Triplet Superconductivity in a Nutshell
Kazumi Maki
0
1
2
3
Hae Young Kee
0
1
2
3
Yoshifumi Morita
0
1
2
3
0
H.Y. Kee Department of Physics, University of Toronto
,
Toronto
, Ontario M5S 1A7,
Canada
1
K. Maki Department of Physics and Astronomy, University of Southern California
,
Los Angeles, CA 90089-0484, USA
2
K. Maki Max-Planck Institute for the Physics of Complex Systems
, Nthnitzer Str.
38 01187 Dresden, Germany
3
Y. Morita ( ) Faculty of Engineering, Gunma University
, Kiryu, Gunma 376-8515,
Japan
The triplet superconductors have been around us since 1980, when Jerome et al. discovered the Bechgaard salts (TMTSF)2PF6 and (TMTSF)2ClO4. Now there are more than 20 or so triplet superconductors discovered. Recently we found that most of them with the known gap symmetries can be mapped to the superfluid phase of 3He-A and 3He-A1. Further, in most of them, l (the chiral vector, i.e. the quantization axis of the pair angular momentum) is fixed parallel to one of the crystal axes and all the topological defects are considered in terms of d-textures, where d is the spin vector. Then Sr2RuO4, UPt3, PrOs4Sb12 and (TMTSF)2ClO4 belong to type-A, which is an analog of superfluid 3He-A. In these superconductors, a vortex splits into a pair of half quantum vortices (HQVs) at low temperatures. On the other hand, CePt3Si, CeIrSi3, CeRhSi3, UIr and Li2Pt3B (those in non-centrosymmetric crystals) belong to type-A1, which is an analog of superfluid 3He-A1. In all of these triplet superconductors, vortices harbor the zero mode or the Majorana fermions, the implications of which deserves further exploration.
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The first triplet superconductors have been discovered in
1980 by Jerome et al. [1] in Bechgaard salts (TMTSF)2PF6
and (TMTSF)2ClO4 [2, 3]. Since then many triplet
superconductors have been discovered, which is concisely
reviewed by Sigrist and Ueda [4]. Although it has been noticed
that all of these triplet superconductors are nodal (i.e. non
s-wave), the gap symmetry has not been addressed seriously
until around 1994. We now have a proposal on the gap
symmetry of UPt3, Sr2RuO4 and PrOs4Sb12 as shown in Fig. 1
[5, 6]. Here we understand that these triplet superconductors
are characterized by the equal spin pairing (ESP) and with
extremely small spin-orbit coupling energy Eso103 ,
where is the superconducting energy gap [7]. Therefore
the superconducting order parameter is characterized by l
(the chiral vector) and d (the spin vector) as in superfluid
3He-A, except for the gap symmetry [8, 9].
On the other hand, the discovery of a
non-centrosymmetric superconductor CePt3Si by Bauer et al. in 2004 [10]
opened a new door. According to Andersons argument [11],
such a triplet superconductor cannot exist. In order to
clarify the role of the parity-breaking term (e.g. the Rashba
term [12]), Frigeri et al. [13, 14] considered a model with
a Rashba term. They found, first of all, the Fermi surface
is split into the one for up-spin and the another for
downspin, as shown in Fig. 2. Here we have introduced the
Rashba Hamiltonian HR = kzz, different from the one
used by Frigeri et al. [13, 14]. Secondly, they found the
strong admixture of the singlet component. Later, many
noncentrosymmetric superconductors have been discovered in
Fig. 1 Our proposal on gap functions of Y3,2(, ) for UPt3 and
CePt3Si, chiral f-wave for Sr2RuO4, (p + h)-wave for PrOs4Sb12
Fig. 2 The split Fermi surface
for each spin due to the
parity-breaking term
CeIrSi3, CeRhSi3, UIr and Li2Pt3B [1518]. Also the
spinorbit coupling energy Eso or the Rashba term is found
invariably extremely large [15, 18], i.e. Eso103 K in these
systems. Recently in [19], we have proposed that the triplet
superconductors in non-centrosymmetric crystals should be
like the one in 3He-A1, in contrast to the previous
proposals. The superconductivity occupies the Fermi surface
associated with one spin component (say, up-spin) while the
other Fermi surface remains in the normal state. With this
assumption, we can describe many unusual experimental
results of CePt3Si.
2 The Zero Mode and the Majorana Fermion
In this section, we shall summarize the bound-state spectrum
around an Abrikosovs vortex in triplet superconductors. In
both type-A and type-A1 triplet superconductors, we
conclude that vortices harbor the zero mode which behaves as
the Majorana fermion.
First of all, we note that l (the chiral vector) of many
triplet superconductors are fixed parallel to the crystal
c-axis. Secondly, in the absence of a magnetic field, d l is
due to the spin-orbit coupling. However, when the magnetic
field H (parallel to l) is applied in the vicinity of Hc2(T )
(the upper critical field), dl is realized. In this particular
situation, Ivanov [20, 21] have shown that the BdG
equations for the triplet superconductor decouple into the one
for the up-spin component and the other for the down-spin
component. Then these BdG equation have the same
structure as the one for s-wave superconductor first written down
in [22, 23]. Therefore, (...truncated)