Transport in superconductor--normal metal--superconductor tunneling structures: Spinful p-wave and spin-orbit-coupled topological wires
TTransport in superconductor–normal metal–superconductor tunneling structures:Spinful p -wave and spin-orbit-coupled topological wires F. Setiawan, ∗ William S. Cole, Jay D. Sau, and S. Das Sarma
Condensed Matter Theory Center, Station Q Maryland,and Joint Quantum Institute, Department of Physics,University of Maryland, College Park, Maryland 20742, USA (Dated: May 24, 2017)We theoretically study transport properties of voltage-biased one-dimensional superconductor–normal metal–superconductor tunnel junctions with arbitrary junction transparency where the su-perconductors can have trivial or nontrivial topology. Motivated by recent experimental effortson Majorana properties of superconductor-semiconductor hybrid systems, we consider two explicitmodels for topological superconductors: (i) spinful p -wave, and (ii) spin-split spin-orbit-coupled s -wave. We provide a comprehensive analysis of the zero-temperature dc current I and differen-tial conductance dI/dV of voltage-biased junctions with or without Majorana zero modes (MZMs).The presence of an MZM necessarily gives rise to two tunneling conductance peaks at voltages eV = ± ∆ lead , i.e., the voltage at which the superconducting gap edge of the lead aligns withthe MZM. We find that the MZM conductance peak probed by a superconducting lead without aBCS singularity has a nonuniversal value which decreases with decreasing junction transparency.This is in contrast to the MZM tunneling conductance measured by a superconducting lead with a BCS singularity, where the conductance peak in the tunneling limit takes the quantized value G M = (4 − π )2 e /h independent of the junction transparency. We also discuss the “subharmonicgap structure”, a consequence of multiple Andreev reflections, in the presence and absence of MZMs.Finally, we show that for finite-energy Andreev bound states (ABSs), the conductance peaks shiftaway from the gap bias voltage eV = ± ∆ lead to a larger value set by the ABSs energy. Our workshould have important implications for the extensive current experimental efforts toward creatingtopological superconductivity and MZMs in semiconductor nanowires proximity coupled to ordinary s -wave superconductors. I. INTRODUCTION
In recent years there has been great interest in realiz-ing topological superconductors that support Majoranazero modes (MZMs) at boundaries or defects . Thisis driven mainly by the prospect of using MZMs as thebuilding blocks for a fault-tolerant topological quantumcomputer . The simplest model of a topological su-perconductor hosting MZMs is the one-dimensional (1D)spinless p -wave superconductor as originally envisionedby Kitaev . Since electrons carry a spin degree of free-dom, intrinsic spinless p -wave pairing is apparently un-common in nature. However, it can be effectively real-ized in spinful systems by a combination of spin-orbitcoupling and explicitly lifting the Kramer’s degeneracyof the electrons (e.g., by Zeeman spin splitting throughan applied magnetic field). This idea has lead to a num-ber of proposals for realizing topological superconductorin various hybrid structures with conventional s -wave su-perconductors . There are, however, significant dif-ferences between a spinless p -wave superconductor anda spin-split s -wave superconductor with spin-orbit cou-pling although they both can have localized MZMs atthe ends. One way of realizing a spinful 1D topologicalsuperconductor is by proximity-inducing superconductiv-ity in a spin-orbit-coupled semiconducting nanowire in amagnetic field . In this setup, the system can betuned from a topologically trivial to a nontrivial regimeby raising the Zeeman field above a certain critical value where the system undergoes a topological quantum phasetransition with the effective induced superconductivity inthe nanowire changing from an s -wave (trivial) charac-ter to a p -wave (topological) character. As the MZMexists as a zero-energy edge mode in the topological su-perconductor, tunneling conductance spectroscopy pro-vides a simple way of detecting the MZM. In a normalmetal–superconductor (NS) junction, the MZM medi-ates a perfect Andreev reflection at zero energy, whichin turn gives rise to a quantized 2 e /h zero-bias con-ductance value , as long as the two MZMs at thewire ends are far from each other with exponentiallysmall overlap between the MZM wave functions (the so-called “topologically protected regime”). This quantizedconductance is robust against changes to the junctiontransparency. Several experimental groups have observedthe appearance of zero-bias tunneling conductance peaksin the semiconductor–superconductor heterostructure asthe Zeeman field is raised beyond a certain value, whichindeed indicates the existence of zero-energy states .Nevertheless, the observed zero-bias conductance is sub-stantially less than the MZM canonical quantized con-ductance value. A plausible source for this discrepancyis thermal broadening in the normal-metal lead, whichreduces the zero-bias conductance value and widens thepeak , although other possibilities such as dissi-pation and MZM overlap may also be responsible .The ubiquitous absence of the predicted quantized zero-bias conductance peak in topological NS junction (inspite of there often being a weak zero-bias conductance a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y peak) is the central quandary in this subject, making itunclear whether 1D topological superconductivity withlocalized MZMs has indeed been realized experimentallyor not.To mitigate the effect of thermal broadening, one coulduse a superconducting lead instead of a normal leadin probing the MZM tunneling conductance . In asuperconducting lead, thermal quasiparticle excitationsare exponentially suppressed by the superconducting gap ∼ exp( − ∆ lead /T ), which in turn suppresses the broad-ening effect. Peng et al. found that for a conven-tional s -wave superconducting lead, the MZM conduc-tance measured in the tunneling limit (or small junctiontransparency) appears as two symmetric peaks with aquantized value G M = (4 − π )2 e /h at the gap-bias volt-age eV = ± ∆ lead , i.e., when the BCS singularity of theprobe lead aligns with the MZM. The quantized value G M is the conductance due to a single Andreev reflec-tion from the MZM. In Ref. , it was shown that in thepresence of multiple Andreev reflections (MAR), whichare generically present when the junction transparency is not small, the conductance at the voltage eV = ± ∆ lead is no longer quantized at G M . This indicates that un-like the universally quantized 2 e /h zero-bias conduc-tance value for a normal metal–topological superconduc-tor junction, the quantized value G M of the MZM tun-neling conductance for a superconductor–normal metal–superconductor (SNS) junction is not a topologically pro-tected robust quantity with the conductance value beingdependent on the details and thus making it difficult toidentify MZMs using SNS tunneling spectroscopy. Theseresults prompt further exploration of transport proper-ties of various SNS junctions involving different modelsof topological superconductors where signatures of MZMcan be fully investigated and characterized. This is thegoal of the current work where we provide comprehen-sive details on the tunneling transport properties of SNSjunctions involving topological superconductors in orderto guide future experimental work in the subject.In this paper, we calculate the dc current-voltage ( I - V ) relation and corresponding differential conductance( G = dI/dV ) of 1D SNS junctions, invoking two mod-els for topological superconductors, i.e., the spinful p -wave superconductor ( p SC) and the spin-orbit-coupled s -wave superconducting wire (SOCSW). Specifically, weconsider several possible combinations for the junction,where each superconductor can be either topologicallytrivial or nontrivial. We find that unlike the case of s -wave superconducting probe lead with BCS singularity(where (cid:80) σ = ↑ , ↓ | u σ | = (cid:80) σ = ↑ , ↓ | v σ | at the gap edge with u and v being the electron and hole component of the su-perconducting wavefunction at the gap edge), the MZMtunneling conductance measured using a superconduct-ing lead without a BCS singularity has a nonuniversalvalue, which decreases with decreasing junction trans-parencies. Our detailed theoretical and numerical resultsfor the transport properties of various types of SNS junc-tions should be a useful guide for future experimental work on the tunneling spectroscopy of topological SNSjunctions.The paper is organized as follows. In Sec. II, wepresent a general scattering matrix formalism, which canbe used to calculate the transport properties of a gen-eral SNS junction. In Sec. III, we discuss the subhar-monic gap structure (SGS). In the following sections, westudy in detail the transport in SNS junctions involving p SC (Sec. IV) and SOCSW (Sec. V) and compare the dccurrent and conductance of junctions with and withoutMZMs. In Sec. VI, we discuss the conductance due toAndreev bound states (ABSs) in the SOCSW model inorder to distinguish between MZM and ABS signaturesin the tunneling experiment. Finally, we give the conclu-sion in Sec. VII.
II. SCATTERING MATRIX FORMALISM
We begin by modeling the SNS junction by two semi-infinite superconducting regions connected by a normalregion with a delta-function barrier of strength Z , asshown in Fig. 1. The normal region is assumed to beinfinitesimally short with large chemical potential suchthat the propagating modes in this region have constantgroup velocity independent of energy. Quasiparticles canbe injected from the left or right superconducting leadwhich become electrons or holes (depending on their en-ergy) when they enter the normal region. Due to thevoltage bias, these electrons (holes) will then gain (lose)an energy eV as they are accelerated from the left (right)to the right (left). As a result, after each Andreev reflec-tion at an NS interface, an incoming electron with anenergy E will be reflected as a hole back into the sameregion with an energy E + 2 eV . The quasiparticle re-flects repeatedly inside the normal region until it gainsenough energy to be transmitted into the superconduc-tors. This mechanism is termed “multiple Andreev re-flections” (MAR) . Zδ ( x ) J − NL J + NL J in L J out L J − NR J + NR J in R J out R FIG. 1. (Color online) Schematic diagram of asuperconductor–normal metal–superconductor (SNS) junc-tion with a delta-function potential barrier of strength Z . The scattering processes in the SNS junction can besplit among three spatial regions: (i) the left NS inter-face, (ii) tunnel barrier, and (iii) right NS interface. Weexpress these processes in terms of the scattering matri-ces as (cid:18) J out L,ν ( E n ) J + NL,ν ( E n ) (cid:19) = S L ( E n ) (cid:18) J in L,ν ( E n ) δ n δ ν, → J − NL,ν ( E n ) (cid:19) , (1a) (cid:18) J − NL,ν ( E n ) J + NR,ν ( E n ) (cid:19) = (cid:88) n (cid:48) S N ( E n , E n (cid:48) ) (cid:18) J + NL,ν ( E n (cid:48) ) J − NR,ν ( E n (cid:48) ) (cid:19) , (1b) (cid:18) J out R,ν ( E n ) J − NR,ν ( E n ) (cid:19) = S R ( E n ) (cid:18) J in R,ν ( E n ) δ n δ ν, ← J + NR,ν ( E n ) (cid:19) , (1c)where E n = E + neV are the energies of prop-agating modes with n being an integer, J ρ(cid:96),ν =( j e, ↑ ,ρ(cid:96),ν , j e, ↓ ,ρ(cid:96),ν , j h, ↑ ,ρ(cid:96),ν , j h, ↓ ,ρ(cid:96),ν ) T is the current amplitude vec-tor for region (cid:96) = L (left superconductor), N L (normalregion to the left of the tunnel barrier),
N R (normalregion to the right of the tunnel barrier), and R (rightsuperconductor) with ρ = + / − and ρ = in / out being theright/left-moving modes and incoming/outgoing modesindices, respectively, and ν = (cid:29) denoting whether theincoming quasiparticle is from the left or right supercon-ductor. We note that the scattering matrix formalismpresented above is completely general and can be uti-lized to study the transport properties of any kind of SNSjunctions with arbitrary topological properties (includingtrivial superconductors). Moreover, it can be easily in-terfaced with the numerical transport package Kwant ,which can be used to calculate the scattering matricesof the left ( S L ) and right ( S R ) NS interfaces . Fordetails on our numerical simulations, see the Appendix.The scattering matrix S N ( E n , E (cid:48) n ) in Eq. (1b) incor-porates the scattering processes at the tunnel barrier andthe increase (decrease) of the electron (hole) energy by eV each time the electron (hole) passes from the left tothe right. In terms of the electron ( S eN ) and hole ( S hN )component, it can be written as S N ( E n , E n (cid:48) ) = S eN ( E n , E n (cid:48) ) ⊗ σ ⊗ τ + + S hN ( E n , E n (cid:48) ) ⊗ σ ⊗ τ − , (2)where σ is the identity matrix in the spin subspace, τ ± = τ x ± iτ y are the Pauli matrices in the particle-holesubspace. The scattering matrices S eN and S hN are givenby S eN ( E n , E n (cid:48) ) = (cid:18) rδ n,n (cid:48) tδ n,n (cid:48) +1 tδ n,n (cid:48) − rδ n,n (cid:48) (cid:19) ,S hN ( E n , E n (cid:48) ) = (cid:18) r ∗ δ n,n (cid:48) t ∗ δ n,n (cid:48) − t ∗ δ n,n (cid:48) +1 r ∗ δ n,n (cid:48) (cid:19) , (3)where r = − iZ/ (1 + iZ ) and t = 1 / (1 + iZ ) are the re-flection and transmission coefficients, respectively, withthe amplitudes depending on the delta-function barrierstrength Z . Note that Z should be considered an ef-fective barrier strength determining the interface trans-parency represented by a delta-function potential whichis an unknown parameter in the theory (as in the well-known Blonder-Tinkham-Klapwijk (BTK) formalism ). In principle, Z can be calculated from first principles ifall information about the interface is available, but inpractice Z should be determined by comparing theoryand experiment. We change the junction transparencyin the simulation by tuning Z . Since sharp changes ofparameters across the junction, such as mismatch in theFermi level, spin-orbit coupling, p -wave pairing potentialetc., also effectively create barriers for the current, weuse a parameter-independent quantity G N to character-ize the junction transparency, where G N is the normal-ized conductance of the SNS junction at high voltages (inthe unit of G = e /h ), which is the conductance of thecorresponding normal-normal (NN) junction. We notethat G N , which subsumes Z and other possible unknownmicroscopic parameters, can be directly measured exper-imentally allowing experiment and theory to be quanti-tatively compared at arbitrary voltages. We refer to G N as the “junction transparency” in the rest of this papersince it denotes the conductance for the correspondingNN junction. Note that G N = 2 or G N = 1 denotes per-fect transparency (corresponding to Z =0) depending onthe specific tunnel junction one is considering.Solving the coupled linear equations [Eq. (1)], we ob-tain the current amplitudes J ρ(cid:96),ν . The total current canbe calculated by adding up the contribution from theleft- and right-moving modes of the electrons and holesfor the incoming quasiparticles from the left and rightsuperconductors, i.e., I ν ( V ) = 2 eh (cid:90) −∞ dE Tr (cid:32)(cid:88) n ρ z τ z J NL,ν ( E n ) J † NL,ν ( E n ) (cid:33) , (4)where J NL,ν = (cid:16) j e, ↑ , + NL,ν , j e, ↓ , + NL,ν , j h, ↑ , + NL,ν , j h, ↓ , + NL,ν ,j e, ↑ , − NL,ν , j e, ↓ , − NL,ν , j h, ↑ , − NL,ν , j h, ↓ , − NL,ν (cid:17) T (5)is the current amplitude vector in the normal region tothe left of the barrier. It was proven in Ref. that thecurrent is nonnegative for positive V . The differentialconductance ( G = dI/dV ) can be computed by directlydifferentiating the current I with respect to the voltage V . In general, we observe that the differential conduc-tance is particle-hole asymmetric except for sufficientlysmall transparencies.In this paper, we apply this scattering matrix formal-ism to calculate the zero-temperature dc current and con-ductance of junctions composed of spinful p SC (Sec. IV)and SOCSW (Sec. V), considering scenarios where none,one, or both of the superconductors are topologicallynontrivial.
III. SUBHARMONIC GAP STRUCTURE
In general, for SNS junctions with asymmetric gap(∆ L (cid:54) = ∆ R ), where ∆ L,R are the superconducting gap
TABLE I. Voltages at which the subharmonic gap structureappears for an asymmetric SNS junction.SGS voltage e | V | Range of n ∆ L /n n ≥ L + ∆ R ) / (2 n − n ≥ R /n n ≥ of the left and right superconductors, when the junctiontransparency is not small, there will be nonanalyticitiesin the I - V curve or conductance at specific voltages,which is termed the “subharmonic gap structure” (SGS).The sharp change in the conductance happens at voltagesat which there is a change in the number of Andreev re-flections required to transfer charge from the occupied tothe empty band. For incoming quasiparticles from theleft superconductor, this number of Andreev reflectionschanges when e | V | = ∆ L n , n ≥ , (6)and e | V | = ∆ L + ∆ R n − , ≤ n ≤ ∆ R ∆ R − ∆ L , (7)while for incoming quasiparticles from the right super-conductor, the change happens at voltages given inEq. (7) and e | V | = ∆ R n , ≤ n ≤ ∆ R ∆ R − ∆ L . (8)Without loss of generality, in the above we assume ∆ R > ∆ L . The range of n in Eqs. (6)-(8) gives the voltage rangefor “strong” SGS where all Andreev reflections happeninside the superconducting gap. The SGS that occursoutside this range of n is termed “weak” SGS because theAndreev reflections that happen outside the gap have, ingeneral, small amplitude.The SGS (including both weak and strong) happensat the voltages given in Table I (Refs. ), where ∆ L,R can be any of the gap values in the left and right su-perconductors when there are multiple superconductinggaps. In general, the SGS is not apparent for near-perfecttransparency junction and becomes sharper in the inter-mediate range of transparencies. Decreasing the trans-parency further into the tunneling limit will diminish theSGS at small voltages.
IV. SPINFUL p -WAVE SUPERCONDUCTORJUNCTIONS In this section, we consider junctions between an s -wave superconductor ( s SC) and a p SC as well as junc-tions between two p SCs, where the p SC can be topolog-ically trivial or nontrivial, depending on the strength ofthe chemical potential and Zeeman field (i.e., the system momentum , k E n e r g y , E (a) momentum , k E n e r g y , E momentum , k (b)(c) E n e r g y , E FIG. 2. (Color online) Energy spectrum of a spinful p SC fordifferent parameter regimes: (a) | V Z | < | µ p | and µ p < p -SC with no topological channel), (b) | V Z | > µ p and µ p > p -SC with one topological channel), (c) | V Z | < µ p and µ p > p -SC with two topological channels). is below or above the topological quantum phase transi-tion). Within the Bogoliubov-de Gennes (BdG) formal-ism, we can write the Hamiltonian of the system as H j ( x ) = 12 (cid:90) dx Ψ † j ( x ) H j Ψ j ( x ) , (9)where Ψ j ( x ) = (cid:16) ψ j ↑ ( x ) , ψ j ↓ ( x ) , ψ † j ↓ ( x ) , − ψ † j ↓ ( x ) (cid:17) T areNambu spinors, and ψ † jσ ( x ) and ψ jσ ( x ) are the creationand annihilation operators of an electron of spin σ for thesuperconductor of type j = s, p ( s -wave or p -wave). TheBdG Hamiltonians for the s SC and p SC are given by H s = (cid:18) − (cid:126) ∂ x m − µ s (cid:19) τ z + ∆ s τ x , (10a) H p = (cid:18) − (cid:126) ∂ x m − µ p (cid:19) τ z + V Z σ z − i ∆ p ∂ x τ x σ x , (10b)respectively. Here, m is the electron effective mass (forthe numerical simulations done in this paper, we set m = 0 . m e , which corresponds approximately to InSbnanowires , where m e is the bare electron mass), µ s and µ p are the chemical potentials of the s SC and p SC, V Z is the Zeeman field, ∆ s and ∆ p are the s SC and p SCpairing potentials, and τ x,y,z ( σ x,y,z ) are Pauli matricesacting in the particle-hole (spin) subspace. The effectivechemical potential in each spin channel of the p SC( µ p ± V Z ) determines whether that channel is topologicalor not. The channel is topological if its chemicalpotential is positive; otherwise, it is nontopological .The spinful p SC can have zero, one, or two topologicalchannels, depending on the values of V Z and µ p , i.e.,(a) | V Z | < | µ p | and µ p <
0, no topological channel,(b) | V Z | > µ p and µ p >
0, one topological channel,(c) | V Z | < µ p and µ p >
0, two topological channels.Throughout this paper, we denote the p SC in thesethree different regimes as p i , where i = 0 , , , refers tothe number of topological channels in the p SC. Sincethe spinful p SC is essentially made up of two uncoupledspinless p SCs, the spectrum of the spinful p SC thenconsists of the spectrum of two spinless p SCs witheffective chemical potential µ p ± V Z as shown in Fig. 2.In the following we will denote the smallest gap in thespectrum of the p i -SC by ∆ p i . A. sNp junction We begin by considering the s -wave superconductor–normal metal– p superconductor ( sN p ) junction. The p -SC is a spinful p -wave superconductor with no topo-logical channel: it has negative chemical potential ( µ p <
0) and small Zeeman field | V Z | < | µ p | . Its spectrum has agap at k = 0 with value | µ p |±| V Z | where the smallest gapis ∆ p = | µ p | − | V Z | as shown in Fig. 2. In general,the current and conductance for SNS junctions involving p -SC, e.g., the sN p junction discussed here, increasewith the p -SC pairing potential ∆ p . Since the p -SCis essentially an insulator, the current for this junctionis generally small and the SGS is strongly suppressed ascan be seen in Fig. 3. At high voltages ( | V | (cid:29) ∆ s , ∆ p ),the conductance approaches the conductance G N of thecorresponding NN junction (which we define as the junc-tion transparency throughout this paper). The currentand conductance decrease with decreasing junction trans-parency G N as can be seen in Fig. 3. In the weak tun-neling or small transparency limit where MAR is sup-pressed, the current starts to flow only when the voltageis e | V | = ∆ s + ∆ p , i.e., the voltage where the supercon-ducting gap edges of both s SC and p -SC line up. B. sNp junction The p -SC has one topological channel with a pair ofMZM: one at each end of a finite wire. The energy spec-trum of the p -SC is given in Fig. 2(b). The plots of thecurrent and conductance for the sN p junction (i.e., withone isolated MZM in the junction) in the limit of largeand small Zeeman field are plotted against bias voltagein Figs. 4 and 5, respectively. The conductance plotsfor the sN p junction have already been given in Ref. ;we include them here for completeness and, more impor-tantly, a comparison with other SNS junctions. In thelarge Zeeman limit [( | V Z | − µ p ) (cid:38) µ p ], the p -SC is effec-tively a spinless topological p SC . In this limit, MARare totally suppressed and only single Andreev reflections G N = G N = G N = G N = G N = (a)(b) C o ndu c t a n ce , G / G C u rr e n t , I / ( ∆ m i n G G N / e ) Voltage, eV / ∆ s Voltage, eV / ∆ s FIG. 3. (Color online) Plots of (a) dc current I and (b)normalized differential conductance G/G versus bias voltage V for an sNp junction with various values of transparen-cies G N . The parameters used for the s SC are µ s = 20 Kand ∆ s = 0 .
01 K. The parameters used for the p -SC are µ p = − .
01 K, V Z = 0 K, ∆ p = 0 . p = 0 .
01 K. The smallest gap in the junction is ∆ min = 0.01 K. are allowed for the sN p junction because the s SC allowsonly spin-singlet Andreev reflections, while the spinless p SC allows only spin-triplet Andreev reflections. This re-sults in a step jump in the conductance from zero to thequantized value G M = (4 − π )2 e /h at the threshold volt-age e | V | = ∆ s as shown in Fig. 3(b). The quantizedvalue G M corresponds to the conductance due to a singleAndreev reflection from the MZM which happens at thevoltage when the BCS singularity and MZM are aligned.In this large Zeeman limit, since MAR are suppressed,the quantized value G M is robust against the junctiontransparency. The conductance, in general, decreaseswith decreasing junction transparency and for sufficientlysmall transparency, the conductance can become nega-tive for voltages near the threshold voltage e | V | = ∆ s .Our results for the sN p junction in the large Zeemanlimit, calculated using the scattering matrix formalism,are similar to those of the s -wave superconductor–normalmetal–spinless p -wave superconductor junctions calcu-lated using the Green’s function formalism . Recently,the conductance of the spinless p -wave superconductor G N = G N = G N = G N = G N = C u rr e n t , I / ( ∆ m i n G G N / e ) C o ndu c t a n ce , G / G Voltage, eV / ∆ s Voltage, eV / ∆ s (a)(b) FIG. 4. (Color online) Plots of (a) dc current I and (b) nor-malized differential conductance G/G versus bias voltage V for an sNp junction with various values of transparencies G N in the limit of large Zeeman field ( V Z = 2 µ p ). The reddashed line at G M = (4 − π )2 e /h is the conductance valuedue to a single Andreev reflection from the MZM. The pa-rameters used for the s SC are µ s = 200 K and ∆ s = 2 . p -SC are µ p = 20 K, V Z = 40K, ∆ p = 0 . p = 4 K.The smallest gap in the junction is ∆ min = 2.5 K. has been measured using an s -wave superconducting tipin a scanning tunneling microscopy experiment . Thereported results are qualitatively consistent with our the-oretical findings. In the limit of small Zeeman field[( | V Z |− µ p ) (cid:28) µ p ], and when the junction transparency isnot small, MAR are allowed. As a result, there is a finitecurrent and conductance with SGS below the thresholdvoltage e | V | = ∆ s . However, the current and conduc-tance near zero voltage are zero due to the differencein the Andreev reflection spin-selectivity of the s SC andMZM, i.e., the s SC allows spin-singlet Andreev reflectionand the MZM favors spin-triplet Andreev reflection .In this limit, due to MAR, the conductance at the voltage e | V | = ∆ s is no longer robust against increasing junc-tion transparency. The current and conductance gener-ally decrease with decreasing junction transparency. Forsufficiently small transparency that only single Andreevreflection contributes, we recover G ( e | V | = ∆ s ) = G M . G N = G N = G N = G N = G N = (a)(b) C u rr e n t , I / ( ∆ m i n G G N / e ) C o ndu c t a n ce , G / G Voltage, eV / ∆ s Voltage, eV / ∆ s FIG. 5. (Color online) Plots of (a) dc current I and (b) nor-malized differential conductance G/G versus bias voltage V for an sNp junction with various values of transparencies G N in the limit of small Zeeman field ( V Z = 1 . µ p ). The reddashed line at G M = (4 − π )2 e /h is the conductance valuedue to a single Andreev reflection from the MZM. The pa-rameters used for the s SC are µ s = 200 K and ∆ s = 2 . p -SC are µ p = 20 K, V Z = 22K, ∆ p = 0 . p -SC being ∆ p = 2 K. The smallestgap in the junction is ∆ min = 2 K. C. sNp junction The p -SC has two topological channels, and thus twoMZMs at each end of a finite wire. The energy spectrumfor the p -SC is shown in Fig. 2(c). The current andconductance plots for the sN p junction are depicted inFig. 6. In the tunneling limit, the conductance for the sN p junction develops a step jump from 0 to 2 G M atthe threshold voltage e | V | = ∆ s due to single Andreevreflections from a Majorana Kramers pair, with each spinchannel contributing a conductance of G M . For large orintermediate transparencies, due to MAR, the conduc-tance at e | V | = ∆ s is no longer quantized at 2 G M andthere is an SGS in the current and conductance profile.In contrast to the sN p junction where the current andconductance are zero near zero voltage, when the trans-parency is not small the current and conductance for the sN p junction is nonzero near zero voltage. This is be- G N = G N = G N = G N = G N = Voltage, eV / ∆ s Voltage, eV / ∆ s C o ndu c t a n ce , G / G C u rr e n t , I / ( ∆ m i n G G N / e ) (a)(b) FIG. 6. (Color online) Plots of (a) dc current I and (b) nor-malized differential conductance G/G versus bias voltage V for an sNp junction with various values of transparencies G N . The red dashed line at 2 G M = (4 − π )4 e /h is the con-ductance value due to a single Andreev reflection from twoMZMs. The parameters used for the s SC are µ s = 20 Kand ∆ s = 0 .
01 K. The parameters used for the p -SC are µ p = 20 K, V Z = 0 K, ∆ p = 2 × − eV˚A, where the gapis ∆ p = 6 . × − K. The smallest gap in the junction is∆ min = ∆ p = 6 . × − K. cause unlike the case of the sN p junction where there isonly one MZM which facilitates the spin-triplet Andreevreflection in one spin channel, there are two MZMs in sN p junctions facilitating Andreev reflections in bothspin channels. As a result, the MAR are not suppressednear zero voltage. The SGS associated with MAR devel-ops at specific voltages as given in Table I. Similar to theconventional s -wave superconductor–normal– s -wave su-perconductor junction , in the perfectly transpar-ent limit ( G N = 2), the current at small voltages for the sN p junction asymptotically approaches I ( V →
0) = 4 e ∆ min h , (11)which corresponds to the transfer of a charge of 2 e acrossthe junction where ∆ min = min(∆ s , ∆ p ) is the smallestgap in the junction. D. p Np junction G N = G N = G N = G N = G N = (a)(b) C o ndu c t a n ce , G / G C u rr e n t , I / ( ∆ m i n G G N / e ) Voltage, eV / ∆ min Voltage, eV / ∆ min FIG. 7. (Color online) Plots of (a) dc current I and (b)normalized differential conductance G/G versus bias volt-age V for a p Np junction with various values of trans-parencies G N . The parameters used for both p -SCs are µ p = 20 K, V Z = 0 K, ∆ p = 2 × − eV˚A, where the gapis ∆ p = 6 . × − K. The smallest gap in the junction is∆ min = 6 . × − K. For a p N p junction, both superconductors have twotopological channels with two MZMs at each end (4MZMs in the junction). The plots of the current andconductance for this junction are depicted in Fig. 7. Inthe perfectly transparent limit ( G N = 2), the current atsmall voltages asymptotically approaches I ( V →
0) =4 e ∆ min /h , where ∆ min is the smallest gap in the junc-tion. This asymptotic value of the dc current is the sameas the value obtained for the conventional s -wave-normal- s -wave superconductor junction . As V →
0, thecurrent in the p N p junction is transferred via a Ma-jorana Kramers pair where each of the MZMs transfersa charge of unit e giving a total charge of 2 e , the sametotal amount of charge as that carried by a Cooper pair.As a result, the current I ( V →
0) is the same as that forthe conventional SNS junction . Away from perfecttransparency ( G N (cid:54) = 2), the dc current approaches zeroas the voltage approaches zero.The SGS associated with the MAR develops at specificvoltages given in Table. I where for the p N p junctionwith symmetric gaps, the voltages are | V | = ∆ p /en asshown in Fig. 7. The SGS is suppressed in the tunnel-ing limit and the current becomes nonzero only whenthe voltage is above the threshold voltage | V | = ∆ p /e ,i.e., when the quasiparticles have sufficient energy to un-dergo single Andreev reflections from the MZMs. Thisis contrary to the case of the junction between two non-topological superconductors where the tunneling currentcan flow only when the voltage is above | V | = 2∆ /e ,i.e., when the gap edge of the unoccupied band lines upwith that of the occupied band. Since the p -SC does nothave a BCS singularity, the conductance at | V | = ∆ p /e in the tunneling limit is not quantized at G M . Instead, ithas a nonuniversal value, which decreases with decreasingjunction transparency. E. p Np junction G N = G N = G N = G N = G N = (a)(b) C u rr e n t , I / ( ∆ m i n G G N / e ) C o ndu c t a n ce , G / G Voltage, eV / ∆ min Voltage, eV / ∆ min FIG. 8. (Color online) Plots of (a) dc current I and (b) nor-malized differential conductance G/G versus bias voltage V for a p Np junction with various values of transparencies G N . The parameters used for the p -SC are µ p = 20 K,∆ p = 2 × − eV˚A, and V Z = 40 K, where the gap is∆ p = 0.011 K. The parameters for the p -SC are µ p = 20K, ∆ p = 2 × − eV˚A, V Z = 0 K, where the gap is∆ p = 6 . × − K. The smallest gap in the junction is∆ min = ∆ p = 6 . × − K. The current and conductance for a p N p junctionare depicted in Fig. 8. For the p N p junction inthe perfectly transparent limit ( G N = 1), the currentnear zero voltage approaches I ( V →
0) = 2 e ∆ min /h ,which is half of the current for the p N p or s -wavesuperconductor − normal metal − s -wave superconductorjunction. The reason is that the p -SC has only one MZMwhich can transfer a charge in the unit of e in one spinchannel. The SGS appears at voltages given in Table I.Since the p N p junction considered here has asymmetricgap, the current and conductance in the weak tunnelinglimit develop jumps at the voltages | V | = ∆ p /e and | V | = ∆ p /e (which correspond to the voltages wherethe MZMs are aligned with the p -SC and p -SC super-conducting gap edge). The conductance values at thesejumps have nonuniversal values, which decrease with thejunction transparency. F. p Np junction C o ndu c t a n ce , G / G C u rr e n t , I / ( ∆ s G G N / e ) (a)(b) Voltage, eV / ∆ min Voltage, eV / ∆ min G N =1.0 G N =0.829 G N =0.495 G N =0.08 G N =0.009 FIG. 9. (Color online) Plots of (a) dc current I and (b)normalized differential conductance G/G versus bias volt-age V for a p Np junction with various values of trans-parencies G N . The parameters used for both p -SCs are µ p = 20 K, ∆ p = 2 × − eV˚A, and V Z = 40 K wherethe gap is ∆ p = 0.011 K. The smallest gap in the junctionis ∆ min = ∆ p = 0 .
011 K.
Figure 9 displays the current and conductance plotsfor a p N p junction. Similar to the p N p junction,for a perfectly transparent p N p junction ( G N = 1)the current at small voltages asymptotically approaches I ( V →
0) = 2 e ∆ min /h . This is due to the fact that acharge of e is transferred between the MZMs on bothsides of the junction. For a symmetric p N p as consid-ered here, the SGS develops at voltages | V | = ∆ p /ne .In the weak-tunneling limit, there is a step jump in theconductance at | V | = ∆ p /e to a nonuniversal value thatdecreases as the junction transparency decreases. G. p Np junction G N = G N = G N = G N = G N = (a)(b) C u rr e n t , I / ( ∆ m i n G G N / e ) C o ndu c t a n ce , G / G Voltage, eV / ∆ p Voltage, eV / ∆ p FIG. 10. (Color online) Plots of (a) dc current I and (b)normalized differential conductance G/G versus bias voltage V for a p Np junction with various values of transparencies G N . The parameters used for the p -SC are µ p = − .
01 K, V Z = 0 K, ∆ p = 0 . p = 0 .
01 K.The parameters used for the p -SC are µ p = 20 K, V Z = 0 K,∆ p = 2 × − eV˚A, where the gap is ∆ p = 6 . × − K.The smallest gap in the junction is ∆ min = 6 . × − K. The current and conductance plots for the p N p junc-tion are given in Fig. 10. The MAR for this junction aresuppressed since p is essentially an insulator. There isa conductance peak at | V | = ∆ p /e , which correspondsto a single Andreev reflection from the MZMs. However, unlike the case of the sN p junction, the tunneling con-ductance at the threshold voltage | V | = ∆ p /e assumesa nonquantized value which decreases with decreasingjunction transparency. We note again that the MZMtunneling conductance quantization G M = (4 − π )2 e /h holds only if the superconducting probe has a BCS sin-gularity (as derived in Ref. ). H. p Np junction G N =0.266 G N =0.199 G N =0.113 G N =0.018 G N =0.002 C u rr e n t , I / ( ∆ m i n G G N / e ) C o ndu c t a n ce , G / G (a)(b) Voltage, eV / ∆ p Voltage, eV / ∆ p Voltage, eV / ∆ p Voltage, eV / ∆ p FIG. 11. (Color online) Plots of (a) dc current I and (b)normalized differential conductance G/G versus bias voltage V for a p Np junction with various values of transparencies G N . The parameters used for the p -SC are µ p = − .
01 K, V Z = 0 K, ∆ p = 0 . p = 0 .
01 K.The parameters used for the p -SC are µ p = 20 K, V Z = 40K, ∆ p = 2 × − eV˚A, where the gap is ∆ p = 0 .
011 K. Thesmallest gap in the junction is ∆ min = 0 .
01 K.
The current and conductance plots for the p N p junc-tion are given in Fig. 11. The conductance for this junc-tion looks similar to those of the p N p junction. TheMAR for this junction are suppressed and in the tunnel-ing limit, the conductance has a step jump at the thresh-old voltage e | V | = ∆ p to a nonquantized value, whichdecreases with decreasing junction transparency.0 I. p Np junction G N =0.4027 G N =0.0651 G N =0.0253 G N =0.0036 G N =0.0004 C u rr e n t , I / ( ∆ m i n G G N / e ) (a)(b) C o ndu c t a n ce , G / G Voltage, eV / ∆ min Voltage, eV / ∆ min FIG. 12. (Color online) Plots of (a) dc current I and (b)normalized differential conductance G/G versus bias voltage V for a p Np junction with various values of transparencies G N . The parameters used for both p -SCs are µ p = − . V Z = 0 K, ∆ p = 0 . p = 0 .
01 K.The smallest gap in the junction is ∆ min = 0 .
01 K.
For the p N p junction, the plots of the current andconductance versus the bias voltage are displayed inFig. 12. Since the p N p junction is essentially a junctionbetween two insulators, the current and conductance forthis junction are generally small and MAR are stronglysuppressed. In the limit of small transparencies, the cur-rent and conductance for a symmetric p N p junctionrise to a nonzero value at e | V | = 2∆ p , i.e., when thedensity-of-state singularity of the occupied band of one p -SC is aligned with the singularity of the empty bandof the other p -SC. V. SPIN-ORBIT-COUPLEDSUPERCONDUCTING WIRE JUNCTIONS
Pure spinless or spinful p -wave topological supercon-ductors as considered above do not exist in nature al-though they could be approximate models for some realsystems. It is, however, known that effective 1D or 2D (a) (b)(c) (d)momentum , k momentum , k E n e r g y , E E n e r g y , E FIG. 13. (Color online) Energy spectrum of SOCSW for dif-ferent parameter regimes: (a) V Z = 0 (nontopological), (b) V Z < (cid:112) µ + ∆ (nontopological), (c) V Z = (cid:112) µ + ∆ (tran-sition), (d) V Z > (cid:112) µ + ∆ (topological). topological superconductors closely mimicking spinlesstopological superconductors can be artificially engineeredby combining s -wave superconductivity with spin-orbitcoupling and Zeeman splitting . We therefore nowconsider a more physically realistic model for topologicalsuperconductors, namely, a spin-orbit-coupled 1D semi-conducting nanowire placed in proximity to an s -wave su-perconductor in the presence of magnetic field . The s -wave superconductor induces superconductivity in thenanowire through proximity effect, and this proximity-induced nanowire superconductivity is converted to topo-logical superconductivity by the Zeeman splitting in thenanowire (provided it is large enough to overcome the in-duced s -wave superconductivity) in the presence of spin-orbit coupling. The BdG Hamiltonian for the SOCSWis H SOCSW = (cid:18) − (cid:126) ∂ x m − µ (cid:19) τ z − iα∂ x τ z σ y + V Z σ x + ∆ τ x , (12)where µ is the chemical potential of the nanowire, α isthe strength of the spin-orbit coupling, V Z is the Zeemanfield, and ∆ is the proximity-induced s -wave pairing po-tential. The Hamiltonian above is written in the samebasis as that in Eq. (9). The SOCSW can be tuned fromthe nontopological to the topological regime by simplychanging the Zeeman field V Z or chemical potential µ .The critical value V Z = (cid:112) µ + ∆ marks the topologicalquantum phase transition between the topologically triv-ial ( V Z < (cid:112) µ + ∆ ) and topologically nontrivial phase( V Z > (cid:112) µ + ∆ ). In the topological regime, there isone MZM at each end of the nanowire (if the wire is longenough with well-separated MZMs, the system is in thetopologically protected regime). The BdG spectrum ofthe SOCSW is given in Fig. 13. In what follows, we aregoing to denote the minimum gap in the SOCSW spec-1trum by ∆ SOCSW . Below we calculate the current andconductance of several SNS junctions between two SOC-SWs where the SOCSW can be either in the nontopolog-ical or topological regime. The results given in the sub-sections below are our most relevant theoretical resultsfor the currently ongoing MZM experiments in the liter-ature, which mostly involve semiconductor nanowires.
A. Nontopological–nontopological SOCSWjunction G N = G N = G N = G N = G N = Voltage, eV / ∆ nontopoSOCSW Voltage, eV / ∆ nontopoSOCSW Voltage, eV / ∆ nontopoSOCSW C u rr e n t , I / ( ∆ m i n G G N / e ) C o ndu c t a n ce , G / G (a)(b) FIG. 14. (Color online) Plots of (a) dc current I and (b)normalized differential conductance G/G versus bias voltage V for a nontopological–nontopological SOCSW junction withvarious values of transparencies G N and no Zeeman field. Theparameters used for both SOCSWs are µ = 0 K, V Z = 0 K,∆ = 0 .
01 K, α = 0 . nontopoSOCSW = 0.01K. The smallest gap in the junction is ∆ min = 0 .
01 K.
In this subsection, we consider the junction betweentwo SOCSWs where both of them are in the nontopo-logical regime (i.e., V Z < (cid:112) µ + ∆ ). As shown inFig. 14, the current and conductance of this junctionwith no Zeeman field ( V Z = 0) is the same as that ofan s -wave superconductor–normal metal– s -wave super-conductor junction . The SGS for the symmetricnontopological–nontopological SOCSW junction occursat voltages | V | = 2∆ nontopoSOCSW /ne . For a perfectly trans- parent junction ( G N = 2), the current at small voltagesapproaches the value I ( V →
0) = 4 e ∆ min h . (13)In the limit of small transparency the current and con-ductance develop a step jump at | V | = 2∆ nontopoSOCSW /e forjunctions with symmetric gaps.Figure 15 shows the current and conductance for thenontopological–nontopological SOCSW junction in thepresence of Zeeman field. Increasing the Zeeman fieldsmooths out the SGS. In the limit of small transparencies,the conductance has a smooth rise from zero instead ofa step jump at the threshold voltage. G N = G N = G N = G N = G N = C o ndu c t a n ce , G / G (b) C u rr e n t , I / ( ∆ m i n G G N / e ) (a) Voltage, eV / ∆ nontopoSOCSW Voltage, eV / ∆ nontopoSOCSW FIG. 15. (Color online) Plots of (a) dc current I and (b)normalized differential conductance G/G versus bias voltage V for a nontopological–nontopological SOCSW junction withvarious values of transparencies G N and finite Zeeman field.The parameters used for both SOCSWs are µ = 0 K, V Z =0 .
002 K, ∆ = 0 .
01 K, α = 0 . nontopoSOCSW = 0.008 K. The smallest gap in the junction is ∆ min = 0 . B. Nontopological–topological SOCSW junction
Here, we consider junctions between a nontopologicaland a topological SOCSW. The current and conductance2 G N = G N = G N = G N = G N = C u rr e n t , I / ( ∆ m i n G G N / e ) C o ndu c t a n ce , G / G Voltage, eV / ∆ nontopoSOCSW Voltage, eV / ∆ nontopoSOCSW (a)(b) FIG. 16. (Color online) Plots of (a) dc current I and (b) nor-malized differential conductance G/G versus bias voltage V for a nontopological–topological SOCSW junction with var-ious values of transparencies G N . The red dashed line at G M = (4 − π )2 e /h is the conductance value due to a sin-gle Andreev reflection from the MZM. The nontopologicalSOCSW is not subjected to any Zeeman field and the topolog-ical superconductor has a small Zeeman field. The parametersused for the nontopological SOCSW are µ = 0 K, V Z = 0K, ∆ = 0 . α = 0 . nontopoSOCSW = 0 . µ = 0 K, V Z = 15 . = 10 . α = 0 .
05 eV˚A, where the gapis ∆ topoSOCSW = 0 .
75 K. The smallest gap in the junction is∆ min = 0 . for such junctions are given in Figs. 16-18. We first con-sider the case of the junction with the nontopologicalSOCSW having no Zeeman field where the energy spec-trum for this nontopological SOCSW has the minimumgap at the Fermi momentum with a BCS singularity [asshown in Fig. 13(a)]. As shown in Figs. 16 and 17, theconductance in the tunneling limit for this junction de-velops a step jump from 0 to G M = (4 − π )2 e /h atthe gap-bias voltage e | V | = ∆ nontopoSOCSW similar to the caseof sN p junction. This quantized value G M is due toa single Andreev reflection from the MZM of an elec-tron coming from the gap edge with a BCS singularity.In the limit where the Zeeman field in the topologicalSOCSW is small, for intermediate and large transparen-cies, there are MAR and the conductance below the volt-age e | V | = ∆ nontopoSOCSW is nonzero except for small voltages G N = G N = G N = G N = G N = Voltage, eV / ∆ nontopoSOCSW Voltage, eV / ∆ nontopoSOCSW C u rr e n t , I / ( ∆ m i n G G N / e ) (a) C o ndu c t a n ce , G / G (b) FIG. 17. (Color online) Plots of (a) dc current I and (b) nor-malized differential conductance G/G versus bias voltage V for a nontopological–topological SOCSW junction with var-ious values of transparencies G N . The red dashed line at G M = (4 − π )2 e /h is the conductance value due to a sin-gle Andreev reflection from the MZM. The nontopologicalSOCSW is not subjected to any Zeeman field and the topolog-ical superconductor has a large Zeeman field. The parametersused for the nontopological SOCSW are µ = 0 K, V Z = 0K, ∆ = 0 . α = 0 . nontopoSOCSW = 0 . µ = 0 K, V Z = 60 . = 10 . α = 0 .
05 eV˚A, where the gapis ∆ topoSOCSW = 0 .
42 K. The smallest gap in the junction is∆ min = 0 .
42 K. (see Fig. 16). Near zero voltage, the current and conduc-tance vanish due to a mismatch in the Andreev reflec-tion spin-selectivity between the nontopological SOCSWand the MZM. In the limit of large Zeeman field in thetopological SOCSW, where the MAR are suppressed andonly single Andreev reflections are allowed, the conduc-tance for this junction develops a step jump from 0 to G M = (4 − π )2 e /h independent of the junction trans-parency. We note that this result is similar to the casewhere the nontopological SOCSW is replaced by an s -wave superconductor .For the case where there is Zeeman field in the non-topological superconductor, the gap edge of the super-conductor no longer has the BCS singularity. As a re-sult, the MZM tunneling conductance measured using3 G N = G N = G N = G N = G N = Voltage, eV / ∆ nontopoSOCSW Voltage, eV / ∆ nontopoSOCSW C u rr e n t , I / ( ∆ m i n G G N / e ) C o ndu c t a n ce , G / G (a)(b) FIG. 18. (Color online) Plots of (a) dc current I and (b) nor-malized differential conductance G/G versus bias voltage V for a nontopological–topological SOCSW junction with vari-ous values of transparencies G N . The nontopological SOCSWhas a finite Zeeman field and the topological superconductorhas a small Zeeman field. Note that the MZM tunneling con-ductance is not quantized at G M = (4 − π )2 e /h . The pa-rameters used for the nontopological SOCSW are µ = 0 K, V Z = 0 . = 0 . α = 0 . nontopoSOCSW = 0 . µ = 0 K, V Z = 15 . = 10 . α = 0 .
05 eV˚A, wherethe gap is ∆ topoSOCSW = 0 .
75 K. The smallest gap in the junctionis ∆ min = 0 . this nontopological superconductor will not be quantizedat G M for the gap-bias voltage e | V | = ∆ nontopoSOCSW . Instead,the tunneling conductance assumes a nonuniversal valuewhich decreases with decreasing junction transparency asshown in Fig. 18. C. Topological–topological SOCSW junction
The current and conductance plots for a topological–topological SOCSW junction are shown in Fig. 19. Ourresults for this junction, calculated using the scatteringmatrix formalism, are identical to previous results for thesame SNS junction calculated using a Green’s functionmethod and similar to the results obtained in Ref. for a topological Josephson junction between supercon-ductors connected through the helical edge states of a 2D topological insulator in the presence of a magneticbarrier.Similar to the p N p junction, in the limit of perfecttransparency ( G N = 1), the current for a topological–topological SOCSW junction asymptotically approaches I ( V →
0) = 2 e ∆ min h , (14)which is half the value of the current in the conventionalSNS junction. The reason is because there is only oneMZM at both sides of the junction which transfer chargesin unit of e . The SGS for this junction happens at volt-ages | V | = ∆ min /ne . In the weak tunneling limit, thereis a step jump in the conductance at | V | = ∆ min /e . Wenote, however, that since there is no BCS singularity inthe superconducting lead, the conductance at the voltage | V | = ∆ min /e is not quantized at G M = (4 − π )2 e /h . G N = G N = G N = G N = G N = C u rr e n t , I / ( ∆ m i n G G N / e ) C o ndu c t a n ce , G / G Voltage, eV / ∆ min Voltage, eV / ∆ min (a) (b)(a) (b)(a) (b)(a)(b) FIG. 19. (Color online) Plots of (a) dc current I and (b)normalized differential conductance G/G versus bias voltage V for a topological–topological SOCSW junction with variousvalues of transparencies G N . The parameters used for bothSOCSWs are µ = 0 K, V Z = 15 K, ∆ = 1 .
17 K, α = 0 . topoSOCSW = 0 .
01 K. The smallest gapin the junction is ∆ min = 0 .
01 K. G N =1.161 G N =0.398 G N =0.156 G N =0.015 G N =0.001 eV/ ∆ s G / G G N =1.896 G N =0.625 G N =0.213 G N =0.021 G N =0.002 G N =1.343 G N =0.419 G N =0.153 G N =0.014 G N =0.001 eV/ ∆ s G / G s SC topo non-topo SOCSW eV/ ∆ s G / G Voltage, eV/ ∆ s Voltage, eV/ ∆ s Voltage, eV/ ∆ s C o ndu c t a n ce , G / G C o ndu c t a n ce , G / G C o ndu c t a n ce , G / G E n e r g y [ K ] Chemical potential, µ topo [K](a)(b) (c)(d) (e) (c) (d) (e) FIG. 20. (Color online) (a) Schematic diagram of an s SC–SOCSW junction with a pair of ABS (one at each end of the topologicalregion). The chemical potential of the topological and nontopological regions are | µ topo | < (cid:112) V Z − ∆ and | µ nontopo | > (cid:112) V Z − ∆ , respectively. The parameters used for the s SC are µ s = 50 K, and ∆ s = 0.67 K. The SOCSW parameters are µ nontopo = 211.18 K, V Z = 15 K, ∆ = 10 K, α = 0.05 eV˚A, and length of the topological region, L topo = 0 . µ m. We usea dissipation term i Γ τ ⊗ σ in the BdG Hamiltonian of both the left and right superconductors with a dissipation strengthΓ = 0 . K to broaden the van-Hove singularity. (b) The energy of the Andreev bound state closest to zero energy versusthe chemical potential µ topo in the topological region. The red, green and purple dots indicate the value of the topologicalchemical potential used in (c),(d), and (e), respectively. Normalized differential conductance G/G for the SOCSW for severalchemical potential values in the topological region: (c) µ topo = 0 K, (d) µ topo = 1.697 K, and (e) µ topo = 4.5 K. Inset: theABS conductance in the weak tunneling limit which is the conductance for the smallest transparency in the main plot. VI. ANDREEV BOUND STATES
In this section we compare the conductance of an MZMwith that of an ABS. We mention that the possible ex-istence of ABS in the system can never be ruled out apriori , and it is therefore important to take into accounttheir possible effects on transport properties. In particu-lar, we consider the ABS that may arise in the SOCSWmodel with a finite topological region and a semi-infinitenontopological region as shown in the right side of theSNS junction in Fig. 20(a). This model can happen nat-urally in an SOCSW with varying chemical potential,where the chemical potential varies from the topologi-cal regime to the nontopological regime resulting in thedomain walls between the topological and nontopolog-ical regions . The ABSs can be found at the end ofthe topological region. For simplicity, here we considera step jump in the chemical potential in going from thetopologically nontrivial ( | µ | < (cid:112) V Z − ∆ ) to the topo-logically trivial value ( | µ | > (cid:112) V Z − ∆ ) keeping all theother parameters in these two regions to be the same. The ABS closest to zero energy in this model has energyoscillating with the chemical potential in the topologi-cal region as shown in Fig. 20(b), where the zero-energyABS can be found at specific values of system parame-ters . In this section, we compare the conductance ofan MZM with that of an ABS. We mention that thepossible existence of ABS in the system can never beruled out a priori , and it is therefore important to takeinto account their possible effects on transport proper-ties. In particular, we consider the ABS that may arisein the SOCSW model with a finite topological region anda semi-infinite nontopological region as shown in the rightside of the SNS junction in Fig. 20(a). This model canhappen naturally in an SOCSW with varying chemicalpotential where the chemical potential varies from thetopological regime to the nontopological regime resultingin the domain walls between the topological and non-topological regions . The ABSs can be found at the endof the topological region. For simplicity, here we considera step jump in the chemical potential in going from thetopologically nontrivial ( | µ | < (cid:112) V Z − ∆ ) to the topo-5logically trivial value ( | µ | > (cid:112) V Z − ∆ ) keeping all theother parameters in these two regions the same. The ABSclosest to zero energy in this model has energy oscillatingwith the chemical potential in the topological region asshown in Fig. 20(b), where the zero-energy ABS can befound at specific values of system parameters .We consider this SOCSW in a junction with an s -wavesuperconducting lead. To calculate the conductance here,we first introduce a dissipation term − i Γ τ ⊗ σ intothe BdG Hamiltonian. The dissipation term is used tobroaden the van Hove singularity of the BdG spectrum,so that we do not need to use a very fine energy grid inthe numerical calculation. This dissipation term has beenused previously to calculate conductance in topologicalNS junctions , though for different reasons. Our usinga dissipation here could either be physically motivated asin Ref. or simply a technical artifice in handling the vanHove singularity. Figures 20(c)-(e) show the conductanceof the SOCSW calculated for several chemical potentialvalues in the topological region with all other parame-ters the same. The conductance for the zero-energy ABSmay resemble the MZM tunneling conductance, i.e., ithas a sharp rise at the voltage e | V | = ∆ s to a peakwith a value near G M = (4 − π )2 e /h (see the inset inFig. 20(d) or Ref. ). One needs to be careful, therefore,in interpreting experimental data since accidental near-zero-energy ABS would produce tunneling conductancesignatures quite similar to MZM themselves. For nonzeroenergy ABS, the ABS tunneling conductance peak shiftsaway from the threshold voltage e | V | = ∆ s (where ∆ s isthe s -wave superconducting gap) toward a larger voltagevalue by the ABS energy normalized by the tunnel cou-pling between the lead and the system; see Figs. 20(c)and (e). VII. CONCLUSION
In this paper, we have calculated the zero-temperaturedc current and conductance in various 1D voltage-biasedSNS junctions involving topological and nontopologicalsuperconductors, considering both ideal spinful p -waveand realistic spin-orbit-coupled s -wave superconductingwires. For junctions with small transparencies, the pres-ence of an MZM gives rise to a jump in the current andconductance at the gap-bias voltage e | V | = ∆ lead wherethe superconducting gap edge is aligned with the MZM.If the superconducting lead has a BCS singularity at thegap edge then the tunneling conductance at the gap-biasvoltage takes the value G M = (4 − π )2 e /h due to a singleAndreev reflection from the MZM. However, this quan-tization no longer holds if the superconducting lead gapedge does not have the BCS singularity, e.g., p -wave su-perconductor or SOCSW with finite magnetic field. ForSNS junctions where both of the superconductors aretopological (i.e., with one or two MZMs at each end),there is SGS in the I - V curve or conductance profile dueto MAR. However, for nontopological–topological super- conductor junctions where the topological superconduc-tor has only one MZM at each end, the SGS at smallvoltages is suppressed due to the mismatch in Andreevreflection spin-selectivity of the superconducting lead andthe MZM.In contrast to the conventional SNS junction, whereCooper pairs are transferred across the junction with acharge of 2 e , for the topological SNS junction, the chargeis transferred via the MZM in the units of e . As a re-sult, for a perfectly transparent junction with an MZMat each end, the MZM contributes to a near zero-voltagecurrent I ( V →
0) = 2 e ∆ min /h where ∆ min is the small-est gap in the junction. We note that this MZM near-zero voltage current is by no means universal or quan-tized because of the generic presence of the gap ∆ min which surely varies from junction to junction. The sameis also true for the case where there are two MZMs onone side and one MZM on the other side. This near zero-voltage dc current is half of the value for the conven-tional s -wave superconductor–normal– s -wave supercon-ductor junction. However, for the case where there aretwo MZMs on both sides of the junction, the near zero-voltage current is I ( V →
0) = 4 e ∆ min /h because eachMZM can exchange a charge of e between each other.For the case where there is a conventional s -wave super-conductor on one side and one MZM on the other sideof the junction, the current is zero because of the dif-ference in the Andreev-reflection spin selectivity of the s -wave superconductor and MZM, i.e., the s -wave super-conductor allows only opposite-spin Andreev reflectionsand MZM favors equal-spin Andreev reflections. How-ever, for the junction between a conventional s -wave su-perconductor and a Majorana Kramers pair the near-zerocurrent for a perfect transparent junction is not zero butit is I ( V →
0) = 4 e ∆ min /h . This is due to the fact thatthe MZM pair can facilitate Andreev reflections in bothspin channels.We also calculated the conductance with an ABS inthe SOCSW model arising from a finite topological anda semi-infinite nontopological region. For this junction,the energy of the ABS closest to zero energy oscillateswith the chemical potential in the topological region. Forthe parameters where the ABS is at zero energy, the tun-neling conductance may resemble that of Majorana, i.e.,it has a step jump to a value G M at the gap-bias voltage e | V | = ∆ lead . However, when the energy of the ABS isnonzero, the conductance peak shifts away from the gap-bias voltage towards a larger voltage value by the ABSenergy.In conclusion, the tunneling conductance peaks fora conventional SNS junction occur at voltages eV = ± (∆ nontopoL + ∆ nontopoR ), where ∆ nontopoL , R are the super-conducting gaps of the left and right nontopological su-perconductors . For an SNS junction with anMZM at one side of the junction, the tunneling con-ductance peaks occur at voltages eV = ± ∆ nontopo45–48 and for an SNS junction with one MZM on both sides ofthe junction, the tunneling conductance develops peaks6at eV = ± ∆ topoL and eV = ± ∆ topoR where ∆ topoL and∆ topoR are the superconducting gaps of the left and righttopological superconductors . For an SNS junctionwhere both of the superconductors are identical SOC-SWs, in the nontopological regime close to the topolog-ical phase transition, as the Zeeman field increases thezero-momentum gap (∆ = (cid:112) µ + ∆ − V Z ) shrinks andthe tunneling conductance peaks move towards zero volt-age with a rate d ( e | V tcp | ) /dV Z = −
2, where V tcp is thevoltage at which the tunneling conductance peak occurs.In the topological regime near the transition, as the zero-momentum gap (∆ = V Z − (cid:112) µ + ∆ ) reopen, the tun-neling conductance peaks move towards larger voltagevalues with a rate d ( e | V tcp | ) /dV Z = 1. This change inthe dependence of the position of the tunneling conduc-tance peaks with Zeeman field near the topological phasetransition can serve as an evidence for the appearance ofMZMs in the system.To this end, we would like to highlight the new findingof our paper. First, we find that the tunneling conduc-tance of the MZM probed using a superconducting leadwithout a BCS singularity assumes a nonuniversal value,which decreases with decreasing junction transparency.We explicitly show this nonquantized conductance valuefor the case where the superconducting probe lead is ei-ther a topological or nontopological p -wave superconduc-tor or SOCSW with finite magnetic field. Second, wealso show that for the case where the superconductingprobe lead is a p -wave superconductor with no topolog-ical channel, MAR are strongly suppressed due to factthat a nontopological p -wave superconductors is essen-tially an insulator with small Andreev reflection ampli-tudes. Third, we show that for the case where the su-perconducting probe lead is an s -wave superconductorand there is a Majorana Kramers pair in the topologi-cal superconductor, in the high transparency regime, thecurrent and conductance near zero-voltage is not zerobecause there are two MZMs, which facilitate equal-spinAndreev reflections in two different spin channels. Thisis in contrast to the case of an SNS junction between an s -wave superconductor and a topological superconductorwith one MZM at the end. In this junction, MAR arestrongly suppressed near zero voltage because of the dif-ference between the Andreev-reflection spin selectivity ofthe s -wave superconductor and MZM.Our theoretical results should serve as a definitiveguide to future experiments on MZM using tunnelingspectroscopy of topological SNS junctions. We believethat such SNS experiments are now necessary since tun-neling spectroscopy of NS junctions in nanowires hasfailed so far (in spite of > ACKNOWLEDGMENTS
Appendix A: Remarks on Numerical Simulation
The scattering matrices at the left ( S L ) and right NSinterfaces ( S R ) [Eq. (1)] can be calculated numericallyfrom Kwant by constructing the tight-binding modelsfor the corresponding NS junctions. Since the scatteringmatrices given by Kwant are calculated using the currentamplitudes with arbitrary phases at each energy, one canfix the phases by setting the largest element of the currentamplitudes for every energy to be real.We note that Eqs. (1a) and (1c) are invariant underthe transformation: t in L,R ( E ) → t in L,R ( E ) U † L,R ( E ) , J in L,R ( E ) → U L,R ( E ) J in L,R ( E ) , (A1)where t in L,R ( E ) are the transmission matrices at the leftand right NS interfaces, U L,R ( E ) are unitary matri-ces, and J in L,R ( E ) are the input current amplitudes fromthe left and right NS interfaces. By polar decomposi-tion, there exists a unitary matrix U L,R ( E ) such that t in L,R ( E ) = (cid:101) t in L,R ( E ) U † L,R ( E ), where (cid:101) t in L,R ( E ) = (cid:113) t in L,R ( E )[ t in L,R ( E )] † = (cid:113) − r L,R ( E ) r † L,R ( E ) , (A2)with r L,R being the reflection matrices at the left andright NS interfaces. For computational efficiency, we ob-tained only the reflection matrices r L,R from Kwant andused Eq. (A2) to calculate the transmission matrix.For the numerical evaluation of Eq. (4), we used anenergy cutoff E c in the summation over energy where E c is chosen such that the calculation converges for eachvoltage V . The introduction of the energy cutoff sets thefollowing constraint on the scattering matrix: S eN ( E, E + eV ) = S hN ( − E, − ( E + eV )) = − , (A3)for all E > E c . The above constraint is required for theunitarity of the scattering matrices to hold. ∗ [email protected] J. Alicea, Rep. Prog. Phys. , 076501 (2012). M. Leijnse and K. 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