta_{k}$ $\displaystyle=$
$\displaystyle-\frac{[K_{1}^{2}+K_{2}^{2}]^{1/2}}{[\epsilon_{1}^{2}+\epsilon_{2}^{2}]^{1/4}[(1-\epsilon_{1})^{2}+\epsilon_{2}^{2}]^{1/2}}\sin\Theta,$
(6)

where
$\Theta=\tan^{-1}(\frac{K_{2}}{K_{1}})-\frac{1}{2}\tan^{-1}(\frac{\epsilon_{2}}{\epsilon_{1}})-\tan^{-1}(\frac{-\epsilon_{2}}{1-\epsilon_{1}})$.
The calculated real and imaginary parts of K and complex Kerr rotation angles
for different concentrations of Ni-NBT are shown in Fig 8.

As discernible from the panel (d), Kerr rotation shows the same qualitative
behavior for all concentrations at higher energies, above $\sim 2$ eV. There
are substantial peaks in both negative and positive $y$-axes, which are
signatures of clockwise and anticlockwise polarizations, respectively, in the
material. In the low energy range, below $\sim 2$ eV, however, we observed a
mix trend for $\theta_{k}$ at different concentrations. In the negative
$y-$axis, the most significant peak of amplitude 0.58∘ is observed at 3.3 eV
for 6.5% concentration. The amplitude of the peaks is observed to decrease
with Ni-concentration. In the positive $y$-axis, however, a $\theta_{k}$
reaching up to 0.7∘ around 10.8 eV is observed for 25% concentration. Unlike
the trend for negative $\theta_{k}$, the peak amplitude for positive
$\theta_{k}$ increases with Ni-concentrations.

Fig. 8(e) shows the Kerr ellipticity data as function of energy. Like the
trend of $\theta_{k}$, all concentrations show a similar qualitative behavior
at higher energies, whereas a mix trend for amplitudes is observed in the low
energy range. We observed a significant peak of amplitude 0.72∘ around 10 eV
energy in the negative $y$-axis for 6.25% concentration. Unlike $\theta_{k}$,
there is not much variation in the amplitude of this peak with Ni-
concentrations. Consistent with the trend of $\theta_{k}$, apart from this
primary peak, we also observe few secondary peaks of amplitudes 0.33∘, 0.56∘,
and 0.54∘ at 4.0, 5.25 and 6.7 eV, respectively. The significant Kerr signals
obtained from our simulations for Ni-NBT suggest it as a potential candidate
for magneto-optical applications.

##  IV Conclusions

In conclusion, with the help of density functional theory based first-
principles calculations, we examined the effect of transition metal
substitution on electronic, ferroelectric, magnetic, optical and magneto
optical properties of NBT. In agreement with literature, our simulations on
electronic properties show NBT as a direct band semiconductor. Our computed
bandgap 2.56 eV is within the range of previous theory calculations and
experiments. For transition metal substituted-NBT, we observed an emergence of
half-metallic ferromagnetism in the system. Our simulation show, while
minority spin exhibits a wide bandgap, there are nonzero states at Fermi
energy for majority spin. The reason for this could be attributed to the shift
in the energy levels of majority spin states due to hybridization between O
$2p$ and Ni $3d$ states. This asymmetry in the two spin channels lead to an
emergence of nonzero permanent magentic moment in the material. We obtained a
magnetic moment of 1.5 $\mu_{\rm B}/{\rm f.u.}$ for 20% of Ni concentration.
For optical properties of NBT, our simulation results are consistent with the
available experimental and other theory results. For Ni-NBT, however, we
observed a diverging nature of static dielectric constant in the infrared
region, which could be attributed the metallic nature of the material. Our
data on MOKE show significant values of Kerr angles in Ni-NBT, which suggests
transition metal substituted-NBT as potential candidates for magneto-optical
applications.

## Acknowledgments

The authors wish to thank Ravi Kumar, Mohd Zeeshan and Indranil Mal for useful
discussions. C. K. V. acknowledges the funding support from Council of
Scientific & Industrial Research, India (Grant No.
$09$/$086$($1297$)/$2017$-EMR-I). B. K. M. acknowledges the funding support
from SERB, DST (CRG/2022/003845). The results presented in the paper are based
on the computations using the High Performance Computing cluster, Padum, at
the Indian Institute of Technology Delhi, New Delhi

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