Realizing DIII Class Topological Superconductors using $d_{x^2-y^2}$-wave Superconductors

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by  L. M. Wong

Realizing DIII Class Topological Superconductors using d -wave Superconductors
x2−y2
L. M. Wong, K. T. Law
Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
In this work, we show that a quasi-one-dimensional d -wave superconductor with Rashba
x2−y2
spin-orbitcouplingisatopologicalsuperconductor(TS).Thistime-reversalinvariantDIIIclassTS
supports two topologically protected zero energy Majorana fermions at each end of the system. In
contrast to proposals using s-wave superconductors [4–13] in which a strong magnetic field and the
fine tuning of the chemical potential are needed to create Majorana fermions, in our proposal, the
topologicallynon-trivialregimecanbereachedintheabsenceofamagneticfieldandinawiderange
of chemical potential. Experimental signatures and realizations of the proposed superconducting
2
state are discussed.
1
0
2 Introduction—A Majorana fermion is a real fermion that a DIII class TS can be realized when electrons in a

n which has only half the degrees of freedom of a usual quasi-one-dimensional wire with spin-orbit coupling ac-
a Dirac fermion. It was first pointed out by Read and quire a d -wave pairing. In the topologically non-
x2−y2
J
Green [1] that zero energy Majorana fermion modes ex- trivial regime, two zero energy Majorana fermion modes

0 ist at the vortex cores of a 2D p +ip superconductor appear at each end of the wire. In our proposal, Majo-
x y
3
and these Majorana fermions are non-Abelian particles ranafermionscanbecreatedintheabsenceofanexternal

[2]. Soon after, Kitaev constructed a spinless fermion magnetic field and in a wide range of chemical potential,
]
n model and showed that a single Majorana fermion ex- e.g. thereisnoneedtotunethechemicalpotentialtothe
o ists at each end of a p-wave superconducting wire [3]. RCP. In the presence of an external magnetic field, the
c Recently, several groups proposed that effective p-wave system is in the D class and a single Majorana end state
-
r superconductors can be realized when an s-wave pairing appearsateachendofthewire. Experimentalsignatures
p
is induced in systems with spin-orbit coupling [4–13]. andrealizationsofthisDIIIclassTSwillbediscussedat
u
Particularly, the (quasi)-one-dimensional effective p- the end.
s
. wave superconductors attracted much attention [6–13] Single-channel model— Before studying the more re-
t
a due to the fact that Majorana fermion end states can alistic quasi-one-dimensional quantum wires, in this sec-
m exist in the absence of vortices and the energy separa- tion, we first consider a strictly one-dimensional DIII
- tion between the Majorana zero energy mode and other class Hamiltonian which supports double Majorana end
d finite energy fermionic modes is relatively large, on the states in the absence of an external magnetic field. We
n
order of the p-wave pairing gap [10]. construct the following model
o
TheexistenceofMajoranafermionsintheabovemen-
c H =H +H +H +H
[ toifontheedHsyasmteimltsonisiapnrowfohuicnhd.dIetcriisbreeslattheedstyostthemes.ymAcmcoetrrdy- H1tD=(cid:80)j,tα−2t(SψOj†+1,αψSjCα+h.Zc.)−µψj†,αψjα
v2 ing to symmetry classification of Hamiltonians [14–16], HSO =(cid:80)j,α,β−2iαRψj†+1,α(σy)α,βψj,β +h.c. (1)
aBdG Hamiltonianwith particle-holesymmetry, broken H =(cid:80) 1∆ (ψ† ψ† −ψ† ψ† )+h.c.
5 SC j 2 0 j+1,↑ j,↓ j+1,↓ j,↑
7 time-reversal symmetry and broken SU(2) spin rotation H =(cid:80) V (ψ† ψ −ψ† ψ ),
5 symmetry, falls into the D class. In one spatial dimen- Z j z j↑ j↑ j↓ j↓
4 sion, a D class Hamiltonian is classified by a Z topolog- where ψ is a fermion operator at site j, α and β are the
2 j
. ical number. A system described by a BdG Hamiltonian spin indices, t is the hopping amplitude, α is the spin-
0 R
1 with a non-trivial Z2 topological number possesses Ma- orbit coupling strength, ∆0 is the superconducting pair-
1 jorana end states. ing amplitude, and σ is a Pauli spin matrix. V denotes
y z
1 Tobeintheproposedtopologicallynon-trivialregime, the strength of the Zeeman term. Without the pairing
: itrequiresaRashbaspin-orbitcouplingtobreakthespin terms, the above model describes a wire with spin-orbit
v
i degeneracy, a magnetic field to break the Krammers de- coupling. If an s-wave (on-site superconducting) pair-
X
generacy at the Rashba-band crossing point (RCP), fine ing is induced on the wire, as it is done in Ref.[6–13],
r tuning the chemical potential to the RCP and finally no Majorana fermions can be created without breaking
a
induce an s-wave superconducting pairing at the Fermi time-reversal symmetry. In the following, we show that
energy. However, tuning the chemical potential to the our model supports double Majorana end states in the
RCP, which is near the electronic band bottom, reduces presence of time-reversal symmetry.
the electron density severely and electrons can be easily The energy spectrum of H with V = 0 is shown
1D z
localized by disorder in this regime. The strong mag- in Fig.1a. Due to Krammers degeneracy, every energy
netic field required can also suppress superconductivity. level in Fig.1a is doubly degenerate. It is evident from
A schematic picture of this proposal is shown in Fig.2a. the energy spectrum that zero energy modes exist when
A DIII class Hamiltonian respects both time-reversal the chemical potential satisfies |µ| < |α |. The sum of
R
and particle-hole symmetry and breaks SU(2) spin- the two ground-state wavefunctions is shown in Fig.1b
rotation symmetry [14–16]. In this Letter, we point out to comfirm that the zero energy modes are end states.
2
a) b)1 a) E b) E
0.6 2|Ψ2 k k
+| 2 α
2|Ψ1 R
|
∆00.4 0 100 200 3x00 400 500 600 µ
E/ 0.2 2| Ψc2)1 k k1 k2 k
+| 2 V
2|Ψ1 z
|
0 µ RCP
-2 -1 0 1 2 0 100 200 300 400 500 600
µ/α x
R
FIG. 2: a) The energy versus momentum of a typical one-
FIG.1: a)Excitationenergyversuschemicalpotential. The
dimensional Rashba band with a finite Zeeman term. In s-
parametersofH are: L=600,t=12,∆ =2andα =4,
1D 0 R wave-pairing proposals, a strong magnetic field is needed to
whereListhenumberofsites. Zero-energymodesexistinthe
break the Krammers degenercy at the RCP. The chemical
regionboundedbytheverticaldottedlineswhere|µ|<|α |.
R potential needs to be tuned to the topologically non-trivial
All the bands shown in the figure are doubly degenerate due
regime which is bounded by the horizontal dashed lines near
totime-reversalsymmetry. b)|Ψ |2+|Ψ |2 versusxwherex
1 2 theelectronicbandbottom. Electrondensityinthisregimeis
isthesitelabel. Theground-statewavefunctionsarelocalized
low. b) In our one-dimensional model with V =0, topologi-
at the edge in the topologically non-trivial regime µ=α /2. z
R cally non-trivial regime is located at the middle of the band
c) In the topologically trivial regime, e.g. µ = 1.5α , the
R which is bounded by the dashed lines. In this regime, the
ground-state wavefunctions are predominantly in the bulk.
pairing amplitudes ∆ cosk and ∆ cosk at the two Fermi
0 1 0 2
points, k and k , have opposite signs.
1 2
In other words, there are two Majorana fermions at each
endofthewire. Inthetopologicallytrivialregimewhere
Inshort,inordertoreachthetopologicallynon-trivial
|µ|>|α |, the ground state wavefunctions are predomi-
R regime, we need to break the spin degeneracy by the
nantly in the bulk as shown in Fig.1c.
Rashbatermandinduceak-dependentpairingsuchthat
TounderstandhowtheDIIItopologicalsuperconduct-
there can be an odd number of negative pairing ampli-
ingstateisachievedinourmodel,wenotethatageneral
tudes for positive k at the Fermi energy. It is important
criteria for realizing a one-dimensional DIII class TS is
tonotethatthereisnoneedtotunethechemicalpoten-
to have an odd number of negative pairing amplitude at
tial to the RCP which is near the band bottom. If the
the Fermi points with Fermi momentum between 0 and
induced pairing is s-wave [6–13], the topologically non-
π [17]. We show that this is indeed the case for H .
1D trivial phase is not accessible.
In the momentum space, Hamiltonian H can be
1D
The topologically non-trivial state can be further ver-
written as
ified by calculating the topological invariant of H (k)
1D
H (k)=(cid:80) Ψ†[−(tcosk+µ)σ +α sinkσ ]Ψ + [17]. The topological invariant can be written as
1D k k 0 R y k
∆ coskψ† ψ† +h.c.
0 k↑ −k↓ Pf[Tq(k =π)] 1(cid:90) π
(2) N = exp{− dkTr[q†(k)∂ q(k)]},
where Ψ† = (ψ† ,ψ† ). The Hamiltonian has spectrum DIII Pf[Tq(k =0)] 2 0 k
k (cid:112)k↑ k↓ (4)
E(k) = ± ((−tcosk−µ)±α sink)2+(∆ cosk)2
R 0 where Pf denotes the Pfaffian, T = iσ is the time-
y
and it is generally gapped unless |µ| = α at which
points topological phase transitions take plaRce. In the reversal operator, and q(k) = 21[eiθ−(k)(σ0 − σy) +
basis which diagonalize the Rashba term, the Hamilto- eiθ+(k)(σ0+σy)]whichisanoff-diagonalblockoftheflat-
band Hamiltonian [17, 18] derived from H (k). Here,
nian can be written as 1D
eiθ± = √−tcos(k)−µ±αRsin(k)+i∆0cos(k) . From Eq.4,
H˜ (k)=(cid:80) [−(tcosk+µ)+a|α sink|]ψ˜† ψ˜ + [−tcos(k)−µ±αRsin(k)]2+[∆0cos(k)]2
1D k,a=± R ka ka the topological invariant number can be found to be
sgn(k)∆ coskψ˜† ψ˜† +h.c.,
0 ka −ka trivial (NDIII = 1) when |µ| > αR and non-trivial
(3)
(N =−1) when |µ|<α .
DIII R
where ψ˜ka denotes a fermion in the new band basis. Single-channel model with finite Vz—When Vz is fi-
When |µ|<|αR|, there are two Fermi points k1,k2 with nite, the energy spectrum of H1D(k) becomes E(k) =
0<k1,k2 <π. In this regime, it can be shown that one ±(cid:113)F(k)±2(cid:112)G(k), where F(k) = (tcosk + µ)2 +
and only one of the pairing amplitudes of the two bands
at the Fermi level ∆ cosk and ∆ cosk is always neg- α2 sin2k+∆2cos2k+V2 andG(k)=(tcosk+µ)2V2+
0 1 0 2 R 0 z z
ative. Therefore, the superconductor is in the topologi- (tcosk +µ)2α2 sin2k +V2∆2cos2k. From the energy
R z 0
cally non-trivial regime. A schemetic picture is shown in spectrum, we note that the energy gap closes when
(cid:112)
Fig.2b. When |µ| > |α |, we have cosk cosk > 0 and (µ±t)2 =V2−∆2 and |µ|= V2+α2.
R 1 2 z 0 z R
the pairing amplitudes at the two Fermi points have the Moreover, the V term breaks time-reversal symme-
z
same sign and the system is in the topologically trivial try and changes the Hamiltonian from DIII class to D
regime. class. It is known that a 1D Hamiltonian in D class
3
0.2
0
∆
E/ 0.1
0
-6 -4 -2 0 2 4 6
a1 b1 c1 µ/αR c2 b2 a2 FIG. 4: A wire with strong spin-orbit coupling in proxim-
ity to a d -wave superconductor. Double Majorana end
FIG.3: Excitationenergyasafunctionofchemicalpotential. x2−y2
states may appear in the absence of an external magnetic
The parameters of H are: t = 12, ∆ = 1, α = 4 and
1D 0 R field.
V =10. Points a , b and c (with i=1,2) denote gap closing
z i i i
(cid:112)
points with chemical potential µ = ∓| V2−∆2 +t|, µ =
z 0
(cid:112) (cid:112)
∓| V2+α2|andµ=∓| V2−∆2−t|respectively. Double In the quasi-one-dimensional case, the Hamiltonian
z R z 0
Majorana fermion end states exist when µ is between c1 and can be written as:
c . SingleMajoranafermionendstatesexistwhenµisinthe
2
regions between a and c , c and a . H = H +H +H +H ,
1 1 2 2 q1D t SO SC Z
H = (cid:80) −1t(ψ† ψ +h.c.)−µψ† ψ
t R,d,α 2 R+d,α Rα R,α Rα
H = (cid:80) −iα ψ† zˆ·((cid:126)σ ×d)ψ +h.c.
may support single Majorana end states [3, 14]. The SO R,d,α,β 2 R R+d,α αβ R,β
energy eigenvalues of H versus the chemical potential H = (cid:80) 1[∆ (ψ† ψ† −ψ† ψ† )−
1D SC R 2 0 R+dx,↑ R,↓ R+dx,↓ R,↑
pca2roeiansltlhosowwwsintdhoinuµbFl=ieg.M∓3.a|(cid:112)jIonVraF2ni−ag.3f∆e,r2tmh−ieotnr|esgrweioshpneerbceteitcvw1eelayennd[1c9c12].aanIrdne HZ = (cid:80)∆0R(ψVR†z(+ψdR†y,↑↑ψψRR†↑,↓−−ψψR†R†↓+ψdRy↓,↓)ψ. R†,↑)+h.c.]
z 0 (7)
the regions between a and c , c and a , single Majo-
1 1 2 2
Here, R denotes the lattice sites, d denotes the two
ranafermionendstatesemerge. Here,a anda indicate
(cid:112) 1 2 unitvectorsd andd whichconnectsthenearestneigh-
points with µ = ∓| V2−∆2 +t| respectively. When x y
(cid:112) z 0 bor sites in the x and y directions respectively. This
|µ| > | V2−∆2 +t|, the Hamiltonian is topologically
z 0 model is the same as the tight-binding model in Ref[12]
trivialandtheMajoranaendstatesdisappear. Atpoints
(cid:112) except for the superconducting pairing terms. The pair-
b and b where µ=∓ V2+α2, the energy gap closes
1 2 z R ingtermsinH canbewrittenas∆ [cos(k )−cos(k )]
but there are no topological phase transitions at these q1D 0 x y
in the momentum space. Therefore, H describes a
points. q1D
quantumwirewithspin-orbitcouplingandad -wave
The even and odd number of Majorana end states in x2−y2
superconducting pairing. A schematic picture of the ex-
Fig.3 can be verified by calculating the Z Majorana
2 perimental setup is shown in Fig.4.
number M of H (k). Following Refs [3, 11] the Ma-
1D The energy spectrum of the Hamiltonian with V = 0
jorana fermion number can be defined as z
is shown in Fig.5a. The length of the wire is chosen
M=sgn[PfB(0)]sgn[PfB(π)]=±1. (5) to be much larger than the superconducting coherence
length L (cid:29) t/∆ and the width is comparable to the
0
The matrix B(k) is defined as B(k)=H (k)(σ ⊗σ ).
1D x 0 coherence length W ≈ t/∆ . In sharp contrast to the
0
B(k) is anti-symmetric and its Pfaffian is well-defined.
s-wave-pairing proposals in which the topologically non-
M = ±1 indicates the even and odd number of Majo-
trivial regime can be reached only when the chemical
rana fermions at one end of the wire respectively. In
potential is near the RCP, in our proposal, zero energy
terms of the parameters of the Hamiltonian, the Majo-
Majorana modes live all over the full band even when
rana number can be written as
the Fermi level is far away from the band bottom. It is
sgn{[(t+µ)2−(V2−∆2)][(−t+µ)2−(V2−∆2)]}. (6) also evident from Fig.5a that the zero energy Majorana
z 0 z 0
modes are separated from other fermionic states by a
The Majorana numbers calculated according to Eq.6 are large energy gap which is of order ∆ . Due to time-
0
consistent with the results shown in Fig.3. The Majo- reversal symmetry, all energy levels in Fig.5a is doubly
(cid:112)
rana number is M = −1 when | V2−∆2−t| < |µ| < degenerate. Therefore, in the topologically non-trivial
z 0
(cid:112)
| V2−∆2+t| and M=1 otherwise. regime, there are two Majorana fermions at each end of
z 0
Multi-channel case—In realistic situations, multiple the wire. It can be shown that these double Majorana
transverse sub-bands of a wire are occupied and it is im- end states are robust to disorder.
portanttoshowthatMajoranafermionsexistinthissit- InthepresenceoftheV term,time-reversalsymmetry
z
uation. In this section, we show that Majorana fermions is broken and the Hamiltonian is in D class. To break
exist in quasi-one-dimensional wires. Importantly, the some incidental degeneracy, a random on-site potential
(cid:112)
quasi-one-dimensional model can be realized by induc- V with strength (cid:104)V2(cid:105)−(cid:104)V(cid:105)2 = ∆ /2 is introduced
0
ing d -wavesuperconductivityona wirewithstrong into the system. The resulting energy spectrum of the
x2−y2
spin-orbit coupling. model with V = 3 is shown in Fig.5b. Without time-
z
4
a) createasuperconductingwirewithdoubleMajoranaend
states even though single Majorana fermions can be cre-
atedinthepresenceofanexternalmagneticfield. Satoet
00.2
∆ al. studied a d-wave superconductor with Rashba terms
E/
in the presence of an external magnetic field but no DIII
class TS phase is reported [28].
0 Second,itispointedoutrecentlythatsingleMajorana
-8 -6 -4 -2 0
BB µ/α fermion end states can be created at arbitary chemical
R
potential if a magnetic field is applied along the wire in
b) the s-wave pairing case [20, 27]. However, the magnetic
field B(cid:126) required to reach the topologically non-trivial
0.1
0 stateisstillstrongcomparingtotheinducedpairinggap
∆
E/ ∆ , namely, µ |B(cid:126)| > ∆ where µ is the effective mag-
S 0 S 0
netic moment of electrons.
Third, for simplicity, we assumed that the wire is
0
-9 -8 -7 -6 -5 -4 aligned along the x-direction. Tilting the wire with re-
BB µ/α
R spect to the x-axis slightly has no major effect on the
Majoranaendstatessincethiskindofperturbationdoes
FIG.5: a)Excitationenergyversuschemicalpotential. The
not change the symmtry class of the system.
parameters in H are V = 0, t = 18, ∆ = 2 and α =
q1D z 0 R
Fourth, one possible realization of H is by induc-
4. The width of the wire is W = 6 and the length is L = q1D
600. Allthestatesshownaredoublydegenerate. Zeroenergy ingdx2−y2-wavesuperconductingpairingontheAu(111)
Majorana modes, denoted by the red lines, can appear in a surface state. It has been shown that the Au (111) sur-
widerangeofchemicalpotential. Thenormalstateelectronic face has a Rashba band with Rashba energy of about
band bottom is denoted as BB b) Excitation energy versus 60meV[29]. TheproximitygapinducedonAubyLSCO
chemical potential in the presence of random potential. The canreach10meV[30]. TheselargeenergyscalesmakeAu
parameters in Hq1D is the same as in a) except Vz =3. The on LSCO a promising candidate of realizing topological
topologically non-trivial regimes with an odd number of zero
superconducting states. However, the induced proxim-
energy modes are bounded by dotted lines.
ity pairing shown in the recent experiment may not be
d-wave due to the presence of strong disorder in the Au
layer.
reversal symmetry, the spectrum is no longer doubly de-
generate. From Fig.5b, it is evident that zero energy Another candidate material of a DIII class TS is a
modes which represent single Majorana end states ex- layered heavy fermion superconductor CeCoIn5. Bulk
ist. As expected, the single Majorana modes are more CeCoIn5 is a dx2−y2-wave superconductor. Recently, su-
robustwhenthechemicalpotentialisnearthebandbot- perconducting thin films of CeCoIn5 with only several
tom. Majorana fermions exist when the chemical poten- atomic layers thick can be fabricated [31]. Suppose we
ital is further away from the band bottom, however, the have a thin film of CeCoIn5, due to the strong spin-orbit
mini-gap,whichistheenergygapbetweentheMajorana coupling and the broken of inversion symmetry on the
modes and other finite energy fermionic modes, is much surface,thetoplayerofCeCoIn5 acquiresaRashbaterm
smaller as in the s-wave pairing case [20]. [32]. Therefore, the top layer of CeCoIn5 can be de-
It is shown previously that a single Majorana fermion scribed by Hq1D. In a separate work, we show in detail
induces a zero bias conductance peak of G=2e2 in An- that a thin film of CeCoIn5 is a DIII class TS.
h
dreevreflectionexperiments[21,22]whenanormalmetal Conclusion— We show that a quasi-one-dimensional
leadcouplestoaMajoranaendstate. Ontheotherhand, dx2−y2-wavesuperconductorwithRashbaspin-orbitcou-
thedoubleMajoranaendstatesinDIIIclassTSinducea plingtermsisaDIIIclassTSwhichsupportsdoubleMa-
zero bias conductance peak of G = 4e2 [23, 24] instead. joranaendstates. Inthepresenceofamagneticfield,sin-
h
Therefore, the two topological superconducting phases gle Majorana end states appear. These two topological
canbedistinguishedfromeachotherandfromtrivialsu- superconducting states can be probed using Andreev re-
perconducting states through Andreev reflection experi- flectionexperiments. WesuggestthatAuwiresonLSCO
ments. andthinfilmsofCeCoIn5 arecandidatematerialsofthis
Discussion—Afewimportantcommentsfollow. First, new topological superconducting state.
the DIII class topological superconductor proposed in Acknowledgments—WethankC.H.Chung,C.Y.Hou,
this work is truly different from the time-reversal in- J. Liu, T.K. Ng, B. Normand and A. Potter for in-
variant superconducting state obtained by inducing an sightful discussions. We thank Y. Matsuda for bring-
s-waveorad-wavepairingonthesurfaceofatopological ing the CeCoIn system to our attention. KTL thanks
5
insulator (TI) [25–27]. The TI surface state is described Patrick Lee for being a source of inspiration. The au-
by a single Dirac cone (in the simplest case). With only thors are supported by HKRGC through DAG12SC01
a single species of fermion at the Fermi level, one cannot and HKUST3/CRF09.
5
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[19] DuetothefactthatH isreal,onecanconstructaZ
1D n
topologicalinvarantandshowthatthedoubleMajorana