-
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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