Influence of an embedded quantum dot on the Josephson effect in the topological superconducting junction with Majorana doublets

0.78 MB·English

by  Wei-Jiang Gong

Influence of an embedded quantum dot on the Josephson effect in the
topological superconducting junction with Majorana doublets
Wei-Jiang Gong, Zhen Gao, Wan-Fei Shan, and Guang-Yu Yi
∗
1. College of Sciences, Northeastern University, Shenyang 110819, China
(Dated: January 14, 2015)
One Majorana doublet can be realized at each end of the time-reversal-invariant Majorana
5 nanowires. We investigate the Josephson effect in the Majorana-doublet-presented junction mod-
1 ified by different inter-doublet coupling manners. It is found that when the Majorana doublets
0 couple indirectly via a non-magnetic quantum dot, only the normal Josephson effects occur, and
2 the fermion parity in the system just affects the current direction and amplitude. However, in the
odd-parity case, applying finite magnetic field on the quantum dot can induce the appearance of
n
thefractionalJosephsoneffect. Next,whenthedirectandindirectcouplingsbetweentheMajorana
a
doubletscoexist,nofractionalJosephsoneffecttakesplace,regardlessoffinitemagneticfieldonthe
J
quantum dot. Instead, the π-period current has an opportunity to appear in some special cases.
3
All the results are clarified by analyzing the influence of the fermion occupation in the quantum
1
dot on theparity conservation in thewhole system. Weascertain that this work will be helpfulfor

describing thedot-assisted Josephson effect between theMajorana doublets.
]
l
l
a PACSnumbers: 03.75.Lm,74.45.+c,74.78.Na,73.21.-b
h
-
s I. INTRODUCTION Kramersdoubletandareprotectedby time-reversal
e
symmetry. It is certain that the Majorana doublet
m
can induce some new and interesting transport be-
. Topological superconductor (TS) has received
t haviors. Some groups have reported that in the
a considerable experimental and theoretical atten-
Josephson junction formed by the Majorana dou-
m tions because Majorana zero-energy modes appear
blet, the Josephson currents show different periods
- at the ends of the one-dimensional TS which can in the cases of different FPs.27 However, for com-
d potentially be used for decoherence-free quantum
n computation.1–4 In comparison with the conven- pletely describing the transport properties of the
o Majorana doublet, any new proposals are desirable.
tional superconductor, the TS system shows new
c
[ and interesting properties.5 For instance, in the In this work, we aim to investigate the influence
proximity-coupled semiconductor-TS devices, the of an embedded quantum dot (QD) on the current
2 Majoranazeromodesinducethezero-biasanomaly.6
v properties in the Josephsonjunction contributed by
A more compelling TS signature is the unusual
9 the Majorana doublets. Our motivation is based
2 Josephsoncurrent-phaserelation. Namely,whenthe on the following two aspects. First, QD is able to
5 normals-wavesuperconductornano-wireisreplaced accommodate an electron and the average electron
2 by a TS wire with the Majorana zero modes, the
occupation in one QD can be changed via shifting
0 current-phase relation will be modified to be I
J the QD level. Thus, when one QD is introduced
1. sinφ2 and the period of the Josephson current vs∼φ in the TS junction, the FP can be re-regulated and
0 will be 4π (φ is the superconducting phase differ- the fractional Josephson current can be modified.
5 ence). This is the so-calledthe fractional Josephson
Moreover, some special QD geometries can induce
1 effect.7–9 Such a result can be understood in terms
the typical quantum interference mechanisms, e.g.,
:
v offermionparity (FP). If the FP is preserved,there theFanointerference,28 whicharecertaintoplayan
Xi will be a protected crossing of the Majorana bound important role in adjusting the fractional Joseph-
states at φ = π with perfect population inversion.
son effect. Second, one QD can mimic a quantum
r Asaresult,thesystemcannotremainintheground
a impurity in the practical system, which is able to
state as φ evolves from 0 to 2π adiabatically.10
provide some useful information for relevantexperi-
Recently, a new class of topological superconduc- ments. Our calculations show that when the Majo-
tors with time-reversal symmetry, referred to as a rana doublets couple indirectly via a non-magnetic
DIII symmetry-class superconductor and classified QD, only the normal Josephson effects take place,
by the Z topological invariant,11–15 has attracted irrelevant to the change of the FP. When a finite
2
rapidly growing efforts.16–20 Differently from chiral magnetic field is applied on the QD, the fractional
superconductors, in DIII-class superconductors, the Josephson effect comes into being in the odd-FP
zeromodescomeinpairsduetoKramers’stheorem. case. On the other hand, when the direct and in-
Uptonow,manyschemeshavebeenproposedtore- direct couplings between the Majoranadoublets co-
alizeZ time-reversal-invariantMajoranananowires exist, no fractional Josepshon effect occurs despite
2
using the proximity effects of d-wave, p-wave, s - the presence of magnetic field on the QD, and the
±
wave, or conventional s-wave superconductors.21–26 periodsoftheJosephsoncurrentshaveopportunities
It was shown that at each end of such a nanowire to presentthe π-to-2π transitionwhen the QD level
arelocalizedtwoMajoranaboundstatesthatforma isshifted. Theresultsinthisworkwillbehelpfulfor
2
Majorana nanowires due to the presence of an em-
beddedQD(oraquantumimpurity). Itsexpression
can be given by
HTI = Xε0d†σdσ+R(d†↑d↓+d†↓d↑)+Und↑nd↓
σ
+XVLd†σcL,Nσ+XVRd†σcR,1σ+H.c(.4.)

σ σ
FIG. 1: The Josephson junction formed by the direct
couplingbetween theMajorana doubletsand theirindi- Hered†σ anddσ aretheelectroncreationandannihi-
rect coupling via a QD. lation operators in the QD, and ε0 is the QD level.
Rdenotesthe strengthofaneffectivemagneticfield
appliedontheQD,andU denotestheintradotelec-
describingtheQD-assistedJosephsoneffectbetween tron interaction with n = d d . In addition, V
dσ †σ σ α
the Majorana doublets. is coupling coefficient between the QD and the α-th
Majorana nanowire.
In order to discuss the Josephson effect in this
II. MODEL
junction, we have to deduce an effective Hamilto-
nian that reflects the direct and indirect couplings
TheJosephsonjunctionthatweconsiderisformed
between the Majorana doublets. For this purpose,
by the direct couplings between the Majorana
we define the Majorana operators
nanowires and their indirect couplings via a QD, as
illustrated in Fig.1. The particle tunneling process
inthisjunctioncanbedescribedbyHamiltonianHT γα1 =X[µ(α1j)cαj +µ(α1j)∗c†αj],
with j
H = H +H +H . (1) γ˜α1 =X[µ˜(α1j)c˜αj +µ˜(α1j)∗c˜†αj],
T X αM T0 TI j
α=L,R
H denotes the particle motion in the two Ma- γα2 =X[µ(α2j)cLj +µ(α2j)∗c†αj],
αM
j
jorana nanowires. With the proximity-induced p-
waveands-wavesuperconductingpairs,theeffective γ˜α2 =X[µ˜(α2j)c˜αj +µ˜(α2j)∗c˜†αj], (5)
tight-binding Hamiltonian in the α-th nanowirecan j
be written as27
where c = s c + s c is the renormalized
HαM electronαojperat1orα,ajt↑ the2j-αt,hj↓site in the α-th site
=Xjσ tαjc†α,jσcα,j+1σ +Xj (tα,soc†α,j↑cα,j+1↓+H.c.) wweithcacn˜αsjol=veTthceαjeTle−ct1r.onUospinegrattohresainbotveermfosromfuMlaas-,
jorana and nonzero-energy quasiparticle operators.
+X(∆αpc†α,j c†α,j+1 +∆∗αpc†α,j c†α,j+1 +H.c.) Reexpressing the quasiparticles in terms of electron
↑ ↑ ↓ ↓
j
operators, we can interpret c , c˜ , c , and c˜
LN LN R1 R1
+X(∆αsc†α,j c†α,j +H.c.)−µαXnjσ. (2) by
↑ ↓
j jσ
ac†αti,ojσnaannddcaαn,jnσih(iσlat=io↑n,↓oporer±at1o)rsarfeorthteheelje-ctthrosnitecrien- cLN = µ(L2N)∗γL2−Xj aLjcLj−Xj b∗Ljc†Lj,
the α-th nanowire. tjα is the inter-site hopping en- c˜LN = µ˜(L2N)∗γ˜L2−Xa˜Ljc˜Lj−X˜b∗Ljc˜†Lj,
ergy and t represents the strength of spin-orbit
α,so j j
coupling. ∆ and ∆ denote the energies of the
p-waveandsα-wpavesupαersconductingpairings,respec- cR1 = µ(R11)∗γR1−XaRjcRj −Xb∗Rjc†Rj,
tively. µ is the chemical potential in the α-th j j
α
nanowire. Note that the hopping coefficients and c˜R1 = µ˜(R11)∗γ˜R1−Xa˜Rjc˜Rj −X˜b∗Rjc˜†Rj, (6)
thechemicalpotentialaregenericallyreonormalized
j j
by the proximity effect. The second term H de-
T0
notesthedirectcouplingbetweenthe twoMajorana in which the normalization factor has been ne-
nanowires, which can be expressed as glected. Besides,a ,a˜ andb ,˜b areexpansion
αj αj αj αj
coefficients,originatedfromthequasiparticleopera-
HT0 =XΥc†L,NσcR,1σ+H.c., (3) tors other than the corresponding Majorana mode.
σ Substituting Eq.(6) into the expression of H , we
T
whereΥisthedirectcouplingcoefficient. Next,H can obtain the low-energy effective Hamiltonian of
TI
is to express the indirect coupling between the two H in the case of infinitely-long nanowires,which is
T
3
divided into two parts. The first part is the relations in Eq.(6), we can get the relationship
that aαja˜αj = aαja˜αj and bαj˜bαj = bαj˜bαj e2iφα.
(0) =iΓ cosφ(γ γ γ˜ γ˜ )+ε d d Accordingly, (|1) can b|e written as | |
HT 0 2 L2 R1− L2 R1 s †s s HT
+Unsns˜−iWLe−iφ/2d†sγL2+iWLe−iφ/2d†s˜γ˜L2 HT(1) = iΓ1LsinφγL2γ˜L2−iΓ1RsinφγR1γ˜R1
+εs˜d†s˜ds˜+WRd†sγR1+WRd†s˜γ˜R1+H.c.. (7) +(Γ2Le−iφ+Γ2R)d†sd†s˜+H.c., (10)
The relevant parameters here are defined as fol-
lows: Γ0 = 2Υ|µ(L2N)µ(R11)|, WL = VL|µ(L2N)|, and ianndwhΓich=Γ1L1V=2Υ2|µ.(L2UN)p|2|tGoRn|,owΓ1,Rw=e hΥa2v|eµ(Ro1b1)|t2a|iGnLe|d,
WR = VR|µ(R11)| in which µ(L2N) = i|µ(L2N)|eiφL/2, thelow2α-ener2gyαe|ffGeαc|tiveHamiltonianofsuchastruc-
µ˜(2) = iµ(2) eiφL/2, µ(1) = µ(1) eiφR/2, and ture.
LN − | LN| R1 | R1|
µ˜(1) = µ(1) eiφR/2 (It is reasonable to suppose The phase difference between the two Majorana
R1 | R1|
µ(2) = µ˜(2) and µ(1) = µ˜(1) ). Besides, in the wires will drive finite Josephson current through
| LN| | LN| | R1| | R1| them, which can be directly evaluated by the fol-
above formula d†s = (s∗1d† +s∗2d†), d†s˜ = (−s∗2d† + lowing formula
↑ ↓ ↑
s∗1d†), and εs/s˜ = ε0 ± R with ns = d†sds and
↓ 2e ∂
ns˜F=ordt†s˜hdes˜.secondpart,whenthehighest-orderterms IJ = ~ h ∂HφTi (11)
are neglected, it can be approximated as
with being the thermal average. It is certain
h···i
HT(1) =−Υµ(L2N)γL2(XaRjcRj +Xb∗Rjc†Rj) tthhaetdsiaoglvoinnaglitzhaetioJnosoefphson. current is dependent on
j j HT
In the following, we try to diagonalize the Hamil-
(1)
−ΥµR1γR1(XaLjcLj +Xb∗Ljc†Lj) tonian. First, by defining γ1 = √12(γL2+γ˜R1) and
j j
γ = 1 (γ +γ˜ )withγ˜ = γ 1,wereexpress
−Υµ˜(L2N)γ˜L2(Xa˜Rjc˜Rj +X˜b∗Rjc˜†Rj) H2T as√2 R1 L2 j T jT−
j j
−Υµ˜(R11)γ˜R1(Xa˜Ljc˜Lj +X˜b∗Ljc˜†Lj) HT =i(Γ0cosφ2 +Γ1sinφ)γ1γ2+εsd†sds
j j
φ
−XVαd†s(Xaαjcαj +Xb∗αjc†αj) −i(Γ0cos 2 −Γ1sinφ)γ˜1γ˜2+εs˜d†s˜ds˜
−Xαα Vαds†˜(Xjj a˜αjc˜αj +Xjj ˜b∗αjc˜†αj)+H.c.. (8) ++U√1n2s(n−s˜i+WL(Γe2−Liφe/−2idφ†s++ΓW2RR)dds†˜†s)dγ†s˜1
Iptressheonutldinbtehenoqtueadnttuhmatwsiinrcees,tthheese-lwecatvreonpsaicrαinagnids +√12(iWLe−iφ/2d†s˜+WRd†s)γ2
cc˜αesswlielladfosrtmoaanCeoffoepcetirvpeacioruapnlidngcobnedtweneseen.MThaijsorparnoa- +√12(iWLe−iφ/2d†s˜−WRd†s)γ˜1
zero modes localized at the same end and the finite
1
coupling between the Kramers doublet in the QD. +√2(iWLe−iφ/2d†s+WRds†˜)γ˜2+H.c., (12)
Therefore, up to the second-order perturbation in
the tunneling process, we can express (1) as
HT where Γ1α is supposed to be Γ1. Next, T can be
H
expressed in the normal fermion representation by
1
HT(1) = 2(Υ2µ(L2N)µ˜(L2N)γL2γ˜L2+VR2d†sds†˜)GR is(uf˜pposif˜n)g, γ˜γ1 ==((ff˜++ff˜†)),wγh2er=e fi(f,†f˜−afn)danfd, fγ˜˜1ar=e
+21(Υ2µ(R11)µ˜(R11)γR1γ˜R1+VL2d†sds†˜)GL+H.c.,(9) the†f−ermion2ic creation†and annih†ilat†ion operators.
Accordingly,the matrix formof canbe deduced
T
H
onthebasisof n n n n wheren =f f andn =
wPhjebr∗αej˜bG∗ααjR d=τhTPτc†αjja(ατj)a˜c˜α†αjj(R0)diτhTτwciαtjh(τ)c˜Rαjd(τ0h)·i··+i sf˜t†af˜t.esN,ootnelythtaht|einsFtPs˜heifssytf˜shiteemgowodithqfuMaanjtou†rmannaubmobuf˜enrd.
being a time-ordered integral. In the case
Thus, we should build the Fock state according to
of uniform superconducting pairings in the
the FP. First, in the case of the even FP, the Fock
Majorana nanowires, can be further de-
α
duced as Gα = PjGaαja˜αjPk ξα2k(∆∆α∗αss,∆αp) − sat3a0te10c1an+bae4w1r0i0tt1en+aas5|0Ψ1e1i0=+aa16|01000100i++aa27|01011010i++
ePigjebn∗α-ej˜nb∗αejrgPykofξα2tkh(e∆∆iαsαsos,∆laαtpe)dsiunpewrchoincdhucξtαokr.27isWtihthe ba8e||1w1r1it1tiie.nAass|a resiult, |the miatrix| formi of H|T(e) cian
4
Λ 0 i i
 −0 Λ −A −i A −A iA D D 
B B −B B D D
ε Ω 0 0 0 i i
 ∗ ∗ s˜ 
 −iA Bi −0 ε Ω 0 0 B − A 
HT(e) = A∗∗ − B∗∗ 0 s−0 εs˜+Ω 0 Bi −iA , (13)
 −iA −iB 0 0 0 ε +Ω − B − A 
− A∗ − B∗ s B A 
 i i ε +ε +U Λ 0 
 D∗ D∗ − B∗ B∗ B∗ B∗ s s˜ − 
 i i 0 ε +ε +U +Λ 
∗ ∗ ∗ ∗ ∗ ∗ s s˜
D D A −A A A
where = (W eiφ/2 W )/√2, = (W eiφ/2 + b 0010 +b 0100 +b 1000 +b 0111 +b 1011 +
L R L 2 3 4 5 6
A − B | i | i | i | i | i
WR)/√2, D = Γ2Leiφ + Γ2R, Λ = 2Γ0cosφ2, and b7|1101i+b8|1110iandthe matrixofHT(o) takesthe
Ω = 2Γ1sinφ. Next, for the case of the even FP, form as
the Fock state can be written as Ψ = b 0001 +
o 1
| i | i
Ω 0 i i
 −0 Ω B −i B −A iA D D 
B B A A D D
ε Λ 0 0 0 i i
 ∗ ∗ s˜ 
 iB Bi −0 ε Λ 0 0 A − A 
HT(o) = B∗∗ − B∗∗ 0 s−0 εs˜+Λ 0 −iA −iA . (14)
 −iA Ai 0 0 0 ε +Λ − B − B 
− A∗ − A∗ s B −B 
 i i ε +ε +U Ω 0 
 D∗ D∗ − A∗ −A∗ B∗ B∗ s s˜ − 
 i i 0 ε +ε +U +Ω 
∗ ∗ ∗ ∗ ∗ ∗ s s˜
D D A −A B −B
Fortheextremecaseofstrongmagneticfieldlimit, III. NUMERICAL RESULTS AND
ifE isinthefinite-energyregion,E willbeempty, DISCUSSIONS
s˜ s
andthenonlyonelevelcontributestotheJosephson
effects,respectively. Accordingly,insuchacase,the Following the derivation in the above section, we
matrixes of (e) and (o) will be halved, i.e., next perform the numerical calculation to discuss
HT HT
the detailed properties of the Josephson current
Λ 0 through such a system. As a typical case, only the
(e)  −0 Λ −A −A  zero-temperaturelimitisconsidered. Beforecalcula-
= B −B , (15)
HT  ∗ ∗ εs˜ Ω 0  tion, wewouldliketo reviewthe Josephsoneffectin
−−AA∗ −BB∗ −0 εs˜+Ω  thecaseofVα =0. Insuchacase,HT =(Γ0cosφ2+
Γ sinφ)(2n 1)+(Γ cosφ Γ sinφ)(2n 1),
1 f − 0 2 − 1 f˜−
and and n n are the eigenstates of . The two
| f f˜if HT
even-FP eigenstates are 00 and 11 , and their
f f
Ω 0 | i | i
(o) = −0 Ω BB −AA . (16) Ccoornretrsaproinlyd,intgheGoSdedn-FerPgieesigaernestEatG(eeSs) =are∓21Γ00coasnφ2d.
HT  ∗ ∗ εs˜ Λ 0  | if
−BA∗ AB∗ −0 εs˜+Λ  a|0s1ciofnwcliuthdetdheinGtSheepnreervgiieosusEwG(ooS)rk=s,±272tΓh1esifnraφc.tiJounsatl
Josephson effect occurs in the situation of even FP,
Based on the above analysis, we can understand
otherwise only the normal Josephon effect can be
that at the zero-temperature limit, the Josephson
observed.
current in this structure is dependent on the FP,
In Fig.2 we suppose Γ = Γ = 0 and choose
i.e., 0 1
W =0.25andΓ =0.05toinvestigatetheJoseph-
α 2α
son effect in the case where the Majorana doublets
2e ∂ (e/o) 2e∂E(e/o)
I(e/o) = h HT i = GS . (17) couple indirectly to each other via a QD. The re-
J ~ ∂φ ~ ∂φ
sults are shown in Figs.2-3: Fig.2 corresponds to
the noninteracting results, and Fig.3 describes the
E(e/o)aretheground-state(GS)energiesintheeven influences of the intradot Coulomb interaction on
GS
and odd FP cases, respectively. the Josephson effects in different FPs in the case of
5
-1(a) 0.2 1(b) 0.1 01(a) 0.2 0(b) 0.1
e0 0110 (c) 1 f2/p 3 4 -00.2e0-1010 (d) 1 f2/p 3 4 -000.2.1 e0---3210 2 4 -00.2 e0--210 1 2 3 4 -00.1
0 0 00.2 0 0 0 01(c) 0.2 0(d) 0.1
e -110 (e) 1 f2/p 3 4 0-0.2.2e -110 (f) 1 f2/p 3 4 -00..12 e0---3210 2 4 -00.2 e0--210 1 2 3 4 -00.1
e0 0 0 e0 0 0 01(e) 0.2 0(f) 0.05
-110 (g) 1 2 3 4 0-0.2.2 -110 (h) 1 f2/p 3 4 -00.0.15 e0---3210 2 4 -00.2 e0--210 1 2 3 4 -00.05
e0-100 1 f 2/p 3 4 -00.2e0-010 1 f 2/p 3 4 -00.05 e0---32101 (g) -000.2.2 e0--210 (h) -000.1.1
0 2 4 0 1 2 3 4
f /p f /p
FIG. 2: Josephson current spectra in the case where
the Majorana doublets couple indirectly via a QD. The
FIG. 3: Josephson current in the case where the Majo-
structural parameters are taken to be Wα = 0.25 and ranadoubletscoupleindirectlyviaafinite-CoulombQD.
Γ2α = 0.05. The left and right columns correspond to TheCoulombstrengthisU =2.0andtheothersarethe
the even-FP and odd-FP results, respectively. (a)-(b)
sameasthoseinFig.2. Theleftandrightcolumnscorre-
The case of the non-magnetic QD. (c)-(d) The case of
spond to the even-FP and odd-FP results, respectively.
finite magnetic field on the QD with R = 0.1. (e)-(f)
(a)-(b) The case of the non-magnetic QD. (c)-(d) The
R=0.3. (g)-(h) R=0.5. case of finite magnetic field on the QD with R = 0.1.
(e)-(f) R=0.3. (g)-(h) R=0.5.
U = 2.0. First, in Fig.2(a)-(b) we find that when a
the characteristics of QD. In Fig.3 we consider the
non-magneticQDispresented,itinducesthe occur-
case of finite intradot Coulomb interaction and in-
renceofthenormalJosephsoneffectsandthedepar-
vestigate the effect of the magnetic QD on the
tureofε fromzeroweakensthecurrentamplitudes,
0
Josephson currents in the finite-Coulomb case. In
irrelevant to the FP difference. Also, the FP plays
Fig.3(a)-(b) we first find that in the even-FP case
an importantrole in affecting the Josephsoneffects.
with a non-magnetic QD, the Coulomb interaction
Concretely, the Josephson currents in different FPs
benefits the Josephson effect, since in the region of
flow in the opposite directions for the same φ, and
the amplitude of I(o) is about one half of that of −2.5 < ε0 < 0.5 the current amplitude is relatively
J robustandweaklydependentonthe shiftofthe QD
I(e) and when ε >0.5 I(o) gets close to zero. Be-
J | 0| J level. In contrast, for the odd-FP case, the intradot
sides,atthe pointsofφ=(2n 1)π,inthe even-FP
Coulomb interaction only moves the current max-
−
case the discontinuous change of the Josephsoncur-
imum to the point of ε = 1.0, but it does not
0
rentismorewell-definedcomparedwiththeodd-FP −
vary the current oscillation manner compared with
case. Next, when finite magnetic field is applied on
the noninteracting case. Hence, the Coulomb in-
the QD, the even-FP Josephsoncurrentshows little
teraction only adjusts the effect of the QD level on
change except that its amplitude becomes less de-
the Josephson effects but does not modify the cur-
pendentontheQD-levelshift. However,intheodd-
rent oscillation manner with the change of φ. Next,
FP case, the Josephson current changes completely.
Fig.3(c)-(h) show that regardless of the FP differ-
It can be clearly found that with the strengthening
ence, the current amplitudes are suppressed by the
ofthe magnetic field,the originalcurrentoscillation
application of finite magnetic field on the QD. In
is suppressed. Especially in the vicinity of φ=4nπ,
the even-FPcase,the currentamplitude aroundthe
thecurrentamplitudetendstodisappearatthecase
point of ε = 1.0 undergoes a relatively-apparent
0
of R = 0.5. Thus, it is certain that in the case of −
suppression. For the odd-FP case, except the sup-
odd FP, a nonzero field on the QD can induce the
pression of the current amplitude, the fractional
occurrence of the fractional Josephson current. In
Josephson effect becomes weak but can still be ob-
addition, the increase of R enhances the current os-
served.
cillation around the points of φ = (2n 1)π when
Accordingtotheresultsintheaboveparagraphes,
−
ε departs from = 0. Up to now, we can conclude
0 when the Majorana doublets couple indirectly via
6
that when the Majorana doublets are coupled by a
a magnetic QD, the fractional Josephson effect has
magneticQD,the fractionalJosephsoneffecthasan
an opportunity to come into being in the odd-FP
opportunitytotakeplace,butitisdifferentfromthe
case. Otherwise, only the normal Josephson effect
casewhere the Majoranadoublets couple directly.27
can be observed. In order to explain the results in
Figs.2-3,wewouldliketocomparethecaseofΥ=0
6
Coulomb interaction is a key factor to influence and V = 0 with the case of Υ = 0 and V = 0.
α α
6
6
In the former case, the Josephson effects are only 1(a) 1 2 (c) 0.5
determined by the FP of state |nfnf˜if. Namely, 0.5 0.5 1
when the system is located at the states 00 or 0
f
11 f, the fractional Josephson effect occur|s. Hi ow- e0-0.5 0e0 0 0
| i -1
ever, when the Majorana doublets couple indirectly -0.5 -1
-1.5
via a QD, the Fock space defined by n n just
becomes a subspace of the Fock spac|e fforfm˜ifed by -20 1 f2/p 3 4 -1 -20 1 f2/p 3 4 -0.5
|nsns˜nfnf˜i. Since both Γ0 and Γ1 are equalto zero 1(b) 1 2 (d) 0.6
in such a case, the nonzero contributions of 00 0.5 0.4
and 11 to the Josephson effect are dependen|t oinf 0 0.5 1 0.2
f
their|finiite indirect couplings, i.e., their simultane- e0-0-.51 0e0 0 -00.2
ous couplings to the other states. Thus, the man- -0.5 -1 -0.4
-1.5
nperorspeorfttiehseoinfdtihreecJtocsoeupphlsionngseiffneecvtist.abFloyrriengsutalantcee,thine -20 1 f2/p 3 4 -1 -20 1 f2/p 3 4 -0.6
Eq.(13) we find that in the even-FP case, the states
that include 00 f and 11 f couple indirectly in the FIG. 4: Josephson current spectra in the case where
| i | i
way of left-right symmetry. This leads to the equal the direct and indirect couplings between theMajorana
contributions of 00 f and 11 f to the Josepshon doubletsco-exist. TherelevantparametersareΓ0 =0.5,
effect, and then |thei norma|l Jiosephon effect takes Γ1 = 0.1, Wα = 0.25, and Γ2α = 0.05. (a)-(b) The
place,independentofthepresenceofmagneticfield. even-FPcasewithnon-magneticandmagneticQDswith
On the other hand, in the odd-FP case, Eq.(13) R = 0.3. (c)-(d) The odd-FP results with R = 0 and
R=0.3.
shows that the coupling between the states that in-
clude 00 and 11 is left-rightasymmetricunless
f f
| i | i
ε = ε . Consequently, if finite R is considered,
|0s0i6f ans˜d |11if make different contributions to the einnccreesatshesetπo-tεo0-2=π t−ra1n.s0i,titohnewciutrhretnhtepdeisraiopdpeeaxrpaenrcie-
Josephson effect, which gives rise to the fractional
of the current around the points of φ = 2nπ. Next,
Josephson effect. Also, note that in odd-FP case,
in the regionof 1.0<ε <1.0, the Josephsoncur-
the left-right asymmetric coupling manner weakens − 0
rent varies in 2π period with its maximum in the
the quantum coherence and suppresses the current
vicinity of ε = 0. When ε further increases from
amplitudetosomedegree. WhentheCoulombinter- 0 0
1.0, the Josephson currentrecoversthe π-period os-
action is taken into account, the QD level splits. In
cillation gradually. Fig.4(d) presents the effect of
the even-FP case, the picture in the noninteracting
the magnetic field on the QD in odd-FP case. It
case can be doubled since the left-right symmetric
seemsthatinsuchacase,themagneticfieldcannot
coupling manner. Nevertheless, in the odd-FP case,
induce the fractional Josephson effect, but it tends
only in the case of electron-hole symmetry, the left-
to enhance the current amplitude in the region of
right symmetric coupling manner can be satisfied.
1.0 < ε < 1.0, which is exactly opposite to the
Therefore,the maximalJosephsoncurrentoccursat − 0
case of Γ =0.
the point of ε = 1.0 in the case of U =2.0. 0
0 − Following the above result, we present the influ-
We next proceed to pay attention to the Joseph- ence of the magnetic field on the finite-Coulomb
son effect in the case where the direct and indirect results, as displayed in Fig.5. Here, the Coulomb
couplings between the Majorana doublets coexist. strength is also taken to be U =2.0. First, Fig.5(a)
The results are shown in Figs.4-5 where Γ is taken shows the even-FP result with the non-magnetic
0
to be 0.5 and Γ = 0.1. The noninteracting results QD. We can find that in such a case, the current
1
are presented in Fig.4 and Fig.5 describes the case minimum is shifted to the position of ε 0.25.
0
≈ −
of U = 2.0. In Fig.4, we find that for any ε , the Besides,theCoulombinteractionefficientlyweakens
0
two kinds of Josephson currents show dissimilar os- the Josephson effect, since increasing ε from 2.0
0
−
cillations with the adjustment of superconducting beginstoeliminatethecurrentamplitudegradually.
phase difference. In the even-FP case with R = 0, NextwhenfinitemagneticfieldisappliedontheQD
when ε gets approximately close to 0.25, the am- with R=0.3, it further suppresses the minimum of
0
plitudeofthe Josephsoncurrentisdecreased,other- the Josephsoncurrent,similarto the noninteracting
wise,theJosephsoneffectwillbeenhancedandthen case [See Fig.5(b)]. The odd-FP results are shown
holds. However, the current period always keeps to in Fig.5(c)-(d), where the magnetic field strength is
be2π,asshowninFig.4(a). Next,inFig.4(b)where taken to be zero and 0.3, respectively. In Fig.5(c),
R=0.3, we can see that the only effect of the mag- we see that different from the noninteracting result,
netic field in the QD is to further suppress the min- the 2π-period oscillation of the current occurs from
imum of the Josephson current. Such a result is ε = 3.0. In the region of 3.0 < ε < 1.0, the
0 0
− −
exactlysimilartothe resultofΓ =0. Onthe other Josephsoncurrentvariesin2π periodwithitsmaxi-
0
hand,fortheodd-FPcase,inFig.4(c),itshowsthat muminthe vicinity ofε = 1.0. Besides,itcanbe
0
−
in the region of ε < 1.0, the Josephson current notedthattheCoulombinteractionenhancetheam-
1
−
seems to oscillate in π period. When the QD level plitudeoftheJosephsoncurrent,incomparisonwith
7
1(a) 1 2 (c) (a) (b)
0.6
1
0 0.5 0 00..24 e%s e%s
0-1 0 0-1 0
e e
-2 -0.2
-2 -0.5 -3 -0.4 fp/ fp/
-0.6
-3 -1 -4 (c) (d)
0 1 2 3 4 0 1 2 3 4
f/p f/p
1(b) 1 2 (d) e%s e%s
1
0.5
0 0.5
0
0-1 0 0-1 0 fp/ fp/
e e
-2
-2 -0.5 -3 -0.5 FIG. 6: Josephson currents in the limit of strong mag-
-30 1 f2/p 3 4 -1 -40 1 f2/p 3 4 nscertiibcefitehledeovnenthanedQoDd.dTFhPerleesfutltasn.d(ar)i-g(hbt)cTohluemcanssesdoe-f
Γ0 =0. (c)-(d) Resultsof Γ0 =0.5.
FIG. 5: Josephson current spectra in the case of si-
multaneous direct and indirect couplings between the
Majorana doublets. The Coulomb strength is fixed at action, ε splits into two, i.e., ε and ε +U. Ac-
U = 2.0, and the other parameters are identical with s s s
cordingly, in the energy region of U <ε <0, the
s
those in Fig.4. (a)-(b) The even-FP case with non- −
fermion in the QD is changeable between 0 and 1,
magnetic and magnetic QDs with R= 0.3. (c)-(d) The
odd-FPresults with R=0 and R=0.3. which magnifies the transformation of the Joseph-
son effect caused by the shift of QD level. Since a
finite magnetic field on the QD plays a similar role
in affecting the fermion occupation in the QD, the
the noninteracting current results. For the effect of
magnetic field makes a similar contribution to the
the magnetic field in the odd-FP case, as shown in
Josephsoneffect,comparedwiththeCoulombinter-
Fig.5(d), it is analogous to that in the noninteract-
action. In addition, it should be noticed that the
ing case. Namely, it tends to enhance the ampli-
Fanointerferenceinduces the asymmetricspectraof
tude of the 2π-period current, despite in the region
the Josepshon currents vs ε .
0
of 3.0 < ε < 1.0. Also, it does not induce the
− 0 At last, we would like to pay attention to the ex-
appearance of the fractional Josephson effect.
treme case of strong magnetic field where only one
TheresultsinFigs.4-5canbeexplainedsimilarto level (i.e., E ) contributes to the Josephson effects.
s˜
thediscussionaboutFigs.2-3. InthecaseoffiniteΥ, (e)
In such a case, the matrix dimension of and
the underlying physics that governs the Josephson HT
(o) will be halved, as discussed in the above sec-
effectsbecomescomplicated. Thereasonarisesfrom HT
tion. Thecorrespondingnumericalresultsareshown
two aspects. First, the fermion number of the QD
inFig.6. First,inFig.6(a)-(b)wecanseethatinthe
re-regulatestheFPof n n forconservingtheFP
| f f˜if case of Γ = 0, the Josephson currents in different
in the whole system. Second, the Fano interference 0
FPs are the same as each other, with their 2π pe-
can be induced by the direct and indirect couplings
riod. On the other hand, when the direct coupling
betweentheMajoranadoublets. Basedonthisidea,
between the Majorana doublets is considered, the
we see that in the even-FP case with zero magnetic
JosephsoncurrentsbecomedependentontheFP.As
field, when the QD level is far away from the en-
showninFig.6(c),intheeven-FPcase,increasingε
ergy zero point, both n and n are equal to 1 or 0 s˜
s s˜
canchangethecurrentperiodfromπ to2π withthe
simultaneously. Consequently, the states 00 and
f
| i clear transition region near ε 1.0. However, in
11 contribute equally to the Josephson effect, so s˜
f ≈ −
| i the odd-FP case, similar result occurs when ε de-
the normalJosephsoneffect occurs with the current s˜
creases. These results can certainly be clarified by
amplitude proportional to Γ . Alternatively, in the
0
discussingtheinfluenceofthefermionnumberinthe
odd-FP case with ε 0, the π-period current
s/s˜
| | ≫ QD on the FP of states n n . Also, in Fig.6(c)-
occursandits amplitude isrelatedtoΓ1, due to the | f f˜if
co-contributionof states 10 and 01 . When the (d) we can find that the Fano interference induces
f f
QD level gets close to the| eniergy ze|roipoint, it will the dissimilar transition behaviors of the Josephson
be partly-occupied. In such a situation, 10 and currents in different FPs.
f
| i
01 contribute to the even-FP Josephson current,
f
| i
whereas 00 and 11 devote themselves to the
f f
| i | i
odd-FPcurrent. However,due toΓ Γ ,the sup- IV. SUMMARY
1 0
≪
pressionofI(e) onlyappearsinanarrowregionnear
J
the point of ε0 = 0, while the 2π-period oscillation In summary, we have investigated the Joseph-
of I(o) distributes in a wide region accompanied by soneffectintheMajorana-doublet-contributedjunc-
J
its enhanced amplitude [See Fig.4(a) and Fig.4(c)]. tionmodifiedbythe differentinter-doubletcoupling
Next,inthepresenceoftheintradotCoulombinter- manners. It has been found that an embedded QD
8
in this junction plays a nontrivial role in modify- doubletscoexist,nofractionalJosephsoneffecttakes
ing the Josephson effects, since the tunable fermion place, regardless of finite magnetic field on the QD.
occupation in the QD re-regulates the FP of the Moreover, the π-period current has an opportunity
Majorana doublets for conserving the FP in whole to appear with the shift of the QD level. We be-
system. As a result, the 4π-period, 2π-period, and lieve that this work will be helpful for describing
π-period Josephson currents have opportunities to the QD-assisted Josephson effects between the Ma-
come into being, respectively. To be concrete, when jorana doublets.
the Majorana doublets couple indirectly via a non-
magnetic QD, the normal Josephson effects occur, This work was financially supported by the Fun-
and the FP change just causes the reversal of the damental Research Funds for the Central Universi-
current direction and the variation of the current ties(GrantsNo. N130405009andNo. N130505001),
amplitude. Only in the odd-FP case, can applying theNaturalScienceFoundationofLiaoningprovince
finite magnetic field on the QD induce the appear- of China (Grant No. 2013020030), and the
ance of the fractional Josephson effect. When the Liaoning BaiQianWan Talents Program (Grant No.
direct and indirect couplings between the Majorana 2012921078).
∗
Email address: [email protected] 184516 (2012).
1 M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 19 F. Zhang, C. L. Kane, and E. J. Mele, Phys. Rev.
3045(2010); X.-L. Qi and S.-C. Zhang, Rev. Mod. Lett.111, 056402 (2013).
Phys. 83, 1057 (2011). 20 S. Nakosai, J. C. Budich, Y. Tanaka, B. Trauzettel,
2 C. Nayak, S. H. Simon, A. Stern, M. Freedman, S. andN.Nagaosa,Phys.Rev.Lett.110,117002(2013).
Das Sarma, Rev. Mod. Phys. 80, 1083 (2008). 21 A. Keselman, L. Fu, A. Stern, and E. Berg, Phys.
3 F. Zhang, C. L. Kane, and E. J. Mele, Phys. Rev. Rev.Lett. 111, 116402 (2013).
Lett. 111, 056402 (2013). 22 E.Gaidamauskas,J.Paaske,andK.Flensberg,Phys.
4 F. Zhang, C. L. Kane, and E. J. Mele, Phys. Rev. Rev.Lett. 112, 126402 (2014).
Lett. 111, 056403 (2013). 23 A. Haim, A. Keselman, E. Berg, and Y. Oreg, Phys.
5 F. Hassler, A. R. Akhmerov, C.-Y. Hou, and C. W. Rev.B 89, 220504(R) (2014).
J. Beenakker, New J. Phys. 12, 125002 (2010); K. 24 F.ZhangandC.L.Kane,Phys.Rev.B90,020501(R)
Flensberg, Phys. Rev.B 82, 180516(R) (2010). (2014).
6 L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 25 E. Dumitrescu, J. D. Sau, and S. Tewari,
(2008); E. J. H. Lee, X. Jiang, R. Aguado, G. Kat- arXiv:1310.7938.
saros,C.M.Lieber,andS.DeFranceschi,Phys.Rev. 26 J. Klinovaja and D. Loss, Phys. Rev. B 90, 045118
Lett. 109, 186802 (2012). (2014).
7 B.vanHeck,F.Hassler, A.R.Akhmerov,andC.W. 27 X.J.Liu,C.L.M.Wong,andK.T.Law,Phys.Rev.
J. Beenakker, Phys. Rev. B 84, 180502(R)(2011); P. X 4, 021018 (2014).
Lucignano,F.Tafuri,andA.Tagliacozzo, Phys.Rev. 28 A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar,
B 88, 184512 (2013). Rev.Mod. Phys. 82, 2257 (2010).
8 D. Pekker, C. Y. Hou, V. E. Manucharyan, and E.
Demler, Phys. Rev. Lett. 111, 107007 (2013); L.
Jiang, D. Pekker, J. Alicea, G. Refael, Y. Oreg, and
F. von Oppen,Phys. Rev.Lett. 107, 236401 (2011).
9 P. A. Ioselevich and M. V. Feigel man, Phys. R ev.
Lett. 106, 077003 (2011).
10 P. San-Jose, E. Prada, and R. Aguado, Phys. Rev.
Lett. 108, 257001 (2012); R. M. Lutchyn, J. D. Sau,
and S. Das Sarma, Phys. Rev. Lett. 105, 077001
(2010).
11 A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W.
Ludwig, Phys.Rev.B 78, 195125 (2008).
12 X.-L. Qi, T. L. Hughes, S. Raghu, and S.-C. Zhang,
Phys. Rev.Lett. 102, 187001 (2009).
13 J.C.Y.TeoandC.L.Kane,Phys.Rev.B82,115120
(2010).
14 A. P. Schnyder, P.M. R. Brydon, D. Manske, and C.
Timm, Phys.Rev. B82, 184508 (2010).
15 C.W.J.Beenakker,J.P.Dahlhaus,M.Wimmer,and
A. R.Akhmerov,Phys.Rev.B 83, 085413 (2011).
16 S.Deng,L.Viola,andG.Ortiz,Phys.Rev.Lett.108,
036803 (2012).
17 S. Nakosai, Y. Tanaka, and N. Nagaosa, Phys. Rev.
Lett. 108, 147003 (2012).
18 C. L. M. Wong and K. T. Law, Phys. Rev. B 86,