Magnetically induced superconducting correlations: Bogolyubov–de Gennes calculations
of the gap profile in a superconductor with magnetic order
P. D. Sacramento,
1
V. K. Dugaev,
1,2
and V. R. Vieira
1
1
Departamento de Física and CFIF, Instituto Superior Técnico, Universidade Técnica de Lisboa,
Avenida Rovisco Pais, 1049-001 Lisboa, Portugal
2
Department of Mathematics and Applied Physics, Rzeszów University of Technology,
Aleja Powstańców Warszawy 6, 35-959 Rzeszów, Poland
Received 2 June 2007; published 25 July 2007
A dense distribution of classical spins in a conventional superconductor is expected to destroy the super-
conducting state. Tuning the coupling between the spins and the electronic spin density one finds that, as the
coupling increases, the superconducting order parameter vanishes for a relatively small value of the coupling.
However, for a moderate value of the coupling we find that a phase with superconducting correlations is
stabilized, if we consider a Hubbard-like model with repulsive interactions. We also find that if we keep
increasing the spin coupling, in the regime where the system is fully polarized, a local form of superconducting
order arises, for some configurations of the classical spins. These are associated either to certain magnetic
vortex structures or to a distribution of spins where some sites have no classical spin.
DOI: 10.1103/PhysRevB.76.020510 PACS numbers: 74.25.Dw, 72.15.Qm, 75.60.Ch
Magnetism and superconductivity compete. In general, a
magnetic field, if strong enough, will destroy the supercon-
ducting order. There are various circumstances where this
effect is delayed or even reversed to some extent. An exter-
nal magnetic field induces in type-II superconductors a
mixed phase in which the field penetrates the system through
quantized vortex lines,
1
extending the superconducting phase
to external fields that may be quite high, such as in the high-
T
c
superconductors. In some systems a reentrant supercon-
ducting phase is predicted to occur in the quantum limit
where all the electrons occupy the lowest Landau level.
2
Some systems show coexistence of antiferromagnetism and
superconductivity, particularly heavy fermions where two
types of fermions contribute to the behavior of the system.
3
The magnetic field is pair breaking in the case of singlet
superconductors and it may be convenient, or possible, to
have coexistence of a magnetic field or ferromagnetism if the
pairing is triplet in the spin part. This implies either an odd
symmetry in real space like p-wave pairing or a symmetric
pairing if the pairing is odd in frequency space.
4
In some
situations it is also favorable for the system to separate into
spatial regions where magnetism sets in and others where
superconductivity prevails, such as the cryptoferromagnetic
state.
5
It was also shown that in the presence of a uniform
magnetic field it is possible to pair electrons in a spin singlet
state if the Cooper pair has a finite momentum leading to a
phase where the gap function is nonuniform Fulde-Ferrell-
Larkin-Ovchinnikov phase
6
. Another recently proposed
counterexample are the so called frozen flux superconductors
where, due to the vicinity of magnetic dots to a supercon-
ducting film, the critical current is increased by the external
field, since in its absence the magnetization of the dots is
random and less effective as pinning centers.
7
Also, in the
Jaccarino-Peter effect an external magnetic field may counter
the effect of local exchange fields and lead to enhancement
of superconductivity when the magnetic field is increased.
8
Here we consider a different example of the enhancement
of the superconducting correlations in an s-wave supercon-
ductor due to an increased magnetic interaction inside the
superconductor. The addition of magnetic impurities induces
in-gap bound states in conventional superconductors and in
d-wave superconductor virtual bound states.
9
In their pio-
neering work Abrikosov and Gor’kov
10
considered the prop-
erties of a conventional superconductor with magnetic impu-
rities. They showed that the noninteracting magnetic
impurities suppress the superconductivity at some critical
impurity density, later identified as a quantum critical
point.
11
In the simplest case the magnetic impurities can be
treated as classical spins inserted in the superconductor, act-
ing as local magnetic fields. Theoretically and experimen-
tally the critical temperature decreases fast as the impurity
concentration increases.
10,12,13
In this work we are interested in a situation where the
classical spin distribution is dense. At basically all sites of
the system we place classical spins parametrized like S
l
/S
= cos
l
e
x
+ sin
l
e
z
, where S is the modulus of the spin. Thus
we assume that the spins lie in the x-z plane so that the
orbital effect is absent. The Hamiltonian of the system is
given by
H =-
i, j,
t
i, j
c
i
†
c
j
-
i
c
i
†
c
i
+
i
i
c
i↑
†
c
i↓
†
+
i
*
c
i↓
c
i↑
-
i,l,,'
J
i,l
cos
l
c
i
†
,
'
x
c
i
'
+ sin
l
c
i
†
,
'
z
c
i
'
,
where the first term describes the hopping of electrons be-
tween different sites on the lattice, is the chemical poten-
tial, the third term corresponds to the superconducting s pair-
ing with the site-dependent order parameter
i
, and the last
term is the exchange interaction of the spin density of the
electrons with the magnetic impurities. The hopping matrix
is given by t
i, j
= t
j,i+
, where is a vector to a nearest-
neighbor site and we will take units such that t =1. Note that
both the indices l and i , j specify sites on a two-dimensional
system. The indices i , j =1,..., N, where N is the number of
PHYSICAL REVIEW B 76, 020510R2007
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