Effect of disorder in non-uniform hybrid nanowires with Majorana and fermionic bound states

2.1 MB·

by  Zhen-Hua Wang

Effect of disorder in non-uniform hybrid nanowires with Majorana and fermionic
bound states
Zhen-Hua Wang,1 Eduardo V. Castro,2 and Hai-Qing Lin1
1Beijing Computational Science Research Center, Beijing 100084, China
6 2CeFEMA, Instituto Superior T´ecnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
1
0 Majorana fermions in some model semiconductor nanowires proximity coupled with s-wave su-
2 perconductorhavebeen investigated,as well asotherintragap boundstates intopological phaseof
aRashbananowirewithnon-uniformspin-orbitinteraction(SOI).UsingtherecursiveGreen’sfunc-
n
tionmethod,wepresentadetailedstudyoftheparameterspaceforsemiconductor-superconductor
a
J system focusing on understanding the key experimental conditions required for the realization and
detection of Majorana fermions. Various sources of disorder, such as disorder in semiconductor
0
nanowire,bulksuperconductorandsemiconductor-superconductorinterface,areincludedtocharac-
2
terizetheireffectsonthestabilityofthetopologicalphase. Inthecaseofmultibandsemiconducting

nanowires, the phase diagram of the system as a function of the chemical potential and magnetic
]
n field have been calculated. From the different transverse subbands in the nanowire, a new MF
o coexisting region appears, where istoalarge extentrobust against chemical potentialfluctuations.
c Intwo-bandwireortwocoupledsinglebandwire, thephaseofonebandcanbecontrolled through
- theother band,and theseproperties is critical to build multiband quantumdevice. When theSOI
r
p vectorormagnetic fieldinsemiconductornanowirerapidlyrotatesbytheangle π,changesitssign,
u a topological π junction forms at the junction between two wire sections characterized by different
s directions of the SOI vectors or magnetic field. In this case, some fermionic bound states (FBS)
.
t can emerge inside theproximity gap, which localize at thetopological π junction, and coexist with
a
Majorana bound states localized at the nanowire ends. Changing the position of the topological
m
π junction from one end to the other, the zero-energy FBS moves and can be used as mediator to
- transferquantuminformation betweentwodistanceMFs. Meanwhile,theAndreevspectrumofthe
d
junction is qualitatively phase shift by π compared to usual Majorana weak links. The FBS can
n
act as quasiparticle traps and serve as an effective mediator of hybridization between two distant
o
Majorana boundstates (MBSs).
c
[
PACSnumbers: 73.63.Kv,72.15.Qm,73.63.Fg,73.23.-b

1
v
8 I. INTRODUCTION The quantum information encoded in the degener-
1 ate states of MBSs is topologically protected from local
3
sources of decoherence. Braiding of non-Abelian anyons
5
Bound states arising in a variety of condensed mat- have been implemented using one-dimensional semicon-
0
. tersystemsareexploredintensivelyoverthe lastdecade. ductornanowires,in particularT-junctionwire network.
1 Excitations with non-Abelian statistics, such as Majo- Recently, Flensberg et al. have demonstrated the non-
0
rana bound states (MBSs),1–29 fractional fermions,30–33 Abelian rotations in hybrid topological-spin qubit sys-
6
1 and parafermions34–40 are attractive due to their poten- tems, and coherently transfer quantum information be-
: tial use in topological quantum computation schemes. tween topological qubits and normal spin qubits.43–46
v Thesimplestnon-Abelianexcitationiszero-energyMBS, Also, a composite system of Majorana-hosted semicon-
i
X and recent experimental evidence is consistent with the ductor nanowire and superconducting flux qubits has
r possibility that superconducting nanowires with strong been shown to process quantum information.47,48 As is
a spin-orbit coupling support topological superconductiv- known,MBSsdonothavethestructurenecessarytocon-
ity and MBSs.15–18,29 The key to the experimental real- struct a universal quantum computer. Several methods
ization of Majoranafermions in such system is to satisfy have been suggested to achieve universal operation, in-
a certain set of requirements that ensures the stability cluding merging the quasiparticles for a certain time49
of MBSs and how big the effect is for realistic parame- or combining topologicaland conventionalqubits.47,50,51
ters. The challenging task here is to control fluctuations Moreover, there are many other intragap bound states
of parameters and suppress effects of disorder in the re- in realistic system, such that MBSs can be transformed
alistic systems. In multiband nanowires, the stability of from fractional fermions by tuning parameters from the
thetopologicalnontrivialphaseisenhancedduetostrong topologicaltrivialphasetonontrivialphase.14 Withnon-
interbandmixing, andthe system is most robustagainst uniform spin-orbit interaction, topological π Josephson
chemical potential fluctuations.41,42 Therefore, it is es- junction can support three topological phases protected
sential to identify parameter regimes favorable for the by chiral symmetry.52 However, the bound states coex-
existence of Majorana fermions (MFs) in laboratory, in isting with MBSs in some types of topological π Joseph-
particular, in the presence of various disorder types and son junction has not received much attention, as well as
multiband occupancy. itsimplicationsfortopologicalquantumcomputation. It
2
the system is5–7
p2
H =Z dyΨ+σ[(2m −µ)δσσ′ +2αRpσσxσ′ −Vzσσzσ′]Ψσ′
σy
+i dy∆Ψ+ σσ′Ψ+ +h.c. , (1)
Z σ 2 σ′
whereµ is chemicalpotential,the indices σ,σ′ = , rep-
↑ ↓
resent the spin direction. Ψ (y) annihilates spin-σ elec-
σ
trons at position y, σz (σx ,σy ) are Pauli matrices.
σσ′ σσ′ σσ′
For numerical convenience, it is standard to study the
FIG. 1. Sketch of a non-uniform nanowire of length 2L on BdG equation in a discrete lattice tight-binding approx-
an s-wave superconductor directed along y direction which imation with no loss of generality and with fewer un-
consists of two segments y < 0 and y > 0 in which the cor- known parameters. Under the lattice approximation, we
responding SOI vectors αR and αR, point in opposite di- can map Eq. (1) to a tight-binding model:
rections. The magnetic field B ap−plied perpendicular to the
surface can induce a topological regime, where bound states H =H +H +H +H , (2)
wire 0 Rashba Z SC
are formed inside the gap. Firstly, thereare zero-energy Ma-
jorana bound states localized at the nanowire ends y = L. where H includes nearest-neighbor hopping along the
± 0
Secondly,therearezero-energyfermionicboundstatesatthe wire (y direction),
interface y=0, where theSOI vector changes its sign.
H = [ t(w+ w +w+ w ) µ w+ w ],
0 − i+1,σ i,σ i,σ i+1,σ − LAT i,σ i,σ
X
is still an open question whether other intragap bound i,σ
(3)
states coexisting with MBSs could be used to braid Ma-
where w+ creates an electron with spin index σ on site
joranas, even for universal quantum computing. i,σ
i of the wire. The free electron energy in the lattice
Thegoalofthepresentworkistodeveloptherecursive
model is ε = 2tcosk µ , the band width is 4t.
Green’sfunctionmethodforthe realizationandobserva- k LAT
− −
µ is the chemical potential of the wire in the lattice
tion of the emergent Majoranamode and other intragap LAT
model. Thebandhasaenergy 2tshiftcomparedtothe
bound states in some model nanowire systems, and dis-
cuss the requirements of the stability of MBSs in the continuum Hamiltonian εk = 2−km2 −µ. So, the chemical
presence of various disorder. The multiband nanowires potential in this two model are not the same, they must
canenhancethetopologicalnontrivialphaseandarebet- satisfytherelationµLAT =µ 2t. TheRashbaspin-orbit
−
ter suited for observing the Majorana particle. Under- interaction can be written as,
standing the property of multiband quantum device is
critical to build topological 2D quantum device. With HRashba = −iαRwi++1,σσσyσ′wi,σ′ +h.c. . (4)
non-uniform spin-orbit interaction and magnetic field, iX,σσ′
theFBSscanemergeinthetopologicalphaseofaRashba
The direction of α determines the direction in which
R
nanowireinthe presenceofproximity-inducedsupercon-
the spins are polarized by the SOI. In the case of non-
ductivity gap. Such FBSs can act as quasiparticle traps
uniform SOI,without loss of generality,we choose α to
R
and thus canhave implications for quantum information
bepositivefory <0,andnegativefory >0. Amagnetic
transfer. Specially,theAndreevspectrumisqualitatively
field is applied perpendicular to the surface (z direction)
phaseshiftbyπ comparedtousualMajoranaweaklinks.
causing the Zeeman splitting V ,
Z
With even number of topological junction in one wire,
there is no phase shift on the Andreev spectrum. At T-
junction,bytuningthedirectionofSOIvectoramongthe H = V (w+w w+w ). (5)
three segments, the three MFs at the junction combine Z X Z i↑ i↑− i↓ i↓
i
into a zero-energy Majorana mode and a finite-energy
The s-wave pairing term with superconducting order
fermion.
parameter ∆ is
II. MODEL HSC =∆ (wi+↑wi+↓+wi↓wi↑) . (6)
X
i
The physical system for studying Majorana fermions
includes a strongly spin-orbit coupled semiconductor, To account for the spin-flip due to spin-orbit cou-
proximity-coupled to an s-wave superconductor and im- pling and the superconducting pair potential, we make
posed to a Zeeman field. To analyze the robustness of use of the 4 4 spin Nambu space,53–55 spanned by
theMBSsignatureintherealphysicalsystem,thesingle Ψ =(w+,w+×,w ,w ⊗)T. The matrices of the nanowire
i i↑ i↓ i↑ i↓
MBS is replaced by the whole nanowire. The sketch is for i=1,2,...N are
shown in Fig. 1. The continuous BdG Hamiltonian for
3
µ +V 0 0 ∆ t α 0 0
LAT Z R
− − −
Hii = 00 −µLAT∆−VZ µ −∆ V 00 , Hi,i+1 = α0R −0t 0t α0 =(Hi+1,i)+ .
LAT Z R
 ∆ −0 0− µ +V   0 0 α t 
 LAT Z   R 
−
(7)
For a small system size, the full tight-binding model can HRashba = −iαRwi++1,sσsys′wi,s′ +h.c.
be analyzed directly by exact diagonalization. However, iX,ss′
for sufficiently large system, exact diagonalizationis not
realizable. WewillsolvetheBdGequationbyarecursive HRashba = −iαRd+i+1,sσsys′di,s′ +h.c. (10)
Green’s function method. iX,ss′
In a more realistic system, a critical ingredient for re- The spin-orbit band-mixing Hamiltonian reads as:41,42
alizing Majoranafermions in a solid state hybrid system
iqsutahnetitpartoixviemaistpye-icntdsuocfetdhesuSpCerpcoronxdiumcittiyvietyff.ecTtso, oandedrheasss HS(1O2) =iX,ss′Ebm[wi+,s(iσx)di,s′ −d+i,s(iσx)wi,s]. (11)
to consider specific models for the relevant terms in the
total Hamiltonian:56,57 The multiband proximity-induced SC can be described
as
H =H +H +H +H +H ,
tot SM Rashba Z SC SM−SC
H = [∆ w+w+ +∆ d+d+ +
SC 11 i↑ i↓ 22 i↑ i↓
X
H = (tsc µ δ )a+ a +∆ (a+a++a a ), i
SC iX,j,m ij− sc i,j im jm 0Xi i↑ i↓ i↓ i↑ ∆12wi+↑d+i↓+∆21d+i↑wi+↓+h.c.]. (12)
Here w and d represent fermion annihilation opera-
i,s i,s
H = [t˜σmw+ a +h.c.], (8) torsofthefirstandsecondsubbands. Assumingthatthe
SM−SC i0j0 i0σ j0m
iX0,j0Xm,σ confinement energy along the x direction Esb is larger
than all the relevant energy scales of the Hamiltonian.
where H describes s-wave bulk superconductor and
SC E is the band mixing energy, and the multiband in-
H represents the SM-SC coupling. µ is the bm
SM−SC sc duced SC pairing potential ∆ depend on the micro-
chemical potential at the SC surface, a+ is the creation ij
im scopic details of the SM-SC interface.
operator corresponding to state with spin m localized
near sites i, t˜σm are coupling matrix elements between
i0j0
SM and SC local states. In this case, the SM system is
III. RESULTS AND DISCUSSION
embedded into an extended system by applying a self-
energy term, Σ . Applying the Dyson equation allows
SC A. Effect of disorder in uniform nanowire
ustosimplifyG =(E H Σ )−1,wheretheself-
tot SM SC
energy term is given by−Σ =−V G V+ ,
SC SM−SC SC SM−SC
and V is the coupling between SM-SC. Using the Single-channel semiconductor nanowire have been re-
SM−SC
recursive Green’s function method similar to Fig. 8, the cently proposed as a possible platform for realizing and
Green function of the extended two-dimensional super- observing Majorana physics in solid-state systems.5,6,56
conductor surface can be calculated,58,59 and then we To obtain better insight into the existence of Majorana
can obtain the self-energy term Σ . fermion and other intragap bound states, together with
SC
Recent works have also demonstrated that Majo- varioustypesofdisorderintherealisticsystem,wecalcu-
rana end states can be realized outside the strict one- late the localdensity of states (LDOS), energyspectrum
dimensional limit.10,41,42,56,60 Here we consider a quasi- and electron wave function for several relevant regimes,
one-dimensional system where the topological phase using a set of control parameters.
emerges from different transverse subbands in the
nanowire. An effective two-band model for the semicon-
1. Disorder in uniform nanowire
ductor nanowire is constructed, and the Hamiltonian of
the two decoupled bands are similar to Eq. (3)-(5):
Foracleannanowire,theclear-cutevidencefortheex-
H0 = [−t(wi++1,swi,s+wi+,swi+1,s)−µLATwi+,swi,s] istence of the Majorana zero modes can be obtained by
X
i,s drivingthe systemfromtrivialphase to topologicalnon-
+ V (w+w w+w )+ [ t(d+ d trivial phase. In the topological trivial phase there is a
X Z i↑ i↑− i↓ i↓ X− i+1,s i,s well-definedgapforallexcitations,includingstateslocal-
i i,s
izedneartheendsofthewire. Bycontrast,the topologi-
+d+ d ) (µ E )d+ d ]
i,s i+1,s − LAT − sb i,s i,s calnontrivialphaseischaracterizedbysharpzero-energy
+ V (d+d d+d ) (9) peaks localized near the ends of the wire and separated
X Z i↑ i↑− i↓ i↓ from all other excitations by a well-defined minigap. In
i
4
FIG.3. TheLDOSof thesemiconductornanowire proximity
on a s-wave superconducting surface for various parameters.
(a) The SM-SC coupling t˜σm = 2, and Zeeman field on the
i0j0
SC surface VSC =0, (b) t˜σm =3, VSC =0, and (c) t˜σm =
Z i0j0 Z i0j0
3, VZSC = 3. Other parameters: µSC = 0,tsijc = 10,∆0 =
3,t˜σi0mj0 =2,µSM =−20,αR =10,t=10.
disorder are to narrowthe spectral gap to the lowest ex-
cited states, and to break the parity degeneracy of the
FIG. 2. LDOS for a uniform nanowire with disorder. The excited states.64 In contrast with the clean case, all the
strength of the disorder is r = 0 (top), r = 1 (middle), and low-energy excitations are strongly localized. However,
r = 8 (bottom). Notice that all the low-energy states are for sufficiently strong disorder, the excitation gap closes
strongly localized, but the clear-cut distinction between two and the Majorana modes no longer exist. Adding disor-
phases holds at weak disorder. Throughout, µ=0, ∆=1.0, der induces localization, and the topological non-trivial
αR = 10 and Vz = 3.0, the hopping in the nanowire t = 10 phase transforms into trivial phase, which is character-
corresponds to a band width D = 40, and the wire consists ized by the closing of the gap65–67 and by a spectral
of 1000 sites.
weightdistributedoverawideenergyrange. Meanwhile,
the disorder in the Rashba and Zeeman terms are also
both phases, the LDOS is always symmetric, with the aneffective wayofclosingthe topologicalgap. These re-
DOS at positive energy being exactly the same as for sults are in good agreement with those obtained in Ref.
negative energy and also exactly the same at both ends [56]. We have also crosschecked results using the recur-
of the wire. sion method68 and obtained perfect agreement.
Disorder in the nanowire can have significant ad-
verse effects on the stability of the topological nontriv-
ial phase.56,61–63 In a realistic system, disorder comes in 2. Proximity-induced superconductivity
variouswaysthataffectthetopologicalphaseverydiffer-
ently. In this paper we consider three types of disorder:
Using the recursive Green’s function method, the
Disorder in the SM wire, impurities in the s-wave SC,
proximity-induced superconductivity for realizing Majo-
and random nonuniform coupling between the SM wire
ranafermionsinasolidstatehybridsystemcanbeinves-
andtheSC.FordisorderintheSMwire,wefocusonthe
tigated. We can now include the superconductor surface
sources that are the most relevant experimentally: ran-
self-energyΣ intothe semiconductornanowireHamil-
dompotentialscreatedbychargedimpurities. Wemodel SC
tonianandrealizetheMajoranazero-mode. Fig. 3shows
the random potential by adding to the Hamiltonian the
the LDOS of the SM nanowire for various parameters in
term
the SC surface, including SM-SC coupling t˜σm and Zee-
i0j0
H = V c+c , (13) manfield VZ.The obtainedresultis consistentwith pre-
disorder Xi i i i vious work.56 At low energies (ω ≪ ∆0), the frequency
dependence of the dynamically generated terms in the
where the random set V is normally distributed with self-energy can be neglected, and the effective proximity
i
zero mean and standa{rd}deviation r . We use r gap is independent of ω. With the increase of t˜σm, the
{ } { } i0j0
to control the disorder strength. Fig. 2 shows a seg- self energy Σ increases,and the strong coupling make
SC
ment of the LDOS as a function of lattice sites and en- it would require extremely high magnetic fields to reach
ergy ω for typical disorder realizations at three disorder the topologically nontrivial phase. In the bottom figure
strengths. For weak to moderate disorder strength, the of Fig. 3, both SM and SC are considered to be under
Majorana zero mode survives, and the main effects of thesameZeemanfieldV ,whichispossibleunderexper-
Z
5
FIG.4. LDOSof theSM-SCsysteminthepresenceofdisor-
der. (a)-(b)DisorderfromSC-SMcoupling,t˜=t˜σm +∆t˜,(a)
i0j0
r = ∆t˜, (b) r = 5∆t˜, where ∆t˜ are normally distributed.
{ }
(c) Disorder from SC surface with r = 1. The signature
features of the topological nontrivial phase are preserved for
weak or mediate disorder.
imentalcondition. Itisfoundthatthegapclosesandthe
Majorana zero-mode disappears. That may be one rea-
son why it’s difficult to observe MF in experiment, and
this condition should be avoided.
3. Random nonuniform SM-SC coupling and impurities in
the s-wave SC
Moreover, suppressing effects of disorder and con-
trolling fluctuations of other parameters are challeng-
FIG. 5. Phase diagram for the two-band nanowire model as
ing tasks. The disorder in SM wire has been discussed
a function of thechemical potential µand externalmagnetic
above, and we will investigate the other two types of
dpilsinogrd,et˜r.=Fit˜rσsmtly+, w∆et˜d,iwschuesrseth∆et˜driesporrdeseernitns SthMe-SraCndcooum- fideoltdiVszu.sTedhetoprvoenroifuynctehdezpehroa-sbei,asAp(ωea)k=of−de2tΓeIctmin(gGqR1↓u↓a(nωt)um+
component.i0Aj0s can be seen from Fig. 4, the signature GR2↓↓(ω)). Only the spin-down channel is considered because
of the large Zeeman splitting. Realistic parameters of the
featuresofthe topologicalnontrivialphasearepreserved systemareused,weassumeherem∗ =0.04me withme being
forweakormediatedisorder.ThisisconsistentwithRef. electron mass and α = 0.1eVA˚yielding m∗α2 0.6K. t˜=
[Bena]. Strong disorder induce the disappearance of the 2.5, ∆˜11 = ∆˜22 = 4. (a) E˜sb = E˜bm = ∆˜12 =≈∆˜21 = 0, (b)
zero Majorana modes, and the spectral gets localized. E˜sb = 30, E˜bm = ∆˜12 = ∆˜21 = 0, (c) E˜sb = 30, E˜bm = 5,
Similar phenomenon can be observed in the presence of ∆˜12 =∆˜21 =0, (d) E˜sb=30, E˜bm =5, ∆˜12 =∆˜21 =4.
nonuniform SC surface, H = V a+ a , where
disorder i im im
Pi
V are normally distributed. Therefore, any other sig-
i
{ }
nificanttypeofdisordergeneratesthesameLDOSbehav-
neling of electrons to the ends of the nanowire would
ior, and does not change the key property of Majorana
reveal a pronounced zero-bias peak, in particular, it
bound states before reach a certain size.
is e2/h in the topological nontrivial phase, G =
peak
1/2, in contrast to that for a dot coupled to a regu-
lar fermionic zero mode, G = 0, and its topologi-
peak
4. Multiband semiconductor nanowire cally trivial phase, G = 1.55,69,70 In Fig. 5(a), when
peak
the confinement energy and spin-orbit band mixing en-
In this part, the Majorana end states are discussed ergy E = E = 0, the two bands decouple and Ma-
sb bm
outside the strict one-dimensional limit. Fig. 5 shows jorana fermions emerge from different transverse sub-
the phase diagram for the two-band nanowire model bands at same regions (V > ∆2 +µ2), which is
| Z| 11
as a function of the chemical potential µ and exter- similar to two physically differentpchains. However, the
nal magnetic field V . The emergent region of Majo- muchstrongZeemanfield(V ∆2 +µ2)generates
Z | Z|≫ 11
rana modes in the system and the corresponding phase a transition from topological nontprivial phase to trivial
diagram are obtained by using local tunneling. Tun- phase. The blue R1 region corresponds to nontopologi-
6
band 2 is also in topological nontrivial phase through
the spin-orbit band mixing energy E . In Fig. 6(c)-
bm
(d), the two bands have same parameters except for α
R
(α = 0.167,α = 0.167). In this parameter region,
R1 R2
−
the two isolated bands can be in topological nontriv-
ial phase, respectively. But these two bands will be in
topologicaltrivialphasewhen they couple toeachother.
Therefore, the multiband case or several coupled chains
can provide a new way to control the existence of MFs
and achieve non-Abelian braiding.
B. Effect of disorder in non-uniform nanowires
FIG. 6. The LDOS of band 1 (a,c) and band 2 (b,d) in two
bandnanowire. (a)-(b)αR1 =0.167,αR2 =0,VZ1 =8,VZ2 =
0. (a)-(b) αR1 = 0.167,αR2 = 0.167,VZ1 = 8,VZ2 = 8. 1. Non-uniform Rashba spin-orbit interaction in single
−
Otherparameters are thesame as in Fig. 5(d). band wire
In this section, we study intragap bound states in the
cal phase, Gpeak = 0. With the increase of Esb, see Fig. topologicalphaseofaRashbananowirewithnon-uniform
5(b), the confinement energy is switched on, and Ma- spin-orbit interaction and magnetic field. The zero-
jorana fermions of two subbands emerge inside different energy intragap bound states can serve as a mediator to
regions (|VZ|> ∆211+µ2,|VZ|> ∆222+(Esb−µ)2). achieveMajoranafermionexchangeinstrictlyonedimen-
Fivedistinctphapsescanbeobservedp,nontopological(R1 sional structures. Firstly, we consider a situation where
region), topological trivial phase (red region), topolog- the SOI vector changes its direction along the nanowire
ical nontrivial phase with Majorana fermions originat- axiscreatinganinterfacebetweentwonanowireswithdif-
ing either from first or second subband, and the last ferent SOI vector directions. The non-uniform nanowire
one with two Majorana modes localized on each end of length 2L directed along y direction which consists of
(R2 region). In Fig. 5(c), the spin-orbit band mixing two segments, y <0 and y >0 in which the correspond-
energy Ebm modifies the phase boundary. The most ing SOI vectors are in different directions. In the partic-
interesting parameter regimes is µ Esb/2, two sub- ular case, when the SOI has a sharp discontinuity, such
∼
bands are in topological trivial phase, and two topo- that the SOI vector rapidly rotates by the angle φ = π,
logical nontrivial phases of subbands can’t coexist. In equivalently,changesitssign,thesystempossessesanad-
the presence of strong interband mixing ∆12, as shown ditional symmetry that constrains the fermionic bound
in Fig. 5(d), there is a new window where topological states to be zero-energy states. In Fig. 7, we have cal-
phases from different subbands coexist. At µ = Esb/2 culated the LDOS vs ω and lattice sites i for different
the width of the topologically nontrivial region is given spin-orbit interaction, including strength and direction.
by Esb/2 ∆12 < Vx < Esb/2+∆12. The topologi- TheLDOSsymmetriesbetweenpositiveandnegativeen-
− | |
cal phase around this region is to a large extent robust ergy is preserved, but which is destroyed at two ends of
against chemical potential fluctuations, which now have the wire. TheSOI α inthe topofFig. 7 havethe same
R
tobeδµ Esbtocausethetransitionintothetopological direction but with different strength. There are zero-
∼
trivial state. This regime provides a promising route to energy Majorana bound states localized at the nanowire
realizinga robusttopologicalnontrivialphase. However, ends y = L. Having a little change on the strength of
accordingtoourcalculation,inthiscoexistingregion,the SOI does ±not destroy the signature features of the two
disorder in the spin-orbit band mixing energy Ebm has phase, and the obtained results are consistent with the
a sensitive effect on the stability of topological phases. discussion of disorder. In the middle figure, α =10 for
R
Even for a weak disorder, the topological gap closes and y < 0 and α = 0 for y > 0, the half is in topological
R
theMFdisappears,whichisdifferentfromothertypesof trivial phase in the region where α = 0, and the DOS
R
disorder in systems. It is therefore required strict condi- spreads over the entire energy range as there is no finite
tions to obtain the MF in this region. gap. In the topological nontrivial part, α = 10, one
R
Two coupled transverse subbands is similar to two Majorana zero modes localizes at left end, but the other
physically different chains. If we change the phase of Majoranazeromodesisdelocalizedandspreadsoverthe
one band or chain, the other one will also changes. In y >0partofthewire. Thisisduetohybridizationofthe
Fig. 6, the two bands have two sets of control pa- right hand Majorana with continuum zero energy levels
rameters, and we let the two bands be in topological where the gap is closed. In the bottom figure, the two
trivial phase firstly (α = α = 0,V = V = 0). half wire have two α with same strength but opposite
1 2 Z1 Z2 R
Then the band 1 is tuned in topologicalnontrivialphase directions,andadditionalfermionicboundstatesemerge
(α =0.167,α =0,V =8,V =0),andFig. 6(a)- inside the proximitygap. They arelocalizedatthe junc-
R1 R2 Z1 Z2
(b) corresponds to band 1 and 2. It is found that the tionbetweentwo wiresections characterizedby different
7
FIG.8. The energy spectrum for aclean nanowire with non-
uniform spin-orbit interaction. The in-gap states are Majo-
FIG. 7. LDOS for a clean nanowire with non-uniform spin-
rana zero-energy modes, and n labels the eigenvalues of the
orbit interaction. For y <0, αR =10, y >0, αR =5,0,−10 systemwiththelowestenergy. WithoppositeSOIvector,the
from top to bottom,respectively. As the SOI vector changes
fermionicboundstatesarezero-energy. Otherparametersare
its sign, the fermionic bound states localize at the interface
thesame as in Fig. 7.
x = 0, coexisting with MBSs at the nanowire ends. Other
parameters: µ=0, ∆=1.0, t=10 and Vz =3.0.
energy spectrum and electron wave function have been
calculated. As shown in Fig. 8, the spectra of uniform
directions of the SOI vectors, and they coexist with Ma-
nanowire with an odd number of pairs of zero-energy
joranaboundstateslocalizedatthenanowireends.Inthe
modes characterizes topological SC phases. The in-gap
particular case, when SOI change its sign, the FBSs and
states are Majorana zero-energy modes. When α = 0,
MBSsaredegeneracysincebotharezero-energystates.71 R
the gap disappears, it is because that the half wire with
As discussed above, we only show that the FBS is at
zero α is in topological trivial phase. Therefore, the
R
the interface in the center of the nanowire. If we change
continuousexcitationcancovertheentireenergyregime.
thepositionoftheinterfacefromlefttorightofthewire,
For opposite direction of SOI vector, there are two pairs
the FBS can serve as a mediator to transfer Majorana
of zero-energy states in the gap, which are energy de-
fermion, and even exchange MFs in strictly one dimen-
generate MBSs and FBSs. Note that FBS only exist at
sional structures. The interface is created between two
zero energy if the angle between SOI vectors is π. For
partswithdifferentSOIvectordirections,andweletthe
a different angle FBS are not zero energy, and as this
interface localize at the position of left end of the wire
angle goes to zero, the states merge in the continuum of
firstly. In this case, the FBS is overlapped with MBS of excitations.52,71
leftend,andtherearenoFBSsinthegap. Changingthe
position of the interface from left to right, the informa-
tion of left MBS can be carried to the right. When the
interface localizes at the right end of the wire, the FBS 2. Non-uniform magnetic field
overlaps with the right MBS, and serves as a mediator
to transfer information. By fine tuning the parameter, Next, we replace the uniform magnetic field by non-
thismethodevencanachieveMajoranafermionexchange uniformfield. Auniformmagneticfieldperpendicularto
in strictly one dimensional structures. Franz et.al have SOI vector can induce a topological phase where MBSs
demonstratedthe MFexchangein1Dstructuresusing π are formed at the ends of the wire. The features il-
domain wall by tuning the phase of order parameter ∆. lustrated in Fig. 9 are in non-uniform magnetic field.
The non-uniform Rashba SOI nanowire can also be used Firstly, we let the magnetic field in the two half wire
to implement protected quantum computation. havesamedirectionbutwithdifferentstrength. TheMa-
To get a deep insight into the additional FBSs, the jorana bound states are also localized at the ends of the
8
FIG. 10. The LDOS of the non-uniform single band
nanowire. (a)-(b) Non-uniform SOI case. The strength of
the disorder is r = 0 (a), and r = 8 (b). Other parameters
are in same as Fig. 7. (c)-(e) Non-uniform magnetic field
case. The strength of thedisorder is r=0 (c), r=2 (d)and
r=8 (e). Otherparameters are in same as Fig. 9.
from zero-energy and disappear in the continuum exci-
tation states. Therefore, the intragap bound states are
morestablethanMBSs,andcanbeserveasanpotential
way to transfer quantum information.
FIG. 9. LDOS for a clean nanowire with non-uniform mag-
netic field. For y < 0, Vz = 3, y > 0, Vz = 2,0, 3 from
−
top to bottom, respectively. As the magnetic vector changes 4. Double topological junction and T-junction
its sign, the fermionic bound states localize at the interface
y =0, coexisting with MBSs at the nanowire ends. αR =10 Similarly, here we consider a double topological junc-
and other parameters are thesame as in Fig. 7.
tion composed of two SOI-junction discussed above. In
Fig. 11(a)-(b), there are two FBSs localizing at the two
junctions. Thisbehaviorisverymuchreminiscentofdou-
wire. As Vz =0 for y >0, there is a gap with no states, ble quantum dots,72? which suggests that such double
since this part of the system is in the non-topological
FBSs can serve as a promising platform for conventional
phase. The MBSs localize at the ends of the other half
charge qubits. Embedding such charge qubits in a topo-
wire for y < 0, where the system is in the topological
logical superconductor might be rather well protected
phase. When the two half wire have opposite magnetic
against environmental noise and thus enjoy unusually
field, two additional peaks emerge in the mini-gap, and
large dephasing times. With the increase of topological
coexistwiththeMBSslocalizingattheendsofthewhole
π junction in the SM nanowire, shown in Fig. 11(c), the
wire. These two peaks have finite energies, which are
zero-energy modes increase. When one dot is connected
close to zero-energy. Similar to non-uniform spin-orbit
to two Majorana modes, as shown in Fig. 11(d)-(e), the
interaction, some intragap bound states emerge in the
energetically degenerate states in Flensberg’s qubits can
topologicalphase. ButtheyarenotdegeneratewithMa-
be obtained, and through which the two MBSs reduces
jorana zero-energy modes.
toasingleeffectiveMBS.43,69 InFig. 11(d),thesemicon-
ductor nanowire has uniform SOI vector, and the ener-
getically degenerate states can be achieved as the phase
3. The effect of disorder on FBSs
difference is φ=(1+2n)π, where φ is the magnetic flux
through the loop. In Fig. 11(e), non-uniform spin-orbit
It is also interesting to discuss the effect of disorder interaction is used, the behavior of the dot spectrum is
on FBSs. Here we only show the disorder of random po- nearlythesameexceptfortheπ phaseshift. Inthiscase,
tentials created by charged impurities in SM nanowire onlywhenφ=2nπ,the energeticallydegeneratestatein
with non-uniform Rashba SOI and zeeman field. Differ- Flensberg’s qubit can be obtained. It is because of the
entfromthebehaviorofMBSsindisorderwire,Asshown topological π junction, changing the sign of the Rashba
inFig. 10(a)-(b), the zero-energyFBSnotonlycoexists coupling α is effectively equivalent to imposing a rigid
R
with MBS in weak disorder, but also localizes at the in- phase shift of π to the order parameter. Therefore, the
terfaceinstrongdisorder,wheretheMBSdisappears. In flux qubit in Flensberg’s qubit can be canceled out and
Fig. 10(c)-(e),withtheincreaseofdisorder,thenon-zero the hybrid system becomes more controllable and scal-
energy FBSs caused by opposite zeeman field run away able. If there are even number of junction in the wire,
9
FIG. 12. T-junction viewed as three wire segments with su-
perconducting phases, defined such that the site indices in-
crease upon moving towards the junction, ΦA = ΦB = ΦC.
Two topological regions with same SOI vector αRA = αRB
meetingatthejunctionformatopologicalπ junction,leading
to two pairs MFs existing at the segment ends. Bottom left:
Lattice structure giving rise to the T-junction. The recur-
siveGreen’sfunctionmethodisusedtocalculatedtheLDOS,
where i corresponds to a set of ith slice of the T-junction.
Otherparameters are thesame as in Fig. 7.
one segment is in the topological superconducting state,
thereareonlytwoMFsandonelocalizesatthejunction.
FIG. 11. (a)-(b) TheLDOS and energy spectrum for a clean
nanowire with a double SOI-junction. (c) Three topological In Fig. 12,two topologicalsegments with same SOI vec-
π junction with four pairs zero-energy modes corresponding tor, αRA =αRB, induce two pairsof MFs emerging,and
to one pair MBSs and three FBSs. Other parameters are two MFs exist at the junction, where forming a π junc-
the same as in Fig. 7. (d)-(e) Dot spectral function in the tion. Itisalsointerestingtonotethatthereiszero-energy
more realistic nanowire case. (d) Uniform SOI vector, αR = states in vertical wire, although it is in non-topological
10, (e) Non-uniform SOI vector, y < 0, αR = 10, y > 0, phase. The reasonis thatthe verticalwire formsπ junc-
αR =−10. The dot spectrum is qualitatively phase shift by tionbetweentwohorizontalwires. ButasαRA = αRB,
πcomparedtousualMajoranaweaklinks. φarethemagnetic −
two SOI vector with opposite direction, the two segment
flux through the loops, quantumdot level εd =Vz, dot-MBS end MFs at the junction combine into an finite-energy
coupling Γ=0.2, dot-lead coupling λ=0.5.
ordinary fermion. Finally, consider the case shown in
Fig. 13 where all three segments are topological, there
are three pairs of MFs exist at every segment end when
the phase shift of Andreev spectrum disappears. α = α = α . Tuning the direction of the SOI
RA RB RC
A T-junction, three segments meeting at a point, pro- vector,αRA = αRB =αRC,theT-junctionalwayssup-
−
vides the simplest wire network that enables meaningful ports four Majoranazero modes andone ordinaryfinite-
adiabatic exchangeofMajoranafermions.73 Inthis part, energyfermion,amongwhichoneMBSandthe ordinary
we’ll discuss the intragap bound states at the three seg- fermionlocalizeatthejunction. HeretheHamiltonianof
ment junction as a function of SOI vector. As Fig. 12 thejunctionsimplifiestoHABC ∝ iVNN+1γC(γB γA),
− −
illustrated,the horizontalandverticalwires aretakento where VNN+1 is the hopping matrix between Nth and
consist of 2N +1 and N sites, and the phases of every N +1th slice, and γ indicates the Majorana fermions
{ }
segments are defined such that the site indices increase in each wire. It follows that the linear combination
uponmovingtowardsthe junction,ΦA =ΦB =ΦC.The (γA + γC)/√2 remains a zero-energy Majorana mode,
recursive Green’s function method is used to calculated whileγ and(γ γ )/√2combinedintoafinite-energy
B A C
−
the LDOS,58,59 where i corresponds to a set of ith slice fermion. It is imperative to avoid generating spurious
of the T-junction. There arethree casesto considerthat zero modes at the T-junction as we braid MBSs. In the
one, two, or three of the wire segments emanating from presence of non-uniform Rashba SOI nanowire, the T-
the junction reside in a topological region. When only junction can achieve more quantum operation and be-
10
FIG. 15. The LDOS of every bands in the non-uniform
two-band naowire, (a,c,e) corresponds to band 1, (b,d,f) cor-
respondstoband2. (a)Band1isuniform,αR1 =0.167 (y<
0),αR1 = 0.167 (y > 0),VZ1 = 8 (y < 0),VZ1 = 8 (y > 0),
band 2 αR2 = 0.167 (y < 0),αR2 = 0.167 (y > 0),VZ2 =
−
8 (y < 0),VZ2 = 8 (y > 0). (b) αR2 = 0.167 (y <
FIG. 13. Three topological segments. when αRA = αRB = 0),αR2 = 0.167 (y > 0),VZ2 = 8 (y < 0),VZ2 = 8 (y > 0).
− −
αRC, the three Majorana zero modes at the junction com- (c) Both two bands are non-uniform, αR1 = 0.167 (y <
binetoformafinite-energyfermionandasingletopologically 0),αR1 = 0.167 (y > 0),VZ1 = 8 (y < 0),VZ1 = 8 (y > 0)
−
protected Majorana. It follows that the linear combination αR2 = 0.167 (y < 0),αR2 = 0.167 (y > 0),VZ2 = 8 (y <
(γA+γC)/√2 remains a zero-energy Majorana mode, while 0),VZ2 = 8(y>0). Otherparametersofthetwobandsare
γB and (γA γC)/√2combined intoa finite-energyfermion, thesame a−s in Fig. 5(d).
−
corresponding tothe peak in thegreen circles.
Fig. 14(b), there are two pairs of non-zero energy FBSs
inthetwobandnanowire,whoseresultsaresimilartothe
resultinsingleband. InFig. 14(c),theband1isuniform
but band 2 is not, α (y > 0) = α (y < 0),V (y >
R2 R2 Z2
−
0) = V (y < 0), the topological gap in y > 0 part is
Z2
−
closed, and the right hand Majorana fermion hybridizes
with continuum zero energy levels. The behavior of the
LDOS is the same as that of Fig. 7, where one half
of wire has zero Rashba SOI α . This is because that
R
the right parts of the two bands has Rashba SOI with
opposite direction, and the effect of opposite α is the
R
same as zero Rashba SOI. Therefore, we investigate the
case of non-uniform α and non-uniform zeeman field
FIG. 14. The total LDOS of the two bands in non-uniform R
two-band naowire. (a) Both two bands are in non-uniform in band 2, respectively. It is found that the opposite
SOI case. αR = 0.167 (y < 0),αR = 0.167 (y > 0),VZ = αR can close the gap of right part completely, but the
−
8 (y < 0),VZ = 8 (y > 0). (b) Both two bands are in opposite zeeman field only narrow the gap. Meanwhile,
non-uniform magnetic field case. αR = 0.167 (y < 0),αR = the intragap bound states hybridize with the continuum
0.167 (y > 0),VZ = 8 (y < 0),VZ = 8 (y > 0). (c) Band excitation states.
−
1 is uniform but band 2 not, αR2 = 0.167 (y < 0),αR2 =
0.167 (y > 0),VZ2 = 8 (y < 0),VZ2 = 8 (y > 0). Other
− −
parameters of thetwo bands are thesame as in Fig. 5(d). IV. SUMMARY AND CONCLUSION
come a potential quantum device.
Inthis paper,wehavedevelopedthe recursiveGreen’s
functionmethodfortherealizationandtheobservationof
theemergentnon-AbelainMajoranamodeinsomemodel
5. Non-uniform multiband band nanowire
semiconductor nanowires proximity coupled to an ordi-
nary s-wave superconductor in the presence of various
Thecaseofnon-uniformRashbaSOIandzeemanfield disorder, non-uniform spin-orbit interaction, and non-
is also consider in multiband wire, and the main results uniform Zeeman field. Three types of disorder, such as
are shown in Fig. 14 and 15. Firstly, the two bands disorder in semiconductor nanowire, bulk superconduc-
have non-uniform Rashba SOI α , shown in Fig. 14(a), tor and semiconductor-superconductor interface, are in-
R
a pair of zero-energy FBSs appear in the intragap. In cludedto characterizetheir effects onthe stability ofthe