Novel Majorana mode and magnetoresistance in ferromagnetic superconducting topological insulator

0.66 MB·

by  Hadi Goudarzi

Novel Majorana mode and magnetoresistance in ferromagnetic
superconducting topological insulator
7 H. Goudarzi∗, M.Khezerlou† and S. Asgarifar
1 DepartmentofPhysics,FacultyofScience,UrmiaUniversity,P.O.Box:165,Urmia,Iran
0
2

n
Abstract
a
J Among the potential applications of topological insulators, we investigate theoretically the ef-

8 fectofcoexistenceofproximity-inducedferromagnetismandsuperconductivityonthesurfacestates
2 of 3-dimensionaltopologicalinsulator, where the superconductingelectron-holeexcitationscan be
significantly affected by the magnetization of ferromagnetic order. We find that, Majorana mode
] energy,asaverifiedfeatureofTIF/Sstructure,alongtheinterfacesensitivelydependsonthemag-
l nitudeofmagnetizationm inFSregion,whileitsslopeinperpendicularincidencepresentssteep
l zfs
a andnochange. Sincethesuperconductinggapisrenormalizedbyafactorη(m ),henceAndreev
zfs
h reflection is more or less suppressed, and, in particular, resulting subgap tunneling conductanceis
- moresensitivetothemagnitudeofmagnetizationsinFSandFregions. Furthermore,aninteresting
s
e scenariohappensattheantiparallelconfigurationofmagnetizationsmzf andmzfsresultinginmag-
m netoresistanceinN/F/FSjunction,whichcanbecontrolledanddecreasedbytuningthemagnetization
magnitudeinFSregion.
.
t
a PACS:74.45.+c;85.75.-d;73.20.-r
m
Keywords: topological insulator;ferromagneticsuperconductivity; Andreevreflection;Majoranamode;
- tunneling conductance
d
n
o 1 INTRODUCTION
c
[
Topological insulators (TIs) represent new type of material which has emerged in the last few years as

1 oneofthemostactively research subjects incondensed matterphysics. Theyarecharacterized byafull
v insulating gap in the bulk and gapless edge or surface states, which are protected by the time-reversal
5 symmetry[1, 2, 3, 4]. RegardingBernevigandHughesprediction [3, 5],TIshavebeenexperimentally
7
observed with such properties that host bound states on their surface, e.g. in 3-dimensional topological
2
insulators (3DTI) Bi Te , Bi Se , Sb Te and Bi Sb alloy, and also in the CdTe/HgTe/CdTe
8 2 3 2 3 2 3 x 1−x
quantumwellheterostructure[6, 7, 8]. Thesestatesformaband-gapclosingDiracconeoneachsurface,
0
. and lead to a conducting state with properties unlike any other known electronic systems. In particular,
1 conformityoftheconductionandvalencebandstoeachotherinandaroundDiracpointsinthefirstBril-
0
louin zone, possessing an odd number of Dirac points, description of fermionic excitations as massless
7
two-dimensional chiral Dirac fermions, depending chirality on the spin of electron, having the signif-
1
icant electron-phonon scattering on the surface, owning very low room-temperature electron mobility
:
v are the peculiar properties of electronic structure of TIs. Interestingly, the charge carriers in the surface
i states can behave as massive Dirac fermions [9] due to its proximity to a ferromagnetic material, that
X
the vertical component of the magnetic vector potential may be proportional to the effective mass of
r
Diracfermion. Theexperimentally observedproximity-induced superconductivity onthesurfacestateis
a
anotherinteresting dynamicalfeatureoccuring in3DTI,seeRefs. [10, 11, 12].
More importantly, the coexistence of superconductivity and ferromagnetism as one of potential in-
terestsforspintronics andhighmagneticfieldapplications hasfirstlybeenpredictedbyFuldeandFerrel
[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 po-
larization 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 supercon-
ductor/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
1
isregained as soon astheferromagnetic exchange fieldexceeds ∆ /√2atzero temperature. Tobeem-
0
pirically, the ErRh B [21] has been discovered to be the first ferromagnetic superconductor, which
4 4
superconductivity is found to occur in a small temperature interval with adjusted ferromagnetic phase.
Also, superconductivity is detected in itinerant ferromagnetic UGe in a limited range of pressure and
2
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
processandresultingsubgapconductance weproceed,inthispaper,totheoretically studythedynamical
properties of Dirac-like charge carriers in the surface states of 3DTI under influence of both supercon-
ducting 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 anenergy gapattheFermilevelinthe3DTI.Itwillbeparticularly
interesting toinvestigate thetopological insulator superconducting electron-hole excitations inthepres-
enceofaexchangefield. WeassumethattheFermilevelisclosetotheDiracpoint,andtheferromagnet
has a magnetization M < µ. The chirality conservation of charge carriers on the surface states in the
| |
presence ofmagnetization (due toopening theband gap)allows touse afinitemagnitude of M . Inthe
absenceoftopologicalinsulator,thespin-splittingcausedbymagnetizationgivesrisetolimitin|gt|hemag-
nitude of M inaFSstructure. These excitations, therefore, are found to play acrucial role inAndreev
reflection|(A|R) 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
ofFShybridstructure. Wepresent,insection2,theexplicitsignatureofmagnetizationinlow-energyef-
fectiveDirac-Bogogliubov-de Gennes(DBdG)Hamiltonian. Theelectron(hole) quasiparticle dispersion
energyisanalyticallycalculated,whichseemstoexhibitqualitativelydistinctbehaviorinholeexcitations
(k < k )by varying themagnitude of magnetization. Byconsidering the magnetization is everless
fs F
th|anc|hemicalpotential inFSregion, thesuperconducting wavevectorandcorresponding eigenstates are
derivedanalytically. Section3isdevotedtounveiltheabovekeypointofFSenergyexcitation,Majorana
mode energy, Andreev process and resulting tunneling conductance in N/F/FS junction and respective
discussions. Inthelastsection, themaincharacteristics 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 [37] that is included the
effectiveexchangefieldMfollowsfrom:
1
H = − tρρ′cˆ†ρscˆρ′s+ 2 Uρρ′ss′nˆρsnˆρ′s′ + cˆ†ρs(σ·M)cˆρs′, (1)
′ ′ ′ ′
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 [38], 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)
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) =
2
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 [18]:
∆(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-Chandrasekharlimiting[19, 20]. 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 [39]. 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[9]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 − fs zfs | FS| | 0| µfs | 0| − µfs
(cid:18) r (cid:19) (cid:18) (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
exchange fieldas m = 0(see Ref. [9]), = 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
3
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 FSinterplay attheTI 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 prox-
imity effect. The wide topological insulator junction is taken along the x-axis with the FS region for
x > L,Fregionfor0 < x < LandNregionforx < 0. Thesuperconducting order parameter vanishes
identically in N and F regions, and we can neglect its spatial variation in the FS region close to the in-
terface. Themagnetization vectorsofbothsections istaken, ingeneral, m (i f,fs),whichcanbeat
zi
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 ǫ = k2 +m2 µ below the superconducting gap, transmitted (normal reflected)
F ± Ff zf − f
from the N region aqnd reflected (Andreev reflected) at the FS interface. Thus, the leftover 2e charge is
transferredintotheFSregionasaCooperpairatFermilevel. Atenergyexcitationabovethenormalized
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 ARcan 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
maybeaccordingly obtained from thefact ofconservation oftransverse wavevector under quasiparticle
scattering attheinterface:
µ sinθ
n
θ = arcsin , (7)
fs
 µ2 m2 
fs− zfs
q 
whereµ andθarethechemicalpotentialandincidenceangleinNregion,respectively. Asanimportant
n
point,theelectron(hole)angleofincidenceinallregionsmaybespantherangefrom0toπ/2aroundthe
normal axis. Regarding the Eqs. (7), the angle θ needs to be meaningful when the chemical potential
fs
of FS region takes a magnitude greater than its value in N region (µ > µ ). On the other hand, we
fs n
previously applied the condition m < µ , as an experimentally used manner to calculate the wave
zfs fs
functions Eq. (6).
Byintroducing thenormalrandAndreevr reflectioncoefficients andthescattering coefficients in
A
Fregion, thetotalwavefunction insidetheNandFregioncanbewrittenas:
ΨN = eiknyy ψNe+eiknxx+rψNe−e−iknxx+rAψNh−eiknxx ,
(cid:16) (cid:17)
ΨF = eikfyy aψFe+eikfxex +bψFe−e−ikfxex+cψFh+e−ikfxhx+dψFh−eikfxhx , (8)
(cid:16) (cid:17)
where the eigenvectors ψ 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
regionisdefinedasΨ = teψe +thψh . Finally,wefindthefollowinganalyticalexpressionsforthe
FS FS FS
reflectioncoefficients, thattheauxiliary quantities isdescribed inAppendix A:
r = teeiβ(2 1)+the−iβ(2 1) (isin(kxeL))+
1 2 f
M − M −
h i
teeiβ +the−iβ cos(kxeL) 1,
f
−
h i
4
r = teeiθfs(2 1) the−iθfs(2 1) (isin(kxeL))
A 2 1 f
M − − M − −
h i
teeiθfs the−iθfs cos(kxeL). (9)
f
−
h i
The reflection amplitudes measurements under the BTK formalism enables us to capture the tunneling
conductance through thejunction:
θc
G(eV)= G dθecosθe 1+ r 2 r 2 , (10)
0 A
| | −| |
Z0 (cid:16) (cid:17)
wherethecriticalangleofincidence θ isdetermineddepending onthedopingofFregion. Thequantity
c
G isarenormalization factorcorresponding totheballistic conductance ofnormalmetallicjunction.
0
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 supress. The signature
0
of the≪se v|alenc|e band excitations can be clearly shown in AR, where the Majorana mode may also be
formedatthe3DTIF/FSinterface [23, 25].
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) ,
0
2i υ
(cid:20) 2 (cid:21)
p
wherewedefine
υ = 4isinkxeLcosθ +2e−ikfxeLcosθ .
1(2) f 2(1) 1(2) 1(2) 2(1) 1(2)
M A A −B A
We have checked numerically that sign of Λ is changed by sgn(m ). Thus, the sign of Andreev res-
zf
onance states may be changed by reversing the direction of m , and it corresponds to the chirality of
zf
Majoranamodeenergies. AsshowninFig. 3,theslopeoftheenergycurvesof˜ǫ(θ)around˜ǫ(θ = 0) = 0
become steep and show nochange withtheincrease ofm /µ forfixedm ,while itexhibits signif-
zfs n zf
icantly decreasing behavior with the increase of m /µ for fixed m . The dispersion of Majorana
zf n zfs
modes along the interface (θ = π/2) decreases with the increase of both magnetizations of FS and F
regions. Note that, due to the presence of m it needs to consider the Fermi level mismatch between
zfs
normal and FS sections, i.e. µ = µ . Then, the above contributions can be considerable in Andreev
n fs
processandresulting subgaptunn6 eling conductance.
3.2 Tunneling conductance
Fromtheangle-resolvedAndreevandnormalreflectionprobabilitiesusingEq. (9),weseefromFig. 4(a)
the main contribution of AR belongs to the angle of incidence θ < 0.15π in zero bias. It, therefore, is
expectedtoachievethelowerzerobiasconductance, asshowninFig. 4(b)and(c). Furthermore,varying
m has no significant influence on AR in zero bias ǫ(eV) = 0 owing to the very small decrease of
zfs
the renormalized superconducting gap with the increase of the m , while the increasing m results
zfs zf
in more suppression of AR. The latter can be understood by the increase of band gap in Dirac point in
5
F region. The resulting normalized angle-averaged tunneling conductance curves are reported in Figs.
4(b)and(c)fortwoparallel andantiparallel configurations ofmagnetizations inFSandFregions. Zero
bias conductance peak disappears with the decrease of the m , and instead of it a high conductance
zf
peak appears in bias ǫ = η∆ . This result should be compared to that is obtained in Ref. [9, 24].
0
Interestingly, by increasing the m the magnitude of subgap bias ǫ/η∆ < 1, for which the new
zfs 0
peak takes place, is limited, as seen from Fig. 4(c). Thus, parameter η = 1 (m /µ )2 can be
zfs s
−
considered as a “bias-limitation coefficient”. These features have been obtained when the direction of
magnetizationsinFandFSregionsareattheparallelconfiguration. Thefundampentallydistinctscenarios
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 is
zf
invertedwe,indeed,meetwithaninverseenergygapinDiracpointof3DTIgivingrisetoenhancing the
conductance peakrespective biasenergy ǫ/∆ = η inlowvalues ofm . Onecanexpress thatthezero
0 zf
bias conductance originates from the chiral Majorana mode, which significantly depends on the m .
zf
Thechirality actually corresponds tothe sign ofm ,while themagnitudes ofzero biasconductance at
zf
the parallel and antiparallel configurations arethe same. Hence, the both deep and peak ofconductance
curvesinantiparallel casearesignificance beinginfluenced bytheinverted gapcausedbythe m .
zf
Remarkably, the importance of above findings can be featured by the capture of magneto−resistance
(MR)ofthetopologicalinsulatorjunction. Themagnetization(specialyinFSregion)dependenceofMR
is presented in Fig. 5, where we observe a considerable MR peak for extra values of m (e.g. 0.9µ
zf n
infigure). Importantly, increasing them weakenstheMRpeak,since,regardingthesuperconducting
zfs
excitations in Fig. 2, the AR is more or less suppressed in the presence of m and Fermi wavevec-
zfs
tor mismatch also causes to decrease the η∆ -bias conductance peak at the antiparallel configuration.
0
Accordingtotheconductance curves,thereisnoMRinzerobias.
4 CONCLUSION
Insummary, wehave investigated the influence offerromagnetic superconducting orders coexistence in
the surface state of topological insulator. The topological insulator superconducting electron-hole exci-
tations in the presence of magnetization have led to achieve qualitatively distinct transport properties in
tunneling N/F/FSjunction. Oneofkeyfindings ofthepresent workisthatthe resulting subgap conduc-
tancehasbeenfoundtobestronglysensitivetotheparallelorantiparallelconfigurationofmagnetization
directions inFSandFregions. Thus,thisfeature hasactually ledtopresent themagnetoresistance peak
for bias energy close to the renormalized superconducting gap ǫ(eV) = η∆ , which the bias limitation
0
coefficient η includes the magnetization ofFSregion m . Particularly, wehavefound thepresence of
zfs
Majorana modeattheF/FSinterface tobecontrolled bythetuning ofmagnetizations magnitude. How-
ever, these results have been obtained in the case of m ,m < µ and µ ∆ , which is relevant to
zfs zf 0
theexperimental 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
movingelectron andholespinorsinFandNregion:
T T T
ψe+ = 1,eiθ,0,0 , ψe− = 1, e−iθ,0,0 , ψh− = 0,0,1, e−iθ ,
N N − N −
h i h i h i
T T T T
ψe+ = 1,αeiθf,0,0 , ψe− = 1, αe−iθf,0,0 , ψh+ = 0,0,1,αeiθf , ψh− = 0,0,1, αe−iθf
F F − F F −
h i h i h i h i
where we define α = µf−mzf. By matching boundary conditions on Ψ and Ψ at x = 0 and Ψ
µf+mzf N F F
and ΨFS at x = L, theqreflection amplitudes are obtained. We introduce auxiliary quantities in Eq. (9)
as:
2 cosθ 2 cosθ
te = A2 , th = − A1 .
eiβ e−iβ eiβ e−iβ
1 2 2 1 1 2 2 1
B A −B A B A −B A
with
1(2) = e(−)iθfs ( ) 1( 2(1) 1)eikfxeL (+) 2 2(1)e−ikfxeL ,
A − N M − − N M
h −ikxeL ikxeL i
1(2) = 1( 1(2) 1)e f + 2 1(2)e f ,
B −N M − N M
αeiθf (+)e(−)iθfs
(2) = ( )αe(−)iθf +e−iθ, = − .
1 1(2)
N − M 2αcosθ
f
6
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∗[email protected] ;[email protected]
†[email protected]
8
Figurecaptions
Figure 1 (color online) Sketch of the topological insulator-based N/F/FS junction. The magnetization
vectorsinFandFSregionscanbeattheparallelorantiparallel configuration.
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), (d) (color online) (a) Probability of the normal and Andreev reflections as a func-
tion of electron incidence angle at the interface in zero bias ǫ(eV)/η∆ = 0 with m = 0.5µ and
0 zfs n
µ /µ = 1.5. The plots show the results for different values of m . (b) Normalized tunneling con-
fs n zf
ductance versus bias voltage eV and magnetization of F region. We set m = 0.5µ (c) Normalized
zfs n
tunneling conductance versus bias voltage and magnetization of FS region. We set m = 0.2µ (d)
zf n
The tunneling conductance as a function of bias voltage for two signs of m , corresponding to the
zf
parallel and antiparallel configurations in F and FS regions. The±solid lines correspond to +m and
zf
markerdashedlinescorrespond to m . Wesetm = 0.2µ .
zf zf n
−
Figure 5 (color online) The magnetoresistance spectra as function of bias voltage, where the influence
of m and m is indicated, separately. We have set µ /µ = 1.2 in the resulting conductance and
zfs zf fs n
magnetoresistance spectra.
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:
(a) 1
y
bilit0.8 Normal
a
b
o0.6 m =0.1µ
Pr zf n
n m =0.9µ
o0.4 zf n
ecti Andreev
efl0.2
R
0
0 0.1 0.2 0.3 0.4 0.5
θe/π
10