Probing vortex Majorana fermions and topology in semiconductor-superconductor heterostructures
PProbing vortex Majorana fermions and topology in semiconductor-superconductor heterostructures
Kristofer Bj¨ornson and Annica M. Black-Schaffer Department of Physics and Astronomy, Uppsala University, Box 516, S-751 20 Uppsala, Sweden (Dated: October 19, 2018)We investigate the local density of states, spectral function, and superconducting pair amplitudes forsignatures of Majorana fermions in vortex cores in ferromagnetic and spin-orbit coupled semiconductor-superconductor heterostructures. We show that the Majorana fermion quasiparticle momentum distribution isalways symmetrically distributed at a finite radius around a high symmetry point, thereby providing a necessarycondition for a low-energy state to be a Majorana fermion. In real space profiles of the local density of statesthrough the vortex core the Majorana fermion, together with other finite-energy vortex states, form a characteris-tic x-shape structure only present at non-trivial topology. Moreover, we find that the Mexican hat band structureproperty of the topologically non-trivial phase translates into multiple high-intensity band edges and also vortexcore states located above the superconducting gap in the local density of states. Finally, we find no strong cor-relation between odd-frequency pairing and the appearance of Majorana fermions, but odd-frequency pairingexists as soon as ferromagnetism is present. In fact, we find that the only vortex superconducting pair amplitudedirectly related to any phase transition, is the appearance of certain spin-triplet p -wave pairing components inthe vortex core at a pre-topological vortex core widening transition. PACS numbers: 74.90.+n, 03.65.Vf, 74.20.Rp, 74.25.Uv, 74.55.+v
I. INTRODUCTION
In recent years the study of topological phases has lead tothe prediction that so-called Majorana fermion quasiparticlescan appear in certain types of topological superconductors.
These Majorana fermions are of great interest for two quitedistinct reasons. First of all, Majorana fermions are hypo-thetical particles long sought in particle physics, but so farwithout any conclusive evidence in favor of their existence. Majorana fermion quasiparticles in the solid state, althoughnot fundamental particles, are analogous to their fundamentalcounter-parts, and may therefore provide an independent wayto discover Majorana fermions. The second reason they areof interest is because when they emerge in condensed mattersystems, typically localized on various defects, they do so ina way which makes the ground state degenerate. These dif-ferent ground states can in two dimensions be continuouslydeformed into each other by braiding the defects around eachother. Such operations are predicted to be non-Abelian, andbraiding Majorana fermions may thus be utilized for the im-plementation of robust topological quantum computing.
One place where Majorana fermions are expected to emergeis in vortex cores in certain two-dimensional (2D) topolog-ical superconductors. A prominent example of such a sys-tem is provided by a heterostructure of a thin layer of Rashbaspin-orbit coupled semiconductor, sandwiched between a fer-romagnet and a conventional s -wave superconductor. Ma-jorana fermions are also expected to occur at the end pointsof 1D wires of similar composition.
Possible signaturesof Majorana fermions has already been reported for such 1Dsystems, although some results are still debated.
Evenif braiding in principle is possible for 1D systems through theuse of wire networks, vortices lend themselves more natu-rally to be braided, for example through the use of magneticforce microscopy. In the light of the potential versatilityof vortex Majorana fermions it is of large interest to make athorough investigation of different types of signatures for Ma- jorana fermions in superconducting vortices.Analytical and numerical results on vortices have al-ready predicted that Majorana fermions appear in thetopologically non-trivial phase of the above mentionedheterostructures.
In this work we carefully investi-gate experimentally relevant signatures of both the Majoranafermions and the topologically non-trivial phase. In particu-lar, we focus on signatures in the spectral function, local den-sity of states (LDOS), and superconducting pair amplitude. Asimultaneous investigation of these three quantities is benefi-cial, as many signatures in one or another of these are closelyrelated to particular features also in the other properties.More specifically, we find that the Bogoliubov-de Gennes(BdG) quasiparticle spectrum for a Majorana fermion isstrictly required to be symmetrically distributed at a finite ra-dius around a high symmetry point, which thus provides anecessary condition for any candidate Majorana state. Wealso find that the vortex Majorana fermion and the genericfinite-energy Caroli-Matricon-de Gennes vortex states arewell separated in energy. Together they form a characteris-tic x-shape structure in subgap LDOS profiles right throughthe vortex core only in the topological phase. The Majoranamode is well-localized to the center of the core, while thefinite-energy states disperse further out from the center. Be-yond the occurrence of Majorana fermions, the topologicallynon-trivial phase can also clearly be distinguished by a Mexi-can hat shaped band structure, which can be probed with mo-mentum space probes such as angle-resolved photoemissionspectroscopy (ARPES). The Mexican hat shaped band struc-ture further gives rise to multiple band edges, showing up asdouble peaks in the DOS. The existence of the double peaksenables experimental techniques sensitive to the DOS to actas a probe of the topological phase, possibly most relevant forscanning tunneling spectroscopy (STS). The two band edgesin the double peak also behave differently in the presence ofa vortex. While one edge collapses to give rise to a Majoranafermion and the generic Caroli-Matricon-de Gennes subgap a r X i v : . [ c ond - m a t . s up r- c on ] A p r vortex core states, the other band edge is instead pushed up inenergy and there gives rise to a second set of vortex core statesthat appears as a rising band edge in the LDOS spectrum of thevortex core. We note that these types of STS measurable sig-natures are of particular interest, as the feasibility of such ex-periments recently have been demonstrated in a related setupconsisting of an s -wave superconductor with vortices coatedby a topological insulator. Finally, we show that there is nodistinct onset of odd-frequency pairing as a result of the ap-pearance of Majorana fermions, but odd-frequency pairing ispresent as soon as there is finite magnetism, independent onthe topological phase. However, we find a strong correlationbetween a pre-topological vortex core widening transition andthe onset of p -wave pair amplitudes in and around the vortexcore. II. MODEL
We here consider a 2D topological superconductor withthe essential building blocks being s -wave superconductiv-ity, Rashba spin-orbit interaction, and Zeeman ferromagneticterm. To achieve self-consistent microscopic details for a vor-tex core we study the system on a square lattice. The free pa-rameters of the model are then the nearest neighbor hopping t (setting the kinetic energy), chemical potential µ , Zeemanfield V z , Rashba spin-orbit interaction α , and superconduct-ing pair potential V sc . All energies can be measured relativeto the kinetic term, which we do by setting t = 1 . The Hamil-tonian describing this system can be written as H = H kin + H V z + H SO + H sc , (1) H kin = − t (cid:88) (cid:104) i , j (cid:105) ,σ c † i σ c j σ − µ (cid:88) i ,σ c † i σ c i σ , H V z = − V z (cid:88) i ,σ,σ (cid:48) ( σ z ) σσ (cid:48) c † i σ c i σ (cid:48) , H SO = − α (cid:88) i (cid:104) ( c † i − ˆ x ↓ c i ↑ − c † i +ˆ x ↓ c i ↑ )+ i ( c † i − ˆ y ↓ c i ↑ − c † i + ˆy ↓ c i ↑ ) + H . c . (cid:105) , H sc = (cid:88) i ∆ i ( c † i ↑ c † i ↓ + H . c . ) . Here i and j are site indices on the square lattice, σ is thespin index, and c † i σ ( c i σ ) is the electronic creation (annihila-tion) operator. We are primarily interested in a lightly hole-doped semiconductor, which is achieved by setting µ = 4 .The superconducting order parameter ∆ i enters as a param-eter in the Hamiltonian, but is determined self-consistentlyusing a superconducting pair-potential V sc . This is done bysolving Eq. (1) within the BdG formalism, and re-calculatingthe order parameter using ∆ ( m +1) i = − V sc (cid:104) c i ↓ c i ↑ (cid:105) ( m ) = − V sc (cid:88) E ν < v ( m ) ∗ ν i ↓ u ( m ) ν i ↑ . (2) E V z I'II III
FIG. 1: (Color online). Energy spectrum for a superconducting vor-tex as a function of the Zeeman field (red and green lines), strength ofthe superconducting order parameter (thick blue line), and absolutevalue of the Zeeman field (black dashed line). The I, I’, and II regionscorresponds to the trivial phase, trivial phase with wide vortex core,and topologically non-trivial phase, respectively. Four representativesample points are also marked, used in other figures.
Here u ν i ↑ ( v ν i ↓ ) is the electron up (hole down) component onsite i , and m is the iteration step. We are able to study a singlevortex by specifying the initial order parameter configuration ∆ (0) ( r, θ ) = | ∆ (0) ( r, θ ) | e − iθ , but then letting the supercon-ducting amplitude and phase fully relax. This allows for afully self-consistent order parameter profile to be obtained un-der the single requirement that the phase winds π around thevortex core. A. Phase transitions
We have previously shown that close to the topo-logical phase an unrelated phase transition can takeplace, due to the competition between ferromagnetism andsuperconductivity. This phase transition manifests itself inthe sudden widening of the vortex core, together with a jumpin the magnetization in the core. This means that the phasediagram for the system can be divided into three differentregions which we label I, I’, and II. These are the topologi-cally trivial, topologically trivial but with a wide vortex core,and topologically non-trivial phases, respectively. In Fig. 1the energy spectrum as a function of the Zeeman ferromag-netic field is reproduced, and the three phases are marked withcorresponding labels. In the topologically non-trivial phaseII, zero-energy Majorana states appear (red line). However,states at or close to E = 0 , which are not Majorana fermions,also appears in the I’ region. It is therefore directly clear that,if looking for a single signature such as a state at E = 0 , thereis a significant risk of mistakenly identifying a state as a Ma-jorana fermion even though it is not. In this work we thereforecarefully investigate several different experimental signaturesof both the Majorana fermions directly, as well as signaturesrelated to the different phases. Most of these signatures canbe directly accessed with real space and band structure probessuch as STS and ARPES, respectively, while a few other sig-natures are of at least important conceptual value. III. BULK BAND STRUCTURE
Before turning to the results for a vortex, we begin witha few important remarks about the bulk band structure and the topological phases of the system. First of all, becausewe study a lightly hole-doped semiconductor, the relevantcondition for being in the topologically non-trivial phase is (4 t − µ ) + | ∆ | < V z < µ + | ∆ | . We note that in thebulk there is no difference between the two topologically triv-ial phases I and I’. The difference between these two phasesonly become apparent in self-consistent vortex calculations.Further, the band structure of the system is given by E n ( k ) = ± (cid:114) (cid:15) ( k ) + α L ( k ) + V z + | ∆ | ± (cid:113) (cid:15) ( k ) α L ( k ) + ( (cid:15) ( k ) + | ∆ | ) V z , (3)where n is the band index, (cid:15) ( k ) = − t (cos( k x ) + cos( k y )) − µ the kinetic energy, and L ( k ) = (sin( k y ) , − sin( k x )) thespin-orbit coupling. In Fig. 2, we plot the band structure forrepresentative points of the trivial and non-trivial phases, aswell as points where the band gap closes. It is clear from theplot for phase II that in the topologically non-trivial phase thetwo bands closest to E = 0 form two oppositely facing Mex-ican hats centered around the point k = ( π, π ) . [More gener-ally this could also be at the other high symmetry point: e.g. ina lightly electron-doped semiconductor the Mexican hats arecentered around k = (0 , ]. The trivial band structure inphase I instead takes the form of two ordinary parabolas. Fur-thermore, the two plots (D) and (M) shows two different waysthrough which the bulk band gap of the non-trivial phase canbe closed. At the topological phase transition (I ↔ II), theband gap closes by forming a Dirac cone. On the other hand,the band gap can also be closed by letting ∆ → , in whichcase the system becomes metallic. In the top left figure aschematic view of the two different routes through which thegap can be closed is shown. From a topological point of view,the only important thing is that the I and II regions are sepa-rated by a gap closing, and the two types of gap closings canbe considered topologically equivalent. We will however seethat by distinguishing between the two types of gap closings,the origin of important experimental signatures of Majoranafermions as well as the topological phase can be understood. A. Edge modes
In the semi-classical limit it is possible to treat real and re-ciprocal coordinates as independent of each other. The edgestates that appear at interfaces of topologically non-trivial phases can then be understood as a consequence of the bulkband structure being required to go through a gap closingwhen passing from the topologically non-trivial phase insideto the trivial phase outside. Zero-energy modes on impuri-ties can be understood to appear for similar reasons. However,in reality interface and impurities are fairly abrupt in nature,and it is therefore not obvious that a semi-classical treatmentrelying on the bulk band structure reliably predicts the proper-ties of such defects. Nevertheless, we know that the predictionof zero-energy states at interfaces is still valid also for abruptdefects, when the forced gap closing is associated with a jumpin a topological invariant, such as the Chern number. Thishave been demonstrated for many systems both numericallyand analytically, and can in some cases be formally justifiedthrough the use of an appropriate index-theorem, see e.g. 27.
1. Semi-classical prediction
In what follows we will repeatedly make use of the semi-classical limit to explain various signatures of Majoranafermions and the topological phase. To demonstrate themethod, we begin by determining the momentum space distri-bution of the Majorana fermions in the vortex core and at theedge of the system. First, we note that the edge of a topolog-ically non-trivial superconductor can be seen as an interfacebetween the topologically trivial vacuum, and the non-trivialbulk. It is therefore a transition from region I to II, similar tothat along the dashed line in Fig. 2. In the semi-classical limitwe thus expect the zero-energy spectrum on the edge to berelated to a gap closing of type D, that is, the Majorana edgestate will be located at the high symmetry point, k = ( π, π ) .On the other hand, the vortex core is a region where the most I II IDM |4t- μ | | μ | MDII I E -22 E -22 E -22 E -22 k x k y0 2 π π k x k y0 2 π π k x k y0 2 π π k x k y0 2 π π FIG. 2: (Color online). [Top left] Schematic figure of the topologically trivial (I) and non-trivial (II) phases as functions of | V z | and | ∆ | . At thetopological phase transition following the dashed grey line, the band gap is closed by the formation of a Dirac cone (D). The band gap can alsobe closed by following the full line down to ∆ = 0 , at which point the system becomes metallic (M). [I, II, D, M] The two bands closest to theFermi level plotted in the topologically trivial (I) and non-trivial phase (II), at the topological phase transition where a (D)irac cone appears,and for ∆ = 0 where the system becomes (M)etallic. essential feature is that ∆ → . The Majorana state in thevortex core should therefore be expected to instead be relatedto a gap closing of type M, which produces zero-energy statesin a circle at a certain radius from k = ( π, π ) .
2. Tight-binding confirmation
Having deduced the expected behavior from the semi-classical limit, we now look at the actual Majorana fermionsthat appear in a fully self-consistent solution. To do so wepick the eigenstates corresponding to the two zero-energy lev-els at point 4 in Fig. 1, and rotate these into the basis wherethe two resulting Majorana fermions γ M and γ M becomesclearly localized in the core and on the edge, respectively. InFig. 3 we plot the square of the real space wave function givenby | γ M , ( x ) | = (cid:88) σ | u M , x σ | + | v M , x σ | . (4)We also we calculate the momentum space distribution of thestates | γ M , ( k ) | = (cid:88) σ | u M , k σ | + | v M , k σ | , (5)where u M , k σ and v M , k σ are the Fourier transforms of u M , x σ and v M , x σ , respectively. The result is displayed in the recipro-cal space plots in Fig. 3. It is clear that the Majorana fermionat the edge is mainly built up from k -components at k =( π, π ) , while the vortex core Majorana fermion mainly con-sists of k -components a finite radius away from k = ( π, π ) . This is in complete agreement with our expectations from thesemi-classical limit.Apart from showing that important features of the Majoranafermions can be derived directly from the bulk band struc-ture by considering the semi-classical limit, this result alsohas important experimental implications; a necessary indica-tor of the vortex core state being a Majorana fermion is that theFourier transform of the state is distributed along a circle offinite radius, and centered at the high symmetry point wherethe bulk band gap closes at the topological phase transition.However, we note that this is not an exclusive signature of aMajorana fermion, e.g. ordinary Caroli-Matricon-de Gennesvortex states can be expected to have a similar momentumspace distribution. The reason being that also these states area consequence of the same local collapse of the superconduct-ing gap. IV. PARTICLE-HOLE SYMMETRY
Before moving on to further results, we here make a fewgeneral remarks related to particle-hole symmetry. The BdGHamiltonian for a general and homogeneous bulk supercon-ductor can be written as a × matrix equation: H ( k ) = (cid:20) H ( k ) ∆ ( k ) − ∆ ∗ ( − k ) − H T ( − k ) (cid:21) , (6)in the basis ( c k ↑ , c k ↓ , c †− k ↑ , c †− k ↓ ) T , and we write the eigen-states as | γ ν k (cid:105) = (cid:2) u ν k ↑ u ν k ↓ v ν k ↑ v ν k ↓ (cid:3) T . From theequation H ( k ) | γ (cid:105) = E | γ (cid:105) and the complex conjugate of H ( − k ) | γ (cid:105) = − E | γ (cid:105) , it can be seen that these two equa-tions are in fact the same set of equations. This leads to a Vortex core Edge R e a l s p a c e π π π x xk k x x yk y M o m e t u m s p a c e FIG. 3: (Color online). Majorana fermions in the vortex core and onthe edge. Real space distribution (top) and Fourier transform of thesame states (bottom). The vortex core Majorana fermion is mainlybuilt up of momentum components at a certain radius away from k =( π, π ) , while the Majorana state at the edge is consists predominantlyof states at k = ( π, π ) . particle-hole symmetry for eigenstates ν and ¯ ν of oppositeenergy E ν = − E ¯ ν . Identification of coefficients in the twoequations also leads to u ν k σ = v ∗ ¯ ν − k σ . (7)Equation (7) is of particular interest to us because it tells usthat the states at ( E ν , k ) and ( − E ν , − k ) are related through γ ν k = (cid:88) σ (cid:16) u ν k σ c k σ + v ν k σ c †− k σ (cid:17) = γ † ¯ ν − k . (8)This pairwise correspondence is a consequence of the artificialdoubling of degrees of freedom which occurs when the prob-lem is treated using a × BdG formulation. The two statesare therefore not distinct, but rather the occupation of one isnecessarily accompanied by the deoccupation of the other.
A. Consequences for Majorana fermions
The set of γ ν k provides a complete basis for the solutionof any not homogeneous problem, with a general eigenstatewritten as γ λ = (cid:88) ν k a λν k γ ν k = (cid:88) k σ (cid:16) u λ k σ c k σ + v λ k σ c †− k σ (cid:17) , (9)where u λ k σ = (cid:80) ν a λν k u ν k σ and v λ k σ = (cid:80) ν a λν k v ν k σ . Notethat the eigenstates are now labeled by their superscript λ , while the subscript ν becomes a summation index. Ifnow γ M is a Majorana fermion, it also satisfies the relation γ M = γ M † , which using Eqs. (8)-(9) leads to (cid:80) ν k a Mν k γ ν k = (cid:80) ν k a M ∗ ¯ ν − k γ ν k . In particular, this implies u M k σ = (cid:88) ν a Mν k u ν k σ = (cid:88) ν a M ∗ ¯ ν − k u ν k σ = v M ∗− k σ . (10)It is therefore clear that the apparent symmetry of the momen-tum distribution in Fig. 3 is not an accident. Rather, Eq. (10)guarantees that the momentum distribution of the Majoranafermion is inversion symmetric. However, we note that boththe real space and momentum distribution of the Majoranafermions expressed here takes into account both the electronand hole components of the eigenstates. This strict symmetryis therefore not necessarily present for a physical probe onlymeasuring the electron component. B. Electronic and BdG expressions
The discrepancy just mentioned between what can be statedabout the Majorana fermion in the BdG formulation and whatcan be seen experimentally reflect a general conceptual diffi-culty with regard to topological superconductors. The topo-logical properties are derived from the BdG band structure,so the consequences for these properties are most easily un-derstood in relation to it. However, physical quantities arecalculated in a way that can obscure their relation to the BdGband structure. A common prescription for calculating physi-cal quantities such as the electronic LDOS, DOS, and spectralfunction involves summations of the form (see e.g. Ref. [28]): (cid:88) E ν > (cid:0) | u νρ | δ ( E − E ν ) + | v νρ | δ ( E + E ν ) (cid:1) , (11)where ρ is some index and the summation runs over positiveenergies to avoid over-counting due to the artificial doublingof degrees of freedom in the BdG formulation. We note thatif the δ -function had entered as δ ( E − E ν ) in front of the v ’s,then the v ’s would appear as just another orbital, and suchexpressions can be compared straightforwardly with the BdGband structure. For this reason it is beneficial to follow an-other equivalent prescription for calculating physical quanti-ties. Using Eq. (7), or the same expression transformed to realspace (depending on the type of index ρ ): u ν x σ = v ¯ ν ∗ x σ , it ispossible to replace terms involving v ’s in expressions of theform in Eq. (11), by u terms of the opposite energies. We thenwrite (cid:88) E ν (cid:0) | u νρ | + (cid:8) | v νρ | (cid:9)(cid:1) δ ( E − E ν ) , (12)where the second term in brackets is to be ignored in order forthe expression to correspond to physical, or electronic, quan-tities. On the other hand, if the second term is kept, the re-sulting properties can be straightforwardly related to the BdGband structure. The latter expression, which we call BdG-type, therefore provides an important conceptual bridge be-tween the theoretical formulation of (topological) supercon-ductivity and experimentally measurable quantities. In fact,we already used the BdG-type expression in Eq. (4)-(5) andFig. 3.
17 lattice sites4 lattice sitesEdgeCoreCore
FIG. 4: (Color online). Eigenstates are classified as edge, vortexcore, or bulk states according to in which region they are mainlylocated. Eigenfunctions are classified as edge or vortex core states ifmore than 50% of their density are located in the edge or vortex coreregion, respectively. Other states are classified as bulk states.
V. SPECTRAL FUNCTION
In Section III A we related the bulk band structure to themomentum space structure of the Majorana fermions. Herewe continue our study of the band structure by solving the realspace Hamiltonian in Eq. (1) with a vortex, and then Fouriertransform the results to arrive at the spectral function. In thelimit of an infinite homogeneous sample, this is equivalent tocalculating the analytical bulk band structure. However, whenthe sample is finite and includes defects such as edges andvortices, the result will be distorted. The spectral function canbe calculated as A ( k , E ) = (cid:88) E ν ,σ (cid:0) | u ν k σ | + (cid:8) | v ν k σ | (cid:9)(cid:1) δ ( E − E ν ) . (13)It is useful to divide A ( k , E ) into parts consisting of edge,vortex core, and bulk contributions. For this reason we definethe state classification functions C e ( ν ) = (cid:26) if (cid:80) x ∈ X e ,σ (cid:0) | u ν x σ | + | v ν x σ | (cid:1) > , otherwise ,C c ( ν ) = (cid:26) if (cid:80) x ∈ X c ,σ (cid:0) | u ν x σ | + | v ν x σ | (cid:1) > , otherwise ,C b ( ν ) = (cid:26) if C e = C c = 0 , otherwise , (14)where X e and X c are the set of points classified as edge andcore sites according to Fig. 4. Letting L = e, c, b denote edge,core, and bulk, respectively, we define A L ( k , E ) = (cid:88) E ν ,σ C L ( ν ) (cid:0) | u ν k σ | + (cid:8) | v ν k σ | (cid:9)(cid:1) δ ( E − E ν ) , (15)where A ( k , E ) = (cid:80) L A L ( k , E ) . A peculiarity of this is thatthe numerical zero-energy states, representing the Majoranafermions before they are rotated into the Majorana basis, areclassified as bulk states. This follows since they are locatedboth at the edge and in the vortex core, with additional tailsstretching into the bulk. This conveniently allows us to extractfrom A ( k , E ) the contribution from the bulk band and Ma-jorana fermions separately from the contributions from edgestates and other vortex core states.In Fig. 5 we plot the BdG-type A ( k , E ) and A L ( k , E ) along k y = π , at the topologically non-trivial point 4 in Fig. - - = Full Edge Vortex Bulk + MF k x0 2 π E -1.50 k x0 2 π k x0 2 π k x0 2 π FIG. 5: (Color online). The BdG-type spectral function A ( k , E ) and its edge, core, and bulk components plotted along k y = π forparameters corresponding to the topologically non-trivial phase atpoint 4 in Fig. 1. A ( k , E ) is seen to be dominated by contributionsfrom the bulk band structure. However, topological edge states, aswell as vortex core states also contribute. When the edge and or-dinary vortex core states are subtracted from A ( k , E ) , the resulting A b ( k , E ) consists of only bulk bands and the two zero-energy Majo-rana fermion states. Because the scaling behavior is different for thedifferent types of states, the color scale is arbitrary but set to enhancethe relevant features.
1. First of all, we see the two dispersive (at non-zero energy)branches of topological edge states in A e ( k , E ) . Similarly, awealth of vortex core states are visible in A c ( k , E ) . We notethat vortex core states predominantly originate from statesaround the band edges. This can be understood by consideringthat at the band edges the DOS is high, which makes it easyto hybridize these states into states localized in the vicinity ofthe vortex. We also see that the ordinary vortex core statesclosest to E = 0 are formed mainly from k -components at adistance away from k = ( π, π ) , while higher energy vortexstates are located notably closer to k = ( π, π ) . The former,which are subgap states, are the ordinary Caroli-Matricon-deGennes states, and their momentum distribution is in agree-ment with that expected from our discussion in Section III A,where we mentioned that they, just like the vortex core Ma-jorana fermions, result from a local (M)-type collapse of thebulk gap. In addition, it is clear that A b ( k , E ) contains thebulk band structure, as well as faint signals of the Majoranafermions at E = 0 .Although it is possible to image the electronic A ( k , E ) di-rectly using e.g. ARPES, it may be difficult to extract the edgeand vortex core features from such data, due to limited in-tensity. However, the bulk features such as the Mexican hatshaped band structure in Fig. 5 scale with the area of the bulk,and can therefore be imaged directly with ARPES. On theother hand, it should be feasible to image the vortex core con-tribution by Fourier transforming data from local probes suchas STS. Note, however, that it is not the BdG-type spectralfunction in Fig. 5 that is physically measured. Rather it is thecorresponding electronic spectrum which results from drop-ping the hole-part in Eq. (15). Having seen how various fea-tures of the spectral function can be understood by splitting itdecomposing into edge, vortex core, and bulk states contribu-tions, we define A b + c ( k , E ) = A b ( k , E ) + A c ( k , E ) . This isthe spectral function with contributions from edge states ex-cluded, which are artificial effects introduced by the finitessize of our sample. We will use this spectral function in therest of this article. VI. LOCAL DENSITY OF STATES
Having investigated signatures in momentum space, wenow turn to a discussion of the LDOS, given by ρ ( x , E ) = (cid:88) ν,σ (cid:0) | u ν x σ | + (cid:8) | v ν x σ | (cid:9)(cid:1) δ ( E − E ν ) . (16)We will see that a comparison between the LDOS and spec-tral function is helpful for revealing important signatures ofthe topological phase. In addition we also compare with thebulk DOS calculated from a pure bulk solution with an equiv-alent | ∆ | . In Fig. 6 the LDOS, bulk DOS, and A b + c ( k , E ) areplotted side by side at the points labelled 1-4 in Fig. 1. Boththe electronic and BdG-type results are shown here. In whatfollows we use the BdG-type results for the interpretation, butthe electronic results are to be considered when comparingto experiments. When the Zeeman field is zero (1), the bandstructure can be seen to consist of two Rashba spin-orbit splitparabolas crossing at a high symmetry point. The band edgesshows up as sharp features in both the LDOS and DOS. Vor-tex core states below the band edges can also be observed. Asthe Zeeman field is turned on (2), the two bands are split offfrom each other, and the upper band becomes irrelevant forthe low-energy spectrum making the system effectively ”spin-less”. Still vortex core states form below both band edges.Inside the I’ region (3), a vortex core state appears at or closeto E = 0 , and there gives rise to a non-Majorana fermionzero bias peak. It is clear from the spectral function that thetopologically non-trivial phase has not yet been entered, asthe two parabolic bands closest to the Fermi level do not givea Mexican hat shaped spectrum. However, this can not beseen from the LDOS at E = 0 alone, and care thus has tobe taken to not casually interpret every zero-energy peak inthe LDOS spectrum as a Majorana fermion. We further notethat all vortex core states in these first three cases are asso-ciated with faint signals in the spectral function around thehigh-symmetry point and below the band edges. This is par-ticularly important for the zero-energy states in region I’, asthis make their momentum distribution clearly distinct fromthose of the Majorana vortex core states.Once the two low-lying bands becomes inverted and ac-quires Mexican hat like forms, the topologically non-trivialphase is entered, as exemplified by point (4). Because of thecomparatively rather flat nature of the Mexican hat like struc-ture, the band edges acquires a large density of states, whichgenerates a strong intensity in the (L)DOS. The observation ofstrong band edges in the LDOS spectrum therefore providesevidence for the system being in the topologically non-trivialphase. However, this only distinguishes the topologically non-trivial phase from the trivial phase, and not from a conven-tional superconductor where strong band edges also appears. We also note that the bulk DOS clearly shows that the bandedge in fact consists of two edges, one from the Mexican hatslower edge away from k = ( π, π ) , and one from its edge athigher energy at k = ( π, π ) . This provide a clear distinctionalso from the band edge behavior of a conventional supercon-ductor. Additionally, we see that the zero-energy Majoranafermion is accompanied by a wealth of other vortex states,showing up as a distinct x-shape structure in the low-energyLDOS. These subgap states are the ordinary Caroli-Matricon-de Gennes states already discussed.Another important piece of evidence for the topologicallynon-trivial phase is provided by an apparently larger band gapinside the vortex core. From the LDOS it appears as if theband gap is increased around the vortex, leaving a strong vor-tex core signal above the bulk band edge. This is due to thefact that the vortex core states now behave differently than thevortex core states observed so far, which always appear belowthe bulk band edge. We can understand this as a consequenceof the two different band edges of the Mexican hat shapedbulk band. The lower band edge away from k = ( π, π ) col-lapses in the core because it is due to superconductivity. How-ever, the second band edge does not collapse. Rather it can beseen from Eq. (3) that at k = ( π, π ) the energy is given by E n = ± (cid:16) | V z | − (cid:112) (4 t − µ ) + | ∆ | (cid:17) . It is further clear thatin the topologically non-trivial phase the square root term isnecessarily smaller than | V z | . Thus, as | ∆ | → inside thevortex core, the second band edge is pushed away from theFermi level. This result in a set of vortex core states abovethe band edge. For more details of this process we provide azoomed-in plot of the LDOS at point (4) in Fig. 7. First of all,the Majorana fermion is clearly visible as a state at E = 0 , andhas a clear energy-separating from the other Caroli-Matricon-de Gennes states, making it possible to resolve. Shown is alsothe BdG-type DOS, obtained by summing the bulk and vor-tex core DOS of the self-consistent solution, the vortex coreDOS for the self-consistent solution, as well as the DOS of thepure bulk solution. A comparison between the self-consistentbulk+core DOS and the DOS of the pure bulk solution clearlyshow how the lower band edge essentially collapses aroundthe vortex and gives rise to sub gap states. On the other hand,the upper band edge is pushed up in energy, giving rise to thevortex core states above the band edge, most clearly visible asa half x-shape structure above the band edge in the electronicspectrum.We finally also point out that the Majorana fermion is welllocalized in the center of the vortex core, and that the arms ofthe x-shape structure formed by the subgap vortex core statesmeet at the center of the vortex. We put this in contrast to re-cent STS experiments, where a spatially extended zero-energypeak, together with the arms of the x-shape structure pointingtowards zero a finite distance away from the center of the vor-tex, were assumed to provide evidence of Majorana fermionsin a superconductor-topological insulator heterostructure. Our results clearly show that Majorana fermions do not pro-duce such signatures in a spin-orbit coupled semiconductor.However, it should be noted that these experiments were doneon an s -wave superconductor coated by a topological insula-tor, and may therefore be sufficiently different that a direct E -1.51.5 E -1.5 1.5 E -1.51.5 E -1.5 BdGElectronic BdGElectronic k x k x k x k x x x x x DOS DOS DOS DOS0 39 0 2 π π π π k x k x k x k x x x x x DOS DOS DOS DOS0 39 0 2 π π π π
00 00
FIG. 6: (Color online). LDOS for a cut through the vortex core (left), bulk DOS (center), and spectral function at k y = π (right), for the foursample points shown in Fig. 1, using both the electronic and BdG-type expressions. The position of the vortex core states in the LDOS andspectral function are indicated in the DOS plots by dotted lines, and appears in grey if they are associated with the two higher-energy bands. DOS x π E -1.50 x π BdGElectronic b u l k + c o r e p u r e b u l k c o r e FIG. 7: (Color online). Same LDOS as in (4) in Fig. 6, but withhigher resolution around E = 0 . The vortex core states are nowseen to be clearly separated by finite energies. The DOS is plottedto the right. Leftmost DOS consists of bulk and core contributionsfrom vortex calculation. The middle DOS contains only vortex corestates. Rightmost DOS is from a pure bulk calculation. comparison is not possible. VII. PAIR AMPLITUDES
It has recently been suggested that the appearance of Ma-jorana fermions is closely related to the presence of odd-frequency pairing.
In addition, the appearance of uncon-ventional pair amplitudes in vortices has attracted great in-terest in general.
For this reason we here also provide adetailed investigation of the superconducting pair amplitude F ( R , r , σ, σ (cid:48) ) = (cid:104) c R + r ,σ c R − r ,σ (cid:48) (cid:105) = (cid:88) E ν < v ∗ ν, R + r ,σ u ν, R − r ,σ (cid:48) , (17)which we decompose into singlet and triplet components F s ( R , r ) = 12 ( F ( R , r , ↑ , ↓ ) − F ( R , r , ↓ , ↑ )) ,F m =1 ( R , r ) = F ( R , r , ↑ , ↑ ) ,F m =0 ( R , r ) = 12 ( F ( R , r , ↑ , ↓ ) + F ( R , r , ↓ , ↑ )) ,F m = − ( R , r ) = F ( R , r , ↓ , ↓ ) . (18)Here R is the center-of-mass coordinate for a pair of electrons,while r and − r points to the two electrons. The pair ampli-tude can further be classified as s - p - d -wave, and so forth, ac-cording to its angular dependence around the center-of-masscoordinate. For this purpose the pair amplitude is further pro-jected onto e ilθ to obtain the s -, p + -, p − -, d + -, and d − -wave s s p p d d ext + - + - FIG. 8: Orbital wave functions defined by the projection operatorsin Eq. (19). The values on the sites are the complex conjugate ofthe value of the projector when r points from the central site to thecorresponding site. pair amplitudes for l = 0 , , − , , − , respectively. In gen-eral, there is a certain degree of freedom in how to performthis projection, because the pair amplitude also has a radialdependence. In practice, however, the pair amplitude is onlyexpected to be sizable within a few coherence lengths, and itis therefore only the components of the pair amplitude whichcorresponds to small r that is of interest. Focusing on thepair amplitude components for minimal r (without becomingtrivially zero) for the various orbital moments, the appropriateprojectors are P s = δ r0 ,P s ext = 14 (cid:88) a ∈ A δ ra ,P p ± = 14 (cid:88) a ∈ A δ ra e ∓ iθ ( r ) ,P d ± = 18 (cid:88) b ∈ B δ rb e ∓ i θ ( r ) , (19)with A = { (1 , , (0 , , ( − , , (0 , − } , B = A ∪{ (1 , , ( − , , ( − , − , (1 , − } , and θ ( r ) the angular co-ordinate of r . The orbital wave functions onto which the pairamplitude is projected are displayed in Fig. 8. The projectedpair amplitudes can now be written as F SO ( R ) = (cid:88) r P O F S ( R , r ) , (20)where we study O ∈ { s, s ext , p ± , d ± } and S ∈ { s, , , − } ,while r runs over all possible lattice vectors. A. Even-frequency pairing
In Fig. 9 we plot both the maximum value as a function ofthe Zeeman field, as well as representative real space profilesof the pair amplitudes. All pair amplitudes are normalized bythe bulk value of the conventional s -wave (singlet) pair am-plitude at zero Zeeman field. The conventional s -wave pairamplitude has been excluded from the plots as it is directlygiven by the relation | ∆( R ) | = | − V sc F ( R , , ↓ , ↑ ) | . Fromthe real space profiles it is clear that three types of behaviorcan be identified. Namely, the pair amplitudes can be seen asoriginating in the bulk, vortex core, or region surrounding thevortex core, which here will be referred to as the pre-core re-gion. All pair amplitudes have their maximum value in one of these three regions, and decays to zero in the other two. It isalso clear that the three most important pair amplitudes in thebulk, apart from the s -wave (singlet) pair amplitude, are theextended s -wave (singlet), p + ( m = − ), and p − ( m = 1 )components. These all have a total angular momentum z -axisproject J z = 0 , as expected if angular momentum is to beconserved. The appearance of extended s -wave (singlet) isa direct consequence of conventional s -wave superconductiv-ity; it only gives further details about the size of the pair am-plitude at a finite radius r . The p -wave pair amplitudes in thebulk appear because of the Rashba spin-orbit interaction andis in agreement with previous results. Our results do how-ever provide further details showing that the p + ( m = − )component is largest when the Zeeman field is positive, while p − ( m = 1 ) dominates for a negative Zeeman field. This canbe understood since essentially all electron levels are occupiedin a lightly hole-doped semiconductor. Thus the band cross-ing the Fermi level is the band which is pushed up in energyby the Zeeman field and contains predominantly spins anti-aligned with the Zeeman field. This is also the band that isgapped by superconductivity, explaining the overweight of su-perconducting pairing among spins anti-aligned with the Zee-man field. We note that the argument would be reversed for alightly electron-doped semiconductor.Next, we note that the existence of two p − states, in thevortex core ( m = 0 ) and in the pre-core region ( m = − ),are strongly correlated with the onset of the wide vortex coreregion I’. For negative V z , for which no vortex core wideningoccurs, we already have p − ( m = 1 ) preferred in the bulk.The rotation direction of the p − orbital part agrees with therotation direction of the vortex. For positive V z , on the otherhand, the p + ( m = − ) state is preferred in the bulk but ithas an orbital motion directed opposite to that of the vortex.This appears to lead to a widening of the vortex core. Finally,we note that in the bulk and vortex core the total angular mo-mentum is J z = 0 and J z = − , respectively. The angularmomentum J z = − of the vortex core pairing can be ex-plained by the vortex winding n = − being absorbed intothe orbital part of the pair amplitude. On the other hand,the pre-core pair amplitudes respect neither the bulk nor thevortex core angular momentum conservation rule.
B. Odd-frequency pairing
Having described the behavior of the regular pair ampli-tudes we now turn to an investigation of odd-frequency pairamplitudes. The possibility of odd-frequency pairing arisewhen r is considered to not only be a coordinate in space, butalso in time, which we denote by ˜r = ( r , t/ . The ordinaryrequirement on the pair amplitude to be odd under the simul-taneous interchange of position and spin (cid:104) c R + ˜r ,σ c R − ˜r ,σ (cid:48) (cid:105) = −(cid:104) c R − ˜r ,σ (cid:48) c R + ˜r ,σ (cid:105) is then transformed to the requirement thatthe pair amplitude is odd under the simultaneous interchangeof position, spin, and time (frequency). The pair amplitudesinvestigated so far are all even in frequency, as only the evenfrequency components can be non-zero for t = 0 . To identifypair amplitudes which are odd in time, we have to consider0 sppdd ext+--+ -0.64 0.64 V z III III ' sppdd ext+--+ x FIG. 9: (Color online). Even frequency pair amplitudes. [Left] maximum absolute value of the pair amplitudes as a function of V z . [Right]representative profiles for of the pair amplitudes along the x -axis, through the vortex core. All profiles are plotted for V z = 0 . , just to theright of the transition from I to I’, except the m = 1 profiles where V z = − . , but all pair amplitudes retain their profile characteristics as V z is varied. All values are normalized by the bulk value of the spin-singlet s -wave pair amplitude at zero Zeeman field, F ss ( bulk, . the derivative of the pair amplitude with respect to time dF ( R , ˜r , σ, σ (cid:48) ) dt = ddt (cid:104) c R + ˜r ,σ c R − ˜r ,σ (cid:48) (cid:105) = ddt (cid:88) E ν < v ∗ ν, R + r ,σ u R − r ,σ (cid:48) e − i (cid:126) E ν t . (21)Preforming the decomposition of the total pair amplitude intoits spin and orbital components as outlined in Eq. (17-20) al-lows us to also consider the amplitudes dF SO ( R , t ) /dt .In Fig. 10 the odd frequency equivalent of Fig. 9 is dis-played. Most notable is the appearance of an s -wave ( m = 0 )pair amplitude in the bulk when the Zeeman field is non-zero.Likewise, the extended s -wave ( m = 0 ) component is foundto have a similar behavior. All the pair amplitudes associatedwith the bulk have J z = 0 .Next, we note four non-zero pair amplitudes in the core.These are the s ( m = − ), extended s ( m = − ), p − (sin-glet), and d − ( m = 1 ), which all have J z = − . The exis-tence of these can once again be understood as a consequenceof the vortex winding being rotated into the pair orbital part.However, although small, an anomalous d + ( m = 1 ) compo-nent also appear in the core, violating the otherwise seeminglyperfect agreement with vortex core pair amplitudes having J z = − . This is due to the fact that each site has four nearestneighbors, while four of the sites included in the projectiononto the d -wave pair functions are second-nearest neighbors.On the nearest neighbor sites e i θ ( r ) = e − i θ ( r ) , so with re-spect to these site alone the angular momentum can only bedefined modulo 4, and consequently J z = 3 mod − .This is an example of how a radial variation in the pair am-plitude, in this case the d − ( m = 1 ) component, can give riseto seemingly angular momentum violating components whencalculated on a discrete lattice. Unfortunately it is impossible to define a completely satisfactory projection procedure on adiscrete lattice. Finally, we note the existence of the pre-coretype pair amplitudes s ( m = 1 ), extended s ( m = 1 ), p + (singlet), and d + ( m = − ). These, just as the pre-core typeeven-frequency p ± , all have J z satisfying neither J z = 0 nor J z = − . They do, however, all have in common that theyeither have a bulk or core counter-part differing only in spinor orbital rotation direction. We therefore interpret these asbeing secondary in nature, being induced from their bulk andcore counter-parts.Before ending the discussion of pair amplitudes we alsonote that although the odd frequency s -wave ( m = − ), aswell as even frequency p + ( m = − ) and p − ( m = 1 ) pairamplitudes all increase as function of the Zeeman field to-wards the non-trivial phase, they contain no specific signatureof the non-trivial phases themselves. We put this in contrast torecent results relating the existence of odd-frequency s -wavepair amplitudes with the appearance of Majorana fermions in1D. Likewise, the various pair amplitudes which appear inthe vortex core provides little clue to the topological phase orexistence of Majorana fermions, but appear in the whole phasediagram. On the other hand, several of the pair amplitudeschanges abruptly at the I ↔ I (cid:48) transition, thereby reflectingthe sudden widening of the vortex core. A summary of thedescribed pair amplitudes is provided in Table I. VIII. SUMMARY
We have in this work investigated the local density ofstates (LDOS), band structure, and superconducting pair am-plitude for signatures of the non-trivial topological phaseand Majorana fermions in vortex cores in spin-orbit coupled1 ssppdd ext+--+ -0.64 0.64m=1m=0m=-1singlet V z III III ' ssppdd ext+--+ x FIG. 10: (Color online). Same as Fig. 9, but for odd frequency. All real-space profiles are for V z = 0 . .Orbital Frequency Spin J z Origin Max (%) p + even m = 1 2 pre-core 2 p + even m = 0 1 pre-core 1 s ext even singlet bulk 10 p + even m = − bulk 14 p − even m = 1 0 bulk 14 p − even m = 0 − core 7 p − even m = − − pre-core 5 d − even singlet − pre-core 1 d + odd m = 1 3 core 2 s odd m = 1 1 pre-core 1 s ext odd m = 1 1 pre-core 1 p + odd singlet pre-core 2 d + odd m = − pre-core 1 s odd m = 0 0 bulk 41 s ext odd m = 0 s odd m = − − core 6 s ext odd m = − − (pre-)core 6 p − odd singlet − core 10 d − odd m = 1 − core 5TABLE I: Even and odd pair amplitudes, ordered according to their J z quantum number. J z = 0 and J z = − pairing is seen to orig-inate in the bulk and core, respectively, while other pair amplitudesonly appears in the pre-core region. The d + , odd frequency, m = 1 pair amplitude is anomalous in that it originates in the core regionalthough it has J z = 3 . As explained in the text, it is an artifact ofthe pair amplitudes being calculated on a lattice. Likewise, the s ext ,odd frequency, m = − pair amplitude is anomalous in the widevortex core region, as it there originates in the pre-core region ratherthan in the core, in spite of having J z = − . The maximum relativestrength as compared to the s -wave, even frequency, spin-singlet pairamplitude at zero Zeeman field is tabulated at the right. semiconductor-superconductor heterojunctions. A necessaryindicator of a zero-energy vortex core state being a Majo-rana fermion has been identified to be a momentum distribu-tion centered at a finite radius away from the high symme-try point k = ( π, π ) , [ k = (0 , in the case of a lightlyelectron-doped semiconductor]. Moreover, the vortex Ma-jorana fermion and finite-energy Caroli-Matricon-de Gennesvortex states are found to be well separated in energy and inthe topological phase they together form a characteristic x-shape structure in the subgap LDOS when scanning throughthe vortex core. The Majorana mode is very well-localized inthe center of the core, while the finite-energy states dispersefurther out from the center, although the x-shape structure isstill centered at the core center.Furthermore, we show that a clear signature in the spec-tral function of the topological phase itself is the Mexicanhat shaped band structure, which also gives rise to doubleband edges, very clearly visible in the DOS due to their highconcentration of states. These double band edges also giverise to the existence of a second class of vortex core states,distinct from the ordinary Caroli-Matricon-de Gennes vortexcore states and the Majorana fermion. These vortex states ap-pear beyond the superconducting gap and forces locally largerband gap in the vortex core region, both features that are ex-perimentally measurable.Finally, we have also investigated the superconducting pairamplitude, showing that multiple pair amplitudes with total J z = 0 , both even and odd in frequency, develops in the bulkbecause of the finite spin-orbit interaction and magnetic field.In the vortex core we instead find pair amplitudes which havea total J z = − , where the vortex core momentum ( − ) hasbeen rotated into the orbital part of the pair amplitudes. De-spite multiple unconventional pairing amplitudes developingin the core, we find no amplitude that signals the onset of non-trivial topological order. Specifically, the appearance of a Ma-jorana fermion does not imply any noticeable increase in the2odd-frequency pairing. However, we find a strong correlationbetween the transition from narrow to a wide vortex core andthe development of even-frequency p − components. In sum-mary, these results provide multiple specific characteristicsfor the non-trivial topological phase and its vortex Majoranafermion in spin-orbit coupled semiconductor-superconductorheterostructures. These distinct indicators provide both addedphysical understanding of the topological phenomena in theseheterostructures and are in many cases directly experimentallymeasurable. Acknowledgments
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