# Novel Majorana mode and magnetoresistance in ferromagnetic superconducting
topological insulator

H. Goudarzi∗, M. Khezerlou† and S. Asgarifar
Department of Physics, Faculty of Science, Urmia University, P.O.Box: 165,
Urmia, Iran

###### Abstract

Among the potential applications of topological insulators, we investigate
theoretically the effect of coexistence of proximity-induced ferromagnetism
and superconductivity on the surface states of 3-dimensional topological
insulator, where the superconducting electron-hole excitations can be
significantly affected by the magnetization of ferromagnetic order. We find
that, Majorana mode energy, as a verified feature of TI F/S structure, along
the interface sensitively depends on the magnitude of magnetization $m_{zfs}$
in FS region, while its slope in perpendicular incidence presents steep and no
change. Since the superconducting gap is renormalized by a factor
$\eta(m_{zfs})$, hence Andreev reflection is more or less suppressed, and, in
particular, resulting subgap tunneling conductance is more sensitive to the
magnitude of magnetizations in FS and F regions. Furthermore, an interesting
scenario happens at the antiparallel configuration of magnetizations $m_{zf}$
and $m_{zfs}$ resulting in magnetoresistance in N/F/FS junction, which can be
controlled and decreased by tuning the magnetization magnitude in FS region.

PACS: 74.45.+c; 85.75.-d; 73.20.-r
Keywords: topological insulator; ferromagnetic superconductivity; Andreev
reflection; Majorana mode; tunneling conductance

##  1 INTRODUCTION

Topological insulators (TIs) represent new type of material which has emerged
in the last few years as one of the most actively research subjects in
condensed matter physics. They are characterized by a full insulating gap in
the bulk and gapless edge or surface states, which are protected by the time-
reversal symmetry [1, 2, 3, 4]. Regarding Bernevig and Hughes prediction [3,
5], TIs have been experimentally observed with such properties that host bound
states on their surface, e.g. in 3-dimensional topological insulators (3DTI)
$Bi_{2}Te_{3}$, $Bi_{2}Se_{3}$, $Sb_{2}Te_{3}$ and $Bi_{x}Sb_{1-x}$ alloy, and
also in the $CdTe/HgTe/CdTe$ quantum well heterostructure [6, 7, 8]. These
states form a band-gap closing Dirac cone on each surface, and lead to a
conducting state with properties unlike any other known electronic systems. In
particular, conformity of the conduction and valence bands to each other in
and around Dirac points in the first Brillouin zone, possessing an odd number
of Dirac points, description of fermionic excitations as massless two-
dimensional chiral Dirac fermions, depending chirality on the spin of
electron, having the significant electron-phonon scattering on the surface,
owning very low room-temperature electron mobility are the peculiar properties
of electronic structure of TIs. Interestingly, the charge carriers in the
surface states can behave as massive Dirac fermions [9] due to its proximity
to a ferromagnetic material, that the vertical component of the magnetic
vector potential may be proportional to the effective mass of Dirac fermion.
The experimentally observed proximity-induced superconductivity on the surface
state is another interesting dynamical feature occuring in 3DTI, see Refs.
[10, 11, 12].

More importantly, the coexistence of superconductivity and ferromagnetism as
one of potential interests for spintronics and high magnetic field
applications has firstly been predicted by Fulde and Ferrel [13], and Larkin
and Ovchinnikov [14] as FFLO state. This effect can be in compliance with
standard BCS theory for phonon-mediated $s$-wave superconductivity, because
the ferromagnetic exchange field is expected to prevent spin-singlet Cooper
pairing, (see, Ref. [15] as a prior work). The magnetic polarization of a pair
electron caused by a ferromagnetic material can lead to the different momentum
of Cooper pair occurring in a ferromagnetic superconducting (FS) segment. It
seems to be in contrast to the formation of a typical cooper pair, where two
electrons may be in opposite spin direction with the same momentum. However,
Bergeret et.al. [16] and Li et.al. [17] have studied the effect of
superconductor/ferromagnetic bilayer on the critical Josephson current, where
the orientation of ferromagnetic exchange field strongly affects the critical
current. Also, the effect of superconductivity in coexistence with
ferromagnetism has been studied on the superconducting gap equation for two
case of singlet $s$-wave and triplet $p$-wave symmetries [18]. The authors
have reconsidered the Clogston-Chandrasekhar limiting [19, 20]. According to
the Clogston criterion in the conventional FS mixture, the normal state is
regained as soon as the ferromagnetic exchange field exceeds
$\Delta_{0}/\sqrt{2}$ at zero temperature. To be empirically, the
$ErRh_{4}B_{4}$ [21] has been discovered to be the first ferromagnetic
superconductor, which superconductivity is found to occur in a small
temperature interval with adjusted ferromagnetic phase. Also,
superconductivity is detected in itinerant ferromagnetic $UGe_{2}$ in a
limited range of pressure and temperature [22].

Regarding several works in the recent few years concerning with the
topological insulator-based junctions [23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
33, 34, 35, 36], which are related to the Andreev process and resulting subgap
conductance we proceed, in this paper, to theoretically study the dynamical
properties of Dirac-like charge carriers in the surface states of 3DTI under
influence of both superconducting and ferromagnetic orders via the introducing
the proper form of corresponding Dirac spinors, which are principally distinct
from those given in Ref. [29]. The magnetization induction opens a gap at the
Dirac point (no inducing any finite center of mass momentum to the Cooper
pair), whereas the superconducting correlations causes an energy gap at the
Fermi level in the 3DTI. It will be particularly interesting to investigate
the topological insulator superconducting electron-hole excitations in the
presence of a exchange field. We assume that the Fermi level is close to the
Dirac point, and the ferromagnet has a magnetization $|\textbf{M}|<\mu$. The
chirality conservation of charge carriers on the surface states in the
presence of magnetization (due to opening the band gap) allows to use a finite
magnitude of $|\textbf{M}|$. In the absence of topological insulator, the
spin-splitting caused by magnetization gives rise to limiting the magnitude of
$|\textbf{M}|$ in a FS structure. These excitations, therefore, are found to
play a crucial role in Andreev reflection (AR) process leading to the
tunneling conductance below the renormalized superconducting gap.
Particularly, we pay attention to the formation of Majorana bound energy mode,
as an interesting feature in topological insulator ferromagnet/superconductor
interface, depending on the magnetization of FS hybrid structure. We present,
in section 2, the explicit signature of magnetization in low-energy effective
Dirac-Bogogliubov-de Gennes (DBdG) Hamiltonian. The electron(hole)
quasiparticle dispersion energy is analytically calculated, which seems to
exhibit qualitatively distinct behavior in hole excitations ($|k_{fs}|<k_{F}$)
by varying the magnitude of magnetization. By considering the magnetization is
ever less than chemical potential in FS region, the superconducting wavevector
and corresponding eigenstates are derived analytically. Section 3 is devoted
to unveil the above key point of FS energy excitation, Majorana mode energy,
Andreev process and resulting tunneling conductance in N/F/FS junction and
respective discussions. In the last section, the main characteristics of
proposed structure are summarized.

##  2 THEORETICAL FORMALISM

###  2.1 Topological insulator FS effective Hamiltonian

In order to investigate how both superconductivity and ferromagnetism
induction to the surface state affects the electron-hole excitations in a 3DTI
hybrid structure, we consider magnetization contribution to the DBdG equation.
Let us focus first on the Hubbard model Hamiltonian [37] that is included the
effective exchange field $\mathbf{M}$ follows from:

${\cal
H}=-\sum_{\rho\rho^{{}^{\prime}}s}t_{\rho\rho^{{}^{\prime}}}\hat{c}^{\dagger}_{\rho
s}\hat{c}_{\rho^{{}^{\prime}}s}+\frac{1}{2}\sum_{\rho\rho^{{}^{\prime}}ss^{{}^{\prime}}}U_{\rho\rho^{{}^{\prime}}ss^{{}^{\prime}}}\hat{n}_{\rho
s}\hat{n}_{\rho^{{}^{\prime}}s^{{}^{\prime}}}+\sum_{\rho
ss^{{}^{\prime}}}\hat{c}^{\dagger}_{\rho
s}(\mbox{\boldmath$\sigma$}\cdot\mathbf{M})\hat{c}_{\rho s^{{}^{\prime}}},$
(1)

where $U_{\rho\rho^{{}^{\prime}}ss^{{}^{\prime}}}$ denotes the effective
attractive interaction between arbitrary electrons, labeled by the integer
$\rho$ and $\rho^{{}^{\prime}}$ with spins $s$ and $s^{{}^{\prime}}$. The
matrices $t_{\rho\rho^{{}^{\prime}}}$ are responsible for the hopping between
different neighboring sites, and $\hat{c}_{\rho s}$ and $\hat{n}_{\rho s}$
indicate the second quantized fermion and number operators, respectively.
Here, $\mbox{\boldmath$\sigma$}(\sigma_{x},\sigma_{y},\sigma_{z})$ is the
vector of Pauli matrix. Using the Hartree-Fock-Gorkov approximation and
Bogoliubov-Valatin transformation [38], the Bogoliubov-de Gennes Hamiltonian
describing dynamics of Bogoliubov quasiparticles is found. In Nambu basis,
that electron(hole) state is given by
$\Psi=\left(\psi_{\uparrow},\psi_{\downarrow},\psi^{\dagger}_{\uparrow},\psi^{\dagger}_{\downarrow}\right)$,
the BdG Hamiltonian for a $s$-wave spin singlet superconducting gap in the
presence of an exchange splitting can be written as:

${\cal H}_{SF}=\left(\begin{array}[]{cc}h(\mathbf{k})+M&\Delta(\mathbf{k})\\\
-\Delta^{\ast}(-\mathbf{k})&-h^{\ast}(-\mathbf{k})-M\end{array}\right),$ (2)

where $h(\mathbf{k})$ denotes the non-superconducting Schrodinger-type part,
and $\Delta(\mathbf{k})$ is superconducting order parameter. In the simplest
model, $\Delta(\mathbf{k})$ can be chosen to be real to describe time-reversed
states. The effective exchange field by rotating our spin reference frame can
be gain as $\left|\mathbf{M}\right|=\sqrt{m^{2}_{x}+m^{2}_{y}+m^{2}_{z}}$. The
four corresponding levels of a singlet superconductor in a spin magnetic field
is obtained
$E_{s}(\mathbf{k})=\sqrt{\epsilon^{2}_{\mathbf{k}}+\left|\Delta(\mathbf{k})\right|^{2}}+s\left|\mathbf{M}\right|$
with $s=\pm 1$, where $\epsilon_{\mathbf{k}}$ is the normal state energy for
$h(\mathbf{k})$. However, dependence of superconducting order parameter on the
exchange energy can be exactly derived from self-consistency condition [18]:

$\Delta(\mathbf{k})=-\frac{1}{4}\sum_{\mathbf{k}s}U_{s-s}(\mathbf{k})\frac{\Delta_{0}(\mathbf{k})}{\sqrt{\epsilon^{2}_{\mathbf{k}}+\left|\Delta_{0}(\mathbf{k})\right|^{2}}}\tanh\left(\frac{\sqrt{\epsilon^{2}_{\mathbf{k}}+\left|\Delta_{0}(\mathbf{k})\right|^{2}}+s\left|\mathbf{M}\right|}{2k_{B}T}\right),$
(3)

where $\Delta_{0}(\mathbf{k})$ is the conventional order parameter in absence
of ferromagnetic effect, $k_{B}$ and $T$ are the Boltzmann constant and
temperature, respectively. The exchange splitting dependence of
superconducting gap indicates that equation (3) has no functionality of
$\mathbf{M}$ at zero temperature. This takes place under an important
condition known as Clogston-Chandrasekhar limiting [19, 20]. According to this
condition, if the exchange splitting becomes greater than a critical value
$\left|\mathbf{M}_{c}\right|=\left|\Delta(T=0)\right|/\sqrt{2}$, then the
normal state has a lower energy than the superconducting state. This means
that a phase transition from the superconducting to normal states is possible
when the exchange splitting is increased at zero temperature.

We now proceed to treat such a ferromagnetic superconductivity coexistence at
the Dirac point of a 3DTI. It should be stressed that the dressed Dirac
fermions with an exchange field in topologically conserved surface state have
to be in superconducting state. Here, the influence of exchange field
interacts in a fundamentally different way comparing to the conventional
topologically trivial system, where the exchange field splits the energy bands
of the majority and minority spins. A strong TI is a material that the
conducting surface states at an odd number of Dirac points in the Brillouin
zone close the insulating bulk gap unless time-reversal symmetry is broken.
Candidate Dirac-type materials include the semiconducting alloy
$Bi_{1-x}Sb_{x}$, as well as HgTe and $\alpha-Sn$ under uniaxial strain [39].
In the simplest case, there is a single Dirac point in the surface Fermi
circle and general effective Hamiltonian is modeled as $h^{TI}_{N}=\hbar
v_{F}(\mbox{\boldmath$\sigma$}\cdot\mathbf{k})-\mu$, where $v_{F}$ indicates
the surface Fermi velocity, and $\mu$ is the chemical potential. Under the
influence of a ferromagnetic proximity effect, the Hamiltonian for the two-
dimensional surface states of a 3DTI reads as:

$h^{TI}_{F}=\hbar
v_{F}(\mbox{\boldmath$\sigma$}\cdot\mathbf{k})-\mu+\mathbf{M}\cdot\mbox{\boldmath$\sigma$},$

where the ferromagnetic contribution corresponds to an exchange field
$\mathbf{M}=(m_{x},m_{y},m_{z})$. It has been shown [9] that transverse
components of the magnetization on the surface $(m_{x},m_{y})$ are responsible
to shift the position of the Fermi surface of band dispersion, while its
perpendicular component to the surface induces an energy gap between
conduction and valence bands.

In what follows, we will employ the relativistic generalization of BdG
Hamiltonian, which is interacted by the effective exchange field to obtain the
dispersion relation of FS dressed Dirac electrons in a topological insulator:

${\cal
H}^{TI}_{FS}=\left(\begin{array}[]{cc}h^{TI}_{F}(\mathbf{k})&\Delta(\mathbf{k})\\\
-\Delta^{\ast}(-\mathbf{k})&-h^{TI\ast}_{F}(-\mathbf{k})\end{array}\right).$
(4)

The superconducting order parameter now depends on both spin and momentum
symmetry of the Cooper pair, that the gap matrix for spin-singlet can be given
as $\Delta(\mathbf{k})=i\Delta_{0}\sigma_{y}e^{i\varphi}$, where $\Delta_{0}$
is the uniform amplitude of the superconducting gap and phase $\varphi$
guarantees the globally broken $U(1)$ symmetry. By diagonalizing this
Hamiltonian we arrive at an energy-momentum quartic equation. Without lose of
essential physics, we suppose the component of magnetization vector along the
transport direction to be zero $m_{x}=0$ for simplicity. Also, we set
$m_{y}=0$, since the analytical calculations become unwieldy otherwise. The
dispersion relation resulted from Eq. (4) for electron-hole excitations is
found to be of the form:

${\cal
E}_{FS}=\zeta\sqrt{\left(-\tau\mu_{fs}+\sqrt{m^{2}_{zfs}+\left|\mathbf{k}_{FS}\right|^{2}+\left|\Delta_{0}\right|^{2}(\frac{m_{zfs}}{\mu_{fs}})^{2}}\right)^{2}+\left|\Delta_{0}\right|^{2}\left(1-(\frac{m_{zfs}}{\mu_{fs}})^{2}\right)},$
(5)

where, the parameter $\zeta=\pm 1$ denotes the electron-like and hole-like
excitations, while $\tau=\pm 1$ distinguishes the conduction and valence
bands. We might expect several anomalous properties from the above
superconducting excitations, which is investigated in detail in the next
section. Equation (5) is clearly reduced to the standard eigenvalues for
superconductor topological insulator in the absence of exchange field as
$m_{z}=0$ (see Ref. [9]), ${\cal
E}_{S}=\zeta\sqrt{\left(-\tau\mu_{s}+\left|\mathbf{k}_{S}\right|\right)^{2}+\left|\Delta_{0}\right|^{2}}$.
The mean-field conditions are satisfied as long as $\Delta_{0}\ll\mu_{fs}$. In
this condition, the exact form of superconducting wavevector of charge
carriers can be acquired from the eigenstates
$k_{fs}=\sqrt{\mu^{2}_{fs}-m^{2}_{zfs}}$.

The Hamiltonian Eq. (4) can be solved to obtain the electron (hole)
eigenstates for FS topological insulator. The wavefunctions including a
contribution of both electron-like and hole-like quasiparticles are
analytically found as:

$\psi^{e}_{FS}=\left(\begin{array}[]{cc}e^{i\beta}\\\
e^{i\beta}e^{i\theta_{fs}}\\\
-e^{i\theta_{fs}}e^{-i\gamma_{e}}e^{-i\varphi}\\\
e^{-i\gamma_{e}}e^{-i\varphi}\end{array}\right)e^{i(k_{fs}^{x}x+k_{fs}^{y}y)},\
\ \psi^{h}_{FS}=\left(\begin{array}[]{cc}1\\\ -e^{-i\theta_{fs}}\\\
e^{i\beta}e^{-i\theta_{fs}}e^{-i\gamma_{h}}e^{-i\varphi}\\\
e^{i\beta}e^{-i\gamma_{h}}e^{-i\varphi}\end{array}\right)e^{i(-k_{fs}^{x}x+k_{fs}^{y}y)},$
(6)

where we define

$\cos{\beta}=\frac{{\cal E}_{FS}}{\eta\left|\Delta_{0}\right|}\ ;\ \
\eta=\sqrt{1-(\frac{m_{zfs}}{\mu_{fs}})^{2}}\ ,\ \
e^{i\gamma_{e(h)}}=\frac{\Delta(\mathbf{k})}{\left|\Delta(\mathbf{k})\right|}.$

Note that, the solution is allowed as long as the Zeeman field being lower
than chemical potential $m_{zfs}\leq\mu_{fs}$.

###  2.2 FS interplay at the TI interface

We consider Andreev reflection in a hybrid N/F/FS structure formed on the
surface of a 3DTI which coexistence between ferromagnet and superconductor is
assumed to be induced by means of the proximity effect. The wide topological
insulator junction is taken along the $x$-axis with the FS region for $x>L$, F
region for $0<x<L$ and N region for $x<0$. The superconducting order parameter
vanishes identically in N and F regions, and we can neglect its spatial
variation in the FS region close to the interface. The magnetization vectors
of both sections is taken, in general, $m_{zi}(i\equiv f,fs)$, which can be at
the parallel or antiparallel configuration, as shown in Fig. 1. In the
scattering process follows from the Blonder-Tinkham-Klapwijk (BTK) formula
[40], we find the reflection amplitudes from the boundary condition at the
interface. In ferromagnetic case, right- and left- moving electrons (holes)
with energy excitation $\epsilon_{F}=\pm\sqrt{k_{Ff}^{2}+m_{zf}^{2}}-\mu_{f}$
below the superconducting gap, transmitted (normal reflected) from the N
region and reflected (Andreev reflected) at the FS interface. Thus, the
leftover $2e$ charge is transferred into the FS region as a Cooper pair at
Fermi level. At energy excitation above the normalized superconducting gap
resulted from Eq. (5) (see, in particular, Fig. 2) quasiparticle states can
directly tunnel into the superconducting section. The reflected hole leading
to AR can be actually controlled by the doping level in order to take place
possible specular Andreev reflection. Particularly, we have to determine (via
the dynamical features of system) the allowed values of Fermi energy in three
regions. We set the Fermi energy to zero in F region. The electron(or hole)
transmitted to the FS region angle may be accordingly obtained from the fact
of conservation of transverse wavevector under quasiparticle scattering at the
interface:

$\theta_{fs}=\arcsin\left(\frac{\mu_{n}\sin{\theta}}{\sqrt{\mu^{2}_{fs}-m^{2}_{zfs}}}\right),$
(7)

where $\mu_{n}$ and $\theta$ are the chemical potential and incidence angle in
N region, respectively. As an important point, the electron(hole) angle of
incidence in all regions may be span the range from $0$ to $\pi/2$ around the
normal axis. Regarding the Eqs. (7), the angle $\theta_{fs}$ needs to be
meaningful when the chemical potential of FS region takes a magnitude greater
than its value in N region ($\mu_{fs}>\mu_{n}$). On the other hand, we
previously applied the condition $m_{zfs}<\mu_{fs}$, as an experimentally used
manner to calculate the wave functions Eq. (6).

By introducing the normal $r$ and Andreev $r_{A}$ reflection coefficients and
the scattering coefficients in F region, the total wave function inside the N
and F region can be written as:

$\Psi_{N}=e^{ik^{y}_{n}y}\left(\psi^{e+}_{N}e^{ik^{x}_{n}x}+r\psi^{e-}_{N}e^{-ik^{x}_{n}x}+r_{A}\psi^{h-}_{N}e^{ik^{x}_{n}x}\right),$
$\Psi_{F}=e^{ik^{y}_{f}y}\left(a\psi^{e+}_{F}e^{ik^{xe}_{f}x}+b\psi^{e-}_{F}e^{-ik^{xe}_{f}x}+c\psi^{h+}_{F}e^{-ik^{xh}_{f}x}+d\psi^{h-}_{F}e^{ik^{xh}_{f}x}\right),$
(8)

where the eigenvectors $\psi$ can be found in Appendix A. The probability
amplitude of reflections in Eq. (8) are calculated from the continuity of the
wavefunctions at the interface. The wave function in FS region is defined as
$\Psi_{FS}=t^{e}\psi^{e}_{FS}+t^{h}\psi^{h}_{FS}$. Finally, we find the
following analytical expressions for the reflection coefficients, that the
auxiliary quantities is described in Appendix A:

$r=\left[t^{e}e^{i\beta}(2\mathcal{M}_{1}-1)+t^{h}e^{-i\beta}(2\mathcal{M}_{2}-1)\right](i\sin{(k^{xe}_{f}L}))+$
$\left[t^{e}e^{i\beta}+t^{h}e^{-i\beta}\right]\cos{(k^{xe}_{f}L})-1,$
$r_{A}=\left[t^{e}e^{i\theta_{fs}}(2\mathcal{M}_{2}-1)-t^{h}e^{-i\theta_{fs}}(2\mathcal{M}_{1}-1)\right](i\sin{(k^{xe}_{f}L}))-$
$\left[t^{e}e^{i\theta_{fs}}-t^{h}e^{-i\theta_{fs}}\right]\cos{(k^{xe}_{f}L)}.$
(9)

The reflection amplitudes measurements under the BTK formalism enables us to
capture the tunneling conductance through the junction:

$G(eV)=G_{0}\int^{\theta_{c}}_{0}d\theta^{e}\cos{\theta^{e}}\left(1+\left|r_{A}\right|^{2}-\left|r\right|^{2}\right),$
(10)

where the critical angle of incidence $\theta_{c}$ is determined depending on
the doping of F region. The quantity $G_{0}$ is a renormalization factor
corresponding to the ballistic conductance of normal metallic junction.

##  3 RESULTS AND DISCUSSION

###  3.1 Energy excitation and Majorana mode

In this section, we proceed to analyze in detail the dynamical features of
Dirac-like charge carriers in 3DTI with ferromagnetic and superconducting
orders deposited on top of it. We assume that the Fermi level controlled by
the chemical potential $\mu$ is close to the Dirac point. In this case, it is
expected the signature of $m_{zfs}<\mu_{fs}$ to be significant. In Fig. 2, we
demonstrate the FS 3DTI electron-hole excitations. A net superconducting gap
$\Delta_{0}$ is obtained in Dirac points (for $|k_{fs}|=k_{F}$, where $k_{F}$
is Fermi wavevector) when we set $m_{zfs}=0$. Increasing $m_{zfs}$ up to its
possible maximum value results in three outcomes: i) the superconducting
excitations, which is renormalized by a factor
$|\Delta(\textbf{k})|\sqrt{1-(m_{zfs}/\mu_{fs})^{2}}$, disappear in hole
branch ($|k_{fs}|<k_{F}$). It means that for the greater magnetizations, if we
consent the superconductivity in FS 3DTI still exists, there is almost
vanishing quantum state for reflected hole by Andreev process in the valence
band, ii) Dirac point is shifted towards smaller FS quasiparticle electron-
hole wavevectors, iii) the superconducting gap decreases slowly, where the
variation of net gap is very low $\delta\Delta_{0}\ll|\Delta(\textbf{k})|$.
The Andreev process, therefore, is believed to inconsiderably supress. The
signature of these valence band excitations can be clearly shown in AR, where
the Majorana mode may also be formed at the 3DTI F/FS interface [23, 25].

As a verified result, considering the topological insulator interface between
the ferromagnetic insulator and conventional superconductor leads to the
appearance of the chiral Majorana mode as an Andreev bound state. In other
words, the Majorana mode and Andreev reflection are strongly related to each
other. The latter can be realized by the fact of looking for bound energies
produced by the perfect AR, which yields the following solution:

$\tilde{\epsilon}(\theta)=\eta\Delta_{0}sgn\left(\Lambda\right)/\sqrt{1+\Lambda^{2}}\
;\ \ \
\Lambda=\tan\left[\frac{1}{2i}\ln(\frac{\upsilon_{1}}{\upsilon_{2}})\right],$

where we define

$\upsilon_{1(2)}=4i\sin{k^{xe}_{f}L}\cos{\theta}\mathcal{M}_{2(1)}\mathcal{A}_{1(2)}+2e^{-ik^{xe}_{f}L}\cos{\theta}\mathcal{A}_{1(2)}-\mathcal{B}_{2(1)}\mathcal{A}_{1(2)}.$

We have checked numerically that sign of $\Lambda$ is changed by
$sgn(m_{zf})$. Thus, the sign of Andreev resonance states may be changed by
reversing the direction of $m_{zf}$, and it corresponds to the chirality of
Majorana mode energies. As shown in Fig. 3, the slope of the energy curves of
$\tilde{\epsilon}(\theta)$ around $\tilde{\epsilon}(\theta=0)=0$ become steep
and show no change with the increase of $m_{zfs}/\mu_{n}$ for fixed $m_{zf}$,
while it exhibits significantly decreasing behavior with the increase of
$m_{zf}/\mu_{n}$ for fixed $m_{zfs}$. The dispersion of Majorana modes along
the interface ($\theta=\pi/2$) decreases with the increase of both
magnetizations of FS and F regions. Note that, due to the presence of
$m_{zfs}$ it needs to consider the Fermi level mismatch between normal and FS
sections, i.e. $\mu_{n}\neq\mu_{fs}$. Then, the above contributions can be
considerable in Andreev process and resulting subgap tunneling conductance.

###  3.2 Tunneling conductance

From the angle-resolved Andreev and normal reflection probabilities using Eq.
(9), we see from Fig. 4(a) the main contribution of AR belongs to the angle of
incidence $\theta<0.15\pi$ in zero bias. It, therefore, is expected to achieve
the lower zero bias conductance, as shown in Fig. 4(b) and (c). Furthermore,
varying $m_{zfs}$ has no significant influence on AR in zero bias
$\epsilon(eV)=0$ owing to the very small decrease of the renormalized
superconducting gap with the increase of the $m_{zfs}$, while the increasing
$m_{zf}$ results in more suppression of AR. The latter can be understood by
the increase of band gap in Dirac point in F region. The resulting normalized
angle-averaged tunneling conductance curves are reported in Figs. 4(b) and (c)
for two parallel and antiparallel configurations of magnetizations in FS and F
regions. Zero bias conductance peak disappears with the decrease of the
$m_{zf}$, and instead of it a high conductance peak appears in bias
$\epsilon=\eta\Delta_{0}$. This result should be compared to that is obtained
in Ref. [9, 24]. Interestingly, by increasing the $m_{zfs}$ the magnitude of
subgap bias $\epsilon/\eta\Delta_{0}<1$, for which the new peak takes place,
is limited, as seen from Fig. 4(c). Thus, parameter
$\eta=\sqrt{1-(m_{zfs}/\mu_{s})^{2}}$ can be considered as a “bias-limitation
coefficient”. These features have been obtained when the direction of
magnetizations in F and FS regions are at the parallel configuration. The
fundamentally distinct scenarios we find for the case of antiparallel
configuration of magnetizations. In this case, first, the tunneling subgap
conductance is enhanced, secondly, the zero bias conductance peak presented in
parallel case is replaced by a deep, see, in detail, Fig. 4(d). Dynamically
description, when the direction of $m_{zf}$ is inverted we, indeed, meet with
an inverse energy gap in Dirac point of 3DTI giving rise to enhancing the
conductance peak respective bias energy $\epsilon/\Delta_{0}=\eta$ in low
values of $m_{zf}$. One can express that the zero bias conductance originates
from the chiral Majorana mode, which significantly depends on the $m_{zf}$.
The chirality actually corresponds to the sign of $m_{zf}$, while the
magnitudes of zero bias conductance at the parallel and antiparallel
configurations are the same. Hence, the both deep and peak of conductance
curves in antiparallel case are significance being influenced by the inverted
gap caused by the $-m_{zf}$.

Remarkably, the importance of above findings can be featured by the capture of
magnetoresistance (MR) of the topological insulator junction. The
magnetization (specialy in FS region) dependence of MR is presented in Fig. 5,
where we observe a considerable MR peak for extra values of $m_{zf}$ (e.g.
$0.9\mu_{n}$ in figure). Importantly, increasing the $m_{zfs}$ weakens the MR
peak, since, regarding the superconducting excitations in Fig. 2, the AR is
more or less suppressed in the presence of $m_{zfs}$ and Fermi wavevector
mismatch also causes to decrease the $\eta\Delta_{0}$-bias conductance peak at
the antiparallel configuration. According to the conductance curves, there is
no MR in zero bias.

##  4 CONCLUSION

In summary, we have investigated the influence of ferromagnetic
superconducting orders coexistence in the surface state of topological
insulator. The topological insulator superconducting electron-hole excitations
in the presence of magnetization have led to achieve qualitatively distinct
transport properties in tunneling N/F/FS junction. One of key findings of the
present work is that the resulting subgap conductance has been found to be
strongly sensitive to the parallel or antiparallel configuration of
magnetization directions in FS and F regions. Thus, this feature has actually
led to present the magnetoresistance peak for bias energy close to the
renormalized superconducting gap $\epsilon(eV)=\eta\Delta_{0}$, which the bias
limitation coefficient $\eta$ includes the magnetization of FS region
$m_{zfs}$. Particularly, we have found the presence of Majorana mode at the
F/FS interface to be controlled by the tuning of magnetizations magnitude.
However, these results have been obtained in the case of $m_{zfs},m_{zf}<\mu$
and $\mu\gg\Delta_{0}$, which is relevant to the experimental regime.

## APPENDIX A: Normal and Andreev reflection amplitudes

To complete calculation of probability of reflections in N/F/FS junction, we
write down right and left moving electron and hole spinors in F and N region:

$\psi^{e+}_{N}=\left[1,e^{i\theta},0,0\right]^{T},\ \
\psi^{e-}_{N}=\left[1,-e^{-i\theta},0,0\right]^{T},\ \
\psi^{h-}_{N}=\left[0,0,1,-e^{-i\theta}\right]^{T},$
$\psi^{e+}_{F}=\left[1,\alpha e^{i\theta_{f}},0,0\right]^{T},\
\psi^{e-}_{F}=\left[1,-\alpha e^{-i\theta_{f}},0,0\right]^{T},\
\psi^{h+}_{F}=\left[0,0,1,\alpha e^{i\theta_{f}}\right]^{T},\
\psi^{h-}_{F}=\left[0,0,1,-\alpha e^{-i\theta_{f}}\right]^{T}$

where we define $\alpha=\sqrt{\frac{\mu_{f}-m_{zf}}{\mu_{f}+m_{zf}}}$. By
matching boundary conditions on $\Psi_{N}$ and $\Psi_{F}$ at $x=0$ and
$\Psi_{F}$ and $\Psi_{FS}$ at $x=L$, the reflection amplitudes are obtained.
We introduce auxiliary quantities in Eq. (9) as:

$t^{e}=\frac{2\mathcal{A}_{2}\cos{\theta}}{\mathcal{B}_{1}\mathcal{A}_{2}e^{i\beta}-\mathcal{B}_{2}\mathcal{A}_{1}e^{-i\beta}},\
\ \
t^{h}=\frac{-2\mathcal{A}_{1}\cos{\theta}}{\mathcal{B}_{1}\mathcal{A}_{2}e^{i\beta}-\mathcal{B}_{2}\mathcal{A}_{1}e^{-i\beta}}.$

with

$\mathcal{A}_{1}(2)=e^{(-)i\theta_{fs}}\left[(-)\mathcal{N}_{1}(\mathcal{M}_{2(1)}-1)e^{ik^{xe}_{f}L}-(+)\mathcal{N}_{2}\mathcal{M}_{2(1)}e^{-ik^{xe}_{f}L}\right],$
$\mathcal{B}_{1}(2)=-\mathcal{N}_{1}(\mathcal{M}_{1(2)}-1)e^{-ik^{xe}_{f}L}+\mathcal{N}_{2}\mathcal{M}_{1(2)}e^{ik^{xe}_{f}L},$
$\mathcal{N}_{1}(2)=(-)\alpha e^{(-)i\theta_{f}}+e^{-i\theta},\ \ \
\mathcal{M}_{1(2)}=\frac{\alpha
e^{i\theta_{f}}-(+)e^{(-)i\theta_{fs}}}{2\alpha\cos{\theta_{f}}}.$

## References

  * [1] C.L. Kane, E.J. Mele, Phys. Rev. Lett. 95 (2005) 146802.
  * [2] B.A. Bernevig, S.C. Zhang,Phys. Rev. Lett. 96 (2006) 106802.
  * [3] M.Z. Hasan, C.L. Kane, Rev. Mod. Phys. 82 (2010) 3045.
  * [4] X.L. Qi, S.C. Zhang, Rev. Mod. Phys. 83 (2011) 1057.
  * [5] B.A. Bernevig, T.L. Hughes, S.C. Zhang, Science 314 (2006) 1757.
  * [6] Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y.S. Hor, R.J. Cava, M.Z. Hasan, Nature Physics 5 (2009) 398.
  * [7] Y.L. Chen, J.G. Analytis, J.H. Chu, Z.K. Liu, S.K. Mo, X.L. Qi, H.J. Zhang, D.H. Lu, X. Dai, Z. Fang, S.C. Zhang, I.R. Fisher, Z. Hussain, Z.X. Shen, Science 325 (2009) 178.
  * [8] M. Konig, S. Wiedmann, C. Brune, A. Roth, H. Buhmann, L.W. Molenkamp, X.L. Qi, S.C. Zhang Science 318 (2007) 766.
  * [9] J. Linder, Y. Tanaka, T. Yokoyama, A. Sudbo, N. Nagaosa. Phys. Rev. B 81 (2010) 184525.
  * [10] J. Wang, C.Z. Chang, H. Li, K. He, D. Zhang, M. Singh, X.C. Ma, N. Samarth, M. Xie, Q.K. Xue, M.H.W. Chan, Phys. Rev. B 85 (2012) 045415.
  * [11] G. Koren, T. Kirzhner, E. Lahoud, K.B. Chashka, A. Kanigel, Phys. Rev. B 84 (2011) 224521.
  * [12] P. Zareapour, A. Hayat, S.Y.F. Zhao, M. Kreshchuk, A. Jain, D.C. Kwok, N. Lee, S.W. Cheong, Zh. Xu, A. Yang, G.D. Gu, Sh. Jia, R.J. Cava, K.S. Burch, Nature communications 3 (2012) 1056.
  * [13] P. Fulde, R.A. Ferrell, Phys. Rev. 135 (1964) A550.
  * [14] A.I. Larkin, Y.N. Ovchinnikov, Sov. Phys. JETP. 20 (1965) 762.
  * [15] N.F. Berk, J.R. Schrieffer, Phys. Rev. Lett. 17 (1966) 433.
  * [16] F.S. Bergeret, A.F. Volkov, K.B. Efetov Phys. Rev. Lett. 86 (2001) 3140.
  * [17] X. Li, Zh. Zheng, D.Y. Xing, G. Sun, Zh. Dong, Phys. Rev. B 65 (2002) 134507.
  * [18] B.J. Powell, J.F. Annett, B.L. Gyorffy, J. Phys. A: Math. Gen. 36 (2003) 9289.
  * [19] A.M. Clogston, Phys. Rev. Lett. 9 (1962) 266.
  * [20] B.S. Chandrasekhar, Appl. Phys. Lett. 1 (1962) 7.
  * [21] S.K. Sinha, et al., Superconductivity in Magnetic and Exotic Materials, Springer Berlin Heidelberg, (1984).
  * [22] S.S. Saxena, P. Agarwal, K. Ahilan, F.M. Grosche, R.K.W. Haselwimmer, M.J. Steiner, E. Pugh, I.R. Walker, S.R. Julian, P. Monthoux, G.G. Lonzarich, A. Huxley, I. Sheikin, D. Braithwaite, J. Flouquet, Nature 406 (2000) 587.
  * [23] L. Fu, C.L. Kane, Phys. Rev. Lett. 100 (2008) 096407.
  * [24] J. Linder, Y. Tanaka, T. Yokoyama, A. Sudbo, N. Nagaosa, Phys. Rev. Lett. 104 (2010) 067001.
  * [25] Y. Tanaka, T. Yokoyama, N. Nagaosa, Phys. Rev. Lett. 103 (2009) 107002.
  * [26] K.T. Law, P.A. Lee, T.K. Ng, Phys. Rev. Lett. 103 (2009) 237001.
  * [27] M. Snelder, M. Veldhorst, A.A. Golubov, A. Brinkman, Phys. Rev. B 87 (2013) 104507.
  * [28] J. Nussbaum, T.L. Schmidt, Ch. Bruder, R.P. Tiwari, Phys. Rev. B 90 (2014) 045413.
  * [29] P. Burset, B. Lu, G. Tkachov, Y. Tanaka, E.M. Hankiewicz, B. Trauzettel, Phys. Rev. B 92 (2015) 205424.
  * [30] T. Choudhari, N. Deo, Physica E 85 (2017) 238-247.
  * [31] G. Gupta, H. Lin, A. Bansil, M. Bin Abdul Jalil, G. Liang, Physica E 74 (2015) 10-19.
  * [32] M. Snelder, A.A. Golubov, Y. Asano, A. Brinkman, Journal of Physics: Condensed Matter 27 (2015) 315701.
  * [33] R. Vali, H.F. Khouzestani, Physica E 68 (2015) 107-111.
  * [34] M. Khezerlou, H. Goudarzi, Physica C 508 (2015) 6.
  * [35] T. Yokoyama, Sh. Murakami, Physica E 55 (2014) 1-8.
  * [36] H. Goudarzi, M. Khezerlou, J. Alilou, J. Super. and Novel Mag. 26 (2013) 3355.
  * [37] J. Hubbard, Proc. Roy. Soc. (London) A 276 (1963) 238.
  * [38] P.G. de Gennes, Superconductivity of Metals and Alloys, W. A. Benjamin, New York, (1966).
  * [39] L. Fu, C.L. Kane, Phys. Rev. B 76 (2007) 045302.
  * [40] G.E. Blonder, M. Tinkham, T.M. Klapwijk, Phys. Rev. B 25 (1982) 4515.

<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS>

Figure captions
Figure 1 (color online) Sketch of the topological insulator-based N/F/FS
junction. The magnetization vectors in F and FS regions can be at the parallel
or antiparallel configuration.
Figure 2 (color online) The ferromagnetic superconducting excitation spectra
on the surface state of 3DTI for several values of $m_{zs}$, calculated from
Eq. (5). We set the net value of superconducting gap $|\Delta_{S}|=0.5\;eV$
(this value of pair potential is taken only to more clarify the behavior of
spectra in Dirac point, although it does not further need to use it in our
calculations, since $\mu_{fs}/|\Delta_{S}|\gg 1$ is supposed.
Figure 3 (color online) The dispersion of Majorana modes as a function of the
electron incident angle for several values of magnetizations in FS and F
regions. The solid lines correspond to $m_{zf}=0.2\mu_{n}$ and the dashed
lines to $m_{zfs}=0.2\mu_{n}$.
Figure 4(a), (b), (c), (d) (color online) (a) Probability of the normal and
Andreev reflections as a function of electron incidence angle at the interface
in zero bias $\epsilon(eV)/\eta\Delta_{0}=0$ with $m_{zfs}=0.5\mu_{n}$ and
$\mu_{fs}/\mu_{n}=1.5$. The plots show the results for different values of
$m_{zf}$. (b) Normalized tunneling conductance versus bias voltage $eV$ and
magnetization of F region. We set $m_{zfs}=0.5\mu_{n}$ (c) Normalized
tunneling conductance versus bias voltage and magnetization of FS region. We
set $m_{zf}=0.2\mu_{n}$ (d) The tunneling conductance as a function of bias
voltage for two $\pm$ signs of $m_{zf}$, corresponding to the parallel and
antiparallel configurations in F and FS regions. The solid lines correspond to
$+m_{zf}$ and marker dashed lines correspond to $-m_{zf}$. We set
$m_{zf}=0.2\mu_{n}$.
Figure 5 (color online) The magnetoresistance spectra as function of bias
voltage, where the influence of $m_{zfs}$ and $m_{zf}$ is indicated,
separately. We have set $\mu_{fs}/\mu_{n}=1.2$ in the resulting conductance
and magnetoresistance spectra.

Figure 1:  Figure 2:  Figure 3:  Figure 4: (a),(b),(c),(d) Figure 5: