Supercurrent detection of topologically trivial zero-energy states in nanowire junctions
SSupercurrent detection of topologically trivial zero-energy states in nanowire junctions
Oladunjoye A. Awoga, Jorge Cayao, ∗ and Annica M. Black-Schaffer Department of Physics and Astronomy, Uppsala University, Box 516, S-751 20 Uppsala, Sweden (Dated: September 16, 2019)We report the emergence of zero-energy states in the trivial phase of a short nanowire junctionwith strong spin-orbit coupling and magnetic field, formed by strong coupling between the nanowireand two superconductors. The zero-energy states appear in the junction when the superconductorsinduce a large energy shift in the nanowire, such that the junction naturally forms a quantum dot,a process that is highly tunable by the superconductor width. Most importantly, we demonstratethat the zero-energy states produce a π -shift in the phase-biased supercurrent, which can be usedas a simple tool for their unambiguous detection, ruling out any Majorana-like interpretation. Majorana bound states (MBSs) in topological super-conductors have generated remarkable interest due totheir potential applications in fault tolerant quantumcomputation [1–3]. A promising route for engineeringthe topological phase is based on nanowires (NWs) withstrong Rashba spin-orbit coupling (SOC) and proximity-induced s -wave superconductivity, with MBSs emergingat the NW ends for sufficiently large magnetic fields [4–6].Initial issues, such as a soft superconducting gap [7–13],of the first experiments [14–19] have been solved throughthe fabrication of high quality interfaces between the NWand external superconductors (SCs) [20–30].Despite the advances, there is still no consensuswhether MBSs have been observed or not. In fact,recent reports show that trivial zero-energy Andreevbound states (ABSs) from e.g. chemical potential inho-mogeneities, appearing well outside the topological phase[31–34], can also lead to a 2 e /h quantized conductance[35, 36], a feature previously attributed solely to MBSs[37]. This controversy can at least partially be attributedto oversimplified models used to describe the experi-ments. Indeed, a common treatment of superconductiv-ity has been to simply add an induced superconductinggap into a one-dimensional (1D) NW model, ignoring allother effects caused by coupling a SC to a NW.A more accurate approach is to study the wholeNW+SC system, since the achieved high-quality inter-faces result in a strong coupling between NW and SCand thus the SC generates both an induced gap and af-fect other NW parameters. Importantly, the NW ener-gies are shifted when the coupling between the SC andNW is strong due to the lowest states having a largeweight in the SC [38–42]. This results in an effectivechemical potential µ eff in the NW, which regulates whenthe NW reaches the topological phase. Therefore usinga NW+SC model is crucial for gaining further insightsinto the experimental situation.In this Letter we study the whole NW+SC systemand find trivial zero-energy ABSs spontaneously emerg-ing in a NW strongly coupled to two SCs forming a shortsuperconductor-normal-superconductor (SNS) junction.The zero-energy ABSs appear in the junction when theSCs induce a large µ eff in the NW, such that the junction forms natural quantum dot (QD). The QD formation oc-curs at regular intervals, every Fermi wavelength incre-ment in SC width, and is thus predictable. By simplyregulating the width of the SCs, we can tune the NWfrom an ideal regime with no energy shifts, to forminga QD or even a potential barrier (PB) at the junction.The formation of the QD and its zero-energy ABSs istherefore very different from previous situations wherethe QD was simply put in by hand [36, 43–48]. Most im-portantly, we find that the trivial zero-energy QD statesproduce a π -shift in the phase-biased supercurrent, whileMBSs appearing in the topological phase do not. Thusthe Josephson effect in short SNS junctions offers a re-markably powerful, yet simple tool for distinguishing be-tween trivial zero-energy states and MBSs. Model .— We use a 1D NW with strong SOC with theright (R) and left (L) parts strongly coupled to the middleof two 2D conventional SCs, leaving only the central partof the NW uncoupled and forming a short SNS junction,see Fig. 1(a). By varying a magnetic field parallel to theNW we easily tune the topology of the junction. TheHamiltonian is thus H = H NW + H L SC + H R SC + H S − W , with H NW = L NW (cid:88) x =1 ,σσ (cid:48) d † xσ ( ε NW δ σσ (cid:48) + Bσ xσσ (cid:48) ) d xσ (cid:48) − L NW − (cid:88) x =1 ,σ d † xσ ( t NW δ σσ (cid:48) − iα NW σ yσσ (cid:48) ) d x +1 ,σ (cid:48) + H.c. , H R / L SC = (cid:88) i , j ,σ c † i σ (cid:104)(cid:0) ε sc δ ij − t sc δ (cid:104) i , j (cid:105) (cid:1) c j σ + ∆ R / L sc (i) c † i ↑ c † i ↓ (cid:105) + H . c ., H S − W = − Γ (cid:88) i L NW (cid:88) x =1 ,σ c † i σ d xσ δ i y , Ly +12 δ i x ,x + H . c ., where d xσ is the destruction operator for a particle withspin σ at site x in the L NW long NW, while c i σ is thedestruction operator at site i = ( i x , i y ) in the 2D SCswith length L x , width L y . Here, (cid:104)· · · (cid:105) implies nearestneighbor sites, t represents the nearest neighbor hoppingand µ the chemical potential, such that the on-site en-ergies ε NW = 2 t NW − µ NW , ε sc = 4 t sc − µ sc . In the NW α NW = α R / a is the SOC, with α R the SOC strengthand a the lattice constant, and B is the effective Zeemancoupling caused by the magnetic field with σ ν a Pauli a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p (b)(a) + FIG. 1. (a) 1D NW (cyan) coupled to the middle of two 2DSCs (red) by Γ. A short central region of the NW is left un-coupled, giving a short SNS junction with a φ superconduct-ing phase difference. (b) Effective chemical potential profiledeep into the S parts of the NW as a function of SC width L y . Markers are representative points at which three casesare studied: ideal (triangle), PB (cross), and QD (dot). matrix. The SCs have an onsite s -wave superconductingorder parameter ∆ R / L sc (i) = | ∆ sc | e iφ R / L , with φ R / L beingthe SC phase. Finally, H S − W is the NW-SC tunnelingHamiltonian with finite coupling strength Γ, wheneverthe NW touches either SCs.We solve the Hamiltonian within the Bogoliubov-deGennes framework [49] using parameters in units of t sc : µ sc = 0 . µ NW = 0 . t NW , α NW = 0 . t NW , t NW = 4, whichaccounts for the small NW effective mass and mismatch-ing Fermi wavevectors in NW and SC, and being closeto realistic values. We also set ∆ sc (i) = 0 . φ R = 0,and φ L = φ . Here, the strong coupling regime, withthe induced gap in the NW close to ∆ sc , is reachedaround Γ = 0 .
7. For smaller ∆ sc and µ sc , a smaller Γachieves strong coupling. Further, we use L x = 520 a , L NW = 1000 a , and keep the N-junction 2 a long, to reachrealistic sizes with the outer ends of the NWs well withinthe SCs. The width of the SC, L y , is varied in order totune the influence of the SC on the NW [39–42]. We haveverified that our results remain qualitatively unchangedfor ∆ sc and Γ both being smaller (or even larger), aswell as when ∆ sc (i) is calculated self-consistently [50–54]. Our results also do not depend on L x , L NW , junctionlength, provided L x , L NW are longer than the supercon-ducting coherence length and the junction is short, seeSupplementary Material (SM) for more information [55].As a result of strong coupling to the SC, all inherentNW parameters are renormalized [38–42, 56]. Most im-portant is an energy shift of the NW bands [42]. Weencode this by an effective chemical potential µ eff , whichwe define as the energy of the bottom of the hybridizedsubband closest to the Fermi energy (since superconduc-tivity occurs around the Fermi energy). We extract µ eff deep in the S regions of the NW and find that it oscil-lates as a function of L y , see Fig. 1(b). The oscillationsare due to a mismatch between the SC and NW bands,with period (here 8 a ) given by the SC Fermi wavelength.Thus, by changing L y we can easily tune through a rangeof µ eff . Low-energy spectrum .— When the S regions of the NWget a non-zero µ eff , the properties of the SNS junction (a)(b)(c) (e) TopologicalTrivial (f) + QDPB + ideal (d) TrivialTopologicalTrivial Trivial QD FIG. 2. Zeeman field-dependent spectrum at φ = 0 for ideal( L y = 41 a ) (a), PB ( L y = 11 a ) (b) and QD ( L y = 21 a )(c) cases. Vertical dashed green lines in (a-d) mark topolog-ical phase transition, while red arrow in (c) marks start ofzero-energy levels. (d) Local spin projection at the junction S ( x ) x = L NW / in lowest level E for cases (a-c) as a function of B/B c . Cyan/magenta marks spin up/down while marker sizedenotes magnitude. (e,f) Color plot of E as a function ofΓ and B for L y = 41 a (e) and L y = 21 a (f) cases. Greenline marks topological phase transition, red line start of thesupercurrent π -shift, and dotted white line Γ = 0 .
7. Filledcircles in (f) denotes colored markings in (c). change. We show this first by studying the Zeeman de-pendent low-energy spectrum at φ = 0 for three valuesof the SC width L y , see Fig. 2(a-c). The common char-acteristic in all three cases is that the spectrum exhibitsa sizable gap at zero B , indicating the presence of super-conductivity, which then closes and reopens at the criticalfield B c signaling the topological phase transition (greendashed line). By calculating the topological invariant fora NW coupled to a single SC [57] we verify that the gapclosure in Fig. 2(a-c) matches the topological phase tran-sition point. In the topological phase the SNS systemhosts a pair of MBSs, with zero energy, one at each endof the NW (outer MBS), for all cases. Since µ eff changesthe NW properties, we find that B c also changes some-what with L y .Remarkably, there is a very strong effect of L y onthe low-energy spectrum inside the junction, resulting inthe emergence of additional low-energy states below B c .These can be understood when comparing µ eff in the Sregions of the NW to the native chemical potential µ NW ,which is still the relevant energy in the N region. In fact,in Fig. 2(a) the low-energy spectrum does not exhibitany unusual features, since here µ eff ≈ µ NW (triangle inFig. 1(b)). We refer to this regime as the ideal case. How-ever, when µ eff < µ NW (cross in Fig. 1(b)), the junctionacts as a potential barrier (PB) and we see in Fig. 2(b)that such PB junction can host discrete low-energy lev-els in the trivial phase. Finally, when µ eff > µ NW (dot inFig. 1(b)), there is instead a quantum dot (QD) profilein the junction. Remarkably, this QD accommodates aclear single zero-energy crossing in the trivial phase, seeFig. 2(c).We here stress that the QD with a zero-energy crossingin the trivial phase emerges spontaneously at the junc-tion, just due to strong NW-SC coupling and tuning L y .We have numerically verified that the QD zero-energystates occur for ˜ α < B < B c , where ˜ α is the renor-malized SOC in the NW (dependent on L y and Γ, here˜ α ≈ . α NW ), see SM [55]. Zero-energy states have previ-ously been reported in simple 1D models with a QD putin by hand [32, 36, 44, 47, 48, 58], producing signaturessimilar to MBSs and thus challenging attempts trying todistinguish between such trivial zero-energy levels andMBSs [36, 42, 59, 60]. In our work the QD instead devel-ops naturally and we also find that the trivial zero-energycrossings appear solely in the QD regime, not in the PBor ideal regimes.Further insights can be obtained from the local spinprojection along B (i.e. the x -component), in the lowestlevel E states, which is given by S ( x ) x = v ∗ x ↑ u x ↓ + u ∗ x ↓ v x ↑ ,and superscript/subscript denotes component/positionand u xσ , v xσ are the wave function amplitudes at posi-tion x [61–64]. In Fig. 2(d) we show S ( x ) x at the junction,i.e. x = L NW /
2, with marker size denoting the magni-tude. S ( x ) L NW / vanishes in the topological phase as thelowest level, E , is then the outer MBSs. However, inthe trivial phase the zero-energy crossing in the QD caseis accompanied by an exchange of spins in the occupiedstate. Such spin exchange does not occur in the othercases, leading to a fundamental difference in the spinproperties of the QD and PB cases, even if they both hostdiscrete low-energy states below the quasi-continuum.We finally analyze the size of the regime where trivialzero-energy QD states are observed. In Fig. 2(e,f) we plot E as a function of Γ and B for the cases in Fig. 2(a,c),respectively. From the low-energy spectrum, we iden-tify the topological phase transition (green line) and thebeginning of the zero-energy state QD regime (red line).The QD regime forms a triangular region which is clearlyenlarged with Γ. Remarkably, Fig. 2(e) shows that evenwide SCs can host a QD regime with trivial zero-energystates for strong enough couplings (white dotted linemarks Γ = 0 . (a) (b)(c) (d) + (e) (f) FIG. 3. (a-d) Phase-dependent low energy spectrum in theQD case ( L y = 21 a ), obtained at the color-marked B val-ues in Fig. 2(c). Inset in (a): scaled free energy ˜ F =( F − F min ) / ( F max − F min ) for (a-d), with F min / max the min-imum/maximum of F in each case. Probability density of thelowest state, | Ψ | · , in (a-d) at φ = 0 (e) and φ = π (f). conclude that SNS junctions readily form natural QDshosting trivial zero-energy states in the strong couplingregime. Phase-dependence .— Next we allow for a finite phase φ across the SNS junction. In particular, we study thephase-dependent energy spectrum for the QD case inFig. 2(c) at the B -values identified by the colored bars.At very low B (blue) we find ABSs detached from thequasi-continuum and exhibiting the usual cosine behav-ior [65, 66], see Fig. 3(a). These lowest energy statesare localized at the junction for both φ = 0 , π , see blueline in Figs. 3(e,f). On the other hand, in the topolog-ical phase at very large B (green) four MBSs appear inthe system: two dispersionless outer MBSs and at φ = π also two MBSs located in the junction (inner MBSs), seeFig. 3(d) for the energy spectrum and Figs. 3(e,f) for thewave function probabilities. In both the low B trivialand high B topological regimes, the lowest level reachesmaximum negative energy at φ = 0. The SNS junc-tion is therefore in the 0-state because the free energy, F = (cid:80) n< E n , is minimized at φ = 0, see blue and greenlines in the inset of Fig. 3(a).It is at intermediate B in the trivial phase that dra-matic changes takes place. First, the ABSs move to-wards zero energy with increasing B and start to cross,see Fig. 3(b). As a consequence, the free energy, plotted (c) quasi-continuum (dashed) (d)(a) (b) FIG. 4. (a) Colorplot of supercurrent for QD case ( L y = 21 a )as a function of φ and B . Topological phase transition (greendashed line), beginning of ABS crossings in phase-dependentenergy spectrum (magenta), and zero-energy crossing at φ =0, i.e. red arrow in Fig. 2(c) (white). Total supercurrent (b),with contributions from E (c) and E (d) energy levels atthe color-marked B values Figs. 2(c), repeated in (a). in gold in the inset in Fig. 3(a), has a global minimumat φ = 0 and a local minimum at φ = π . The junctionis thus in a 0 (cid:48) -state [67]. Further increasing B we findthat the global and local minima interchanges, eventuallyreaching the situation in Fig. 3(c). Here the zero-energycrossing is at φ = 0, implying that a full π -shift has oc-curred in the low-energy spectrum. As a consequence,this junction is in a π -state, since the minimum of F isnow at φ = π , see red Fig. 3(a) inset. At φ = 0 theABSs are localized at the junction, as in all other casesin the trivial phase, while at φ = π the lowest energystate is completely delocalized because of mixing withthe quasi-continuum, see red in Fig. 3(e,f).We find that the π -state always emerges when the SNSjunction hosts a pair of QD states with zero-energy cross-ings. In essence this is because the QD forces the ABSto be at or close to zero energy for φ = 0. We alsonote that the QD introduces a phase-dependence for thequasi-continuum, unlike in conventional short junctions[68]. We have also verified that the ideal and PB cases donot exhibit any π -states, see SM [55]. Thus, the phase-dependent energy spectrum offers a remarkably clear dif-ferentiation between topologically trivial zero-energy QDlevels and MBSs. Current-phase relationship .— To perform a direct de-tection of the QD trivial zero-energy states we considerthe junction supercurrent I ( φ ), obtained from I ( φ ) = I ∂F∂φ , where I = e/ (cid:126) . Figure 4(a) shows a color plot of¯ I = I ( φ ) /I as a function of φ and B for the QD casein Fig. 2(c). For a complete understanding of how the QD levels contribute to I ( φ ), we also plot both the totalcurrent and the contributions from the lowest ( E ) andfirst excited ( E ) energy levels in Figs. 4(b,c,d) for thesame B values analyzed in Fig. 3.At low B in the trivial phase I ( φ ) displays the usualsin( φ )-like behavior. This is the 0-state, where E givesthe dominating contribution to the supercurrent, albeit E also give a small positive contribution, see blue linein Fig. 4(b,c,d). Beyond the topological phase transition(green dashed line) the situation is also easy to under-stand. Here I ( φ ) has a characteristic sawtooth profile at φ = π due to the special zero energy behavior of the innerMBS at φ = π , which has been proposed as a signatureof true MBSs in short SNS junctions [65, 66].Between the magenta and white lines in Fig. 4(a), wefind a region with a discontinuous I ( φ ), which is causedby the ABS crossings in Fig. 3(b). Here, the E levelsare strongly dispersive with φ leading to the largest con-tributions to I ( φ ), see gold in Fig. 4(c). Finally, betweenthe dashed white and green lines in Fig. 4(a), we find afull sign-reversal for the supercurrent, with the white linecorresponding to the red arrow in Fig. 2(c) indicating thezero-energy crossing at φ = 0. This π -shifted supercur-rent arises from the special behavior of the low-energyspectrum: the lowest ABSs exhibit maximum energy at φ = π , see Fig. 3(c), instead of a minimum as is the casefor conventional junctions [68]. Thus the E level con-tributes strongly to the π -shifted supercurrent, as alsoseen in red in Fig. 4(c). Due to the presence of the QDlevels, the quasi-continuum also gives a π -shifted contri-bution to I ( φ ). For the ideal and PB junctions, the ABSenergy spectrum only exhibits 0, 0 (cid:48) , π (cid:48) -states, but neverthe π -state and thus we never see a π -shifted supercur-rent. Some signatures of the QD and PB junctions canalso be captured by the critical current but not as clearas the π -shift, see SM [55].For SNS junctions with trivial zero-energy crossingswe always find a π -shifted supercurrent, independent onany zero-energy pinning after the crossing. These zero-energy levels, appearing in the QD regime, are howeversomewhat sensitive to SOC [44], with very large SOCinducing level repulsion, which gaps the spectrum andthus destroys the supercurrent π -shift, see SM [55]. Inter-estingly, QD levels in clearly non-topological Josephsonjunctions have previously been shown to change the stateof the junction from 0 to π with increasing magnetic fieldand also associated with a spin exchange [19, 67, 69–75],fully consistent with our findings.In conclusion, we demonstrate the emergence of zero-energy states in the trivial phase of short SNS NW junc-tions, due to strong NW-SC coupling causing a QD for-mation in the NW and tunable by the SC width. Mostsignificantly, these zero-energy states produce a π -shiftin the phase-biased supercurrent, making them easilydistinguishable from MBSs appearing in the topologicalphase.We thank C. Reeg and C. Schrade for useful discus-sions and M. Mashkoori for helpful comments on themanuscript. 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Estrada Salda˜na, R. ˇZitko, J. P. Cleuziou, E. J. H.Lee, V. Zannier, D. Ercolani, L. Sorba, R. Aguado, andS. De Franceschi, “Charge localization and reentrant su-perconductivity in a quasi-ballistic inas nanowire coupledto superconductors,” Sci. Adv. (2019), 10.1126/sci-adv.aav1235. upplemental Material: Supercurrent detection of topologically trivial zero-energystates in nanowire junctions Oladunjoye A. Awoga, Jorge Cayao, ∗ and Annica M. Black-Schaffer Department of Physics and Astronomy, Uppsala University, Box 516, S-751 20 Uppsala, Sweden (Dated: August 29, 2019)
In this supplementary material we provide details to further support the results and conclusions of the main text.We first give detailed demonstrations of the effective chemical potential, µ eff , and the renormalization of the spin-orbitcoupling (SOC). Then we show how different QD systems, produced by different SC widths or junction lengths, canhave varying energy spectra, but still always result in a π -shifted supercurrent in the short to intermediate junctionregime. However, the π -shift in the supercurrent is destroyed when SOC is extremely large. Then we turn to thepotential barrier (PB) case, and show explicitly that the phase-dependent energy spectrum and supercurrent do notsupport π -shift. We also show that the trivial zero or near-zero energy levels leave traces in the critical currents,albeit not as clear as the π -shifted supercurrent. Further, by changing the model parameters we show that our resultsare robust even at realistic parameters and under self-consistent calculations. Finally, we show that an 1D effectivemodel with spatially inhomogeneous effective chemical but put-in by hand exhibits the same π -shift feature in thesupercurrent. Effective chemical potential
In this section we provide more details in order to understand the origin of µ eff , which is plotted in Fig. 1(b) in themain text. For simplicity we consider only one half of the system, that is, instead of a NW coupled to two SCs we onlytake one of the SCs since the coupling is the same for both SCs. The Hamiltonian is the same as that presented in themain text except now we have a single SC. We further assume that the SC and the NW are infinitely long, with theNW terminated at the edges of the SC, as schematically illustrated in Fig. S1(a). Then the system is translationallyinvariant along the NW direction so that we can Fourier transform with k x being a good quantum number, while inthe direction perpendicular to NW, L y is still finite. This approach grants access to the individual reciprocal spacesubbands of the NW and SC (even after hybridization), making it easy to extract renormalized parameters. Since∆ sc is a small parameter, we can extract all renormalized parameters in the normal state. (a) (b) (c) (d) FIG. S1. (a) Schematics of a 1D NW, with length L NW and parallel Zeeman field B coupled to a single 2D SC, of width L y and length L x , with coupling strength Γ. Energy subbands for L y = 11 a and B = 0 for Γ = 0 (b) and Γ = 0 . µ eff . Here dashed red line marks the bottom of the NW band before hybridization. Minimumof the effective band gives the spin-orbit energy, while the momentum at this point is the renormalized SOC momentum ˜ k so ,indicated by red marker in (d). Subband colors in (b-d) are set by the eigenstate weight in the NW for each subband. At Γ = 0, the energy spectrum of the 2D SC consists of a discrete set of doubly (spin) degenerate subbands, due tothe finite width, shown for L y = 11 a in Fig. S1(b). The NW, on the other hand, has a pair of non-degenerate bandsdue to the spin mixing effect of the SOC, except at k x = 0 where the SOC vanishes. This is the situation beforeany hybridization or renormalization. At finite coupling, Γ = 0, all the parameters of the NW are renormalized as aconsequence of NW-SC hybridization [1–4]. In particular, the subbands of the SC and NW are hybridized and shiftedin energy with respect to the subbands at Γ = 0, see Fig. S1(c), where the eigenstate weight in the NW of each band2is coded in color. We find that the band shift directly modifies the native chemical potential of the wire, µ NW , leadingto an effective chemical potential µ eff . To a good first approximation µ eff can be measured at k x = 0 as the distanceto the bottom of the subband closest to the Fermi energy (here E = 0), since the superconducting gap opens aroundthe Fermi energy, see Fig. S1(c,d). By changing L y , the number of subbands changes, which in turn modifies thespacing between the subbands and thus µ eff . Hence, µ eff depends heavily on L y . In Fig. 1(b) in the main text wepresent µ eff as a function of L y for Γ = 0 . µ eff should be as close as possible to the Fermi energy in order to easily be able to tune the system intothe topological phase [5]. In reality it is however difficult to know a priori the exact L y that gives such a desired µ eff .Instead we likely obtain any of the three cases discussed in the main text, namely, the ideal, potential barrier (PB),and quantum dot (QD) cases, when considering a short SNS junction with different SC widths. To further explorethese three cases, we fix L y to the values that gave each of the three cases in the main text and plot in Fig. S2 thebehavior of µ eff as a function of coupling strength. Specifically, we use L y = 41 a (triangle) giving the ideal junctionin the main text, L y = 11 a (cross) giving the PB junction, and L y = 21 a (dot) giving the QD junction for Γ = 0 . µ eff . In the strong coupling, however, µ eff is very sensitive to L y , thus creating the threedifferent cases. The magnitude of µ eff also increases with Γ, thus increasing the possibility of a QD spontaneouslyforming in the SNS junction. This is in agreement with the results in Fig. 2(e) in the main text. Note that in theweak coupling regime µ eff is the bottom of the NW subband. FIG. S2. Effective chemical potential µ eff as a function of Γ, for the SC widths L y = 41 a (triangle), L y = 11 a (cross) and L y = 21 a (dot), giving the ideal, PB and QD cases for Γ = 0 . Effective spin-orbit coupling
In the main text we give the condition for the existence of a QD solution as ˜ α < B < B c and quoted ˜ α ≈ . α NW forour choice of parameters, with ˜ α being the renormalized SOC strength. Here we demonstrate how ˜ α is numericallyextracted and show that the SOC in the NW is generally reduced in the NW+SC system. FIG. S3. Renormalized SOC strength ˜ α as a function of Γ for the SC widths L y = 41 a (triangle), L y = 11 a (cross) and L y = 21 a (dot), giving the ideal, PB, and QD cases at Γ = 0 . . α ≈ . α NW . α , let us first consider the non-hybridized Γ = 0 case, when ˜ α = α NW . The NW band then has E NW as itsminimum at k so . Therefore, dE NW dk x | k so = 0. By solving the resulting equation we obtain α NW t NW = tan ( k so a ). Since k so a is small we can write α NW t NW ≈ k so a . At finite Γ = 0, the minimum of the effective NW subband is instead at ˜ k so , see redmarker in Fig. S1(d). Since the effective NW band is similar to the NW band at low momentum before the coupling,the expression for the SOC in the effective band is the same as before but now with renormalized parameters suchthat ˜ αt NW ≈ ˜ k so a , which directly give us ˜ α .In Fig. S3, we show ˜ α for three different L y as a function of Γ. As Γ increase, ˜ α is very clearly reduced. Thereis also a smaller reduction in ˜ α when increasing L y . This means SOC is weakened when coupling the NW to a SC.Note also that since the junction in our system is very short, the SOC at the junction itself must be renormalized inalmost the same manner as in the superconducting parts of the wire. Interestingly, this simple analysis provided heregives the same qualitative result as a more elaborate recent work [6]. Variation of quantum dot levels with superconductor width
In the main text we show that the value of the effective chemical potential µ eff is crucial for the emergence ofzero-energy QD levels. When µ eff is appreciably positive, a QD is induced at the junction and the QD levels can evencross zero energy in the topologically trivial phase. However, the QD level depends L y since µ eff is highly tunablewith L y . In Fig. S4(a-d) we show the energy spectrum of the first four L y values that results in the peak positivevalues in µ eff in Fig. 1(b) in the main text, as representative QD systems. (a) (b) (c) (d)(e) (f) (g) (h) FIG. S4. (a-d) Zeeman field-dependent energy spectrum at φ = 0 (only lowest 30 levels) for the first four L y values that resultsin a µ eff peak (results in (c) are the same Fig. 2(c) in the main text). Vertical green lines mark the topological phase transition.The QD levels can be pinned around zero as in (c,d), but can also oscillate around zero as in (a,b). (e-h) Phase-dependentspectrum taken at the B values marked by color bars in (b), showing a π -shift in the lowest level (g). Despite varying behavior of the QD levels when changing L y , we always find a π -shift in the phase-dependentspectrum, E ( φ ), and thus in the supercurrent I ( φ ). This can be understood when considering that the ABS spectrumresults in states close to zero energy at φ = 0 but close to the band gap edge at φ = π . The spin exchange takingplace at the first zero-energy crossing further supports the existence of the π -shift as it means the ground state of thejunction is changed, consistent with results from other non-topological QD junctions [7–12]. To explicitly demonstratethe π -shift we show in Fig. S4(e-h) the phase-dependent spectrum E ( φ ) for the case of L y = 13 a , taken at the colormarkings in Fig. S4(b). Although the QD levels are not at all pinned to zero energy at φ = 0 (in contrast to the L y = 21 a case in the main text), there is still a π -shift in the lowest level in the trivial phase, see Fig. S4(g). This π -shift in the lowest level gives rise to a π -shift in the supercurrent, as demonstrated in Fig. 4 in the main text.4 Increased junction length
In the main text we used a short junction with length of 2 a , to be compared to the the superconducting coherencelength ξ ≈ a for our choice of parameters. In Figs. S5(a,b) we display the spectrum of the L y = 21 a QD case fortwo longer junctions. Interestingly, zero-energy QD levels persist even in very long junctions, albeit they do not staypinned to zero energy. We find, due to the evolution of the spectrum with φ and coinciding with the exchange of thespin state at every zero-energy crossing of the QD levels, that the state of the junction fluctuates between π and 0,with 0 and π -states in-between. This can numerically be quantified by finding the phase φ min at which the E levelhas its minimum, i.e. E ( φ min ) is the minimum of E ( φ ), which we plot in Figs. S5(c,d). As seen, there are no direct0 to π transitions, rather the 0 − π transitions pass through intermediate 0 and π states, giving rise to the slightlyslanted lines. Still the π -state is the dominating state across the QD regime. (a) (b)(d)(c) FIG. S5. Zeeman field-dependent energy spectrum at φ = 0 (only lowest 300 levels) for the L y = 21 a QD case for different longjunction lengths, 6 a ≈ ξ (a) and 14 a ≈ ξ (b), with the phase at the minimum of the lowest level E ( φ ), φ min , (c,d). Verticalgreen lines mark the topological phase transition, while red arrows indicate the original starting point of QD levels for the 2 a short junction, i.e. same red arrow as in Fig. 2(c) in the main text. Insets show phase dependent spectrum of the lowest levelsat the color markings in (a,b). Furthermore we find that the phase-dependent spectra, insets in Figs. S5(a,b), exhibit π -shifts in the lowest statebut with flatter dispersions compared with the short junction, Fig. 3 in the main text. Very long junctions even allowfor multiple levels, with counter-dispersing phase dependences on the energy, see in insets of Fig. S5(b), which weakenthe π -shift in the supercurrent. Thus the π -shift is most reliable as a tool to distinguish QD zero-energy levels inshort SNS junctions. Effects of spin-orbit coupling
We here further clarify the effects of SOC on the QD zero-energy levels. We consider L y = 21 a which gives QDlevels as shown in Fig. 2(c) in the main text. The effect of SOC is shown in Fig. S6. First, we set α NW = 0. In thiscase there is no opening of the topological gap since there is no SOC, see Fig. S6(a). However, the QD levels still existbefore the putative gap closure. The π -shift in the supercurrent also remains (not shown), beginning at the B -valuefor the zero-crossing of the QD levels. Note that the condition for QD levels leading to π -shift in the supercurrent˜ α < B < B c clearly holds in this case.Next we set α NW = 0 . t NW , i.e. twice the value in the main text. We here find that the QD levels do not reachzero energy, see Fig. S6(b). Instead, the large α NW value introduces a large hybridization between the QD levels5 (a) (b) FIG. S6. Zeeman field-dependent energy spectrum (only lowest 30 levels) for α NW = 0 (a) and α NW = 0 . t NW (b) for the L y = 21 a QD case. In (a) there is no topological phase transition, while in (b) the QD levels do not reach zero energy due to a too large α NW , results to be compared with Fig. 2(c) in the main text, where α NW = 0 . t NW . leading to anti-crossings. In this case, the π -shift is also not observed in the system, but we only find a discontinuoussupercurrent. For this very large α NW the renormalized value of SOC ˜ α in the NW is also large and violates thecondition for zero energy QD levels, ˜ α < B < B c . This behavior of non-zero-energy QD states at high SOC is alsoconsistent with earlier reports of QD levels in non-topological SOC systems [13]. Potential barrier case
As stated in the main text we do not find a π -shift in the phase-dependent energy spectrum or supercurrent forthe potential barrier (PB) case. For completeness we present in Fig. S7(a-d) the phase-dependent spectrum at fourdifferent B values for the PB case occurring at L y = 11 a (see Fig. 2(b) in main text for the Zeeman-field dependentspectrum at φ = 0, with the B values chosen by the corresponding color indicated values in Fig. 2(c)). As seen inFig. S7(b,c), the lowest states never cross zero energy close to φ = 0 so there cannot be a π -shift in the lowest levels.As expected from the features of the phase-dependent spectrum, the supercurrent does not either have have a π -shift, see Fig. S7(e,f). The individual contributions to the current from the E and E levels are for completenessalso shown in Fig. S7(g,h). Note that close to φ = π , for B -values between the magenta and green dashed lines inthe trivial phase, there is a sign change in the supercurrent in Fig. S7(e). However, this sign change does not occurthroughout the whole φ range, and it therefore does not correspond to a π -shifted current. Instead, this sign changeonly occurs in a limited range of φ , which only renders the supercurrent discontinuous, a consequence of the ABScrossings in Fig. S7(b,c). Critical currents
Beyond the current-phase relationship of the current, another experimentally relevant quantity is the critical current, I c , which represent the maximum supercurrent that flows across the junction. Critical currents can provide additionalinsight to current-phase relationships that we present in the main text. Technically we extract the I c by maximizingthe supercurrent I ( φ ) with respect to the superconducting phase difference φ . The supercurrents for the PB andQD cases are shown in Fig. S8. In an ideal situation the critical current decreases with increasing B and traces outthe topological gap closure [14]. Remarkably, in the PB and QD cases the low-lying states in the trivial phase alsomodifies I c . In the PB case the critical current is low in much of the trivial phase, see Fig. S8(a). This can beunderstood from having close lying levels, see Fig. S7(b,c), with counter-dispersive phase-dependent energy spectrum,thereby contributing destructively to the total supercurrent, hence, giving a low critical current. Note how the junctiontransitions between intermediate states but never reaches the full π -state such that only a discontinuous current-phaserelationship is found in the PB case.For the QD case we instead find a bump in I c within the QD region, between the black and green lines in Fig. S8(b).This bump is a telltale of the zero-energy QD levels carrying most of the supercurrrent. The dip at the topologicaltransition comes from the energy gap closing. Note also the transition between intermediate states before reaching the π -state. In general, we find that the critical current shows a kink or plateau whenever there is a transition betweendifferent states of the junction.6 (a) (b) (c) (d)(f) (g) (h)(e) FIG. S7. (Top row) Phase-dependent energy spectrum for the L y = 11 a PB junction obtained at the color-marked B values inpanel (e). Compare with Fig. 3 and 4 for the QD case in the main text. (Bottom row) Colorplot of supercurrent as a functionof φ and B (e), total supercurrent (f), with contributions from the E (g) and E (h) energy levels taken at B values markedwith the corresponding color bars in (e). Note that there is no π -shift in the supercurrent. (a) (b)
0' 000 0'0' 0
FIG. S8. Zeeman field-dependent critical current I c for L y = 11 a PB (a) and L y = 21 a QD (b) cases. Magenta and greendashed lines mark the beginning of the ABS zero-crossing (at φ = π ) and topological phase transition, respectively. Blackdashed lines mark transitions to intermediate states 0 and π . TOWARDS MORE REALISTIC PARAMETERS
In the main text we used an somewhat overestimated value of the superconducting order parameter, ∆ sc , in theparent SC and as a consequence a large coupling strength Γ is needed to reach the strong coupling regime. We herereduce the order parameter by a factor of 5, such that ∆ sc = 0 .
02, in order to be close to realistic values. For thisvalue of ∆ sc , strong coupling is achieved already around Γ = 0 .
3, see Fig. S9(b), when also using the reduced chemicalpotentials, µ sc = 0 . µ NW = 0 to keep close to realistic values. As a consequence of the longer superconductingcoherence length in the SC we also increase the system size and junction length. Since µ eff depends on the Fermimomentum of the SC, we find that µ eff changes with the chemical potential of the SC but the overall behavior doesnot change, compare Fig. S9(a) with Fig. 1(b) in the main text.Similar to the main text we consider three SC widths yielding different µ eff , see markers in Fig. S9(a), representingthe ideal, PB, and QD junction behaviors. We find the same behavior in the low energy spectrum as in Fig. 2 in themain text: the topological phase transition point changes with L y and there are trivial zero-energy states for the QDcase and also near zero-energy states in the PB case, see Fig. S11(b,c).The phase dependent energy spectrum for the QD case ( L y = 49 a ) is shown in Fig. S11. We find that the π -shiftpersists in the QD regime, see Fig. S11(c). This π -shift in the lowest level gives a π -shift in the supercurrent (notshown). Furthermore, we find only 0 , π states but no π state in the PB case (not shown), despite the energy levels7 (a) (b) + FIG. S9. Effective chemical potential profile deep into the SC parts of the NW as a function of L y for fixed Γ = 0 . L y = 55 a (triangle), L y = 47 a (cross) and L y = 49 a (dot), representative of the ideal,PB, and QD cases, respectively. Here ∆ sc = 0 . t sc , µ s = 0 . t sc , µ NW = 0, where t sc = 25meV and junction length = 4 a . (c)(a) (b) FIG. S10. Zeeman field-dependent energy spectrum (only lowest 30 levels) at φ = 0 for ideal ( L y = 55 a ) (a), PB ( L y = 47 a )(b), and QD ( L y = 49 a ) (c) cases, given by the markers in Fig. S9. Vertical dashed green lines mark the topological phasetransition. Here Γ = 0 . t sc , L x = 775 a , L NW = 1500 a , with other parameters the same as Fig. S9. Compare with Fig. 2(a-c) inthe main text. (a) (b) (c) (d) FIG. S11. Phase-dependent energy spectrum (only lowest 30 levels) in the QD case ( L y = 49 a ) obtained at the color-marked B values in Fig. S10(c). Compare with Fig. 3 in the main text. almost reaching zero energy close to the topological phase transition for the PB case. SELF-CONSISTENCY
Hitherto we have taken ∆ sc to be simple constant. It would however be more accurate to calculate ∆ sc self-consistently in the SC through the self-consistency equation ∆ s (i) = − V sc h c i ↑ c i ↓ i , which allows a site dependence forsuperconducting order parameter, since the NW will also affect the SC [15, 16]. In this calculation, the pair potential V sc is chosen to be constant as it represents a constant tendency for pairing in the SC. To make self-consistentcalculations computationally feasible we have to limit ourselves to smaller systems and thus we need to increase ∆ sc such that the superconducting coherence length becomes smaller than the total system size. This is done by using a V sc value that results in large ∆ sc .We find that, qualitatively, self-consistency yields the same result as a constant ∆ sc . Except for slightly different8topological phase transition points and QD regimes both calculations match well, as seen in Fig. S12(a). We havealso verified that the supercurrents exhibit the π -shift in the QD regime (not shown). We also study the inducedsuperconducting correlations in the NW. In Fig. S12(b) we plot the spin-singlet pair correlation in the NW normalizedwith that of the SC: F ↓↑ = h d i ↑ d i ↓ i / |h c ↑ c ↓ i| , where h c ↑ c ↓ i is calculated deep into the SC where ∆ sc is constant.As seen, there is a strong suppression of the NW pair correlations in the junction region, with disturbances reachingreasonably far into the S regions. Notably there is also a strong variation with φ . We also study the same NW paircorrelations for the non-self-consistent case and find qualitatively the same results, despite the order parameter ∆ sc being constant in the SC. (b)(a) FIG. S12. (a) Zeeman field-dependent energy spectrum at φ = 0 for self-consistent (black) and non-self-consistent (red)calculations for the QD ( L y = 21 a ) case. Vertical dashed green line marks the self-consistent topological phase transition.In the topological phase the zero energy states (MBSs) for the self-consistent (black) calculation are behind the red lines.Magnitude (b) of the induced superconducting pair correlations in the NW at B = 0 normalized by the correlation in theparent SC, F ↓↑ . Dashed magenta line mark the junction region. Here V sc = 0 . sc = 0 . , in the bulk SC, µ s =0 . , µ NW = 0 . , Γ = 0 . , L x = 45 a and L NW = 80 a . EFFECTIVE 1D MODEL
In the main text and so far in this supplementary material, we have considered a NW coupled to two 2D SCs andsolved for the full system. Now, we consider the equivalent setup for a pure 1D junction: a NW with Rashba SOC,parallel magnetic field B , and use an assumed induced onsite superconducting order parameter, ∆ , which we putby hand in the left and right sectors of the NW along with a superconducting phase difference φ across the junction.We again consider three different cases, ideal, PB, and QD junctions, that correspond to the ones investigated in theNW+SC model. Here we however have to put by hand the values of the chemical potentials in the left/right andcentral regions of the 1D system that give these three cases. Specifically, the cases are: (1) The chemical potential inthe wire is constant, ideal case. (2) The chemical potential at the junction is higher than that of the superconductingparts, PB case. (3) The chemical potential at the junction is lower than that of the superconducting parts, QD case.The Hamiltonian for this pure 1D system is H = L NW X x =1 ,σσ (cid:2) ε NW ( x ) d † xσ d xσ + Bd † xσ σ xσσ d xσ + ∆( x ) d x ↓ d x ↑ (cid:3) + L NW − X x =1 ,σ h − t NW d † xσ d x +1 ,σ + α NW (cid:0) d x ↑ d † x +1 , ↓ − d † x, ↓ d x +1 ↑ (cid:1)i +H . c . , where ε NW ( x ) = 2 t NW + µ NW ( x ), and t NW with µ NW the nearest neighbor hopping and chemical potential, respectively,while α NW is the SOC strength and B the Zeeman field. The assumed induced order parameter ∆ and the effectivechemical potential µ eff are further defined as: { ∆( x ) , µ NW ( x ) } = {| ∆ | e iφ , µ eff } x < L s { , } L s < x ≤ L s + d {| ∆ | , µ eff } x > L s + d , d = 4 a is the length of the junction, φ the phase difference across the junction and L s the length of each of thesuperconducting parts, assumed to be large. That is, this setup has a superconducting order and a modified chemicalpotential in the superconducting regions of the NW, modeling what actually happens in the NW+SC system. Inparticular, we use values of µ eff extracted from Fig. 1(b) in the main text in order to as closely as possible modelstrong coupling regime of the NW+SC. (a) (b) FIG. S13. (a) Zeeman field-dependent energy spectrum at φ = 0 for PB (a) and QD (b) cases for a purely 1D system. Here∆ = 0 . α NW = 0 . t NW and L NW = 2000 a . Vertical dashed green line marks the topological phase transition. (a) (b) (c) (d) FIG. S14. Phase-dependent energy spectrum for the QD case obtained at the color-marked B values in Fig. S14. We proceed by numerically diagonalize H within the Bogoliubov-de Gennes formalism, as in the main text. First,we calculate the evolution of the energy spectrum with magnetic field B for the PB and QD cases, see Fig. S13. Asseen, we obtain the same result as for the full NW+SC system, with the trivial phase hosting zero energy statesin the QD case. We also find the π -shift in the phase-dependent energy spectrum for the QD case, see Fig. S14,but not in the PB case (not shown). Thus the existence of the QD zero-energy levels and the associated π -shiftedsupercurrent are not due to modeling the full NW+SC system per see, although the QD is produced automaticallywhen considering the full NW+SC, while it has to be put in by hand in the 1D system.It is here also worth mentioning that we did not find the QD behavior for experimentally realistic values of α NW and ∆ . We instead had to use a weak α NW = 0 . t NW to obtain the QD behavior in Fig. S14(b). This is because, asdiscussed earlier, SOC is significantly weakened in the NW+SC system. 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