PHYSICAL REVIEW B 87, 035401 (2013)

Vortex lattices in the superconducting phases of doped topological insulators and heterostructures

Hsiang-Hsuan Hung,1,2 Pouyan Ghaemi,3 Taylor L. Hughes,3 and Matthew J. Gilbert1,2

1Department of Electrical and Computer Engineering, University of Illinois, Urbana, Illinois 61801, USA2Micro and Nanotechnology Laboratory, University of Illinois, 208 N. Wright St, Urbana Illinois 61801, USA

3Department of Physics, University of Illinois, Urbana, Illinois 61801, USA(Received 27 September 2012; published 2 January 2013)

Majorana fermions are predicted to play a crucial role in condensed matter realizations of topological quantumcomputation. These heretofore undiscovered quasiparticles have been predicted to exist at the cores of vortexexcitations in topological superconductors and in heterostructures of superconductors and materials with strongspin-orbit coupling. In this work, we examine topological insulators with bulk s-wave superconductivity in thepresence of a vortex lattice generated by a perpendicular magnetic field. Using self-consistent Bogoliubov–deGennes calculations, we confirm that beyond the semiclassical, weak-pairing limit the Majorana vortex statesappear as the chemical potential is tuned from either side of the band edge so long as the density of statesis sufficient for superconductivity to form. Further, we demonstrate that the previously predicted vortex phasetransition survives beyond the semiclassical limit. At chemical potential values smaller than the critical chemicalpotential, the vortex lattice modes hybridize within the top and bottom surfaces, giving rise to a dispersivelow-energy mid-gap band. As the chemical potential is increased, the Majorana states become more localizedwithin a single surface but spread into the bulk toward the opposite surface. Eventually, when the chemicalpotential is sufficiently high in the bulk bands, the Majorana modes can tunnel between surfaces and eventuallya critical point is reached at which modes on opposite surfaces can freely tunnel and annihilate leading to thetopological phase transition previously studied in the work of Hosur et al. [Phys. Rev. Lett. 107, 097001 (2011)].

DOI: 10.1103/PhysRevB.87.035401 PACS number(s): 71.10.Fd, 75.10.Jm, 71.10.Pm, 75.40.Mg

I. INTRODUCTION

Majorana fermions, quasiparticle excitations which aretheir own antiparticle, were originally proposed in high-energy physics but1 have now arrived at the forefront ofcondensed matter physics. It is predicted that the braidingof multiple Majorana modes will cause nontrivial rotationswithin the degenerate many-body Hilbert space and forms thebackbone of many proposed topological quantum computingarchitectures.2–8 Within condensed matter physics, there existmany candidate systems which are predicted to harbor Majo-rana fermions. One of the earliest of such candidates is the frac-tional quantum Hall effect at filling factor ν = 5

2 (Ref. 9), thephysics of which may be described by the Moore-Read Pfaffianwave function.10 While this state is yet to be experimentallyconfirmed, tantalizing evidence observed in tunneling in quan-tum constrictions points to the fact that the ν = 5

2 fractionalquantum Hall state does possess non-Abelian statistics aswould be necessitated by the presence of Majorana fermions.11

Beyond the fractional quantum Hall states, other possiblesystems thought to contain Majorana fermions are the px + ipy

superconductors2,7,10,12,13 where the relevant Majorana modesare predicted to appear as bound states on exotic half-quantumvortices, which were recently observed in magnetic forcemicroscopy experiments performed in Sr2RuO4.14 In additionto fractional quantum Hall states and superconductors, therehave been an abundance of proposals to realize Majoranafermions in materials with strong spin-orbit coupling. Notableexamples are: proximity-induced superconductivity in three-dimensional (3D) topological insulators (TIs),15 bulk super-conductivity in doped TIs,16 and semiconductors proximitycoupled to s-wave superconductors.17–20 Indeed, the latterproposals have led to exciting measurements in high-mobilityquantum wire s-wave superconductor systems.21

In this article, we will focus on the two mechanismsproposed in TI materials. As is now well known, TIs arematerials which possess an insulating bulk but contain robustmetallic states that are localized on their surfaces.22–32 Wewill consider time-reversal-invariant 3D topological insulatorswhich harbor an odd number of massless Dirac cones on eachsurface. As mentioned, there currently exist two proposals thatutilize topological insulators as a platform for the observationof Majorana fermions. The first of which is to consider an s-wave superconductor/topological insulator heterostructure inwhich a superconductor is coupled to the topological insulatorvia the proximity effect and subjected to a vortex-producingmagnetic field.15 In Fu and Kane’s pioneering work,15 theyshow that in the s-wave superconductor/TI heterostructure,the interface between the TI and the superconductor behavessimilar to a spinless chiral p-wave superconductor yet withoutbreaking time-reversal symmetry. As such, Majorana fermionswill reside in the vortex cores15,33 so long as the quantizedmagnetic flux lines penetrating the system are broad.34

On the other hand, another strategy to realize Majoranafermions is to consider vortex bound states in a 3D TI withbulk s-wave superconductivity.5,16,35 Bulk superconductivityin doped topological insulators has been observed in recentexperiments that dope Bi2Se3 with copper.36–38 For thisparticular material, the nature of the order parameter is stillunder debate, but the two most probable options are s waveor an interorbital topological pairing parameter.39 There is anopportunity to observe Majorana fermions in both cases, butfor the purpose of this work we will only consider the s-wavecase. Recent work16 reveals that, while doped topologicalinsulators that develop s-wave pairing may harbor Majoranabound states in the vortices, the Majorana fermions do notsurvive for all doping levels. Specifically, there exists a critical

035401-11098-0121/2013/87(3)/035401(11) ©2013 American Physical Society

HUNG, GHAEMI, HUGHES, AND GILBERT PHYSICAL REVIEW B 87, 035401 (2013)

chemical potential μc at which point the system undergoesa topological (vortex) phase transition. This phase transitioncan be regarded as a topology change in the 1D electronicstructure of vortex lines from a system which supports gaplessend states to one that does not. Therefore, only at chemicalpotentials below the critical value, the doped superconductingBi2Se3 supports Majorana modes at the vortex ends (placeswhere vortex lines intersect the surface). The work of Ref. 16provided a semiclassical treatment in the infinitesimal pairinglimit, but such an approach might be inadequate to captureimportant quantum effects. One such effect that can not bedetermined this way is the zero-point energy contribution ofthe vortex core states which could shift the states away from thezero-energy gapless point and invalidate the previous analysis.In the weak-pairing limit, the physics is determined by thestructure exactly at the Fermi surface, and it is possible thatthe energy spectrum away from the Fermi level can serve torenormalize the location of the critical point. If the critical pointis sufficiently shifted toward the band edge, it could be thatthere is never a viable doping range over which the Majoranafermions can be observed. Calculations provided in this articlego beyond the semiclassical, and the infinitesimal weak-pairing limit is thus essential to confirm the previous results.

In both TI-based approaches we have thus far discussed,there is an additional assumption underlying the resultantphysical predictions, namely, that the vortices are completelyisolated. However, this may not be the most appropriate or ex-perimentally relevant picture for the observation of Majoranastates in either type of TI approach. Thus, in this article, weexamine the behavior of 3D topological insulators with bulks-wave superconductivity in the vortex lattice limit as a func-tion of doping level. The paper is organized in the followingfashion: In Secs. II and III, we introduce the 3D topological in-sulator model Hamiltonian which is used for each of the subse-quent calculations, and the background for the self-consistentcalculations, respectively. In Sec. IV, we present the results ofour calculations for three separate geometries: (a) periodicboundary conditions with vortex rings, (b) open boundaryconditions with vortex lines terminating on the TI surface,and (c) an inhomogeneously doped heterostructure with openboundary conditions. We find that as the chemical potential ismoved from the gap into the bulk bands, the Majorana statesform when the density of states is large enough to support awell-formed superconducting gap. As the chemical potentialmoves past this onset value, we find that the vortices are local-ized on the surfaces but hybridize with neighboring vorticeson the same surface, giving rise to a dispersive low-energyquasiparticle spectrum. As the chemical potential is pushedfurther into the bands, we find a critical chemical μt at whichintersurface tunneling is enabled through a gapless channel onthe vortex line. The value is renormalized from that stated inRef. 16, but we find that even for strong attractive interactions

that μt remains at a finite value of the bulk doping. After thechemical potential exceeds μt , we find that a gap opens inthe spectrum and there are no longer any low-energy localizedmodes remaining. Additionally, in Sec. V, we evaluate the su-perconducting gap equation in order to determine the relevanttemperature scale on which these effects may be observed.Finally, in Sec. VI, we summarize our findings and conclude.

II. MODEL HAMILTONIAN OF A 3D TOPOLOGICALINSULATOR

We will use a minimal, four-band Dirac-type model which,with the proper choice of parameter values, captures the bulk,low-energy physics of known TI materials such as Bi2Se3

(Refs. 40 and 41):

H =∑

�r

{�

†�r Hm��r +

∑�δ

�†�r H�δ��r+�δ

}, (1)

Hm = M�0, H�δ =∑

�δ

b�0 + iγ �δ · ��2a2

, (2)

where ��r = (cA,↑,�r cA,↓,�r cB,↑,�r cB,↓,�r )T is a four-componentspinor with A/B and ↑ / ↓ labeling orbital and physicalspin, respectively, so that c

†α,σ,�r is the creation operator

for an electron with spin σ in orbital α at position �r,�δ = ±ax, ± ay, ± az are vectors that connect nearestneighbors on a simple cubic lattice with lattice constanta, the vector �� = �xx + �yy + �zz with �α = τ x ⊗ σα

and �0 = τ z ⊗ I, where α = x, y, z; τα and σα are 2 × 2Pauli matrices acting on orbital and spin degrees of freedom,respectively. We also define I as the 2 × 2 identity matrixand M = m − 3b/a2 as the mass parameter which controlsthe magnitude of the bulk band gap. The TI/trivial insulatorphase depends on the chosen values for the parameters m andb and the TI phase has m/b > 0 while the trivial phase hasm/b < 0. By tuning the material parameters γ, b, m, and a

in Eq. (1), one can model the low-energy effective model forthe common binary TI materials.40,41 Since we only addressthe qualitative effects stemming from the TI phase, wewill fix the parameters to be b = a2(1 eV), γ = a(1 eV),and m = 1.5 eV in terms of the lattice constant a so thatM = −1.5 eV, thereby ensuring that we are in the TI phase.

With translation invariance and periodic boundary condi-tions in all x, y, and z directions, it is often more convenientto work in momentum space. In this case, we expect tosee no gapless states due to the lack of a boundary. TheFourier-transformed Dirac Hamiltonian in momentum spacemay be written as

H =∑

�k�

†�kH0(�k)��k, (3)

where H0(�k) in Eq. (3) is a 4 × 4 matrix which we may write as

H0(�k) =

⎛⎜⎜⎜⎜⎝

M + g(�k) 0 sin kz sin kx − i sin ky

0 M + g(�k) sin kx + i sin ky − sin kz

sin kz sin kx − i sin ky −[M + g(�k)] 0

sin kx + i sin ky − sin kz 0 −[M + g(�k)]

⎞⎟⎟⎟⎟⎠ , (4)

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VORTEX LATTICES IN THE SUPERCONDUCTING PHASES . . . PHYSICAL REVIEW B 87, 035401 (2013)

with g(�k) = cos kx + cos ky + cos kz. If we instead chooseopen boundaries along one direction, there will be robustgapless edge states on those boundary surfaces for the samechoice of model parameters.

III. TOPOLOGICAL INSULATORS WITH BULK S-WAVESUPERCONDUCTIVITY

In this work, we are interested in the properties of dopedTIs which become intrinsically superconducting at low tem-perature. In a doped topological insulator, like any other metal,when the chemical potential is in the conduction or valenceband, an attractive interaction will lead to the formationof superconductivity and generate a superconducting gapat the Fermi surface. In order to study the formation ofsuperconductivity in doped TI, we add an attractive Hubbard-type density-density interaction to the Hamiltonian inEq. (1):

Hint = −|U |∑

�rn↑,�rn↓,�r , (5)

where nσ,�r = c†A,σ,�rcA,σ,�r + c

†B,σ,�r cB,σ,�r and the parameter

−|U | represents the attractive intraorbital interaction.At the mean-field level, the interaction term may be

decoupled as42

−|U |∑α,�r

{∗α,�rcα,↓,�r cα,↑,�r + α,�rc

†α,↑,�r c

†α,↓,�r − |α,�r |2},

where α,�r = 〈cα,↓,�r cα,↑,�r〉 is the standard intraorbital s-wavepairing order parameter. Combining this with Eq. (1), we getthe Bogoliubov–de Gennes (BdG) Hamiltonian

HBdG =∑

�r�

†�r

(Hm − μ�r (�r)

(�r)† −H ∗m + μ�r

)��r

+∑�r,�δ

�†�r

(H�δ 0

0 −H ∗�δ

)��r+�δ, (6)

where ��r = (��r ,�†�r )T is now an eight-component Nambu

spinor, and (�r) denotes a 4 × 4 pairing matrix. In thisexpression, the interaction −|U | has been absorbed into thepairing matrix (�r), which we write as

(�r) = −|U |

⎛⎜⎜⎜⎝

0 A,�r 0 0

−A,�r 0 0 0

0 0 0 B,�r0 0 −B,�r 0

⎞⎟⎟⎟⎠ . (7)

To study the bulk superconductivity, we will assume μ�r = μ

is uniform throughout the material for simplicity.The BdG Hamiltonian of Eq. (6) can be diagonalized by

applying a Bogoliubov transformation as42

(��r�

†�r

)=

∑n

(un,�r −v∗

n,�rvn,�r u∗

n,�r

) (γn

γ†n

), (8)

where n labels the eigenstate index. Plugging the transforma-tion into Eq. (6), we have

HBdG

∑n

(un,�r −v∗

n,�rvn,�r u∗

n,�r

)

=∑

n

(En 0

0 −En

) (un,�r −v∗

n,�rvn,�r u∗

n,�r

). (9)

This indicates that the eigenvectors associated with En (−En)of the above BdG equations are (un,�r ,vn,�r )T [(−v∗

n,�r ,u∗n,�r )T ].

The mean-field pairing order parameters are obtained via

α,�r = 〈cα,↓,�r cα,↑,�r〉=

∑n

un,�rv∗n,�r tanh

βEn

2, (10)

where β = 1/kBT . Once the pairing order parameter is deter-mined initially, it is plugged back into the BdG Hamiltoniangiven in Eq. (6) and then HBdG is diagonalized again asshown in Eq. (9). The process continues until we reachself-consistency and we have a convergent α,�r for all �r . Wenote that in our numerical calculations we use a small, nonzerotemperature in order to avoid divergences, but this temperatureis much smaller than the superconducting gap so as not to affectthe physical results.

In Fig. 1, we show the self-consistently determined intraor-bital pairing order parameter in the bulk as a function of |μ| atdifferent |U |, where, due to translation invariance, α,�r = α.

In this paper, we will only consider p doping (μ < 0), butthe particle-hole symmetry of the model Hamiltonian inEq. (1) makes the electron-doped case similar in nature. WithM = −1.5 eV in Eq. (1), the top of the bulk valence band islocated at μv = −0.5 eV and the total size of the insulating gapis 1.0 eV. We see from Fig. 1 that when the chemical potentialis in the gap where there is no carrier density with whichto form Cooper pairs, the resulting pairing potential is zero.When the chemical potential enters the valence band, a Fermisurface develops, and low-energy states become available to

FIG. 1. (Color online) Intraorbital pairing order parameters A

(solid symbols) and B (hollow symbols) vs |μ| at different |U |.|μ| and |U | are in units of eV. Via our definition for pairing inEq. (10), remains unitless. The mass term M = −1.5 eV. Thesystem contains periodic boundary conditions in x, y, and z directions.The simulations are performed on a lattice grid of size 80a × 80a ×10a.

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HUNG, GHAEMI, HUGHES, AND GILBERT PHYSICAL REVIEW B 87, 035401 (2013)

pair. However, when the density of states at the Fermi level isinsufficient, the size of the pairing potential will continue to beexponentially small. As we see in Fig. 1, a significant pairingpotential does not form until |μ| is well above the valence bandedge |μv|.

This result matches standard BCS phenomenology andrepresents the point of inception for the remainder of the paper.To be specific, in Ref. 16, Hosur et al. used a semiclassicaltreatment to show that a vortex in the superconducting phaseof a doped topological insulator exhibits a topological phasetransition as the chemical potential is tuned through a criticalvalue. The two phases separated by this transition are gappedand differ by the presence or absence of Majorana modes atthe ends of the vortex, i.e., where the vortex line intersectsthe surface of the TI. At the transition point, the vortexline becomes gapless and provides a channel which allowsthe Majorana modes to annihilate one another by tunnelingin-between the opposing surfaces. In their treatment, however,there is an assumption of adiabaticity (i.e., the continuoustransformation of the system Hamiltonian from insulating stateinto the superconducting state without closing the band gap)as it is always assumed that at any chemical potential otherthan the critical chemical potential, there is no gapless channelto hybridize the Majorana mode. This seems innocuous, butone has to remember that the arguments rely on the adiabaticconnection between a gapped insulating phase and a gappedsuperconducting phase. This assumption becomes importantwhen one considers the behavior of the system as the chemicalleaves the insulating gap and enters the bulk bands. Althoughthis is a reasonable assumption within which to theoreticallystudy the vortex phase transition, one may then ask whathappens in the region where the chemical potential is not largeenough to form a significant pairing potential, and there isfinite density of gapless modes in the bulk. In other words,how does the Majorana mode emerge out of the bulk gaplessstates? This question is certainly relevant for experimentswhere a finite-size TI sample is used. Our self-consistentsolution of the BdG equations in the vortex lattice in thestrong-pairing limit (i.e., when the magnitude of attractivepotential is comparable with the bandwidth) can present aclearer picture of the appearance of the Majorana modes andthe vortex phase transition than the previous semiclassicalanalysis.

IV. VORTEX LATTICES IN SUPERCONDUCTING PHASEOF DOPED TOPOLOGICAL INSULATORS

The self-consistent BdG formalism is in a real-space basisand thus can be also used to study the vortices in thesuperconducting phase where the order parameter will benonuniform. To induce vortices, we consider the system undera uniform magnetic field �B = Bz. When electrons are hoppingon the xy plane, this generates a Peierls phase factor, and theBdG Hamiltonian becomes34,43

HBdG =∑

�r�

†�r

(Hm − μ�r (�r)

†(�r) −H ∗m + μ�r

)��r

+∑�r,δ

�†�r

(Hδe

−iη�r 0

0 −H ∗δ eiη�r

)��r+δ, (11)

where η�r denotes the extra phase given by the vector potential�A(�r) induced by the magnetic field through �B = ∇ × A(�r):

η�r = π

�0

∫ �r+δ

�r�A�r ′ · d�r ′, (12)

where �0 is the superconducting flux quantum; �0 = h2e

. Inthe following discussion, we choose the Landau gauge, i.e.,�A(�r) = (Ax,Ay) = (0,Bx).

We will treat the system as a type-II superconductor in avortex lattice state. In each magnetic unit cell, the amountof magnetic flux is 2�0, so that each unit cell carries twosuperconducting vortices.44 We designate the size of eachmagnetic unit cell as lxa × lya × lza using the integers li todenote the number of lattice sites in each spatial direction. Forour choice of geometry, we will use square vortex lattices, andfix lx = ly/2. The corresponding magnetic field magnitude is

B = 2�0

lx lya2. (13)

From this relation, we can observe that stronger magnetic fieldsbring smaller magnetic unit cells, in which vortices are closerto each other. Therefore, the dilute vortex limit comes fromapplying very weak magnetic fields. As is standard for latticecalculations with uniform field, in order to see experimentallyreasonable field sizes, we would need to use a very largenumber of unit cells as, for example, the case when lx = ly = 1gives a magnetic field on the order of thousands of Tesla. Forour system sizes, we have an unphysically large magnetic fieldon the order of 103 T, assuming a lattice constant of 1 A. This,however, will not affect the qualitative physics in which we areinterested and we will not worry about this issue any further.

We choose the entire system size as Lxa × Lya × lza suchthat there are Nx × Ny magnetic unit cells, where Nx = Lx/lxand Ny = Ly/ly and Nx = 2Ny . Since each magnetic unitcell carries two vortices, the Lxa × Lya × lza vortex latticecontains 2NxNy vortices. In Fig. 2, we show a schematicillustration of a 4 × 4 square vortex lattice. By tuning sizesof the magnetic unit cells, we can study the vortex lattice atdifferent external magnetic fields. In this paper, we set lx > 8to avoid strong overlap between vortices, but lx � 12 due tocomputational limitations.

We consider a system with periodic boundary conditionsalong the x and y directions. Although the vortices breaklattice translation invariance, we still have magnetic periodicboundary conditions for vortex lattices. In addition to the

FIG. 2. A 4 × 4 vortex lattice. In this example, the number ofmagnetic unit cells is Nx × Ny = 4 × 2. Each black solid circledenotes a vortex location and each magnetic unit cell contains twovortices.

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VORTEX LATTICES IN THE SUPERCONDUCTING PHASES . . . PHYSICAL REVIEW B 87, 035401 (2013)

phases given by vector potentials eiη�r , the magnetic periodicboundary conditions also contribute another phase factor whenelectrons are hopping across unit-cell boundaries.45,46 Supposethat in a 2D vortex lattice the translation vector in units ofa is written as �R = (Xlx,Yly), where X = 0, . . . ,Nx − 1 andY = 0, . . . ,Ny − 1 are integers. The coordinate of an arbitrarylattice site can be expressed as �r + �R, where �r = (x,y) denotesthe coordinate in units of the lattice site a, within a magneticunit cell, i.e., 1 � x � lx and 1 � y � ly . Under the magneticperiodic boundary conditions, we can define the relation ofthe magnetic Bloch wave functions47 which have a periodicstructure written as47,48

(un(�r + lx x)

vn(�r + lx x)

)= eikx

(e

2πiy

ly un(�r)

e−2πi

y

ly vn(�r)

),

(un(�r + ly y)

vn(�r + ly y)

)= eiky

(un(�r)

vn(�r)

). (14)

Here, kx = 2πXNx

and ky = 2πYNy

represent the x and y com-

ponents of the magnetic Bloch wave vector. The phases eikx

and eiky arise from hopping to neighboring cells. Addition-

ally, e±2πi

y

ly is provided by the magnetic periodic boundaryconditions or quasiperiodic boundary conditions.47 The BdGeigenstates (un,�r ,vn,�r )T satisfy magnetic translation invarianceunder Eq. (14).

The onsite pairing potential can be expressed as α,�r =|α,�r |eiφ(�r) with a phase eiφ(�r) and amplitude |α,�r |. In thepresence of vortices, both the pairing potential and the phaseare site dependent. The superconducting order parameters aresuppressed near the vortex cores, and are restored to the bulkvalues away from the vortex cores. The spatial form of thepairing order parameters α,�r are determined self-consistently.We consider different attractive Hubbard interaction strengths|U | and uniform doping levels |μ| distributed through theentire bulk.

We study two different geometries for the vortex lattice.First, we consider vortices oriented in the z direction (alongthe applied magnetic field) with periodic boundary conditionsalong the x,y,z directions which yield vortex rings loopingaround the z direction. In this geometry, we study the vortexphase transition where the vortex modes become gapless alongthe vortex rings. The second geometry we consider has openboundaries in the z direction. In this case, the vortex linesterminate at the open surfaces perpendicular to the z axisand we can study the Majorana modes that can appear at thevortex ends in the topological phase. We compare these results,which are neither in the semiclassical nor infinitesimallyweak-pairing limits, to the results studied in Ref. 16, whichare in these limits.

A. Periodic vortex rings in vortex lattices

With periodic boundary conditions in all spatial directions,we can not directly study the Majorana modes that mightappear at the vortex ends. However, we can indirectly studythem by identifying the vortex phase transition through astudy of the low-energy modes along the vortex lines. Asthe chemical potential is tuned deeper into the band, thepoint where one of these modes becomes gapless signals the

location of a critical point. For this geometry, the magneticunit-cell sizes we use are lx × ly × lz = 12 × 24 × 10. Wechoose Nx × Ny = 10 × 5 unit cells so that there are 100vortices in the vortex lattice.

In Fig. 3, we present the evolutions of the low-energystates versus |μ| for different interaction strengths |U |. Wecan identify two distinctly different doping regimes. In thefirst regime, the chemical potential lies in the valence bandbut below a value we call |μo|, which signals the onsetof a well-formed superconducting gap discerned from ournumerics. It should be noted that |μo| has no real intrinsicmeaning (as it is strongly finite-size dependent) and only servesto indicate a common feature shared by all of our spectrumplots. Clearly, before the chemical potential hits the top ofvalence band, there is no density of states to generate the

FIG. 3. (Color online) The energy spectra of the low-energy statesvs |μ| for periodic boundary conditions along the z axis (vortex rings)for (a) |U | = 2 eV, (b) |U | = 2.5 eV, and (c) |U | = 2.8 eV. Thesystems have a Nx × Ny = 10 × 5 (100 vortices) vortex lattice andthe size of the magnetic unit cell is lx × ly × lz = 12a × 24a × 10a.

We note that in this work, we assume that the Zeeman splitting dueto the applied magnetic field is negligible.

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superconducting gap, and all the states are gapped by the bulkinsulating band gap. After the chemical potential hits the topof valence band, the superconducting pairing starts to form butit is exponentially small in magnitude. Comparing the pairingstrengths without vortices (i.e., Fig. 1) to the vortex latticecase in Fig. 3, our numerics show that in the vortex lattice thepairing is more poorly formed over a larger range of doping.That is, the states at the Fermi level remain gapless with nosuperconducting gap formation. Note that μo decreases withincreasing |U |, which indicates that this point is sensitive to thepoint where the exponentially suppressed superconducting gapwould turn on. In this regime, any localized Majorana modes orlow-energy vortex core states are difficult to distinguish fromthe extended gapless metallic states in the bulk. The detailsof this regime are dominated by strong finite-size effects.One hindrance is that for cases where only a tiny pairingpotential would form, it is numerically challenging to generatea convergent, self-consistent solution with vortices present. Inthe thermodynamic limit, we would expect to see a nonzerobut exponentially small pairing gap as soon as the chemicalpotential hits the valence band. Here, the situation is not soclear, and unfortunately, due to the computational limitations,we can not glean a great deal of physical information fromthis regime except that it is not obvious that the picture ofan “adiabatic” continuation from the gapped insulating stateimmediately to a gapped superconducting state would be validin a real sample. We will attempt to address this issue fromanother direction by studying a heterostructure geometry inSec. IV C in which we can generate a convergent, vortex latticesolution by inhomogeneously doping the system, i.e., highdoping on the surface and low doping in the bulk.

In the second distinct regime, once |μ| is tuned beyond |μo|,then significant s-wave pairing begins to develop. Because ofthe particle-hole constraint of the BdG quasiparticle spectrum,the energies appear in ±E pairs. The lowest-energy branchesare nearly (2 × Nx × Ny)-fold degenerate. This degeneracyclearly indicates that these states are in-gap vortex statesas there is essentially one for each vortex. As the chemicalpotential is pushed more into the valence band, the lowest-energy branch approaches zero energy, and at critical chemicalpotential |μt |, the particle and hole branches cross indicatingthe location of the vortex phase transition. In the weak-pairingtreatment, the critical chemical potential is independent of thevalue of the attractive potential |U | and if we repeat theiranalysis for our choice of parameters, we find a weak-pairingestimate of |μt | = 1.35 eV. In our case, as the interactionstrength is quite large, we are not in the weak-pairing limitand the critical chemical potential depends on the attractivepotential. At |U | = 2, 2.5, and 2.8 eV, |μt | 1.26, 1.22, and1.2 eV, respectively. A stronger |U | gives a smaller value of |μt |and it approaches to the weak-pairing limit as we decrease themagnitude of interaction. Since the phenomenon survives theweak-pairing limit, it is possible that the vortex topologicalphase transition could also be observed in a strong-pairingatomic limit which is realizable in ultracold optical lattices.49

In Fig. 4, we show the self-consistent vortex profiles in asingle unit cell as a function of |μ| for |U | = 2.8 eV. It isevident that in all cases around the vortex cores, the pairingorder parameters are suppressed. Away from the vortex cores,the pairing order parameters are restored to A = 0.21, 0.213,

FIG. 4. (Color online) The spatial pairing order parameter profiles|A,�r | within a unit cell at |U | = 2.8 eV and at (a) |μ| = 0.96 eV, (b)|μ| = 0.98 eV, and (c) |μ| = 1.3 eV. The vertical axis represents thepairing order parameter magnitudes and the horizontal plane is the xy

plane. Note the variation of the pairing order parameter magnitudesat the unit-cell center center =

A,�r∈unit cell center for differentμ: (a) center 0, (b) center 0.12, and (c) center 0.28.center 0 indicates that the vortex core stays onsite, whereascenter �= 0 means that the vortex core moves off the lattice verticesand into the plaquette. We note that the profiles for |B,�r | look similar.

and 0.29, which are roughly equal to the corresponding bulkvalues at |μ| = 0.96, 0.98, and 1.3 eV, respectively (cf. Fig. 1).In the bulk superconducting TI, at larger |μ|, stronger Cooperpairing is induced, and the strong superconductivity leads toa shorter coherence length ξ0 (ξ0 = hvF

π),42 and thus a smaller

vortex size. Therefore, in Figs. 4(b) and 4(c), we see flatterorder parameter profiles. However, we find unusual behavior inthese figures associated with chemical potentials of |μ| = 0.98and 1.3 eV. At these chemical potentials, the vortices do not

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VORTEX LATTICES IN THE SUPERCONDUCTING PHASES . . . PHYSICAL REVIEW B 87, 035401 (2013)

seem to be as well formed as they are when |μ| = 0.96 eV. Thisis due to the numerical discreteness in our simulations. In oursystem, because of the nontrivial order parameter winding dueto the vortex, there must be a place where the order parametermagnitude vanishes. One can see that in Fig. 4 this onlyhappens for |μ| = 0.96. What is happening is that the vortexcore moves from being centered at a lattice vertex to theinterior of a plaquette. The order parameter then vanishes in theplaquette interior (which of course is not seen on our discretelattice spatial sampling). In fact, we find that at particularvalues of the chemical potential, it becomes energeticallyfavorable for the vortex to move its core off of a lattice vertexand into the center of a plaquette. This is seen in the energyspectra in Fig. 3 where a kink in the spectrum a appears wherethis vortex shift occurs, namely, around |μk| = 0.97 eV. Webelieve this is simply an artifact of our numerical techniqueand does not represent any real physics.

B. Open vortex lines in vortex lattices

Next, we turn to the case with open boundary conditionsalong the z direction such that the vortices terminate onthe surfaces. This setting is directly relevant for possibleexperiments where the Majorana vortex modes are presentat the end of vortex lines. Due to the open boundaries, theself-consistent BdG calculations must be performed in threespatial dimensions as we can not exploit any translationsymmetry in the z direction. The magnetic unit cells arelx × ly = 8 × 16 sized with lz layers (usually lz = 6) and thereare Nx × Ny = 40 × 20 magnetic unit cells chosen so thatwe are simulating 1600 vortices in the vortex lattice. In thissection, we only consider |U | = 2.8 eV and |μ| > |μo| wherethe superconducting gap is formed and the value of |μo| isestimated from the periodic boundary condition case. Here, weself-consistently determine the BdG quasiparticle spectrum inthe vortex lattice state.43,47,50 For the square vortex lattice withNx × Ny magnetic unit cells, there are NxNy magnetic Blochwave vectors �k analogous to the wave vectors in the Brillouinzone of a Nx × Ny square lattice.

In Fig. 5, we present the dispersion of vortex modes atfour different chemical potentials as has been done previouslyfor s- and d-wave superconductors.43,50 The high-symmetrypoints of the square lattice are at � = (0,0), X = (π,0), andM = (π,π ), as indicated in the inset of Fig. 5(c). There arefour low-energy “Majorana” modes at each momentum whichare contributed by the two vortices per cell and the two endsof each vortex line. For a single magnetic unit cell, we wouldthus expect to see one Majorana mode on the two ends ofeach of the two vortex lines giving rise to a total of fourvortices per cell. In this context, we put the word Majoranain quotes because, strictly speaking, the low-lying energystates only have true Majorana character if they are strictlyat zero energy. In Fig. 5(a), we study the quasiparticle bandsat |μ| = 0.6 eV and we find that the vortex modes are clearlydispersing. Although the superconducting gap is formed, itremains small and the vortex modes of different vortices on thesame surface can tunnel laterally and hybridize, which leadsto the dispersion of the vortex core states. As we increase thedoping level, the lowest-energy quasiparticle band flattens asis clear in the dispersion plot for |μ| = 0.9 eV in Fig. 5(b).

FIG. 5. (Color online) The quasiparticle band structure for openvortex lines in the bulk superconducting TI. The interaction strengthis chosen at |U | = 2.8 eV and the chemical potentials are (a) |μ| =0.6 eV, (b) |μ| = 0.9 eV, (c) |μ| = 1.0 eV, and (d) |μ| = 1.3 eV. Theinset in (c) denotes the magnetic Brillouin zone for the square vortexlattice. The magnetic unit-cell sizes are lx × ly × lz = 8a × 16a × 6a.

This happens because of the increasing bulk superconductinggap, indicated in Fig. 3(c). The vortex core size shrinks, whichleads to smaller overlap of the modes localized in differentvortices and suppresses the quasiparticle dispersion. Thiseffect (i.e., increase of superconducting gap by increasing thedoping) stabilizes the Majorana modes. The low-energy, flatquasiparticle bands contain 4NxNy nearly degenerate statescoming from the 2NxNy vortex Majorana modes on the twodistinct surfaces.

In Fig. 5, we see two clear gaplike behaviors. One typein Fig. 5(a) shows gaps at low energy but with strongdispersion, while Figs. 5(c) and 5(d) show clear gaps butwith flat dispersion. For the flat-dispersing cases, we studiedthe dependence of the energy splitting δE on the samplethickness. An exponential dependence would indicate that thedispersionless gap is a result of the hybridization of the modesat the end of the vortices between two surfaces. Figure 6 showsthe energy splitting δE has an exponential decreasing relationwith the thickness lz described as3

δE ∝ e− lz

ξm , (15)

where ξm denotes the characteristic decay length for the Ma-jorana modes. A smaller ξm means more localized Majorana

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FIG. 6. (Color online) The energy splitting δE vs thickness lz.The magnetic unit cells are 8 × 16 × lz at |μ| = 1 and 1.1 eV. TheHubbard interaction is |U | = 2.8 eV.

bound states. By linear fitting from Fig. 6, the characteristiclength at |μ| = 1 eV is ξm 5.46a and at |μ| = 1.1 eV isξm 9.49a. This suggests that the Majorana modes are stillexponentially localized on the surface even though there isa gap in our finite-size numerics. Therefore, although theMajorana modes may tunnel to the opposite surface, in thethermodynamic limit (lz → ∞), the Majorana modes are stillbound to the surface. Although we do not show it here, wenote that this is not the case for |μ| = 1.3 eV where the gapdoes not decrease exponentially, which is expected since thisis the trivial regime where it should have a power-law decaywith inverse thickness due to finite-size splitting.

The nature of the Majorana modes may be further illustratedby studying real-space probability distributions of the in-gapmodes. Figures 7(a)–7(d) depict side-view spatial slices of

the probability density for the lowest-energy modes andFigs. 7(e)–7(h) show the order parameter distributions in realspace. The plots are cut on the yz surface at x = ±2a, wherethe vortex cores are approximately located. The Majoranamodes (indicated by bright regions) are observed and localizedaround the vortex cores close to the surfaces in Figs. 7(a) and7(b). However, the Majorana mode in Fig. 7(a) spreads morewidely along the surface than that in Fig. 7(b). This shows thatat |μ| = 0.6 eV, neighboring vortices have larger overlap thanthat at |μ| = 0.9 eV, which corroborates with our quasiparticlespectra that indicate stronger dispersion for the former caseas shown in Fig. 5(a) due to the intrasurface hybridizationresulting from the increased lateral overlap of the Majoranamodes. It is also interesting to see that around |μ| = 0.9 eV, themini-gap size of the vortex lines is maximum [see Fig. 3(c)],which is where Fig. 7(f) shows strong, straight-line vortexstructures.

At first, further increases in the chemical potential flattenthe dispersion and strengthen the localization of the Majoranamodes. However, further increases in the chemical potentiallead to another tunneling mechanism for the Majorana modes.As we have already shown for periodic vortex rings [e.g., seeFig. 3(c) as |μ| > 1], the mini-gap of the vortex core statesalong the vortex line eventually begins to decrease as thecritical point is approached. For open boundary conditions, thisleads to increased intersurface hybridization of the modes atthe two ends of the vortex lines. This results in the formation ofgaps due to Majorana mode annihilation on opposite surfaces(for thin samples), and in Fig. 5(c) we can see that thereexists a δE splitting in the Majorana modes. As mentioned andshown in Fig. 6, δE decreases exponentially in the thicknessof the sample. An important feature to note is that, as isclear from the dispersion for |μ| = 0.9 eV, even though thegap increases, the bandwidth of quasiparticles decreases andgets flatter. This is an indication that intrasurface tunnelingis weakening (no 2D hopping on the same surface) and that

FIG. 7. (Color online) Spatial slices (side views in the yz plane at x = ±2a) of the probability density for the Majorana modes and orderparameter density in the bulk superconducting TI at different μ. Upper panels: (a)–(d) show the evolution of the Majorana mode distributions.Brighter regions represent the higher probability density. Lower panels: (e)–(h) show the distribution of pairing order parameters (|A + B |).The chemical potentials are |μ| = 0.6 eV in (a) and (e), |μ| = 0.9 eV in (b) and (f), |μ| = 1 eV in (c) and (g), and |μ| = 1.3 eV in (d) and (h),respectively.

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VORTEX LATTICES IN THE SUPERCONDUCTING PHASES . . . PHYSICAL REVIEW B 87, 035401 (2013)

intersurface tunneling is becoming stronger. We can see thisin Fig. 7(c), where the Majorana bound states begin to leak tothe opposite surface. Furthermore, in Fig. 7(d), in which casethe system is topologically trivial, the lowest-energy modes,which are no longer Majorana in nature and gapped by thevortex mini-gap of order 2/μ, completely penetrate throughthe bulk at |μ| = 1.3 eV and lie along the vortex lines.

C. Superconductor–topological insulator heterostructure

As mentioned in the Introduction, another method to realizethe Majorana modes is through the proximity effect of atopological insulator and an s-wave superconductor. Using ourself-consistent BdG method, we can also study this geometry.By modeling such a structure by an inhomogeneous dopinglevel, we can directly address the effect of the penetrationof superconducting gap into the bulk and the self-consistentformation of Majorana bound states in vortices. We imaginea similar proximity-induced superconductivity as was firstsuggested by Fu and Kane.15 We choose an inhomogeneoussystem where μ(�r) in Eq. (11) is layer dependent. We choosethe surface chemical potential μ(�r) = μS = −1 eV, and thechemical potential μ(�r) = μB = −0.55 eV in other bulklayers so that both the bulk and surface have nonzero densityof states. We investigate the case where superconductivity isinduced primarily on the surfaces by turning on the sameattractive interaction across the entire sample. The inhomo-geneous μ(�r) will generate a much stronger order parameteron the surfaces than in the bulk due to the large differencein chemical potentials. Again, we choose a uniform magneticfield along the z direction which generates the vortex lattice.

There is another reason to consider this system beyondsimply the presence of Majorana fermions. One of the majorobstacles in our bulk calculation is the nonconvergence of astable vortex solution when the order parameter magnitude isvery small. As mentioned, when |μv| < |μ| < |μo| despite thepresence of gapless electrons at the Fermi level, the density ofstates is not large enough to form a sizable superconductinggap (at least for system sizes we consider) and the dopedTI remains a gapless metal. We can counteract this problem

FIG. 8. (Color online) The spatial side view of the pairing orderparameter distribution of a six-layer s-wave/TI heterostructure. Theinteraction strength is chosen at |U | = 2.8 eV, and the surfaceand bulk chemical potentials are |μS | = 1 eV and |μB | = 0.55 eV,respectively. The dark blue tubes indicate the region that pairing issuppressed and form vortex lines. The magnetic unit-cell sizes are8a × 16a × 6a.

FIG. 9. (Color online) (a) The quasiparticle band spectrum forthe six-layer s-wave/TI heterostructure. (b) The spatial side view ofthe Majorana modes. The interaction strength is chosen at |U | =2.8 eV and |μS | = 1 eV and |μB | = 0.55 eV.

by using the superconductor–TI heterostructure geometrywhich acts to pin the vortices with strong superconductivityat the surface (highly doped region), thereby stabilizingthe solution. Figure 8 shows the spatial side view of theresulting self-consistent order parameter profile of a six-layerheterostructure. We see no evidence of superconductivityin the bulk and roughly uniform superconductivity in thesurface which is interrupted near the vortices. The resultingcalculation in Fig. 9(b) shows that Majorana surface modesstill remain even though the superconducting order parameterin the bulk is exponentially small compared to the surface.The six-layer heterostructure can roughly approximate thecase of two vortices existing in a four-layer bulk TI whichis uniformly doped with |μ| = 0.55 eV. This was a region ofinterest that we could not access in our bulk calculation due tofinite-size complications and which we can, admittedly onlyroughly, learn about by stabilizing the vortex solution usinghigher-surface doping.

In Fig. 9(a), we present the quasiparticle band spectrum forthe superconductor–TI heterostructure with a square vortexlattice. Within the superconducting gap, there exist twoprominent low-energy modes which are doubly degeneratewhose energies are split away from zero energy. Although

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not shown here, we find that the energy splitting δE alsohas an exponential decay with increasing sample thickness lz.

This indicates that the low-energy modes are exponentiallylocalized at the superconducting surface and the Majoranafermions can stably reside at the surfaces in the thermodynamiclimit, i.e., lz → ∞. This is indicative that that the vortices in thelow-doping regime (|μv| < |μ| < |μo|) also support Majoranafermions in the bulk superconducting TI. In Fig. 9(b), weshow the probability density of the low-energy modes and seethe tight localization of the resulting modes on the surfaceas one would expect for Majorana modes formed within awell-formed superconducting gap.

V. VORTEX MAJORANA MODES AT FINITETEMPERATURES

In our previous analysis, we have neglected the role oftemperature in our analysis. In this section, we provide arough estimate of the temperature at which one could observethe Majorana fermions experimentally. In the BCS theory,the temperature dependence of the gap size is determinedthrough42

1

N (0)V=

∫ hωc

0

tanh[√

ξ 2+(T )2

2kBT

]√

ξ 2 + (T )2dξ. (16)

Here, N (0) is the density of states at the Fermi level, U is theinteraction coupling, and ωc is the Debye frequency. The finite-temperature gap (T ) may only be determined numerically.The combination of N (0)U can be estimated from the criticaltemperature Tc and the Debye frequency ωc for Cu-dopedBi2Se3 via

N (0)U = −1

ln[

kBTc

1.13hωc

] . (17)

For Bi2Se3, the critical temperature is Tc = 3.8 K,36,37

and the Debye temperature hωc/kB is 180 K.51 With N (0)Udetermined from Eq. (17), one can calculate the temperaturedependence of the gap numerically using Eq. (16). To observethe Majorana modes at finite temperature, the mini-gap size ofthe vortex lines δm(T ) should be stable against the thermalfluctuations: δm(T ) < kBT . The temperature at which thisoccurs may be estimated as

T0 = δm(T )

kB

= π(T )2

2kBδεF

, (18)

where δεF = |μ − μb| where μb denotes the bulk band edge.When T < T0, the Majorana modes can stably exist on the sur-faces and can be detected experimentally. The numerical resultis shown in Fig. 10. From Ref. 52, in Bi2Se3, δεF ∼ 0.25 eVwhich results in an estimate for the critical temperature for theobservation of Majorana modes to be T0 ∼ 0.025 K. Therefore,we can provide a rough estimate that at T � 0.025 K,the Majorana modes can stably exist on the surface ofthe doped topological insulators and may be detectableexperimentally. This number is quite small and indicates onewould need to optimize materials properties in order to hopefor observation. The results for Heusler materials or materialswith similar electronic structure to bulk HgTe may provide

FIG. 10. (Color online) The comparison between the mid-gapsizes δm(T ) and kBT . The intersection occurs at T = T0 ∼ 0.025 K,indicating that as T < T0, the Majorana modes can stably exist onthe surface of the doped topological insulators.

more promising alternatives53 due to the differences in thesustainable levels of doping.

VI. CONCLUSION

In summary, we performed self-consistent Bogoliubov–deGennes calculations to study properties of vortices in dopedtopological insulators that become superconducting. Throughthe use of our numerics, we studied the physics of Majoranafermions in vortex lattices beyond the strict weak-couplinglimit, and the resulting vortex phase transitions between atopological and trivial state. We have shown that the quasipar-ticle band spectra offer evidence that there exists an optimalregime in chemical potential where the Majorana fermions canstably reside even in a finite thickness system. There also existother regimes where the Majorana fermions do not stably existon the system surfaces because of intrasurface and intersurfacehybridization between the vortex modes. Furthermore, we alsoshowed that, through the use of the analogous s-wave–TIheterostructure, TIs with bulk superconductivity containingfinite carrier density but insufficient superconducting pairingstrength can host Majorana fermions on the surface. Similarto the bulk superconducting case, the Majorana modes canalso leak into the bulk and annihilate with the other surface.However, the tunneling behavior exhibits the usual exponentialdecay with thickness and we conclude that the Majoranafermions can survive for thick samples. Unfortunately, thesimple estimates we made for a viable temperature rangein which Majorana modes may be observed indicate thatsuperconducting Cu-Bi2Se3 may not provide a good candidateeven if the doping level can be tuned to the topological vortexphase.

ACKNOWLEDGMENTS

H.H.H. is grateful for helpful discussions with C.-K Chiu.P.G. is thankful for useful discussions with E. Fradkinand P. Goldbart and for support under the Grant No. NSFDMR-1064319. This work was partially supported in part bythe National Science Foundation under Grant No. NSF-OCI

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1053575. T.L.H. acknowledges support from U. S. DOE,Office of Basic Energy Sciences, Division of MaterialsSciences and Engineering under Award No. DE-FG02-07ER46453. M.J.G. and H.H.H. acknowledge support from

the AFOSR under Grant No. FA9550-10-1-0459. We acknowl-edge support from the Center for Scientific Computing at theCNSI and MRL: Grants No. NSF MRSEC (DMR-1121053)and No. NSF CNS-0960316.

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