Interaction-induced zero-energy pinning and quantum dot formation in Majorana nanowires
IInteraction-induced zero-energy pinning and quantum dot formation in Majoranananowires
Samuel D. Escribano, Alfredo Levy Yeyati, and Elsa Prada ∗ Departamento de F´ısica de la Materia Condensada C3 and Departamento de F´ısica Te´orica de la Materia Condensada C5,Condensed Matter Physics Center (IFIMAC) and Instituto Nicol´as Cabrera,Universidad Aut´onoma de Madrid, E-28049 Madrid, Spain
Majorana modes emerge in non-trivial topological phases at the edges of some specific materials,like proximitized semiconducting nanowires under an external magnetic field. Ideally, they are non-local states that are charge neutral superpositions of electrons and holes. However, in nanowires ofrealistic length their wave functions overlap and acquire a finite charge that makes them susceptibleto interactions, specifically with the image charges that arise in the electrostatic environment. Con-sidering a realistic three-dimensional model of the dielectric surroundings, here we show that, undercertain circumstances, this interaction leads to a suppression of the Majorana oscillations predictedby simpler theoretical models, and to the formation of low-energy quantum dot states that interactwith the Majorana modes. Both features are observed in recent experiments on the detection ofMajoranas and could thus help to properly characterize them.
Keywords: Hybrid Superconductor-Semiconductor Nanowire; Interactions; Majorana Bound States; Quan-tum Dot.
I. INTRODUCTION
Semiconducting nanowires with strong spin-orbit in-teraction, like InAs or InSb, are becoming ideal systemsfor the artificial generation of topological superconduc-tivity [1–3]. In addition to its fundamental interest, suchnanowires, that may host Majorana bound states (MBSs)at their ends or interfaces [4, 5], constitute promisingplatforms for Majorana-based quantum computing de-vices [6–9]. Progress in fabrication techniques have al-lowed to induce a hard superconducting gap in InAs[10] or InSb [11] nanowires with epitaxially deposited Allayer. Moreover, last generation devices exhibit a verylow degree of disorder which allows them to almost reachthe ballistic limit [12–14].In spite of these advances, the experimental signaturesof MBSs in the nanowire devices deviate significantly inseveral aspects from the theoretical predictions of mini-mal models. This is the case, for instance, regarding thebehavior of the subgap conductance through the prox-imitized nanowire, which has been addressed in severalexperiments [10, 12–19]. In a long wire (whose lengthis much greater than the induced coherence length) thepresence of MBSs manifests in the appearance of a zerobias conductance peak whose width is controlled by thenormal state conductance [20]. However, for the typicalwire lengths explored in actual experiments (which areof the order of a few µm ) it is expected that the overlapbetween MBSs located at both ends of the wire gives riseto conventional Andreev bound states which deviate fromzero energy, leading to an oscillatory pattern in the lowbias conductance as a function of Zeeman field, chemicalpotential or wire length [21–23]. Conspicuously, in most ∗ Corresponding author: [email protected] (meV) -0.300.3 E ( m e V ) z x Surroundingmedium ( ! a =1) SC shell( ! SC (cid:1) Nanowire ( ! =17.7) W ρ(x) V Z y NormalLead ( ! M →∞ ) Substrate ( ! d =3.9) NormalLead ( ! M →∞ ) (a)(b) FIG. 1. (a) Schematic representation of the setup analyzedin the present work. A nanowire of rectangular cross-section(green) lying on an insulating substrate (grey) and in contactto a thin metallic layer in one of its faces (light blue), cor-responding to the parent superconductor, and two normal-metal leads at its ends (orange) separated by tunnel barriers(brown). Typical values for the dielectric constants for eachregion are indicated. (b) Low energy spectrum versus chemi-cal potential µ for a wire of thickness W = 100nm and length L = 1 µ m. Other parameters are the spin-orbit coupling α = 20nm · meV, the induced pairing energy ∆ = 0 . V Z = 2meV. Electrostatic environment-induced zero energy pinned regions between Majorana oscil-lations are indicated in red. Quantum dot levels (in blue),originated at the wire’s edges due to the interaction withthe bulk contacts, anticross with Majorana levels and removetheir zero-energy pinning. a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p of the available experimental data the emergence of a ro-bust zero bias conductance peak is observed above somecritical Zeeman field without the expected oscillatorypattern [12, 19, 24, 25]. Several mechanisms have beenproposed to account for the reduction or lack of oscil-lations, such as smooth confinement [21, 26–28], strongspin-orbit coupling [29], position dependent pairing [30],orbital magnetic effects [31], Coulomb repulsion amongthe carriers in the nanowire [22], or the presence of thenormal drain lead connected to the hybrid wire [32].Another source of Majorana oscillation suppressionwas put forward by some of us in a recent work [33]. Thekey realization is that MBSs in a finite-length wire possesa finite charge, typically distributed uniformly along thewire [34], which can be susceptible to electrostatic inter-actions with the surrounding medium. We considered thecase of a grounded parent superconductor, thus avoidingthe effect of a charging energy associated to the Cooperpairs, and showed that, in such case, a residual effect ofinteractions may arise from the image charges inducedin the electrostatic environment of the nanowire. Usinga simple model for the induced potential we concludedthat, in typical experimental setups, interactions wouldlead to a pinning of the MBSs to zero energy around par-ity crossings and, thus, to more robust zero bias conduc-tance peaks than predicted by the non-interacting mod-els.The aim of the present work is to test the validity ofthe predictions of Ref. [33] for the case of more realisticcalculations of the induced electrostatic potential, takinginto account the actual three dimensional (3D) geometryas well as the effect of nearby metallic leads. We considerthe geometry depicted in Fig. 1(a), where a nanowire ofrectangular cross section lies on an insulating substrate(typically SiO ) and is contacted to a thin superconduct-ing (SC) layer on one of its faces and to two bulk normalleads at both ends, separated by thin insulating barri-ers. In Fig. 1(a) we indicate the characteristic dielectricconstants of each region, which are relevant for the cal-culation of the induced potential through the Poisson’sequation (discussed bellow). Our aim is to solve thisequation together with the Bogoliubov de Gennes equa-tion for determining self-consistently the charge density ρ ( x ) along the nanowire. For this purpose we derive ageneralized method of image charges which allows us tocalculate the induced potential in rather general condi-tions, taking into account a 3D electrostatic environmentas the one shown in Fig. 1(a).We find two main effects coming from this interactionand exemplified in Fig.1(b). One is, as stated before,the suppression of Majorana oscillations around paritycrossings (zero energy crossings where the total fermionparity of the wire changes), both as a function of theZeeman energy V Z and the wire’s chemical potential µ .This effect is produced because, at each parity crossing, afinite Majorana charge Q M enters the wire from the reser-voir in an abrupt fashion. If the electrostatic screeningis smaller inside the wire than in the reservoirs, a repul- sive interaction is produced between the incoming chargeand its images, preventing its entrance. This translatesinto finite regions in parameter space [in red in Fig.1(b)]where Majorana modes are pinned to zero energy withina finite range of V Z or µ proportional to the Majoranacharge Q M and the strength of the interaction. This wasalready shown in Ref. [33] but for a simplified dielectricprofile where the presence of the superconducting shellhad been ignored. We here include it and find that thesize of the pinned regions decreases but the pinning effectis still present under certain conditions that we discussin detail below. Moreover, we explain the incompressiblebehavior of the electron liquid within these pinned re-gions in terms of the Majorana wave functions and theircharge.Another important effect of the electrostatic environ-ment unexplored before is the creation of deep potentialwells at the ends of the wire close to the bulk metallicelectrodes. These wells, obtained explicitly here throughthe self-consistent calculation, are similar to the con-finement potentials typical of quantum dots. Localizedquantum dot-like energy levels in these regions dispersewith magnetic field (or chemical potential) and appearbelow the induced gap in the wire spectrum [in blue inFig.1(b)]. In the topological regime, dot-like levels inter-act with Majorana states, anticrossing them when theyapproach zero energy. Similar phenomenology can be ob-served in some experiments [14, 19], and has been likelyfound in other occasions but discarded by experimental-ists looking for the simpler picture. Interestingly, it hasbeen shown that the shape of these anticrossings can beused to quantify the degree of Majorana wave functionnon-locality [35, 36], a prediction that has been experi-mentally demonstrated recently [25]. Here, we show thatif the dot-Majorana levels interaction occurs in a pin-ning region, Majorana levels are forced to depart fromzero energy, revealing the existence of a finite wave func-tion overlap between them in spite of their zero energy.We analyze this behavior again in terms of the Majorana(and dot) wave functions and their charge.The paper is organized as follows: in the followingsection (Sec. II) we provide insight into the theoreti-cal model used to treat interactions. In the next section(Sec. III A) we analyze the case in which the influenceof the bulk normal leads can be neglected, recoveringthe pinning effect found in Ref. [33] for a repulsive elec-trostatic environment. However, we focus here on theelectrostatic environment effects on the Majorana wavefunction, rather than on its spectral properties. In thenext section (Sec. III B) we study the effect of includingthe bulk normal leads of Fig.1(a), finding that they giverise to the formation of quantum dot-like bound states.We further analyze the interplay of such states with theMBSs. Finally, we present the conclusions of our work atthe end (Sec. IV). The robustness of the pinning effectis analyzed in detail in Supporting Information (SI) 4. II. MODEL AND THEORETICAL APPROACH
We model the electronic states along the proximitizedRashba nanowire of length L using the following singlechannel Hamiltonian [4, 5] H = 12 (cid:90) dx Ψ † ( x ) H ( x )Ψ( x ) , H = (cid:20) − (cid:126) m ∗ ∂ ∂x − µ + eφ ( x ) (cid:21) σ τ z − iασ y τ z ∂∂x ++ V Z σ x τ z + ∆ σ y τ y , (1)where Ψ † = (cid:16) ψ †↑ , ψ †↓ , ψ ↑ , ψ ↓ (cid:17) is a Nambu bi-spinor, ψ ↑ , ↓ ( x ) are electron annihilation operators, and σ and τ are the Pauli matrices in spin and Nambu space, respec-tively. The model is defined by setting the parameters m ∗ , µ , α , V Z and ∆, corresponding to the effective mass,the chemical potential, the spin-orbit coupling, the Zee-man energy caused by an external magnetic field, andthe induced SC pairing potential.In Eq. (1) we also include the electrostatic poten-tial φ ( x ) felt by charges in the nanowire, which can bedecomposed as φ ( x ) = φ int ( x ) + φ b ( x ), where φ int isthe potential that arises from the free charges inside thenanowire, while φ b corresponds to the potential createdby bound charges that emerge in the electrostatic envi-ronment. We compute the electrostatic potential usingthe Poisson equation (cid:126) ∇ · (cid:104) (cid:15) ( (cid:126)r ) (cid:126) ∇ φ ( (cid:126)r ) (cid:105) = (cid:104) ρ ( (cid:126)r ) (cid:105) , (2)where (cid:15) ( (cid:126)r ) is the non-homogeneous dielectrical permit-tivity of the entire system and (cid:104) ρ ( (cid:126)r ) (cid:105) is the quantum andthermal average of the charge density of the nanowireobtained with Eq. (1). The intrinsic part φ int ( x ) ofthe potential satisfies an analogous equation with a uni-form (cid:15) equal to that of the nanowire. The geometrydepicted in Fig. 1(a) is taken into account through apiecewise (cid:15) ( (cid:126)r ) function where each material is charac-terized by a different dielectric constant, so that (cid:15) ( (cid:126)r )changes abruptly at the interfaces. Then, assuming thatthe charge density in the nanowire is located along itssymmetry axis (x-axis), we obtain the electrostatic po-tential φ b ( x ) using the method of image charges, as ex-plained in detail in SI 1. More precisely, φ b is given by φ b ( x ) = (cid:82) dxV b ( x, x (cid:48) ) (cid:104) ˆ ρ ( x (cid:48) ) (cid:105) , where V b ( x, x (cid:48) ) is a kerneldetermined in order to satisfy the proper boundary condi-tions. We find analytical expressions for V b ( x, x (cid:48) ). Theyare simple but rather lengthy and are given in the SI fortwo different cases: neglecting the effect of the bulk nor-mal leads at the wire ends and including it. The resultsfor these two cases are analyzed in the following sections.The obtained potential φ ( x ) on the nanowire axisshould be plugged back into Eq. (1). The combinedPoisson-Schr¨odinger problem must then be iterated un-til it achieves self-consistency. As shown in Ref. [33], the φ int ( x ) part of the electrostatic solution (i.e. the in-trinsic electron-electron interaction part of the problem),treated at the Hartree-Fock level, has a negligible effecton the low energy spectrum in the topological regime. Wemay therefore concentrate only on the self-consistencywith φ b ( x ). In SI 2 we explain in detail the self-consistentnumerical method used to compute the electrostatic po-tential profile as well as the eigenvalues and eigenvectorsof Eq. (1). For completeness, in SI 3 we also show theeffect of including the intrinsic interaction from φ int ( x ),proving that its effect is small and that the main contri-bution stems from φ b .In the following calculations, we consider the dielec-tric constants shown in Fig. 1(a): for the dielectricsmaterials (the wire, the substrate and the surroundingmedium), we use typical values [37] ( (cid:15) = 17 . (cid:15) d = 3 . (cid:15) a (cid:39)
1, respectively); while for the metallic leads weassume that, because they are bulky, they screen exter-nal electric fields perfectly (i.e. (cid:15) M → ∞ ). This maynot be the case for the SC shell, whose capability forscreening external electric fields may be weaker due toits small thickness and unavoidable presence of possibledisorder [38]. If this is the case, it is then characterizedby a finite effective dielectric permittivity which dependson the SC shell width as well as its composition, as weshow in SI 1. Some experiments [39] have reported thatfor ultrathin metallic layers ( ∼ − (cid:15) SC (cid:39) III. RESULTS AND DISCUSSIONA. Results without bulk normal leads
It is convenient to start by analyzing the simpler casein which we neglect the effect of the bulk normal leads inthe induced potential φ b . As an example we consider ananowire of width W = 100nm, length L = 1 µ m and thefollowing choice of realistic parameters: m ∗ = 0 . m e , α = 20nm · meV, ∆ = 0 . µ = 0 . T =10mK. These could correspond, for example, to a InSb V Z (meV) V Z + - e b ( L / ) ( m e V ) T r i v i a l pha s e Non-interactingInteracting x/L -1012 e ? b ( x ) ( m e V ) V Z =0 V Z =5meVV Z =0.9meV EV Z µV Z + µ ✓ L ◆ k x V Z = 0 V Z ≠ 0 (c)(a) V Z + µ e b V Z + µ e b V Z (meV) Q t o t / e Non-interactingInteracting (d)(b)
FIG. 2. Majorana nanowire subject to interactions from theelectrostatic environment (ignoring the influence of the bulknormal leads at its ends). (a) Schematic of the nanowire’sdispersion relation in the absence and in the presence of theZeeman field. (b) Self-consistent induced potential energy eφ b ( x ) along the wire’s length for increasing values of theZeeman splitting. Wire parameters as in Fig. 1(b) andwith µ = 0 . V Z + µ − eφ b ( L/ Q tot of the nanowireas a function of V Z for the non-interacting (dashed) and in-teracting (solid line) cases. Red curves highlight parameterregions for which there is interaction-induced zero-energy pin-ning in the spectrum. nanowire in contact to an Al superconducting shell [14],but similar results are obtained for InAs wire parameters[19]. For an infinite wire, a schematic representation ofthe energy bands is shown in Fig. 2(a) in the absenceand in the presence of the Zeeman field. At zero tem-perature, the occupied states below the Fermi level arethose between the horizontal dashed line and the bandbottom. Apart from a small contribution coming fromthe spin-orbit energy, the position of the band bottomis controlled by the wire’s chemical potential µ , the Zee-man energy V Z and the induced potential energy eφ b .The magnetic field lowers the band bottom, charging thewire, whereas the induced potential energy, coming fromelectrostatic repulsion, tends to compensate that trend.In the finite-length wire, the evolution of the inducedpotential profile along the nanowire length ( x -axis) fordifferent Zeeman fields is shown in Fig. 2(b). As canbe observed, the induced potential tends to expel chargefrom the center of the wire, where it is positive, while itbends downwards at its ends. On the other hand, theevolution of the potential with Zeeman field exhibits astep-like behavior with regions where it increases linearlywith V Z (red curves), screening the magnetic field effects,and regions where it remains almost constant as V Z in-creases (grey curves). This causes the electron fluid to behave in an incompressible or compressible manner, re-spectively. This different behavior can be clearly seenin Fig. 2(c) where the electrochemical potential at thecenter of the wire, given by V Z + µ − eφ b ( L/ V Z > (cid:112) ∆ + µ , corresponding roughly tothe critical field for the bulk topological transition. Wealso obtain the typical energy oscillations produced byoverlapping Majorana wave functions due to the finitelength of the wire [21–23]. More insight can be obtainedby analyzing the evolution of the total wire’s charge Q tot = (cid:82) L dx (cid:104) ρ ( x ) (cid:105) as well as the Majorana charge Q M ,whose absolute value is given by | Q M | = | Q +1 − Q − | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e (cid:90) L dxu L ( x ) u R ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3)Here, Q ± are the charge corresponding to the even/oddlowest energy eigenstates ψ ± , and u L,R are the elec-tron components of the Majorana wavefunctions γ L = ψ +1 + ψ − and γ R = − i ( ψ +1 − ψ − ). The total chargeincreases in general with magnetic field but, for finitelength wires, it does so by jumping abruptly a quantityequal or smaller than e at each parity crossing (where theMajorana oscillations cross zero energy and the electronparity of the wire changes from even to odd or viceversa),as shown in Fig. 2(d), dashed curve. This abrupt changein charge is actually injected into the fermion state cre-ated by the two overlapping Majoranas and is given by | Q M | at the parity crossings. The (oscillatory) evolutionof | Q M | with magnetic field is given in Fig. 3(b). Strik-ingly, | Q M | is maximum at the parity crossing, where theenergy is zero, and goes to zero at the oscillation cusps.As the length of the wire tends to infinity, Q M tends tozero (not shown). Indeed, the finite value of Q M at theparity crossings is a direct measurement of the Majoranaoverlap, as shown in Ref. [33]. Note that the Majoranaoverlap is defined similarly to the right-hand side of Eq.(3), but with the absolute value inside the integral.The behavior of the Majorana wavefunctions is illus-trated in Figs. 3(c-f). The probability density for theleft and right Majorana wave functions exhibits an over-all decay towards the center of the wire controlled by thelength ξ ∼ (cid:126) v F / ∆ and an oscillatory pattern controlledby λ F [41, 42]. Moreover, the number of oscillations thatfit in L increases by one with Zeeman field at each par-ity crossing. Interestingly, we observe that the left-rightoscillatory patterns are out of phase for the cases wherethe splitting of the MBSs is maximum, panels (c) and (e).This minimizes the left-right wave function overlap andthe Majorana charge goes to zero. On the other hand, x/L | L , R | x/L | L , R | x/L | L , R | x/L | L , R | V Z (meV) -0.300.3 E ( m e V ) T r i v i a l pha s e (a) (c) (d)(e) (f) V Z (meV) | Q M / e | (b) FIG. 3. Majorana wave functions in the non-interacting case: Energy levels (a) and the absolute value of the Majorana charge Q M (b) vs Zeeman energy. Panels (c-f) show the wave-function probability profiles of the two lowest energy states in theMajorana basis at selected values of the Zeeman field within the topological region. When the splitting is maximum (greencircles and yellow triangles) the left and right Majorana wave-function oscillations are out of phase, whereas when the splittingis zero (orange square) they are in phase. the oscillations are in phase (d) when the energy split-ting is zero, at the parity crossings, producing a maxi-mum in | Q M | and overlap. Although the Majorana wavefunctions are more strongly located at the wire edges,we note that the charge density of this fermionic stateis uniform across the wire [34] and, thus, it is uniformlyaffected by the interaction with the environment whenthis is present.In the presence of interactions with the image charges,the single-point parity crossings versus V Z in the spec-trum are replaced by extended regions where the subgapstates remain pinned at zero energy, indicated by the redlines in Fig. 4(a). The abrupt jumps in Q tot in the non-interacting case are replaced by a linear increase alongthe V z -values where the zero-energy pinning occurs, seeFig. 2(d). This is a consequence of the repulsive en-vironment that inhibits the entrance of charge in thewire where the electron liquid behaves in an incompress-ible manner. On the other hand, the Majorana chargeremains basically constant at the pinning plateaux, asshown in Fig. 4(b). The finite value of Q M in these re-gions indicates that zero-energy does not imply absenceof overlap between the left and right Majorana states.This is actually a common misconception that we wouldlike to point out here. The Majorana overlap, which isa measurement of the degree of non-locality of the twoMajorana wave functions, mostly depends on the lengthof the nanowire (and to a lesser extent on other param-eters, such as the induced superconductor gap and theRashba coupling), but it is not necessarily correlated to the Majorana energy splitting. Different mechanisms canreduce this splitting, such as interactions with the envi-ronment as studied here, smooth potential or gap profiles[21, 26–28, 30], or orbital magnetic effects [31], and stillleave the Majorana overlap unaffected. The behavior ofthe Majorana wavefunctions in this case is illustrated inFigs. 4(c-f). In the pinning regions the Majorana wavefunctions remain practically frozen and in phase. This inturn explains why | Q M | is maximum in these regions.The generality of these results with wire parametersis analyzed in the SI 4. There we show how the widthof the pinning plateau evolves with V Z when we changethe chemical potential, the dielectric permittivity or thewidth of the SC shell, and the aspect ratio of the nanowiresection. We find that pinning survives for any chemicalpotential, while it vanishes when the attractive contribu-tion of the SC shell becomes dominant over the dielectricrepulsion. B. Effect of bulk normal leads
In this section we analyze the effect of including thebulk normal leads in the calculation of the induced po-tential φ b . Fig. 5(a) illustrates the evolution of φ b withincreasing Zeeman field for the same choice of parame-ters as in the previous section but including the normalcontacts. As can be observed, while in the central regionof the wire a similar repulsive step-like evolution with V z is found (corresponding to compressible/incompressible x/L | L , R | x/L | L , R | x/L | L , R | x/L | L , R | (c) (d)(e) (f) V Z (meV) -0.300.3 E ( m e V ) (a) V Z (meV) | Q M / e | (b) FIG. 4. Same as Fig. 3 but for the interacting case (without leads). In the pinned regions the Majorana wave functions remainin-phase as a function of Zeeman field and the Majorana charge (b) freezes at its local maximum value (in red), instead ofcontinuing the oscillation like in the non-interacting case (dashed curve). electron fluid behavior), significant attractive regions ap-pear at the wire ends produced by the metallic charac-ter ( (cid:15) M → ∞ ) of the adjacent leads. As we discuss be-low, these attractive regions give rise to the formation ofquantum-dot (QD) like bound states that may interactwith the low energy subgap states of the Majorana wire.The evolution of the spectral properties and of the to-tal charge Q tot in this case are shown in Fig. 5(c) and(e), respectively. On the one hand, we observe that thepinning plateaus around each parity crossing (in red) arestill present although with a smaller width. On the otherhand, the main effect of the presence of the attractivepotential regions is the appearance of an additional fourenergy levels (two per contact, in blue) that approachzero energy for V Z around 2 . − e ), see Fig. 5(e). We can associatethese additional levels with QD-like bound states arisingin the attractive regions of the induced potential that an-ticross with the Majorana levels when their energies areon resonance [35, 36, 43, 44].To demonstrate the validity of this interpretation weshow in Fig. 5(d) and (f) the spectral properties and thetotal charge evolution for an isolated wire with a simpledouble potential well taken to mimic the effect of theelectrostatic environment, shown in (b). Notice that inthis case we do not attempt a self-consistent calculationbut rather include the Zeeman field as a rigid shift ofthe two spin bands (like in the non-interacting case butwith an inhomogeneous potential profile). Although thezero-energy pinning is not captured by this model, onecan clearly observe the presence of four levels coming down towards zero-energy for V Z around 2 . − φ b or φ fixed ) V cZ = (cid:113) ( µ − eφ b,fixed ) + ∆ , (4)which is not constant along the wire because φ b (or φ fixed ) depends on x . For the shown values of V Z , onlythe central part of the wire is in the topological regime( V Z > V cZ ), corresponding to an effectively shorter Majo-rana wire, whereas the outer parts are trivial ( V Z < V cZ ),corresponding to two effective QDs attached to it. Spe-cific details of how QD-levels interact with Majoranananowire ones can be found in Ref. [35, 36, 43–45].Further evidence on the nature of the low energy statesat V Z ∼ V Z (meV) Q t o t / e V Z (meV) Q t o t / e V Z (meV) -0.300.3 E ( m e V ) V Z (meV) -0.300.3 E ( m e V ) x/L -8-6-4-202 e f i x ed ( m e V ) x/L -8-6-4-202 e ? b ( x ) ( m e V ) V Z =5meVV Z =0.5meV (a) (b)(c) (d)(e) (f) V Z > V cZ V Z < V cZ V Z > V cZ V Z < V cZ FIG. 5. Majorana nanowire subject to interactions from theelectrostatic environment (including the influence of the bulknormal leads at its ends). (a) Self-consistent induced poten-tial energy eφ b ( x ) along the wire’s length for increasing valuesof the Zeeman splitting. Same wire parameters as in Fig. 2.Note that the main effect of the bulk normal leads is to cre-ate confining potential wells at the wire edges. (b) Barrier-likepotential energy profile used to mimic the self-consistent so-lution. Spectra of the Majorana nanowire as a function of V Z in the (c) interacting case and in the (d) non-interacting casebut using the potential profile model of (b). (e,f) Evolution ofthe total charge Q tot with Zeeman splitting for the two pre-vious cases, respectively. Red color indicates incompressibleelectron fluid behavior as before, while blue color indicatesQD-like behavior due to the metallic contacts. whereas the usual overlapping behavior of the MBSs isrecovered but with the Majoranas bound to the effectivetopological edges (yellow triangle). The absolute valueof the Majorana charge vs. V Z is shown in Fig. 6(b),calculated considering only the two lowest energy states(as before). At the anticrossing the Majorana charge os-cillation is distorted, see blue region, but the area belowthe curve is conserved. The missing charge in Fig. 5 (b)does not come from the Majorana states, but from thedot ones. At the anticrossing region, the two QD states(one per potential well) that were occupied (below theFermi level) move upwards in energy as the Zeeman fieldincreases and cross the Fermi level, emptying themselves.This is why in the blue regions of Fig. 5 (e) and (f) thewire’s total charge does not increase at the correspondingparity crossing, but instead decreases loosing effectivelytwice the charge of the electron e . Finally, we would like to point out that, when the dotlevels anticross the Majorana ones in a pinning region,the Majorana states detach from zero energy. This canbe seen in Fig. 5(c) and Fig. 1(b). The reason is that,although in the pinning regions the Majorana energy iszero, their wave function overlap is not. It is actuallymaximum, as explained when discussing Fig. 4. EachQD acts as a local probe (one couples to the left topo-logical region of the wire, the other to the right). If thewire’s length were large (much bigger than the coherencelength), left and right Majoranas would be disconnectedfrom each other, and a local probe coupled to one of themwould not be able to change its energy or perturb it. Thisis actually the core manifestation of their topological pro-tection. However, when the wire’s length is finite and theMajoranas overlap, each QD couples to both Majoranasat either end and their energies are modified. The typicalshapes of the anticrossing were recently analyzed and canbe used to quantify the degree of Majorana non-locality[35, 36]. IV. CONCLUSIONS
In this work we have studied the low energy charac-teristics of Majorana nanowires when including its in-teraction with a realistic 3D electrostatic environment.This is done by solving self-consistently the Bogoliubov-de Gennes equation together with the Poisson’s equation.Typically, the total charge of the wire in equilibrium withthe reservoirs increases with magnetic field (or the wire’schemical potential). However, if the electrostatic screen-ing is smaller inside the wire than at the contacts, a repul-sive interaction arises that leads to zero energy pinningaround parity crossings in the wire’s spectrum. Whilethe screening due to the parent SC shell tends, in gen-eral, to reduce this pinning effect, we find that it stillpersists depending on the quality of the SC layer and thelocation of the charge density within the nanowire. Thepinning mechanism could help explain the precise shapeof the Majorana oscillations (or lack thereof) observed insome dI/dV experiments, which exhibit substantial de-viations from the predictions of simple models for finitelength wires.On the other hand, and more importantly, the selfconsistent solution of the electrostatic potential variesnon-homogeneously along the wire. It is relatively flatin the central region but, due to the screening from theleft/right metallic contacts, it becomes strongly negativeat the edges. This creates potential wells that confineQD-like states at the ends of the wire, which appear inthe spectrum as discrete states within the induced gapthat disperse with Zeeman energy or chemical potential.These QD levels interact with the Majorana states in aspecific way which is strongly dependent on the Majoranawavefunction, and particularly on its degree of spatialnon-locality. The pinning mechanism and the couplingto QD-like states compete against each other, so that x/L | , | x/L | L , R | x/L | , | x/L | L , R | (c) (d)(e) (f) V Z (meV) -0.300.3 E ( m e V ) (a) V Z (meV) | Q M / e | (b) FIG. 6. Evolution with Zeeman field of the spectrum (a) and the absolute value of the Majorana charge Q M (b) for thebarrier-like potential model of Fig. 5(b). Panels (c) and (e) show the wave-function probability profile (in the Majorana basis)of the two lowest energy states at the V Z values indicated in (a). Panels (d) and (f) show the same but for the second andthird energy states (QD-like states). At the dot-Majorana levels anticrossing, the Majorana wave function leaks into the dotregions leaving the topological region of the wire practically void. This is manifested in | Q M | by two consecutive zeros, one perdot level (around V Z = 2 . the pinned zero-energy plateaux may become lifted atresonance with the dot states, thus revealing their elec-trostatic origin (as opposed to true wavefunction non-locality). ACKNOWLEDGMENTS
We thank P. San-Jose and R. Aguado for valuablediscussions. Research supported by the Spanish Min-istry of Economy and Competitiveness through GrantsFIS2014-55486-P, FIS2016-80434-P (AEI/FEDER, EU),the Ram´on y Cajal programme Grants RYC-2011-09345and the Mar´ıa de Maeztu Programme for Units of Excel-lence in R&D (MDM-2014-0377).
SUPPORTING INFORMATIONSI 1: EXPRESSIONS FOR THE INDUCEDPOTENTIAL USING THE IMAGE CHARGEMETHODIntroduction
The electrostatic potential φ b produced by the envi-ronment is calculated from the interaction between thenanowire charge density and bound charges it createsat the surrounding medium shown in Fig. (1) of the main text. To do this we use the method of the im-age charges. We model the nanowire as a semiconductorrod of square section of relative permittivity (cid:15) , length L and rectangular section of width W = 2 R , being R the half-width. We assume that the charge density ρ ( (cid:126)r )of the nanowire is located along its symmetry axis ( x -axis) as a linear charge density. The nanowire faces arein contact with different dielectric materials of permit-tivities (cid:15) (substrate), (cid:15) (superconducting shell), (cid:15) and (cid:15) (surrounding medium); while two metal leads of per-mittivities (cid:15) M and (cid:15) M are placed at both ends of thenanowire. In order to give more insight on the solutionof this problem, we solve first some simpler cases. In thefirst section we obtain the electrostatic potential createdby a linear charge density placed before one, betweentwo, and before two infinite planes. These problems canbe understood as 1D problems. In the second section weobtain the potential with all four surrounding media butwithout the bulk leads, which can be treated as a 2Dproblem. Finally, in the third section we obtain the fullmodel including also the interaction with the leads. Section I: One dimension
A. One infinite plane
The solution of this problem can be found in textbookson electromagnetism [46]. When one charge q is placed (a) (b)(c) (d) κ κ κ κ κ κ κ κ κ κ κ κ κ κ κ !! ! ! ! !!! RRR FIG. 7. Configuration of the image charges produced by theoriginal charge (in black) studied in different Sections. Theoriginal charge is in a medium of dielectric constant (cid:15) (greencolor in the plots) that extends in the third direction. (a) Sec.IA: a charge in front of an infinite semi-space of permittivity (cid:15) at a distance R . (b) Sec. IB: a charge between two semi-infinite and parallel regions of permittivities (cid:15) and (cid:15) . (c)Sec. IIA: a charge close to the intersection between two mediaof permittivities (cid:15) and (cid:15) . (d) Sec. IIB: a charge betweenfour different media. As before, the black point depicts thereal charge, while the red, purple and orange points depict thefirst, second and third image charges found in the first, secondand third steps of the image method procedure, respectively.For (a) and (c) all the image charges are shown, whereas for(b) and (d) there is an infinite number of them and only thefirst ones are shown. Black solid lines are interfaces, and blackdashed lines are image interfaces. (at the origin of coordinates) in a medium of dielectricconstant (cid:15) and at a distance R from the interface betweenthis and another medium of permittivity (cid:15) , a boundcharge of magnitude κ q appears spread at the interface,where κ ≡ (cid:15) − (cid:15) (cid:15) + (cid:15) . (5)The effect of this bound charge is equivalent to that ofa point charge of the same magnitude and located ata specular distance with respect to the plane from theoriginal charge, see Fig. 7(a). This is why it is called an image charge. The classical electrostatic potential due tothe interaction between the original and the image chargethrough the Coulomb’s law takes the form φ b ( x ) = 14 π(cid:15)(cid:15) κ q (cid:113) (2 R ) + x , (6) where (cid:15) is the vacuum permittivity. Because φ b is linearin q , this result can be generalized to an arbitrary 1Ddensity charge ρ ( x ): φ b ( x ) = 14 π(cid:15)(cid:15) (cid:90) κ ρ ( x (cid:48) ) (cid:113) (2 R ) + ( x − x (cid:48) ) dx (cid:48) . (7)As argued in Ref. [33], since the bound charges aredistinguishable from the nanowire charges (cannot tun-nel in and out), this potential can be directly transformedinto a quantum operator as a Hartree interaction withoutany Fock correction. Assuming a purely local polarizabil-ity (Thomas-Fermi limit), the density operator may betransformed as ρ ( x (cid:48) ) → (cid:104) ρ ( x (cid:48) ) (cid:105) , which is perfectly equiv-alent to the above classical equation. Then, the potentialtakes the form φ b ( x ) = (cid:90) V b ( x, x (cid:48) ) (cid:104) ˆ ρ ( x (cid:48) ) (cid:105) dx (cid:48) , (8)where the kernel of the interaction V b ( x, x (cid:48) ) ≡ κ / π(cid:15)(cid:15) (cid:113) (2 R ) + ( x − x (cid:48) ) encodes the geometrical in-formation of the interaction.We note that Eq. (8) is general for any geometry andcharge density. For this reason, in the following sectionswe first obtain the kernel function V b for a given geome-try using a single original charge, and then we generalizeour results to an arbitrary density ρ ( x ) and to its corre-sponding quantum expression using Eq. (8). B. Two opposite infinite planes
We now consider a charge between two infinite andparallel planes separating the nanowire’s dielectric con-stant (cid:15) from two other media of permittivities (cid:15) and (cid:15) [see Fig. 7(b)]. In order to try to satisfy the boundaryconditions imposed by the Gauss’ law, in a first step twoimage charges κ q and κ q are required in each dielectricmedium at the same distance R from the interface [see reddots in Fig. 7(b)]. However, each image charge does notsatisfy the boundary condition with respect to the oppo-site interface. For this reason, in a second step, two (ac-cidentally equals) additional image charges of magnitude κ κ q are required at a distance 3 R from each interface[see purple dots in Fig. 7(b)]. But, again, these addi-tional image charges do not satisfy the boundary condi-tions with respect to the opposite interfaces and anotherstep must be taken. It is possible to see that for eachimage charge q ( n ) α in the dielectric α created at the n -thstep of the image charge method, another image charge q ( n +1) β = κ β q ( n ) α (9)is required in the opposite β dielectric at a distance2 ( n + 1) R from the original charge q in order to sat-isfy the boundary conditions, with q (0) α,β = 1. Thus, the0interaction kernel takes the form of an infinite series V b ( x ) = 14 π(cid:15)(cid:15) ∞ (cid:88) n =1 q ( n )1 + q ( n )3 (cid:113) x + (2 nR ) . (10)Note that each term of the sum decreases with n as V ( n ) b ∼ κ n /n . Since | κ | < κ M = − (cid:15) M → ∞ ), then thekernel converges to V b ∼ ln (1 − κ ) in spite of the infinitesummation. C. Two consecutive infinite planes
Finally, we consider the case depicted in Fig. 8(a)where a (real) charge q is placed in a dielectric materialcharacterized by a permittivity (cid:15) A . This charge is at adistance R from a first interface with a material of dielec-tric constant (cid:15) B and width W , and at a distance R + W from a second (parallel) interface with another mediumof permittivity (cid:15) C . As we know, the real charge q cre-ates an image charge κ B q at a distance 2 R from it insidethe (cid:15) B medium. However, the potential created by bothcharges is only valid inside the (cid:15) A medium. The poten-tial created inside the (cid:15) B medium can be found using theimage method [46]: it is the same potential than that cre-ated by an image charge of magnitude q [2 (cid:15) A / ( (cid:15) A + (cid:15) B )]located at the same position of the real charge. Fromthis point, the same steps explained in the previous sub-section can be followed, and once the infinite series isobtained, the potential can be transformed back to the (cid:15) A medium. Thus, one can prove that the bound chargespotential is given by V b = q π(cid:15) A (cid:15) (cid:18) (cid:15) A + (cid:15) B (cid:19) (cid:34) (cid:15) A − (cid:15) B (cid:112) x + (2 R ) ++ 4 (cid:15) A (cid:15) B (cid:15) B − (cid:15) A ∞ (cid:88) n =1 (cid:16) ( (cid:15) B − (cid:15) A )( (cid:15) B − (cid:15) C )( (cid:15) B + (cid:15) A )( (cid:15) B + (cid:15) C ) (cid:17) n (cid:112) x + (2 R + 2 nW ) . (11)Since this expression is rather complex, it is convenientto replace the effect of both media (cid:15) B and (cid:15) C by just onecharacterized by an effective permittivity (cid:15) eff which, froman electrostatic point of view, is equivalent. Hence, thebound charges potential would be given by Eq. 6 with (cid:15) → (cid:15) A and (cid:15) → (cid:15) eff . Comparing both equations, theeffective permittivity (at x = 0) is (cid:15) eff = (cid:15) A − κ eff κ eff , (12)where κ eff = 1 (cid:15) A + (cid:15) B [ (cid:15) A − (cid:15) B ++ 4 (cid:15) A (cid:15) B (cid:15) B − (cid:15) A ∞ (cid:88) n =1
11 + n WR (cid:18) ( (cid:15) B − (cid:15) A )( (cid:15) B − (cid:15) A )( (cid:15) B + (cid:15) C )( (cid:15) B + (cid:15) C ) (cid:19) n (cid:35) . (13) tf S C W SC =8nmW SC =16nm C A (a) W B R (b) FIG. 8. (a) System studied in Sec. IC. A charge (black dot)is in a medium of dielectric constant (cid:15) A (the nanowire) andplaced at a distance R from a thin material of permittivity (cid:15) B and thickness W (the SC shell). Above the thin film there isanother semi-infinite space of permittivity (cid:15) C (the vacuum).When all the permittivities are finite, bound charges arise inboth surrounding mediums. (b) Effective SC permittivity (cid:15) SC calculated using Eq. 12 vs the SC thin film permittivity (cid:15) tf for two different film thicknesses W SC . Parameters are thesame as in the main text. We note that this system corresponds to the nanowire-SC shell-vacuum double interface of Fig. 1(a). There,the effect of the shell finite width and the vacuum on tophas been condensed in an effective SC permittivity. Thismeans that, in Eq. 12, (cid:15) eff → (cid:15) SC , (cid:15) A → (cid:15) , (cid:15) C → (cid:15) a and (cid:15) B is the true SC thin film permittivity (cid:15) tf .In Fig. 8(b) we show the effective SC permittivity (cid:15) SC considered in the main text as a function of (cid:15) tf for twodifferent shell widths W SC . Notice that, as the SC shellbecomes thinner (red curve corresponds to 8nm), theeffective permittivity gets more renormalized. A value (cid:15) SC ∼
100 corresponds to (cid:15) tf ∼ Section II: Two dimensions
A. One rectangular corner
This is also a textbook problem [46]. A charge q insidea dielectric with permittivity (cid:15) is placed at a distance R from dielectric (cid:15) and at a different distance R formdielectric (cid:15) , which are perpendicular to one another [seeFig. 7(c)]. To satisfy the boundary conditions, two im-age charges κ q and κ q are required in each dielectricat ( x, − R ,
0) and ( x, , − R ), see red dots. Becauseof these, another image charge of magnitude κ κ q is re-quired at ( x, − R , − R ), purple dot. In this case, anddue to the closed geometry of the problem, three imagecharges are enough to satisfy the boundary conditions.1The kernel function takes thus the form V b ( x ) = 14 π(cid:15)(cid:15) κ (cid:113) x + (2 R ) + κ (cid:113) x + (2 R ) ++ κ κ (cid:113) x + (2 R ) + (2 R ) . (14) B. Four rectangular corners: interaction without the leads
Now a charge q is placed inside an infinite wire of rect-angular section and of permittivity (cid:15) , see Fig. 7(d). Thischarge is at a distance R from two parallel flat dielectricswith permittivities (cid:15) and (cid:15) , and at a distance R fromanother two parallel dielectrics with permittivities (cid:15) and (cid:15) which are perpendicular to the previous ones. Com-bining what we have learned in the previous Sections,to satisfy the boundary conditions an infinite ensembleof image charges have to be placed in the different di-electrics as shown in Fig. 7(d). For each image charge in one dielectric, another one appears at a specular dis-tance from the opposite interface, while for each two im-age charges placed in two perpendicular media, just onemore appears at the corner. The electrostatic potentialdue to all these charge is given by V b ( x ) = 14 π(cid:15)(cid:15) ∞ (cid:88) n,m =1 (cid:16) q ( n )1 + q ( n )3 (cid:17) (cid:16) q ( m )2 + q ( m )4 (cid:17)(cid:113) x + (2 nR ) + (2 mR ) ++ ∞ (cid:88) n =1 q ( n )1 + q ( n )3 (cid:113) x + (2 nR ) + q ( n )2 + q ( n )4 (cid:113) x + (2 nR ) , (15)where: q ( n +1)1 = κ q ( n )3 , q ( n +1)2 = κ q ( n )4 ,q ( n +1)3 = κ q ( n )1 , q ( n +1)4 = κ q ( n )2 ,q (0) α = 1 ∀ α = { , , , } . (16)If the wire’s section is square ( R = R ), then thekernel function can be rewritten in a more compact way: V b ( x ) = 14 π(cid:15)(cid:15) ∞ (cid:88) n,m =0 (cid:16) q ( n )1 + q ( n )3 − δ n, (cid:17) (cid:16) q ( m )2 + q ( m )4 − δ m, (cid:17)(cid:113) x + (2 nR ) + (2 mR ) (1 − δ n + m, ) . (17)Note that the number of charges increases as 4 n at each n -th step of the image charge method, whereas the restof the expression inside the brackets decreases as κ n /n as before. Thus, each term of the sum goes as V ( n ) b ∼ κ n and the infinite sum is proportional to V b ∼ κ/ (1 − κ )if | κ | <
1, so the convergence of the kernel is ensured inthis case as well.
Section III: The full-model
Finally, we solve the full system of Fig. (1) of the maintext. Now, apart from the four dielectric media in each of the four faces of the square section, there are another twofaces in the x -direction in contact to metallic regions. Weconsider that the nanowire has a square section of semi-width R . First, we assume that the charge q is placed atthe coordinates origin and at the same distance R fromeach metallic region M and M . Following the sameprocedure as before we obtain V b ( x ) = 14 π(cid:15)(cid:15) ∞ (cid:88) n,m,k =0 (cid:16) q ( n )1 + q ( n )3 − δ n, (cid:17) (cid:16) q ( m )2 + q ( m )4 − δ m, (cid:17) q ( k ) M (cid:113) ( x + 2 kR ) + (2 nR ) + (2 mR ) ++ (cid:16) q ( n )1 + q ( n )3 − δ n, (cid:17) (cid:16) q ( m )2 + q ( m )4 − δ m, (cid:17) (cid:16) q ( k ) M − δ k, (cid:17)(cid:113) ( x − kR ) + (2 nR ) + (2 mR ) (1 − δ n + m + k, ) , (18)where (cid:40) q ( n +1) M = κ M q ( n ) M , q ( n +1) M = κ M q ( n ) M ,q (0) α = 1 ∀ α = { M , M } . (19) If now the charge q is placed at an arbitrary position2 x (cid:48) inside the nanowire, and the metal M interface is at x = 0 and the M interface is at x = L , then the kernelfunction is given by V b ( x ) = 14 π(cid:15)(cid:15) ∞ (cid:88) n,m,k =0 (cid:16) q ( n )1 + q ( n )3 − δ n, (cid:17) (cid:16) q ( m )2 + q ( m )4 − δ m, (cid:17) q ( k ) M (cid:114)(cid:16) x − ( − k (cid:16) floor ( k +1 ) L − L + x (cid:48) (cid:17)(cid:17) + (2 nR ) + (2 mR ) ++ (cid:16) q ( n )1 + q ( n )3 − δ n, (cid:17) (cid:16) q ( m )2 + q ( m )4 − δ m, (cid:17) (cid:16) q ( k ) M − δ k, (cid:17)(cid:114)(cid:16) x + ( − k (cid:16) floor ( k +12 ) L − x (cid:48) (cid:17)(cid:17) + (2 nR ) + (2 mR ) (1 − δ n + m + k, ) . (20)If L is large enough compared to 2 R , we can take intoaccount only the lowest order of the image charges at themetals, q ( k =0 , M i , and the number of charges at each n -thstep of the image charge method increases only as ∼ n .Then, the system follows the same convergence criterionas in the previous Section. If L ∼ R , Eq. (20) convergesas well, but the demonstration is longer. SI 2: FURTHER DETAILS ON NUMERICALMETHODSMean field approximation to treat electron-electroninteractions
We want to solve the energy spectrum of the nanowireHamiltonian ˆ H when the interaction between electrons ˆ φ is included. In general, this interaction can be written insecond quantization asˆ φ = (cid:88) α,β ˇ c † α ˇ c α V αβ ˇ c † β ˇ c β , (21)where ˇ c † α , ˇ c α are defined as the Nambufied vector of op-eratorsˇ c † α = (cid:16) c † ↑ , c † ↓ , c † ↑ , ..., c † N ↓ , c ↑ , c ↓ , c ↑ , ..., c N ↓ (cid:17) . (22)Here, c † iσ and c iσ are electron creator/annihilation oper-ators with quantum numbers α . V αβ above encodes theelectronic interaction. Therefore, Greek indexes α, β en-code both particle/hole character, spin ( ↑ , ↓ ), and anyother indexes the electron might have, such as site index i = 1 , ..., N in a tight-binding description.To treat the quartic interaction we resort to amean field approach called the Hartree-Fock-Bogoliubov(HFB) approximation. Using the Wick’s theorem andneglecting fluctuations and constant terms we can writeˆ φ eff = (cid:88) α,β V αβ (cid:104)(cid:10) ˇ c † α ˇ c α (cid:11) ˇ c † β ˇ c β + (cid:68) ˇ c † β ˇ c β (cid:69) ˇ c † α ˇ c α + (cid:10) ˇ c † α ˇ c β (cid:11) ˇ c † β ˇ c α + + (cid:68) ˇ c α ˇ c † β (cid:69) ˇ c † α ˇ c β − (cid:68) ˇ c † α ˇ c † β (cid:69) ˇ c α ˇ c β − (cid:104) ˇ c α ˇ c β (cid:105) ˇ c † α ˇ c † β (cid:105) . (23)The first two terms are known as Hartree terms, whichinclude information about direct (repulsive/attractive)interaction between electrons. The second and thirdare the Fock terms, which include the exchange inter-action due to the electron indistinguishable properties.These terms ensure that non-physical self-interactions in-troduced by the first terms are cancelled. The last twoare known as Bogoliubov terms, which include possiblepairing correlations between electrons.We want to rewrite Eq. (23) in a more compact man-ner. In order to do that, we define two matrices: • The lambda matrix:Λ ≡ I space ⊗ I × ⊗ σ x , (24)where I space is the identity matrix in real space(for a one-dimensional tight-binding model with N sites, this matrix is the N × N identity), I × is theidentity matrix in spin space, and σ x is the Pauli x -matrix in Nambu space. This matrix satisfies theproperty ˇ c † = Λˇ c . • The density matrix: ρ αβ ≡ (cid:10) ˇ c † α ˇ c β (cid:11) . (25)We can express this matrix in terms of the eigen-vectors γ n of the Hamiltonian ˆ H + e ˆ φ eff , which arerelated to the ˇ c α ’s through a unitary transforma-tion Ψ as γ n = Ψ nα ˇ c α . Then ρ αβ = (cid:10) ˇ c † α ˇ c β (cid:11) = Ψ ∗ αn (cid:10) γ † n γ m (cid:11) Ψ mβ = (cid:0) Ψ † F Ψ (cid:1) αβ , (26)where F nm ≡ (cid:10) γ † n γ m (cid:11) = f F D ( (cid:15) n ) δ nm is the Fermi-Dirac distribution matrix.3Using these two matrices, Eq. (23) can be rewritten as φ eff = 2 D [ V · d { ρ } ] + Λ · ( V (cid:63) ρ ) · Λ +
V (cid:63) (Λ · ρ · Λ) −− Λ · ( V (cid:63) ( ρ · Λ)) − ( V (cid:63) (Λ · ρ )) · Λ , (27)where we assume a symmetric potential V αβ = V βα . Herewe have used the notation D [ v ] as the diagonal matrixwith vector v in its diagonal, d { A } as a column vectorwhose elements are the diagonal elements of matrix A ,the dot product A · B as a matrix product, and the starproduct A (cid:63) B as an element wise product between ma-trices.However, Eq. (27) does not have Nambu structurebecause in general V iστ,iστ (cid:54) = V iσ ¯ τ,iσ ¯ τ , so Bogoliubov-deGennes formalism cannot be applied. We symmetrizethis expression by doingˇ c † φ eff ˇ c = ˇ c † (cid:18) φ eff − Λ φ t eff Λ2 (cid:19) ˇ c + cnst ., (28)where we have used the anticommutation relation (cid:8) c, c † (cid:9) = 1, the property Λ t = Λ and we neglect con-stant terms again. Thus, the general interaction betweenelectrons in the HFB approximation can be written as φ HFB = 12 ˇ c † (cid:0) φ eff − Λ φ t eff Λ (cid:1) ˇ c, (29)where φ eff is given by Eq. (27). Inclusion of the intrinsic interaction
The interaction between the electrons inside thenanowire ( intrinsic interaction) is given, in principle, bythe bare Coulomb potential in one dimension. Takinginto account the finite radius (half-width) R of the wire,a more precise form for the interaction is [47] V ( x ) = √ π π(cid:15)(cid:15) R e x /R Erfc (cid:18) | x | R (cid:19) , (30)where x is the distance between electrons.When we consider this potential only at the Hartreelevel, we find zero energy pinning around parity crossings,just like we did with the extrinsic interaction. However,this pinning is unphysical since it comes from spuriousself-interaction terms introduced by the Hartree approx-imation [33]. For the intrinsic interaction it is thus nec-essary to include the Fock correction due to the indistin-guishability of electrons in the nanowire.If we consider the bare interaction, Eq. (30), in theFock terms, an overcompensation of the pinning effect isfound and unphysical jumps appear at each parity cross-ing. To cure this problem, we introduce screening in thequasi-static Thomas-Fermi limit so that the potential inthe Fock terms acquires an additional exponential decay e −| x | /λ TF that depends on the Thomas-Fermi length λ T F (which should be of the order of the Fermi wavelength). In this case, we find that the parity crossings inducedby the intrinsic interaction are suppressed, in agreementwith the self-interaction argument.If the nanowire is discretized using a thigh-bindingmodel, the interaction can be written as V α,β = √ π π(cid:15)(cid:15) R exp (cid:40)(cid:18) i − jaR (cid:19) − | i − j | aλ T F (cid:41) ·· Erfc (cid:18) | i − j | aR (cid:19) [1 − δ α,β ] , (31)where the indexes α = { i, σ, τ } and β = { j, σ (cid:48) , τ (cid:48) } in-clude all the quantum numbers of the electrons, a is thedistance between two neighboring sites (lattice constant),and the term [1 − δ α,β ] ensures that an electron cannotinteract with itself. As stated before, the above equationis only valid for the Fock terms, while for the Hartreeterms it is the bare interaction (same expression with λ T F → ∞ ). Then, one can obtain the potential in theHFB approximation using Eq. (29).The electron-electron interaction between the nanowireand the bound charges of the dielectric environment ( ex-trinsic interaction) is found in SI 1, and it may be im-plemented following the same procedure by substituting x → ( i − j ) /a . We note that now the term [1 − δ α,β ]should not be included since electron α is always in-side the nanowire while β is outside, in the surroundingmedium (or the other way around). Finally, the poten-tial in the HFB approximation can be computed usingEq. (29), but now the Fock and the Bogoliubov termshave to be ignored as we argued in the SI 1, so that thelast four terms of Eq. (27) are not considered. Numerical self-consistent method used to solve theeigenspectrum
We note that ˆ H + e ˆ φ HFB depends on its own eigen-vectors (see Eq.(26)), and thus it has to be solved self-consistently. We solve this problem numerically usingthe following procedure: in the first iteration of the self-consistent method, we find the density-matrix ρ usingthe eigenvectors of the Hamiltonian ˆ H as a test solu-tion. In the next step we obtain a new ρ diagonalizingˆ H + e ˆ φ HFB where the interaction has been obtained us-ing the previous density matrix. In the following steps,the density-matrix is found using a linear combinationof the eigenvectors of ˆ H + e ˆ φ HFB in the two previousiterations. This is done to introduce damping in the it-eration procedure in order to ensure the convergence ofthe self-consistent method. In each step, we compare theeigen-energies of ˆ H + e ˆ φ HFB with those of the previousstep. We repeat this procedure until convergence. Weconsider the iteration has converged when the differencebetween both energy-spectra is much smaller than themain energy scale of our problem (i.e. the superconduc-tor gap ∆).4 x/L -8-6-4-202 e b ( x ) ( m e V ) V Z (meV) -0.300.3 E ( m e V ) V Z (meV) -0.300.3 E ( m e V ) x/L -8-6-4-202 e b ( x ) ( m e V ) (c)(a) (d)(b) FIG. 9. Majorana nanowire in the presence of interactions(including the influence of the bulk normal leads at its ends).Self-consistent induced potential energy eφ b ( x ) along thewire’s length for increasing values of the Zeeman splittingignoring (a) and including (b) the electron-electron interac-tions inside the nanowire. (c) and (d) are their correspondingenergy spectra. Wire parameters are the same as in the maintext, and the Thomas-Fermi length is λ TF = L/ SI 3: NANOWIRE SPECTRUM INCLUDINGTHE INTRINSIC INTERACTION
Here we show that the features studied in the main text(zero-energy pinning and QD formation) remain qualita-tively unaltered when including electron-electron interac-tions φ int inside the nanowire. The intrinsic interactionintroduces small quantitative changes in the spectrum,but the qualitative behavior stays the same. Followingour previous work [33], we treat this interaction at themean field level, within the Hartree-Fock-Bogoliubov ap-proximation, and assume a bare Coulomb interaction forthe Hartree terms and a screened Coulomb interaction inthe quasi-static Thomas-Fermi limit for the Fock terms(see SI 2 for more details).In Fig. 9 we show the bound charges electrostatic po-tential along the nanowire and the energy spectrum ver-sus the Zeeman field ignoring (a and c) and including(b and d) the intrinsic interaction. Note that here wedo not include the Bogoliubov correction since this termjust renormalizes the gap as shown in Ref. [33]. In gen-eral, both spectra are qualitatively similar and thus wecan conclude that the intrinsic interactions do not alterthe features studied in the main text. However, there aresome quantitative differences. First, the dispersive QDlevels approach zero energy at a slightly smaller Zeemanenergy as a result of small changes in the induced poten-tial φ b , as can be seen in Fig. 9(d). Second, the positionof the gap closing and thus, the topological phase tran-sition, shifts to a different magnetic field. This is alsoa consequence of small changes in φ b , as well as smallchanges in the Zeeman energy induced by the Fock terms, that modify the value of the critical Zeeman field throughEq. (4) of the main text. Finally, the energy splitting ofthe Majoranas is larger due to the renormalization of theFermi momentum induced by the Fock terms as well. SI 4: ROBUSTNESS OF THE PINNING EFFECT
We want to test the validity of our results when vary-ing different parameters of the electrostatic environment.Fig. 10 provides various phase-diagrams indicating theoccurrence of Majorana bound states zero-energy pin-ning (in red) as a function of the different parameters.Although we have taken µ = 0 . µ and V Z ,the non-interacting lines of (a) corresponding to point-like parity crossings transform into incompressible finitewidth stripes in the interacting case (b). Pinning regionsare bigger for lower chemical potentials and for highermagnetic fields, since the repulsive interaction is largertoo. It can also be observed that the onset of the topo-logical phase is different in the interacting system thanin the non-interacting one (black dashed line), at leastfor positive µ . This is because the electrostatic potentialrenormalizes the chemical potential [48] and thus it mod-ifies the value of the bulk critical magnetic field, given inEq. (4) of the main text.However, pinning is not general for all kind of envi-ronments. Figure 10(c) shows the zero energy regionsacross the V Z − (cid:15) SC space (the µ − (cid:15) SC diagram exhibitsa similar behavior). For (cid:15) SC (cid:38)
300 the pinning plateauwidth shrinks into points because the electrostatic envi-ronment turns into an attractive one. This means thatbound charges of the opposite sign arise in the dielectricmedium at these large permittivities, so that the entranceof charge at each parity crossing is no longer suppressed.Note that (cid:15) SC represents the effective SC permittivityof the system composed by a SC thin film (epitaxiallygrown over the nanowire) of intrinsic permittivity (cid:15) tm ,finite width W SC and covered by vacuum, as we argue inSI 1(IC). For a film width of 8nm, an effective (cid:15) SC (cid:38) (cid:15) SC (cid:38) V Z and r yz , where r yz = W y /W z is the aspect ratio of the nanowire section.When the distance between the SC shell and the oppo-site side is large (large r yz ), pinning is bigger. This isbecause the relative coverage of the wire by the SC shelldecreases and so does its attractive contribution.Finally, if we consider perfect metallic screening by theSC shell, i.e., (cid:15) SC → ∞ , we can also study the appearanceof the pinned regions depending on the distance betweenthe nanowire charge density and the SC shell. In Fig.4(e) we study the phase diagram as a function of V Z and5 (e) (f) FIG. 10. Phase diagrams indicating the parameter regionswhere the Majorana bound states are pinned to zero-energy(in red). In the upper panels the phase diagram is calculatedas a function of Zeeman field and chemical potential for thenon-interacting case (a) and interacting case(without leads)with (cid:15) SC = 100 (b). The central panels correspond both tothe interacting case, but (c) considers variations in the effec-tive dielectric constant of the thin superconducting layer, (cid:15) SC ,whereas (d) explores different aspect ratios of the nanowire’ssection r yz = W y /W z , where W y,z are the y and z widths ofthe nanowire faces. In the lower panels we consider perfectscreening by the SC shell, (cid:15) SC → ∞ , but we vary the dis-tance between the transversal Majorana charge density andthe SC shell. In (e) y/R is the position of the Majorana wavefunction across the nanowire section. When y = 0 the chargedensity is at the center of the nanowire, whereas when y = R it is a the opposite face of the SC. In (f) the wave functionis fixed at the center of the nanowire section, but we vary itsaspect ratio. Here µ = 0 . W z = 100nm, as in themain text. the position of the charge density across the nanowiresection, y/R , where R is the wire’s half width. 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