Majorana Edge States in Superconductor/Noncollinear Magnet Interfaces

Wei Chen1,2 and Andreas P. Schnyder21Theoretische Physik, ETH-Zurich, CH-8093 Zurich, Switzerland

2Max-Planck-Institut fur Festkorperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany(Dated: April 10, 2015)

Through sd coupling, a superconducting thin film interfaced to a noncollinear magnetic insulatorinherits its magnetic order, which may induce unconventional superconductivity that hosts Majoranaedge states. From the cycloidal, helical, or (tilted) conical magnetic order of multiferroics, or theBloch and Neel domain walls of ferromagnetic insulators, the induced pairing is (px + py)-wave,a pairing state that supports Majorana edge modes without adjusting the chemical potential. Inthis setup, the Majorana states can be separated over the distance of the long range magneticorder, which may reach macroscopic scale. A skyrmion spin texture, on the other hand, induces a(pr + ip)-wave-like state, which albeit nonuniform and influenced by an emergent electromagneticfield, hosts both a bulk persistent current and a topological edge current.

PACS numbers: 73.20.-r, 71.10.Pm, 73.21.-b, 74.45.+c

Introduction.- Motivated by possible applications innon-abelian quantum computation [1], the search for Ma-jorana fermions in condensed matter systems has wit-nessed a boost recently [26]. Indeed there has beenmuch effort to design and fabricate one-dimensional het-erostructures in which topological p-wave superconduc-tivity is proximity induced [712]. One particularlypromising proposal are chains of magnetic atoms withnoncollinear spin texture on the surface of a conventionalsuperconductor (SC) [1322]. The presence of these mag-netic adatoms induces Shiba bound states [2325], whoselow-energy physics is equivalent to a one-dimensional(1D) p-wave SC with Majorana modes at its ends. Scan-ning tunneling measurements of zero-bias peaks at theends of Fe chains deposited on superconducting Pb havebeen interpreted as evidence of Majorana modes [26], al-though no general consensus has been reached regardingthe definitive existence of Majorana states in these sys-tems [27].

Since thin films are generally easier to manufacturethan adatoms, it is intriguing to ask if such 1D pro-posals can be generalized to two-dimensional (2D) sys-tems, where a superconducting thin film is coupled to anoncollinear magnet. The work by Nakosai et al. [28]first shed light on this issue. It was found that a spiralspin texture in proximity to an s-wave superconductorinduces a (px + py)-wave state, while a skyrmion crystalspin configuration gives rise to (px + ipy)-wave-like pair-ing. The former paring state exhibits bulk nodes andMajorana flat-band edge states, whereas the latter one ischaracterized by a full gap with chirally dispersing Ma-jorana edge states that carry a quantized Hall current.In this Letter, we show that these phenomena occur in amuch broader class of superconductor/noncollinear mag-net interfaces. In particular, we find that a great vari-ety of noncollinear magnets, including multiferroic insu-lators with helical, cycloidal, and (tilted) conical order,as well as magnetic domain walls of ferromagnetic (FM)

insulators, interfaced with an s-wave superconductor in-duce a (px + py)-wave pairing state with Majorana flatbands at the boundary. We derive the general criterionfor the Majorana edge state for any wave length and di-rection of the noncollinear magnetic order, show that theMajorana modes occur without fine-tuning of the chem-ical potential, and demonstrate that the translation in-variance along the direction perpendicular to the non-collinear order greatly enhances the chance to observeMajorana states. Furthermore, we investigate a singleskyrmion spin texture coupled to an s-wave superconduc-tor and shown that at this interface an inhomogeneous(pr + ip)-wave-like pairing is induced, which coexistswith the emerging electromagnetic field resulted from thenoncoplanar spin texture.SC/multiferroic interface.- Evidence from the Tc re-

duction in magnetic oxide/SC heterostructures [29, 30]suggests that s d coupling Si generally exists atthe interface atomic layer between an SC and an insu-lating magnetic oxide [31], where is the conductionelectron spin and Si the local moment. This leads usto consider the following model for the SC/multiferroicinterface, which is the 2D generalization of the 1D pro-posals of Refs. 13, 14,

H =i,,

tififi+ + t

ifi+fi

i,

fifi

+i,,

(Bi ) fifi +i

0

(fif

i + fifi

),

(1)

where i = {ix, iy} is the site index, ={a, b

}is the

planar unit vector, and is the spin index. In the ab-sence of spin-orbit interaction, the majority of the non-collinear order discovered in multiferroics, for instance inperovskite rare earth manganites [3235], can be generi-cally described by the conical order

Bi = (Bxi , B

yi , B

zi ) =

(B sin i, B, B cos i

), (2)

arX

iv:1

504.

0232

2v1

[con

d-ma

t.sup

r-con

] 9 A

pr 20

15

2as far as their effect on the SC is concerned, since thechoice of coordinate for is arbitrary. For instance, cy-cloidal and helical order are equivalent by trivially ex-change two components of , and conical orders withany tilting angle are equivalent. Here B = |Si| andB = |Si| are the planar and out-of-plane componentsof the local moment, and the planar angle i = Q riis determined by the spiral wave vector Q and the pla-nar position ri. The cycloidal order is the case whenBy = 0. Note that a three-dimensional conical orderprojected to a surface cleaved at any direction is still thatdescribed by Eq. (2), so our formalism is applicable to athin film or the surface of a single crystal multiferroic inany crystalline orientation, assuming no lattice mismatchwith the SC and a constant |Si| = |S|.

We perform two consecutive rotations to align the Bifield along z,(fifi

)=

(cos i2 sin i2sin i2 cos

i2

)(cos 2 i sin

2

i sin 2 cos2

)(gigi

)= Ui

(gigi

), (3)

where sin = B/B0 and B0 =B2 +B

2 = |S|,

under which one obtains

H =i,,,

tiigigi+ + t

i (

i) g

i+gi

+i,,

(Bz I

)gigi +

i

0

(gig

i + gigi

),

i = Ui Ui+ =

(i ii

i

),

i = cos Q 2 i sin sin Q

2,

i = cos sin Q 2

. (4)

In what follows, we consider hopping to be real andisotropic ti = t

i = t.

In the limit B0 || {t,0} with < 0, onecan construct an effective low energy theory for the spinspecies near the Fermi level, which is the spin down band.This is done by introducing a unitary transformationH = eiSHeiS to eliminate the spin mixing terms orderby order [13]. At first order, the pairing part

Heff, =i,

[(1

B 1

)0t

igigi+ + h.c.

](5)

resembles a spinless p-wave superconductor withanisotropic nearest-neighbor and next-nearest-neighborhopping. From Eq. (4) one has i = cos sinQ /2, sothe induced pairing is of (px + py)-wave symmetry, withthe magnitude of the gap determined by the wave lengthof the planar component of the conical order.

To derive the criterion for the appearance of Ma-jorana edge states, we introduce Majorana fermionsgi =

12 (bi1 + ibi2), g

i =

12 (bi1 ibi2) with{bim, bim} = 2iimm , and express the Hamil-

tonian H = (i/4)

q bqA(q)bq in terms of the basis

bq = (bq1, bq1, bq2, bq2)[14]. We choose open bound-ary condition (OBC) along x and periodic one (PBC)along y, such that pi < qy < pi is a good quantumnumber. At a particular qy, only when qx satisfies

a sin qx + b sin qy = 0 , (6)

is the Majorana Hamiltonian A(qx, qy) skew symmetric.The two solutions qx1 and qx2 are the high symmetrypoints at which the Pfaffian is calculated. The topologi-cal index [14] is now a function of qy

M(qy) = Sign (Pf [A(qx1, qy)]) Sign (Pf [A(qx2, qy)]) . (7)

When M(qy) = 1, or equivalently20 + (| 2tb cos qy|+ |2ta cos qx1|)2 > B

>

20 + (| 2tb cos qy| |2ta cos qx1|)2 , (8)

where = ( + ) /2, the Majorana edge state with

momentum qy appears. Setting b = 0 and qx1 = 0recovers the well known 1D result [14].

Equation (8) is the general criterion for the Majoranastate to appear at momentum qy for any given spiral orconical order. It is supported by numerically solving theBogoliubov-de Gennes (BdG) equation with the bound-ary conditions we choose. A spin-generalized Bogoliubovtransformation gi =

n, uinn + v

in

n is intro-

duced to diagonalize Eq. (4). Figure 1(b) shows a typicaldispersion E(n, qy), which display Majorana zero-energystates in the qys that satisfy Eq. (8). Note that Equation(8) can be satisfied even if = 0, so adjusting chemicalpotential is generally not needed. Consequently, the edgestates can occur in an isolated sample without attachingany leads. The 2tb cos qy factor greatly enlarges thenumber of qys that can satisfy Eq. (8), hence increasesthe chance to observe Majorana Fermions, as one cansee from the phase diagram shown in Fig. 1(a) that hasmuch larger topologically nontrivial phase in the B0-space than the 1D cases, where the (weak) topologicalphase [36] is judged by whether any qy satisfies Eq. (8)at a given (B0, ).

The localized Majorana edge states can be seen asthe zero bias peaks (ZBPs) in the local density of states(LDOS) along x direction, as plotted in Fig. 1(c). Threegap-like features show up in the LDOS. The two sym-metric in come from the bulk gap whose position isshifted by the s d coupling B0 as its effect is similarto a magnetic field, and typically has a magnitude [37]0.01 1eV so B0 > . The gap-like feature near zero en-ergy represents the induced (px+py)-wave gap in Eq. (5).

3(a) (b)

B0

(c) qy

E(n,q y)

/L=0.2 /L=0.4 /L=1 /L=2

FIG. 1: (color online) (a) The topologically nontrivial phase(blue region) of the 2D model Eq. (1) with = 0 as a functionof s d coupling B0 and a for (1, 0) spiral. For com-parison, the nontrivial phase of 1D model with = 0.2 and0.4 are shown as the region inside light blue dots and whitedots, respectively. (b) Energy levels of a system with sizeL = 80a subject to a spiral of wave length = 2pi/|Q| = 16aat B0 = 0.3. The orange regions are the qys at which thetopological criterion Eq. (8) is satisfied, where one also seesthe Majorana zero energy edge state. (c) the LDOS along xfor different values of /L. Red, green, and blue lines are thefirst, second, and third site away from the edge. The /L = 2case represents a Neel or Bloch domain wall in a ferromagneticinsulator. Other parameters are t = 1, = 0.05.

The weight of ZBP decreases as the spiral wave length = 2pi/|Q| increases. The /L = 2 case represents aNeel or Bloch magnetic domain wall joining two regionsof opposite spin orientations in a ferromagnetic insulator,since it can be viewed as a spiral with half a wave length,and the interface to a SC can be described by Eqs. (1) to(8). In such case, the ZBP is small but still discernable.

For a thin film with finite thickness but the s d cou-pling only at the interface atomic layer, the Majoranastate extends over few layers away from the interface, soone may need a SC film of few atomic layers thicknessto observe the Majoranas by any surface probe such asscanning tunneling microscope (STM). Even if the mul-tiferroic contains domains of different spiral chirality, orthe spin texture is not perfectly periodic, the Majoranaedge state still exits at the edge and the boundary be-tween domains. The large single domain of multiferroics,currently of mm size [38], may help to separate the Ma-jorana fermions over a distance of macroscopic scale.

SC/skyrmion interface.- Skyrmion spin textures havebeen observed in thin film insulating multiferroics [39, 40]

(a) (b) (c)

(d) (e)

(f) NM/SK, B0=0.24 (g) NM/SK, B0=0.6 (h) SC/SK, B0=0.24 (i) SC/SK, B0=0.6

q q

E(n, q

)

r! r!

FIG. 2: (color online) (a) The spin texture of the pedagogicalmodel, which is closely related to a single nonchiral skyrmionin (b) a polar lattice and (c) a square lattice. (d) The dis-persion of the pedagogical model with OBC in r and PBC in, with Nr = N = 40, and (e) if the emerging EM field ismanually turned off, at B0 = 0.24. The spontaneous currentpattern at the interface to a single nonchiral skyrmion of size9 9 is shown in (f), (g) for the normal state ( = 0) and(h), (i) for the SC state. Parameters are t = 1, = 0.2, = 0.1.

at temperatures approaching the typical SC transitiontemperature, with a small magnetic field that presum-ably has negligible effect on the SC. To gain more un-derstanding about the SC/skyrmion insulator interface,disregarding external magnetic fields, we first consider aclosely related pedagogical model defined on a square lat-tice, whose low-energy sector can be studied analytically.The spin texture of this model is that shown in Fig. 2(a),yielding a magnetic field on its interface to an SC

Bi = B (sin i cosi, sin i sini, cos i) (9)

at position (ri, i) on the square lattice, where i =piri/R and R is the width in r direction. We assume OBCalong the r direction and PBC along the direction. TheHamiltonian is described by Eq. (1) with = {r, }. Toalign the spin texture along z, one performs the rotationin Eq. (3) with Ui defined by

Ui =

(cos i2 sin i2 eiisin i2 e

ii cos i2

), (10)

4which yields Eq. (4) with

i = 1 + 2ieipi/N sin2

(i2

)sin

pi

N, ir = cos

pi

Nr,

i = i sin iei(i+i+)/2 sin

pi

N ei(i+i+)/2i ,

ir = eii sin

pi

Nr eii ir , (11)

where Nr and N are the number of sites in each direc-

tion. After gauging away the extra phase i i byeii/2gi gi and eii/2gi gi, the Hamiltonian istranslationally invariant along but not along r becausei sin i = sin (piri/R). In the B || {t,0}limit, using Eq. (5) of the low-energy effective theory, theinduced gap along r and are proportional to ir andi, and therefore of (pr + ip)-wave-like symmetry.

The spin-conserved iti and spin-flip iti hoppingin the gi basis contain an emergent electromagnetic(EM) field [41, 42] coming from the spatial dependence ofthe unitary transformation [43]. This becomes evident inthe continuous limit i , i , Ui U , and using ar

0drr = 2pi/Nr 1 and

a0

d = 2pi/N 1.The {, } contain the phase gained over one latticeconstant

=

(

)= eiq

a0 dA , (12)

where A = i (~/q)UU . The factor of i differencebetween Ar and A eventually leads to the induced(pr+ip)-wave-like gap. Figure 2(d) shows the dispersionof this pedagogical model, and Fig. 2(e) shows the dis-persion when the emergent EM field is manually turnedoff by setting i = 1 in Eq. (11). Without the emer-gent EM field, the dispersive edge bands expected for theinduced (pr + ip)-wave-like gap are evident, whereas inthe presence of it the bulk gap is diminished, althoughthe trace of edge bands can still be seen in som cases(compare q 0 and E(n, q) 0 regions in Fig. 2(d)and (e)).

The relevance of this pedagogical model is made clearby shrinking the spins at the ri = 0 edge (green arrows inFig. 2(a)) into one single spin, which results in a nonchi-ral skyrmion on a polar lattice shown in Fig. 2(b), withthe same interface magnetic field described by Eq. (9).Certainly these two lattices cannot be mapped to eachother exactly, but their low energy sectors display simi-lar features.

The pedagogical model indicates that theSC/skyrmion interface hosts a complex interplaybetween (i) the s-wave gap, (ii) the induced (pr + ip)-wave-like gap, (iii) the emergent EM field, and (iv) thes d coupling B0. Motivated by the STM generatedsingle skyrmion [44] (although with an external magneticfield), we proceed to study the SC/skyrmion interfaceon a single open square (SC/SK), whose spin texture

Bi at position ri = (xi, yi) = |ri| (cosi, sini), asshown in Fig. 2(c), is that described by Eq. (9), but withi = pi|ri|/R(ri). Here, R(ri) is the length of the straightline that passes through ri connecting the center of thesquare with the edge. The spontaneous current at site ican be calculated from Eq. (1) by

Ji = i

tfi+fi . (13)

As shown in Fig. 2(f) and (g), in the normal state(NM/SK) there is a persistent current whose vorticitystrongly depends on the emergent EM field, the s dcoupling B0, and finite chemical potential. Because thepersistent current also has edge component, the topolog-ical edge current alone is hard to be quantified in theSC state from the pattern shown in Fig. 2(h) and (i).At B0 , the current pattern in the SC state recov-ers that of the NM state, as can be seen by comparingFig. 2(g) and (i). For the chiral skyrmion seen in mostexperiments, the form of Bx and By in Eq. (9) are ex-changed, which gives the same result as x and y can betrivially exchanged for spin-independent quantities suchas Ji. These results suggest a vortex-like state at theNM/skyrmion lattice or SC/skyrmion lattice interface,whose vorticity depends on material properties.

In summary, we propose that a broad class of non-collinear magnetic orders, including a large part of thosediscovered in multiferroic insulators, and the Bloch andNeel domain walls in collinear magnetic insulators, canbe used to practically generate Majorana edge states attheir interface to a conventional SC. The advantages ofthese systems include a much larger parameter space tostabilize the edge states compared to 1D proposals, thelonger range magnetic order may help to separate theedge states over a macroscopic distance, and adjustingchemical potential is not necessary. The proximity to askyrmion induces an inhomogeneous (pr + ip)-wave-likepairing in the SC under the influence of an emergent elec-tromagnetic field, and consequently a vortex-like statethat features both a bulk persistent current and an edgecurrent.

We thank P. W. Brouwer, Y.-H. Liu, and F. von Oppenfor stimulating discussions.

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