# Signature of Anomalous Andreev bound states in magnetic Josephson junction
of noncentrosymmetric superconductor on a topological insulator

Saumen<EMAIL_ADDRESS>and Umananda Dev
<EMAIL_ADDRESS>Department of Physics, Dibrugarh University,
Dibrugarh 786 004, Assam, India,

###### Abstract

We study the Josephson effect in a clean noncentrosymmetric
superconductor/half-metal/noncentrosymmetric superconductor junction, which is
grown on the surface of a three-dimensional Topological Insulator (TI) in the
ballistic limit. We find the signature of anomalous Andreev Bound States (ABS)
and band splitting for a spin-active barrier whose barrier magnetic moment is
misaligned with the bulk moment. The chiral Majorana mode and 4$\pi$ periodic
ABS are found to exist on the surface of TI for parallel orientation of the
moments in the normal incidence condition. But for anti-parallel misalignment,
we observe the 2$\pi$ periodic ABS. There exist a gap in ABS for oblique
incidence. We find the splitting of Andreev levels in the presence of RSOC and
also for unequal mixing of singlet-triplet correlations present in NCSC. The
Majorana mode, ABS and Josephson supercurrent can be controlled by the ratio
of barrier magnetic and non-magnetic moments. The critical current is found to
be maximum for singlet or triplet dominated NCSC, while it is minimum in the
equal mixing condition. The ABS is found to be barrier thickness dependent and
is suppressed for an opaque barrier. We observe a monotonic decay in critical
current with a finite length of the junction for all the singlet-triplet
mixings and magnetic moments. The current-phase relation is found to be
sinusoidal with no phase shift in the half-metallic limit, however, for
different orientations of the bulk moment, an anomalous characteristic is also
observed.

###### pacs:

67.30.hj, 85.75.-d, 74.90.+n

##  I Introduction

Topological Insulators (TI) are a new class of material that exhibit non-
trivial band topology due to a strong spin-orbit coupling fu1 ; fu2 ; moore2 ;
roy ; moore1 ; hasan1 ; qi ; hasan2 . They remain insulating in bulk, but the
most intriguing characteristic of them is the presence of gapless conducting
edge states (in two dimensions) and surface states (in three dimensions). In
3D TIs, the spin of a moving particle is helically locked to the momentum at
their surface beidenkopf ; roushan ; liu ; dyrdal ; li ; yang ; chen12 . Due
to spin-momentum locking, it is possible to induce new phases when a magnetic
or superconducting element is brought in close proximity with the 3D TIs.
Previous works indicate that when a conventional s-wave superconductor is
coupled with 3D TIs then the superconducting correlation is induced at their
surface fu3 ; fu4 ; sochnikov ; kohda . In contrast to the conventional spin-
singlet Cooper pairing, the induced superconductivity is an admixture of both
singlet and triplet correlations due to spin lifted degeneracy at their
surface gorkov1 ; tkachov ; gong . Moreover, the Andreev Bound States (ABS)
formed in superconductor-topological insulator-superconductor (S$|$TI$|$S)
Josephson junctions can exhibit zero energy crossing, when the phase
difference between the superconductor is $\pi$. Thus it can host Majorana
quasiparticle gorkov1 ; tkachov ; sochnikov ; kohda ; gong ; fu3 ; fu4 ;
tanaka1 , which is a fermionic mode of its own antiparticle. Though in
conventional S$|$N$|$S junctions, a weak backscattering can splits the spin
degeneracy at $\pi$, however, due to non-trivial topology, S$|$TI$|$S
junctions can support $4\pi$-periodic Josephson supercurrent. Recently, their
is a growing interest to understand the robust phenomena like, zero-energy
Majorana modes gorkov1 ; tkachov ; sochnikov ; kohda ; gong ; fu3 ; fu4 ;
tanaka1  and topological superconductivity frolov ; chen11 ; linder11 ; hsieh
; nakhmedov ; trang  on the surface of 3D TIs. Experimentally it is very
challenging to distinguish topologically trivial and non-trivial $4\pi$ modes.
It is due to the reason that most of the TIs discovered till now are not
perfectly insulating in bulk. But recently, it is reported that the majority
of helical supercurrent is due to its surface and only a feeble contribution
arises due to the bulk veldhorst ; cho ; stehno . Moreover, some exotic TIs
are also discovered which display insulating bulk at low temperatures kayyalha
; jauregui ; xu11 ; xu12 .

Many theoretical and experimental efforts are made to demonstrate helical ABS,
Current Phase Relation (CPR) and temperature dependence of critical current in
various S$|$TI$|$S Josephson junctions kwon ; olund ; burset ; ghaemi ;
tkachov1 ; snelder1 ; snelder2 ; kurter ; schrade ; linder12 . However, the
ABS and the supercurrent formed on the helical surface states due to Non
Centrosymmetric Superconductors (NCSC) are yet to be understood. The reasons
for considering NCSC are: (i) Coexistence of spin-singlet and spin-triplet
superconducting correlations and (ii) the existence of strong Antisymmetric
Spin-Orbit Coupling (ASOC) due to the absence of inversion symmetry bauer1 ;
bauer2 ; bauer3 ; motoyama ; kawasaki ; akazawa ; anand1 ; smidman ; anand2 ;
yuan ; togano ; badica ; matthias ; singh ; pecharsky ; hillier ; bonalde ;
yogi ; ali ; xu ; acharjee2 ; acharjee3 ; acharjee1 ; linder15 ; frigeri .
Earlier studies indicate that due to the presence of the Rashba Spin-Orbit
Coupling (RSOC) the pairing symmetry and hence the transport properties
drastically get influenced in NCSC heterostructure yuan ; togano ; badica ;
acharjee2 ; acharjee3 ; linder15 ; frigeri . Moreover, it is also reported
that the RSOC and singlet-triplet mixing ratio of NCSC play very significant
roles in the formation of ABS and supercurrent vorontsov ; klam ; zhang11 ;
smidman1 ; duckheim ; sun ; iniotakis . So it is necessary to study the
interplay of RSOC and singlet-triplet correlation in helical ABS and also to
understand the Majorana modes in NCSC Josephson junctions.

Josephson junctions with a magnetic material as an intermediate layer can
exhibit unconventional spin-triplet proximity effect bobkova ; saxena ; aoki ;
pfleiderer ; huy ; buzdin ; rahnavard ; zutic . As spin-triplet correlations
are long-ranged keizer1 ; eschrig ; khaire ; linder14 , so the supercurrent in
the magnetic element of such heterostructures can be carried by spin-triplet
Cooper pairs. The long-range nature of spin-triplet correlation is due to a
combination of two mechanisms: (i) spin-flip scattering eschrig2 ; galaktionov
; eschrig  and (ii) spin-active interfacial phase shifts of the wave functions
of electrons millis ; hubler ; galaktionov . Thus the superconductor-
ferromagnet heterostructures gained a lot of research interest in the present
era. Also, with the development of spintronic devices, Half Metallic (HM)
ferromagnets have attracted a particular attention due to high spin
polarization eschrig2 ; galaktionov ; eschrig ; millis ; hubler . Previously,
the dc Josephson effect and its dependence on spin-flip parameters have been
studied considering a spin active barrier in S$|$HM$|$S Josephson junctions
galaktionov . In the present work, we consider an NCSC$|$HM$|$NCSC Josephson
junction grown on the surface of 3D TI (See Fig. 1). The goal of this work is
to understand the magnetic-superconducting correlations, the interplay between
spin-dependent interfacial properties, RSOC and superconducting pairing in the
bulk NCSC with regard to (i) helical ABS, (ii) Josephson supercurrent and
(iii) Majorana modes on the surface of 3D TI.

The paper is organized as follows: In Section II, we present our theoretical
model of the proposed system and present an explicit form of the Hamiltonian.
We calculate the ABS and supercurrent both analytically and numerically in
Section III. In Section IV, we discuss the numerical as well as some
approximated analytic results. Finally, we summarize our work with conclusions
in Section V.

##  II Model and formulation

We employ Bogoliubov-de Gennes (BdG) formalism to study the bound states and
Josephson supercurrent (JSC) for the proposed setup as shown in Fig. 1. In
Nambu basis

$\Psi=\\{{\psi_{\uparrow},\psi_{\downarrow},\psi_{\uparrow}^{\dagger},\psi_{\downarrow}^{\dagger}}\\}$
(1)

the Hamiltonian for the surface of a 3D TI with induced superconducting and
magnetic proximity due to NCSC and HM can be defined as snelder1 ; snelder2 ;
linder12 ; tanaka1

$\hat{\mathcal{H}}=\left(\begin{array}[]{cc}\hat{\mathcal{H}}_{0}(\mathbf{k})+\hat{M}&\hat{\Delta}_{\alpha\beta}\\\
\hat{\Delta}_{\alpha\beta}^{\dagger}&-\hat{\mathcal{H}}^{\dagger}_{0}(\mathbf{-k})-\hat{M}^{\dagger}\\\
\end{array}\right)$ (2)

where, $\hat{\mathcal{H}}_{0}(\mathbf{k})$ and $\hat{M}$ are defined as

$\hat{\mathcal{H}}_{0}(\mathbf{k})=v_{F}(\hat{\sigma}_{x}k_{x}+\hat{\sigma}_{x}k_{y})-\mu+V_{int},$
(3) $\hat{M}=\mathbf{m}.\bm{\sigma}\Theta(x)\Theta(L-x)$ (4)

Figure 1: (a) Schematic illustration of the proposed NCSC$|$HM$|$NCSC
Josephson junction on the surface of 3D TI. We consider the bound states at
the surface of a TI which is in strong proximity with the NCSC and a HM. The
current is supposed to flow on the surface of the TI. (b) Representation of a
spin active barrier. We consider the bulk magnetization in the HM is along the
z-direction and the barrier magnetic moment is misaligned with the bulk
magnetization by angles ($\chi_{m}$, $\xi_{m}$).

Figure 2: Energy band spectrum at the surface of TI. The plots in the left and
middle respectively represent the band spectrum for parallel alignment of
barrier and bulk magnetization with $\rho=0$ and $\rho=0.3$, while the plot in
the right is for anti-parallel orientation of barrier and bulk magnetic moment
with $\rho=0.3$.

where, $v_{F}$ is the Fermi velocity, $\mu=\mu_{i}[\Theta(-x)+\Theta(x-L)]$
are the chemical potentials of the respective layers,
$\mathbf{m}=(m_{x},m_{y},m_{z})$ is the magnetization of the bulk HM and
$\bm{\sigma}=$ ($\sigma_{x},\sigma_{y},\sigma_{z}$) are Pauli’s spin matrices.
The last term of Eq. (3) represents the interacting potential defined as

$V_{int}=\big{[}V_{0}\hat{I}+\bm{\sigma}.\mathbf{V}_{m}+V_{R}\hat{e}_{x}.(\hat{\bm{\sigma}}\times\mathbf{k})\big{]}\big{[}\delta(x)+\delta(x-L)\big{]}$
(5)

where, $V_{0}$ gives the intrinsic spin independent barrier potential while
$\mathbf{V}_{m}$ represent the spin dependent the barrier potential. We define
the spin dependent potential as linder12

$\displaystyle\mathbf{V}_{m}$ $\displaystyle=(V_{x},V_{y},V_{z})$
$\displaystyle=(\rho V_{0}\cos\xi_{m}\sin\chi_{m},\rho
V_{0}\sin\xi_{m}\sin\chi_{m},\rho V_{0}\cos\chi_{m})$

where, $\rho=|\mathbf{V}_{m}|/V_{0}$ represent the effective ratio between the
barrier magnetic and non-magnetic moments. The last term in the first square
backet of Eq. (5) is the contribution due to the RSOC, with $V_{R}$ being the
strength of the RSOC and it arises due to the asymmetry of the interface as
well as due to NCSC acharjee2 ; acharjee3 ; badica ; linder15 ; frigeri .

Since both even and odd frequency correlations are present in NCSC at the same
time, so the gap matrix $\hat{\Delta}_{\alpha\beta}$ appearing in Eq. (2) can
be represented as a superposition of the singlet and triplet components
acharjee2 ; acharjee3 ; linder15 , i.e.

$\hat{\Delta}_{\alpha\beta}=\left(\begin{array}[]{cc}\Delta^{T}_{k\uparrow\uparrow}&\Delta^{S}_{k\uparrow\downarrow}+\Delta^{T}_{k\uparrow\downarrow}\\\
-\Delta^{S}_{k\uparrow\downarrow}+\Delta^{T}_{k\uparrow\downarrow}&\Delta^{T}_{k\downarrow\downarrow}\\\
\end{array}\right)\\\
\times\big{[}\Theta(-x)e^{i\phi_{L}}+\Theta(x-L)e^{i\phi_{R}}\big{]}$ (6)

where, $\Delta^{T}_{k\sigma\sigma^{\prime}}$ and
$\Delta^{S}_{k\sigma\sigma^{\prime}}$ are contributions due to the singlet (S)
and triplet (T) components respectively. Here, $\phi_{L}$($\phi_{R}$)
represents the macroscopic phase of left(right) superconductor. Considering
the RSOC vector for CePt3Si linder15 ; frigeri ; acharjee2
$\mathbf{g_{k}}=V_{R}(k_{y},-k_{x},0)^{T}$, the gap vectors can be defined as

$\Delta^{T}_{k\uparrow\uparrow}(\Delta^{T}_{k\downarrow\downarrow})=\mp\frac{\Delta_{t}}{2|\mathbf{k}|}(k_{y}\pm
ik_{x})\big{[}\Theta(-x)e^{\mp i\phi_{L}}\\\ +\Theta(x-L)e^{\mp
i\phi_{R}}\big{]}$ (7)

and $\Delta^{S}_{k\uparrow\downarrow}=\Delta_{k}$. We define a parameter
$\delta$ to characterize singlet-triplet mixing as
$\Delta_{s}=\Delta_{0}\delta$ and $\Delta_{t}=\Delta_{0}(1-\delta)$. So, in
view of Eqs. (3), (4), (5) and (6), the Hamiltonian (2) can be written as

$\hat{\mathcal{H}}=\left(\begin{array}[]{cccc}h_{z}+\mathcal{H^{\prime}}&g_{k_{-}}+\Gamma_{xy}&\Delta_{k\uparrow\uparrow}&\Delta_{k}\\\
g_{k_{+}}+\Gamma^{*}_{xy}&-h_{z}+\mathcal{H^{\prime}}&-\Delta_{k}&\Delta_{k\downarrow\downarrow}\\\
{\Delta_{k\uparrow\uparrow}}^{\dagger}&-{\Delta_{k}}^{\dagger}&-h_{z}-\mathcal{H}^{\prime}&g_{k_{+}}-\Gamma^{*}_{xy}\\\
{\Delta_{k}}^{\dagger}&{\Delta_{k\downarrow\downarrow}}^{\dagger}&g_{k_{-}}-\Gamma_{xy}&h_{z}-\mathcal{H}^{\prime}\\\
\end{array}\right)$ (8)

where, $\mathcal{H}^{\prime}=-\mu_{i}[\Theta(-x)+\Theta(x-L)]+(V_{0}+\sigma
V_{z})[\delta(x)+\delta(x-L)]$, $\Gamma_{xy}=v_{F}(k_{x}-ik_{y})+h_{xy}$,
$h_{xy}=(m_{x}-im_{y})[\Theta(x)\Theta(L-x)]+(V_{x}-iV_{y})[\delta(x)+\delta(x-L)]$,
$h_{z}=m_{z}[\Theta(x)\Theta(L-x)]$ and $g_{k\pm}=V_{R}(k_{x}\mp
ik_{y})[\Theta(-x)e^{\mp i\phi_{L}}+\Theta(x-L)e^{\mp i\phi_{R}}]$.

Assuming $\theta$ as the incident angle of the quasi particle in the left
NCSC, the trajectory of the particle can be expressed using the momentum
components, $k_{x}=|\mathbf{k}|\cos\theta$ and $k_{y}=|\mathbf{k}|\sin\theta$.
Thus the total wave function of an electron injected in the left NCSC with an
angle $\theta$ can be written as

$\Psi_{\text{T}}(x)=\big{[}\Psi^{\text{L}}_{\text{NCSC}}(x)\Theta(-x)+\Psi_{\text{HM}}(x)\Theta(x)\Theta(L-x)\\\
+\Psi^{\text{R}}_{\text{NCSC}}(x)\Theta(x-L)\big{]}e^{ik_{y}y}$ (9)

where, $k_{y}$ is the momentum parallel to the interface and
$\Psi^{\text{L}}_{\text{NCSC}}(x)$, $\Psi_{\text{HM}}(x)$,
$\Psi^{\text{R}}_{\text{NCSC}}(x)$ are the quasi particle wave functions in
the left NCSC, half metal and right NCSC respectively. The wave functions in
different layers are obtained by diagonalizing the Hamiltonian (8) and can be
written as acharjee3

$\Psi^{\text{L}}_{\text{NCSC}}(x<0)=\\\ \tau[u_{\pm}\hat{\eta}_{1}\pm
u_{\pm}e^{-i\phi_{L}}\hat{\eta}_{2}\mp
v_{\pm}e^{-i\phi_{L}}\hat{\eta}_{3}+v_{\pm}\hat{\eta}_{4}]e^{ik^{+}_{x\pm}x}\\\
a_{1}[u_{+}\hat{\eta}_{1}+u_{+}e^{-i\phi_{L}}\hat{\eta}_{2}-v_{+}e^{-i\phi_{L}}\hat{\eta}_{3}+v_{+}\hat{\eta}_{4}]e^{-ik^{+}_{x+}x}\\\
+a_{2}[u_{-}\hat{\eta}_{1}-u_{-}e^{-i\phi_{L}}\hat{\eta}_{2}+v_{-}e^{-i\phi_{L}}\hat{\eta}_{3}+v_{-}\hat{\eta}_{4}]e^{-ik^{+}_{x-}x}\\\
+a_{3}[v_{+}\hat{\eta}_{1}+v_{+}e^{-i\phi_{L}}\hat{\eta}_{2}-u_{+}e^{-i\phi_{L}}\hat{\eta}_{3}+u_{+}\hat{\eta}_{4}]e^{ik^{-}_{x+}x}\\\
+a_{4}[v_{-}\hat{\eta}_{1}-v_{-}e^{-i\phi_{L}}\hat{\eta}_{2}+u_{-}e^{-i\phi_{L}}\hat{\eta}_{3}+u_{-}\hat{\eta}_{4}]e^{ik^{-}_{x-}x}.$
(10)

The wave function in the right NCSC region in a similar way can be written as

$\Psi^{\text{R}}_{\text{NCSC}}(x>L)=\\\
c_{1}[u_{+}\hat{\eta}_{1}+u_{+}e^{-i\phi_{R}}\hat{\eta}_{2}-v_{+}e^{-i\phi_{R}}\hat{\eta}_{3}+v_{+}\hat{\eta}_{4}]e^{ik^{+}_{x+}x}\\\
+c_{2}[u_{-}\hat{\eta}_{1}-u_{-}e^{-i\phi_{R}}\hat{\eta}_{2}+v_{-}e^{-i\phi_{R}}\hat{\eta}_{3}+v_{-}\hat{\eta}_{4}]e^{ik^{+}_{x-}x}\\\
+c_{3}[v_{+}\hat{\eta}_{1}+v_{+}e^{-i\phi_{R}}\hat{\eta}_{2}-u_{+}e^{-i\phi_{R}}\hat{\eta}_{3}+u_{+}\hat{\eta}_{4}]e^{-ik^{-}_{x+}x}\\\
+c_{4}[v_{-}\hat{\eta}_{1}-v_{-}e^{-i\phi_{R}}\hat{\eta}_{2}+u_{-}e^{-i\phi_{R}}\hat{\eta}_{3}+u_{-}\hat{\eta}_{4}]e^{-ik^{-}_{x-}x}.$
(11)

Here we define, $\hat{\eta}_{1}=(1,0,0,0)^{T}$,
$\hat{\eta}_{2}=(0,1,0,0)^{T}$, $\hat{\eta}_{3}=(0,0,1,0)^{T}$ and
$\hat{\eta}_{4}=(0,0,0,1)^{T}$. Here, $a_{1}$($a_{2}$) are the reflection
coefficients for up(down) spin electrons, while $a_{3}$($a_{4}$) are the
reflection coefficients for the up(down) spin holes. $c_{1}$($c_{2}$) are the
transmission coefficients for up(down) spin electrons, while $c_{3}$($c_{4}$)
are the transmission coefficients for the up(down)spin holes in right NCSC
region. $\phi_{L}$ and $\phi_{R}$ are the superconducting phase factor for the
left and right NCSC respectively. $k^{+(-)}_{x\sigma}$ gives the momenta of
the electron (hole) in NCSC with $\sigma=\pm 1$ represent spin up and down
particles. The momenta of the electron and the hole are defined as

$k^{\pm}_{x\sigma}=\sqrt{2(\mu_{S}\pm\Omega_{\pm})-\sigma Z_{R}}$ (12)

where, $\mu_{S}$ is the chemical potential and $Z_{R}$ is strength of RSOC in
NCSC region. The factor $\Omega_{\pm}$ is defined as

$\Omega_{\pm}=\sqrt{E^{2}-|\Delta_{s}\pm\frac{\Delta_{t}}{2}|^{2}}.$ (13)

The quasi particle amplitudes $u_{\pm}$ and $v_{\pm}$ appearing in Eqs. (10)
and (11) are defined as

$\displaystyle u_{\pm}=\frac{1}{\sqrt{2}}\sqrt{1+\frac{\Omega_{\pm}}{E}}$ (14)
$\displaystyle v_{\pm}=\frac{1}{\sqrt{2}}\sqrt{1-\frac{\Omega_{\pm}}{E}}$ (15)

In a similar way the wave function in the half metallic region is linder12

$\Psi_{\text{HM}}(0<x<L)=b_{1}\hat{\eta}_{1}e^{i\kappa^{+}_{x+}x}+b_{2}\hat{\eta}_{2}e^{i\kappa^{+}_{x-}x}\\\
+b_{3}\hat{\eta}_{1}e^{-i\kappa^{-}_{x+}x}+b_{4}\hat{\eta}_{2}e^{-i\kappa^{-}_{x-}x}$
(16)

where, $b_{1}$, $b_{2}$, $b_{3}$, $b_{4}$ are the transmission coefficients
for electron and hole in half metallic region. The quasi particle momenta in
this region is given by

$\kappa^{\pm}_{x\sigma}=\pm\mu_{TI}\mp\sqrt{M_{z}^{2}+(h_{x}+k_{x}v_{F})^{2}+(h_{y}+k_{y}v_{F})^{2}}$
(17)

where, $M_{z}=h_{z}+V_{z}$ and $\mu_{TI}$ is the chemical potential of TI. In
most of the TIs, chemical potential lies closer to the valence band as they
are not perfectly insulating. So in experiment beenakker , the chemical
potential has larger value than the superconducting gap. The band dispersion
at the surface of topological insulator is plotted using Eq. (17) and
displayed in Fig. (2). It is noted that there exist a gap in the band energy
when both bulk and barrier magnetic moment are parallel to each other along
z-direction with $\rho=0.3$. However, the energy gap disappears for the anti-
parallel orientation. This signifies that the band dispersion at the surface
of 3D TIs depends on the alignment angle of the magnetic moment due to bulk HM
and barrier. The boundary conditions that are to be satisfied by the
wavefunctions at two junctions are as follows:

$\displaystyle\big{[}\Psi^{\text{L}}_{\text{NCSC}}(x)-\Psi_{\text{HM}}(x)\big{]}\big{|}_{x\,=\,0}$
$\displaystyle=0,$ (18)
$\displaystyle\big{[}\Psi_{\text{HM}}(x)-\Psi^{\text{R}}_{\text{NCSC}}(x)\big{]}\big{|}_{x\,=\,L}$
$\displaystyle=0,$ (19)
$\displaystyle\partial_{x}\big{[}\Psi_{\text{HM}}(x)-\Psi^{\text{L}}_{\text{NCSC}}(x)\big{]}|_{x\,=\,0}$
$\displaystyle=\big{(}\hat{\tau}+\hat{\Lambda}_{m}\big{)}\Psi^{\text{L}}_{\text{NCSC}}(x)\big{|}_{x\,=\,0},$
(20)
$\displaystyle\partial_{x}\big{[}\Psi^{\text{R}}_{\text{NCSC}}(x)-\Psi_{\text{HM}}(x)\big{]}|_{x\,=\,L}$
$\displaystyle=\big{(}\hat{\tau}+\hat{\Lambda}_{m}\big{)}\Psi_{\text{HM}}(x)\big{|}_{x\,=\,L}$
(21)

where, $\hat{\tau}$ is defined as klam

$\hat{\tau}=\left(\begin{array}[]{cccc}Z_{-}&0&0&0\\\ 0&Z_{+}&0&0\\\
0&0&Z_{+}&0\\\ 0&0&0&Z_{-}\\\ \end{array}\right)$ (22)

with $Z_{\pm}=Z_{0}\pm Z_{R}\sin\theta$ where,
$Z_{0}=\frac{2V_{0}}{k_{F}\cos\theta}$ and $Z_{R}=2V_{R}\tan\theta$ are the
parameters characterizing spin independent barrier strength and RSOC. The
matrix $\hat{\Lambda}_{m}$ gives the contribution due to spin active
interacting potential and has the following form linder16 :

$\hat{\Lambda}_{m}=\left(\begin{array}[]{cccc}\lambda_{1}&\lambda_{2}&0&0\\\
\lambda_{2}^{*}&-\lambda_{1}&0&0\\\ 0&0&\lambda_{1}&\lambda_{2}^{*}\\\
0&0&\lambda_{2}&-\lambda_{1}\\\ \end{array}\right)$ (23)

where, $\lambda_{1}=2\rho V_{0}\cos\chi_{m}$ and $\lambda_{2}=2\rho
V_{0}\sin\chi_{m}e^{-i\xi_{m}}$.

##  III Andreev Bound States and Josephson Supercurrent

The wave functions must satisfy the boundary conditions defined in Eqs. (18),
(19), (20) and (21). This leads to a system of $16$ linear equations
connecting the coefficients defined as $\hat{A}\hat{x}=0$, where $\hat{A}$ is
a $16\times 16$ matrix and
$\hat{x}=(a_{1},a_{2},a_{3},a_{4},b_{1},b_{2},b_{3},b_{4},b_{1}^{\prime},b_{2}^{\prime},b_{3}^{\prime},b_{4}^{\prime},c_{1},c_{2},c_{3},c_{4})^{T}$.
The Andreev Bound States (ABS) can be calculated by the condition
$det(\hat{A})=0$. The solution for these equations needed several lengthy
calculations. However, under some limiting cases they are expressible in
analytic form. Considering the limiting cases, like normal interface
($\theta=0$), transparent barrier ($Z_{0}=0$), Rashba free case ($Z_{R}=0$)
and small length scale ($L/L_{0}<<1$), the Andreev levels can be expressed as

$E_{\sigma\tau}=\tau\beta\cos\left(\frac{\phi+\sigma\chi_{m}}{2}\right)$ (24)

where, $\tau=\pm 1$ for left and right moving Cooper pairs respectively,
$\sigma$ is the helicity index and
$\beta=\sqrt{\Gamma_{1}^{2}+\Gamma_{2}^{2}+\Gamma_{3}^{2}}$. The parameters
$\Gamma_{1}$, $\Gamma_{2}$ and $\Gamma_{3}$ are defined as

$\Gamma_{1}=\big{[}\rho\\{\cos\chi_{m}(\rho\cos\chi-2)-\rho\sin^{2}\xi_{m}\sin^{2}\chi_{m}\\}\\\
+2\big{]}\Delta_{-}^{2}-\sqrt{2}\Delta_{+}^{2}\sqrt{\Gamma_{4}\rho^{2}\sin^{2}\xi_{m}\sin^{2}\chi_{m}},$
(25)
$\Gamma_{2}=2\Delta_{-}^{2}\rho\sin\xi_{m}\sin\chi_{m}-\sqrt{2}\Delta_{+}^{2}\times\\\
\sqrt{\Gamma_{4}\rho^{3}\sin^{3}\xi_{m}\sin^{4}\chi_{m}},$ (26)
$\Gamma_{3}=\frac{2\Gamma_{4}\rho^{2}\sin^{2}(\xi_{m}\sin^{2}\chi_{m}}{2\rho\cos\chi_{m}(\rho\cos\chi_{m}+1)+1}$
(27)

with

$\Gamma_{4}=\rho^{2}\cos 2\chi_{m}+\rho^{2}-4\rho\cos\chi_{m}+4.$ (28)

Using the approximated analytical expression in Eq. (24), we study the
formation of ABS and its dependence on the parameters like barrier spin-
dependent parameters ($\rho$, $\chi_{m}$, $\xi_{m}$) and singlet-triplet
mixing parameter ($\delta$). However, if we consider the barrier thickness
($Z_{0}$), RSOC strength ($Z_{R}$) and oblique incident angle ($\theta$), then
the calculations become a too cumbersome and the analytical expression for the
same is too hard to obtain. So numerical approach is followed to obtain ABS
and the results are also displayed in the later sections.

The angle-averaged Josephson supercurrents of the ABS are evaluated using the
standard expression linder12 ,

$\frac{J}{J_{0}}=\int_{-\pi/2}^{\pi/2}d\theta\cos\theta\tanh\left(\frac{E_{\sigma\tau}}{2k_{B}T}\right)\frac{dE_{\sigma}/\Delta_{\sigma\tau}}{d\phi}$
(29)

where, $J_{0}=\frac{eN\Delta_{\sigma}}{\hbar}$. We obtain the numerical plots
of Josephson supercurrents by considering $T/T_{c}=0.001$ with $T_{c}$ being
the critical temperature. We set, $v_{F}=1$, $h_{z}=1$, $h_{xy}=0$ and as
$\Delta_{0}<<\mu_{S,TI}$ beenakker , so we consider $\mu_{S}=150\Delta_{0}$
and $\mu_{TI}=100\Delta_{0}$ for all our analysis.

Figure 3: ABS spectra for different choices of angle of incidence ($\theta$)
and polar angle ($\chi_{m}$) considering $\rho=0.01$, $\xi_{m}=0.5\pi$,
$\delta=1$, $Z_{R}=0$ and $Z_{0}=0.1$. The plots in the top panel are for
$L/L_{0}=0.01$ while the plots in the bottom panel are for $L/L_{0}=0.05$.

##  IV Results and Discussions

###  IV.1 Andreev Bound States

First, we consider the ABS corresponding to the different angles of incidence
of the injected particles. The results are displayed in Fig. 3 for different
misalignment angles considering two different length scales viz.,
$L/L_{0}=0.01$ and $0.05$. We consider $Z_{0}=0.1$, $Z_{R}=0$ and $\delta=1$
for this analysis. We find a gapless dispersion with $4\pi$ periodic ABS for
perpendicular incidence as represented by the solid curves of Fig. 3(a). It
indicates the presence of Majorana at $\phi=\pi$ for perpendicular incidence.
More interestingly result in Fig. 3(a) signifies that Majorana is present in a
spin-active barrier. However, for oblique incidence, a gap is always observed
for all cases, as shown by the dotted and dashed lines of Figs. 3(a)-3(j). It
is to be noted that the gap opens up further for larger the angle of incidence
as seen for $\theta=\frac{\pi}{3}$ indicated by the dotted lines. The presence
of a gap at oblique incidence is due to non-zero backscattering at a finite
angle. To realize the impact of barrier magnetic moment, we have plotted ABS
in Figs. 3(a)-3(j) for different values of $\chi_{m}$ with $\rho=0.01$ and
$\xi_{m}=0.5\pi$. It is seen that the Andreev levels split corresponding to
spin helicity $\sigma$ and for non-vanishing values of $\chi_{m}$. The wave
vector mismatch occurs for the misalignment of the polar angle of magnetic
moments. Thus the two degenerate energy bands split into two more branches, as
seen in Figs. 3(b)-3(d) and Figs. 3(g)-3(i). We also find that the two sets of
ABS depart away from each other with the increase in misalignment angle
$\chi_{m}$. This results in the formation of a $\phi$ junction for non-
vanishing values of $\chi_{m}$ corresponding to normal incidence. Similar
behavior is also observed for oblique incidence. However, the gap remains
constant in this situation also. It is seen that the $4\pi$ periodicity of the
ABS will not present for non-vanishing values of $\chi_{m}$ and finally for
anti-parallel orientation $\chi_{m}=\pi$, the system resides in $2\pi$-ABS
branches as seen from Figs. 3(e) and 3(j). Thus, interaction with the barrier
magnetic moments results in the system to reside in the lower Andreev energy
branches. The Andreev levels are found to display similar characteristics in
Figs. 3(f)-3(j) for $L/L_{0}=0.05$ in normal incidence condition. However, for
oblique incidence, the gap is significantly reduced for this length scale. It
may be due to the reason that with increasing length, induced magnetic
proximity on the surface of 3D TI gets weaker and hence the effect of barrier
and bulk moment is reduced. Thus, the system is found to display $4\pi$
periodicity for the parallel alignment while it shows anomalous
characteristics and $\phi$ periodicity for the misalignment of the barrier and
bulk moments.

In Figs. 4(a)-4(j), we show the dependence of ABS on the ratio of barrier
magnetic to non-magnetic moment ($\rho$) in perpendicular incidence condition.
We study the ABS spectra for different polar angles ($\chi_{m}$). The plots in
the top panel are for azimuthal angle $\xi_{m}=0.5\pi$ while the plots in
bottom panel are for $\xi_{m}=\pi$. We find that the Majorana modes appear at
$\phi=\pi$ in parallel configuration for all choices of $\rho$ and $\xi_{m}$
as seen in Figs. 4(a) and 4(f). Andreev levels shrinks with the increasing
values of $\rho$ and maximum contraction is observed for $\rho=1$ as seen from
the dashed curves of Figs. 4(a) and 4(b). However, for $\chi_{m}=0.5\pi$,
$0.75\pi$ and $\pi$ the ABS grows more for $\rho=1$ and a opposite
characteristic is seen. It signifies that for anti-parallel alignment ABS
energy becomes maximum when magnitude of barrier magnetic moment approaches to
non-magnetic moment. This behaviour is seen in Figs. 4(c)-4(e) for
$\xi_{m}=0.5\pi$. The ABS depart again for non-vanishing values of $\chi_{m}$.
However, the Andreev levels for $\rho=0.01$ and $0.5$ are found to display
nearly same characteristics for $\chi_{m}=0.5\pi$. The ABS shrinks further for
$\chi_{m}=0.75\pi$ with $\rho=0.01$ as seen from Fig. 4(d). Though a similar
characteristic is seen for $\chi_{m}=0$ and $0.25\pi$ with $\xi_{m}=\pi$, but
for $\chi_{m}=0.5\pi$ the Andreev energy levels are found to be independent of
$\rho$ as seen from Fig. 4(h). For $\chi_{m}=0.75\pi$ and $\pi$ with
$\xi_{m}=\pi$, the Andreev levels grows for all choices of $\rho$. Thus,
although the Andreev energy levels becomes anomalous for nonzero values of
$\chi_{m}$ but the magnitude of the Andreev levels can significantly change
for different azimuthal angle as seen from Fig. 4. This indicate that the ABS
not only depends on the polar angle $\chi_{m}$ but also on azimuthal angle
$\xi_{m}$.

Figure 4: ABS spectra for different choices of $\rho$ and polar angle
($\chi_{m}$) considering $\delta=1$, $\theta=0$, $Z_{R}=0$ and $Z_{0}=0.1$.
The plots in the top panel are for $\xi_{m}=0.5\pi$ while the plots in the
bottom panel are for $\xi_{m}=\pi$.

We study the effect of barrier transparency on ABS as shown in Figs. 5(a)-5(e)
considering a $\delta$-like barrier at $x=0$ and $x=L$. We make this analysis
by considering a quasi-magnetic barrier with $\rho=0.01$ and the barrier
moment is misaligned by an angle $\chi_{m}=\xi_{m}=0.3\pi$ for all plots of
Fig. 5. We also set $L/L_{0}=0.01$, $Z_{R}=0$ and $\delta=1$. For a nearly
transparent barrier the Andreev levels are found to be nearly similar to that
obtained in Figs. 4(b), 4(g), 5(a) and 5(b). In this scenario, the ABS depart
again due to non-vanishing values of $\chi_{m}$. For $Z_{0}=0.5$, the Andreev
levels shrinks but the splitting is still present as seen in Fig. 5(b). With
the further increase in barrier transparency $Z_{0}$, the energy of the ABS
significantly decreases as seen from the Figs. 5(c)-5(e). It is seen that the
pattern of ABS is found to be significantly different in Figs. 5(a) and Fig.
5(e) characterize by an anticrossing at $\phi=0$ and is due to finite value of
$Z_{0}$. However, the splitting of the Andreev levels still present in the
system.

The plots in the middle panel of Fig. 5 display the effect of RSOC on Andreev
energy levels. In the presence of RSOC, the four degenerate energy bands of
the TI further split into four branches, as seen in Figs. 5(f)-5(j). In this
situation, the wave vectors of the NCSC are different from that of TI. Due to
the mismatch of the wave vector, an additional gap appears at $\phi=\pi$.
However, it is seen from Figs. 5(g)-5(i) that the four energy branches grow
while the other four shrinks simultaneously with an increase in the strength
of RSOC. For $Z_{R}=1$, four Andreev energy branches grow to their maximum
value while the other four branches shrink completely, as seen in Fig. 5(j).
In this scenario, thus the supercurrent is carried by only one branch of
Andreev energy levels. Furthermore, it is noted that though the RSOC splits
the energy bands, it does not create a phase shift. It indicates that there
exist Majorana modes at $\phi=\pi$ for normal incidence even in the presence
of RSOC. Thus, the main effect of RSOC on the Andreev levels are, (i) shift of
energy bands and (ii) presence of an additional gap at $\phi=\pi$. So, it can
be concluded that RSOC plays a very significant role on the ABS spectrum.

We show ABS spectra for different values of singlet-triplet mixing parameter
$\delta$ in the bottom panel of Fig. 5. We set $L/L_{0}=0.01$, $Z_{R}=0$ and
$Z_{0}=0$ for this analysis. The mixing parameter $\delta$ plays a very
significant role in NCSC with $\delta=0$ correspond to majority of singlet and
$\delta=1$ correspond to majority of triplet correlations. We find that the
system exhibit exactly same behaviour for $\delta=0$ and $1$ as seen from
Figs. 5(k) and 5(o). We encounter band splitting again for unequal mixing of
the singlet-triplet components. For unequal mixing, the Andreev levels
corresponding to $\Delta_{-}$ shrinks while the levels corresponding to
$\Delta_{+}$ remains unchanged as seen from Fig. 5(l) and 5(n). Moreover, the
helical ABS spectra for $\delta=0.25$ is found to be exactly same as that of
$\delta=0.75$. For equal mixing, the Andreev energy levels corresponding to
$\Delta_{-}$ die out completely while the levels corresponding to $\Delta_{+}$
remains unchanged as seen from Fig. 5(m). Thus the supercurrent is only due to
the $\Delta_{+}$ component, while the contribution of the supercurrent from
$\Delta_{-}$ is zero for equal mixing conditions. We find that the magnitude
of ABS energy is dependent on gap $\Delta_{-}$ while it is totally independent
of $\Delta_{+}$. Furthermore, the crossing of the energy levels and the phase
are totally independent of the gap parameter $\Delta_{\pm}$. It signifies that
Majorana modes can exist at the surface of 3D TI in a $\pi$-NCSC Josephson
junction.

Figure 5: ABS spectra for different choices of $Z_{0}$, $Z_{R}$ and $\delta$.
We consider $\rho=0.01$, $\chi_{m}=0.3\pi$, $\xi_{m}=0.3\pi$ and
$L/L_{0}=0.01$ for all plots. The plots in the top panel are for different
choices of barrier strength $Z_{0}$ with $Z_{R}=0$ and $\delta=1$. The plots
in the middle panel are for different values of RSOC parameter $Z_{R}$
considering $Z_{0}=0$ and $\delta=1$, while the plots in the bottom panel are
for different singlet triplet mixing parameter $\delta$ considering $Z_{0}=0$
and $Z_{R}=0$.

###  IV.2 Josephson Supercurrent

The magnetic tunability of the Andreev energy levels and its effects on the
superconducting phase indicates the presence of a non-trivial Josephson
current in the proposed setup. Experimentally, Josephson junctions made up of
Superconductor-Topological Insulator-Superconductor (S$|$TI$|$S) has been
realized recently veldhorst . The work indicates the anomalous characteristics
of the supercurrent. Moreover, presently there is a great interest in the
experimental realization of ferromagnetic topological insulator hybrids (FTI).
It can be achieved by random doping of transition metal elements like Cr or V
on the surface of 3D TI chang11 ; chang12 . This result in magnetic proximity
on the surface of TI. Recently it is found that due to the presence of
magnetic material in the NCSC Josephson junction, the system displays
anomalous Josephson current zhang11 . It arises due to the different
orientation of the exchange field. However, the Josephson current in an NCSC
corresponding to an HM ferromagnet is not studied yet. Also, as
NCSC$|$HM$|$NCSC Josephson junctions hold Majorana modes at $\phi=\pi$ in
parallel orientation for half-metallic limit, so it is necessary to understand
the behaviour of supercurrent flowing at the surface of 3D TI. We initially
calculated the Josephson supercurrent for an HM with bulk magnetic moments
$\mathbf{m}=(0,0,|m|)$ and the barrier magnetic moments
$\mathbf{V}_{m}=(V_{x},V_{y},V_{z})=(\rho V_{0}\cos\xi_{m}\sin\chi_{m},\rho
V_{0}\sin\xi_{m}\sin\chi_{m},\rho V_{0}\cos\chi_{m})$. The results in the
half-metallic limit are displayed in Figs. 6, 7, 8 and 9. However, we also
plotted the supercurrent for a general ferromagnet in Fig. 10. For simplicity
of our calculations, we ignore the continuum states of asymmetric contact
junction and we only focus on the discrete states. This approximation does not
drastically change the effective current phase relation and hold good to
simplify the calculations as seen from the previous works klam .

Figure 6: Variation of Josephson supercurrent with $\phi$ for different
choices of $\rho$ and $\chi_{m}$. The plots in the top panel are for
$\xi_{m}=0.01\pi$ while the plots in the bottom panel are for
$\xi_{m}=0.5\pi$. All the plots are for $Z_{0}=0$, $Z_{R}=0$ and $\delta=0$
with $L/L_{0}=0.01$.

Figure 7: Variation of Josephson supercurrent with $\phi$ for different
choices of $\delta$ $\chi_{m}$ considering $\rho=0.3$, $\xi_{m}=0.5\pi$,
$Z_{0}=0$, $Z_{R}=0$ with $L/L_{0}=0.01$.

In Fig. 6, we have plotted the Josephson supercurrent for different values of
the effective ratio of magnetic and non-magnetic barrier potentials. We find
that for parallel orientation of the barrier and bulk moments with
$\xi_{m}=0.01\pi$, the magnitude of supercurrent is maximum for $\rho=0.01$.
It decreases with the increasing values of $\rho$ as seen from Fig. 6(a). This
signifies that in this situation, a non-magnetic barrier offers more
supercurrent than the magnetic barrier. It is also noted that the supercurrent
is found to be 2$\pi$ periodic. If the barrier and the bulk moments are
misaligned to an angle $\chi_{m}=0.25\pi$, then the supercurrent significantly
reduces for all choices of $\rho$. However, in this condition also non-
magnetic barrier offer more supercurrent than the magnetic barrier and the
supercurrents still found to display $2\pi$ periodic phase as seen from Fig.
6(b). This is due to the presence of symmetric branches in ABS spectra in HM
limit, which is significantly different for a bulk ferromagnet zhang11 . The
magnitude of the supercurrent further decreases for the perpendicular
misalignment of the barrier and bulk moments characterize by
$\chi_{m}=0.5\pi$. In this case, all the magnetic barrier corresponding to
non-zero values of $\rho$ display exactly the same characteristics. As the
misalignment angle $\chi_{m}$ increases further to $0.75\pi$ and $\pi$, the
supercurrents display $\pi$ phase shift. But in this case, the component
$\rho=1$ display critical current while it is found to be minimum for
$\rho=0.01$. Thus the magnetic barriers enhance the supercurrent while the
non-magnetic barriers suppress the supercurrent, as seen from Figs. 6(d) and
6(e). It is to be noted that the magnitude of the supercurrent is found to be
maximum for the anti-parallel misalignment of the barrier and the bulk
moments. To understand the effect of azimuthal angle on the supercurrent, we
plotted the CPR for $\xi_{m}=0.5\pi$ in Figs. 6(f)-6(j). It is seen that the
nature of the supercurrent remains quite similar for the $\chi_{m}=0$ case as
seen from Fig. 6(f). However, for $\chi_{m}=0.25\pi$ we find that the
supercurrent corresponding to $\rho=0.01$ and $0.2$ enhances while it
suppressed further for $\rho=0.8$ and $0.1$ as seen from Fig. 6(g).
Supercurrent is found to be independent of the strength of the magnetic
barrier for $\chi_{m}=0.5\pi$. However, it is still found to be maximum for
non-magnetic barrier that is seen from Fig. 6(h). The CPR for
$\chi_{m}=0.75\pi$ from Fig. 6(i) is found to drastically different from Fig.
6(d). In this condition, the magnitude of the supercurrent corresponding to
the magnetic barrier drastically increases while it suppressed for a non-
magnetic barrier. It indicates that CPR is also dependent on the azimuthal
angle of the barrier moment. For the anti-parallel misalignment, we find
exactly similar characteristics from Figs. 6(e) and 6(j). Thus these results
indicate that supercurrent is dependent on the misalignment of barrier and
bulk magnetic moments. Depending upon the orientations, both magnetic and non-
magnetic barriers offer critical current in certain orientations.

We study the CPR of the total supercurrent for different singlet-triplet
correlations in Fig. 7. We set $\rho=0.3$, $\xi_{m}=0.5\pi$, $Z_{0}=0$,
$Z_{R}=0$ and $L/L_{0}=0.0$1 for this analysis. We find that the CPR
corresponding to $\delta=0$ coinciding with that for $\delta=1$ which is in
accordance with Fig. 5. It is to be noted that the critical current is
observed for the majority of either singlet or triplet components represented
by $\delta=0$ and $1$ respectively. The critical current is found to be
minimum for equal mixing of the singlet-triplet parameter for all possible
misalignment angles. It is due to the reason that for equal mixing, the
supercurrent is only due to the $\Delta_{+}$ component while it is completely
suppressed for the $\Delta_{-}$ component represented by solid black lines of
Fig. 7. However, the critical current display a moderate value for unequal
mixing of the singlet-triplet parameter represented by solid orange lines of
Fig. 7. In this condition, though the ABS is suppressed due to the
$\Delta_{-}$ component, some supercurrent still presents due to the non-
vanishing value of $\Delta_{-}$. Moreover, the CPR is found to execute nearly
sinusoidal characteristics as the contribution from the higher harmonics are
very much suppressed.

Figure 8: Variation of Josephson supercurrent with $\rho$ for different
choices of $\chi_{m}$ considering $\xi_{m}=0.1\pi$, $Z_{0}=0$, $Z_{R}=0$ with
$L/L_{0}=0.01$.

Figure 9: Plot of length dependence of critical current for different choices
of $\rho$ with $\chi_{m}=0.01\pi$, $\xi_{m}=0.5\pi$, $Z_{0}=0$ and $Z_{R}=0$.

From Figs. 6 and 7, we have seen that the characteristics of Josephson
supercurrent are dependent on the effective ratio between magnetic and non-
magnetic moments, misalignment angle between the barrier and bulk magnetic
moment and singlet-triplet mixing parameter. So to understand the interplay
between them, we have plotted supercurrent as a function of $\rho$ in Fig. 8
for different singlet-triplet mixing parameter. We find that for $\chi_{m}=0$,
the critical current is observed at $\rho=0$ and the supercurrent decreases
monotonically with the increasing values of $\rho$ for all choices of singlet-
triplet mixing parameter as seen from Fig. 8(a). The maximum critical current
is observed for $\delta=0$ and $1$ while the minimum critical current is
observed for $\delta=0.5$ as seen earlier from Figs. 6 and 7. The
characteristics of the critical current remain the same for $\chi_{m}=0.25\pi$
as seen from Fig. 8(b). However, the critical current is found to be decreased
in all mixing scenarios for $\chi_{m}=0.25\pi$. For $\chi_{m}=0.5\pi$, the
Josephson current is found to be totally independent of $\delta$ for all
$\rho$ values as seen from Fig. 8(c). We find an exactly opposite behaviour
characterized by the negative critical current for anti-parallel misalignment.
In this case also $\delta=0$ and $\delta=1$ display maximum critical current
as seen from Fig. 8(d).

We study the dependence of the critical Josephson current on the length of the
Josephson junction in Fig. 9. It is a quantity that represents the maximum
supercurrent and often measured in magnetic Josephson junctions linder12 ;
snelder1 ; snelder2 . We consider $\xi_{m}=0.5\pi$, $Z_{0}=0$ and $Z_{R}=0$
for this analysis. We find that the critical current monotonically decays for
all choices of mixing parameter $\delta$. The misalignment angle can only
split the energy bands but not transfer any finite centre of mass momentum to
the Cooper pairs. This results in a monotonous decay of the critical current.
From Fig. 9 we see that $\rho=0.1$ provides the maximum critical currents in
all three scenarios, while minimum critical current is observed for $\rho=1$.
It is also seen that the magnitude of critical current is dependent on the
mixing parameter $\delta$. The critical current is found on decaying rapidly
in an equal mixing scenario as seen from Fig. 9(a). For unequal mixing
$\delta=0.75$ and $1$, the critical current decays comparatively slower than
that for $\delta=0.5$ as seen from Figs. 9(b) and 9(c). The decay rate of
critical current is found to be minimum for $\delta=1$. Thus it can be
concluded that the decay rate of the critical current is dependent on the
singlet-triplet mixing parameter.

Figure 10: ABS energy levels $E$ as a function of incident angle $\theta$ and
phase $\phi$. The plot (a) corresponds to no barrier and bulk magnetization
i.e., $h_{xy}=0$, $h_{z}=V_{z}=0$ while plot (b) is for $h_{xy}=0.1V_{z}$ and
plot (c) is for $h_{xy}=V_{z}$. The corresponding ABS spectra and Josephson
supercurrent are shown in plots (d) and (e) respectively.

Figs. 4, 5 and 6 indicate that the presence of a spin-active barrier
drastically changes the characteristics of Andreev energy levels and Josephson
supercurrent in an HM ferromagnet. For all our analysis, we consider the
magnetization of the bulk HM ferromagnet is along z-direction and contribution
due to xy-component is zero, i.e., $h_{xy}=0$. However, it is natural to
address the scenario of a general ferromagnet which can be represented by
random orientation of the bulk moments. Though many similar works tanaka1 ;
zhang11 ; linder12  is done before but the interplay of the bulk moment with
spin-active barrier moments on the surface of a TI is yet to be studied. So,
we have plotted ABS energy levels as a function of $\theta$ and $\phi$ in Fig.
10. We consider three different bulk moments. In Fig. 10(a), we consider the
scenario where no barrier and bulk moment is present, i.e., $h_{xy}=0$,
$h_{z}=V_{z}=0$. In this scenario, we observe a 4$\pi$ periodic ABS and a band
crossover at $\phi=-\pi$ and $\pi$ specif by the darker region of the density
plot of Fig. 10(a). In this condition, the Andreev energy levels are found to
be symmetric with respect to $\phi=0$ and Majorana modes are observed at
$\phi=\pi$. However, for $h_{xy}=0.1V_{z}$, the band crossing will not appear
at $-\pi$ and $\pi$. Thus the ABS spectra are found to be asymmetric in this
condition seen from Fig. 10(b). In Fig. 10(c), we display the ABS for
$h_{xy}=V_{z}$. In this condition, the crossover of the Andreev energy bands
will appear at $\phi=0$, which signify the $2\pi$-periodic ABS. The ABS
spectra and CPR for all the choices of $h_{xy}$ are shown in Figs. 10(d) and
10(e). We find that there exists a phase shift in Josephson supercurrent for
random orientation of the bulk magnetic moments.

It is very worthwhile to mention that experimentally transport properties of
heterostructures are studied in the diffusive limit. In the diffusive limit,
one needs to use quasi-classical Usadel equations to obtain the Andreev levels
and supercurrent. However, an analytic equation for ABS energy considering all
contributing parameters is too hard to obtain for heterostructure like
NCSC$|$HM$|$NCSC. So in that case also we need to use a numerical approach.
Though we perform all our calculations considering BdG equations in the
ballistic limit, our results are quite adequate in diffusive limits also
because of two reasons: (1) The anomalous Andreev energy levels and the
$\phi$-states are generated due to chiral spin symmetry breaking along with
scattering from the spin-active barrier. It is to be noted that these effects
pertain uniquely to both ballistic and diffusive limits. (2) We find that CPR
for low and moderate interface transparency is close to sinusoidal in nature.
It is due to the suppression of the higher harmonics. As in diffusive limit,
first harmonics give the most important contribution. So our results will be
quite adequate in diffusive limits also. There may be some difference in
ballistic and diffusive limits qualitatively. But it is expected to have no
dramatic alteration of the results in diffusive limit.

##  V Conclusions

In summary, we have studied the transport in NCSC$|$HM$|$NCSC Josephson
junction grown on a 3D TI. Our model predicts the formation of the helical
Andreev Bound States, Majorana mode and Josephson supercurrent on the surface
of a 3D TI. We employ Bogoliubov de-Gennes equations to obtain ABS and CPR. We
investigated the role of the spin-active barrier by considering different
alignments of barrier magnetic moment on Andreev levels and supercurrent. It
is seen that a 4$\pi$-periodic ABS is formed for parallel alignment of barrier
and bulk magnetic moments for all incidence angles. However, for anti-parallel
alignment, the system resides in 2$\pi$ in Andreev levels. It is seen that
Majorana modes exist at $\pi$ for parallel alignment in normal incidence,
while they disappear in oblique incidence condition. We find that the
degenerate Andreev bands split into their corresponding branches for
misalignments of the barrier and bulk moments. In this condition, a phase
shift is observed, which results in $\phi$-junction. It is seen that the
Andreev energy levels are suppressed for a strongly magnetic barrier in
parallel orientation conditions, while a reverse scenario is observed for
anti-parallel orientation. The magnitude of ABS energy decreases with the
increase in barrier width. It is found that there exists an anti-crossing at
$\phi=0$ in the ABS spectra for a finite barrier thickness. In the presence of
RSOC, the degenerate sub-bands further split into their respective branches.
It is seen that with the increasing strength of RSOC, the Andreev levels
corresponding to positive spin helicity grow, while the other shrinks. For an
NCSC with very strong RSOC, the ABS branches corresponding to negative spin
helicity shrinks completely result in no supercurrent for these branches. We
find that the band splits for unequal mixing of singlet-triplet parameter even
in the absence of RSOC. However, no splitting is observed for equal mixing or
singlet/triplet dominated scenarios. We also find that for parallel alignment
of the moments, a non-magnetic barrier offers more supercurrent than a
magnetic barrier. However, an opposite scenario is observed in anti-parallel
orientation. So depending on the orientations, both magnetic and non-magnetic
barriers offer critical current in certain orientations. The decay rate of the
critical current is found to be dependent on the singlet-triplet mixing
parameter. We also find an anomalous characteristic of Josephson supercurrent
for random orientation of the bulk magnetic moment, but it is not observed in
the half-metallic limit. These results would be a significant guiding
background for innovating novel TI based sprintronics appliances.

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