Dominant Majorana bound energy and critical current enhancement in ferromagnetic-superconducting topological insulator

0.49 MB·

by  Maryam Khezerlou

Dominant Majorana bound energy and critical current
enhancement in ferromagnetic-superconducting topological
insulator
7
M.Khezerlou∗, H. Goudarzi†,and S. Asgarifar
1
0 DepartmentofPhysics,FacultyofScience,UrmiaUniversity,P.O.Box:165,Urmia,Iran
2

n
a
J
Abstract
8
2 Among the potential applications of topological insulators, we theoretically study the coexis-
tence of proximity-inducedferromagnetic and superconductingorders in the surface states of a 3-

] dimensionaltopologicalinsulator.Thesuperconductingelectron-holeexcitationscanbesignificantly
l affectedbythemagneticorderinducedbyaferromagnet. Inonehand,thesurfacestateofthetopo-
l
a logical insulator, protected by the time-reversal symmetry, creates a spin-triplet and, on the other
h hand, magnetic order causes to renormalize the effective superconductinggap. We find Majorana
- modeenergyalongtheferromagnet/superconductorinterfacetosensitivelydependonthemagnitude
s
ofmagnetizationm fromsuperconductorregion,anditsslopearoundperpendicularincidenceis
e zfs
m steep with verylowdependencyonmzfs. Thesuperconductingeffectivegapis renormalizedby a
factorη(m ), andAndreevboundstateinferromagnet-superconductor/ferromagnet/ferromagnet-
. zfs
t superconductor(FS/F/FS) Josephsonjunctionismoresensitiveto themagnitudeofmagnetizations
a
of FS and F regions. In particular, we show that the presence of m has a noticeable impact on
m zfs
thegapopeninginAndreevboundstate,whichoccursinfiniteangleofincidence. Thisdirectlyre-
- sultsinzero-energyAndreevstatebeingdominant.Byintroducingtheproperformofcorresponding
d DiracspinorsforFSelectron-holestates,wefindthatviatheinclusionofm ,theJosephsonsuper-
zfs
n currentisenhancedandexhibitsalmostabruptcrossovercurve,featuringthedominantzero-energy
o Majoranaboundstates.
c
[ PACS:74.45.+c;85.75.-d;73.20.-r

Keywords: topological insulator; Josephson junction; Majorana bound state; superconductor ferromag-
1
v netproximity
7
7
1 INTRODUCTION
2
8
0 Three-dimensional topological insulator (3DTI), which has been predicted theoretically [1] and discov-
. ered experimentally [2, 3, 4, 5, 6] is characterized by gapless surface states and represents fully insu-
1
0 lating gap in the bulk. Particularly, coincidence of the conduction and valence bands to each other in
7 Dirac point, description of fermionic excitations as massless two-dimensional chiral Dirac fermions in
1 thefirstBrillouinzone,dependingchiralityonthespinofelectron,havingthesignificantelectron-phonon
: scatteringonthesurfaceandowningverylowroom-temperature electronmobilityarethepeculiarprop-
v
erties of electronic structure of the 3DTI. Due to spin-orbit interaction, the surface states are protected
i
X by the time-reversal symmetry, which are robust against perturbations. The spin of two-dimensional
r chiral Dirac-like charge carriers is tied to the momentum direction. It is highly desirable to investigate
a proximity-induced superconducting and ferromagnetic orders onthe surface of a3DTI[7, 8, 9, 10, 11].
The broken spin rotation symmetry of the chiral surface states creates the spin-singlet component in
topological insulator from conventional s-wave superconductor [7, 12, 13, 14]. On the other hand, the
long-rang proximity effectinaconventional superconductor ferromagnet hybrid [21]features theexotic
odd-frequency spin-triplet component, which is odd under the exchange of time coordinates and even
in momentum. This new superconducting condensation can be induced to the topologically conserved
systems[22]. Asaremarkablepoint,observing Majoranafermions,whichhavebeendetected inneutral
systems, e.g. Sr RuO and 3He[18, 19, 20], isofexperimentally importance [15, 16]. Inthis regard,
2 4
theinterplay between aferromagnet andsuperconductor ontopofa3DTIactually makes asense toen-
gineer the chiral Majorana mode [8, 17]. Therefore, magnetic order can directly influence the transport
manifestation inthesuperconducting 3DTI.
∗[email protected]
†Correspondingauthor,e-mailaddress:[email protected]
1
ThesearchfortransportpropertiesofdifferenthybridstructuresincludingMajoranafermionshasled
topublish impressive number ofguiding theoretical studies forexperimental measurements [7,8, 9,17,
23,24,25]. Inthesesystems,magnetizationfromaferromagnetplacedontopofthe3DTIcandrastically
affect the Andreev bound states. For instance, the magnetic order causes no 0 π oscillations in the
critical current, leading toanomalous current-phase relation [8],incontrast tothe−metallic topologically
trivial similar systems [26]. Also, the direction of magnetization seriously influences the Josephson
current,suchthatin-planecomponentofmagnetizationcreatesanintermediatephaseshift,i.e. seeRefs.
[8, 17]. Recently, Burset et.al. [27] have proposed normal/superconductor hybrid system deposited on
topofthe3DTI,wherethewholejunctionisexposedtoauniformmagneticorder. However,itcanbeof
particular interest that the magnetization induction to the superconducting and ferromagnetic regions in
aN/Sjunction maybeseparately appliedwithadifferentmagnitude foreachregion.
Regarding severalworksintherecent fewyearsrelating tothetopological insulator-based junctions
[24,25,27,28,29,30,31,32,33,34,35],whicharerelatedtotheAndreevprocessandresultingcurrent-
phase relation, we proceed, in this paper, to theoretically study the dynamical properties of Dirac-like
charge carriers inthe surface states ofthe 3DTIunder influence ofboth superconducting and ferromag-
netic orders. Weintroduce a proper form of corresponding Dirac spinors, which are principally distinct
from those in Ref. [27]. 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 creates
an energy gap at the Fermilevel, which isrelated to the chemical potential µ. It is particularly interest-
ing toinvestigate thetopological insulator superconducting electron-hole excitation inthe presence ofa
magnetic order. We assume that the Fermi level is close to the Dirac point, and the ferromagnet has a
magnetization M µ. In the presence of magnetization, the chirality conservation of charge carriers
in the surface s|tat|es≤(due to opening the band gap in Dirac point) allows to use a finite magnitude of
M . Insimilarsystemswithouttopological insulator, thespin-splitting arisingfrommagnetization gives
| |
rise to limiting the magnitude of M in a FS hybrid structure [36, 37]. This excitation, therefore, is
found to play a crucial role in And|ree|v process leading to the formation of Andreev bound state (ABS)
between two superconducting segments separated by a weak-link ferromagnetic insulator. Particularly,
we pay attention to the formation of Majorana bound mode as an interesting feature of the topological
insulatorF/Sinterface. Wepresent,insection2,theexplicitsignatureofmagnetizationinlow-energyef-
fectiveDirac-Bogogliubov-de Gennes(DBdG)Hamiltonian. Theelectron(hole) quasiparticle dispersion
energyisanalyticallycalculated,whichseemstoexhibitqualitativelydistinctbehaviorinholeexcitations
(k < k )byvaryingthemagnitudeofmagnetization. Byconsideringthemagnetizationbeingafinite
fs F
v|alue|lessthanchemicalpotentialinFSregion,thecorresponding propereigenstates areanalytically de-
rived. Section3isdevotedtounveiltheabovekeypointofFScorrelationsandrespectivediscussionina
proposed FS/F/FSJosephson junction. Inthelast section, the maincharacteristics ofproposed structure
aresummarized.
2 THEORETICAL FORMALISM
2.1 TopologicalinsulatorFSeffective Hamiltonian
In order to investigate how both superconductivity and ferromagnetism induction to the surface state
affects the electron-hole excitations in a 3DTIhybrid structure, weconsider magnetization contribution
to the DBdG equation. Let us focus first on the Hubbard model Hamiltonian [38] that is included the
effectiveexchangefieldMfollowsfrom:
1
= t cˆ† cˆ + U nˆ nˆ + cˆ† (σ M)cˆ , (1)
H − ρρ′ ρs ρ′s 2 ρρ′ss′ ρs ρ′s′ ρs · ρs′
Xρρ′s ρXρ′ss′ Xρss′
where U denotes the effective attractive interaction between arbitrary electrons, labeled by the in-
ρρ′ss′
teger ρ and ρ′ with spins s and s′. The matrices t are responsible for the hopping between differ-
ρρ′
ent neighboring sites, and cˆ and nˆ indicate the second quantized fermion and number operators,
ρs ρs
respectively. Here,σ(σ ,σ ,σ )isthevectorofPaulimatrix. UsingtheHartree-Fock-Gorkov approxi-
x y z
mation and Bogoliubov-Valatin transformation [39], the Bogoliubov-de Gennes Hamiltonian describing
dynamics of Bogoliubov quasiparticles is found. In Nambu basis, that electron(hole) state is given by
† †
Ψ = ψ ,ψ ,ψ ,ψ ,theBdGHamiltonian foras-wavespinsinglet superconducting gapinthepres-
↑ ↓ ↑ ↓
enceo(cid:16)fanexchange s(cid:17)plitting canbewrittenas:
h(k)+M ∆(k)
HSF = ∆∗( k) h∗( k) M , (2)
(cid:18) − − − − − (cid:19)
2
whereh(k)denotesthenon-superconducting Schrodinger-type part,and∆(k)issuperconducting order
parameter. In the simplest model, ∆(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 M = m2 +m2 +m2.
| | x y z
The four corresponding levels of a singlet superconductor in a spin magnetic field isqobtained Es(k) =
ǫ2k+ ∆(k) 2 + s M with s = 1, where ǫk is the normal state energy for h(k). However, de-
| | | | ±
pqendence ofsuperconducting order parameter ontheexchange energy canbeexactly derived from self-
consistency condition [40]:
∆(k) = 1 U (k) ∆0(k) tanh ǫ2k+|∆0(k)|2+s|M| , (3)
−4 Xks s−s ǫ2k+|∆0(k)|2 q 2kBT 
q  
where ∆ (k) is the conventional order parameter in absence of ferromagnetic effect, k and T are the
0 B
Boltzmannconstant andtemperature, respectively. Theexchange splitting dependence ofsuperconduct-
inggapindicates thatequation (3)hasnofunctionality ofMatzerotemperature. Thistakesplaceunder
animportantconditionknownasClogston-Chandrasekhar limiting[36,37]. Accordingtothiscondition,
if the exchange splitting becomes greater than a critical value M = ∆(T = 0) /√2, then the nor-
c
malstatehasalowerenergy thanthesuperconducting state. Thi|sme|ans|thataphase| transition fromthe
superconductingtonormalstatesispossiblewhentheexchangesplittingisincreasedatzerotemperature.
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
conservedsurfacestatehavetobeinsuperconductingstate. Here,theinfluenceofexchangefieldinteracts
in a fundamentally different way comparing to the conventional topologically trivial system, where the
exchangefieldsplitstheenergybandsofthemajorityandminorityspins. AstrongTIisamaterialthatthe
conductingsurfacestatesatanoddnumberofDiracpointsintheBrillouinzoneclosetheinsulatingbulk
gapunlesstime-reversalsymmetryisbroken. CandidateDirac-typematerialsincludethesemiconducting
alloy Bi Sb , as well as HgTe and α Sn under uniaxial strain [41]. In the simplest case, there is
1−x x
−
a single Dirac point in the surface Fermi circle and general effective Hamiltonian is modeled as hTI =
N
~v (σ k) µ, where v indicates the surface Fermi velocity, and µ is the chemical potential. Under
F F
theinflu·ence−ofaferromagneticproximityeffect,theHamiltonianforthetwo-dimensionalsurfacestates
ofa3DTIreadsas:
hTI = ~v (σ k) µ+M σ,
F F · − ·
where the ferromagnetic contribution corresponds to an exchange field M = (m ,m ,m ). It has
x y z
beenshown[8]thattransversecomponentsofthemagnetizationonthesurface(m ,m )areresponsible
x y
to shift the position of the Fermi surface of band dispersion, while its perpendicular component to the
surfaceinduces anenergy gapbetweenconduction andvalencebands.
In what follows, we will employ the relativistic generalization of BdG Hamiltonian, which is inter-
acted by the effective exchange field toobtain the dispersion relation of FSdressed Dirac electrons ina
topological insulator:
hTI(k) ∆(k)
TI = F . (4)
HFS ∆∗( k) hTI∗( k)
(cid:18) − − − F − (cid:19)
ThesuperconductingorderparameternowdependsonbothspinandmomentumsymmetryoftheCooper
pair, that the gap matrix for spin-singlet can be given as ∆(k) = i∆ σ eiϕ, where ∆ is the uniform
0 y 0
amplitude of the superconducting gap and phase ϕ guarantees the globally broken U(1) symmetry. By
diagonalizing this Hamiltonian we arrive at an energy-momentum quartic equation. Without lose of
essential physics, wesuppose thecomponent ofmagnetization vector alongthetransport direction tobe
zero m = 0 for simplicity. Also, we set m = 0, since the analytical calculations become unwieldy
x y
otherwise. The dispersion relation resulted from Eq. (4) for electron-hole excitations is found to be of
theform:
m 2 m
= ζ τµ + m2 + k 2+ ∆ 2( zfs)2 + ∆ 2 1 ( zfs)2 , (5)
EFS s(cid:18)− fs r zfs | FS| | 0| µfs (cid:19) | 0| (cid:18) − µfs (cid:19)
where, the parameter ζ = 1 denotes the electron-like and hole-like excitations, while τ = 1 dis-
tinguishes the conduction an±d valence bands. We might expect several anomalous properties fr±om 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
3
exchange fieldasm = 0(see Ref. [8]), = ζ ( τµ + k )2+ ∆ 2. Themean-field conditions
z S s S 0
E − | | | |
are satisfied as long as ∆0 µfs. In this condiqtion, the exact form of superconducting wavevector of
≪
chargecarriers canbeacquired fromtheeigenstates k = µ2 m2 .
fs fs− zfs
The Hamiltonian Eq. (4) can be solved to obtain the eqlectron (hole) eigenstates for FS topological
insulator. The wavefunctions including a contribution of both electron-like and hole-like quasiparticles
areanalytically foundas:
eiβ 1
ψFeS =  eiθefisβee−iiθγfese−iϕ ei(kfxsx+kfysy), ψFhS =  eiβe−−iθfes−ei−θfiγshe−iϕ ei(−kfxsx+kfysy), (6)
− e−iγee−iϕ eiβe−iγhe−iϕ
   
   
wherewedefine
m ∆(k)
cosβ = EFS ; η = 1 ( zfs)2 , eiγe(h) = .
η ∆ − µ ∆(k)
0 fs
| | r | |
Notethat,thesolutionisallowedaslongastheZeemanfieldbeinglowerthanchemicalpotentialm
zfs
µ . ≤
fs
2.2 FS/F/FSJosephsonjunction
In what follows, we consider a line ferromagnetic junction of width L between two FS sections in the
coordinate rangesx < 0andx > LincontactwithTIsurfacestates. Theferromagnetic regionlengthis
assumedtobesmallerthanthesuperconducting coherence length. Themagnetizations oftwosupercon-
ducting semi-finite regions are taken to be equal and the same direction, see Fig. 1. Coupling between
electron and hole wave functions at the interface leads to scattering matrix, and it is necessary to solve
the system of equations at each interface. Welook for the energy spectrum for the ABSs,which can be
obtain from a nontrivial solution for the boundary conditions ΨL = Ψ at x = 0 and Ψ = ΨR at
FS F F FS
x = L, where the eigenvectors ψ can be found in Appendix A. Solving the system of equations con-
cerning to 8 electron-hole reflected and transmitted amplitudes coefficients results in a 8 8-scattering
matrix: ×
Ξ = 0; Ξ = te,th,a,b,c,d,te ,th . (7)
S L L R R
h i
The analytical expression of scattering matrix is introduced in Appendix A. The phase difference
∆ϕ = ϕ ϕ is introduced by assuming thatSthe phase of the left and right superconducting regions
R L
is ϕ and ϕ− , respectively. When determinant of vanishes, then the nontrivial solution leads to an
L R
analytical expression forABSintermofmacroscopiSc phasedifference ∆ϕ
1 Γ(∆ϕ)
ǫ(∆ϕ) = η∆ 1 , (8)
0
s2 − 2Ω
(cid:18) (cid:19)
wherewehavedefined
Γ(∆ϕ) = κ cos∆ϕ+κ sin2(kxL), Ω = κ +κ cos(2kxL).
1 2 f 3 4 f
The explicit form of κ (i = 1,2,3,4) is given in Appendix A. The analytical progress can be made in
i
the limit of thin and strong barrier, which the barrier strength parameter is then defined as Z = kxL.
f
Otherwise, the results of analytical calculations can be presented as a function of length of junction.
In short junction limit, the length of the junction is smaller than the superconducting coherence length
ξ = ~v /∆ .
F 0
We now analyze the supercurrent flowing through the Josephson junction. To calculate the nor-
malized Josephson current in the short junction case which carried mostly by the ABS, the standard
expression isintroduced:
π/2 ǫ(∆ϕ) dǫ(∆ϕ)
(∆ϕ) = dθ cosθ tanh , (9)
0 fs fs
I I 2K T d∆ϕ
Z−π/2 (cid:18) B (cid:19)
where = (e k W∆ )/(π~)isthenormalcurrentinasheetofTIofwidthW. Notethatthecritical
0 FS 0
I | |
current can be measured as = max( (∆ϕ)). In the framework of such model, the ferromagnetic
c 0
I I I
4
quasiparticlesincidenceanglemaybereal. Theconservationoftransversecomponentofthewavevectors
allows us tofind the propagation angle ofelectron or hole in the middle region, which isobtained to be
ofthefollowingform:
µ2 m2
θ = arcsin fs− zfs sinθ . (10)
f v µ2 m2 fs
u f − zf
u
t 
It is worth noting the Eq. (10) indicates that the chemical potential in FS region may be lower than its
magnitude inFregion(µ > µ ).
f fs
3 RESULTS AND DISCUSSION
3.1 Energy excitationandMajorana 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. Weassume that the Fermi
levelcontrolledbythechemicalpotentialµisclosetotheDiracpoint. Inthiscase,itisexpectedthesig-
natureofm < µ tobesignificant. InFig. 2,wedemonstratetheFS3DTIelectron-hole excitations.
zfs fs
Anetsuperconductinggap∆ isobtainedinDiracpoints(for k = k ,wherek isFermiwavevector)
0 fs F F
whenwesetm = 0. Increasing m uptoitspossible m|axim|um valueresults inthreeoutcomes: i)
zfs zfs
thesuperconducting excitations,whichisrenormalizedbyafactor ∆(k) 1 (m /µ )2,disappear
zfs fs
| | −
in hole branch (k < k ). It means that for the greater magnetizations, if we consent the supercon-
fs F
ductivity in FS 3|DT|I still exists, there is almost vanishing quantum statepfor 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
δ∆ ∆(k). The Andreev process, therefore, is believed to inconsiderably suppress. The signature
0
of the≪se v|alenc|e band excitations can be clearly shown in AR,and as wellin ABS,where the Majorana
modemayalsobeformedatthe3DTIF/FSinterface [7,17].
Asaverifiedresult,consideringthetopologicalinsulatorinterfacebetweentheferromagneticinsula-
torandconventional superconductor leadstotheappearance ofthechiralMajoranamodeasanAndreev
boundstate. Inotherwords,theMajoranamodeandAndreevreflectionarestronglyrelatedtoeachother.
The latter can be realized by the fact of looking for bound energies produced by the perfect AR, which
yieldsthefollowingsolution:
1 υ
ǫ˜(θ)= η∆ sgn(Λ)/ 1+Λ2 ; Λ= tan ln( 1) , (11)
0
2i υ
(cid:20) 2 (cid:21)
p
wherewedefine
υ = 4isinkxeLcosθ +2e−ikfxeLcosθ .
1(2) f M2(1)A1(2) A1(2)−B2(1)A1(2)
Theaxiliarly parameters , and are given inApendix B.Wehave checked numerically
2(1) 1(2) 2(1)
M A B
that sign ofΛischanged bysgn(m ). Thus, the sign ofAndreev resonance states maybechanged by
zf
reversingthedirectionofm ,anditcorresponds tothechirality ofMajoranamodeenergies. Asshown
zf
in Fig. 3, the slope of the curve of ǫ˜(θ) around ǫ˜(θ = 0) = 0 shows no change with the increase of
m /µ forfixedm ,whileitexhibitssignificantly decreasing behavior withtheincrease ofm /µ
zfs n zf zf n
for fixed m . The dispersion of Majorana modes along the interface (θ = π/2) decreases with the
zfs
increase of both magnetizations of FS and F regions. Note that, due to the presence of m it needs
zfs
to consider the Fermi level mismatch between normal and FS sections, i.e. µ = µ . Then, the above
n fs
contributions canbeconsiderable inAndreevprocess andresulting Josephson su6 percurrent.
3.2 Current-phase relation
InordertostudytheinfluenceofferromagneticorderinFSregiononthesupercurrentpassingthroughthe
FS3DTIJosephsonjunction,weproceedtofocusontheABS.Asausualresultinsimilarsystems,the4π
periodic gapless bound energy corresponding tochiral Majorana bound isobtained in normal incidence
θ = 0forfinitemagnitude ofm < µ . Insharp contrast tothe previous results [8], whichincrease
fs zfs s
ofthemagnetization magnitude ofFregionm causes toflattening theABS,theopening ofthegapin
zf
finiteangles ofincidence presented in∆ϕ = π issuppressed withtheincrease ofthe m . Also, the
zfs
minimumofgapedABSscomesthrough theh±igher angleofincidence. Thesearedemonstrated inFigs.
5
4(a) and (b). According to Majorana mode of Sec. 3.1, the gapless Andreev Majorana bound along the
interface decreases withtheincrease ofm ,e.g. itreachesavalue0.6ǫ˜ (∆ϕ)form = 0.8µ .
zfs max zfs s
However,thechirality ofMajorana ABS,whichisobserved inFigs. 3and4,canbeprovided bythe
factthattheFS3DTIHamiltoniansolutions maybeprotectedbytime-reversal symmetry. Itisnoweasy
toseethatthewavefunctions Eq. (6)ψe andψh areconnected bythetime-reversal operation as
FS FS
ψh (θ = 0) = τ iσ ψe∗ (θ = 0),
FS fs 0 y FS fs
where τ denotes a unit matrix in Nambu space. Moreover, ψe (θ = 0) and ψh (θ = 0) are the
0 FS fs FS fs
eigenstates ofthehelicitymatrixτ σ satisfying theeigenvalue equation
z x
e(h) e(h)
τ σ ψ (θ = 0) = ψ (θ = 0),
z x FS fs ± FS fs
and orthogonal to each other. They, therefore, immune to spin-independent potential scattering and,
hence,theferromagnetic exchange fieldm canplayanimportantroleintransport properties.
zfs
Inwhatfollows,weconsider theangle-averaged supercurrent originated fromAndreevbound states
in which the zero-temperature limit is assumed in the following plots. The 2π periodic current-phase
curve is found for different values of m , as shown in Fig. 4(c), with a shape far from sinusoidal.
zfs
One can say that almost abrupt crossover curve is seen from Josephson current originating from the
zero energy states (ZES), whereas the supercurrent peak does not appear in maximum phase difference
∆ϕ = π. The latter originates from the presence of ferromagnetic-proximity in topological insula-
tor. Consequently, in the presence of high magnetization of FS region (e.g. m = 0.8µ ) and low
zfs s
magnetization of F region (m < 0.2µ ), the Majorana bound energy may dominate the formation of
zf n
angle-averaged Josephson current. Thisisinagreement withtheMajorana bound energy, showninFig.
4(b). Finally, to see the effect of m on the critical current, weplot the width L/ξ of F region depen-
zfs
dence of critical current, where ξ is the superconducting coherence length. Increasing the m causes
zfs
to increase the maximum of supercurrent. The critical current exhibits oscillatory function in terms of
length of junction, where the oscillation-amplitude is considerable and decreases with the decrease of
m .
zfs
4 CONCLUSION
In summary, wehave investigated the influence of ferromagnetic and superconducting orders proximity
atthesametimeonthesurfacestateofatopologicalinsulator. Thesuperconductingtopologicalinsulator
electron-hole excitations in the presence of magnetization led to achieve qualitatively distinct transport
properties intheFS/F/FSJosephson junction. Ithasbeenshownthatformagneticorderm µ ,the
zfs s
spin-triplet component becomesdominant. Oneofkeyfindingsofthepresent workistheappe≈arance of
novelMajoranaboundmodeattheF/FSinterface,whichcanbecontrolledbythetuningofmagnetization
magnitude ofFSregion. The4π periodic gapless Andreev bound states, corresponding totheMajorana
bound energy, were created for finite magnitude of m µ . The current-phase relation curve has
zfs s
been found to be far from sinusoidal, and its critical valu≤e shows increasing with the increase of the
magnetizationofFSregion. Notethat,theseresultshavebeenobtainedinthecase,whenm ,m µ
zfs zf
andµ ∆ ,whichisrelevanttotheexperimental regime. ≤
0
≫
APPENDIX A: Josephson scattering matrix
Byintroducing thescattering coefficients inFregionwewritedownthetotalwavefunction insidetheF
region:
ΨF = eikfyy aψFe+eikfxex +bψFe−e−ikfxex+cψFh+e−ikfxhx+dψFh−eikfxhx ,
whererightandleftmoving(cid:16)electron andholespinors canbewrittenas: (cid:17)
T T
ψe+ = 1,αeiθf,0,0 , ψe− = 1, αe−iθf,0,0 ,
F F −
h i h i
T T
ψh+ = 0,0,1,αeiθf , ψh− = 0,0,1, αe−iθf .
F F −
Also, in this appendix, we chharacterize thie 8 8 matrihx in the form oif four 4 4 matrices which
is used to calculate the Andreev energy bound st×ates and coSrresponding Josephson×supercurrent in the
FS/F/FSjunction:
= S1 S2 ,
S 3 4
S S
(cid:16) (cid:17)
6
where
eiβ e−iβ 1 1 0 0 0 0
eiβe−iθfs e−iβeiθfs α−eiθf αe−−iθf 0 0 0 0
S1 =  e−−iθfse−iϕL eiθfse−iϕL − 0 0 ; S2 =  1 1 0 0 ;
e−iϕL − e−iϕL 0 0 α−eiθf αe−−iθf 0 0
   − 
   
0 0 eikfxeL e−ikfxeL
= 0 0 α−eiθfeikfxeL αe−−iθe−ikfxeL ;
S3  0 0 − 0 0 
0 0 0 0
 
 
0 0 eiβeikfxsL e−iβe−ikfxsL
0 0 eiβeiθfseikfxsL e−−iβe−iθfse−ikfxsL
S4 =  e−ikfxhL eikfxhL eiθfse−iϕReikfxsL −e−iθfse−iϕRe−ikfxsL .
 α−eiθfe−ikfxhL αe−−iθfeikfxhL − e−iϕ2eikfxsL e−iϕ2e−ikfxsL 
 − 
 
Thedefinitionofκ quantities inAndreevboundstateenergyaregivenby:
i
κ = 16α2cos2θ cos2θ ,
1 fs f
−
κ = 16αsinθ sinθ (1+α2) 16α2 +4cos2θ (1+α4)+8α2cos2θ ,
2 fs f fs f
−
κ = 4αsinθ sinθ (1+α2)+4α2+(1+α4)+2α2cos2θ cos2θ ,
3 f fs f fs
−
κ =4αsinθ sinθ (1+α2) 4α2sin2θ +2α2cos2θ (1+α4).
4 f fs fs f
− −
APPENDIX B: Majorana mode energy
Theparameters relatedtotheMajoranamodeenergyofEq. (11)readasfollowing:
1(2) = e(−)iθfs ( ) 1( 2(1) 1)eikfxeL (+) 2 2(1)e−ikfxeL ,
A − N M − − N M
h i
1(2) = 1( 1(2) 1)e−ikfxeL + 2 1(2)eikfxeL,
B −N M − N M
αeiθf (+)e(−)iθfs
(2) = ( )αe(−)iθf +e−iθ, = − .
1 1(2)
N − M 2αcosθ
f
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8
Figurecaptions
Figure1(coloronline)Sketchofthetopological insulator-based FS/F/FSJosephsonjunction. Themag-
netization vectorsinFandFSregionsareprependicular tothesurface oftopological insulator.
Figure 2 (color online) The ferromagnetic superconducting excitation spectra on the surface state of
3DTI for several values of m , calculated from Eq. (5). We set the net value of superconducting gap
zs
∆ = 0.5 eV (thisvalueofpairpotential istakenonlytomoreclarify thebehavior ofspectrainDirac
S
| |
point,although itdoesnotfurtherneedtouseitinourcalculations, sinceµ /∆ 1issupposed.
fs S
| |≫
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 = 0.2µ
zf n
andthedashedlinestom = 0.2µ .
zfs n
Figure 4(a), (b), (c) (color online) Plot of the Andreev bound state energy versus phase difference and
superconductor quasiparticle angleofincidenceinJosephsonFS/F/FSjunction. Theroleofthemagneti-
zationofFSregionm isdemonstrated. Curve(a)represents m = 0.1µ and(b)m = 0.8µ .
zfs zfs fs zfs fs
We set µ = 100∆ , µ = 2µ and m = 0.2µ and L/ξ = 0.05. Plot (c) represents the nor-
fs 0 f fs zf fs
malized Josephson supercurrent as a function of the phase difference with respect to varying m =
zfs
0.8,0.5,0.1µ when m = 0.4µ . The critical current J /J is plotted versus lenght of junction for
fs zf fs c 0
twomagnitudes ofm = 0.8µ (solidline)and0.2µ (dashedline).
zfs fs fs
9
Figure1:
1.2
∆ =0.5 eV mzs=0.01 eV
s m =0.1 eV
s zs
n 1 m =0.35 eV
o Electron branch zs
ati mzs=0.5 eV
t
ci0.8
x
E

I
T k k
F0.6 F F
S
δ∆
s Hole branch
0.4
−2 −1 0 1 2
|k|
Figure2:
1
m =0.1µ
gy mzfs=0.5µn
ner 0.5 mzfs=0.9µn
e zfs n
e m =0.9µ
d zf n
mo 0 mzf=1.5µn
a
n
a
or−0.5
aj
M
−1
−0.5 −0.3 −0.1 0.1 0.3 0.5
θe/π
Figure3:
10