Quantized and unquantized zero-bias tunneling conductance peaks in Majorana nanowires: Conductance below and above 2 e 2 /h
Haining Pan, Chun-Xiao Liu, Michael Wimmer, Sankar Das Sarma
QQuantized and unquantized zero-bias tunneling conductance peaks in Majoranananowires: Conductance below and above e /h Haining Pan, ∗ Chun-Xiao Liu, ∗ Michael Wimmer, and Sankar Das Sarma Condensed Matter Theory Center and Joint Quantum Institute,Department of Physics, University of Maryland, College Park, Maryland 20742, USA Qutech and Kavli Institute of Nanoscience, Delft University of Technology, Delft 2600 GA, The Netherlands
Majorana zero modes can appear at the wire ends of a one-dimensional topological superconductorand manifest themselves as a quantized zero-bias conductance peak in the tunnel spectroscopy ofnormal-superconductor junctions. However, in superconductor-semiconductor hybrid nanowires,zero-bias conductance peaks may arise owing to topologically trivial mechanisms as well, mimickingthe Majorana-induced topological peak in many aspects. In this work, we systematically investigatethe characteristics of zero-bias conductance peaks for topological Majorana bound states, trivialquasi-Majorana bound states and low-energy Andreev bound states arising from smooth potentialvariations, and disorder-induced subgap bound states. Our focus is on the conductance peak value(i.e., equal to, greater than, or less than 2 e /h ), as well as the robustness (plateau- or spike-like)against the tuning parameters (e.g., the magnetic field and tunneling gate voltage) for zero biaspeaks arising from the different mechanisms. We find that for Majoranas and quasi-Majoranas,the zero-bias peak values are no more than 2 e /h , and a quantized conductance plateau formsgenerically as a function of parameters. By contrast, for conductance peaks due to low-energyAndreev bound states or disorder-induced bound states, the peak values may exceed 2 e /h , anda conductance plateau is rarely observed unless through careful fine-tunings. Our findings shouldshed light on the interpretation of experimental measurements on the tunneling spectroscopy ofnormal-superconductor junctions of hybrid Majorana nanowires. I. INTRODUCTION
Majorana zero modes (MZM), which are the fun-damental non-Abelian units for topological quantumcomputing, have been extensively studied in exper-iments on superconductor-semiconductor (SC-SM) hy-brid nanowires over the past decade.
Owing to theimproved sample quality of hybrid nanowires, severalhallmarks of Majorana zero modes have been observedin tunneling conductance spectroscopy measurements,even the apparent quantized zero-bias conductance peak(ZBCP). Recently, Refs. 12 and 19 have shown an almost-quantized conductance of 2 e /h with plateaus robustagainst the Zeeman field or the gate voltage to somedegree. However, these experiments are inconclusive toconfirm the MZM due to the lack of the manifestation ofother hallmarks of MZM, which should be accompaniedby the observation of the quantized ZBCP at the topolog-ical quantum phase transition (TQPT), such as the bulkgap closing and reopening features and the growingMajorana oscillations with an increasing magnetic fieldstrength. Also, no nonlocal correlation measurementshave been reported as expected for a topological state.Therefore, it is necessary to understand the origin of theplateau in the conductance data before making any con-clusions.MZM can induce a quantized conductance of2 e /h in a long wire due to perfect Andreev reflec-tion in the local conductance measurement in the tunnelspectroscopy experiment. However, this “quantized con-ductance of 2 e /h ” is only a necessary condition but nota sufficient one to deduce the topological MZM. Manyother mechanisms can also trivially induce seemingly quantized ZBCPs arising from subgap Andreev boundstates (ABS). For example, inhomogeneous chemi-cal potential and random disorder are common mecha-nisms that may induce the trivial ABS with ∼ e /h ZBCPs. However, unlike the robust topological MZM-induced quantized ZBCPs, these trivial ZBCPs are usu-ally either unstable or unquantized. An important issuein this context is the extent to which these trivial ZBCPscould accidentally produce, perhaps through some fine-tuning of parameters, somewhat stable-looking 2 e /h ap-parent quantization, consequently misleading the exper-imentalists into thinking that topological MZMs mighthave been observed. The crucial question is whether onecan assert that the observation of a fine-tuned appar-ently stable quantized ZBCP automatically implies theexistence of topological MZMs.The answer to this question turns out to be negative.The inhomogeneous smooth confining potential is a typi-cal counter-example that has the ability to manifest triv-ial stable quantized ZBCPs with a nonzero probabilitygiven appropriate parameters. Namely, the inhomoge-neous potential should be barrier-like, which, as a result,confines the quasi-Majoranas that are partially-separatedin the nanowire so that they are highly-overlapped spa-tially (and, therefore, not isolated MZMs, only quasi-MZMs). The nonlocality of Majoranas is not preservedin such a situation; the two Majoranas can both couple tothe same normal lead at the end of the nanowire. There-fore, it manifests a robust ZBCP with the conductanceslightly below the theoretical quantized value of 2 e /h .Even if quasi-Majoranas are not the topological MZM,if the experimentalists observe such robust ZBCP withthe conductance slightly below 2 e /h , it is still a positive a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b development because it may indicate the real MZMs arenot far away—one only has to reduce the inhomogene-ity along the nanowire to separate the two overlappingMZMs in order to see the topological MZMs. In ad-dition, finite temperature and dissipation in experimentscause an inevitable broadening, which lowers the conduc-tance peak. Therefore, it is conceivable that the expectedtopological ZBCP has a robust conductance peak that isslightly below 2 e /h because of broadening, even if it ispurely induced by the real MZM. In this sense, it maynot be too worrisome to observe a robust plateau with theconductance slightly below 2 e /h in experiments. Such apeak may arise from topological MZMs or quasi-MZMs,and the quasi-MZMs should generate topological MZMsif the two overlapping MZMs can be spatially separated.However, what if the observed conductance is slightlyabove 2 e /h ? Is it possible that an inhomogeneous-potential-induced robust ZBCP with conductanceslightly above 2 e /h could arise from trivial quasi-MZMs? In this work, we show that it is, in principle, pos-sible to find trivial ZBCPs induced by an inhomogeneouspotential that have conductances above 2 e /h , but theyare all in the form of spikes and not robust as a functionof parameters. We discuss two such explicit examplesof the inhomogeneous potential, and find that they areboth dip-like inhomogeneous potentials (i.e., a potentialwell rather than a potential barrier, which we call a ‘dip’throughout) in contrast to the barrier-like smooth confin-ing potential which can induce a robust plateau that mayhave a conductance slightly below 2 e /h . Therefore, thisfinding leads to the fact that, if the observed ZBCP man-ifests a robust conductance plateau that is slightly above2 e /h , it is very unlikely to be induced by the inhomo-geneous potential, irrespective of the shape— barrier ordip— because, otherwise, its conductance should be be-low 2 e /h or in the form of spikes instead of a robustplateau. Such a ‘spiky’ ZBCP rises quickly to 4 e /h andthen falls rapidly to almost zero without manifesting anystability as a function of parameters in contrast to the ro-bust 2 e /h (or slightly below it) ZBCP often induced bythe barrier-type potential inhomogeneity. We establishthat a ZBCP plateau slightly above 2 e /h cannot there-fore arise from quasi-MZMs or topological MZMs whichwould produce ZBCPs at or slightly below 2 e /h .Finally, we show that even if such a robust plateau witha conductance above 2 e /h cannot be induced by the in-homogeneous potential, it does not preclude the possibil-ity of a robust plateau with a conductance above 2 e /h arising from other mechanisms. We show that such a sce-nario may occur in the presence of on-site random disor-der in the chemical potential through fine-tunings of therandom configuration. However, even if we allow carefulfine-tunings of the random configuration, the robustnessis still delicate. Unlike the inhomogeneous smooth con-fining potential, which can manifest a robust conductanceplateau (but not above 2 e /h ) in a large region of the pa-rameter space without extensive fine-tuning, the robustregion induced by the random disorder is rather limited: It is only robust against a small range of energy intervalof the Zeeman field, and is not robust against all the gatevoltages, e.g., it may manifest robustness against the Zee-man field to some degree but it becomes uncontrollablewhen the tunneling gate voltage is tuned. Therefore, ifsuch a plateau with the conductance above 2 e /h occursin the laboratory, it indicates that the observed ZBCPcannot be the bad ZBCP induced by inhomogeneous po-tential, as introduced in Ref. 39; instead, it must be theugly ZBCP arising from disorder, and such a ZBCP witha value above 2 e /h is unlikely to be very robust. It issignificant in this context to point out that the recentexperiments reporting 2 e /h conductance quantiza-tion invariably find a conductance slightly above 2 e /h indicating that the current experimental nanowires aredisorder-limited.The remainder of the paper is organized as follows. InSec. II, we introduce the Hamiltonian of the SC-SM hy-brid nanowire, and how the inhomogeneous and disorderpotentials are modeled. In Sec. III, we present our mainresults of the robust (almost-) quantized conductanceplateau in the presence of the inhomogeneous potentialin 1D single-subband nanowire, and also in the more real-istic 3D model using Thomas-Fermi-Poisson method. InSec. IV, we present the opposite scenario, where the con-ductance does not manifest the robustness and becomesarbitrary in the presence of the dip-like inhomogeneouspotential. In Sec. V, we show the conductance spectra inthe presence of random disorder. We discuss the impli-cation of our results in Sec. VI. We conclude in Sec. VII.For completeness, we also provide two movie files showingresults of our simulations while scanning system param-eters. II. MODEL AND METHOD
In much of this work, we model a finite-lengthsuperconductor-semiconductor hybrid nanowire usingthe one-dimensional minimal model, as shown schemati-cally in Fig. 1. The corresponding Bogoliubov-de Gennes(BdG) Hamiltonian is H = 12 (cid:90) L dx ˆΨ † ( x ) H BdG ˆΨ( x ) ,H BdG = H pristine + H V ,H pristine = H SM + H Z + H SC = (cid:18) − (cid:126) m ∗ ∂ x − iα∂ x σ y − µ (cid:19) τ z + E Z σ x − γ ω + ∆ τ x (cid:112) ∆ − ω ,H V = V ( x ) τ z (1)under the basis of ˆΨ = (cid:16) ˆ ψ ↑ , ˆ ψ ↓ , ˆ ψ †↓ , − ˆ ψ †↑ (cid:17) (cid:124) . Here H pristine is the Hamiltonian for a pristine Majorana nanowire inwhich all the physical parameters are spatially homoge-neous, while H V represents the potential inhomogene-ity inside the nanowire. L is the length of the hybrid Lead SMSC V ∆( x ) = ∆ V g V ( x ) = 0(a) Lead SMSC V ∆( x ) = ∆ V g Gaussian(b) Lead SMSC V ∆( x ) = ∆ V g Sinusoidal(c) Lead SMSC V ∆( x ) = ∆ V g Random V imp ( x )(d) FIG. 1. Schematics for the one-dimensional NS junctionmodel (a) of a pristine wire; (b) in the presence of Gaus-sian inhomogeneous potential; (c) in the presence of sinusoidalpotential; (d) in the presence of random disorder. V g is thepotential barrier at the NS junction. nanowire. We describe each term in the Hamiltonian be-low. A. Pristine nanowire
The Hamiltonian for the pristine nanowire is H pristine = H SM + H Z + H SC in Eq. (1). H SM describes thebare spin-orbit-coupled semiconducting nanowire, with m ∗ being the effective mass of the conduction band elec-trons, α the strength of the Rashba spin-orbit coupling,and µ the chemical potential. (cid:126) σ and (cid:126) τ are vectorsof Pauli matrices acting on the spin and particle-holespaces, respectively. H Z describes the induced Zeemansplitting due to an externally applied magnetic field par-allel to the nanowire, with the field strength E Z . H SC de-scribes the superconducting proximity effect induced bya conventional s -wave superconductor. The frequency-dependent self-energy term in H SC is obtained by in- tegrating out the degrees of freedom in the parent su-perconductor, with γ being the SC-SM interfacial cou-pling strength producing the proximity effect, and ∆ be-ing the pairing potential of the parent superconductivity.Near zero energy ( ω → γτ x . In the numerical calculations, unless other-wise stated, we choose the parameters corresponding tothe InSb-Al hybrid nanowire: m ∗ = 0 . m e ( m e is therest electron mass), ∆ = 0 . α = 0 . γ = 0 . L = 3 µ m, phenomenological tun-neling barrier height V g = 10 meV, and assume the zero-temperature limit. Our generic conclusions should notdepend on the choice of these parameters at all. B. Potential inhomogeneity
The effect of potential inhomogeneity inside the hybridnanowire is described by H V in Eq. (1). In realistic junc-tion devices, such an inhomogeneity may arise from theeffect of gates controlling various voltages in the hybridnanowire, from charge impurities, or unintentional disor-ders due to imperfect sample quality. There could alsobe unintentional effective quantum dots at the wire endscreating an inhomogeneous potential. Although the pre-cise profile of the potential inhomogeneity is not known apriori , here, without loss of generality, we consider threedifferent types of potential profiles, representing the typ-ical scenarios for the potential inhomogeneity in realisticdevices.The first scenario for the potential inhomogeneity is aGaussian-like profile near the NS junction: V ( x ) = V max exp (cid:18) − x σ (cid:19) (2)with σ being the width, and V max the peak of the in-homogeneity, as shown in Fig. 1(b). Note that thepotential V ( x ) acts as a confinement potential barrierwhen V max >
0, while it becomes a potential dip when V max <
0. Confinement- or dip-like potential profile havequalitatively different effects on the conductance spec-troscopy of the NS junction.In the second scenario, we consider a sinusoidal-function-like potential profile near the NS junction: V ( x ) = V max cos (cid:16) xπ σ (cid:17) θ ( σ − x ) − V (cid:48) max sin (cid:18) x − σσ (cid:48) π (cid:19) θ ( x − σ ) θ ( σ + σ (cid:48) − x ) , (3)where θ ( x ) is the Heaviside step-function. The sinusoidalpotential contains barrier-like and dip-like parts simulta-neously, as shown in Fig. 1(c). V max and σ denote theheight and width of the barrier-like potential, while V (cid:48) max and σ (cid:48) represent the depth and width of the dip-like po-tential.The third scenario is a disorder-induced random po-tential V ( x ) = V imp ( x ) , (cid:104) V imp ( x ) (cid:105) = 0 , (cid:104) V imp ( x ) V imp ( x (cid:48) ) (cid:105) = σ µ δ ( x − x (cid:48) ) . (4)Here V imp ( x ) is a short-ranged random potential drawnfrom an uncorrelated Gaussian distribution with zeromean and standard deviation σ µ . Note that the numeri-cal results including the disorder-induced random poten-tial is based on one particular configuration of V imp ( x ),as shown in Fig. 1(d), without ensemble average. Notethat while Eqs. (2) and (3) are deterministic and smooth,the disorder potential defined by Eq. (4) is random andhence nondeterministic, distinguishing the smooth inho-mogeneous potential from the non-smooth random dis-order potential.We mention that the disorder strength discussed in thispaper should not be too large compared to the chemicalpotential µ . Because, otherwise, it cannot be properly de-scribed by the BdG Hamiltonian in (1). Instead, such astrong disorder regime should be studied using the frame-work of random matrix theory in class D ensemble. Topological MZMs do not exist at all in such highly dis-ordered systems, and we do not study strong disorder inthe current work.
C. Numerical method
To numerically calculate the conductance spectra ofthe NS junction, we apply the python scattering matrixtransport package KWANT. Following the standardprocedure, we first discretize the continuum Hamiltonianin Eq. (1) into a tight-binding model, and then calculatethe corresponding S matrix and the conductance as afunction of Zeeman field and chemical potential (or gatevoltage).
III. SMOOTH-POTENTIAL-INDUCEDZERO-BIAS CONDUCTANCE PEAKS
In this section, we focus on the situation where asmooth confinement potential is present near the NSjunction inducing a quasi-Majorana bound state in thetopologically trivial regime. We analyze the features ofthe quasi-Majorana-induced ZBCP and discuss how todistinguish it from the topological Majorana counter-part. Furthermore, we perform a self-consistent Thomas-Fermi-Poisson calculation of the electrostatic potentialin realistic three-dimensional devices, and show that asmooth confinement potential and the induced quasi-Majorana bound state are likely to appear in realisticdevices in a smooth potential. − . . . V b i a s ( m e V ) (a) (b) E Z =0.8 meV0 2 4 G ( e /h )0 1 E Z (meV)02 G ( e / h ) V bias =0 0 1 2 3 V max (meV) V bias =0 FIG. 2. (a) NS conductance as a function of E Z and V bias for a1D nanowire subject to a Gaussian inhomogeneous potentialnear the NS junction. The height and width of the potentialis V max =1.2 meV and σ = 0 . µ m, respectively. The reddashed vertical line indicates TQPT. The lower panel showsthe corresponding line-cut for the conductance at zero bias.(b) Conductance as a function of V max and V bias at a fixedZeeman field E Z = 0 . e /h . A. Basic properties of smooth-potential-inducedZBCPs
We first investigate the basic properties of the smooth-potential-induced ZBCPs. The smooth potential has aGaussian-function profile with the spatial width beingfixed at σ = 0 . µ m which is roughly 10% of the wirelength. In Fig. 2, we show the calculated conductanceas a function of both the Zeeman field strength E Z andsmooth potential height V max .Figure 2(a) shows the tunnel conductance as a func-tion of the bias voltage V bias and the field strength E Z ata fixed smooth potential height V max = 1 . e /h within the range of 0.8 meV < E Z < E Zc = (cid:112) µ + γ = 1 .
02 meVfor TQPT (labeled as the red dashed line). Further-more, as indicated in the lower panel of Fig. 2(a), thesmooth-potential-induced trivial conductance plateau atzero bias is quite stable as a function of E Z , and is evenmore stable than the Majorana-induced counterpart for E Z > .
02 meV. For Majorana bound states (MBS), thetwo Majoranas at the opposite ends of the nanowire over-lap and cause oscillation in the ZBCP, while for quasi-Majoranas, both MBS are formed at one wire end over-lapping strongly with each other.This smooth-potential-induced quantized trivial zero-bias peak is stable not only against the Zeeman fieldstrength, but also against the potential height. InFig. 2(b), we plot the conductance as a function of the po-tential height V max , with the field strength being fixed at E Z = 0 . < V max < . V max < V max > . e /h is not an exclusive signature of real MZMs; it mayalso be falsely mimicked by the trivial quasi-Majoranasin the presence of a smooth confinement potential withsuitable parameters. Note that at finite temperatures,the ZBCP will have a conductance slightly lower than2 e /h . B. Distinguishing between real and quasi-MZMsby correlation measurements
We now consider a correlation measurement in thethree-terminal setup for distinguishing between realtopological MZMs and trivial quasi-Majorana boundstates. The correlation measurement is able to detect thenonlocality of the bound-state wavefunction. For Majo-rana bound states, a pair of Majoranas appear at bothends and manifest themselves simultaneously as quan-tized zero-bias peaks in the tunnel spectroscopy. By con-trast, quasi-Majoranas guarantee the conductance peakat only one wire end because its wavefunction is localizedonly at that end. To illustrate this correlated scheme, weconsider three different scenarios: (i) a pristine nanowire,(ii) a wire with a smooth potential on the left end, and(iii) a wire with smooth potentials on both ends. The cal-culated left- and right-end conductance results are shownin Fig. 3.The top row of Fig. 3 shows the topological Majo-rana conductance of a pristine nanowire, where the leftand right conductance spectra are identical. The ZBCPsappear above the same value of Zeeman field strength( E Z = E Zc ) at the two ends thus showing perfect corre-lation, because both ends of the nanowire are occupiedby the real MBS above the TQPT (labeled as the reddashed line).In the middle row of Fig. 3, we show the conduc-tance for a nanowire in the presence of a smooth confine-ment potential localized only at the left wire end and theother end free of any inhomogeneity. In the weak fieldregime below TQPT ( E Z < E Zc = 1 .
02 meV), a sta-ble essentially quantized zero-bias conductance plateau − . . . V b i a s ( m e V ) MZM(a) 0 2 4 G L ( e /h ) MZM(b) 0 2 4 G R ( e /h ) − . . . V b i a s ( m e V ) q-MZM(c) MZM(d)0 1 2 E Z (meV) − . . . V b i a s ( m e V ) q-MZM(e) 0 1 2 E Z (meV)q-MZM(f) FIG. 3. The correlation measurements for three differentphysical scenarios. Here the conductance is measured as afunction of the bias voltage and the strength of the Zee-man field. The left and right column show the left-endconductance G L and right-end conductance G R . (a, b) Apair of MZMs with one MZM being localized at each end.(c, d) One smooth-potential-induced quasi-Majorana boundstate at the left end and one MZM at the right end. Thesmooth potential at the left end of the nanowire has theparameter of ( σ, V max ) = (0 . µ m , . σ, V max ) = (0 . µ m , . σ, V max ) = (0 . µ m , . of 2 e /h appears only in the left conductance results [seeFig. 3(c)], while no ZBCP is observed in the right con-ductance [see Fig. 3(d)]. This is because the localizedquasi-Majorana bound state below TQPT appears onlyat the left end in this scenario of the inhomogeneous po-tential being only at the left end. However, when the Zee-man field strength is above the critical value, the wholenanowire becomes topological, and a perfect end-to-endcorrelation between the left and right conductance spec-troscopies appears.This means that whenever a quasi- − . . . V b i a s ( m e V ) (a) (b) 0 2 4 G ( e /h )0 1 E Z (meV)02 G ( e / h ) V bias =0 0 1 2 E Z (meV) V bias =0 FIG. 4. The stability of conductance peaks against the Zee-man field. (a) The conductance for a smooth potential-induced quasi-MZM, which is quite stable against the Zee-man field. Refer to Fig. 2(a) for the parameters. (b) Theconductance for a real MZM in the pristine nanowire. Thezero-bias conductance shows prominent oscillations between 0to 2 e /h as a function of the Zeeman field strength, owing tothe wavefunction overlap between two Majorana zero modesat the opposite ends of a finite-length nanowire. Majorana situation appears below TQPT, increasing themagnetic field must necessarily lead to the topologicalMZMs at higher field strength. There is no experimentalreport of such an observation in the literature.In the third scenario, two slightly different smooth con-finement potentials are present at the left and right wireends, inducing quasi-Majoranas at each wire end in thetopologically trivial regime below TQPT. As shown inthe bottom row of Fig. 3, although zero-bias conduc-tance peaks are observed in both the left and right tun-nel conductance individually below TQPT, they are notcorrelated, e.g., the values of E Z at which ZBCP startsto emerge at the two ends are different. This is be-cause these ZBCPs at E Z < E Zc are induced by quasi-Majoranas, the properties of which depend sensitivelyon the details of the smooth confinement potential. Fortopological MZMs [Fig. 3(a)], correlated ZBCPs appearat the same E Z for both ends, which is the TQPT point. C. Distinguishing between real and quasi-MZMsby stability against Zeeman field
We now consider another method for distinguishingbetween real and quasi-Majorana bound states; which is,by testing the stability of conductance quantization atzero bias against the Zeeman field strength. In a real-istic topological Majorana nanowire, due to the finite-size effect, the wavefunctions of the two localized Ma-jorana bound states at opposite wire ends overlap andlead to an oscillation of the zero-bias conductance peakas a function of the increasing Zeeman field strength. Bycontrast, for topologically trivial quasi-Majorana boundstates, since its wavefunction is localized at one end of the nanowire, the corresponding zero-bias conductancepeak does not show oscillatory behaviors. In Fig. 4,we present the conductance calculations for both trivialquasi-Majoranas (left panels) and topological Majoranabound states (right panels). The ZBCP induced by thetopological MBS in Fig. 4(b) oscillates prominently aboveTQPT with the Zeeman field strength ( E Z > < E Z < .
02 meV. There-fore the observation of an increasing oscillation in ZBCPwith the Zeeman field strength can support the presenceof Majorana bound states over quasi-Majorana boundstates in the hybrid nanowire. In some situations wheresmooth potential and q-MZMs appear on both sides ofthe nanowire, as discussed in the previous subsection,the ZBCP induced by q-MZM would also oscillate [seeFigs. 3(e) and 3(f)]. But the oscillation amplitude is stillmuch smaller than the MBS ZBCPs, because q-MZMsform in the much weaker Zeeman field regime. This alsonecessitates the presence of q-MZMs on both ends of thewire which cannot be ruled out, but is not a generic sit-uation.
D. Conductance for realistic 3D NS junctions
In this subsection, we go beyond the one-dimensionalminimal model and consider the realistic three-dimensional NS junction, focusing on whether thesmooth confinement potential and the induced quasi-Majorana bound state can appear in realistic device ge-ometries with realistic gate voltage values. Instead ofadding the potential profile manually simply as a modelpotential term V ( x ) in Eq. (1), we now calculate the elec-trostatic potential profile inside the nanowire by solv-ing the three-dimensional Thomas-Fermi-Poisson equa-tion self-consistently, including all the ingredients in atypical NS junction device— the dielectric layers, themetallic aluminum layer, the normal lead, and the gates.As we will show, when the values of the gate voltagesare appropriately chosen, a smooth confinement poten-tial can appear near the junction, giving rise to a quasi-Majorana-induced ZBCP in the 3D realistic device at afinite magnetic field. This also provides a justification forthe effective 1D model of Eq. (1).The model system, as shown in Fig. 5, represents atypical NS junction device in the laboratory, and is usedhere for the electrostatic potential calculation. The self-consistent Thomas-Fermi-Poisson equation is ∇ · [ ε r ( r ) ∇ φ ( r )] = − ρ [ φ ( r )] ε , (5)with φ being the electrostatic potential, and ε r ( ε ) therelative (vacuum) permittivity. ρ is the charge den- Normal leadInSbSubstrateBack gate Tunnel gateSuper gate AlDielectric layer0 100 200 300 400axial distance (nm) − B a nd e d g e ( m e V ) FIG. 5. The schematic of the 3D NS junction. Top panels(left to right): view from the normal lead end, from side, andfrom the hybrid nanowire end. Middle panel: view from top.Bottom panel: the view of band edge profile focuses on first400 nm of the wire, where the inhomogeneity is prominent. sity of the mobile electrons in the InSb semiconductingnanowire, which in the Thomas-Fermi approximation is ρ e ( φ ) = − e π (cid:18) m ∗ eφθ ( φ ) (cid:126) (cid:19) / , (6)where m ∗ = 0 . m e and θ ( x ) is the Heaviside stepfunction. We did not include any free holes in the va-lence band within the Thomas-Fermi approximation be-cause they are irrelevant in the parameter regime of gatevoltages in this work, where the Fermi level is always inthe conduction band. The effects of gates are includedas Dirichlet boundary conditions inside the gate regions,with the potential values being fixed as the gate voltages.After obtaining the electrostatic potential inside thesemiconducting nanowire, we turn to the quantum me-chanical problem by calculating the band edge profile andthe conductance in the three-dimensional NS junction.The BdG Hamiltonian for conductance calculation is asfollows H =[ − (cid:126) m ∗ ( ∂ x + ∂ y + ∂ z ) + α R ( − i∂ x σ z + i∂ z σ x ) − eφ ( r )] τ z + E Z σ x + ∆( r ) τ x . (7)Here α R = 0 . y -direction. φ ( r ) is the three-dimensional electrostatic po-tential profile obtained from the Thomas-Fermi-Poissonself-consistent calculation. ∆( r ) is the induced super-conducting gap in the regions covered by the aluminumlayer. Figure 6(a) shows the calculated conductance as afunction of the field strength and the bias voltage forthe 3D NS junction with a particular set of device pa-rameters. The electrostatic potential profile is fixed forFig. 6(a) with the gate voltages being: V Al = 0.02 V(band offset between InSb and Al), V supergate = 0.03 V, V tunnel = 0.015 V, and V backgate = 0.0 V. We further vi-sualize the 3D electrostatic potential by plotting the 1Dband edge profile in the vicinity of the junction along thewire axis in the bottom panel of Fig. 5. The band edgeprofile is for the lowest subband in the nanowire, that is,the hybrid nanowire is in the single-subband limit. Notethat a smooth confinement potential is naturally presentbetween 100 nm < x <
300 nm, owing to the combinedeffect of the tunnel gate, the InSb-Al band offset, andthe supergate above the nanowire. The smooth potentialhere is present by virtue of the electrostatics of the 3DNS junction without having to be put in manually as amodel potential as in the effective 1D model of Eq. (1).The nanowire is 1 . µ m in total length, and the potentialvalue reaches an asymptotic value for x >
400 nm; thatis we take φ ( x >
400 nm , y, z ) = φ ( x = 400 nm , y, z ).So, the smooth potential extends over roughly 10% ofthe wire near the tunnel junction end. In Fig. 6(a), aquasi-Majorana-induced nearly-quantized zero-bias con-ductance peak appears in the trivial regime (to the left ofthe red dashed line), due to the presence of the smoothconfinement potential. The stable quantized zero-biasconductance peak becomes oscillatory after the nanowireenter the topological phase at a stronger Zeeman field (tothe right of the red dashed line). We then fix the strengthof the Zeeman field at E Z = 2 meV, which is below theTQPT, such that the ZBCP is now induced by the quasi-Majorana bound state, and sweep the voltage of the tun-nel gate. The resulting conductance profile as a functionof the tunnel gate voltage is shown in Fig. 6(b). Note thatthe quasi-Majorana-induced ZBCP is stable and nearly-quantized only in the regime 5 mV < V tunnel <
15 mV.Outside this range of tunnel gate voltages, the confine-ment potential at the junction is either too sharp or tooflat to induce a quasi-Majorana bound state. Figure 6,therefore, shows that a smooth confinement potential isindeed possible in a realistic 3D NS junction device withappropriate gate voltage values, and that such a poten-tial can induce a quasi-Majorana bound state with a sta-ble almost-quantized ZBCP in the tunnel spectroscopy.Thus, the observation of a ‘stable’ and ‘quantized’ ZBCPis not sufficient to establish the existence of topologicalMBS in the system.Another important finding here is that the conduc-tance profiles obtained from 3D simulations (Fig. 6)closely resemble those from the effective 1D model(Fig. 2), especially the conductance line-cuts at the zerobias. This indicates that the 1D minimal model is a goodapproximation of the 3D realistic nanowire model whenthe nanowire is in the single-subband limit. This justi-fies all the calculations using the 1D minimal model inthis and other previous works. Given the very high com- − . . . V b i a s ( m e V ) V tunnel =15 mV (a) (b) E Z =2 meV G ( e /h )0 1 2 3 E Z (meV)02 G ( e / h ) V bias =0 − − − V tunnel ( ×
10 mV) V bias =0 FIG. 6. The conductance for realistic 3D NS junctions. Herethe electrostatic potential profile is numerically calculated us-ing the self-consistent Thomas-Fermi-Poisson method. (a) NSconductance as a function of E Z and V bias with the tunnelgate being fixed at V tunnel = 15 mV. The lower panel of (a)shows the zero-bias conductance as a function of the Zeemanfield strength. The TQPT is labeled by the red dashed ver-tical line. (b) NS conductance as a function of V tunnel and V bias with the Zeeman field being fixed at E Z = 2 meV.The lower panel of (b) shows the zero-bias conductance asa function of the tunnel gate voltage. Note that the strongsimilarity between this figure and Fig. 2 indicates that thephysical scenario of smooth potential-induced ZBCP is possi-ble in realistic situations, and that simulations based on the1D effective model is a good approximation for the 3D modelin the single-subband limit. putational cost of the 3D simulations and the relativesimplicity of the 1D calculations, it makes sense to baseconductance calculations on the 1D model. IV. DIP-POTENTIAL-INDUCED ZERO-BIASCONDUCTANCE PEAKS
In this section, we consider, within the 1D effectivemodel of Eqs. (1)-(3), a quantum-dot like potential nearthe NS junction, i.e., a dip in the potential, rather thanthe smooth barrier-like potential considered above. Wefocus on the peak values and the conductance robust-ness of the induced ZBCPs. In particular, the quantumdot considered here is either a Gaussian-function poten-tial with a negative amplitude (a dip in the potentialprofile), or a sinusoidal-function potential (armchair-likepotential profile), as shown in Figs. 1(b) and 1(c). Un-der these conditions, low-energy Andreev bound statesappear inside the potential inhomogeneity near the endof the hybrid nanowire at a finite strength of Zeemanfield. These trivial Andreev bound states can induce(near-) zero-bias conductance peaks, mimicking topolog-ical Majorana bound states. However, a major differ-ence between Andreev and Majorana bound states forthe dip-potential is that the zero-bias conductance pro-files of trivial peaks are sharp spike-like in the parameterspace, e.g., in E Z or µ , and the peak values may exceed − . . . V b i a s ( m e V ) (a) (b) E Z =0.46 meV0 2 4 G ( e /h )02 G ( e / h ) V bias =0 V bias =0 − . . . V b i a s ( m e V ) (c) (d) E Z =0.67 meV0 1 E Z (meV)02 G ( e / h ) V bias =0 0 1 2 3 | V max | (meV) V bias =0 FIG. 7. The conductance for diple-like potential-induced An-dreev bound states. One generic feature of the conductancein this physical scenario is that the ZBCP as a function of thesystem parameter shows a spike-like conductance peak whichcan be greater than 2 e /h . (a) The conductance as a functionof E Z and V bias for a nanowire which has a negative chemicalpotential and is subject to a dip potential near the NS junc-tion. Here it takes the form of Gaussian potential with the pa-rameters of ( σ, V max ) = (0 . µ m , − . µ = − V max and V bias for a nanowire with a negative chem-ical potential at E Z = 0 .
46 meV [labeled in the green dashedline in (a)]. (c) Similar to (a), but now the dip-like potentialis in the form of the sinusoidal potential, with the parameterbeing ( σ, V max , σ (cid:48) , V (cid:48) max ) = (0 . µ m , , . µ m , µ = 1 meV. (d)Similar to (b) for the sinusoidal potential. The strength ofthe Zeeman field is fixed at E Z = 0 .
67 meV [labeled in thegreen dashed line in (c)]. e /h in extremely narrow ranges of parameters, whilethose of topological peaks are plateau-like with peak val-ues never more than 2 e /h .Figure 7(a) shows the calculated ABS-induced triv-ial conductance as a function of bias voltage and Zee-man energy. The quantum-dot potential is dip-like withdepth V max = − . σ = 0 . µ m, and thebulk chemical potential µ = − .
45 meV (cid:46) E Zc (cid:46) E Z ≈ .
46 meV, the value of the zero-bias conductance peakexceeds 2 e /h , which is a unique feature that distin-guishes these dot-induced trivial ABSs from topologicalMBSs. We also calculate the conductance as a functionof the dot potential amplitude V max at a fixed Zeemanfield strength, which qualitatively models the tunnel-gate-dependence of conductance in realistic junctions.As shown in Fig. 7(b), an oscillatory conductance peaknear zero bias appears in the range of − . 67 meV and V D ≈ e /h .One way to understand the features of dip-potential-induced ‘spiky’ ZBCPs is to decompose the low-energyAndreev bound states into a pair of overlapping Majo-rana wavefunctions forming the q-MZMs. The presenceof the dip potential enhances the coupling between thetwo Majoranas wavefunctions, and thus the energy of theAndreev bound state crosses the zero energy instead ofsticking. Therefore we only see spike-like ZBCP ratherthan a conductance plateau in this scenario. Moreover,owing to the dip potential, both Majoranas couple effec-tively with the external normal lead, which causes thetunneling peak value to be greater than 2 e /h . In fact,our simulations show (see the animations presented inthe Supplementary Materials ) that the resulting zero-bias conductance quickly rises to 4 e /h , corresponding tothe combined conductance of two Majoranas, and thenquickly becomes zero as the ABS is no longer a zero en-ergy state. An extremely fine-tuned dip-potential maygive rise to a zero-bias conductance above 2 e /h , but itis unlikely to be experimentally observable as a ZBCPbecause of the strongly spiky nature of the conductancepeak. This is in sharp contrast to the barrier-like po-tential where a conductance plateau at 2 e /h may stickto zero bias for finite parameter regimes both in Zeemanfield and chemical potential. V. DISORDER-INDUCED ZERO-BIASCONDUCTANCE PEAKS We now consider the scenario in which the potentialinhomogeneity is in the form of a nondeterministic ran-dom potential, i.e., disorder. In realistic devices, such apotential can originate from the growth imperfection atthe superconductor-semiconductor interface or from un-intentional (and hence unknown) impurities in the wireor the superconductor or the substrate. Our focus is onwhether it is possible to find a disorder-induced apparent − . . . V b i a s ( m e V ) (a) (b) E Z =0.83 meV0 2 4 G ( e /h )02 G ( e / h ) V bias =0 V bias =0 − . . . V b i a s ( m e V ) (c) (d) E Z =0.78 meV0 1 E Z (meV)02 G ( e / h ) V bias =0 0 10 20 V g (meV) V bias =0 FIG. 8. Conductance for a Majorana nanowire in the pres-ence of a random disorder potential in the bulk. (a) Conduc-tance as a function of E Z and V bias . The disorder potentialcan induce ZBCP when the Zeeman field is strong enough,and the zero-bias conductance as a function of the Zeemanfield strength shows a spike-like profile with the peak valuegreater than 2 e /h . (b) The peak conductance at zero bias(blue solid line) and normal conductance above SC gap (greensolid line) as a function of V g and V bias at fixed E Z = 0 . E Z and V bias underanother set of disorder configuration which also manifests thesame conductance peaks. trivial quantized conductance plateau whose peak valueis greater than 2 e /h .Figure 8 shows the calculated tunneling conductancein the presence of disorder with two distinct random con-figurations. The first row of Fig. 8 uses one specificrandom configuration. We notice that the gap closes at E Z ∼ . E Z axis and fo-cus on only the nearby region of the plateau, we can stillmanually create a “robust” plateau, although this triv-ial plateau is a deceptive visual effect because of fine-tuning the parameter range. Such a trivial disorder-induced ZBCP ‘plateau’ may misleadingly masqueradeas evidence supporting the presence of topological MBSsince experimentally the TQPT location is not known, and a priori there is no way to know whether one is0below or above the TQPT based just on NS tunnelingmeasurements.However, this artificial plateau, which to some extentis robust against Zeeman field strength, is not robustagainst the tunnel barrier. In Fig. 8(b), we show theconductance dependence on the tunnel barrier height ata fixed field strength E Z = 0 . 78 meV [corresponding tothe green solid line in Fig. 8(a)]. The conductance de-creases from 3 e /h to 1 . e /h monotonically as V g in-creases. This instability indicates that the trivial ZBCPdoes not have a truly stable quantized conductance witha value higher than 2 e /h against both the Zeeman fieldand the tunnel barrier. The important point to note isthat the disorder-induced trivial ZBCP could produce asomewhat-stable plateau with conductance above 2 e /h .In the second row of Fig. 8, we present the conduc-tance results for another random disorder configuration.A trivial ZBCP with a peak value well above 2 e /h isobserved at E Z ∼ . E Z ∼ . < V g < V g [see Fig. 8(d)]. Note that the ZBCPs in Fig. 8for both random configurations are obtained by carefulfine-tunings.These examples demonstrate that disorder couldgenerically introduce trivial ZBCPs at conductance val-ues higher than 2 e /h which, with some fine tuning, mayreflect some limited apparent ‘robustness’ in both the ap-plied magnetic field and applied gate voltage dependingon the details of the disorder. Of course, disorder canalso induce trivial ZBCPs with conductance values at orbelow 2 e /h , as has recently been studied in the litera-ture. VI. DISCUSSION Strictly speaking, the confirmation of topological su-perconductivity and MZMs should be based on variousexperimental measurements probing both the ends andthe bulk of the wire simultaneously. For example, theMajorana-induced robust quantized ZBCP should be ac-companied by the topological gap closing and reopeningin the vicinity of TQPT, and the ZBCP oscillations withan increasing Zeeman field strength. However, in prac-tice, the topological bulk gap features are difficult to ob-serve using the local probe in NS tunnel spectroscopies,and the strength of Zeeman field is constrained by thecritical field of the parent superconducting layer abovewhich all bulk superconductivity vanishes, thus destroy-ing the topological superconductivity in the process. Thefundamental problem underlying local tunneling spec-troscopy is its inability to confirm any signature of non-locality, which is the hallmark of topological Majoranamodes. In most of the recent experimental measurements, weonly see the formation of a ZBCP just before the clos-ing of the parent superconducting gap. This apparentgap closing feature in the NS tunneling spectrum may,however, have nothing to do with a bulk gap closing, butonly a manifestation of Andreev bound states comingtogether leading to a trivial ZBCP. Thus, the questionthat motivates this work is: What can we learn if we havemerely the local tunneling measurements on the Majo-rana nanowires, and can we infer the underlying physicalmechanisms for the ZBCP based on their peak profiles?With extensive numerical simulations, we find thattrue- or quasi-Majoranas do (not) induce ZBCPs of apeak value equal to or smaller (greater) than 2 e /h in thezero-temperature limit. By contrast, for those ZBCPsdue to dip-potential-induced ABSs and disorder-inducedbound states, the peak value can exceed 2 e /h generi-cally, although such dip-induced trivial ZBCPs have verysharp spiky structures as a function of system parame-ters (e.g., Zeeman field, gate potentials) and are there-fore only observed when tunnel/temperature/dissipationbroadening effect is large.We further note that the disorder scenario has muchricher physics than realized before, because through care-ful fine-tunings, a particular random disorder configura-tion may induce a ZBCP (above or below or close to2 e /h ) which shows robustness either against the fieldstrength or the tunnel barrier height. So any NS tunnel-ing observation by itself, e.g., a nearly-quantized ZBCPat or around 2 e /h , or a robust peak against one systemparameter, is not sufficient to conclude topological Majo-ranas or even trivial quasi-Majoranas in superconductor-semiconductor nanowires. Only with a combination ofconductance quantization at 2 e /h at zero temperature,robustness against both Zeeman field strength and gatevoltages, the observation of Majorana oscillations withincreasing field strength, closing/reopening of the bulkgap just as the ZBCP develops, and correlation measure-ments on both wire ends can give convincing evidenceto Majorana zero modes. One needs to be particularlymindful of the fact that disorder-induced trivial ZBCPsmay have observable features above, at, or below 2 e /h ,perhaps even manifesting some apparent (but mislead-ing) limited robustness in Zeeman field and chemical po-tential.We also emphasize that all the calculations in this workare based on the single-subband assumption. The num-ber of subbands in the devices in the laboratory is notknown a priori . When multiple subbands come belowthe Fermi level of the hybrid nanowire due to gating, theZBCP may exceed 2 e /h in principle. However, evenin the multi-subband model, a conductance plateau isobserved only at half-integer multiples of 4 e /h , and aplateau of conductance slightly above 2 e /h is unlikelyfor MZM.Before concluding, we now provide a brief discussion ofthe existing experimental results on Majorana nanowiresin the context of our theoretical findings. Early Majo-1rana experiments during 2012 to 2016 were all plaguedby very strong disorder in the systems leading to softgaps indicating the presence of considerable disorder-induced subgap fermionic states, and although weakZBCPs ( ∼ . − . e /h ) often manifested in the tun-nel conductance spectra, it is manifestly clear that theseexperimental observations are not conclusive at all forthe existence of topological Majorana modes. The firstexperiment involving a hard gap was by Deng et al . which saw the emergence of ZBCPs (but with peak height (cid:28) e /h ) arising from the merging of Andreev boundstates. A detailed theoretical analysis established theseobservations as most likely the manifestation of trivialZBCPs with ABSs becoming almost zero-energy statesmimicking as MZMs. Two influential later papers, byZhang et al . and Nichele et al . , reported the observa-tion of ZBCPs with peak heights at 2 e /h in nanowireswith hard gaps, creating considerable excitement thatperhaps the topological MZMs have been seen. It wassoon realized, however, that the Nichele experiment seeZBCPs with peak heights higher than 2 e /h even at fi-nite temperatures (and the T = 0 peak height would bestill higher), indicating that the observation is inconsis-tent with topological MZMs. Very recent experimen-tal work now indicates that the same is true for theZhang experiment also, with the ZBCP height actuallybeing 2 . e /h already at T = 25 mK. In addition, thenew data and analysis indicate that the original exper-imental conductance plateau reported in Ref. 19 is anartifact and careful consideration of charge jumps in thesystem eliminates the ZBCP stability. Therefore, bothof these experiments, Nichele and Zhang , report un-stable ZBCPs with conductance values somewhat above2 e /h . Our work clearly demonstrates that these obser-vations are inconsistent with either the topological MZMor the trivial q-MZM interpretation. Thus, the only pos-sible conclusion, which has also been reached recentlyin other theoretical publications, is that even thebest experimental samples of today still have consider-able disorder in them, and in all likelihood, ZBCPs be-ing observed experimentally arise from disorder and aretrivial. Given that semiconductor nanowire systems areundoubtedly the cleanest platforms for Majorana experi-ments, it is reasonable to conclude that all ZBCP-impliedconclusion about the observation of Majorana zero modesare actually observing disorder-induced trivial zero-biaspeaks. This is in fact an encouraging scenario for thefuture of Majorana experiments— all one needs to do isto remove disorder and produce cleaner samples in orderto obtain topological Majorana zero modes. VII. CONCLUSION To conclude, we have systematically investigatedZBCPs in the NS tunnel spectroscopy arising from dif-ferent physical mechanisms, focusing on the conductancepeak values and its robustness against system parame- ters. We find that for true topological Majoranas andsmooth-potential-induced quasi-Majoranas, the inducedZBCP does not exceed 2 e /h . In a realistic finite-lengthnanowire, the Majorana-induced ZBCP oscillates with anincreasing Zeeman field strength due to the wavefunctionoverlap between the Majoranas at the opposite ends ofthe wire, while the quasi-Majorana-induced ZBCP doesnot oscillate because its wavefunction is localized at theend of the inhomogeneity. Both MZMs and q-MZMsshow a conductance plateau as a function of the tunnelbarrier transparency.Beyond the simple single-subband calculation, we usethe self-consistent Thomas-Fermi-Poisson method, andshow that a smooth confinement potential is indeed likelyto form near the NS junction due to the combined effectsof gates and superconductor-semiconductor band offset.It also indicates that the realistic 3D model is qualita-tively equivalent to the widely-used effective 1D minimalmodel in the single-subband limit.If there is a dip or well inside the potential inhomo-geneity, the ZBCP can exceed 2 e /h , and the profile ofthe ZBCP then becomes spike-like as a function of eitherZeeman energy or tunnel barrier height. We ascribe thespike-like conductance to the enhanced inter-Majoranacoupling in the dip-like potential inhomogeneity. Such aspiky conductance rises and falls too quickly as a func-tion of parameters (we provide animations in the Sup-plementary Information ) to be experimentally relevantin our view. It is of course possible that very carefulfine-tuning plus effects of temperature and dissipationmay lead to just the correct experimentally observed con-ductance slightly above 2 e /h arising from a quan-tum dot induced dip potential, but such fine-tuning ismore likely to lead to a ZBCP with conductance below2 e /h . In any case, the dip potential would not leadto any robustness in the ZBCP, and cannot be construedto belong to the q-MZM category.Finally, we present conductance calculations for therandom disorder potential which also induces trivialZBCP, with its peak value which may exceed 2 e /h , andhaving certain limited robustness against either Zeemanfield or tunnel gate, if any, through a careful fine-tuningprocedure. Such fine-tuned somewhat robust ZBCPswith conductance values above 2 e /h have been observedin recent experiments, and we believe that these areall disorder-induced trivial ZBCPs.In summary, our message to the experimentalists isthat the presence of a conductance plateau approachingquantized 2 e /h from below without much fine-tuningsmay indicate the existence of real or quasi-Majoranazero modes. By contrast, a conductance peak above2 e /h , whether the profile is spike-like or plateau-like,would be from disorder- or dip-induced trivial Andreevbound states. Producing cleaner samples with less dis-order should be the highest priority for progress in thefield.The definitive evidence for the existence of topologi-cal MZMs in the tunneling spectroscopy must minimally2satisfy the following five criteria: (1) generic observa-tion of 2 e /h (or slightly below) ZBCP without exten-sive fine-tuning; (2) stability of the ZBCP in the Zeemanfield, tunnel barrier potential, and chemical potential;(3) ZBCP manifesting some oscillatory behavior with in-creasing Zeeman field; (4) observation of end-to-end tun-neling nonlocal correlations in the ZBCP; (5) observationof a bulk gap closing and reopening concomitant with theappearance of the ZBCP.Finally, we comment on the nonexistence of any meso-scopic conductance fluctuations in Majorana nanowires,particularly in view of our conclusion that even the bestavailable samples are currently disorder-dominated. Itis worthwhile to mention that our exact phase-coherenttunnel conductance calculations in the presence of ran-dom disorder in the nanowire do not manifest any meso-scopic fluctuations in the system. In particular, thetheoretical tunnel conductance does not fluctuate ran-domly at all (let alone by O ( e /h )) as a function of biasvoltage, chemical potential, and Zeeman field, insteadvarying smoothly according to the spectral evolution ofthe BdG equation with parameter variation. In fact, nomesoscopic fluctuations are expected here because of theexistence of the superconducting gap and the nature of tunneling transport as would have happened in a similardisordered normal system for coherent diffusive trans-port. 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