Geometry-induced topological superconductivity
Po-Hao Chou, Chia-Hsin Chen, Shih-Wei Liu, Chung-Hou Chung, Chung-Yu Mou
GGeometry-induced topological superconductivity
Po-Hao Chou , Chia-Hsin Chen , Shih-Wei Liu , Chung-Hou Chung , and Chung-Yu Mou , , Physics Division, National Center for Theoretical Sciences, P.O.Box 2-131, Hsinchu, Taiwan, R.O.C. Center for Quantum Technology and Department of Physics,National Tsing Hua University, Hsinchu,Taiwan 300, R.O.C. Electrophysics Department, National Chiao-Tung University, Hsinchu,Taiwan 300, R.O.C. and Institute of Physics, Academia Sinica, Nankang, Taiwan 115, R.O.C.
Intrinsic topological superconductors with p -wave pairing are rare in nature. Its underlying reasonis due to the fact that it is usually difficult to change the relative strength between the singlet andtriplet channels for the electron-electron interaction in material. Here we show that by consideringsuperconductivity occurring on surfaces of topological insulators (TIs), the relative strength betweenthe singlet and triplet channels can be changed by geometry and sizes of TIs. Specifically, we showthat pairing of electrons at different locations on the surface of a topological insulator generally tendsto favor the triplet pairing and can induce topological superconductivity by controlling the surfacecurvature and size of the topological insulator. We illustrate the effects in two configurations, thinfilm geometry and the spherical geometry with a sphere or a hemisphere, and find that topologicalsuperconductivity arises with the p ± ip pairing symmetry dominated in nanoscale size of the TI. Asa consequence, vortices can spontaneously form on surfaces of topological insulators with roughnessof appropriate curvature. These vortices support a Majorana zero mode inside each core and canbe used as a platform to host Majorana zero modes without invoking real magnetic fields. Ourtheoretical discovery opens a new route to realize topological superconductivity in material. I. INTRODUCTION
The discovery of the topological insulators (TIs) hasnaturally led to consider superconductors with nontrivialtopology . Due to additional particle-hole symmetryimposed by superconductivity, quasi-particles in topolog-ical superconductors can realize the Majorana fermionwhich is its own anti-particle with quasi-particle opera-tors satisfying γ † = γ . In particular, the localized Ma-jorana zero mode is considered as an important buildingblock for constructing topological quantum computers .To realize topological superconductivity, a number ofproposals have been put forth. Fu and Kane proposed torealize topological superconductivity through the prox-imity of the surface states in TIs to s-wave supercon-ductors, where in the presence of magnetic fields, thesurface Dirac cone can be used to simulate the chi-ral p x + ip y pairing . This idea was further developedto use other materials with spin-orbit interactions inproximity to both ferromagnetic insulators and s-wavesuperconductors . While much experimental progresshas been made along this approach , the confinementof Majorana zero modes to the interface of the TI anda superconductor limits experimental access for manip-ulations. On the other hand, materials that develop in-trinsic topological superconductivity generally requiresthe presence of the odd-parity pairing symmetry . Inreal materials, singlet and triplet pairing channels aregoverned by the same electron-electron interactions butprojected to different channels. For the singlet pairingchannel, the wavefuction of the Cooper pair needs to besymmetric in space; while for the triplet pairing chan-nel, the wavefuction for the Cooper pair has to be anti-symmetric in space. The Coulomb interaction energy isreduced when the orbital wavefunction is antisymmetric in space; while the energy for the attractive interactionbetween electrons due to the electron-phonon interactionis reduced when the orbital wavefucntion is symmetricin space. For typical superconductors, the attractive in-teraction between electrons dominates so that the sum-mation of electron-phonon interaction and the Coulombinteraction yields a net attraction between electrons nearthe Fermi surface. As a result, the condition for the odd-parity superconductivity to emerge is usually difficult tobe satisfied. Hence natural topological superconductorsare rare. The well-known old candidates are superfluid He and Sr RuO . Recent intensive search has ledto the discovery of a number of new possible candidatessuch as Cu x Bi Se , In x Sn − x Te, PrOs Sb , β -PdBi ,UTe , and etc. Although there are experimentalevidences of unconventional superconductivity in thesematerials, due to possible alternative scenario to accountfor experimental observations , unambiguous signa-ture of topological superconductivity has not been firmlyestablished.In this paper, by considering the surface superconduc-tivity that was recently found on TIs , we would liketo explore an alternative way to realize the topologicalsuperconductivity. By exploiting the competition of theCoulomb interaction and the electron-electron attractiondue to phonons on surfaces, we show that the relativestrength of the Coulomb interaction and the electron-electron attraction due to phonons can be tuned by cur-vatures and sizes of TI. Specifically, we show that throughpairing of electrons at different locations on surfaces ofa TI, surfaces of TIs can be turned into topological su-perconductors by controlling surface curvature and sizesof TIs. We illustrate the mechanism by first consideringcompetition of pairing of electrons on the same surfaceand paring between top and down surfaces (interlayer) in a r X i v : . [ c ond - m a t . s up r- c on ] J a n the thin film geometry, and we then extend the illustra-tion to the same competition on a curve surface by usingthe spherical geometry with a sphere and a hemisphere.In the thin film geometry, we find that the inter-layerpairing with p ± ip pairing symmetry emerges when thethickness decreases to the nanoscale. In the sphericalgeometry, we show that p ± ip pairing dominates andvortices can spontaneously form without invoking realmagnetic fields. These vortices generally supports a Ma-jorana zero mode inside each vortex core. Our resultsthus indicate that TIs with appropriate geometry (suchas surface roughness) can be used as a platform to hostMajorana zero modes on their surfaces. II. THEORETICAL MODEL
We start by considering electrons in a general manifold M occupied by a topological insulator. The low energyphysics is characterized by interacting Dirac Hamiltonian H = (cid:82) M d(cid:126)r ψ † ( (cid:126)r ) (cid:18) ( m (cid:126)r − µ ) I λ so (cid:126)σ · (cid:126)pλ so (cid:126)σ · (cid:126)p ( − m (cid:126)r − µ ) I (cid:19) ψ ( (cid:126)r )+ (cid:82) M d(cid:126)rd(cid:126)r (cid:48) V ( (cid:126)r, (cid:126)r (cid:48) ) ψ † ( (cid:126)r ) ψ ( (cid:126)r ) ψ † ( (cid:126)r (cid:48) ) ψ ( (cid:126)r (cid:48) ) . Here ψ † ( (cid:126)r ) = ( C † (cid:126)r,A ↑ , C † (cid:126)r,A ↓ , C † (cid:126)r,B ↑ , C † (cid:126)r,B ↓ ) with C † (cid:126)r,τσ be-ing the creation operator of conduction electrons with or-bital τ = A, B and spin σ = ↑ , ↓ at position (cid:126)r . µ is thechemical potential. m (cid:126)r = m + m (cid:126)p is the effective Diracmass with (cid:126)p being the momentum operator, m >
0, and m <
0. The interaction V ( (cid:126)r, (cid:126)r (cid:48) ) includes the phonon-mediated attractive interaction V ph ( (cid:126)r, (cid:126)r (cid:48) ) and the repul-sive Coulomb interaction V c ( (cid:126)r, (cid:126)r (cid:48) ). We shall assumethat the superconductivity occurs in topological pro-tected surface states so that H will be projected ontothe curved surface of M and become an effective sur-face Hamiltonian H eff . The energy dispersion of H eff includes Dirac cones with particle-hole symmetry. Herethe chemical potential µ and the superconducting pairingenergy cutoff will be set to lie in lower Dirac cone. III. TOPOLOGICAL p -WAVE PAIRINGINDUCED IN THIN FILM GEOMETRY We first analyze superconductivity due to the compe-tition between V ph and V c in the thin film geometry.Let the thickness of the thin film be L . As shown inFig. 1(a), electronic states at top and down surfaces aredescribed by the Dirac cone states when L is large. Thepossible pairing of electrons are intra-surface and inter-surface pairings as indicated in 1(b) and (c). Clearly,the combined intra-surface pairing of top ( t ) and bottom( b ) surfaces is singlet-pairing, represented by ∆ s ; whilethe inter-surface pairing is triplet-pairing, represented by∆ t . The Hamiltonian for describing the surface supercon-ductivity is the specialization of Eq.(1) to the thin film FIG. 1: (a) Schematic plot of Dirac cone states on top (t) andbottom (b) surfaces in a thin film of thickness L. (b) Pairingof electrons in the same surface, ∆ s (top surface) and − ∆ s (bottom surfaces), is singlet. (c) Pairing of electrons betweentop and bottom surfaces, ∆ t , is triplet. geometry and can be written as H = (cid:88) k z Ψ † k z ( H k z − ε F )Ψ k z +12Ω (cid:88) kk (cid:48) q ,zz (cid:48) V q ,zz (cid:48) (Ψ † k − q z Ψ k (cid:48) z )(Ψ † k (cid:48) + q z (cid:48) Ψ k z (cid:48) ) , (1)where ε F is the Fermi energy, Ω is the surface area,A and B are indices for atomic orbits, z is the in-dex for quintuple layers along z-axis, k = ( k x , k y ) isthe in-plane momentum along the surface, and Ψ † k z =( C † k z,A ↑ , C † k z,A ↓ , C † k z,B ↑ , C † k z,B ↓ ) is the creation operatorfor surface electrons. The Hamiltonian matrix H k z isgiven by the following form H k z = ( m + m p z + m k ) τ z +[ λ zso p z σ z + λ so ( k x σ x + k y σ y )] τ x , (2)where τ α ( α = x, y, z ) are Pauli matrices in the pseudospace formed by atomic orbits A and B and parame-ters are given as follows: m = −
175 (meV), m = 200(meV · nm ), m = 500 (meV · nm ), λ so = 330 (meV · nm)and λ zso = 330 (meV · nm). The interaction V q ,zz (cid:48) is asummation of bare Coulomb interaction and phonon-mediated interaction, V q ,zz (cid:48) = V c q ,zz (cid:48) + V ph q ,zz (cid:48) , with V c q ,zz (cid:48) = 4 πe κ e − q | z − z (cid:48) | q ,V ph q ,zz (cid:48) = − ( Z (cid:126) qV cq ) M a (cid:88) ε φ ε ( z ) φ ε ( z (cid:48) )( v p q ) + ε . (3)Here V cq is the average of V c q ,zz (cid:48) , Z = 3 . e is the renor-malized ion charge, M = 124 . u is the average ion mass, a = 0 . v p = 2 km/s is the speed of sound, ε is the phonon disper-sion calculated by 1D phonon model with open boundary ( a ) s * t * s t ( b ) s * t * s t ( c ) s * t * s t L (QL) T c ( K ) T sc T tc L (QL) T c ( K ) T sc T tc L (QL) T c ( K ) T sc T tc FIG. 2: Thickness dependence of electron-phonon coupling λ α , strength of Coulomb interaction µ ∗ α , and T c for three dif-ferent Fermi energies : (a) 0 . ε F (b) 0 . ε F (c) ε F , where ε F is the bulk Fermi energy of Sb Te . Here L is number ofquintuple layers(QL, 1 QL ∼ s and t denoteintra-surface and inter-surface. condition along z-axis, and φ ε ( z ) is the correspondingwave function of phonon.There are four surface energy eigenstates for eachFourier mode k , which are related to the real space basisby a unitary transformation UC † k ,vα = (cid:88) zτσ U τσ ∗ k ,vα ( z ) C † k z,τσ . (4)Here v = ± specifies the spin rotation direction of the sur-face state, i.e. the chirality of the surface state, α = p, n specifies sign of the eigen-energy(positive or negative), τ = A, B specifies atomic orbits, and σ is the index forspin. We shall assume that only the negative energy sur-face states lie on the Fermi level, hence only the α = n channel is kept and we shall drop the index for α . Thescreened Coulomb interaction and phonon-mediated in-teraction, V c q ,zz (cid:48) and V ph q ,zz (cid:48) , can be obtained from thez-axis charge screening as done in Ref[31]. The resultingBCS Hamiltonian is then given by H eff = (cid:88) k (cid:88) v = ± ( ε k − µ ) C † k ,v C k ,v + 12Ω × (cid:88) kk (cid:48) v ,v ,v ,v =+ , − Γ v v v v kk (cid:48) C † k ,v C †− k ,v C − k (cid:48) ,v C k (cid:48) ,v . (5)Here Ω is the surface area of the film, ε k = − (cid:113) λ so k + ∆ g,k with ∆ g,k being the hybridized gap oftop and bottom surface states, v = ± is the index forchirality, and Γ v v v v kk (cid:48) is the projected effective interac-tion between quasi-particles of different chiralities. Thepairing amplitudes, ∆ s and ∆ t , are determined by theeffective potential by g s, kk (cid:48) = 2(Γ ++++ kk (cid:48) − Γ ++ −− kk (cid:48) ) and g t, kk (cid:48) = 2(Γ + −− + kk (cid:48) + Γ + − + − kk (cid:48) ). In the mean-field approxi-mation, the pairing term in H eff becomes H ∆ = (cid:88) k (cid:110) − i ∆ s ( C † k ↑ C †− k ↓ − C † k ↓ C †− k ↑ ) + ∆ t ( k x − ik y ) C † k ↑ C †− k ↑ − ∆ t ( k x + ik y ) C † k ↓ C †− k ↓ + h.c. (cid:111) . (6)As a result, the energy of quasi-particles is given by E k = (cid:112) ξ k + ∆ s + ∆ t with ξ k = ε k − µ and the gap equationsare given by ∆ α = − (cid:80) k (cid:48) g α, kk (cid:48) ∆ α E k (cid:48) tanh βE k (cid:48) , where α = t or s and β = 1 /k B T . Clearly, the gap equation cannotbe simultaneously satisfied for both ∆ t and ∆ s . Hence∆ t and ∆ s cannot coexist for a given thickness L . Sinceinteractions between quasi-particles crucially depend on L , their competition may lead to the change of pairingsymmetry as the thickness changes. An estimation ofthe transition temperature can be found based on theempirical McMillan formula T c = Θ D .
45 exp (cid:20) − . λ ) λ − µ ∗ (1 + 0 . λ ) (cid:21) . (7)[hbtp] Here Θ D is the Debye temperature, λ is FIG. 3: Phase diagram of superconducting orders for thinfilm geometry with the Fermi energy being in the lower surfaceDirac cone. Here L is the thickness of the thin film in the unitof number of quintuple layers, n h is the surface hole density,and n h, is the bulk hole density of Sb Te . the electron-phonon coupling constant, and µ ∗ is theCoulomb pseudo-potential. λ and µ ∗ are obtained bysetting k, k (cid:48) ∼ k F and performing the average over theangle of k (cid:48) , φ k (cid:48) ,for the effective potential due totheelectron-phonon coupling g phα, kk (cid:48) and the Coulomb inter-action g cα, kk (cid:48) respectively λ α = (cid:90) π dφ k (cid:48) g phα, k F k (cid:48) F ,µ α = (cid:90) π dφ k (cid:48) g cα, k F k (cid:48) F ,µ ∗ α = µ α µ α ln ( W/ Θ D ) , (8)where α = s or t , and W ∼ ε F is the Coulomb interac-tion cut off . Fig. 2(a) shows λ α and µ ∗ α versus L withvalues λ s at large L being consistent with experiments .In Fig. 2(b), we find that the triplet pairing wins overfor L ∼ ε F , indicating that pairingbetween parallel surfaces may enhance the triplet pairingand turn a TI into a surface topological superconductorin thin film geometry. Finally, the resulting phase di-agram of superconducting orders in thin films with theFermi energy being in the lower surface Dirac cone isgiven in Fig. 3. IV. EFFECTIVE MAGNETIC FIELD INDUCEDBY CURVATURE
The results of thin film geometry imply that pairing ofelectrons at different locations on a curved surface mayalso enhance the triplet pairing. Hence we consider su-perconductivity on a general surface. It is well knownthat the Dirac equation on a curved space acquires thespin connection as a gauge field . In particular, it isfound that a spin-1/2 Fermion on a sphere in the localframe will see an effective magnetic monopole ± / ∓ respectively . We havegeneralized this result to a general surface S and provedthat the effective local magnetic field (cid:126)B eff is proportionalto the Gauss curvature K as (cid:126)B eff = ± K ˆ n, (9)where ˆ n is the unit vector normal to the surface and ± FIG. 4: The total magnetic flux due to the spin-connectiongauge field that passes through a surface S is a topologicalinvariant, which depends on the genus g of the surface as (cid:72) S (cid:126)B eff · d(cid:126)a = ± π (1 − g ), where ± are the local spin directions(down and up) of electrons. are the local spin directions of electrons. As a result, ac-cording to the Gauss-Bonnet theorem, the total magneticflux through a surface S , (cid:72) S (cid:126)B eff · d(cid:126)a , is a topological in-variant that depends only the genus g of S as illustratedin Fig. 4. For a general surface with genus g , consid-ering possible formation of vortices, the Poincar´e-Hopftheorem further implies that the minimum number ofvortices on the surface is 2(1 − g ). The formation ofvortices is determined by (cid:126)B eff and energetics with theconfiguration of the lowest free energy being realized. V. CURVATURE INDUCEDSUPERCONDUCTING VORTEX STATES
To consider effects of effective magnetic fields on sur-face superconductivity, we shall consider the simplest sur-face with g = 0. In particularly, we shall consider thesurface on a sphere or a semi-sphere with different cur-vature. As we shall see, the results allow one to furtherestimate the formation of vortices on general surfaces. Inthe presence of effective magnetic fields induced by cur-vature, since electrons with different spin directions seeopposite monopole charges, we expect that only Cooperpairs of the triplet pairing, ++ and −− , are affected bynon-vanishing B eff , while the singlet pairing would notbe affected by B eff . The effective Hamiltonian is givenby H eff = (cid:90) d ΩΨ † Ω ( H surf − µ )Ψ Ω +12 (cid:90) (cid:90) d Ω d Ω (cid:48) V (Ω , Ω (cid:48) )(Ψ † Ω Ψ Ω )(Ψ † Ω (cid:48) Ψ Ω (cid:48) ) . (10)Here Ψ † Ω ≡ ( C † Ω , + , C † Ω , − ) is the creation operator for thelocal spinor at the solid angle Ω. H surf is the projectiveHamiltonian describing the free local surface states H surf = λ so R (cid:32) D − D + (cid:33) , (11)where R is the radius of sphere and D ± = e ± iφ ( ± ∂ θ + i sin θ ∂ φ ∓ tan θ ) . V (Ω , Ω (cid:48) ) is the effective interactionthat includes the screened Coulomb interaction adaptedto the sphere, V c (Ω , Ω (cid:48) ) = V C R sin ∆Ω2 exp( − α c | sin( ∆Ω2 ) | )and the attractive phonon-mediated interaction V ph (Ω , Ω (cid:48) ) = − V ph exp( − α ph (∆Ω) ) with ∆Ω = Ω − Ω (cid:48) .Here the interactions are modeled by a Gaussian form and the Yukawa potential form with α ph = Rξ ph and α c = Rξ C ( ξ ph and ξ C are the corresponding decay lengthand screening length). Finally, as the attractive phononinteraction is cutoff in energy, we assume that H surf iscutoff in energy near the chemical potential with thecutoff Λ.The mean-field pairing amplitude is given by ∆ s,s (cid:48) Ω , Ω (cid:48) = V (Ω , Ω (cid:48) ) (cid:104) C Ω ,s C Ω (cid:48) ,s (cid:48) (cid:105) so that the pairing Hamiltonian is H ∆ = (cid:82) (cid:82) d Ω d Ω (cid:48) (cid:80) ( s,s (cid:48) ) ∆ s,s (cid:48) Ω , Ω (cid:48) C † Ω ,s C † Ω (cid:48) ,s (cid:48) + h.c. . To solvethe self-consistent gap equation on the sphere, we needto go to the eigen-basis of H surf with the eigen-energiesbeing E ± = ± ( λ so /R )( j + 1 /
2) with j = 0 , , , · · · .The corresponding annihilation operators are denotedby C jm,E ± , which are related to the local spinor C Ω ± by C jm,E ± = (cid:82) d Ω( Y m ∗− / ,j C Ω+ ± Y m ∗ / ,j C Ω − ). Here Y mq,j are the monopole harmonics with q being the monopolecharge and ( j, m ) being the angular momentum quantumnumber. By keeping the E − operator and dropping thesubindex E ± , we obtain C † jm = √ (cid:82) d Ω( Y m − / ,j (Ω) C † Ω , + − Y m / ,j (Ω) C † Ω , − ) ,C † Ω , + = √ (cid:80) jm C † jm Y m ∗− / ,j (Ω) ,C † Ω , − = − √ (cid:80) jm C † jm Y m ∗ / ,j (Ω) . (12)The interaction V (Ω , Ω (cid:48) ) can be re-expressed in this ba-sis. First, we set γ = Ω − Ω (cid:48) and apply the additiontheorem of spherical harmonics to write V (Ω , Ω (cid:48) ) as V (Ω , Ω (cid:48) ) = (cid:88) l πV l l + 1 (cid:88) ¯ m ( − ¯ m Y − ¯ ml (Ω) Y ¯ ml (Ω (cid:48) ) , (13)where V l = 2 l + 12 (cid:90) − V ( γ ) P l (cos γ ) d cos γ, (14) P l ( x ) is the l th Legendre polynomial, and Y ¯ ml is thespherical harmonics. By using Eq.(12), the interactionterm in Eq.(10) can be expressed as (cid:90) (cid:90) d Ω d Ω (cid:48) V (Ω , Ω (cid:48) )(Ψ † Ω Ψ Ω )(Ψ † Ω (cid:48) Ψ Ω (cid:48) )= 12 (cid:90) (cid:90) d Ω d Ω (cid:48) V (Ω , Ω (cid:48) )( C † Ω , + C Ω , + + C † Ω , − C Ω , − ) × ( C † Ω (cid:48) , + C Ω (cid:48) , + + C † Ω (cid:48) , − C Ω (cid:48) , − )= 12 (cid:88) j j j j m m m m V m m m m j j j j C † j m C † j m C j m C j m , (15)where V m m m m j j j j = (cid:90) (cid:90) d Ω d Ω (cid:48) (cid:88) l πV l (2 l + 1) (cid:88) ¯ m ( − ¯ m Y − ¯ ml (Ω) Y ¯ ml (Ω (cid:48) ) × (cid:88) q,q (cid:48) = ± / Y m ∗ q,j (Ω) Y m q,j (Ω) Y m ∗ q (cid:48) ,j (Ω (cid:48) ) Y m q (cid:48) ,j (Ω (cid:48) ) . (16)The integrals involved between three harmonics are givenby (cid:90) d Ω Y ¯ ml (Ω) Y m ∗ q,j (Ω) Y m (cid:48) q,j (cid:48) (Ω)= ( − l + j + j (cid:48) − q − m (cid:20) (2 l + 1)(2 j + 1)(2 j (cid:48) + 1)4 π (cid:21) / × (cid:18) l j j (cid:48) ¯ m − m m (cid:48) (cid:19) (cid:18) l j j (cid:48) − q q (cid:19) , (17)where (cid:18) l j j (cid:48) ¯ m − m m (cid:48) (cid:19) is the 3 − j symbol. Furthermore,to ensure that the anti-commuting properties of fermionoperator are preserved in the interaction, we defineΓ m m m m j j j j = 14 ( V m m m m j j j j − V m m m m j j j j − V m m m m j j j j + V m m m m j j j j ) . (18) Eq.(10) is then turned into the following form H surfeff = (cid:88) jm [ − λ so /R ( j + 1 / − µ ] C † jm C jm + 12 (cid:88) j j j j m m m m Γ m m m m j j j j C † j m C † j m C j m C j m , (19)while the corresponding mean-field equations are givenby ∆ mm (cid:48) jj (cid:48) = (cid:88) j j m m Γ mm (cid:48) m m jj (cid:48) j j (cid:104) C j m C j m (cid:105) , (20)where Γ m m m m j j j j is the interaction V (Ω , Ω (cid:48) ) expressedin the ( j, m ) basis.To find the superconducting state, we first solve ∆ mm (cid:48) jj (cid:48) .The solution is then converted to the solid-angle basis,∆ s,s (cid:48) Ω , Ω (cid:48) . The gap function generally depends on solid an-gles Ω and Ω (cid:48) of two electrons in a Cooper pair and canbe decomposed into the superposition of products of theamplitude at the center of mass point ¯Ω = (Ω+Ω (cid:48) ) / s,s (cid:48) Ω , Ω (cid:48) = ∆ S ( ¯Ω) S (Ω , Ω (cid:48) ) +∆ P ( ¯Ω) P (Ω , Ω (cid:48) ) + · · · , where S (Ω , Ω (cid:48) ) , P (Ω , Ω (cid:48) ) , · · · de-note pairing symmetries of s-wave, p-wave, · · · withthe corresponding amplitudes ∆ s ( ¯Ω) , ∆ p ( ¯Ω) , · · · at thecenter of mass point. The s-wave, S (Ω , Ω (cid:48) ), is a con-stant. The local pairing symmetry, p x ± ip y , on a sphereis taken to be in the form and their higher harmon-ics in solid angles : ( P + iP )(Ω , Ω (cid:48) ) = αβ (cid:48) − βα (cid:48) and( P − iP )(Ω , Ω (cid:48) ) = αβ (cid:48)∗ − βα (cid:48)∗ , where ( α , β ) denotes thelocal spinor . Numerical values of the pairing ampli-tude can be obtained by first taking the limit Ω − Ω (cid:48) → . The coefficients that gowith different pairing symmetries are the amplitudes atthe center of mass point when Ω = Ω (cid:48) .In Fig. 5, we show numerical solutions of generic super-conducting states found on spheres and semi-spheres. Itis found that vortices of two different charges ( Q = 1 and Q = 2) are spontaneously formed in different regimes .Fig. 5(a) shows ∆ ++ with the pairing symmetry p x + ip y in color-scale for the vortex solution with one vortex ofcharge Q = 1 at the north pole and the other vortex ofcharge Q = 1 at the south pole. Phase with such vor-tices is termed as the vortex phase. A similar solutionwith minor component exhibiting the vortex solution ofcharge Q = 2 is also found. Phase with such vortices istermed as the vortex* phase. This is consistent with thePoincar´e-Hopf theorem which states that the total wind-ing number (= Q ) of a vector field on a sphere shouldbe two. In Fig. 5(b), we show that the spontaneously-formed vortex solution survives on a semi-sphere, whichsimulates a bump on the surface. This solution impliesthat vortices can also form on bumps with appropri-ate curvature on surfaces of TIs. Fig. 5(c) shows that FIG. 5: Self-consistent vortex solution on the surface of a spherical TI and a semi-spherical TI. Here λ so = 35meV · nm, ξ C = 4nm, ξ ph = 50nm, R = 50 nm, V ph = 7 . V C = 50 meV · nm, µ = − . . . × − nm − . (a) The superconducting amplitude, ∆ ++ (meV) (pairing symmetry p x + ip y ), on a sphere forthe vortex solution with vortex charge Q = 1 in the vortex phase. Here the color-scale decreases from 0.001meV(yellow) to0(blue). Contours and arrows mark directions of supercurrents. (b) The superconducting amplitude, ∆ ++ (meV), for a vortexsolution on a semi-sphere. (c) The quasi-particle energy spectrum (in the unit of meV) inside the vortex core exhibits theMajorana zero mode (indicated by the arrow, tolerance is 0.02meV). Here m and n are z-component of angular momentum ofthe quasi-particle. (d) Relative strength of the singlet pairing to the triplet pairing (∆ ++ − ) versus Ω − Ω (cid:48) in the Meissner phase.Here V C = 70 meV and T = 1 . ++ and ∆ −− for vortex phase (e) and vortex*phase (f). Here θ is the azimuthal angle relative to the north pole, red circles represent ∆ +++ , blue squares represent ∆ −−− , andblack dots represent ∆ ++ − (=∆ −− + ). each vortex in the vortex phase support a Majorana zeromode in the core. For the vortex* phase, since the vor-tex solution only exists in the minor component (∆ ++ − ,see Fig. 5(f)), the vortex core does not host Majoranazero mode . Fig. 5(d) shows the relative strength of sin-glet pairing to triplet pairing versus Ω − Ω (cid:48) in the Meiss-ner phase in which there is no vortex. Here only ∆ ++ and ∆ −− (= ∆ ++ ) are non-vanishing triplet componentsin the Meissner phase. Clearly, due to the momentum-locking of Dirac fermions at different solid angles, thetriplet pairing has higher chance of occurrence and dom-inates. This results in the topological superconductivity.The triplet component for the vortex phase can be gener-ally expressed as ∆ ++ = ∆ ++ − ( p x − ip y ) + ∆ +++ ( p x + ip y )and ∆ −− = ∆ −−− ( p x − ip y ) + ∆ −− + ( p x + ip y ). The com-ponents for different pairing symmetries are shown inFig. 5(d) and (e). These components ∆ sss (cid:48) can be ex-pressed in terms of monopole harmonics.As the local magnetic field is determined by 1 /R , theinterplay between the Meissner phase and vortex phaseson a sphere is generally controlled by the radius and thestrength of Coulomb interaction. Fig. 6 shows typicalphase diagrams we find for superconducting states on asphere. It is seen that the vortex phase generally existsat low temperature on surfaces with radius of curvature being around the nanoscale (smaller than 50 nm). T (mev) ( = R ) ( n m ! ) Meissner
Vortex Vortex $ ( a ) T (mev) -101234 l og ( V C = ) MeissnerVortexVortex $ Normal ( b ) FIG. 6: (a) The phase diagram of superconducting stateson a sphere: 1 /R versus temperature T . Here 1 /R actsimilar as the local magnetic field. (b) The phase diagramof superconducting states on a sphere: V C versus T . Here ξ C = 4nm, ξ ph = 50nm, V ph = 7 . µ = − . . R → ∞ , which can be accessed in (b) with V C → VI. DISCUSSION AND CONCLUSION
Although the emergence of vortices with Majoranazero mode is mainly illustrated on a perfect sphere, ourresults are not restricted to a perfect sphere. This is il-lustrated in Fig. 5(b) by a semi-sphere, which simulates abump on a plane. The solution also contains a Majoranazero mode inside the vortex core. It implies that vorticescan also spontaneously form on bumps with appropriatecurvature on surfaces of TIs. On general surfaces, thevortex state is competing with the Meissner phase in en-ergy. At the critical curvature when the vortex state witha single vortex wins over the Meissner phase, the free en-ergy for the vortex state is equal to that of the Meissnerphase. Similar to lower critical field H c for conventionalsuperconductors, this determines the critical curvature K c for the formation of a single vortex as K c = ± π(cid:15) Φ , (21)where (cid:15) is the free energy per unit length of the vor-tex and Φ is the flux quantum. Hence there is a criticradius of curvature for forming a vortex. Below the crit-ical radius of curvature, the Meissner phase wins. Thenumerical values for K c can be estimated by using thephase digram shown in Fig. 5(a), where at different tem-peratures, the critical curvature for a vortex to form isdetermined by the boundary between the Meissner phaseand the vortex phase. Our results can be thus appliedto an ellipsoid or bumps on surface of topological insula-tor in which vortices with Majorana modes could emergeat places where the local curvature ( 1 /R ) exceeds thethreshold as marked in Fig. 5(a).The connection of the thin-film geometry to the spher-ical geometry is in the fact that they all belong to the g = 0 surfaces. In this case, the Poincar´e-Hopf theoremimplies that the minimum number of vortices is two, butgenerally, number of vortices in a vortex state with charge 1 can be 2, 4,6, . . . . The extra vortices that deviate from2 contain vortex pairs of ± g = 0. Dependingon curvatures of the corners, if they exceed the thresholdcurvature, vortices can form. The formation of vorticeswill appear at corners. However, minimum number ofvortices that can appear is still two. In this case, onestill has to control the curvature of the corner to enablethe formation of vortices.In conclusion, we have shown that pairing of electronson different locations of a TI can generally induce topo-logical superconductivity through the control of surfacecurvature and size of the TI. These are illustrated bothin the thin film geometry and in the spherical geometry.The generic phase diagram of topological superconduc-tivity on a sphere is constructed. For general surfaces,our results imply that vortices can spontaneously formedon bumps with appropriate curvature on surfaces of TIs.These vortices generally host a Majorana zero mode in-side each core, which is shown to be robust against thepresence of the disorder potential. We expect that theexperimental detection can be performed on surfaces ofSb Te by using STM and one looks for surface rough-ness of size around 50 nm or less at temperatures below9K. The surface of a TI can be thus used as a platformto host Majorana zero modes without invoking real mag-netic fields. Acknowledgments
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