Majorana dc Josephson current mediated by a quantum dot
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec Majorana dc Josephson current mediated by a quantum dot
Luting Xu, Xin-Qi Li, ∗ and Qing-Feng Sun
2, 3, † Center for Advanced Quantum Studies and Department of Physics,Beijing Normal University, Beijing 100875, China International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
The Josephson supercurrent through the hybrid Majorana–quantum dot–Majorana junction isinvestigated. We particularly analyze the effect of spin-selective coupling between the Majoranaand quantum dot states, which emerges only in the topological phase and will influence the cur-rent through bent junctions and/or in the presence of magnetic fields in the quantum dot. Wefind that the characteristic behaviors of the supercurrent through this system are quite counterin-tuitive, remarkably differing from the resonant tunneling, e.g., through the similar (normal phase)superconductor–quantum dot–superconductor junction. Our analysis is carried out under the influ-ence of full set-up parameters and for both the 2 π and 4 π periodic currents. The present study isexpected to be relevant to future exploration of applications of the Majorana-nanowire circuits. PACS numbers:
I. INTRODUCTION
Majorana fermions (MFs) are exotic self-Hermitianparticles with non-Abelian statistics, hold a prop-erty with themselves as their own antiparticles andpromise robust building blocks for topological quantumcomputation . Remarkable insight predicts that MFscan emerge as novel excitations of Majorana zero modesor Majornana bound states in condensed matter systems,e.g., from the non-Abelian excitations in a 5/2 fractionalquantum Hall effect in semiconductor heterostructures ,and based on exotic superconductors where MFs corre-spond to zero-energy states of an effective Bogoliubov-deGennes Hamiltonian .More recent proposals employ the proximity effectfrom a conventional superconductor, either in nanowiresin the presence of strong spin-orbit interaction and Zee-man spliting , or in topological insulators . Theseefforts bring the MFs closer to experimental realizationand predict more reliable experimental signatures of theirpresence. Among the signatures include such as the half-integer conductance quantization , the zero-bias peak inthe tunneling conductance , and the 4 π Josephson ef-fect in superconductor-superconductor junctions .In particular, in order to distinguish MFs from otherquasi-particle states, some interest in recent years turnsto the spin selective
Andreev reflection .In this work we consider the hybrid system ofMajorana–quantum dot–Majorana junction which canbe realized from semiconductor nanowires in proximity-contact with s -wave superconductors, as schematicallyshown in Fig. 1. Similar systems of Majorana nanowirecoupled to quantum dots (QDs) have been investigatedfor phenomena such as teleportation , anomaly of con-ductance peak , characteristic signatures in currentnoise spectrum , and featured Josephson current . Ourpresent interest is the dc Josephson current under the in-fluence of spin-selective coupling between the Majoranaand QD states, which is most relevant to bent junctions and the presence of magnetic field in the QD area.Due to the helical property of MF, the MF at the endof the nanowire only couples to a unique spin state inthe normal region, e.g., the spin-up QD state as shownin Fig. 1. Actually, this spin-selective coupling is theorigin of the spin-dependent Andreev reflection . Inprevious studies, the set-up configuration is usually as-sumed to be either a single Majorana nanowire coupledto normal leads or QDs, or a straight Majorana–normalregion–Majorana junction with bent angle θ = 0. In bothcases, only the spin-up states (in the normal parts) coupleto the MF and contribute to the current. However, if theMajorana–QD–Majorana junction is not straight (with abent angle θ = 0 between the nanowires), the two MFsat the ends of the nanowires (see Fig. 1) will couple tospin states in the QD with different orientations, leadingthus to both the spin-up and spin-down states partic-ipating in the transport. This is anticipated to resultin different current behaviors of the straight and bentjunctions. On the application aspect (e.g. in the topo-logical quantum computations), the Majorana nanowireswill possibly have an orientational angle, or one wouldlike to employ this orientational angle to modulate thecharge transfer properties. This makes thus the bendingstructure studied in this work relevant to possible realcircuits.We will also analyze the effect of magnetic fields inthe QD area. This is motivated by viewing that, in or-der to induce the emergence of MF, magnetic fields areneeded to apply in the nanowires, which must spill overin the QD area owing to its close separations from thenanowires. Similar to the consequence of junction bend-ing, we expect that the non- z -axial direction magneticfield will make the spin-down state involved in the trans-port as well. We will show that, remarkably, owing to thespin-selective coupling, both the magnetic field in the QDarea and the junction bending will result in some coun-terintuitive behaviors of the Josephson current. For in-stance, in the case of QD level aligned with the Fermi en-ergy (under resonant tunneling), the Josephson current isto be strongly suppressed by the magnetic field and junc-tion bending. However, as the QD level deviates fromthe Fermi level (violates the resonant-tunneling condi-tion), the oscillation amplitude of the Josephson currentalways shows an enhanced value, together with robustjumps in the current-phase curves. q x y z z z ¢ QD SOC NW S O C N W L g L g R g R g S-wave SC surface S-wave SC surface
FIG. 1: (color online) Set-up sketch of the Majorana–QD–Majorana junction, realized by semiconductor nanowires incontact with the s -wave superconductors. The two nanowiresmay have a mutual orientation angle which is expected toaffect the supercurrent via the unique spin-selective couplingbetween the Majorana and QD states. II. MODEL AND METHODSA. Set-up Model
In this work we consider the simple setup as shown inFig. 1, where two semiconductor nanowires are connectedthrough a QD. The nanowires are in proximity contactedwith an s -wave superconductor. Then, the proximity-effect-caused superconductivity, together with the strongRashba spin-orbit coupling (SOC) and Zeeman splittinginside the nanowires, can possibly induce emergence of apair of MFs at the ends of each nanowire , as denotedin Fig. 1 by γ L , for the left wire and γ R , for the rightone.The tunnel-coupled system can be modeled by an ef-fective low-energy Hamiltonian H = H M + H dot + H dL + H dR , more explicitly with H M = iε L γ L γ L + iε R γ R γ R , (1a) H dot = X σ ε d d † σ d σ + ( d †↑ , d †↓ ) ~σ · ~B (cid:18) d ↑ d ↓ (cid:19) , (1b) H dL = ( λ L d ↑ − λ ∗ L d †↑ ) γ L , (1c) H dR = i ( λ R ˜ d ↑ + λ ∗ R ˜ d †↑ ) γ R . (1d)Here H M is the effective low energy Hamiltonian for thetwo pairs of Majorana states emerged at the ends of thetwo nanowires. Each Majorana pair may have nonzerocoupling energy, i.e., ε L ∼ e − l L /ξ L and ε R ∼ e − l R /ξ R ,where l L ( R ) and ξ L ( R ) are, respectively, the length ofthe nanowire and the superconductor coherence length. H dot denotes the QD Hamiltonian, with a single spa-tially quantized level ε d (tunable by gate voltage), and possibly affected via the Zeeman effect by magnetic field B = ( B x , B y , B z ) (Here we assume an arbitrary direc-tion of magnetic field). d † σ and d σ are the creation andannihilation operators of the QD electron with spin σ . H dL and H dR describe the tunnel coupling betweenthe dot and the nearby Majorana states. In general,the coupling amplitudes can be expressed as λ L ( R ) = | λ L ( R ) | e iφ L ( R ) / , where the phase factors are determinedby the phase of the substrate superconductors and theirdifference will result in the famous Josephson current.Another important issue to be noted is that, due to thehelical property of MF, the MF only couples to the spin-up state in the QD (defined in the same z -representationof the associated nanowire) . As mentioned in the in-troduction, our special interest in this work is to considerthe two nanowires not aligned in the same orientation,but with an angle θ , as shown in Fig.1. Thus, the leftand right MFs would couple only to the QD state spin-polarized, respectively, along the z and z ′ axes, with theassociated electron operators connected by the followingunitary transformation: (cid:18) ˜ d ↑ ˜ d ↓ (cid:19) = (cid:18) cos θ sin θ − sin θ cos θ (cid:19) (cid:18) d ↑ d ↓ (cid:19) . (2)This implies that, if the bent angle θ = 0, the spin-downstate also couples to the MFs.In practice, it would be convenient to convert theMFs to regular fermion representation, via the sim-ple transformation: c L = ( γ L + iγ L ) / √
2, and c R =( γ R + iγ R ) / √
2. The creation operators are their Her-mitian conjugate and satisfy { c L ( R ) , c † L ( R ) } = 1. Nowlet us apply the generalized Nambu representation, byintroducing the field creation and annihilation opera-tors as ψ † = ( d †↑ , d †↓ , d ↓ , d ↑ , c † L , c L , c † R , c R ), and ψ =( d ↑ , d ↓ , d †↓ , d †↑ , c L , c † L , c R , c † R ) T . Then, the Hamiltonian ofthe whole system can be rewritten as H = 12 ψ † H ψ , (3)where the Hamiltonian matrix has a block form given by H = H dd H dL H dR H Ld H LL H Rd H RR , (4)and each sub-matrix reads, respectively, H dd = ε d + B z B x − iB y B x + iB y ε d − B z − ε d + B z − B x + iB y − B x − iB y − ε d − B z , (5a) H LL = (cid:18) ε L − ε L (cid:19) , (5b) H RR = (cid:18) ε R − ε R (cid:19) , (5c) H dL = 1 √ − λ ∗ L − λ ∗ L λ L λ L , (5d) H dR = 1 √ λ ∗ R cos θ − λ ∗ R cos θ λ ∗ R sin θ − λ ∗ R sin θ λ R sin θ − λ R sin θ λ R cos θ − λ R cos θ . (5e)The other two off-diagonal sub-matrices are given by H Ld = H † dL and H Rd = H † dR . B. Josephson current
From the description of the above low-energy effectiveHamiltonian, it seems that this system, which accom-modates discrete energy levels, can support only unitaryevolution (quantum oscillations). However, this is nottrue. Note that the MFs at the ends of the nanowire areinduced via contact with superconductor which has theCooper pair reservoir. Even under zero bias voltage, theboth superconductors (see Fig. 1) can support station-ary dc Josephson current, if a phase difference betweenthe superconductors is maintained. This understandingallows us to apply the quantum transport formalism ofnonequilibrium Green’s function (nGF) to the presentset-up with only discrete energy levels.Following the nGF technique outlined in Ref. [37,38],the Josephson current reads I = eh Z d ǫ Re Tr { ˜ σ z [ ˜Σ( ǫ ) G d ( ǫ )] < } . (6)This result is expressed in the Nambu representation.Accordingly, ˜ σ z = diag { , , − , − } , which makes theKeldysh equation [ ˜Σ G d ] < = ˜Σ < G ad + ˜Σ r G In the numerical investigations, results will be cal-culated under the influence of a couple of set-up pa-rameters: the gate voltage, which would affect the QDlevel ε d ; the bent angle θ of the junction; the magneticfield in the QD area; and the overlap strength of theMajorana wavefunctions ( ε L/R ). We would like to set | λ L | = | λ R | = λ = 1 and scale all energies by λ . Also,we denote the Fermi level of the entire set-up as reference(zero) energy, the phase difference between the supercon-ductors as ∆ φ = φ R − φ L , and assume zero temperatureand identical nanowires ( ε L = ε R ). -2-1012 C u rr en t ( e / h ) d = 0 d =0.5 d =1.0 d =1.5 d =2.0 (a) E ne r g y d (b) FIG. 2: (color online) (a) Josephson current versus the phasedifference between the two superconductors for the straightMajorana–QD–Majorana junction ( θ = 0), under modulationof the QD energy level. The parameters ε L = ε R = 0 and themagnetic field B = 0. (b) Example of the associated in-gapenergy diagram of the Josephson junction in the topologicalphase, which can help us understand the current behaviorshown in (a). A. Effect of Dot Level Modulation We first display the result for the straight junction with θ = 0. In this case, the unique Majorana feature is re-vealed, as shown in Fig. 2(a), by the robust “jump” ofthe Josephson current at ∆ φ = π and the non-vanishingcurrent when modulating the dot level ε d (via modula-tion of the gate voltage applied). We see that, with theincrease of ε d (even far away from the Fermi energy, i.e., ε d = 0), the Josephson current only reduces by a smallamount, and the jump at ∆ φ = π always survives there,being robust against the variation of ε d . For instance, for ε d = 2 . 0, the amplitude of the Josephson current is onlyreduced to about 70% of the value at ε d = 0.The both features revealed here are entirely differentfrom the normal superconductor–QD–superconductorjunction. In the normal (trivial) case, the jump onlyappears at ε d = 0 and the current is to be stronglysuppressed when ε d deviates far away from the Fermienergy . We thus conclude that the features revealed inFig. 2(a) are closely associated with the superconductor-proximity-induced nontrivial (topological) phase of thenanowire where the MF emerges. In this case, the struc-ture of the energy diagram is as shown in Fig. 2(b), wherewe find the remarkable zero-energy crossing points at∆ φ = ± π which are responsible for the current “jumps”as observed in Fig. 2(a).Qualitatively speaking, the current is the sum of allcontributions from the occupied energy levels, witheach individual proportional to the derivative of theassociated energy curve (with respect to ∆ φ ) . Inthis work, we calculate the current using Eq. (8). Weintegrate the energy from −∞ to the Fermi level, im-plying that the system is always in thermal equilibriumespecially when ∆ φ passes through π . This treatmentcorresponds to certain relaxation mechanism involved,as a consequence of particle addition/loss from/to thesurrounding environment. As a consequence, we obtainthe 2 π periodic current (as a function of ∆ φ ), Fig. 2(a).Only under the fermion-number-parity conservation, theso-called 4 π periodic current can be expected. We willaddress this issue later in more detail. B. Effect of Magnetic Fields We now consider the possible effects of magnetic fieldin the dot area. For the Zeeman splitting of the dotlevel caused by the magnetic field along the z axis, wecan understand that the effect is the same as the electricgate-modulation of the dot level, as shown in Fig. 2(a).The effect of B x (magnetic field along the x axis) isshown in Fig. 3. For the ideal configuration with ε d = 0,the jump at ∆ φ = π disappears and evolves to a roundedtransition behavior. And, with the increase of B x , theJosephson current quickly decreases and vanishes at last.For ε d = 0, similar modulation effect of B x on the am-plitude of the Josephson current is caused by the spin-selective coupling between the Majorana and QD states,by noting that the magnetic field B x has a role of rotat-ing the electron spin, leading thus to a change of the z - component of the spin. However, in this more “relaxed”case ( ε d = 0), the jump behavior at ∆ φ = π survives.This indicates that the Majorana characteristic jump be-havior is robust against the deviation of the dot level fromthe Fermi energy, even under the influence of magneticfield in the QD area.Moreover, out of simple expectation, if the dot level ε d is far away from the Fermi energy (e.g. ε d = 1),the Josephson current is reduced by B x much moremodestly, than in the case of ε d = 0, see, e.g., theresults in Fig. 3(a) and (d). Indeed, this behavior is verycounterintuitive, since in usual case the largest currentalways appears at resonant tunneling , i.e., the dotlevel ε d at the Fermi energy. -6-30360 0.5 1.0 1.5 2.0(0)-6-3036 0.5 1.0 1.5 2.0 B x =0 B x =0.5 B x =1.0 B x =1.5 B x =2.0 (a) (b) C u rr en t ( e / h ) (c) (d) FIG. 3: (color online) Current-phase behavior under the influ-ence of the x -component of the magnetic field in the dot area.Results for several locations of the dot level are displayed: (a) ε d = 0; (b) ε d = 0 . 2; (c) ε d = 0 . 5; and (d) ε d = 1 . C. Effect of Junction Bending As remarked in the Introduction, the spin-selectivecoupling can be manifested also by bending the junc-tion, i.e., by altering the mutual angle ( θ ) between thenanowires. The results are displayed in Fig. 4, which ac-tually resemble what we observed in Fig. 3. That is, the“jump” at ∆ φ = π is rounded in Fig. 4(a) for ε d = 0,while in Fig. 4(b), (c), and (d) for ε d = 0, the “jump”survives and the maximum (amplitude) of the currentdecreases with the increase of the bent angle θ . Again,out of expectation, for finite θ (e.g. θ = π , π and π ),the amplitude of the Josephson current would increasewith the deviation of the dot level from the Fermi en-ergy. This means that, for a bent Majorana-Josephsonjunction, the stronger the resonance condition is violated,a larger supercurrent will flow through the junction. Wefinally mention that, for all cases, the parameter θ = π simply means rotating the right wire to the same side ofthe quantum dot and in parallel to the left wire, whichwould result in a completely vanished current. (b) (c) C u rr en t ( e / h ) (a) (d) FIG. 4: (color online) Effect of junction bending on thecurrent-phase behavior. Displayed are results for several lo-cations of the dot level: (a) ε d = 0; (b) ε d = 0 . 2; (c) ε d = 0 . ε d = 1 . D. Energy Diagram Based Interpretation We found through Figs. 2, 3 and 4 that, regardless ofthe magnetic field B and the mutual angle θ , the currentjump can safely survive and the amplitude of the currentmaintains a large value, if the dot level ε d violates theresonance condition. We may further understand the be-haviors as follows, with the help of the energy diagramsof Fig. 2(b) and Fig. 5.First, in Fig. 2(b) for ε d = 0 and θ = 0, the flat zero-energy level are of four-fold degeneracy, i.e., for the states -2-1012 -1 0 1 2-2 -1 0 1 2(-2)-2-1012 (a) E ne r g y (d) (c) (b) FIG. 5: (color online) Energy diagram of the junction fora couple of set-up parameters: (a) ε d = 0 , θ = 0 . π ; (b) ε d = 0 . , θ = 0 . π ; (c) ε d = 0 . , θ = 0 . π ; and (d) ε d =0 . , θ = 0. γ L , γ R , d ↓ and d †↓ . The other four phase-difference(∆ φ ) dependent eigen-energies are from the coupling ofthe states γ L , γ R , d ↑ and d †↑ . To be more specific,the two Majoranas γ L and γ R couple commonly to thespin-up dot state, resulting in the ‘crossing’ structure atzero energy at ∆ φ = ± π , which is similar to the resultof direct coupling of two Majoranas. It is just owing tothis zero-energy crossing (at the Fermi energy) that thelarge Josephson current with abrupt jump is resulted in,in the presence of ‘relaxation’ or thermal equilibrium.Second, for the dot level ε d = 0 but θ = 0 [seeFig. 5(d)], the Majorana states γ L and γ R have alsothe flat zero-energy (independent of ∆ φ ); however, theenergies of d ↓ and d †↓ move to ε d and − ε d , respectively.These would lead to an opening of the high-energy cross-ing at ∆ φ = 0 (or ± π ), but still keeping the zero-energycrossings at ∆ φ = ± π caused by γ L and γ R . As a re-sult, the Josephson current keeps a large value and thejump survives, because the current are dominantly con-tributed by the zero-energy crossing states.Third, if θ = 0 and ε d = 0 [see Fig. 5(a)], we see thatthe zero-energy degeneracy of γ L and γ R at ∆ φ = ± π isremoved. The disappearance of the zero-energy crossingsleads to a strong reduction of the Josephson current androunding the jump to smooth transition, as shown inFig. 4(a).Finally, for the case of both θ = 0 and ε d = 0, as shownin Fig. 5(b) and (c), the absence of efficient energy levelinteraction between the Majorana and QD states (owingto ε d = 0) does not remove the zero-energy degeneracy of γ L and γ R at ∆ φ = ± π . Then, the zero-energy crossingstructure of the energy spectrum at ∆ φ = ± π results in alarge Josephson current with abrupt jumps, despite thatthe dot level ε d deviates far from the Fermi energy. E. Effect of Majorana Interaction Below we show that, in order to observe the featuredbehaviors discussed above, the overlap of the Majoranawavefunctions at the ends of the same nanowire shouldbe negligibly small. Or, equivalently, the Majorana in-teraction should be negligibly small. In Fig. 6 we dis-play the result for ε L = ε R = 0, which correspondsto nonzero coupling between the two Majoranas in thesame nanowire. Indeed, we find that the amplitude ofthe Josephson current is strongly reduced and the jumpdisappears, with the increase of ε L,R [see Fig. 6(a) and(c)]. The basic reason is that the Majorana interactionin the same nanowire destroys the zero-energy Majoranastate.We have also checked that the large Josephson currentwith jump behavior cannot be restored by altering thedot level, even with ε d in resonance with ε L and ε R [seeFig. 6(c) and (d)]. The reason is that, if ε L = ε R = 0, thezero-energy Majorana states γ L and γ R are destroyedand the zero-energy crossings at ∆ φ = ± π disappear. Asa consequence, the jumps at ∆ φ = ± π are replaced byrounded transitions, and the current is strongly reduced.In Fig. 6(b) and (d), taking the current at ∆ φ = π/ θ dependence of current, in the pres-ence of Majorana coupling ( ε L,R = 0). From this resultwe see clearly that, in order to obtain a larger super-current, we should make the nanowire longer than thesuperconductor coherence length to ensure the emergenceof Majorana zero modes at the ends of the nanowire. -2 -1 0 1 2012340.0 0.5 1.0 1.5 2.0-6-3036 01234-6-3036 (d) C u rr en t ( e / h ) C u rr en t ( e / h ) d =1.0 (d) (c) d =1.0 L = R =0 L = R =0.5 L = R =1.0 L = R =1.5 L = R =2.0 (b) d = 0 (a) d = 0 FIG. 6: (color online) (a) and (c): Effect of Majorana inter-action (in the same nanowire manifested by nonzero ε L,R ),which would reduce the Josephson current and destroy thejump behavior at ∆ φ = ± π . (b) and (d): Nanowire-orientation-angle dependence of the current at ∆ φ = 0 . π .Results for dot level ε d = 0 and ε d = 1 are presented, respec-tively. F. π Periodic Current So far we have assumed that the system always relaxesto a thermal equilibrium when we vary the phase dif-ference ∆ φ . In particular, at the zero-energy crossingsat ∆ φ = ± π , this relaxation is accompanied by eitheran addition or a loss of a single particle, which there-fore changes the parity of the particle numbers (i.e., thefermion parity). If such relaxation channel is blocked orthe fermion parity is conserved, rather than the 2 π pe-riodic current we obtained above, a remarkable 4 π peri-odic Josephson current can be expected, which is usuallyregarded as one of the most prominent Majorana signa-tures.The 4 π periodic current can be calculated as well byusing Eq. (8), based on the following technique: When weincrease ∆ φ after passing through ± π , for the state oc-cupation of the levels crossing at zero energy (the Fermilevel) at ∆ φ = ± π , we replace the occupation of the lowerlevel under the Fermi energy by its counterpart above theFermi level (owing to the fermion parity conservation),while satisfying the condition of thermal equilibrium af-ter this replacement.The results of the 4 π periodic current are shown inFig. 7, for one period. Compared with the 2 π periodic current, we find that the jumps at ∆ φ = ± π disappearfor all the 4 π periodic currents. However, for the case of θ = 0 and ε d = 0 similar jumps may appear near (butnot at ) ∆ φ = ± π , as observed in Fig. 7(b) and (c), owingto the accidental energy crossings at the specific phases.Again, along the increase of the angle θ , the amplitudeof the current decreases. However, the current is morestrongly suppressed in the case of ε d = 0, while for largerdeviation of ε d (from the Fermi level) the current is lessreduced. -2 -1 0 1 2(-2)-8-4048-8-4048 -1 0 1 2 (b) (c) C u rr en t ( e / h ) (a) (d) FIG. 7: (color online) The so-called 4 π periodic current fora couple of set-up parameters, i.e., the nanowire-orientation-angles (as shown by the inset) and the dot levels (a) ε d = 0,(b) ε d = 0 . 2, (c) ε d = 0 . 5, and (d) ε d = 1 . IV. SUMMARY To summarize, we have investigated the dc Joseph-son supercurrent through the Majorana–quantum dot–Majorana junction. Our particular interest is the con-sequence of the unique spin-selective coupling betweenthe Majorana and dot states, which emerges only inthe topological phase and will drastically influence thecurrent through bent junctions and/or in the presenceof magnetic fields in the dot area. Differing from thetypical resonant tunneling behavior of the supercurrentthrough similar system in normal phase such as thesuperconductor–quantum dot–superconductor junction,we uncovered some counterintuitive results associatedwith the exotic nature of the Majorana fermion.For instance, even for a straight junction and with-out magnetic field in the dot area, when the dot leveldeviates considerably from the Fermi energy, the Joseph-son supercurrent keeps a large amplitude of oscillationwith the superconductor phase difference ∆ φ and revealsabrupt jumps of current at ∆ φ = ± π . Drastically, thisresult differs from the usual resonant tunneling behav-ior through similar system in normal phase. For a bentjunction and/or in the presence of magnetic field in thedot, richer unexpected behaviors are found. In resonance(the dot level aligned with the Fermi energy), we find thatthe supercurrent is to be strongly reduced by either thejunction bending or the magnetic fields in the dot. At thesame time, the current jumps at ∆ φ = ± π are rounded.However, if the dot level deviates from the Fermi energy(i.e., violates the resonant tunneling condition), the su-percurrent can, on the contrary, maintain a large ampli-tude of current and the current jumps robustly survive at∆ φ = ± π , even under the influence of junction bendingand magnetic fields in the dot. We expect these findingsto be useful in future design of novel circuit devices basedon quantum dots and Majorana nanowires. Acknowledgements This work was supported by NBRP of China (GrantNo. 2015CB921102), NSF-China under Grants No.11675016, No. 11274364 and No. 11574007, the Bei-jing Natural Science Foundation under No. 1164014, theChina Postdoctoral Science Foundation funded project(Grant No. 2016M591103) and the Fundamental Re-search Funds for the Central Universities. ∗ [email protected] † [email protected] E. Majorana, Nuovo Cimento , 171 (1937). F. Wilczek, Nat. Phys. , 614 (2009); M. Franz, Physics , 24 (2010). C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. DasSarma, Rev. Mod. Phys. , 1083 (2008). S. R. Elliott and M. Franz, Rev. Mod. Phys. , 137(2015). G. Moore and N. Read, Nucl. Phys. B , 362 (1991). A. Y. Kitaev, Phys. Usp. , 131 (2001). D. A. Ivanov, Phys. Rev. Lett. , 268 (2001). M. T. Deng, C. Yu, G. Huang, M. Larsson, P. Caroff, andH. Q. Xu, Nano Lett. , 6414 (2012). A. D. K. Finck, D. J. Van Harlingen, P. K. Mohseni, K.Jung, and X. Li, Phys. Rev. Lett. , 126406 (2013). E. J. H. Lee, X. Jiang, M. Houzet, R. Aguado, C. M.Lieber, and S. De Franceschi, Nat. Nano. , 79 (2014). V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A.M. Bakkers, and L. P. Kouwenhoven, Science , 1003(2012). Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. ,177002(2010). R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev.Lett. , 077001(2010). G. Y. Huang, M. Leijnse, K. Flensberg, and H. Q. Xu,Phys. Rev. B , 214507(2014). L. Fu and C. L. Kane, Phys. Rev. Lett. , 096407 (2008). A. R. Akhmerov, J. Nilsson, and C. W. J. Beenakker, Phys.Rev. Lett. , 216404 (2009). C. K. Chiu, M. J. Gilbert, and T. L. Hughes, Phys. Rev.B , 144507 (2011). J.-P. Xu, M.-X. Wang, Z. L. Liu, J.-F. Ge, X. Yang, C.Liu, Z. A. Xu, D. Guan, C. L. Gao, D. Qian, Y. Liu, Q.-H.Wang, F.-C. Zhang, Q.-K. Xue, and J.-F. Jia,Phys. Rev. Lett. , 017001 (2015). S.-Y. Xu, N. Alidoust, I. Belopolski, A. Richardella, C. Liu,M. Neupane, G. Bian, S.-H. Huang, R. Sankar, C. Fang,B. Dellabetta, W. Dai, Q. Li, M. J. Gilbert, F. Chou, N.Samarth, and M. Z. Hasan, Nat. Phys. , 943 (2014). M. Wimmer, A. R. Akhmerov, J. P. Dahlhaus, and C. W.J. Beenakker, New J. Phys. , 053016 (2011). K. T. Law, P. A. Lee, and T. K. Ng, Phys. Rev. Lett. , 237001 (2009). A. R. Akhmerov, J. P. Dahlhaus, F. Hassler, M. Wimmer,and C. W. J. Beenakker, Phys. Rev. Lett. , 057001(2011). K. Sengupta, I. Zutic, H.-J. Kwon, V. M. Yakovenko, andS. Das Sarma, Phys. Rev. B , 144531 (2001). J. Liu, A. C. Potter, K. T. Law, and P. A. Lee, Phys. Rev.Lett. , 267002 (2012). L. Fu and C. L. Kane, Phys. Rev. B , 161408(R) (2009). J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A.Fisher, Nat. Phys. , 412 (2011). S.-F. Zhang, W. Zhu and Q.-F. Sun, J. Phys.: Condens.Matter , 295301 (2013). J. J. He, T. K. Ng, P. A. Lee, and K. T. Law, Phys. Rev.Lett. , 037001 (2014). A Haim, E Berg, F. von Oppen, and Y. Oreg, Phys. Rev.Lett. , 166406 (2015). T. Kawakami and X. Hu, Phys. Rev. Lett. , 177001(2015). H.-H. Sun, K.-W. Zhang, L.-H. Hu, C. Li, G.-Y. Wang,H.-Y. Ma, Z.-A. Xu, C.-L. Gao, D.-D. Guan, Y.-Y. Li, C.Liu, D. Qian, Y. Zhou, L. Fu, S.-C. Li, F.-C. Zhang, andJ.-F. Jia, Phys. Rev. Lett. , 257003 (2016). S. Tewari, C. Zhang, S. Das Sarma, C. Nayak, and D. H.Lee, Phys. Rev. Lett. , 027001 (2008). D. E. Liu and H. U. Baranger, Phys. Rev. B , 201308(2011). Y. Cao, P. Wang, G. Xiong, M. Gong, and X.-Q. Li, Phys.Rev. B , 115311 (2012). M. Lee, J. S. Lim, and R. Lopez, Phys. Rev. B , 241402(2013). Q.-F. Sun and X. C. Xie, J. Phys.: Condens. Matter ,344204 (2009). Q.-F. Sun, B.-G. Wang, J. Wang, and T.-H. Lin, Phys.Rev. B , 4754 (2000). U. Meirav, M. A. Kastner, and S. J. Wind, Phys. Rev.Lett. , 771 (1990). Y. Meir, N. S. Wingreen, and P. A. Lee, Phys. Rev. Lett. , 3048 (1991). P. Wang, J. Liu, Q.-F. Sun, and X. C. Xie, Phys. Rev. B91