# Magnetic exchange interactions at the proximity of a superconductor

Uriel A. Aceves Rodriguez1,2 Filipe Guimarães3 Sascha Brinker1 Samir Lounis1,2
<EMAIL_ADDRESS>1Peter Grünberg Institut and Institute for Advanced
Simulation, Forschungszentrum Jülich & JARA, 52425 Jülich, Germany
2Faculty of Physics & CENIDE, University of Duisburg-Essen, 47053 Duisburg,
Germany
3Jülich Supercomputing Centre, Forschungszentrum Jülich & JARA, 52425 Jülich,
Germany

###### Abstract

Interfacing magnetism with superconductivity gives rise to a wonderful
playground for intertwining key degrees of freedom: Cooper pairs, spin,
charge, and spin-orbit interaction, from which emerge a wealth of exciting
phenomena, fundamental in the nascent field of superconducting spinorbitronics
and topological quantum technologies. Magnetic exchange interactions (MEI),
being isotropic or chiral such as the Dzyaloshinskii-Moriya interactions
(DMI), are vital in establishing the magnetic behavior at these interfaces as
well as in dictating not only complex transport phenomena, but also the
manifestation of topologically trivial or non-trivial objects as skyrmions,
spirals, Yu-Shiba-Rusinov states and Majorana modes. Here, we propose a
methodology enabling the extraction of the tensor of MEI from electronic
structure simulations accounting for superconductivity. We apply our scheme to
the case of a Mn layer deposited on Nb(110) surface and explore proximity-
induced impact on the MEI. Tuning the superconducting order parameter, we
unveil potential change of the magnetic order accompanied with chirality
switching. Owing to its simple formulation, our methodology can be readily
implemented in state-of-the-art frameworks capable of tackling
superconductivity and magnetism. Our findings opens intriguing exploration
paths, where chirality and magnetism can be engineered depending on the
conducting nature of magneto-superconducting interfaces. We thus foresee
implications in the simulations and prediction of topological superconducting
bits as well as in cryogenic superconducting hybrid devices involving magnetic
units.

##  I Introduction

Despite the hostility between the superconducting and magnetic orders,
together they are known to bring to life an abundance of interesting physics,
such as in-gap effects such as Majorana[1, 2, 3], Andreev, and Yu-Shiba-
Rusinov (YSR) states[4, 5, 6, 7, 8]. These phenomena are currently in the
spotlight given their potential applications in the field of topological
quantum computing[9, 10, 11]. In the context of superconducting
spinorbitronics, the interplay of the underlying Cooper pairs with the three
electronic degrees of freedom—spin, charge and spin-orbit interaction—can
trigger tantalizing opportunities for cryogenic quantum technologies.

Since Majorana zero modes are essential for topological quantum computing,
numerous platforms have been proposed for their physical realization: magnetic
islands[12], skyrmions[13] and spin chains[14], among others. In the latter
two examples, the non-collinearity of the magnetic moments is a crucial
ingredient for the emergence of the coveted in-gap states. For non-
collinearity to occur, there must be competition between the magnetic
interactions in the system. We need thus methods to quantify and analyze these
interactions within a realistic description of the electronic structure of the
given systems. Moreover, we need to understand how are these interactions
affected by superconductivity and, in turn, how superconductors are influenced
by the magnetic structures in their proximity.

The microscopic theory to describe conventional superconductivity goes back
more than 60 years, first by the hands of John Bardeen, Leon Cooper, and
Robert Schrieffer (BCS)[15] in 1957. Few years later, the Bogoliubov–de Gennes
(BdG) method[16, 17, 18, 19] has been proposed and is currently an extensively
used framework to investigate superconducting systems with impurities,
superconductor/non-superconductor heterostructures, Josephson junctions, and
topological superconductors, to name some notable examples[20, 21, 22, 23,
24]. The BdG method is a mean-field approximation that relies upon
Bogoliubov–Valatin transformations that take the Hamiltonian from a particle
space into a particle-hole one, and it has been used in a variety of
situations from tight-binding[25, 26] to density functional theory (DFT)[27,
28]. Especially in latter, there have been efforts to computationally analyze
superconductor/non-superconductor heterostructures based on a realistic
description of the electronic structure[29, 21, 30, 31]. On the experimental
front, the activity regarding magnetic/superconductor interfaces has been
intense as well, with much focus on Majorana modes and other end-states on
atomic chains[32, 33, 14, 34, 35, 36]. In this paper, we provide a simple and
detailed demonstration on how to quantify the magnetic exchange interactions
(MEI) from electronic structure simulations of realistic materials accounting
for electron-hole coupling channels originating from the superconducting order
in the BdG method, spin-orbit coupling and multi-orbital hybridization
phenomena. We thus go beyond fundamental basic models suggested in the
past[37, 38]. Furthermore, we apply the proposed methodology to a magnetic
interface with a superconducting substrate: a Mn monolayer deposited on
Nb(110) surface and analyse the different contributions to the MEI emerging
from the proximity to a superconductor.

This paper is organized as follows: In Subsection II.1, we introduce our
multi-orbital tight-binding theory necessary for the incorporation of the BdG
equations. Subsection II.2 contains a discussion and description of the
theoretical formalism enabling the quantification of the bilinear tensor of
magnetic exchanges in the context of the BdG method. We present a simple
formula to calculate the bilinear magnetic exchanges within the Green function
formalism in the particle-hole space, and establish how this formula converges
to the same one as in the metallic case[39, 40] in absence of
superconductivity. In Section III, a prototypical system composed of one
monolayer of Mn (110) on top of a 5-atom-thick slab of Nb (110) is used as a
proof-of-concept to which we apply our theory. We perform self-consistent
calculations to investigate the effect that the superconductivity in Nb has on
the magnetic properties of the Mn atoms and vice-versa. We find the magnetic
ground state of the Mn monolayer to be row-wise antiferromagnetic, agreeing
with recent experimental and theoretical results[41]. We proceed to test a
wide range of electron-phonon coupling strengths that directly influences the
size of the superconducting gap, evaluating the resulting self-consistent gap
parameters and the corresponding magnetic moments in Mn. The effect of
superconductivity in the Heisenberg exchange interactions for different gap
sizes is scrutinized in Subsection III.1. We find that for a realistic value
of the superconducting gap, the change induced by superconductivity is minimal
and therefore does not have important repercussions for the case of
Mn/Nb(110). However, for gap values of the order of the Heisenberg exchange,
we observe a change in the magnetic ground state, from row-wise
antiferromagnetic to ferromagnetic. Finally, in Subsection III.2, we focus on
the Dzyaloshinskii-Moriya interaction (DMI) and observe that the corrections
due to superconductivity are also relatively small in this case. Nevertheless,
we notice that in our case the chirality of the corrective term is opposite to
the one at the non-superconducting case. Utilizing a multiple-scattering
expansion, we identify how the intertwining of the superconducting parameter,
intra-atomic spin-orbit and exchange interactions impact the sign of the
Heisenberg exchange as well as the DMI.

##  II Theoretical Description

To investigate the characteristics and effects of superconducting magnetic
structures, we explored a system consisting of a slab of Nb (110) with a
thickness of 5 layers, with a monolayer of Mn on top, as shown in Figs. 1(a)
and (b). Fig. 1(b) represents the magnetic ground state, which is
antiferromagnetic, found after our self-consistent simulations. We chose Nb as
a substrate given its large superconducting gap[42]
($2\Delta=$3.8\text{\,}\mathrm{meV}$$), critical temperature of
$T_{C}=$9.3\text{\,}\mathrm{K}$$[42], and most importantly, given the
development of recent experimental techniques to fabricate clean surfaces of
Nb (110)[43]. The latter work led to a breakthrough, and since its publication
the Nb(110) surface became a standard playground for the exploration of
potential Majorana and YSR states hosted by adatoms[44, 45, 28, 46],
nanowires[47, 14, 36], two-dimensional diluted structures [48] and thin
films[49, 50, 41]. In the following, we describe the theoretical framework
that we developed to quantify magnetic interactions at the vicinity of a
superconductor.

Figure 1:  (a) Side view of a slab of five Nb (110) layers with a Mn monolayer
on top, blue spheres represent Nb atoms, lilac spheres Mn. (b) Top view. Each
layer has the structure of a centered rectangular lattice. The arrows show the
magnetic ground state which for this case is row-wise antiferromagnetic. (c)
Schematics of different Green functions after perturbative expansion with
respect to the superconducting order parameter. (d) Superconducting gap
parameter as function of $\lambda$, the vertical dotted line is at
$\Delta$=$3.264\text{\,}\mathrm{meV}$ and indicates the lowest value of
$\lambda$ for which TITAN converged to a state with a finite $\Delta$. All
calculations were done at $4.2\text{\,}\mathrm{K}$ with 15000 k-points. (d)
Magnetic moments of the Mn layer as a function of $\lambda$.

###  II.1 Tight-binding and the Bogoliubov–de Gennes method

Before presenting the method to extract magnetic exchange interactions, which
is the focus of our work, we introduce the multi-orbital electronic
Hamiltonian and the Bogoliubov–de Gennes method. The magnetic and
superconducting system may be described by the Hamiltonian

$\displaystyle
H_{S}=\frac{1}{N}\left\\{\sum_{\alpha\beta,\sigma\eta,\mu\nu,\bm{k}}H_{\alpha\beta,\sigma\eta}^{\mu\nu}(\bm{k})c^{\dagger}_{\alpha\mu\sigma}(\bm{k})c_{\beta\nu\eta}(\bm{k})-\sum_{\alpha,\mu\nu,\bm{k}\bm{k}^{\prime}}\lambda_{\alpha\mu}c^{\dagger}_{\alpha\mu\uparrow}(\bm{k})c^{\dagger}_{\alpha\mu\downarrow}(-\bm{k})c_{\alpha\mu\downarrow}(-\bm{k}^{\prime})c_{\alpha\mu\uparrow}(\bm{k}^{\prime})\right\\},$
(1)

with $c^{\dagger}_{\alpha\mu\sigma}(\bm{k})$ and $c_{\beta\nu\eta}(\bm{k})$
being the creation and annihilation operators of electrons with wave vector
$\bm{k}$ and spin $\sigma$ in the orbitals $\mu$ or layer $\alpha$ and spin
$\eta$ in the orbitals $\nu$ or layer $\beta$, respectively. $N$ is the number
of wave vectors in the Brillouin zone. $\bm{k}$ is a reciprocal vector in in
the xy-plane (within each layer) where two-dimensional periodic boundary
conditions are assumed. The second element on the right-hand side corresponds
to the BCS term, allowing electrons to form Cooper pairs and, therefore to
give rise to superconductivity. Its strength, given
$\lambda_{\alpha\mu}\in\mathbb{R}$, originates from the electron-phonon
coupling and may depend on the orbital $\mu$ of layer $\alpha$.
$H_{\alpha\beta,\sigma\eta}^{\mu\nu}(\bm{k})$ is the non-superconducting
Hamiltonian, which can be further separated into

$H^{\mu\nu}_{\alpha\beta,\sigma\eta}(\bm{k})=H_{\alpha\beta}^{0\mu\nu}(\bm{k})\sigma^{0}+\bm{\sigma}\cdot\hat{\bm{e}}_{\alpha}B_{\alpha}^{[\text{xc}]\mu\nu}(\bm{k})\delta_{\alpha\beta}+\bm{\sigma}\cdot\bm{B}_{\alpha}^{[\text{soc}]\mu\nu}(\bm{k})\delta_{\alpha\beta},$
(2)

where $H_{\alpha\beta}^{0\mu\nu}$ is the spin-independent tight-binding term,
the second term comprises the intra-atomic exchange interactions (originating
from a Hubbard-like contribution[51, 52]), and the last term describe the
spin-orbit interaction. The hopping parameters for Nb and Mn were obtained
from first-principles calculations [53].

In the mean-field approximation, Eq. (1) simplifies to

$\displaystyle H_{S}^{\text{MF}}=$
$\displaystyle\frac{1}{N}\sum_{\bm{k}}\left\\{\sum_{\alpha\beta,\sigma\eta,\mu\nu}H_{\alpha\beta,\sigma\eta}^{\mu\nu}(\bm{k})c^{\dagger}_{\alpha\mu\sigma}(\bm{k})c_{\beta\nu\eta}(\bm{k})\right.$
(3)
$\displaystyle-\left.\sum_{\alpha,\mu}\left(\Delta_{\alpha\mu}^{*}c_{\alpha\mu\downarrow}(-\bm{k})c_{\alpha\mu\uparrow}(\bm{k})+\Delta_{\alpha\mu}c_{\alpha\mu\uparrow}^{\dagger}(\bm{k})c_{\alpha\mu\downarrow}^{\dagger}(-\bm{k})\right)\right\\},$

with

$\Delta_{\alpha\mu}=\lambda_{\alpha\mu}\frac{1}{N}\sum_{\bm{k}}\langle
c_{\alpha\mu\downarrow}(-\bm{k})c_{\alpha\mu\uparrow}(\bm{k})\rangle,\quad\Delta_{\alpha\mu}^{*}=\lambda_{\alpha\mu}\frac{1}{N}\sum_{\bm{k}}\langle
c_{\alpha\mu\uparrow}^{\dagger}(\bm{k})c_{\alpha\mu\downarrow}^{\dagger}(-\bm{k})\rangle,$
(4)

$\Delta_{\alpha\mu}$ is known as the superconducting gap parameter. For clean
superconductors it is half of the superconducting gap, as it defines the
necessary energy to scatter Cooper pairs (which live at the Fermi level)[54].
It is important to note that for each choice of $\lambda_{\alpha\mu}$ and
$B_{i}^{[\text{xc}]\mu\nu}$ the final values for $\Delta_{\alpha\mu}$ and the
magnetic moments $\bm{m}^{\mu}$ in the ground state are obtained self-
consistently. This means that even though it seems to be linearly proportional
to $\lambda_{\alpha\mu}$, this is not the case in practice as seen in Fig. 1
(d). This leads to a more realistic characterization of materials as no
determined state is enforced to the system, and they can then evolve after
self-consistency into their ground state. In this work, we restrict ourselves
to only two different values for $\lambda_{\alpha\mu}$, namely, a constant
$\lambda_{\alpha\mu}=\lambda$ for all the orbitals of the Nb layers, and
$\lambda_{\alpha\mu}=0$ for all $\mu$ in the Mn one.

To diagonalize the Hamiltonian in Eq. (3) we use a Bogoliubov–Valatin
transformation[19], thus transferring the original Hamiltonian from an
electron representation to an electron-hole one. The transformation is given
by

$c_{\alpha\mu\sigma}(\bm{k})=\sum_{n}^{{}^{\prime}}u_{\alpha\sigma}^{n}(\bm{k})\gamma_{n}+v_{\alpha\sigma}^{n*}(\bm{k})\gamma_{n}^{\dagger},\quad
c_{\alpha\mu\sigma}^{\dagger}(\bm{k})=\sum_{n}^{{}^{\prime}}u_{\alpha\sigma}^{n*}(\bm{k})\gamma_{n}^{\dagger}+v_{\alpha\sigma}^{n}(\bm{k})\gamma_{n}^{\dagger},$
(5)

where the tilde indicates that the sums run only over the states with positive
energy[19, 55]. This restriction in the sum is done to counteract the doubling
of the degrees of freedom originated from the change of basis. After the
transformation, we land in a system where the new particles (sometimes called
bogolons[56]) are constituted by mixtures of electron and hole operators. The
transformation in Eq. (5) is canonical, this means that the new operators
$\gamma_{n}$ and $\gamma_{n}^{\dagger}$ fulfill the same anticommutation
relations as $c_{n}$ and $c_{n}^{\dagger}$, namely

$\\{\gamma_{n},\gamma_{m}\\}=\\{\gamma_{n}^{\dagger},\gamma_{m}^{\dagger}\\}=0,\quad\\{\gamma_{n}^{\dagger},\gamma_{m}\\}=\delta_{nm}.$
(6)

After the transformation, we arrive at a set of equations of the form[19]

$\sum_{\beta\mu}H_{\text{BdG}}^{\alpha\beta,\mu\nu}(\bm{k})\phi_{\beta\mu}(\bm{k})=E_{n}(\bm{k})\phi_{\alpha\nu}(\bm{k}),$
(7)

where the Bogoliubov–de Gennes Hamiltonian is given by

$H_{\text{BdG}}^{\alpha\beta,\mu\nu}(\bm{k})=\begin{pmatrix}H_{\alpha\beta,\uparrow\uparrow}^{\mu\nu}(\bm{k})-E_{F}&H_{\alpha\beta,\uparrow\downarrow}^{\mu\nu}(\bm{k})&0&-\Delta_{\alpha\mu}\mathbb{I}\\\
H_{\alpha\beta,\downarrow\uparrow}^{\mu\nu}(\bm{k})&H_{\alpha\beta,\downarrow\downarrow}^{\mu\nu}(\bm{k})-E_{F}&\Delta_{\alpha\mu}\mathbb{I}&0\\\
0&\Delta_{\alpha\mu}^{*}\mathbb{I}&-H_{\alpha\beta,\uparrow\uparrow}^{\mu\nu*}(-\bm{k})+E_{F}&-H_{\alpha\beta,\uparrow\downarrow}^{\mu\nu*}(-\bm{k})\\\
-\Delta_{\alpha\mu}^{*}\mathbb{I}&0&-H_{\alpha\beta,\downarrow\uparrow}^{\mu\nu*}(-\bm{k})&-H_{\alpha\beta,\downarrow\downarrow}^{\mu\nu*}(-\bm{k})+E_{F}\end{pmatrix}.$
(8)

It is important to notice that due to the transformation given in Eq. (5), the
hole-space change the wave vector arguments from $\bm{k}$ to $-\bm{k}$. The
eigenvector of Eq. (7) is

$\phi_{i\nu}(\bm{k})=\begin{pmatrix}u_{\alpha\nu\uparrow}(\bm{k})\\\
u_{\alpha\nu\downarrow}(\bm{k})\\\ v_{\alpha\nu\uparrow}(\bm{k})\\\
v_{\alpha\nu\downarrow}(\bm{k})\end{pmatrix}.$ (9)

$E_{F}$ in Eq. (8) is the Fermi energy and it is placed in the diagonal such
that the Fermi level is at zero for the BdG system.

Structurally we can consider $H_{\text{BdG}}^{ij,\mu\nu}$ as subdivided into
four parts; namely, blocks of electron-electron, electron-hole, and hole-hole
interactions. The main diagonal consists of non-hybrid interactions, while the
antidiagonal terms contain only the superconducting gap parameter that
hybridizes electrons and holes. For simplicity in the discussions, we break
down the BdG Hamiltonian as follows

$\displaystyle H_{\alpha\beta,\mu\nu}^{\text{ee}}(\bm{k})$
$\displaystyle=\begin{pmatrix}H_{\alpha\beta,\uparrow\uparrow}^{\mu\nu}(\bm{k})-E_{F}&H_{\alpha\beta,\uparrow\downarrow}^{\mu\nu}(\bm{k})\\\
H_{\alpha\beta,\downarrow\uparrow}^{\mu\nu}(\bm{k})&H_{\alpha\beta,\downarrow\downarrow}^{\mu\nu}(\bm{k})-E_{F}\end{pmatrix},\quad$
$\displaystyle
H_{\alpha\beta,\mu\nu}^{\text{eh}}=\begin{pmatrix}0&-\Delta_{\alpha\mu}\mathbb{I}\\\
\Delta_{\alpha\mu}\mathbb{I}&0\end{pmatrix},$ $\displaystyle
H_{\alpha\beta,\mu\nu}^{\text{hh}}(\bm{k})$
$\displaystyle=\begin{pmatrix}-H_{\alpha\beta,\uparrow\uparrow}^{\mu\nu*}(-\bm{k})+E_{F}&-H_{\alpha\beta,\uparrow\downarrow}^{\mu\nu*}(-\bm{k})\\\
-H_{\alpha\beta,\downarrow\uparrow}^{\mu\nu*}(-\bm{k})&-H_{\alpha\beta,\downarrow\downarrow}^{\mu\nu*}(-\bm{k})+E_{F}\end{pmatrix},$
$\displaystyle
H_{\alpha\beta,\mu\nu}^{\text{he}}=\begin{pmatrix}0&\Delta_{\alpha\mu}^{*}\mathbb{I}\\\
-\Delta_{\alpha\mu}^{*}\mathbb{I}&0\end{pmatrix}.$

To avoid confusion from handling too many indices we will drop them for these
submatrices whenever the context allows it. Thus, we write the BdG Hamiltonian
in the following form:

$H_{\text{BdG}}=\begin{pmatrix}H^{\text{ee}}&H^{\text{eh}}\\\
H^{\text{he}}&H^{\text{hh}}\end{pmatrix}.$ (10)

From Eq. (10), we can obtain the corresponding retarded Green function via

$G_{\text{BdG}}(\bm{k},E+i\eta)=(E-H_{\text{BdG}}(\bm{k})+i\eta)^{-1}.$ (11)

$G_{\text{BdG}}(E+i\eta)$ in turn is a matrix that for the sake of simplicity
we also consider as subdivided into four blocks as in Eq. (10)

$G_{\text{BdG}}=\begin{pmatrix}G^{\text{ee}}&G^{\text{eh}}\\\
G^{\text{he}}&G^{\text{hh}}\end{pmatrix}.$ (12)

When the system is not superconducting $\Delta=0$, the Hamiltonian given in
Eq. (10) becomes diagonal in particle-hole space (i.e., $H^{\text{eh}}$ and
$H^{\text{he}}$ vanish). Consequently $G_{\text{BdG}}$ is also diagonal
($G^{\text{eh}}=G^{\text{he}}=0$), and the non-superconducting system is
described by $G^{\text{ee}}$.

###  II.2 Bilinear magnetic exchange tensor and the BdG method

Mapping the magnetic interactions from the electronic structure simulations of
a realistic system into model Hamiltonians gives us the possibility of
isolating different phenomena and analysing each of them separately depending
on the underlying mechanims. Here we focus on the extended Heisenberg model,
represented by

$H_{\text{Hb}}=-\frac{1}{2}\sum_{ij}\hat{\bm{e}}_{i}\cdot\mathcal{J}_{ij}\cdot\hat{\bm{e}}_{j},$
(13)

where $\hat{\bm{e}}_{i}$ is the direction of the magnetic moment for atom $i$
and $\mathcal{J}_{ij}$ is the bilinear tensor of magnetic exchange
interactions with the magnetic moment for atom $j$. This Hamiltonian can be
divided into three terms with different symmetries[57]

$J_{ij}=\frac{\text{Tr}(\mathcal{J}_{ij})\mathbb{I}_{3}}{3},\quad
J_{ij}^{\text{s}}=\frac{\mathcal{J}_{ij}+\mathcal{J}_{ji}}{2}-\frac{\text{Tr}(\mathcal{J}_{ij})\mathbb{I}_{3}}{3},\quad
D_{ij}=\frac{\mathcal{J}_{ij}-\mathcal{J}_{ji}}{2}.$ (14)

Using these definitions, Eq. (13) can also represented as

$H_{\text{Hb}}=-\frac{1}{2}\sum_{ij}J_{ij}\hat{\bm{e}}_{i}\cdot\hat{\bm{e}}_{j}-\frac{1}{2}\sum_{ij}\hat{\bm{e}}_{i}\cdot
J_{ij}^{\text{s}}\cdot\hat{\bm{e}}_{j}-\frac{1}{2}\sum_{ij}\bm{D}\cdot(\hat{\bm{e}}_{i}\times\hat{\bm{e}}_{j}).$
(15)

The first term on the right-hand side (Heisenberg exchange) favours ferro- or
antiferromagnetic alignments depending on the sign of J. The second one is the
traceless anisotropic part of $\mathcal{J}_{ij}$ induced by spin-orbit
coupling. Finally, the last item in Eq. (15) corresponds to the DMI, which is
finite when inversion symmetry is broken and requires spin-orbit coupling. It
may induce a relative rotation in the magnetic moments of neighbouring atoms
and is vital for the stabilization of magnetic textures such as skyrmions[58,
59, 60]. In this work we focus only on the Heisenberg exchange and the DMI,
since $J_{ij}^{\text{s}}$ is negligible.

The components of the DMI are extracted as follows[57]

$D_{ij}=\begin{pmatrix}0&D_{z}&-D_{y}\\\ -D_{z}&0&D_{x}\\\
D_{y}&-D_{x}&0\end{pmatrix}.$ (16)

Having a realistic description of the magnetic interactions of the system
through these terms is instrumental for the understanding and the development
of upcoming technologies that rely on magnetism, such as spintronic
devices[61] with magnetic domain walls[62], spintronic diodes[63, 64], and
superconductor/ferromagnet systems for quantum computing[65]. Although the
purely magnetic scenario has been intensely investigated, that is no longer
the case when the structure contains a superconductor. Here, Cooper pairs
enter the picture through the coupling between the electronic and hole states,
which is not taken into account by the basic theory, developed exclusively for
metals. To analyze those cases we need to account for the potential mutual
impact of superconductivity and magnetism.

There are several techniques to obtain the tensor $\mathcal{J}_{ij}$ from
electronic structure simulations, one of the most common being the
infinitesimal rotations method[39], which presents a way to map energies from
the electronic structure into energies on an extended Heisenberg model. The
basic idea behind this approach is to perturb the magnetic moments at two
different locations $i$ and $j$, and quantify the resultant change in energy.
To get the bilinear tensor of magnetic exchanges $\mathcal{J}_{ij}$ we take
the second order term of the energy change. For a non-superconducting system,
such change (given a perturbation potential $\delta V$) is represented by[39,
66, 40, 67]

$\delta
E=-\frac{1}{\pi}\text{Im}\int_{-\infty}^{\epsilon_{F}}d\epsilon\sum_{p}\frac{1}{p}\text{Tr}[G(\epsilon)\delta
V]^{p}.$ (17)

where $p$ describes the order of the expansion and Tr is the trace over the
site, spin and orbital spaces. This formula is based on the expansion of the
band energy in a perturbative fashion, which is a well established technique.
In order to follow its derivation, we refer the readers to the aforementioned
literature. The extension of the matrix space by the Bogoliubov–de Gennes
equations imposes changes on the Green functions as well as on the
perturbation potential. Within this formalism Eq. (17) becomes

$\delta
E=\frac{1}{\pi}\text{Im}\int_{-\infty}^{\epsilon_{F}}d\epsilon\sum_{p}\frac{1}{p}\text{
tr}\left\\{\Theta[G_{\text{BdG}}(\epsilon)\delta V]^{p}\right\\}.$ (18)

where the new trace (tr) runs over electron-hole space in addition to the
site, spin and orbital ones. $\Theta$ is a matrix in electron-hole space,
whose function in Eq. (18) is to isolate the purely electronic terms, and it
is given by

$\Theta=\begin{pmatrix}\mathbb{I}&0\\\ 0&0\end{pmatrix}.$ (19)

The tensor of bilinear magnetic exchanges is obtained by taking the second-
order term of the expansion of Eq. (18), that is

$\mathcal{J}_{ij}=-\frac{\partial^{2}E_{ij}}{\partial\hat{\bm{e}}_{i}\partial\hat{\bm{e}}_{j}}$
(20)

as expected from Eq. 13, with

$\delta
E_{ij}=\frac{1}{2\pi}\Theta\text{Im}\text{Tr}\int_{-\infty}^{\epsilon_{F}}d\epsilon
G_{\text{BdG}}(\epsilon)\delta VG_{\text{BdG}}(\epsilon)\delta V.$ (21)

Here, the shape of the dispersion potential $\delta V$ must also be
generalized by varying the magnetization orientation vector $\hat{\bm{e}}_{i}$
in the BdG Hamiltonian in Eq. (8). This results in

$\delta V=\begin{pmatrix}\delta V^{e}&0\\\ 0&\delta V^{h}\end{pmatrix},$ (22)

where

$\delta V^{h}_{i}=-\delta
V^{e*}_{i}=-(B^{[\text{xc}]}\bm{\sigma}\cdot\delta\bm{e}_{i})^{*}.$ (23)

Collecting these results into Eq. (21), we obtain

$\displaystyle\mathcal{J}_{ij}=\delta
E_{ij}=-\frac{1}{2\pi}\mathrm{Im}\mathrm{Tr_{Ls}}\int_{-\infty}^{\epsilon_{F}}$
$\displaystyle
d\epsilon\,\left[B^{\mathrm{[xc]}}\bm{\sigma}\cdot\delta\bm{e}_{i}G^{\text{ee}}_{ij}(\epsilon)B^{\mathrm{[xc]}}\bm{\sigma}\cdot\delta\bm{e}_{j}G_{ji}^{\text{ee}}(\epsilon)\right.$
(24)
$\displaystyle\left.\quad-B^{\mathrm{[xc]}}\bm{\sigma}\cdot\delta\bm{e}_{i}G^{\text{eh}}_{ij}(\epsilon)B^{\mathrm{[xc]*}}\bm{\sigma}^{*}\cdot\delta\bm{e}_{j}G_{ji}^{\text{he}}(\epsilon)\right].$

For non-superconducting systems, $G_{ji}^{\text{he}}(\epsilon)$ and
$G_{ji}^{\text{eh}}(\epsilon)$ vanish and so does the second term in the
right-hand side of Eq. (24), leading to the resulting equation that is the
same as for the non-superconducting case. Nevertheless, it is important to
notice that when $\Delta>0$, not only is the second term finite, but also the
first term gets renormalized by the presence of the superconducting gap. When
analyzing the resulting MEI it is often useful to separate both terms in the
right hand side of Eq. (24), here we will do so in the following form

$\displaystyle\mathcal{J}_{ij}$
$\displaystyle=\mathcal{J}^{ee}_{ij}+\mathcal{J}^{ee}_{ij},$ (25)
$\displaystyle\mathcal{J}^{ee}_{ij}$
$\displaystyle=-\frac{1}{2\pi}\mathrm{Im}\mathrm{Tr_{Ls}}\int_{-\infty}^{\epsilon_{F}}d\epsilon\,B^{\mathrm{[xc]}}\bm{\sigma}\cdot\delta\bm{e}_{i}G^{\text{ee}}_{ij}(\epsilon)B^{\mathrm{[xc]}}\bm{\sigma}\cdot\delta\bm{e}_{j}G_{ji}^{\text{ee}}(\epsilon),$
$\displaystyle\mathcal{J}^{eh}_{ij}$
$\displaystyle=\frac{1}{2\pi}\mathrm{Im}\mathrm{Tr_{Ls}}\int_{-\infty}^{\epsilon_{F}}d\epsilon\,B^{\mathrm{[xc]}}\bm{\sigma}\cdot\delta\bm{e}_{i}G^{\text{eh}}_{ij}(\epsilon)B^{\mathrm{[xc]*}}\bm{\sigma}^{*}\cdot\delta\bm{e}_{j}G_{ji}^{\text{he}}(\epsilon).$

Utilizing perturbation theory, we expect a second order correction to
$G^{\text{ee}}$ and $G^{\text{hh}}$ due to superconductivity, while the
electron-hole parts of the Green function $G^{\text{eh}}$ and $G^{\text{he}}$
would at least experience a first-order correction involving a possible sign
change (see Fig. 1 (c)), which could counteract the electron-electron
contribution to the magnetic exchange interaction. This can be easily grasped
by utilising the perturbative expansion of the Dyson equation that related the
Green function, $g$, associated to the Hamiltonian with $\Delta=0$ to $G$ for
which $\Delta$ is finite. The Dyson equation reads

$G=g+g\Delta G$ (26)

thus

$\displaystyle G^{\text{ee}}$ $\displaystyle\approx$ $\displaystyle
g^{\text{ee}}+g^{\text{ee}}\Delta g^{\text{hh}}\Delta^{*}g^{\text{ee}}$ (27)
$\displaystyle G^{\text{hh}}$ $\displaystyle\approx$ $\displaystyle
g^{\text{hh}}+g^{\text{hh}}\Delta g^{\text{ee}}\Delta^{*}g^{\text{hh}}$ (28)
$\displaystyle G^{\text{eh}}$ $\displaystyle\approx$ $\displaystyle
g^{\text{ee}}\Delta g^{\text{hh}}$ (29) $\displaystyle G^{\text{he}}$
$\displaystyle\approx$ $\displaystyle g^{\text{hh}}\Delta^{*}g^{\text{ee}}.$
(30)

To summarize the theoretical scheme, we first proceed to the electronic
simulations of a given material without including the superconducting order
parameter. Once the ground state is found, we extract the associated Green
functions and compute the tensor of magnetic exchange interactions following
the infinitesimal rotation method. Knowing the interactions, we can determine
the magnetic ground state. The full procedure is repeated once the
superconducting order parameter is included.

##  III Results

To ascertain how the superconducting gap on the material influences the
magnetic states, we selected a large range of values for $\lambda$ on the Nb
slab, ranging from $2.45\text{\,}\mathrm{eV}$ to $8.16\text{\,}\mathrm{eV}$.
The calculations were performed with a broadening of the energy levels of
$0.113\text{\,}\mathrm{meV}$, which should mimic a temperature of
$4.2\text{\,}\mathrm{K}$. For bulk Nb at this temperature and
$\lambda=$2.45\text{\,}\mathrm{eV}$$, we found a superconducting gap parameter
of $\approx$1.8\text{\,}\mathrm{meV}$$, which is close to the realistic values
found by experiments (from $\approx$1.41\text{\,}\mathrm{meV}$$ to
$\approx$1.57\text{\,}\mathrm{meV}$$[68]).

In Fig. 1 (d), we display the resulting self-consistent gap parameters
$\Delta$ at the Nb layer adjacent to the Mn layer attained for each value of
$\lambda$. For this system, we observe that the strength needed to open the
superconducting gap are larger compared to the one in bulk Nb, for which
$\lambda=$2.45\text{\,}\mathrm{eV}$$ already opens a gap. Here, the lowest
electron-phonon coupling strength to produce a superconducting state is
$3.26\text{\,}\mathrm{eV}$, resulting in a gap parameter of
$\Delta=$6.25\text{\,}\mathrm{meV}$$. The resulting growth of the self-
consistent $\Delta$ with respect to $\lambda$ is not linear, and although the
coupling strength is in the order of $\text{\,}\mathrm{eV}$, the resulting
gaps are of the same order of magnitude as the ones reported
experimentally[41]. We note that this is enabled by self-consistency of our
simulations. A one-shot calculation leads to a gap of the same order than
$\lambda$.

We show the immediate impact superconductivity has on the magnetic moments of
the Mn atoms in Fig. 1 (e). The magnetic moment experiences a total change of
about 0.25 $\mu_{B}$ starting from $3.80\text{\,}$ $\mu_{B}$ in the metallic
regime and intriguingly increases to reach a maximum of $4.07\text{\,}$
$\mu_{B}$ when the gap is about $5.30\text{\,}\mathrm{meV}$ before
experiencing a decrease for larger gaps. This observed behavior is induced by
the non-trivial impact of $\lambda$, simulating here the electron-phonon
coupling, on the electronic structure.

###  III.1 Symmetric magnetic exchange

According to our convention in Eq. (15), a positive $J_{ij}$ favours a
parallel alignment as it leads to lower (negative) energies, while negative
$J_{ij}$ favours anti-parallel alignment. The red points in Fig. 2 (a)
represent the MEI for the normal (non-superconducting) case. While the
strongest interaction coming from the nearest neighbour interaction is
antiferromagnetic, the next-nearest neighbour interactions are ferromagnetic.
Since the crystal lattice for Mn(110) is centered rectangular these MEI lead
to a magnetic ground state which is row-wise antiferromagnetic as shown in
Fig. 1 (b). In Fig. 2 (a) we also display the Heisenberg exchange for the
system when superconductivity is present (blue points). In this case,
$\lambda=$3.26\text{\,}\mathrm{eV}$$, the smallest value in the investigated
grid that leads to a finite gap parameter
($\Delta=$6.25\text{\,}\mathrm{meV}$$; as a reference, the measured value for
bulk Nb is in a range from $\approx$1.41\text{\,}\mathrm{meV}$$ to
$1.57\text{\,}\mathrm{meV}$[68]).

The difference between the interactions in the normal and superconducting
systems is presented in Fig. 2 (b). One can notice that the maximum change in
the symmetric part of the MEI is in the order of $\text{\,}\mathrm{meV}$,
which is small in comparison to the values of $J_{ij}$ themselves.
Interestingly, enabling superconductivity at the interface does not have a
uniform impact on the magnetic exchange interactions as function of distance.
At short distances, a small decrease in the absolute value of the magnetic
interaction is identified. In the particularly investigated interface,
however, the magnetic ground state stays unperturbed and the row-wise
antiferromagnetic ordering in the Mn monolayer prevails, even after including
effects derived from superconductivity. Experimentally, Roberto lo Conte et
al.[41] observed a row-wise antiferromagnetic ground state, where they
theoretically obtain a row-wise antiferromagnetic ground state for the non-
superconducting case, and experimentally observe that the magnetic ground
state remains like that even when the Nb(110) slab is superconducting.

Figure 2: (a) Symmetric exchange of the normal (red) vs superconducting
($\Delta=$6.25\text{\,}\mathrm{meV}$$, blue) states. (b) Difference between
the symmetric magnetic exchanges in the normal ($J_{N}$) vs the
superconducting regime ($J_{S}$), when $\Delta=$6.25\text{\,}\mathrm{meV}$$.
(c) Symmetric exchange of the normal vs superconducting
($\Delta=$295\text{\,}\mathrm{meV}$$) states. (d) Difference between $J_{N}$
and $J_{S}$ when $\Delta=$295\text{\,}\mathrm{meV}$$. All calculations were
done at $4.2\text{\,}\mathrm{K}$, with 15,000 k-points.

To uncover the possible effects that superconductivity may cause in the
magnetic states, we artificially increase $\lambda$ to
$8.16\text{\,}\mathrm{eV}$ such that $\Delta=$295\text{\,}\mathrm{meV}$$. Such
a scenario where $\Delta$ is one order of magnitude larger than the nearest
neighboring magnetic interaction looks at first sight unrealistic but it can
correspond to a diluted lattice as realized recently experimentally with Cr
adatoms sitting on Nb(110) surface [48]. In that case, the magnetic
interactions are rather weak since the magnetic atoms are not nearest
neighbors and interact magnetically via the substrate through RKKY-
interactions.

Although the gap is large, the magnetic moments on the Mn layer
($3.75\text{\,}$$\mu_{\text{B}}$) are still on the order of the ones in the
non-superconducting case ($3.80\text{\,}$$\mu_{\text{B}}$. See Fig. 1 (e)).
The resulting Heisenberg interaction is displayed in Fig. 2 (c), both for the
normal (red) and superconducting (blue) states. Their differences can be seen
in Fig. 2 (d), in which we recognize crucial changes—the largest one being the
nearest neighbour interaction that changes sign and switches from
antiferromagnetic to ferromagnetic. This leads to a dramatic impact on the
magnetic ground state, which switches from the row-wise antiferromagnetic to a
ferromagnetic one.

While the previous figures show the extreme cases of low and high
superconductivity in relation to the magnetic interactions, Figs. 3 (a)-(c)
present a complete depiction of how the Heisenberg exchange as a function of
the distance is affected by the value of $\lambda$. The electron-hole
contribution to the Heisenberg exchange for $\Delta=0$ vanishes, as expected,
and hence the total Heisenberg exchange is given only by the electron-electron
one. Additionally, we notice that the electron-hole part is mostly negative,
favouring antiferromagnetism. When focusing on the first two nearest neighbors
(given by the first two columns of each plot), we detect a change in the
magnetic ground state, from antiferromagnetic (blue) alignment to
ferromagnetic (red). It is interesting to note that this does not originate
from the electron-hole contribution, but rather from the changes experienced
by the electron-electron term. These changes are caused by the shift of the
bands as $\Delta$ increases.

Performing a multiple-scattering expansion of the Green functions involved in
defining the magnetic exchange interactions, in the same spirit than the
approach undertaken in [69, 70, 67, 71, 72], we end up with systematic
corrections illustrated in Fig. 4 (a)-(b). The derivations are based on the
Green functions expanded in
Eqs.LABEL:eq_Gee_delta,eq_Ghh_delta,eq_Geh_delta,eq_Ghe_delta. For the
symmetric part, we expect a second order correction due to the superconducting
order parameter. Therefore the electron-electron part of the Heisenberg
exchange is modified by interference effects occurring from the electron-hole
and hole-electron propagation when mediating the magnetic interaction between
two sites $i$ and $j$. The electron-hole part, however, inherently involves
the scattering of a hole at the intra-atomic exchange of a given site $j$.
such a hole-scattering is automatically accompanied with a minus sign, which
could then explain the tendency of the electron-hole part to favor
antiferromagnetism.

Figure 3: (a) Total exchange interaction as given by Eq. (24), as each of its
contributions separately as defined in Eq. (25): (b) electron-electron
contribution $\mathcal{J}^{ee}$ (c) electron-hole contribution
$\mathcal{J}^{eh}$. (d) Polar plot ($\theta$ [Deg] vs $|\bm{D}|$ [meV])
showing the evolution of the total DMI vector coming from Eq. (24) as a
function of $\lambda$ for one of the four nearest neighbours (the remaining
cases are similar). (e) Evolution of the DMI vector derived from the electron-
electron term $\mathcal{J}^{ee}_{ij}$. (f) Evolution of the DMI vector derived
from the electron-hole term $\mathcal{J}^{eh}_{ij}$. We can see that $D^{ee}$
and $D^{eh}$ tend to have opposite orientations. All calculations were done at
$4.2\text{\,}\mathrm{K}$, with 15,000 k-points.

###  III.2 Dzyaloshinskii–Moriya interaction

Even though the possible effects of superconductivity on the Heisenberg
exchange interaction may lead to interesting outcomes, one realizes soon that
in magnetic systems the correction is mostly not sufficient to heavily impact
the magnetic state, given that the original interaction tends to be large. The
DMI in turn is most of the times smaller that the Heisenberg exchange, and the
impact of the superconducting state might be larger there. Although the DMI
has most of the times low values compared to the Heisenberg one, this
interaction is important in the realm of non-collinear magnetism and it is
responsible for stabilizing complex magnetic structures, influencing skyrmion
chirality, as well as being instrumental in the study of spin waves[66]. For
the case of our system when $\Delta=$6.25\text{\,}\mathrm{meV}$$ the DMI
vector is small $|\bm{D}|=$0.37\text{\,}\mathrm{meV}$$ with an even smaller
electron-hole component
$|\bm{D}^{eh}|=$25.7\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$. Therefore, to
further the impact of superconductivity we also analyzed the cases with larger
superconducting gaps. Analyzing the DMI on Mn/Nb(110), we observe that the
out-of-plane component ($D_{z}$) is negligible, ensuing an in-plane $\bm{D}$.

Figure 4:  Duality of magnetic exchange interactions impacted by the
superconducting order parameter. There are several scattering events that
influence the exchange interactions emerging in Eqs. 25. The lines represent
Green functions, thick for finite $\Delta$ and SOC, thin when the latter are
not included. Intra-atomic exchange interactions ($B^{\text{[xc]}}$, and
$-B^{\text{[xc]}}$ for electrons and holes respectively). Scattering from
electron to hole ($\Delta$) and vice versa ($\Delta^{*}$). And finally, for
the DMI there is the spin-orbit interaction ($B^{[soc]}$, and $-B^{[soc]}$ for
electrons and holes respectively). (a) Electron-electron symmetric exchange.
This term gets a second order correction with respect to $\Delta$. (b)
Electron-hole symmetric exchange. This term has a sign change given the intra-
site interaction with the hole. (c) Electron-electron DMI. This term has two
corrections, one of them catches a minus sign coming from the spin-orbit
interaction with a hole. (d) Electron-hole DMI. The first term gets a sign
flip twice, while the second does it just once. (e) Total $\bm{D}$ (red),
$\bm{D}^{ee}$ (golden), and $\bm{D}^{eh}$(blue) for the case with
$\Delta=$$295\text{\,}\mathrm{meV}$.

In Figs. 3 (d)-(f), we examine the in-plane vector $\bm{D}$ for one of the
four nearest neighbours in the Mn monolayer. Here we detect that the electron-
hole term tends to go in the opposite direction to the electron-electron DMI.
Thus, both terms have opposite chiralities, thus, favouring different kinds of
magnetic textures. This effect opens options such as switching between
magnetic states by strengthening or weakening the superconducting order of the
system at hand. In Fig. 4 (e) we show $\bm{D}$ for the system with
$\Delta=$295\text{\,}\mathrm{meV}$$, as well as its electron-electron and
electron-hole components. We notice that the chiralities are opposite, and the
electron-hole term slightly dominates.

Similarly to the symmetric exchange interaction, we scrutinized on the basis
of perturbation theory, how the superconducting order parameter, intra-atomic
exchange and spin-orbit coupling affect the DMI, as illustrated schematically
in Figs. 4 (c) and (d). The DMI emerges from the scattering at a site hosting
spin-orbit coupling. Therefore the propagators in Eq. 25 can be expanded in
first-order with respect to spin-orbit coupling, which leads to the different
processes depicted in Figs. 4 (c) and (d). We notice that besides possible
interference effects induced by the different electron and hole propagators,
the hole-scattering at an intra-atomic exchange interaction can provide a sign
change to the original chirality pertaining to the electron-electron DMI,
while this can happen for the electron-hole part due to hole-scattering at the
intra-atomic exchange and spin-orbit coupling.

##  IV Conclusions

We have introduced a method for extracting the bilinear tensor of magnetic
exchange interactions within the Bogoliubov-de Gennes (BdG) formalism,
utilizing infinitesimal rotation of magnetic moments. This novel approach has
provided insights into the intricate interplay between superconductivity,
magnetism, and spin-orbit interaction, unraveling remarkable potential effects
on both the Heisenberg exchange and the Dzyaloshinskii-Moriya interactions.

Through rigorous self-consistent simulations based on parameters derived from
ab initio calculations, our investigation has captured the intricacies of the
electronic structure in the Mn monolayer on Nb(110) surface system. By tuning
the electron-phonon coupling and thereby the superconducting state, we have
demonstrated the potential pivotal influence of the superconducting state on
the magnetic ground state, specifically through the Heisenberg exchange.
Furthermore, intriguing modifications in the chirality of the Dzyaloshinskii-
Moriya interactions have been unveiled. Within the confines of Mn monolayer
deposited on Nb(110) and for experimentally consistent gap sizes, we have
concluded that the impact of superconductivity on the magnetic ground state
remains minimal, leaving it largely unperturbed. However, the implications of
our findings could be of importance for the case where the superconducting gap
is of the order of magnitude than the magnetic exchange interactions. This
could be realized in the context of diluted magnetic structures, where the
magnetic atoms are not nearest neighbors, as realized recently with Cr adatoms
on Nb(110) surface [36, 48].

The versatility of our method enables its application in various schemes based
on Green function techniques, thereby facilitating exploration of complex
magneto-superconducting interfaces. These endeavors hold immense potential for
uncovering novel and non-trivial effects of superconductivity in topological
magnetic and superconducting materials, where intricate magnetic interactions
may play a decisive role. The exploration of complex magnetic structures, such
as skyrmions or spin chains, in conjunction with superconductivity, opens up
possibilities for the discovery of novel states and quasiparticles with
implications for topological quantum computing, driving the progress of
cutting-edge technological applications.

###### Acknowledgements.

The authors acknowledge funding provided by the Priority Programmes SPP 2137
“Skyrmionics” (Projects LO 1659/8-1) of the Deutsche Forschungsgemeinschaft
(DFG). We acknowledge the computing time provided through JARA on the
supercomputers JURECA[73]. Simulations were also performed with computing
resources granted by RWTH Aachen University under project jara0189.

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# Deflection angle evolution with plasma medium and without plasma medium in a
parameterized black hole

Xiaoling He<EMAIL_ADDRESS>School of Science, Zhejiang University of
Science and Technology, Hangzhou 310023, Zhejiang, China.  Tianyu Xu
<EMAIL_ADDRESS>School of Science, Zhejiang University of Science and
Technology, Hangzhou 310023, Zhejiang, China.  Yun Yu<EMAIL_ADDRESS>Zhejiang University of Science and Technology, Hangzhou 310023, Zhejiang,
China  Anosha Karamat<EMAIL_ADDRESS>Department of Mathematics,
Minhaj University, Lahore-54000, Pakistan  Rimsha Babar
<EMAIL_ADDRESS>Division of Science and Technology, University of
Education, Township, Lahore-54590, Pakistan  Riasat Ali<EMAIL_ADDRESS>Department of Mathematics, GC University Faisalabad Layyah Campus,
Layyah-31200, Pakistan

###### Abstract

Using the Keeton and Petters approach, we determine the deflection angle. We
also investigate the motion of photons around a parameterized black hole in
the presence of non-magnetized cold plasma by using a new ray-tracing
algorithm. In spherically symmetric spacetime, we examine the influence of the
plasma by applying the Hamiltonian equation on the deflection angle as well as
shadow. It is examine to derive the rays analytically from Hamilton’s equation
by separating the metric and the plasma frequency. We study that the presence
of plasma affects the deflection angle as well as shadow for the parameterized
black hole and they depend on plasma frequency. If the plasma frequency is
significantly lower than the photon frequency, the photon sphere and shadow
radius expressions can be linearized around the values found for space light
rays. Furthermore, we have graphically analyze the behavior of shadow for
distinct positions under the effects of plasma frequency as well as low
density plasma medium.

##  I introduction

Black holes are invisible objects that are typically thought to arise in the
gravitational collapse of enormous astronomical objects, as predicted by
Einstein’s general relativity (GR). It is commonly known that photons released
from a lit source behind a black hole (BH) will create the so-called BH
shadow, a two-dimensional dark region in the observer’s sky. The BHs shadow,
which is an impression of the BH, tells us important things about the BH. For
instance, one can determine the BHs charge and spin 1 ; 2 ; 3 ; 4 ; 5 ; 6  as
well as limit certain additional parameters brought about by changing
gravities 7 ; 8 ; 9 ; 10 ; 11  from the shadow of the BH. First, Synge 12  and
later Luminet 13  looked at the perfect circle-shaped shadow that a
spherically symmetric BH casts. They also presented formulas to determine the
shadow size and angular radius, respectively. Bardeen 14  was the first to
examine the shadow produced by a Kerr BH. Because of the dragging action, the
shadow shape is modified. Since then, there has been a lot of research done in
the literature on the BH shadow. For instance, 15  looked at the BH shadow and
photon sphere in dynamically changing spacetimes.

The gravitational lensing effect is the process by which BHs bend the escaping
light beams in addition to capturing them. Gravitational lensing is another
effective approach that gives us a vast amount of data on BHs, including their
location, angular momentum and mass. The geodesic technique 16 ; 17 ; 18 ; 19
; 20  has been intensively researched for gravitational lensing for black
holes, wormholes, cosmic strings and other phenomena since the first
measurement of the deflection angle of light by the sun. Gibbons and Werner
presented a different approach in ref. 21  for calculating the weak deflection
angle of light by a spherically symmetric BH within the framework of optical
geometry using the Gauss-Bonnet theorem and then, Werner 22  extended this
method to stationary BHs by applying Kerr-Randers optical metric.Â
Additionally, whereas the impact of plasma on light rays can often be
disregarded in astronomical contexts, this is not the case for light rays in
the radio frequency spectrum. The effects of the solar corona, which is
thought of as a non-magnetized pressureless plasma, on the time delay 23  and
deflection angle when the light rays propagate close to the Sun 24  are one
well known examples. Later, Perlick studied the impact of a non-homogeneous
plasma on aÂ light deflection in the equatorial plane of the Kerr metric and
in the Schwarzschild metric 25 . Since then, research on how plasma affects
light transmission has become increasingly popular. For instance, refs. 26 ;
27 ; 28 ; 29 ; 30 ; 31  looked at the effects of plasma on gravitational
lensing by BHs and compact objects. In refs. 32 ; 33 ; 34 ; 35 , the shadows
of wormholes and BHs surrounded by plasma were studied.

One of the most important reasons to study quantum gravity is the search for
ultraviolet complete gravitational theories that could prevent space-time
singularity. There are numerous attempts, including string field theory and
quantum gravity, even though we do not yet have a fully formed quantum gravity
theory. According to a common assumption shared by nearly all of these
theories, there should be an intrinsic modified geometry in space-time 36
that is on the order of the Planck length. A configuration with this much
extension suggests that space-time is not local 37 ; 38 ; 39 ; 40 . So, it is
expected that the linear correction to GR is the first-order correction 41 ;
42 ; 43 ; 44 ; 45 ; 46 ; 47 ; 48 ; z14c  of quantum theory. The quantum
gravity and shadow are compulsory components to study BH geometry.

The deflection angle for Einstein-Maxwell-Dilaton-Axion like BH using the
methodology developed by Gibbons and Werner has been studied RA1 . Further
evidence that the gravitational lensing effect is a global and even
topological phenomenon, i.e., there are several light rays converging between
the source and observer, was provided by the fact that the deflection of a
light ray was determined outside of the lensing area. The weak limit approach
RA2  of the Gauss-Bonnet theorem was used to compute the deflection angle of a
null aether like BH by investigating the light beams in a BH gravitational
field. Matsuno calculated RA3  the deflection angles while taking into account
photon movements around a charged squashed Kaluza-Klein like BH particle in an
unmagnetized homogeneous cold plasma medium. The light deflection angle at the
leading order terms was determined using the Gauss-Bonnet theorem and a
straight line approach by RA4  and also examined the gravitational lensing of
the Einstein-Cartan-Kibble-Sciama like BH in the weak field approximation. The
Gauss-Bonnet theorem has been used RA5  to study weak gravitational lensing in
Kerr-Newman-Kasuya like optical spacetime, where optical geometry provides a
more geometrical view. It was found that the deflection angle decreased
linearly with the additional magnetic charge compared to Kerr-Newman BHs.

In our work, we explore the deflection angle by using the Keeton and Petters
technique and ray-tracing algorithm from the photon pathway around a BH in the
presence of non-plasma and plasma with radial density and evaluate the impacts
of plasma on the BH shadow. We have formulated our paper by the following
pattern: In Sec. II, we review the BH metric and study the effects of
deformation parameter to metric function. In Sec. III, using the Keeton and
Petters approach, to determine the deflection angle. In Sec. IV explains the
motion equation of light ray in the scenario of BH with plasma. In Sec. V, we
study the circular light orbits that are very important for the creation of
the shadow. Sec. VI, computes the angular radius of the BH shadow. In Sec.
VII, we graphically discuss the impacts of different parameters on BH Shadow.
Sec. VIII, examines the case of a low-density plasma Shadow. In Sec. IX, we
summarize the results.

##  II Deflection angle analysis with plasma medium

Huang et al. used DA  a new ray-tracing algorithm to examine the photon’s
velocity around a BH in the presence of plasma, whose density is a function of
the radius coordinate, and look into the impact of the plasma on the BHs
shadow. We expect that the spacetime is loaded up with a non-magnetized cold
plasma whose electron plasma frequency $w_{p}$ only seems on the coordinate of
radius ($r$) in the form

$w^{2}_{p}(r)=\frac{4\pi E(r)e^{2}}{m},$ (1)

whereas $E(r)$, $m$ and $e$ are the number of electron’s density in the non-
magnetized plasma, electron mass and electron charge, respectively. This
plasma’s refraction index $i$, is influenced by the photon’s frequency as
measured by a static observer and the radius coordinate $r$ in the following
way

$i^{2}(r,w)=1-\frac{w_{p}^{2}(r)}{w^{2}}.$ (2)

Due to the spherical symmetry, we can limit to the equatorial plane
$\theta=\frac{\pi}{2},~{}u_{\theta}=0$. The Hamiltonian explains the
particle’s motion in a pressure-free and non-magnetized plasma as

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(g^{ik}u_{i}u_{k}+w^{2}_{p}(r)\right),$ (3)
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\frac{u^{2}_{t}}{A(r)}+\frac{u^{2}_{r}}{B(r)}+\frac{u^{2}_{\phi}}{C(r)}+w^{2}_{p}(r)\right).$

Light waves for Hamilton equations imply that

$\dot{u}_{i}=-\frac{\partial
H}{\partial\delta^{i}},~{}~{}~{}~{}\dot{\delta}^{i}=\frac{\partial
H}{\partial{u}_{i}},$ (4)

it can be written as

$\displaystyle\dot{u}_{t}$ $\displaystyle=$ $\displaystyle-\frac{\partial
H}{\partial t}=0,$ (5) $\displaystyle\dot{u}_{\phi}$ $\displaystyle=$
$\displaystyle-\frac{\partial H}{\partial\phi}=0,$ (6)
$\displaystyle\dot{u}_{r}$ $\displaystyle=$ $\displaystyle-\frac{\partial
H}{\partial
r}=\frac{1}{2}\left(-\frac{u^{2}_{t}\acute{A}(r)}{A^{2}(r)}+\frac{u^{2}_{r}\acute{B}(r)}{B^{2}(r)}+\frac{s^{2}_{\phi}\acute{C}(r)}{C^{2}(r)}-\frac{d}{dr}{w^{2}}_{p}(r)\right),$
(7) $\displaystyle\dot{t}$ $\displaystyle=$ $\displaystyle\frac{\partial
H}{\partial u_{t}}=-\frac{u_{t}}{A(r)},$ (8) $\displaystyle\dot{r}$
$\displaystyle=$ $\displaystyle\frac{\partial H}{\partial
u_{r}}=\frac{u_{r}}{B(r)},$ (9) $\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle\frac{\partial H}{\partial u_{\phi}}=\frac{u_{\phi}}{C(r)},$
(10)

for $H=0$, we have

$0=-\frac{u^{2}_{t}}{A(r)}+\frac{u^{2}_{r}}{B(r)}+\frac{u^{2}_{\phi}}{C(r)}+{w^{2}_{p}(r)},$
(11)

here, dot and prime shows differentiation w. r. t an affine parameter and $r$,
respectively. It follows from Eq. (5) and (6) that $u_{t}$ and $u_{\phi}$
represents constants of motion. We may type $w_{0}=-u_{t}$. In case of a
asymptotically flat spacetime, the $w_{0}$ is fixed and $A(r)\rightarrow 1$ as
$r\rightarrow{\infty}$ and the frequency of a static observer represented by
$w(r)$ can be defined by redshift formula as

$w(r)=\frac{w_{0}}{\sqrt{A(r)}}.$ (12)

From the equation (11), the constant motion $w_{0}$ of light beam is specified
to the particular region with

$\frac{w^{2}_{0}}{A(r)}>{w^{2}_{p}(r)}.$ (13)

The photons frequency at that place should be greater than the plasma
frequency, according to Eq. (13). It is true for the light transmission in a
medium of plasma. We consider Eqs. (9) and (10), to get the equation of orbit

$\frac{dr}{d\phi}=\frac{\dot{r}}{\dot{\phi}}=\frac{C(r){u}_{r}}{B(r){u}_{\phi}}.$
(14)

By substituting for $u_{r}$ from equation (11), which is taken by

$\frac{dr}{d\phi}=\pm\frac{\sqrt{C(r)}}{\sqrt{B(r)}}\left(\sqrt{\frac{w^{2}_{0}}{u^{2}_{\phi}}{D(r)^{2}}-1}\right),$
(15)

with

$\frac{w^{2}_{0}}{u^{2}_{\phi}}{D(r)^{2}}-1>0,~{}~{}{D^{2}(r)}=\frac{C(r)}{A(r)}\left({1}-{A(r)}\frac{w^{2}_{p}(r)}{w^{2}_{0}}\right).$
(16)

In this case, the equation (15) sign-in must be suitably followed and the
orbit should be divided into segments at the points $r$ is slowly decreasing
and increasing in connection to $\phi$. The expression for the deflection
angle ($\Psi$) for an infinite light beam with a radius of $Y$ and an
additional infinity is produced by integrating over the orbit, we get

$\pi+\Psi={2}\int_{Y}^{\infty}\frac{\sqrt{{B}(r)}}{\sqrt{C(r)}}\left(\frac{w^{2}_{0}}{u^{2}_{\phi}}{D^{2}(r)}-{1}\right)^{-\frac{1}{2}}{dr}.$
(17)

Since $Y$ is related with the pivot of trajectory, the specified case
$\frac{dr}{d\phi}|_{Y}=0$ must be fulfilled. The above mentioned equation
determines the following relationship between the radius $Y$ and the constant
motion $\frac{u_{\phi}}{w_{0}}$ as follows

${D^{2}(X)}=\frac{u^{2}_{\phi}}{w^{2}_{0}}.$ (18)

In terms of $w$ and $Y$, the angle of deflection for a certain plasma
distribution can be expressed as follows

$\pi+\Psi=2\int_{Y}^{\infty}\left(\frac{\frac{r^{5}}{r^{3}-\eta-2Mr^{2}}-\frac{r^{2}w^{2}_{p}(r)}{w^{2}_{0}}}{\frac{Y^{5}}{Y^{3}-\eta-2MY^{2}}-\frac{Y^{2}w^{2}_{p}(Y)}{w^{2}_{0}}}-1\right)^{-\frac{1}{2}}\frac{dr}{\sqrt{r^{3}-\eta-2Mr^{2}}}.$
(19)

The deflection angle is dependent by the horizon radius, the infinite radius
of the light wave $Y$, the BH mass, the photon and plasma frequency, the BH
geometry, and BH deformation parameter. When $\eta=0$, the equation (19) for
the plasma’s deflection angle on Schwarzschild spacetime was developed in DAN1
. In DAN2 , it was derived using Synge’s method and formulated in terms of an
elliptic integral for a homogeneous plasma. Further, the strong deflection
limit $\Psi\gg 1$ was examined. Moreover, we conclude that for $\eta=0$, we
recover the deflection angle for Schwarzschild BH as calculated in 33 .

##  III Orbits of Light

For photons emitted from the observer’s screen in the previous ray-tracing
method, there are only two possible final states: either the photon has been
absorbed by the BH or it is scattered indefinitely. Once released, the photons
will gradually scatter around the BH, which is where the shadow of a BH is
situated in the observer’s image plane. This section will discuss the
condition of circular light orbits, which is necessary for locating the
shadow. It is important to note that the light is flowing in a circular
direction while the states $\dot{r}=0$ and $\ddot{r}=0$. Eqs. (9) and (11)
with $u_{r}=0$ give us the relation shown below:

$0=-\frac{w^{2}_{0}}{A(r)}+\frac{u^{2}_{\phi}}{C(r)}+{w^{2}_{p}(r)}.$ (20)

As an approach, (9) implies

$\dot{u}_{r}=\frac{d}{d\lambda}(B(r)\dot{r})=\ddot{r}B(r)+\dot{r}^{2}{B^{\prime}}(r).$
(21)

If $\dot{r}=0$, $\ddot{r}=0$ and $\dot{u}_{r}=0$, then Eq. (6) represents the
light orbits second equation

$0=-\frac{w^{2}_{0}A^{\prime}(r)}{A^{2}(r)}+\frac{u^{2}_{\phi}C^{\prime}(r)}{C^{2}(r)}-\frac{d}{dr}{w^{2}_{p}(r)}.$
(22)

For $u^{2}_{\phi}$, we canÂ compute the solution from Eqs. (20) and (22) as

$\displaystyle u^{2}_{\phi}$ $\displaystyle=$ $\displaystyle
C(r)\left(\frac{w^{2}_{0}}{A(r)}-{w^{2}_{p}(r)}\right),$ (23) $\displaystyle
u^{2}_{\phi}$ $\displaystyle=$
$\displaystyle\frac{C^{2}(r)}{C^{\prime}(r)}\left(\frac{w^{2}_{0}A^{\prime}(r)}{A^{2}(r)}+\frac{d}{dr}{w^{2}_{p}(r)}\right).$
(24)

By eliminating these two equations from one another after making a few
straightforward adjustments provides the following result for the radius of a
circular light orbit in the form

$0=r^{2}\Big{(}5r^{3}-5\eta-10r^{2}M-3r^{3}+4r^{6}M\Big{)}-\Big{(}r^{3}-\eta-2r^{2}M\Big{)}^{2}\Big{[}\frac{2w^{2}_{p}(r)}{w^{2}_{0}}-\frac{r^{2}}{w^{2}_{0}}\frac{d}{dr}{w^{2}_{p}(r)}\Big{]},$
(25)

at which the property $D^{2}(r)$ is produced by the formula (16). The photon
sphereÂ radiusÂ can be calculated by using the (25) solution for $r=r_{p}$. A
light moving tangentially into a sphere always goes in a circle with radius
$r=r_{p}$. The furthest photon circle is consistently unstable in terms of
radial disorder if the BH space-time is asymptotically flat with
$w_{p}\rightarrow A~0$ and $r\rightarrow\infty$. This implies that the
asymptotic light beams that reach them could act as limit curves for the
circular photon’s orbits. The exterior photon circle’s radius and the
analytical point of the recently discussed un-important radius $X$ are
identical. A light ray goes to its point of origin when it leaves its
infinity-based origin and goes to a minimum radius $Y$ that is higher than
$r_{p}$. A light beam spirals asymptotically along the direction of a photon
sphere’s circular orbit in the case where $Y=r_{p}$. The fact that the
equation (25) can be reduced to the Atkinson scenario 49  for $w_{p}(r)=0$ is
crucial.

##  IV SHADOW RADIUS

The shadow is a visualization of a spherically symmetric space-time. By taking
into account the description of the expression from the preceding section, we
examine the effects of plasma on an incident light. We discuss the two
classifications of these incident photons. Through the BH, photons of first-
class light are reflected back and move continuously. The photon of second-
class light is going toward the BH’s horizon.

Let’s start by supposing that there isn’t a source of light in the observer’s
sky that would interfere with the BH-observer relationship. The BH’s shadow is
represented by the sphere in the observer’s sky. The starting trajectory of
the photon’s light, which is asymptotically radial in the direction of the
outer photon sphere, determines the shadow radius.

$\cot\Phi=\frac{g_{rr}}{g_{\phi\phi}}\frac{dr}{d\phi}|_{r=r_{0}}=\frac{\sqrt{B(r)}}{\sqrt{C(r)}}\frac{dr}{d\phi}|_{r=r_{0}}.$
(26)

Using the equation (18) to the orbit equation (15), we obtain the minimumÂ ray
of light as it approaches a minimal radius Y.

$\frac{dr}{d\phi}=\pm\frac{\sqrt{C(r)}}{\sqrt{B(r)}}\sqrt{\frac{D^{2}(r)}{D^{2}(X)}-1}.$
(27)

For the angle $\Phi$, we obtain the following:

$\cot^{2}\Phi=\frac{D^{2}(r_{0})}{D^{2}(X)}-1.$ (28)

This suggests

$\sin^{2}\Phi=\frac{D^{2}(r_{0})}{D^{2}(X)}.$ (29)

The limit of shadows is determined by the approximate rotation of light into a
sphere with a radius of $r_{p}$. As a result, sending $X\rightarrow r_{p}$
gives the angular shadow size in Eq. (29) as

$\sin^{2}\Phi=\frac{D^{2}(r_{ph})}{D^{2}(r_{0})},$ (30)

where the equation expression (16) expresses $D(r)$. We can assume that the
observer is situated in a zone of low-density plasma. The equation result (16)
gives

$D^{2}(r_{0})=\frac{C(r_{0})}{A(r_{0})},$ (31)

and equation (30) implies

$\sin^{2}\Phi=\frac{r^{5}_{ph}(r^{3}_{0}-\eta-2Mr^{2}_{0})}{r^{5}_{0}(r^{3}_{ph}-\eta-2Mr^{2}_{ph})}\left[1-\Big{(}1-\frac{\eta}{r^{3}_{ph}}-\frac{2M}{r_{ph}}\Big{)}\frac{w_{p}^{2}(r_{ph})}{w^{2}_{0}}\right].$
(32)

This demonstrates how the plasma is gradually affecting the shadow’s
decreasing radius. It seems to be the case for all spherically symmetric
spacetimes, such as wormhole metrics and compact metrics with an instability
photon sphere. These objects would cast shadows like BHs if the light were not
shining from the direction of the centre object towards the observer. In order
for the light rays in the photon sphere to function as limit curves, it is
absolutely necessary for them to be unstable in relation to radial
perturbations. Because of this, the shadow can be constructed in any static,
spherically symmetric spacetime that accepts an unstable photon sphere. This
includes black holes as well as wormholes. The outermost photon sphere is
always unstable in a static, asymptotically flat, spherically symmetric
spacetime, provided that the plasma density tends to zero for
$r\rightarrow\infty$. It is also worth mentioning here that for $\eta=0$ and
$r_{ph}=3M$ in Eq. (32), we recover the radius of shadow for Schwarzschild BH
33 .

##  V Graphical Analysis for Shadow of Black Hole

This section gives the graphical behavior for shadow of parameterized BH.

Figure 2: The BH shadow $\Phi$ versus radius $r_{0}$ for various $\omega_{p}$,
$\omega_{0}$ and fixed $r_{ph}=0.5=\omega_{0},M=1~{}\text{and}~{}\eta=0.6$ in
the left plot while for fixed
$r_{ph}=0.5,\omega_{0}=0.1,M=1~{}\text{and}~{}\eta=0.2$ in the right plot.

The left plot of Fig. 2 shows the behavior of the angular radius of shadow
with static observer at distinct positions $r_{0}$ for varying $\omega_{p}$.
The $\Phi$ slowly decreases and then, it gets a flat form for rising $r_{0}$.
It is notable that the $\Phi$ rises for rising values of $\omega_{p}$.

In the right plot of Fig. 2, the $\Phi$ represents the behavior with $r_{0}$
for changing values of $\omega_{0}$. The $\Phi$ slowly decreases till
$r_{0}\rightarrow\infty$. The $\Phi$ decreases with rising $\omega_{0}$.

From Fig. 2, we conclude that the BH shadow is smaller for different
variations of $r_{0}$. To get large shadow radius in strong gravity, the
frequency parameter must be extremely small.

Figure 3: The BH shadow angular radius w.r.t $r_{ph}$ for various
$\omega_{p}$, $\omega_{0}$ and fixed $M=1,\eta=50,\omega_{0}=20$.

The left plot in Fig. 3 represents the dependency of $\Phi$ versus $r_{ph}$
for different values of $\omega_{p}$. In the left plot, the $\Phi$
continuously goes on increasing. The $\Phi$ reduces for rising values of
$\omega_{p}$. In the right plot of Fig. 3, the $\Phi$ shows a rising attitude
for rising $r_{ph}$ as well as the radius of shadow increase for rising values
of $\omega_{0}$.

The Fig. 3 demonstrates that the BH shadow is smaller for different variations
of $r_{ph}$ but it shows an increasing behavior. To get large shadow radius in
strong gravity, the frequency parameter must be extremely small.

##  VI A Low Density Plasma’s Shadow

If the plasma frequency is significantly lower than the photon frequency, the
photon sphere and shadow radius expressions can be linearized around the
values found for space light rays. This is addressed by modifying equation
(16) as

$D^{2}(r)=Z(r)(1-\varsigma\alpha(r)),$ (33)

whereas

$Z(r)=\frac{C(r)}{A(r)},~{}~{}\alpha(r)=\frac{A(r)w^{2}_{p}(r)}{w^{2}_{0}}=\frac{w^{2}_{p}(r)}{w^{2}(r)}.$
(34)

We include an accounts parameter $\varsigma$, which is set to unity once all
results have been linearized.Â This consequence of linearity can be deduced by
power law i.e., $\frac{w^{2}_{p}(r)}{w^{2}(r)}=\beta_{0}\frac{M^{k}}{r^{k}}$,
where $\beta_{0}>0$ and $k\geq 0$ that represents dimensionless constants. A
detailed example for Schwarzschild black hole for this consequence is given in
[A1] . Now, the photon sphere result (25) written as

$0=Z^{\prime}(r)(1-\varsigma\alpha(r))-\varsigma Z(r)\alpha^{\prime}(r).$ (35)

It is possible to express the equation’s solution as follows

$r_{ph}=r^{0}_{ph}+\varsigma r^{1}_{ph}+...,$ (36)

in the absence of plasma with a solution $r^{0}_{ph}$ such that

$Z^{\prime}(r^{0}_{ph})=0.$ (37)

We derive the $\varsigma$ coefficients by inserting these formulas into (35)
and after comparing them, we get

$r^{1}_{ph}=\frac{Z^{(}r^{0}_{ph})\alpha^{\prime}(r^{0}_{ph})}{Z^{\prime\prime}(r^{0}_{ph})}.$
(38)

In relation to the values of $\alpha^{\prime}(r^{0}_{ph})$ and
$Z^{\prime\prime}(r^{0}_{ph})$, this equation can be either positive or
negative, so that the plasma can move the light sphere outside or inside. Now,
we implement an expansion (36) to the shadow result (30). Once more set all
components equal to unity while ignoring any quadratic and higher order
factors, we obtain

$\sin^{2}\Phi=\frac{\tilde{r}^{5}(r_{0}^{3}-\eta-2r_{0}^{2}M)}{r_{0}^{5}\big{(}\tilde{r}^{3}-\eta-2\tilde{r}^{2}M\big{)}}\left[1-\Big{(}1-\frac{\eta}{\tilde{r}^{3}}-\frac{2M}{\tilde{r}}\Big{)}\frac{w^{2}_{p}(\tilde{r})}{w_{0}^{2}}+\Big{(}1-\frac{\eta}{r_{0}^{3}}-\frac{2M}{r_{0}}\Big{)}\frac{w^{2}_{p}(r_{0})}{w_{0}^{2}}\right],$
(39)

with $\tilde{r}=r^{0}_{ph}$. Note that the computation for to within this
order no longer takes $r_{ph}^{1}$ into account. From the Eq. (39), the plasma
has less of an impact on the shadow than
$(1-\frac{\eta}{r_{0}^{3}}-\frac{2M}{r_{0}})\frac{w^{2}_{p}(r_{0})}{w_{0}^{2}}<(1-\frac{\eta}{\tilde{r}^{3}}-\frac{2M}{\tilde{r}})\frac{w^{2}_{p}(\tilde{r})}{w_{0}^{2}}$.
The photon sphere radius $r_{ph}$ must be computed for a given metric
(LABEL:sol), a defined photon frequency at infinity $w_{0}$, and a specified
plasma frequency that satisfies the case $w_{p}(r)<<w(r)$. Approximate
$r^{0}_{ph}$ from (37) and $r^{1}_{ph}$ from (38), then estimate $r_{ph}$
allowing to (38) with $\varsigma=1$. Then, write down the functions $Z(r)$ and
$\alpha(r)$ (see (34)). In this procedure, we have to put $r^{0}_{ph}$ into
the expression to obtain the radius of shadow angular $\Phi$ for observer
particular $r_{0}$ position (39).

Figure 4: The plasma shadow $\Phi$ versus radius $r_{0}$ for various
$\omega_{p}$, $\omega_{0}$ and fixed
$\tilde{r}=0.5=\omega_{0},M=1~{}\text{and}~{}\eta=0.6$ in the left plot while
for fixed $\tilde{r}=0.1=\omega_{p},M=1~{}\text{and}~{}\eta=0.2$ in the right
plot.

The left plot of Fig. 4 shows the behavior of the angular radius of shadow
with static observer at distinct positions $r_{0}$ for varying $\omega_{p}$ in
low density medium. The $\Phi$ continuously goes on increasing but the radius
of shadow becomes smaller. The plasma shadow rises for rising values of
$\omega_{p}$. In the right plot of Fig. 4, the $\Phi$ exponentially rises for
rising $r_{0}$. It is also observable that the shadow radius in low density
plasma is smaller as compared to Fig. 3. The Fig. 4 states that the BH shadow
goes on increasing for different variations of distinct observer $r_{0}$. To
get a very large shadow radius in strong gravity, the frequency parameter must
be extremely small.

##  VII Conclusions

In this research, we have examined deformation parameter effect on metric
function by considering the methodology of Keeton and Petters and the
framework ofÂ PPNÂ is a direct way for dealing with all types of gravity
theories in which the weak-deflection limit is stated as a series expansion
withÂ single variable $M$. The deflection angle depends on the deformation
parameter ($\eta$), the impact parameter ($b$) and the mass ($M$). The results
we have obtained for BH solutions in the presence of non-plasma mediums show
that deflection angle has a direct relationship with mass $M$ and a parameter
$\eta$, indicating that BH with greater mass has greater gravitational pull
and bends the light passing by it at large angles. While BH with smaller mass
deflect the light at a smaller angle. Additionally, we observe that deflection
angle has an inverse relationship with impact parameter $b$, demonstrating
that a smaller value of the impact parameter results in a larger deflection
angle and vice versa. Moreover, we conclude that for $\eta=0$, we recover the
deflection angle for Schwarzschild BH.

We have examined the BH shadow radius in plasma as well as low density plasma
medium by using the Hamilton equation to describe photon motion. To do so, by
using a new ray-tracing algorithm, we numerically solved the Hamilton
equations to determine the BH shadow radius. The size and shape of the BH
shadow can alter when there is plasma surrounding a BH because the trajectory
of a photon is dependent on the photon frequency. The photon-absorbing and
scattering actions of plasma electrons are not taken into consideration.
Further, we ignore the plasma particle’s gravitational field. Furthermore, the
size of shadow always shrinks when plasma is present and for $\eta=0$ and
$r_{ph}=3M$ in Eq. (32), we get the radius of shadow for Schwarzschild BH. At
low frequency plasma density, the shadow of a BH is shown to appear on the
observer’s screen. The BH shadow is identical to that BH at high frequency,
and the influence of the plasma is minimal. Furthermore, the presence of a
plasma may result in the formation of a number of structures in the BH image.

Moreover, we have graphically analyzed the behavior of shadow for distinct
positions under the effects of plasma frequency $w_{p}$ and $w_{0}$ in plasma
as well as low density plasma medium. We have observed that $\Phi$ slowly
decreases and then, it gets a flat form for rising $r_{0}$. It is notable that
the $\Phi$ rises for rising values of $\omega_{p}$. For changing values of
$\omega_{0}$, the $\Phi$ slowly decreases till $r_{0}\rightarrow\infty$. The
$\Phi$ decreases with rising $\omega_{0}$. The BH shadow is smaller for
different variations of $r_{0}$. To get large shadow radius in strong gravity,
the frequency parameter must be extremely small. The $\Phi$ reduces for rising
values of $\omega_{p}$ in the region $1\leq r_{ph}\leq 5$. It shows a rising
attitude for rising $r_{ph}$ as well as the radius of shadow increases for
rising values of $\omega_{0}$. In low density medium, the $\Phi$ continuously
goes on increasing but the radius of shadow becomes smaller. The plasma shadow
rises for rising values of $\omega_{p}$. The $\Phi$ exponentially rises for
rising $r_{0}$ for varying $\omega_{0}$. It is also observable that the shadow
radius in low density plasma is smaller as compared to Fig. 3. The BH shadow
is goes on increasing for different variations of distinct observer $r_{0}$.
To get a very large shadow radius in strong gravity, the frequency parameter
must be extremely small. We have simply taken into account the photon motion
in a stationary, non-magnetized, pressure-free plasma in the background of a
BH here as a first step towards understanding the influence of ionised matter
on the various astrophysical compact objects. Future research should naturally
build on these findings to incorporate more complex yet realistic plasma
models.

We anticipate that the observation will provide us with important hints and
surprises regarding space-time and the distribution of matter in the vicinity
of the BH, as revealed by the future results of the Event Horizon Telescope
(EHT). However, expecting that EHT can effectively limit the plasma density
distribution’s properties is suggested. Although the pulling effect extends
the shadow of a revolving BH in this position.

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