Magnetic impurities on the surface of topological superconductor

0.74 MB·English

by  Ting-Pong Choy

Magnetic impurities on the surface of topological superconductor
Ting-Pong Choy,1,2 K. T. Law,1 and Tai-Kai Ng1
1Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay Road, Hong Kong, China
2Institute for Advanced Study, Hong Kong University of Science and Technology, Hong Kong, China
WeconsidertheeffectsofmagneticimpuritiesonthesurfaceofsuperconductingCudopedBi2Se3
in the odd parity pairing phase which support topologically protected Majorana fermions surface
states with linear spectrum. Weshow thata single magnetic impurity on thesurface mayinducea
3 pair of in-gap localized bound states. The energy of the in-gap state is extremely sensitive to the
1 orientation of the magnetic impurity due to the so-called Ising properties of Majorana fermions.
0 The magnetic impurity induced spin-texture, which can be measured using spin sensitive STM, is
2 calculated. We also show that the RKKY interactions between magnetic impurities mediated via
theMajoranafermionsarealwaysferromagneticanddenseenoughmagneticimpuritieswilldevelop
n
long-rangemagneticorderandbreakthetime-reversalsymmetryonthematerialsurfaceeventually.
a
J
PACSnumbers: 75.30.Hx,74.20.Rp,73.20.At,03.65.Vf
0
1

Introduction— Topological superconductors (TSCs) /01
] ! %&’()!
n are new states ofmatter with bulk pairinggapand sym- #
’*+!
o
metry protected gapless boundary modes.[1–7] The zero
c energy surface Andreev bound states (SABSs) are Ma- .!
- -!
r jorana fermions (MFs) as they act as their own anti-
up particles due to particle-hole (PH) symmetry. It has !!!"#$! ,!
s been shown that the self-Hermitian property of MFs is
.
t the origin of several interesting phenomena such as the
a
MFinducedresonantAndreevreflection[8,9]andthe 4π
m
Josephsoneffect [10–15]. Particularly,it has been shown
-
d that MFs boundary states only interact with one spe-
n cific direction the magnetic impurities where the specific
o direction is determined by the pairing symmetry of the
c
bulk superconductor and the surface orientation. This
[
is called the Ising spin properties of MFs. The conse- FIG.1. (a)IllustrationofaTSCwithasingleMajoranacone

1 quences of the Ising spin properties of MFs for the sur- onthesurfacecoupledtoamagneticimpurity. TheIsingspin
v face states of He3 B phase[16] and for edge states of directionoftheMajoranamodesisnormaltothesurface. (b)
8 two-dimensional superconductors[17] have been studied. ThemodulationoftheLDOSρ(R,−0.7∆)aroundamagnetic
6
impuritywithscatteringstrengthU/∆=1whichispolarized
0 It is shown that the magnetic susceptibilities of mag-
2 netic impurities which interact with MFs exhibit strong ateloxntugrtehseatz-eanxeirsgy(nωz==−1.00.)7∆is(sch)oawnnd.ωT=he−c0o.r5r∆esp(odn)dnienagrtphine
. anisotropy due to the Ising properties of MFs. However,
1 magneticimpurityareshown. Thecomponentinthexyplane
0 the Ising spin properties of MFs of a three dimensional is denoted by a vector while thebackground colour indicates
3 TSC are yet to be explored. the magnitude of sz(r,ω). The arrows are normalized to the
1 The recently discovered superconducting topological longest in-plane spin length and a=~vs/∆.
:
v insulator (STI) Cu Bi Se is a possible realization of
x 2 3
i
X a three dimensional TSC.[18] It has been proposed
using spin-resolved STM, is calculated. Moreover, we
that Cu Bi Se is a TSC with odd-parity triplet pair-
r x 2 3
show that the RKKYinteractions between magnetic im-
a ing and fully gapped in the bulk [19] and this super-
purities mediatedbythe IsingMFs arealwaysferromag-
conducting phase support SABSs with linear disper-
netic. Finite density of magnetic impurities can break
sion. Specific heat [20] and point contact spectroscopy
time-reversal symmetry spontaneously and open an en-
measurements[21, 22] of the materialare consistent with
ergy gap on the surface. When this happens, the TS
this odd-parity triplet pairing phase.
becomes a quantum thermal Hall insulator.
In this work, we consider the effects of magnetic im-
MF Ising spin— We start with the Bogoliubov-de
purities coupled to the MFs emerged on the surface of
Gennes (BdG) Hamiltonian of Cu Bi Se given by[19]:
an odd-parity triplet pairing TSC. We show that the x 2 3
magneticimpurityinducedin-gapboundedstatesareex-
tremelysensitiveto the orientationsofthe magneticmo- H =Z d3kξk†H(k)ξk, (1)
ment due to the Ising properties of MFs. The spin tex-
H(k)=[H (k) µ]τ +∆σ s τ . (2)
ture near a magnetic impurity, which can be measured 0 z y z x
−
2
nHoetreedξbky=the(cPk↑a,uclki↓m,ca†−trki↓c,e−sτc†−k↑)Tanadre,Naamrebtuhebealseicstrdoen-
x,y,z
↑ ↓
spinindices. s arePaulimatrices. Atlowenergy,the
x,y,z
bandstructureoftheparentcompoundBi Se iswellde-
2 3
scribed by the k p Hamiltonian:
·
H (k)=mσ +v(k σ s k σ s )+v k σ , (3)
0 x x z y y z x z z y
−
where σ = 1 is an orbital index which denotes the
z
±
two Se p orbitals on the top and bottom layer in each
z
unit cell. When m = 0, the bulk is a single species of FIG.2. Theresonantenergyandthelifetimeofthelocalized
noninteracting three-dimensional Dirac quasiparticles. boundstatebyvaryingthecouplingstrengthUnz. Theblack
The BdG Hamiltonian describes a DIII class [12, 23] solid line is the resonant energy ~ωloc/∆ while the red dash
superconductor which satisfies ΘH(k)Θ 1 = H( k) lineistherelaxationrate(~/∆τloc)oftheassociatedlocalized
−
− state.
and JH(k)J 1 = H( k), where Θ = is K and
− y
− −
J =s τ K are the time-reversalsymmetry and particle-
y y
hole symmetry operators respectively. Interestingly, the
and τ . In this explicit model they are identical to the
y
Hamiltonian satisfy an extra inversion (Z ) symmetry
2 physical spin, (s ,s ,s ). The velocity of the bound-
x y z
σ τ H(k)σ τ =H( k) where σ is an inversion opera-
x z x z − x aryMajoranamodeshasasignchangedepending onthe
tor interchanging two orbitals. In this work,we consider value of m and v v∆2/µ2 as m 0. The signchange
the superconducting pairing denoted by ∆ˆ = ∆σ s τ s ∼ →
y z x corresponds to a structural transition of energy disper-
with σ ∆ˆσ = ∆ˆ which is an odd-parity inter-orbital
x x sion of the surface modes.[24] For positive value of m,
−
tripletpairing. Suchpairingsymmetryisconsistentwith
the energy spectrum of SABS forms a Dirac cone. In-
the crystal point group D of the Cu Bi Se and fully
3d x 2 3 deed,whenmbecomesnegative,asecondcrossingofthe
gappedinthebulk. AsuperconductordescribedbyEq.3
zero energy appears at finite k and this crossing is pro-
cansupporttopologicallyprotectedMFsurfacestatesas
tected by the band-inversion of the parent topological
shown below .[19]
insulator.[25]
Tostudythesurfacestates,weconsiderthecasewhere
The eigenstates of the effective Hamiltonian H form
s
the superconductor is terminated at the z = 0 plane two branches φ (k)= 1 (1, ieiθk)T with energy E =
where the wave function on the topmost layer vanishes, ± √2 ± k
v kwhicharelocalizedontheboundaryz =0. Because
i.e. σzψ|z=0 =−ψ|z=0. Bysolvingthesemi-infiniteBdG ±of tshe PH symmetry, two branches are not independent
equation at k = k = 0, we find a Kramers pair of
x y and satisfy: φ (k)=φ ( k).
+
zero-energy surface Andreev bound states ψ , − −
± TostudytheinteractionbetweentheMFsurfacestates
and a local magnetic impurity, we need to construct the
sin(k z)
ψs(z)=Aeκz(cid:18)sin(kFzF+θ) (cid:19)σ⊗|sz =s,τy =sgn(vz)si, ltoiocnalaexliesctarloonngotpheeraxt-odrisr.ecBtiyonroψtat(inr)g,ψthe(srp),inthqeusaunrtfiazcae-
(4)
→ ←
mode expansion of the local electron operators can be
written as:
where κ = ∆/v , v k = µ2 m2 and eiθ = (m+
z z F
| | −
i µ2 m2)/µ. A is the norpmalziation constant for the ψ (r)
−
HwpaavmeiflutnonctiaionnsH.sBfyorks·uprfathceeoAryn,dwreeeovbbtaoiunntdhsetalotwes-ebnyeregxy- ψψ→←((rr)) = (γkeik·r +γk†e−ik·r)eκz (7)
panding at small momentum, ψ→†† (r)  Xk
 ← 
H (k)=v (k s˜ k s˜ ), (5) cos(θk+π/2)
s s x y− y x sin(k z)  isin(θk2+π/2) 
wnihaenreist,he renormalizedvelocityintheeffective Hamilto- ×(cid:18)sin(kFzF+θ) (cid:19)σ⊗−icsoins((θθkk++2π2π//22))  ,
 − 2 τ
v κ(1 cos2θ)+k sin2θ
s F
= − . (6)
v 2(κ2+k2) κ(1+cos2θ)+k sin2θ where tanθk = ky/kx. The mode expansion satisfies
κ F − F the Majorana-like conditions: ψ (r) = iψ (r) and
†
The effective Hamiltonian describes the gapless Majo- ψ (r) = iψ† (r). For convenienc→e, we defi−ne t→he Majo-
rana fermions on the surface boundary up to a high ra←na operato←rs γα(r) = eiαπ/4ψα(r) with γα†(r) = γα(r)
energy cutoff ∆ within the bulk gap. In general, s˜ = for α = , and the Majorana operators γ ,γ
→ ← { → ←}
(s˜ ,s˜ ,s˜ ) are SU(2) Pauli matrices which describes the transform to γ , γ under the TR transformation.
x y z { ← − →}
couplingbetweentwobranchesψ (k,z)with opposites The effective Hamiltonian can be rewritten in term of
s z
3
a spinless fermion f =(γ +iγ )/√2, be evaluatedanalyticallyandT(r ,r ,ω)= r Tˆ(ω)r
1 2 1 2
→ ← h | | i
becomes,
H =v d2rΨ (r)(i∂ τ +i∂ τ )Ψ(r), (8)
s s † x y y x Un
Z T(r ,r ,ω)= z ( τ +Un g I)δ(r )δ(r ).
1 2 1 U2n2g2 − z z 0 1 2
− z 0
where Ψ(r) = (f(r),f (r))T and τ = (τ ,τ ,τ ) are (12)
† x y z
SU(2) Pauli matrices in the Nambu space. Accordingly,
thephysicalspinSU(2)matricesscantransformintothe Heretheunperturbedon-siteGreenfunctiong0(ω)isreg-
new basis, i.e. (sx,sy,sz)→(τx,−τy,−τz), respectively. ulaTtehde bfuyllaGshreoernt-’sdifsutnanctcieoncuGtoff a=0 ≪G0a≈+~Gv0s/∆TG. 0 =
Because of the Majorananature of the surface modes, ret ret ret ret
neitherthe localdensityoperatorρ(r)= αψα†(r)ψα(r) G0ret +δGret can be computed using the T-matrix and
nor the components of the local spin Pdensity opera- the additional part δGret becomes,
tors parallel to the surface, sˆ = ψ ψ ψ ψ and
sˆy = iψ† ψ iψ† ψ can bxe non→†-triv→ia−l. H←†ow←ever it δGret(r,r,ω)=B(ω)(2if0f1(cosθrτx−sinθrτy)
is possib→le t←o c−onst←ruct→the non-trivial component of the +(f02+f12)τz −(f02−f12)Unzg0I , (13)
spin density operator which is perpendicular to the sur- (cid:1)
where f = f (r,ω) for i = 0,1 and B(ω) =
face, sˆ =ψ ψ +ψ ψ = 2iγ γ . The Ising-type i i
spindeznsity→†isc←rucial←†for→Majo−rana→bou←ndarymodes. Its Unzω2/16π(1−U2n2zg0(ω)2). The pole of B(ω) deter-
mines the position of a localized bound state induced by
strong anisotropy reflects the spin-triple pairing symme-
a magnetic impurity where the imaginary part of its de-
tryandthe spin-orbitalcouplinginthe bulkTSC.Inthe
nominator determines the relaxation rate (1/τ ) of it.
restofthis paper,we will analysethe effects ofmagnetic loc
As shown in Fig.2, we find that there are always two
impurities coupled with this gapless Ising spin density.
localized states with energy ω for large enough cou-
Magnetic impurity induced in-gap state— We ± loc
pling strength Un , and the resonant energy ω goes
firstconsider a static magnetic impurity with largemag- z loc
to zero as Un . On the other hand, the resonant
netic moment S which scatters the Majorana boundary z → ∞
energy will reach the cutoff value W for small enough
modes classically with interaction strength J at the ori-
value of Un , which implies that the magnetic impurity
gin r = 0. The scattering process can be described by z
the perturbation V = drΨ (r) r Vˆ r Ψ(r ) where cannot induce any localized in-gap states if its magnetic
† 1 2 2
h | | i moment pointing close to the material boundary or the
R
r Vˆ r = Un τ δ(r )δ(r ), (9) coupling strength U to the surface modes is sufficiently
h 1| | 2i − z z 1 2 weak.
The local density of state (LDOS) ρ(R,ω) =
where U = JS. Here we have assumed the classical im-
1 TrG (R,R,ω) varies due to the scattering with
purityspinS=Sn,whosedirectionisgivenbythefixed −πℑ ret
the magnetic impurity,
unit vector n =(n ,n ,n ). In contrast, the spin quan-
x y z
tization axis of a quantum impurity spin is determined
ω
by the Ising-spin orientation of the surface modes. ρ(R,ω)= |4| +2ℑB(ω)Unz(f02−f12). (14)
The effect of the magnetic impurity scattering can
be addressed using the T-matrix technique with the T- Thelineardensity ofstatesreflectsthe linearenergydis-
matrix,[26] persion of the Majorana excitations in the absence of
magneticimpurities. AsshowninFig.(3a,b),wefindlow-
1 energyresonancesinLDOSinducedbyamagneticimpu-
Tˆ(ω)=Vˆ +VˆGˆ0 (ω)Tˆ(ω)= Vˆ, (10)
ret 1 VˆGˆ0 (ω) rity. The resonance peaks become sharper and approach
− ret to zero energy with increasing impurity strength, and
where G0(r ,r ,ω) = r G0 (ω)r is the bare Green they decay away from the impurity as 1/R2 which can
1 2 h 1| ret | 2i
function of the effective Hamiltonian H in real space, be determined by the scalingdimension ofthe Majorana
s
operators. Incontrasttothesurfaceofthree-dimensional
iω
G0(R,0,ω)= (f (R,ω)I topologicalinsulator where in-gap states can be induced
0
4 independentofthe impuritymomentorientation[26], the
+f1(R,ω)(τycosθR+τxsinθR)). (11) modulationofthe LDOSδρ(R,ω)showninFig.(3a,b)is
only sensitive to its projection along the z-axis.
Here θ is the angle of vector R from the x-axis.
R Similarly, the energy-resolved spin density averages,
f0(R,ω)= sgn(ω)J0(ω R) iY0(ω R) and f1(R,ω)= s(R,ω) = 1 TrG (R,R,ω)σ, which can be mea-
iJ (ω R)−+sgn(ω)Y (| ω| R)−where|J|(x) , Y (x) are the −πℑ ret 2
1 1 i i sured by the recently developed spin-resolved STM
− | | | |
Besselfunctionsofthefirstandsecondkindrespectively.
technique[27], is found to be
Because of the local form of the scattering poten-
tial Eq.(9) and the unperturbed on-site Green function s(R,ω)= B(ω) (f2+f2)zˆ 2if f (cosθ xˆ+sinθ yˆ) .
ℑ 0 1 − 0 1 R R
G(0,0,ω) = g0(ω)I is diagonal in τz, the T-matrix can (cid:0) (15)(cid:1)
4
is projected perpendicular to the surface. The RKKY
interactionbetweentwomagneticimpuritiescanbeeval-
uatedbyintegratingouttheMajoranamodesusingtheir
real-space Green function,
H =J(r r )n (r )n (r ), (17)
RKKY 1 2 z 1 z 2
−
where J(R) = J2χ (R)/4 and χ (R) =
zz zz
2 0 dω Tr[G0(R,ω)τ G0( R,ω)τ ] is the spin
−π ℑ z − z
suscRe−p∞tibility of the Majorana modes, which can be
evaluated analytically for 2D helical Majorana fermions
a4
χ (R)= . (18)
FIG. 3. LDOS plots showing the low-energy resonances at zz −8π~vs R3
| |
R=0.5a away from a magnetic impurity with (a) Unz/∆=
1.0 and (b) Unz/∆ = 0.5 respectively. Note that the reso- In evaluating the spin susceptibility χzz, a soft cutoff
nance peak is sharper and stronger for larger Unz and the function e−ω/ω0 is used and we take the limit ω0
modulationoftheLDOSatnz =0.5ispronouncedattheen- after performing the integrals[30]. The RKKY int→era∞c-
ergyscalebeyondthehigh-energycutoff∆. Energyresolved
tions can also be derived from the energy-resolved spin
spin density plot at R = (0.5a,0) at scattering strength (c)
average,Eq.(15), if we integrate all the filled states with
Unz/∆ = 1 and (d) Unz/∆ = 0.5 respectively. The black
solid line is sx(R,ω) while the red dash line is sz(R,ω). We energy up to the chemical potential, µ=0.
note that the modulation of the spin density is much weaker From Eq.18. We see that the Ising interactions be-
andextendedbeyondthehigh-energycutoff∆atUnz =0.5. tween two impurities are always ferromagnetic. Inter-
In both cases, sy(R,ω)=0. actions of an ensemble of magnetic impurities give rise
to an ordered ferromagnetic phase with spins pointing
perpendicular to the surface and breaks the TR sym-
In Fig.(1b,c), we show the energy-resolved spin textures
metry spontaneously, driving the surface state into a
aroundamagneticimpuritywhichispointinginthenor-
gapped one. According to Eq.(18), the ordering tem-
mal direction. We see that it induces not only the z-
direction spin LDOS, but also an in-plane one which is perature can be estimated as kBTc ≈ J2a4n3im/2p/~vs,
originating from the helical nature of the surface modes. where nimp is the impurity density. At the mean-field
Ifthemagneticimpuritypointinginotherdirections,the level, the ordering magnetic impurities open a mass gap
induced magnetization is similar to Fig.(1b,c) qualita- m = nimpJS. Such a mass term breaks the TR sym-
tively. However, the magnitude of the induced magneti- metry of the surface modes and exhibits an anomalous
zation linearly proportionalto the magnetic spin projec- quantumHalleffect, providesa halfquantizedHallther-
tion to the Majorana Ising spin direction nz. We notice mal conductance σH = sgn(m)/2h.[28, 29] It is in con-
that the out-of-plane energy-resolved spin density is an trast to the effects of dense magnetic impurities on the
odd function of energy while the in-plane one is even surface of three dimensional TI where the RKKY inter-
as shown in Fig.(3c,d), which are originated from the actions between magnetic impurities mediated via the
PHsymmetryonthe boundarymodes, J(H +V)J 1 = helicalfermionicsurfacemodesarefrustrated,andresult
s −
(H +V) where J = τ K is the effective PH symme- inadisorderspin-glassphaseinwhichtheTRsymmetry
s x
−
try transformation on the surface. Furthermore, the in- is still preserved.[28]
planeenergy-resolvedspindensitiessatisfythesumrules Conclusion— In summary, we find that the local
0
s (R,ω)dω = 0 which are also emerged from the magnetic impurity can induced a pair of localized in-
x,y
MR−a∞jorana nature of the underlying surface excitations. gap states on the 2D surface of three-dimensional DIII
RKKY interaction—Finally, we consider the dy- TSCs. Importantly, the energy of the induced in-gap
namics of magnetic impurities interacting with helical states is sensitive to the orientations between the mag-
Majorana excitations. In particular, we analyse the netic impurity and the MF Ising spin direction. We also
RKKY interactions between magnetic impurities medi- showthatthe RKKY-likeinteractionsbetweenmagnetic
atedviathe Majoranasurfacemodes. Inthe DIII TSCs, impurities are ferromagnetic Ising. Therefore, at large
we have shown that the coupling between the surface densitiesofmagneticimpurities,long-rangemagneticor-
helical Majorana states and the magnetic impurities is dering is developed and TR symmetry on the surface is
effectively Ising for T ∆, broken spontaneously.
≪
Acknowledgement— We thank the discussion with
H = JSnˆ Ψ (0)τ Ψ(0), (16)
ex − z † z X. J. Liu and R. Shindou. This work is supported by
andSˆ isthespinoperatorofamagneticimpuritywhich HKRGC through Grant 605512 and HKUST3/CRF09.
z
5
(2009).
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