Crossover between trivial zero modes in Majorana nanowires
CCrossover between trivial zero modes in Majorana nanowires
Haining Pan and Sankar Das Sarma
Condensed Matter Theory Center and Joint Quantum Institute,Department of Physics, University of Maryland, College Park, Maryland 20742, USA
We consider the superconductor-semiconductor nanowire hybrid Majorana platform (“Majoranananowire”) in the presence of a deterministic spatially slowly varying inhomogeneous chemical po-tential and a random spatial quenched potential disorder, both of which are known to producenon-topological almost-zero energy modes mimicking the theoretically predicted topological Ma-jorana zero modes. We study the crossover among these mechanisms by calculating the tunnelconductance while varying the relative strength between inhomogeneous potential and random dis-order in a controlled manner. We find that the entire crossover region manifests abundant trivialzero modes, many of which showing the apparent ‘quantization’ of the zero-bias conductance peakat 2 e /h , with occasional disorder-dominated peaks exceeding 2 e /h . We present animations of thesimulated crossover behavior, and discuss experimental implications. Our results, when comparedqualitatively with existing Majorana nanowire experimental results, indicate the dominant role ofrandom disorder in the experiments. I. INTRODUCTION
Following the predicted possible realization ofnon-Abelian Majorana zero modes (MZMs) insuperconductor-semiconductor hybrid platforms inthe presence of superconducting proximity effect, spin-orbit coupling, and spin splitting [1–4], a large numberof experiments from many different groups reported theobservation of zero-bias conductance peaks (ZBCPs)in tunneling spectroscopy of InAs and InSb nanowires,presumably as evidence for the predicted topologicalMZM [5–20]. Recent experiments [11, 15] have evenreported ZBCPs with the approximate conductancevalue of 2 e /h , which is the predicted topologicallyquantized value for MZM conductance [21–24]. Thiscreated considerable excitement in the community thatperhaps the elusive non-Abelian MZM has finally beenobserved.It was, however, quickly realized that topologicalMZMs are unlikely to have been observed in these tunnel-ing measurements. First, the current nanowires may besimply too short, most likely shorter than the supercon-ducting coherence length, and thus, the system is mostlikely not in the topological regime. Second, there is noevidence for a bulk gap opening (or more generally, nosign for a topological quantum phase transition to theMZM-carrying topological superconducting phase) whenthe ZBCP shows up, which is a necessary topologicalrequirement by virtue of the bulk-boundary correspon-dence. Third, no signature of the predicted MZM os-cillations [25], associated with the overlap of the Ma-jorana wavefunctions from the two wire ends, has everbeen reported. Fourth, no nonlocal experimental fea-ture, for example, correlated ZBCPs from tunneling atboth ends [26–28], has ever been observed, casting doubton nonlocal topological nature of the observed ZBCPsjust from one end of the nanowire. Fifth, the observedZBCPs are typically not stable as a function of system pa-rameters such as applied tunnel barrier, magnetic field,and gate voltages, casting doubt on their robust topo- logical nature. Although these features indicate seri-ous difficulties with the MZM interpretation of the ob-served ZBCPs, perhaps the most compelling argumentagainst the MZM interpretation of the experimentallyobserved ZBCPs is that two persuasive non-MZM phys-ical mechanisms have been theoretically identified whichproduce non-topological (i.e., trivial) ZBCPs genericallyin nanowires, and these trivial ZBCPs appear consistentwith all the observed features in the tunneling measure-ments, leading to a consensus that the reported ZBCPsso far are most likely trivial and not topological.These two trivial ZBCP mechanisms of nontopolog-ical origin, which we have recently dubbed ‘bad’ and‘ugly’ [29], are, respectively, slowly varying chemical po-tential due to the presence of an inhomogeneous potentialand random spatial disorder arising from unknown im-purities and defects in the system. The possibility thatan inhomogeneous chemical potential could give rise tosubgap fermionic states was pointed out early in the Ma-jorana nanowire literature [30–32], but its importance indetermining the tunnel conductance measurements wasnot immediately appreciated. Following the experimentby Deng et al [10], where a claim was made for the ob-servation of Majorana bound states (a different name forMZMs) from coalescing Andreev bound states (ABS) inthe InAs quantum dot-nanowire-Al hybrid system, it waspointed out that the observations are more consistentwith almost-zero-energy trivial ZBCPs arising from non-topological fermionic subgap states induced by the inho-mogeneous potential associated with the quantum dot.This is the ‘bad’ scenario for ZBCPs, where trivial ABSsproduce rather stable zero energy states in nanowires,often giving rise to ZBCPs with values close to 2 e /h value [33–35]. Similar trivial ZBCPs arise from smoothslowly spatially varying potential along the wire also.We will refer to these inhomogeneous potential inducedZBCPs as ‘bad’ zero modes for notational convenience.(The truly topological MZMs will be referred to as ‘good’following the nomenclature introduced in Ref. 29.)The fact that random disorder by itself could produce a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b ZBCPs in the nanowire mimicking MZM behavior wasalso pointed out early [36–40], but its relevance to theexperimental tunneling spectroscopy has only been ap-preciated recently [29, 41]. In particular, we recently es-tablished that random disorder in the nanowire by itselfcan produce relatively stable trivial ZBCPs with a highprobability of achieving ∼ e /h conductance value. Wewill refer to these disorder induced ZBCPs as ‘ugly’ fordescriptive brevity as in our recent publications [27, 29].They are also sometimes referred to as ‘class D’ peaksalluding to their connection to antilocalization effects insystems breaking time-reversal invariance and spin rota-tional symmetry [41].Since the superconductor-nanowire hybrid systems arelikely to have both inhomogeneous potentials and randomdisorder, neither of which is intentional and therefore notcontrollable, it is important to consider their interplay bytaking into account both mechanisms together. This isprecisely what we do in this work by calculating the tun-nel conductance and the local density of states (LDOS)of the Majorana nanowire including both bad and uglymechanisms and using a tuning parameter to study thecrossover between the two, going from the completelybad situation (with only potential inhomogeneity) to thecompletely ugly situation (with only random disorder)in a controlled manner. We find ubiquitous presence oftrivial zero modes throughout the crossover region, oftenwith ZBCP values ∼ e /h , thus considerably compli-cating the interpretation of experimental results whereZBCPs, particularly with conductance ∼ e /h value,are assumed to be synonymous with the existence oftopological good MZMs. We present, for the sake of a di-rect comparison, results for the pristine ‘good’ situationalso, where neither disorder nor inhomogeneous potentialis present in the nanowire leading, therefore, to the pres-ence of topological MZMs in the system. In addition,we present the calculated tunnel conductance and theLDOS of the nanowire in the crossover between two uglycases arising from different random disorder configura-tions, which also manifests the ubiquitous trivial ZBCPsthroughout the crossover region, for a complete story ofcrossover physics. We also present, in the Supplemen-tary Information [42], animations showing the simulatedconductance and LDOS results with varying amounts ofinhomogeneous potential and random disorder. Thesecrossover results should help better understand experi-mental results, where both inhomogeneous chemical po-tential and random disorder are invariably present.The rest of this paper is organized as follows: In Sec. IIwe describe the two distinct crossover models we use. InSec. III, we briefly describe the underlying theory andthe calculational details, presenting and discussing ourdetailed results for the calculated tunnel conductance asa function of bias voltage and applied magnetic field inSec. IV. We conclude in Sec. V. The appendix providesthe calculated LDOS at the two ends of the nanowire aswell as at the middle of the wire for a comparison with thecorresponding conductance results presented in Sec. IV. The Supplementary Information contains detailed ani-mations for the crossover conductance behavior as a con-tinuous function of the tuning parameter controlling thecrossover [42]. II. CROSSOVER MODEL
In this section, we briefly revisit three types ofZBCPs— the good, the bad, and the ugly— as first in-troduced in Ref. 29 by presenting their Hamiltonians andthe corresponding mechanisms. The experimental devicethat we are theoretically simulating is a three-terminalSC-SM hybrid nanowire as shown in Fig. 1(a). The semi-conductor which is covered by a grounded s -wave pair-ing superconductor is attached with two normal leadswith the bias voltages ( V L and V R ) applying to the leftand right end, respectively. We assume the nanowirehas a spatially constant proximitized superconductivityinduced by the parent SC gap ∆ but may have an in-homogeneous chemical potential V ( x ) as we will discusslater. Lead LeadSMSC V L V R ∆( x ) = ∆ (a) α = α = . α = V ( x ) x α Bad to Ugly(b) α = α = . α = V ( x ) x α Ugly to Ugly(c)
FIG. 1. (a) The schematic for the three-terminal device of theSC-SM hybrid nanowire; (b) The crossover between the badZBCP and the ugly ZBCP controlled by α . V ( x ) transformsfrom an inhomogeneous potential at α = 0 to a random disor-der potential at α = 1; (c) The crossover between two distinctugly ZBCPs controlled by α . V ( x ) transforms from a ran-dom disorder potential at α = 0 to another random disorderpotential at α = 1. A. Good ZBCP
The good ZBCP is the ideal case of the pristinenanowire without any inhomogeneous potential or dis-order. This pristine limit can be described by the stan-dard Bogoliubov de Gennes (BdG) Hamiltonian of thesuperconductor-semiconductor hybrid nanowire [1–4] H nanowire = 12 (cid:90) L dx ˆΨ † ( x ) (cid:20)(cid:18) (cid:126) ∂ x m ∗ − iα R ∂ x σ y − µ (cid:19) τ z + V Z σ x + Σ( ω ) (cid:21) ˆΨ( x ) , (1)where ˆΨ( x ) = (cid:16) ˆ ψ ↑ ( x ) , ˆ ψ ↓ ( x ) , ˆ ψ †↓ ( x ) , − ˆ ψ †↑ ( x ) (cid:17) (cid:124) representsa position-dependent Nambu spinor, (cid:126) σ and (cid:126) τ are vectorsof Pauli matrices that act on the spin space and particle-hole space, respectively. L is the wire length, m ∗ is theeffective mass of the conduction band, α R is the strengthof Rashba-type spin-orbit coupling, and V Z is the Zee-man field applied along the nanowire. The self-energyterm Σ( ω ) accounts for the proximitized superconductiv-ity in the semiconductor by integrating out the degreesof freedom in the parent superconductor [43, 44] isΣ( ω ) = − γ ω + ∆ τ x (cid:112) ∆ − ω . (2)Here, ω is the energy in the retarded Green’s function forthe BdG Hamiltonian, ∆ is the pair energy of the parentsuperconductor, and γ is the effective SC-SM coupling(tunneling) strength producing the SC proximity effectin the SM nanowire.Without loss of generality, we choose the set of pa-rameters corresponding to the experimental platform ofInSb-Al hybrid system [45]: the effective mass m ∗ =0 . m e with m e being the electron rest mass, the par-ent SC gap ∆ = 0 . γ = 0 . α R = 0 . µ = 1 meV. These are typical numbers for Majo-rana nanowire systems in real units. We focus on thelong wire limit with L = 3 µ m to avoid the misleadingfinite-size effect as topological effects cannot manifest atall in short wires. We calculate our theoretical resultsat zero temperature under the aforementioned parame-ters unless otherwise stated. Inclusion of temperatureis straightforward in the theory [46] and is an unneces-sary complication— typical experiments are carried outat 20-25 mK which should be well-represented by ourzero-temperature theory. B. Bad ZBCP
We omit the effect of the electrostatic potential arisingfrom various gate voltages in the pristine ‘good MZM’limit. However, this is too ideal in the realistic scenario:The charge impurity and various gate voltages may in-duce an inhomogeneous smooth confining potential in thesemiconductor [30, 32, 33, 47]. This inhomogeneous po-tential is a common mechanism that gives rise to the badZBCP [29] in tunneling spectroscopy.Since the details of this inhomogeneous chemical po-tential is usually not precisely known in experiments, without loss of generality [32], we model the inhomo-geneous chemical potential V bad ( x ) in the form of theGaussian function V bad ( x ) = V max exp (cid:18) − x σ (cid:19) , (3)where σ and V max define the linewidth and height of thepeak of the inhomogeneous potential. In the followingcalculations, we choose σ = 0 . µ m and V max = 1 . µ − V bad ( x ) inthe Hamiltonian (1). We show an example of V bad ( x )in Fig. 1(b) at α = 0. C. Ugly ZBCP
Besides the bad ZBCP, another type of trivial ZBCPsis the ugly ZBCP arising from random disorder in thechemical potential of the nanowire. Disorder is uninten-tional and unavoidable due to the imperfect sample qual-ity, subband occupation [48], etc. Thus, we model disor-der as the randomness in the chemical potential V ugly ( x )which is drawn from an uncorrelated Gaussian distribu-tion with the mean value of zero and standard deviationof σ µ , i.e., (cid:104) V ugly ( x ) V ugly ( x (cid:48) ) (cid:105) = σ µ δ ( x − x (cid:48) ). Spatialrandom disorder also gives an effective chemical poten-tial µ − V ugly ( x ) in the Hamiltonian (1). We sketch arepresentative random disorder V ugly ( x ) in Fig. 1(b) at α = 1. Note that the inhomogeneous potential produc-ing bad ZBCP is deterministic, varying smoothly spa-tially whereas disorder, producing ugly ZBCP, is spatiallyrandom— although impurities may give rise to both ef-fects, their physical origins are qualitatively different asare their effects on the Majorana physics. We also em-phasize that neither is ‘noise’ in the usual sense, which isthe random temporal variation of current and voltage inelectronic systems.Because weak disorder preserves the topological prop-erties of the nanowire which does not induce the trivialZBCP, and strong disorder completely destroys the hy-brid nanowire model which should be instead describedby a random matrix approach in the class D ensem-ble [36, 41, 49–52], we focus on intermediate disorderwith σ µ /µ ∼
1. We clarify that all following results arepresented just for one particular random configuration,without taking the ensemble average, which is the appro-priate theory for low temperature nanowire experiments.
D. Crossover between the bad and the ugly ZBCP
We introduce two types of crossovers in this work: thecrossover between the bad ZBCP and the ugly ZBCP[Fig. 1(b)], and the crossover between one ugly ZBCP andanother ugly ZBCP with different disorder configurations[Fig. 1(c)]. In Figs. 1(b) and 1(c), we use the tuningparameter α to control the crossover.The first type of crossover [Fig. 1(b)] is the case wherethe nanowire starts with showing the bad ZBCP arisingfrom the inhomogeneous potential denoted by V bad ( x )and ends up with showing the ugly ZBCP arising fromrandom disorder denoted by V ugly ( x ). We interpolate thetwo limits linearly such that the nanowire experiencesan effective potential V ( x ; α ) continuously during thecrossover as per V ( x ; α ) = (1 − α ) V bad ( x ) + αV ugly ( x ) . (4)Therefore, α = 0 corresponds to the inhomogeneous po-tential that gives rise to the strictly bad ZBCP and α = 1corresponds to random disorder in the chemical potentialthat gives rise to the strictly ugly ZBCP. When α ∈ (0 , α = 0 .
5] which is of our interest. In this crossover regime,which is the likely generic experimental situation, theZBCP is neither strictly bad nor strictly ugly, but issome kind of a complicated combined bad-ugly mixturedepending on the value of α . We present a representativeresult at α = 0 . α = 0and 1 in Sec. IV. For a complete process of the crossover,we refer to an animation in the Supplementary Informa-tion to show the crossover continuously [42]. E. Crossover between two distinct ugly ZBCPs
The second type of crossover [Fig. 1(c)] involves twougly ZBCPs arising from two distinct disorder configu-rations denoted by V (1)ugly ( x ) and V (2)ugly ( x ). Similar to thefirst type of crossover in Eq. (4) between the bad ZBCPand the ugly ZBCP, we also interpolate the two limits ofthe ugly ZBCPs linearly. Namely, the nanowire is in thepresence of an effective potential, V ( x ; α ) = (1 − α ) V (1)ugly ( x ) + αV (2)ugly ( x ) , (5)during the crossover. Therefore, α = 0 or 1 correspondsto either one of the predetermined random configura-tions, and α ∈ (0 ,
1) corresponds to an intermediatestate. We also present a representative result at α = 0 . α = 0 and 1 in Sec. IV.The continuous process of the crossover is shown in theform of an animation in the Supplementary Informationas well [42].In fact, the linear interpolation of two predetermineddisorder configurations actually decreases the variance oforiginal disorder from 1 [e.g. Fig. 1(c) at α = 0 and 1]to 1 − α + 2 α [e.g, Fig. 1(c) at α = 0 . a priori . There is noparticular reason why the disorder variance will be pre-served as system parameters change to vary the disorderfrom one to another configuration. For example, the elec-trostatic potential in the nanowire may be different evenwhen the device undergoes a charge jump with all gate voltages returning to the initial state [15]. Theoretically,we are simulating a nanowire which was initially in thepresence of one particular random disorder is now in thepresence of another distinct random disorder. Thus, itis not guaranteed that disorder will preserve its strengththroughout the process of the voltage switch in the lab-oratory. We have, however, carried out variance pre-serving crossover calculations between two ugly scenariosalso, but the results are qualitatively the same. We men-tion that we have modified our crossover interpolationformula to other forms tuned by a single parameter, andthe results are qualitatively very similar to the linear in-terpolation [Eqs. (4) and (5)] results shown here. Thereis no current experimental information on the detailedforms of V bad and V ugly and therefore, of V [Eq. (4)] and V [Eq. (5)]. III. THEORYA. Local density of states
To explicitly show the presence or absence of the gapclosing/reopening feature in the bulk region, and the end-to-end correlation (or lack thereof) at two ends in thegood, bad, and ugly cases, we resort to the local densityof states in the middle of the nanowire and at both endsof the nanowire. We first discretize the BdG Hamiltonianusing a fictitious lattice constant a = 10 nm and replac-ing the derivatives with the finite differences to constructa tight-binding model [53]. Thus, the LDOS correspond-ing to the tight-binding model at the energy ω and posi-tion x i is defined asLDOS( ω, x i ) = − π Im (cid:20) tr σ,τ (cid:18) ω + η − H (cid:19)(cid:21) i,i , (6)where tr σ,τ is a partial trace over the spin and particle-hole space, Im [ . . . ] takes the imaginary part, H is theHamiltonian of the nanowire, the subscript i, i takes the i -th diagonal term in the matrix, and η is a standardpositive infinitesimal required to ensure the causality.We show the LDOS of the Hamiltonian in the pris-tine limit as well as two aforementioned crossovers in theappendix. For distances larger than a our calculatedLDOS corresponds to the continuum system of interest,and the details of the tight binding prescription becomescoarse-grained. B. Tunnel conductance
To simulate the experimental measurements of the tun-neling spectroscopies, we also calculate the tunnel con-ductance based on the BTK formalism [54–56]. We firstattach two semi-infinite normal leads on both ends of thenanowire [Fig. 1(a)], where the Hamiltonian of the leadtakes the same form as that of the nanowire except for − . . . V b i a s ( m e V ) LeftPristine(a) 0 2 4 G ( e /h )0 1 V Z (meV) − . . . V b i a s ( m e V ) Right(b) 0.801.290 2 4 G ( e /h ) 0.801.29 FIG. 2. The differential conductance as a function of Zee-man field V Z and bias voltage V bias from the left end (a) andthe right end (b) in the pristine wire limit. The right col-umn shows the corresponding line-cuts in the trivial regime at V Z =0.8 meV (red) and in the topological regime at V Z =1.29meV (blue). The parameters are: chemical potential µ = 1meV, parent SC gap ∆ = 0 . γ = 0 . α = 0 . L = 3 µ m, dissipation is 10 − meV and zero temperature.The TQPT is at V Z = 1 .
02 meV. The corresponding LDOSis shown in Fig. 7. the superconducting term (i.e., no proximitized super-conductivity in the normal lead). The chemical potentialof the lead is 25 meV, and the tunnel barrier height at theNS junction interface is 10 meV following the choice inRef. 27. Then we assume a propagating wave in the nor-mal leads, and calculate the S-matrix at the NS interface.The calculation of the S-matrix is done with the helpof the Python scattering matrix package KWANT [57].Details of the tunneling conductance calculation are notprovided since they are standard and can be found in theliterature.
IV. RESULTS
In this section, we first present the results of the tun-nel conductance as a function of the Zeeman splittingand bias voltage in the pristine wire limit correspondingto Hamiltonian (1) as shown in Fig. 2. These results cor-respond to the good Majoranas. Figures 2(a) and 2(b)show the tunnel conductances measured from the left andright end, respectively. On the right panels, we show twoline-cuts in the trivial regime at V Z = 0 . V Z = 1 .
29 meV (blue). The con-ductances measured from two ends in this ideal case showa perfect end-to-end correlation because of the nonlocal topological nature of MZMs. The ZBCPs shown hereare all topological good ZBCPs with a robust quantizedplateau of 2 e /h above the TQPT at V Z = 1 .
02 meVwhere the bulk gap closes. The good ZBCP also mani-fests an increasing Majorana oscillation as Zeeman fieldincreases above the TQPT. In addition, we present thecorresponding LDOS at two ends and in the middle ofthe nanowire in Fig. 7 in the appendix. We find the bulkgap closing and reopening features are very prominentshowing up in the middle of the nanowire at the TQPTas shown in Fig. 7(b).However, the pristine wire limit rarely applies in thelaboratory; therefore, a more realistic scenario is thenanowire in the presence of the inhomogeneous poten-tial or random disorder. In Fig. 3, we show a represen-tative result of the tunnel conductance in the crossoverbetween the bad ZBCP arising from the inhomogeneouspotential and the ugly ZBCP arising from random dis-order. In Fig. 3, the top (bottom) panels show the tun-nel conductances measured from the left (right) with thecorresponding line-cuts in the trivial regime (red) andtopological regime (blue), respectively. At α = 0, thenanowire shows a bad ZBCP with a quantized plateauat the left end [Fig. 3(a)] below the TQPT, which is thequasi-Majorana arising from the inhomogeneous poten-tial [58]. The conductance at the right end [Fig. 3(b)]does not show trivial ZBCPs because the right end ofthe nanowire does not have inhomogeneity. At α = 0 . . < V Z < . V Z = 1 . α = 1 [Figs. 3(e) and 3(f)],the nanowire shows the trivial ugly ZBCP with the con-ductance peak above 2 e /h at the left end. At the rightend, there are only sporadic ZBCPs with arbitrary valueof conductances below 2 e /h , unlike the bad ZBCPs at α = 0 whose conductances are almost quantized at 2 e /h ,which explains the terminology of ‘quasi-Majorana’ oftenused to describe this nontopological ZBCP.A generic feature throughout the crossover region from α = 0 to 1 is the absence of any end-to-end correlationbelow the TQPT because the trivial ABS is a fermionicstate which does not have the nonlocal property of topo-logical MZM. Near the ugly region ( α ∼ V Z = 1 .
02 meV) since the topological regimeis suppressed by disorder to some degree. However, nearthe bad region ( α ∼ − . . . V b i a s ( m e V ) Left(a) α = 0 . G ( e /h ) Left(c) α = 0 . G ( e /h ) Left(e) α = 1 . G ( e /h )0 1 V Z (meV) − . . . V b i a s ( m e V ) Right(b) 0 1 V Z (meV)Right(d) 0 1 V Z (meV)Right(f)0.81.2 0.81.65 0.841.690 2 4 G ( e /h )0.81.2 0 2 4 G ( e /h )0.81.65 0 2 4 G ( e /h )0.841.69 FIG. 3. The crossover between the bad ZBCP arising from an inhomogeneous potential (a,b) and the ugly ZBCP arising fromrandom disorder (e,f). The upper (lower) panels show the differential conductances as a function of Zeeman field V Z and biasvoltage V bias from the left end (the right end). (c,d) An intermediate case interpolated by the inhomogeneous potential andrandom disorder. The corresponding line-cuts in the trivial regime (red) and the topological regime (blue) are shown right tothe color plot of the conductance. The inhomogeneous potential has σ = 0 . µ m and V max = 1 . σ µ = 1 meV. Refer to Fig. 2 for other parameters. The corresponding LDOS is shown in Fig. 8. field increases in the inhomogeneous potential. We alsonote that the bad ZBCP arising from the inhomogeneouspotential shows considerable stability with the Zeemanfield even in the presence of some disorder (i.e., α = 0 . α = 0[Fig. 8(b)], we see sharp gap closing and reopening fea-tures at the TQPT ( V Z = 1 .
02 meV), which indicates thesame transition from the trivial ABS to the topologicalMBS as the reappearance of the end-to-end correlation inthe local conductance in Figs. 3(a) and 3(b) does. How-ever, as α increases [Fig. 8(e)], more disorder is blendedwith the inhomogeneous chemical potential. Thus, wenotice the gap closing and reopening features graduallybecoming ambiguous, i.e., the gap closing and reopeningdo not happen at the same field. This ambiguity becomesthe worst in the ugly region at α = 1 [Fig. 8(h)] becausethe gap closes and reopens at different fields, which indi-cates the TQPT is strongly renormalized by disorder.In Fig. 4, we present the tunnel conductance in thecrossover between two ugly ZBCPs. Namely, we startwith the random disorder configuration in Figs. 3(e)and 3(f), and let it transform into another disorder con-figuration. At two limits ( α = 0 and 1), they areboth ugly ZBCPs without the end-to-end correlation,which are similar to the previous crossover between thebad ZBCP and the ugly ZBCP. However, during the crossover, one additional noteworthy feature is the ubiq-uitous presence of the trivial ZBCPs: The conductancepeaks show arbitrary values between 0 and 4 e /h at ran-dom ranges of Zeeman fields in the trivial regime at differ-ent α ’s. The ubiquitous trivial ZBCPs in the crossoverfrom α = 0 to 1 resemble the experimentally observedZBCPs during a cycle of voltage switch or a cycle of heat-up and cool-down [15], which are arising from the slowly-varying disorder inside the sample as the impurities movearound. We present an animation that shows randomappearance and disappearance of the ugly ZBCPs in theSupplementary Information [42] corresponding to Fig. 4.In Fig. 9 of the appendix, we also present the LDOScorresponding to the local conductance in Fig. 4. Simi-lar to the crossover between the bad ZBCP and the uglyZBCP, we also find the absence of the end-to-end cor-relation at two ends, and the gap closing and reopen-ing features are also not sharp (i.e., the bulk gap doesnot reopen immediately after it closes) throughout thecrossover (which can be seen from LDOS in the middleof the nanowire in Figs. 9(b), 9(e), and 9(h)) because ofthe ubiquitous zero energy modes in the trivial regime inthe presence of random disorder.Finally, in Figs. 5 and 6 we provide detailed ugly tougly crossover results for several values of alpha between0 and 1, showing the system continuously going from onedistinct random disorder configuration to another, us-ing variance-conserving and linear interpolations, respec-tively. For the results shown in Fig. 5, we use the follow- − . . . V b i a s ( m e V ) Left(a) α = 0 . G ( e /h ) Left(c) α = 0 . G ( e /h ) Left(e) α = 1 . G ( e /h )0 1 V Z (meV) − . . . V b i a s ( m e V ) Right(b) 0 1 V Z (meV)Right(d) 0 1 V Z (meV)Right(f)0.841.69 1.041.67 0.81.610 2 4 G ( e /h )0.841.69 0 2 4 G ( e /h )1.041.67 0 2 4 G ( e /h )0.81.61 FIG. 4. The crossover between two ugly ZBCPs induced by random disorder following a simple linear interpolation of Eq. (5).The upper (lower) panels show the differential conductances as a function of Zeeman field V Z and bias voltage V bias from the leftend (the right end). (a,b) Random disorder with σ µ = 1 meV at α = 0; (c,d) Interpolated disorder at α = 0 .
5; (e,f) Differentrandom disorder with σ µ = 1 meV at α = 1. The corresponding line-cuts in the trivial (red) regime and the topological regime(blue) are shown right to the color plot of the conductance. Refer to Fig. 2 for other parameters. The corresponding LDOS isshown in Fig. 9. ing variance-conserving interpolation replacing Eq. (5): V ( x ; α ) = √ − αV (1)ugly ( x ) + √ αV (2)ugly ( x ) (7)whereas for Fig. 6 we use the linear interpolation ofEq. (5).The key generic messages of Figs. 5 and 6 are: (1)Disorder could produce ZBCPs with conductance val-ues at, below, above 2 e /h ; (2) disorder-induced ZBCPsare unstable, without manifesting any robustness in theZeeman field; (3) disorder-induced ZBCPs are uncorre-lated for tunneling from the two wire ends; (4) disorder-induced ZBCPs could occasionally suddenly disappearwith small parameter variations, reflecting the so-called‘charge jumps’ or ‘voltage switches’ associated with ran-dom traps in electronic materials; (5) on a qualitativelevel, most existing Majorana nanowire experimental re-sults appear to be consistent with the system manifestingcrossovers between disorder configurations as gate volt-ages are tuned with the observed ZBCPs being ‘ugly’ones. V. CONCLUSION
We theoretically study Majorana nanowires insuperconductor-semiconductor hybrid platforms by fo-cusing on the crossover behavior of tunnel conductanceand density of states arising from inhomogeneous poten-tial and random disorder. The main qualitative find-ing is that the crossover behavior is dominated by trivial zero modes, some of which manifest conductance peakswith values ∼ e /h , mimicking the predicted topologi-cal MZM behavior. For inhomogeneous potential, as hasalready been pointed out in the theoretical literature, the2 e /h trivial ZBCP may be stable as a function of sys-tem parameters even in the presence of some finite dis-order since the emergent Andreev bound states, some-times called quasi-Majoranas in this context, may re-main pinned near zero energy for finite ranges of param-eters (e.g., Zeeman field). The difference between thesecrossover trivial zero modes and their topological MZMcounterparts in pristine nanowires is not necessarily thevalue of conductance or the existence of zero-bias peaks,which are ubiquitous in the trivial situation, even in thepresence of some random disorder, but the facts thatthe topological MZMs manifest generic end-to-end cor-relations (i.e., the observed tunnel conductance peak issimilar from both ends) and the topological MZMs mani-fest generic Majorana oscillations (i.e., the zero-bias con-ductance shows oscillatory behavior with increasing mag-netic field) and that the topological MZMs are necessarilyaccompanied by the opening of a superconducting gap inthe bulk. Another qualitative finding is that disorder typ-ically may produce zero-bias conductance values slightlyhigher than 2 e /h (as observed recently [11, 15]) whereasinhomogeneous smooth potential tends to mostly pro-duce zero-bias peaks with conductance pinned at 2 e /h or below [59] even in the presence of some random disor-der.Based on our results and its qualitative agreement withmuch of the existing Majorana nanowire measurements − . . . V b i a s ( m e V ) Left(a) α = 0 . G ( e /h ) Left(c) α = 0 . G ( e /h ) Left(e) α = 0 . G ( e /h ) − . . . V b i a s ( m e V ) Right(b) Right(d) Right(f) − . . . V b i a s ( m e V ) Left(g) α = 0 . α = 0 . α = 1 .
00 1 V Z (meV) − . . . V b i a s ( m e V ) Right(h) 0 1 V Z (meV)Right(j) 0 1 V Z (meV)Right(l)0.841.69 0.921.65 0.951.650.841.69 0.921.65 0.951.650.951.65 0.921.63 0.81.610 2 4 G ( e /h )0.951.65 0 2 4 G ( e /h )0.921.63 0 2 4 G ( e /h )0.81.61 FIG. 5. The crossover between two ugly ZBCPs induced by random disorder following a variance-conserving interpolation ofEq. (7). The first and third (second and fourth) panels show the differential conductances as a function of Zeeman field V Z andbias voltage V bias from the left end (the right end). The results of α ranging from 0 to 1 with a step of 0.2 are shown in (a,b),(c,d), (e,f), (g,h), (i,j), (k,l), respectively. The corresponding line-cuts in the trivial (red) regime and the topological regime(blue) are shown right to the color plot of the conductance. Refer to Fig. 2 for other parameters. in the literature, we suggest the following five criteria asthe minimal conditions for any future experimental claimof the possible observation of topological Majorana zeromodes in nanowires based just on the tunneling spec-troscopy: (1) There must be stable (both in gate voltageand in magnetic field) zero-bias conductance peak witha value close to (but not above) 2 e /h at the lowest ex-perimental temperature; (2) the conductance value mustsaturate with lowering temperature and varying tunnelbarrier to a value close to 2 e /h (but not above—in fact,the expected Majorana conductance under experimentalconditions should be slightly below 2 e /h ); (3) similarZBCPs with ‘quantized conductance’ must be observed in tunneling from both ends of the wire (the whole tun-neling spectra need not be identical from both ends, butthe ZBCPs must be); (4) there should be some signa-tures for Majorana oscillations as the magnetic field in-creases above the field value where the ZBCP appears;(5) there should be some signatures of a gap reopeningwhen the ZBCP shows up. If these five criteria are notsatisfied, chances are very high, as shown explicitly in thecurrent work, the system is most likely manifesting non-topological Andreev bound state induced zero modes insome complicated crossover behavior between inhomo-geneous chemical potential and random disorder dom-inated trivial regimes (or more likely, simply crossover − . . . V b i a s ( m e V ) Left(a) α = 0 . G ( e /h ) Left(c) α = 0 . G ( e /h ) Left(e) α = 0 . G ( e /h ) − . . . V b i a s ( m e V ) Right(b) Right(d) Right(f) − . . . V b i a s ( m e V ) Left(g) α = 0 . α = 0 . α = 1 .
00 1 V Z (meV) − . . . V b i a s ( m e V ) Right(h) 0 1 V Z (meV)Right(j) 0 1 V Z (meV)Right(l)0.841.69 0.961.69 1.091.690.841.69 0.961.69 1.091.691.091.69 1.041.69 0.81.610 2 4 G ( e /h )1.091.69 0 2 4 G ( e /h )1.041.69 0 2 4 G ( e /h )0.81.61 FIG. 6. The crossover between two ugly ZBCPs induced by random disorder following a simple linear interpolation Eq. (5).The disorder configurations V (1)ugly and V (2)ugly here are identical to the ones in Fig. 4. But we present more crossover results ofdifferent α ’s here. The first and third (second and fourth) panels show the differential conductances as a function of Zeemanfield V Z and bias voltage V bias from the left end (the right end). The results of α ranging from 0 to 1 with a step of 0.2are shown in (a,b), (c,d), (e,f), (g,h), (i,j), (k,l), respectively. The corresponding line-cuts in the trivial (red) regime and thetopological regime (blue) are shown right to the color plot of the conductance. Refer to Fig. 2 for other parameters. among distinct random disorder configurations as sys-tem parameters vary, changing the impurity states in theenvironment). It is important to emphasize that neitherany reported experimental observation nor any of ourtrivial zero mode simulations satisfies these topologicalcriteria although both often manifest zero-bias conduc-tance peaks with approximate 2 e /h value over narrow fine-tuned parameter ranges. The observation of a ZBCP,even with a value close to 2 e /h , is at best a necessarycondition for the existence of topological Majorana zeromodes, satisfying the sufficient conditions, as discussedabove, require substantial more work.This work is supported by the Laboratory for Physi-cal Sciences (LPS) and the University of Maryland HighPerformance Computing Cluster (HPCC). [1] R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Majo-rana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures, Phys. Rev. Lett. , 077001 (2010).[2] J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma,Generic New Platform for Topological Quantum Compu-tation Using Semiconductor Heterostructures, Phys. Rev.Lett. , 040502 (2010).[3] J. D. Sau, S. Tewari, R. M. Lutchyn, T. D. Stanescu, andS. Das Sarma, Non-Abelian quantum order in spin-orbit-coupled semiconductors: Search for topological Majoranaparticles in solid-state systems, Phys. Rev. B , 214509(2010).[4] Y. Oreg, G. Refael, and F. von Oppen, Helical Liquidsand Majorana Bound States in Quantum Wires, Phys.Rev. Lett. , 177002 (2010).[5] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard,E. P. A. M. Bakkers, and L. P. Kouwenhoven, Signa-tures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices, Science , 1003(2012).[6] A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum,and H. Shtrikman, Zero-bias peaks and splitting in anAl–InAs nanowire topological superconductor as a signa-ture of Majorana fermions, Nature Physics , 887 (2012).[7] M. T. Deng, C. L. Yu, G. Y. Huang, M. Larsson,P. Caroff, and H. Q. Xu, Anomalous Zero-Bias Conduc-tance Peak in a Nb–InSb Nanowire–Nb Hybrid Device,Nano Letters , 6414 (2012).[8] H. O. H. Churchill, V. Fatemi, K. Grove-Rasmussen,M. T. Deng, P. Caroff, H. Q. Xu, and C. M. Marcus,Superconductor-nanowire devices from tunneling to themultichannel regime: Zero-bias oscillations and magneto-conductance crossover, Phys. Rev. B , 241401 (2013).[9] A. D. K. Finck, D. J. Van Harlingen, P. K. Mohseni,K. Jung, and X. Li, Anomalous Modulation of a Zero-Bias Peak in a Hybrid Nanowire-Superconductor Device,Phys. Rev. Lett. , 126406 (2013).[10] M. Deng, S. Vaitiek˙enas, E. B. Hansen, J. Danon, M. Lei-jnse, K. Flensberg, J. Nyg˚ard, P. Krogstrup, and C. M.Marcus, Majorana bound state in a coupled quantum-dothybrid-nanowire system, Science , 1557 (2016).[11] F. Nichele, A. C. C. Drachmann, A. M. Whiticar, E. C. T.O’Farrell, H. J. Suominen, A. Fornieri, T. Wang, G. C.Gardner, C. Thomas, A. T. Hatke, P. Krogstrup, M. J.Manfra, K. Flensberg, and C. M. Marcus, Scaling of Ma-jorana Zero-Bias Conductance Peaks, Phys. Rev. Lett. , 136803 (2017).[12] H. Zhang, ¨O. G¨ul, S. Conesa-Boj, M. P. Nowak, M. Wim-mer, K. Zuo, V. Mourik, F. K. de Vries, J. vanVeen, M. W. A. de Moor, J. D. S. Bommer, D. J.van Woerkom, D. Car, S. R. Plissard, E. P. A. M.Bakkers, M. Quintero-P´erez, M. C. Cassidy, S. Koelling,S. Goswami, K. Watanabe, T. Taniguchi, and L. P.Kouwenhoven, Ballistic superconductivity in semicon-ductor nanowires, Nature Communications , 16025(2017).[13] S. Vaitiek˙enas, M.-T. Deng, J. Nyg˚ard, P. Krogstrup,and C. M. Marcus, Effective g Factor of Subgap States inHybrid Nanowires, Phys. Rev. Lett. , 037703 (2018).[14] M. W. A. de Moor, J. D. S. Bommer, D. Xu, G. W.Winkler, A. E. Antipov, A. Bargerbos, G. Wang, N. vanLoo, R. L. M. O. het Veld, S. Gazibegovic, D. Car, J. A.Logan, M. Pendharkar, J. S. Lee, E. P. A. M. Bakkers,C. J. Palmstrøm, R. M. Lutchyn, L. P. Kouwenhoven,and H. Zhang, Electric field tunable superconductor- semiconductor coupling in Majorana nanowires, New J.Phys. , 103049 (2018).[15] H. Zhang, M. W. A. de Moor, J. D. S. Bommer,D. Xu, G. Wang, N. van Loo, C.-X. Liu, S. Gazibegovic,J. A. Logan, D. Car, R. L. M. O. het Veld, P. J.van Veldhoven, S. Koelling, M. A. Verheijen, M. Pend-harkar, D. J. Pennachio, B. Shojaei, J. S. Lee, C. J.Palmstrøm, E. P. A. M. Bakkers, S. D. Sarma, andL. P. Kouwenhoven, Large zero-bias peaks in InSb-Alhybrid semiconductor-superconductor nanowire devices,arXiv:2101.11456 [cond-mat] (2021); H. Zhang, C.-X.Liu, S. Gazibegovic, D. Xu, J. A. Logan, G. Wang,N. van Loo, J. D. S. Bommer, M. W. A. de Moor,D. Car, R. L. M. Op het Veld, P. J. van Veldhoven,S. Koelling, M. A. Verheijen, M. Pendharkar, D. J.Pennachio, B. Shojaei, J. S. Lee, C. J. Palmstrøm,E. P. A. M. Bakkers, S. D. Sarma, and L. P. Kouwen-hoven, Quantized majorana conductance, Nature ,74 (2018), arXiv:1710.10701.[16] J. D. S. Bommer, H. Zhang, ¨O. G¨ul, B. Nijholt, M. Wim-mer, F. N. Rybakov, J. Garaud, D. Rodic, E. Babaev,M. Troyer, D. Car, S. R. Plissard, E. P. A. M. Bakkers,K. Watanabe, T. Taniguchi, and L. P. Kouwenhoven,Spin-Orbit Protection of Induced Superconductivity inMajorana Nanowires, Phys. Rev. Lett. , 187702(2019).[17] A. Grivnin, E. Bor, M. Heiblum, Y. Oreg, and H. Shtrik-man, Concomitant opening of a bulk-gap with an emerg-ing possible Majorana zero mode, Nat Commun , 1(2019).[18] G. L. R. Anselmetti, E. A. Martinez, G. C. M´enard,D. Puglia, F. K. Malinowski, J. S. Lee, S. Choi, M. Pend-harkar, C. J. Palmstrøm, C. M. Marcus, L. Casparis, andA. P. Higginbotham, End-to-end correlated subgap statesin hybrid nanowires, Phys. Rev. B , 205412 (2019).[19] G. C. M´enard, G. L. R. Anselmetti, E. A. Martinez,D. Puglia, F. K. Malinowski, J. S. Lee, S. Choi, M. Pend-harkar, C. J. Palmstrøm, K. Flensberg, C. M. Mar-cus, L. Casparis, and A. P. Higginbotham, Conductance-Matrix Symmetries of a Three-Terminal Hybrid Device,Phys. Rev. Lett. , 036802 (2020).[20] D. Puglia, E. A. Martinez, G. C. M´enard, A. P¨oschl,S. Gronin, G. C. Gardner, R. Kallaher, M. J. Manfra,C. M. Marcus, A. P. Higginbotham, and L. Casparis,Closing of the Induced Gap in a Hybrid Superconductor-Semiconductor Nanowire, arXiv:2006.01275 [cond-mat](2020).[21] K. Sengupta, I. ˇZuti´c, H.-J. Kwon, V. M. Yakovenko, andS. Das Sarma, Midgap edge states and pairing symmetryof quasi-one-dimensional organic superconductors, Phys.Rev. B , 144531 (2001).[22] K. T. Law, P. A. Lee, and T. K. Ng, Majorana FermionInduced Resonant Andreev Reflection, Phys. Rev. Lett. , 237001 (2009).[23] K. Flensberg, Tunneling characteristics of a chain of Ma-jorana bound states, Phys. Rev. B , 180516 (2010).[24] M. Wimmer, A. R. Akhmerov, J. P. Dahlhaus, andC. W. J. Beenakker, Quantum point contact as a probeof a topological superconductor, New J. Phys. , 053016(2011).[25] S. Das Sarma, J. D. Sau, and T. D. Stanescu, Split-ting of the zero-bias conductance peak as smoking gunevidence for the existence of the Majorana mode in a superconductor-semiconductor nanowire, Phys. Rev. B , 220506 (2012).[26] Y.-H. Lai, J. D. Sau, and S. Das Sarma, Presence versusabsence of end-to-end nonlocal conductance correlationsin Majorana nanowires: Majorana bound states versusAndreev bound states, Phys. Rev. B , 045302 (2019).[27] H. Pan, J. D. Sau, and S. Das Sarma, Three-terminalnonlocal conductance in Majorana nanowires: Distin-guishing topological and trivial in realistic systems withdisorder and inhomogeneous potential, Phys. Rev. B ,014513 (2021).[28] T. ¨O. Rosdahl, A. Vuik, M. Kjaergaard, and A. R.Akhmerov, Andreev rectifier: A nonlocal conductancesignature of topological phase transitions, Phys. Rev. B , 045421 (2018).[29] H. Pan and S. Das Sarma, Physical mechanisms for zero-bias conductance peaks in Majorana nanowires, Phys.Rev. Research , 013377 (2020).[30] G. Kells, D. Meidan, and P. W. Brouwer, Near-zero-energy end states in topologically trivial spin-orbit cou-pled superconducting nanowires with a smooth confine-ment, Phys. Rev. B , 100503 (2012).[31] E. Prada, P. San-Jose, and R. Aguado, Transportspectroscopy of NS nanowire junctions with Majoranafermions, Phys. Rev. B , 180503 (2012).[32] C.-X. Liu, J. D. Sau, T. D. Stanescu, and S. Das Sarma,Andreev bound states versus Majorana bound statesin quantum dot-nanowire-superconductor hybrid struc-tures: Trivial versus topological zero-bias conductancepeaks, Phys. Rev. B , 075161 (2017).[33] T. D. Stanescu and S. Tewari, Robust low-energy An-dreev bound states in semiconductor-superconductorstructures: Importance of partial separation of compo-nent Majorana bound states, Phys. Rev. B , 155429(2019).[34] C. Moore, C. Zeng, T. D. Stanescu, and S. Tewari, Quan-tized zero-bias conductance plateau in semiconductor-superconductor heterostructures without topological Ma-jorana zero modes, Phys. Rev. B , 155314 (2018).[35] C. Moore, T. D. Stanescu, and S. Tewari, Two-terminal charge tunneling: Disentangling Majorana zeromodes from partially separated Andreev bound statesin semiconductor-superconductor heterostructures, Phys.Rev. B , 165302 (2018).[36] S. Mi, D. I. Pikulin, M. Marciani, and C. W. J.Beenakker, X-shaped and Y-shaped Andreev resonanceprofiles in a superconducting quantum dot, J. Exp.Theor. Phys. , 1018 (2014).[37] J. D. Sau and S. Das Sarma, Density of states of disor-dered topological superconductor-semiconductor hybridnanowires, Phys. Rev. B , 064506 (2013).[38] D. I. Pikulin, J. P. Dahlhaus, M. Wimmer, H. Schome-rus, and C. W. J. Beenakker, A zero-voltage conductancepeak from weak antilocalization in a Majorana nanowire,New J. Phys. , 125011 (2012).[39] D. Bagrets and A. Altland, Class D Spectral Peak inMajorana Quantum Wires, Phys. Rev. Lett. , 227005(2012).[40] J. Liu, A. C. Potter, K. T. Law, and P. A. Lee, Zero-Bias Peaks in the Tunneling Conductance of Spin-Orbit-Coupled Superconducting Wires with and without Ma-jorana End-States, Phys. Rev. Lett. , 267002 (2012).[41] H. Pan, W. S. Cole, J. D. Sau, and S. Das Sarma, Generic quantized zero-bias conductance peaks insuperconductor-semiconductor hybrid structures, Phys.Rev. B , 024506 (2020).[42] See the Supplementary Information for animations.[43] T. D. Stanescu, J. D. Sau, R. M. Lutchyn, andS. Das Sarma, Proximity effect at the superconductor–topological insulator interface, Phys. Rev. B , 241310(2010).[44] T. D. Stanescu and S. Das Sarma, Proximity-inducedlow-energy renormalization in hybrid semiconductor-superconductor Majorana structures, Phys. Rev. B ,014510 (2017).[45] R. M. Lutchyn, E. P. A. M. Bakkers, L. P. Kouwen-hoven, P. Krogstrup, C. M. Marcus, and Y. Oreg, Majo-rana zero modes in superconductor–semiconductor het-erostructures, Nature Reviews Materials , 52 (2018).[46] F. Setiawan, C.-X. Liu, J. D. Sau, and S. Das Sarma,Electron temperature and tunnel coupling dependenceof zero-bias and almost-zero-bias conductance peaks inMajorana nanowires, Phys. Rev. B , 184520 (2017).[47] C.-X. Liu, J. D. Sau, and S. Das Sarma, Distinguishingtopological Majorana bound states from trivial Andreevbound states: Proposed tests through differential tunnel-ing conductance spectroscopy, Phys. Rev. B , 214502(2018).[48] B. D. Woods, S. Das Sarma, and T. D. Stanescu,Subband occupation in semiconductor-superconductornanowires, Phys. Rev. B , 045405 (2020).[49] C. W. J. Beenakker, Random-matrix theory of quantumtransport, Rev. Mod. Phys. , 731 (1997).[50] T. Guhr, A. M¨uller–Groeling, and H. A. Weidenm¨uller,Random-matrix theories in quantum physics: Commonconcepts, Physics Reports , 189 (1998).[51] P. W. Brouwer, K. M. Frahm, and C. W. J. Beenakker,Distribution of the quantum mechanical time-delay ma-trix for a chaotic cavity, Waves in Random Media , 91(1999).[52] C. W. J. Beenakker, Random-matrix theory of Majoranafermions and topological superconductors, Rev. Mod.Phys. , 1037 (2015).[53] S. Das Sarma, A. Nag, and J. D. Sau, How to in-fer non-Abelian statistics and topological visibility fromtunneling conductance properties of realistic Majoranananowires, Phys. Rev. B , 035143 (2016).[54] G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Tran-sition from metallic to tunneling regimes in supercon-ducting microconstrictions: Excess current, charge im-balance, and supercurrent conversion, Phys. Rev. B ,4515 (1982).[55] S. Datta, Electronic Transport in Mesoscopic Systems ,Cambridge Studies in Semiconductor Physics and Mi-croelectronic Engineering (Cambridge University Press,Cambridge, 1995).[56] M. P. Anantram and S. Datta, Current fluctuations inmesoscopic systems with Andreev scattering, Phys. Rev.B , 16390 (1996).[57] C. W. Groth, M. Wimmer, A. R. Akhmerov, andX. Waintal, Kwant: A software package for quantumtransport, New Journal of Physics , 063065 (2014).[58] A. Vuik, B. Nijholt, A. Akhmerov, and M. Wimmer,Reproducing topological properties with quasi-Majoranastates, SciPost Physics , 061 (2019).[59] H. Pan, C.-X. Liu, M. Wimmer, and S. D. Sarma, Quan-tized and unquantized zero-bias tunneling conductance peaks in Majorana nanowires: Conductance below andabove 2 e /h , arXiv:2102.02218 [cond-mat] (2021). Appendix A: LDOS for the good ZBCP and two crossovers V Z (meV) − . . . V b i a s ( m e V ) Left(a) 10 − − LDOS 0 1 2 V Z (meV)Middle(b) 10 − − LDOS 0 1 2 V Z (meV)Right(c) 10 − − LDOS
FIG. 7. The LDOS of a pristine wire (a) at the left end, (b) in the middle of the wire and (c) at the right end, correspondingto Fig. 2. − . . . V b i a s ( m e V ) Left α = 0 . − − LDOS Left α = 0 . − − LDOS Left α = 1 . − − LDOS − . . . V b i a s ( m e V ) Middle(b) 10 − − LDOS Middle(e) 10 − − LDOS Middle(h) 10 − − LDOS0 1 2 V Z (meV) − . . . V b i a s ( m e V ) Right(c) 10 − − LDOS 0 1 2 V Z (meV)Right(f) 10 − − LDOS 0 1 2 V Z (meV)Right(i) 10 − − LDOS